Computational Materials Science of Polymers

COMPUTATIONAL MATERIALS SCIENCE OF POLYMERS

COMPUTATIONAL MATERIALS SCIENCE OF POLYMERS A NDREY A LEKSANDROVICH ASKADSKII

CAMBRIDGE INTERNATIONAL SCIENCE PUBLISHING

Published by Cambridge International Science Publishing 7 Meadow Walk, Great Abington, Cambridge CB1 6AZ, UK http://www.cisp-publishing.com First published January 2003 © A A Askadskii © Cambridge International Science Publishing

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About the Author Andrey Aleksandrovich Askadskii is a Professor of Chemistry at the Institute of Organo-Element Compounds of the Russian Academy of Sciences. He holds M.S. in Civil Engineering from the Moscow Civil Engineering Institute (1959), M.S. in Chemistry from the Mendeleev Institute of Chemical Technology (1962) and Ph.D. in Physics of Polymers (1968). The main scientific interests of the author are: the development of a physical approach to the quantitative evaluation of the physical properties of linear and network polymers on the basis of their chemical structure; development of computer programs for calculating the properties of polymers and low-molecular liquids and also computer synthesis of polymers with the required properties; experimental examination of the structure of properties of heat-resistant aromatic polymers of different grades; development of new methods of experimental and theoretical analysis of the relaxation properties of polymer materials; production of new types of polymers; production and examination of electrically conducting polymer materials on the basis of heat-resistant polymers and organo-element compounds; development of gradient polymer materials with a variable modulus of elasticity within the limits of the same material and retaining elastic (not viscoelastic) properties at any point of the gradient material. Prof Askadskii is the author of more than 400 scientific studies and 20 books, six of which have been published abroad.

Contents

Preface Introduction Chapter I. Brief information on types of polymes and their chemical structure Chapter II. Packing of macromolecules and polymers density II.1. Increments method and basic physical assumption Chapter III. Temperature coefficient of volumetric expansion Chapter IV. Glass transition temperature of polymers IV.I. Thermomechanical and other methods of evaluation of the glass transition temperature of polymers IV.2. Mechanism of glass transition IV.3. Calculation of the glass transition temperature of linear polymers IV.4. Influence of plasticization on the glass transition temperature of polymers IV.5. Calculation of the glass transition Chapter V. Temperature of transition into the viscous flow state for amorphous polymers V.1. Estimation of temperature of transition into the viscous flow state of polymers V.2. Dependence of Newtonian viscosity on molecular mass of polymer in a wide range of its change Chapter VI. Melting point of polymers Chapter VII. Temperature of onset of intense thermal degradation of polymers Chapter VIII. Optical and opto-mechanical properties of polymers VIII.1. Refractive index VIII. 2. Stress-optical coefficient Chapter IX. Dielectric constant of polymers and organic solvents Chapter X. Equilibrium rubber modulus for polymer networks X.1. Calculation of the equilibrium modulus X.2. Heteromodular and gradient-modulus polymers Chapter XI. Description of relaxation processes in polymers XI.1. Stress relaxation XI. 2. Sorption and swelling processes Chapter XII. Solubility of polymers XII.1. Specific cohesive energy of organic liquids and polymers. Hildebrand solubility parameter XII.2. Solubility criterion XII.3. Influence of molecular mass and degree of macromolecule orientation on solubility Chapter XIII. Surface properties of organic liquids and polymers XIII.1. Surface tension of organic liquids XIII.2. Surface tension of polymers Chapter XIV. Miscibility of polymers Chapter XV. Influence of the end groups on the properties of polymers Chapter XVI. Thermophysical properties of polymers XVI.1. Heat capacity XVI.2. Thermal diffusivity and heat conductivity

3 9 16 16 58 67 67 88 108 322 343 385 385 388 398 408 418 418 426 445 456 456 466 475 475 497 504 504 509 520 527 528 536 547 555 562 562 564

Chapter XVII. Molecular design and computer synthesis of polymers with predermined properties Appendix 1. Examples of solution of direct problems of polymers synthesis Appendix 2. Examples of solving the reverse problem of polymer synthesis Appendix 3. The example of solving the complex problem – analysis of the chemical structure of phenol formaldehyde resin Appendix 4. Application of the approach to multicomponent copolymers Appendix 5. Influence of strong intermolecular interaction occurring between two dissimilar polymers on their miscibility Appendix 6. On formation of super-molecular structure in amorphous polymers 1. Scheme of formation of the super-molecular structure 2. Calculation method of evaluation of dimensions of elements of supermolecular structure of polymers 3. Phase state of polymers as a result of formation of the super-molecular structure by one-cavity bond hyperboloids References Index

567 589 602 607 621 625 645 645

653 669 689

PREFACE Published in the journal “Chemistry and Life”, No. 2, 1981 was the article by me, titled by the editor as “Atom plus atom plus thousand atoms”. This article discussed the possibility of calculating some physical properties of polymers on the basis of the chemical structure of the repeat unit (it was then possible to calculate properties of linear polymers only). In conclusion of the article, titled “A little fantasy”, it was written: “Therefore, many properties of polymer can be predicted, if nothing except the structural formula of the appropriate monomer is known. It is a great progress: nowadays already, such calculations allow chemists to be drawn away from heavy duty to synthesize hopeless monomers. Formerly, under empirical selection of materials, many of such monomers had to be synthesized. Nevertheless, calculations are to be made manually still. Moreover, when they are translated into the machinery language, chalk and blackboard traditional for any chemical dispute can be substituted by an electronic “pencil”. A chemist will draw a formula of the suggested monomer on the screen by it, and the computer will answer immediately if it is useful or not to synthesize it. Another opposite task seems to be much more absorbing. If the computer is able to calculate properties by structural formulae, apparently, it may be taught, vice versa, to calculate the formula of a suitable monomer (or several formulae to choose) by any, even contradictory set of properties, given to it. In this case, it will be able to substitute the chemist in his most problematic part of work, one is able to succeed in on the basis of experience, intuition and luck.” That was a fantasy, and it could be hardly imagined that these ideas would be realized at any time in neat future. However, events were developing very fast, especially after appearance of high-power personal computers. Before discussing stages of this great work, methods of the quantitative estimation of polymer physical properties must be presented in brief performed on the basis of their chemical structure. At the present time, there are three main approaches to this estimation. One of them, developed by Van Krevelen [214], is based on the idea of so-called ‘group contributions’, according to which the simplest empirical expressions of the additive type are written down, the present group, existing in different polymeric units, making one and the same contribution to the calculated characteristic (for example, glass transition temperature, melting, etc.). As the author states, this is just an empirical approach, which allows the physical properties of many of linear polymers to be calculated with high accuracy. Another approach, being developed for a long time by the author of this preface in company with Yu.I. Matveev [28, 128] is semi-empirical. According to it, equations for calculation of the physical properties are deduced on the basis of ideas of physics of solids, and calibration of the method is performed with the help of physical characteristics of polymeric standards, the properties of which are studied well. Consequently, parameters of equations possess a definite physical sense (energy of dispersion interaction, energy of strong intermolecular interaction, including hydrogen bonds, Van-der-Walls volume, etc.). Application of this approach makes possible estimation with enough accuracy of many physical characteristics of polymers (about 60 up to now). Therefore, the number of polymers of various structures is unlimited. The third approach developed by J. Bicerano [133] has appeared recently. It is based on the so-called coherence indexes, reduced in practice to a search for various

2

correlations of physical properties with many rules of obtaining coefficients of correlation dependencies. Discussed in the present monograph are principles of the approach, developed by A.A. Askadskii and Yu.I. Matveev, special attention being paid particularly to computer realization of the current calculation method for physical properties of polymers. The first computer software has been composed by E.G. Galpern, I.V. Stankevich and A.L. Chistyakov - investigators of quantum chemistry laboratory of A.N. Nesmeyanov Institute of Organo-Element Compounds, RAS. Initially, computer “synthesis” of polymers by this software was performed from so-called large procurements representing residues of monomers, involved into the synthesis reaction. In the second variant, computer synthesis was performed from smallest procurements, from which the repeat unit of the polymer was constructed. This broadens significantly capabilities of the software for solving both direct (calculation of the polymer properties from its chemical structure) and reverse task (computer ‘synthesis’ of polymers with preliminarily programmed /assigned/ properties, the ranges of which were set in the computer), because the amount of ‘synthesized’ olymers has increased sharply. Then principally new software was composed by A.F. Klinskikh, in which chemical structure of the repeat unit was ‘constructed’from atoms. Thus, the user needs just to depict chemical structure of the polymer on the computer screen as chemist does it on the paper, and computer lays out all physical properties of polymers, involved in the software (all about 60). This software also provides for calculation of a sequence of properties of low-molecular weight organic compounds, as well as, which is very important, properties of polymeric networks. Solution of the reverse task is also provided. Of special importance is the possibility to calculate properties of copolymers and their mixtures, to predict solubility and compatibility of polymers, to construct dependencies of properties on temperature, molecular mass, crystallinity degree, microtacticity (of special importance are dependences of glass transition temperature and temperature of transition into the viscous flow state on molecular mass). It stands to reason that not all the problems are solved. Accuracy of the calculation and various predictions of polymers behavior at dissolution and mixing with each other must be increased, calculation schemes to estimate new properties of polymers must be developed, and their computer realization must be performed, etc. It is obvious that the present monograph possesses some drawbacks. The authors will be thankful for any notes on the point of the book.

3

INTRODUCTION As mentioned above, the approach to estimation of the physical properties of polymers, discussed in the monograph, is semi-empirical. When estimating the thermal characteristics of polymers, such as glass transition temperature, melting point, it is supposed that the repeat unit is composed of a set of anharmonic oscillators representing atomic pairs, linked by intermolecular physical bonds. The critical temperature of this set of anharmonic oscillators is that determines the abovementioned two transition temperatures. The thermal expansion coefficient is also closely related to these characteristics. In the case of a characteristic as the temperature of the onset of intensive thermal degradation, the polymeric unit is considered as a set of anharmonic oscillators representing atomic pairs, linked by chemical bonds. The critical temperature of such a set of oscillators characterizes the temperature of the onset of intensive thermal degradation at the given rate of heating (clearly at a different rate of heating, the temperature of the onset of intensive thermal degradation will be different, i.e. kinetic effects play a significant role in this case). At first glance, it may seem strange that thermal degradation is considered here not as a kinetic, which is conventional, but as an original phase transition, at which, however, the initial substance cannot be obtained from the products of thermal decomposition by simple cooling down. Equations for calculating other physical characteristics are based on physical approaches, discussed in detail below, and we will not consider them in this part. Common for all these equations is summarizing the sequence of atomic constants, which characterize contributions to the energy of intermolecular interaction, chemical bonds energy, Van-der-Waals volume, etc. Strictly speaking, the present approach cannot be named additive in the common sense of the word, because the calculated properties are not additive in relation to atoms and groups, which compose the repeat unit of polymer. Here additivity is applied to the characteristics which are really additive (Vander-Waals volume, molecular mass, intermolecular interaction energy, etc.). The approach being described allows calculation of their properties of the unlimited number of polymers and conduction of the computer synthesis of polymers with assigned properties with the help of software created and described in the monograph that is not possible using other existing programs. As mentioned above, the approach discussed in the monograph is semiempirical, calibration of the method being based on the so-called polymeric standards, the properties of which are studied in detail and common. Let us consider the essence of calibration on an example of the equation calculating glass transition temperature of a linear polymer, Tg:

Tg =

∑ ∆Vi i

∑ ai ∆Vi + ∑ b j i

j

,

4

where ai are atomic constants; bj are constants bound to the energy of strong intermolecular interaction (dipole-dipole, hydrogen bonds), occurred between polymeric chains at the sacrifice of polar groups existing in them; ∑ ∆Vi is the Vani

der-Waals volume of the polymer repeat unit, summarized from Van-der-Waals volumes of atoms participating in the composition of the unit. Reduce the equation to the following view: 1

∑ ai ∆Vi + ∑ b j = T ∑ ∆Vi . i

g i

j

Basing on this equation, the excessive system of linear equations is composed as follows:   1   a1∆V1,1 + a2 ∆V1,2 + ... + an ∆V1, n + α1b1 + β1b2 + ... + γ1bk = ∆Vi  ; ∑   Tg ,1  i  1   1   ∆Vi  ; a1∆V2,1 + a2 ∆V2,2 + ... + an ∆V2, n + α 2b1 + β 2b2 + ... + γ 2bk = ∑   Tg ,1  i 2   .......... .................... .................... .................... .................... .................... ...........   1  ∆Vi  . a1∆Vm,1 + a2 ∆Vm,2 + ... + an ∆Vm, n + α m b1 + β m b2 + ... + γ m bk = ∑  Tg ,1  i  m Then the matrix of coefficients at the unknowns of this excessive system of equations:  ∆'  ∆'  =    ∆'P



 



∆' ∆'

 

 



∆'P



∆' Q ∆' Q













α α







∆'P Q α P





β β





 





βP



γ  γ     γ P  



and the column matrix of free terms of these equations       ∑ ∆'L       %J  L       ∑ ∆'L     =  %J  L            ∑ ∆'L     %J P  L  P  









are composed. Further on, a transposed matrix à is composed and multiplied by the initial one – ÃA, as well as by the column matrix – ÃB. All this results in obtaining a

5

canonic system of equations. This canonic system is solved, for example, by the Gauss method. The whole procedure of calibration is performed by standard software. Without considering features of such regressive analysis, let us note only that polymers, selected for calibrating the method, must possess experimental values of analyzed physical characteristics in broadest range, and the chemical structure of polymeric standards must be sufficiently different. Usually, an excessive system composed of 30–0 equations is to be solved, which corresponds to 30–40 polymers. Next, the properties of other polymers are calculated from the coefficients obtained. In this case, the energy of weak dispersion interaction, strong dipole–dipole interactions and hydrogen bonds, their relative part and many other physical parameters of the system are determined. We are coming now to a brief description of the contents of individual chapters of the monographs. The first chapter discusses the data of modern classification of polymers and their chemical structure. Of the outstanding importance, induced by the features of the chemical structure and the application field, are interpolymers, dendric and staircase (ladder) polymers. The second chapter discusses the approach to computerized materials technology of polymers on the atomic–olecular level, based on the method of increments. The increments of various atoms and main groups of them are calculated. The main physical ideas about structure of macromolecules of polymers and parameters determining it are displayed. The method for calculating such an important characteristic of the polymer structure, as the coefficient of molecular packing, is given. A connection between the free volume of the polymer, the coefficient of molecular packing and parameters of its porous structures is established. For experimental determination of characteristics of the microporous structure of polymers, the method of positron annihilation, the application of which indicated structural changes in polymers in their relaxation, is used. With consideration of weak dispersion and strong (dipole–dipole and hydrogen bonds), the third chapter gives formulae for calculating the thermal coefficient of the volume expansion in dependence on the chemical structure of the polymer. In this case, the type of atoms in the polymeric chain and type of the intermolecular interaction are estimated by a limited number of corresponding increments, numerical values of which are determined. The fourth chapter describes in detail the thermomechanical method of determination of the glass transition temperature and fluidity of polymers, features of interpreting thermomechanical curves for amorphous and crystalline polymers are analyzed, the calculation method of determination of the mechanical segment from the chemical structure of the polymer is displayed. Two main concepts of the mechanism of vitrification processes of polymers, relaxation and intermolecular, are discussed. The ‘atomistic approach’ which is more universal than the widespread socalled ‘group contributions method’ to calculation of polymer properties from their chemical structure, is considered. This approach was used for deriving an analytical expression to calculate the glass transition temperature of linear and network polymers from their chemical structure. The influence of types of linear polymers branching and the number of units between cross-link points, type and structure of these points, existence and type of the network defects for network polymers on the glass transition temperature of the polymers is analyzed. Given in the fifth chapter is the method for calculating the fluidity temperature of amorphous polymers and the temperature range of the rubbery state of polymers

6

from their chemical structure, and conditions of appearance of the rubbery state in a polymer depending on its molecular mass, as well, which is important for processing of polymers. The sixth chapter describes two approaches to calculating the melting point of polymers from the chemical structure of the repeat unit. The first approach is based on the experimental fact of closeness in parts of the empty volume in melting of a crystalline polymer and in transition of an amorphous polymer of the same structure from the glassy-like into the high-elastic state. The second approach is based on the consideration of the repeat unit of a polymer as a selection of anharmonic oscillators. Discussed in the seventh chapter is the most important characteristic of thermal resistance of polymers — initial temperature of their intensive thermal degradation. The formula to calculate this temperature based on the chemical structure of the polymer was deduced, and necessity to take into account the resulting products of thermal degradation which starts with the decay of end groups in polymer macromolecules, are indicated. In the eighth chapter, Lorenz–Lorentz equations are used for deriving equations for calculation of the refractive index of polymers and copolymers from their chemical structure. To obtain the stress-optical coefficient, empirical and semiempirical approaches are established, in which the contribution of each atom and the type of intermolecular interaction are estimated by an appropriate increment. Using the dependencies obtained for the stress-optical coefficient on the chemical structure of the repeat unit of the polymer, the contribution of various atoms and polar groups to the value of this coefficient is estimated, and a polymer with the properties unique for the method of dynamic photo-elasticity is proposed. The ninth chapter displays a scheme for calculating the dielectric constant of polymers and organic liquids with respect to their chemical structure which is important for both synthesis of polymers with the required dielectric constant and prognosis of polymer solubility in organic liquids. Taking into account not only the contribution of various polar groups to the dielectric constant of polymers and liquids, but also different contributions of a polar group in the present class of liquids resulted in the previously unobtainable agreement in the experimental and calculated values of the dielectric constant for a broad spectrum of organic polymers and liquids. Based on the notion of network polymers as an elastic and rotational–isomeric subsystem and taking into account its structure as linear fragments and cross-linked points, the tenth chapter indicates the deduction of formulae for calculating the equilibrium rubbery modulus and molecular mass of a linear fragment between neighboring cross-linked points. Further analysis of the resultant dependencies allowed the formulation of conditions for obtaining a polymer with unique (unusual) properties – different modulus and gradient polymers characterized by large changes of the equilibrium rubbery modulus within the same article. Existence of these unique properties is confirmed experimentally for synthesized network of polyisocyanurates. The eleventh chapter describes the derivation of analytical expressions for relaxation memory functions, necessary for determining the stress relaxation and creep of the polymers. In this case, the production of entropy of a relaxing system is represented by transition of relaxants (kinetic units of a polymer of different nature) into non-relaxants by means of their interaction or diffusion, the mechanism of interaction of relaxants in stress relaxation being found predominant. The apparatus created for description of relaxation events in polymers is applied in description of sorption and swelling processes. Thus, contrary to stress relaxation, the mechanism of relaxants diffusion is predominant in sorption.

7

The twelfth chapter is devoted to the problem of increasing the accuracy of prediction of polymer solubility in organic liquids. It is shown that the predictive ability of the solubility criterion, calculated with respect to the chemical structure of the polymer and the solvent, sharply increases with consideration for the type of supermolecular structure of the polymer and the degree of its polymerization. Based on the chemical structure of the matter, the thirteenth chapter gives a calculation method for the most important property of organic liquids and polymers, i.e. surface tension. Contrary to the additive scheme for summation of parachors which characterizes the contribution of separate atoms to the surface tension, the approach developed allows estimation of the contribution of polar groups and specific intermolecular interaction to the surface tension value and connection of it with the solubility parameter and density of cohesion energy in substances. Invoking the idea of solubility of a single homopolymer in another one, the fourteenth chapter suggests a criterion for estimating the compatibility of polymers basing on the data of the chemical structure of separate components. The analysis of application of the criterion for compatible, partially compatible or incompatible polymers indicates its high predictive ability. On the example of the calculation of the Van-der-Waals volume, molar refraction, heat capacity and other properties of a number of polymers, chapter fifteen displays the role of the chemical structure of macromolecule end groups and importance of their calculation in the study of regularities of changes in the polymer properties on their molecular mass. The sixteenth chapter indicates a method for calculating the molar heat capacity with respect to the chemical structure of polymers. The method is based on a supposition that the contribution of each atom to heat capacity is proportional to its Van-der-Waals volume. It is noted that the heat capacity, thermal diffusivity and heat conductivity of polymers depend not only on their chemical structure, but also on the physical and phase states of the polymeric body. The seventeenth chapter describes methodological ways of solving the direct problem of computerized determination of the physical characteristics of polymers and low-molecular liquids with respect to their chemical structure and the reverse one — computer synthesis of polymers with the given set of properties. These problems are solved by the methods of fragments and separate atoms. The corresponding software which allows calculation of more than 50 chemical properties of linear and network polymers and copolymers, and a number of the most important properties of low molecular weight liquids, as well, is developed. Discussed is the method of depicting diagrams of polymer properties compatibility, application of which may significantly simplify solution of the direct and, especially, reverse problems of computational materials sciences. Appendices demonstrate abilities of the approach, described in the monograph, to determine the properties of some natural polymers (the example of solving the direct problem of polymers synthesis) with respect to their chemical structure (Appendix 1); to search for chemical structures of polyetherketones (the example of solving the reverse problem of polymer synthesis), the properties of which must lie in a given range (Appendix 2); to solve a mixed problem of polymers synthesis on the example of analyzing the chemical structure of phenoloformaldehyde resin, when the direct problem — estimation of the properties of the ideal structures of such resin with respect to their chemical formulae — and the reverse one — searching for a combination of structures with which the chemical formula of phenoloformaldehyde resin obtained provides experimentally observed values of its

8

properties — are solved consecutively (Appendix 3); to analyze the structure and properties of copolymers, composed of from three to five comonomers (Appendix 4); and the influence of a strong intermolecular interaction appearing between two heterogeneous polymers on their compatibility is analyzed (Appendices 5 and 6).

Chapter I. Brief information on types of polymers and their chemical structure The very large number of existing polymers may be subdivided into three main classes forming the basis of the presently accepted classification. The first class contains a large group of carbochain polymers whose macromolecules have a skeleton composed of carbon atoms. Typical representatively of the polymers of this class are polyethylene, polypropylene, polyisobutylene, poly(methyl methacrylate), poly(vinyl alcohol) and many other. A fragment of a macromolecule of the first of them is of the following structure [–CH2–CH2–]n The second class is represented by a similar large group of heterochain polymers, the main chain of macromolecules of which contains heteroatoms, in addition to carbon atoms (for example, oxygen, nitrogen, sulfur, etc.). Numerous polyethers and polyesters, polyamides, polyurethanes, natural proteins, etc., as well as a large group of elemento-organic polymers relate to this class of polymers. The chemical structure of some representatives of this class of polymers is the following: [–CH2–CH2–O–]n Poly(ethylene oxide) (polyether); (CH2)2

O C O

NH

(C H 2 ) 6

Poly(ethylene terephthalate) (polyester);

C O O

NH

C

(C H 2 ) 4

O

n C O

CH3 Si O CH3

n

Polyamide; n

Polydimethylsiloxane (elemento-organic polymer);

Cl N

Polyphosphonitrile chloride (inorganic polymer).

P

Cl n The third class of polymers is composed of high-molecular compounds with a conjugated system of bonds. It includes various polyacetylenes, polyphenylenes, polyoxadiazoles and many other compounds. The examples of these polymers are: [–CH=CH–]n Polyacetylene

Polyphenylene n

N N C

Polyoxadiazole

C O

n

10

An interesting group of chelate polymers possessing various elements in their composition, able to form coordination bonds (usually, they are depicted by arrows), also relates to this class. The elementary unit of these polymers is often complex, for example: O H3C O

P

CH3 O

Zn O P

H3C

O CH3

O

The most widely used type of material in the large group of polymeric materials are still the materials based on the representatives of the first class of polymers which are carbochain high-molecular compounds. The most valuable materials could be produced from carbochain polymers, for example, synthetic rubbers, plastics, fibers, films, etc. Historically, these polymers have been implemented in practice first (production of phenoloformaldehyde resins, synthetic rubber, organic glass, etc.). Many of carbochain polymers became subsequently the classic objects for investigation and creation of a theory of the mechanical behaviour of polymeric substances (for example, polyisobutylene, poly(methyl methacrylate), poly-propylene, phenoloformaldehyde resin, etc.). Subsequently, materials based on heterochain polymers – polyamide and polyester fibers, films, varnishes, coatings and other materials and articles – became widespread. This has given impetus to investigating the properties and formation of notions, in particular, of anisotropic substances possessing extremely different properties in different directions. A special place in the sequence of these polymers is devoted to high-molecular elemento-organic compounds. Finally, the representatives of the third class – polymers with conjugated system of bonds – were used for the preparation of conducting materials. Considering in general terms the chemical structure of polymers of different classes, we have discussed the structural formula of the repeating unit in the macromolecule. However, the existence of many such units in the macromolecule immediately complicates the situation. Let us begin, for example, with an assumption that each unit in the elementary act of macromolecule growth may be differently attached to the neighbouring one; in this case, we are talking about the ‘head-to-head’, ‘tail-to-tail’ or ‘head-to-tail’ addition. Various variants of the unit addition to the propagating macromolecule are possible for asymmetric monomers of the C H2 C H

R type which possess R substituents on one of carbon atoms. Here, variants of ‘head-tohead’ ... CH2 CH CH CH2 CH2 CH CH CH2 ... R

R

R

R

and “head-to-tail” ...

CH2

CH CH2

CH CH2

CH

R

R

R

...

11

additions are possible. Alternation of the types of addition is possible, i.e. units may be differently attached to each other in a single macromolecule. Existence of a great number of units in the polymeric chain and possibility of only several variants of their attachment gives a huge number of isomers in relation to the whole macromolecule. To put it differently, a polymer may contain (and indeed contains) not only the macromolecules of the same chemical structure, but mixtures of a large number of macromolecules, which, of course, makes the polymer to differ from low-molecular substances, composed of identical molecules only. We will not talk about a rapid increase of the number of possible isomers in the sequence of substituted saturated hydrocarbons with the number of carbon atoms (i.e. with propagation of the molecule); even at a small (compared with polymers) number of them this number reaches a tremendous value. It is easy to imagine that when the number of units becomes tens or hundreds of thousands, the number of possible isomers becomes astronomically high [80]. Let us return to monosubstituted unsaturated hydrocarbons. When a polymeric chain is formed during polymerization, the substituents R may dispose differently in relation to the plane of single bonds. In one of possible cases, these substituents are disposed irregularly in relation to the plane of single bonds; such polymers are called irregular or atactic: H C H

H C R

H H H R H R H H H H R C C C C C C C C C C C C C C H H H H H H H R R R R R R R

H

H

H

In other cases, synthesis may be performed in such a manner that substituents would be disposed either by the same side of the plane of the main bonds H H H H H H H H H H H H H H H H C C C C C C C C C C C C C C C C H H H H H H H H R R R R R R R R or by both sides, but with regular alternation of the substituents direction: R R H H H H H R H H R H H H H H C C C C C C C C C C C C C C C C H R R H H R H R H H H H H H H H The polymers composed of the units with regular alternation of substituents were called stereoregular. If the substituents are disposed on one side of the plane of the main bonds, stereoregular polymers are called isotactic. If they are disposed on both sides of the plane, the polymers are called syndiotactic. The situation is more complicated with polymers synthesized from disubstituted monomers. Already in the monomer, substituents may dispose on the same (cis-isomer) or on both sides (trans-isomer) of the plane of the double bonds: H H H R' C C

C C

R

R H

R'

12

Synthesis of macromolecules from cis-isomers leads to the formation of erythro-diisotactic polymers R R R R' R R' R R' R' R' R' C C C C C C C C C C C C C C C C H H H H H H H H H H H H H H H H and trans-isomers give treo-diisotactic polymers H H H H H H R' H R' R' H R' R' R' R' R' C C C C C C C C C C C C C C C C R R R R R R R R H H H H H H H H Needless to say, other more complex modifications are also possible, which immediately cause a change of properties of polymeric materials. The materials composed from stereoregular polymers are often easily crystallized so that gives their physical structure and properties can be regulated. Here we meet for the first time a modification of the properties of polymeric materials, which is caused by practically any change in the chemical structure of macromolecules and the physical structure of the polymeric substance. Physical modification is often indicated by a change of the chemical structure, and sometimes is completely defined by it. One of the main methods of modification is the synthesis of copolymers, when not a single but several monomers participate in the reaction. That is why the macromolecule becomes composed from different units. These units may alternate continuously: –A–B–A–B–A–B–A–B–A–B– the alternating copolymer; but, most often, they are arranged irregularly: –A–A–B–A–B–B–A–A–A–B– the random copolymer. The units may also be linked in separate blocks which are the linked to each other: –A–A–A–A–A–B–B–B–B–B– the block-copolymer. Obviously, each block may contain a different number of units. This is immediately shown up in the properties of the future polymeric substance. In this case, the copolymerization process becomes regulated. Running ahead, recall that mechanical mixtures of polymers and copolymers of the same molar composition may often possess rather different properties, but sometimes they are practically identical. The considered schemes of addition of units during macromolecule growth indicate the only case of copolymerization of two types of monomers. Even if many combinations are realized in these simplest cases, their number grows immeasurably when three or more monomers (or types of units) are used All the above-discussed chains of polymers represent linear formations. However, branched macromolecular chains could be easily synthesized. For this purpose, it is even unnecessary to introduce multifunctional compounds into the chain composition. Branching also occurs in polymerization of unsaturated hydrocarbons with no functional groups. If no special steps are taken, the products of polymerization of ethylene, propylene, isobutylene and other similar compounds will always contain some amount of chains branched from the main chain. Concerning the products of polycondensation (see the above discussion on polyesters and R

R'

R

R'

R

13

polyamides), introduction of a three-functional compound into the main chain always leads to the formation of branched polymers: ...

A A A A' A A A A A A A

...

A A A A .. . It is self-evident that the polymeric body based on the branched macromolecules will differ in the structure and properties from a substance composed of linear macromolecules. However, we must not hurry in concluding about the type of physical structuring of the branched polymers. At first glance, it seems that the presence of large branches will make obstacles to denser packing of the chains, as well as to the crystallization process or regulation of macromolecules in general. Indeed, this is sometimes the case. In other cases, the opposite situation is observed. It depends upon the chemical structure of the main chain and its branches, which determines the volume of units, interaction forces between them and neighbour chains, etc. Recently, special attention has been paid to the structure and properties of socalled dendric polymers, the macromolecule of which is schematically depicted in Figure 1 [98, 212]. Below, we will discuss in more detail the influence of the types of branchings on the properties of the resulting polymers.

Figure 1. Schematic representation of dendric polymers

Branchings may be composed in different ways. They may contain the same units, which compose the main chain. However, ‘grafted’ polymers have become widely used; they are formed in grafting of previously obtained chains of a definite structure to the main chain with an extremely different structure: ...

A A A A' A A A A A A A B B B B .. .

...

14

Sometimes, such grafting is performed many times. We can now easily pass from the branched to three-dimensional ‘cross-linked’ polymers. This requires just an increase of the concentration of multifunctional compounds in the polymer chain. The chains could also be cross-linked by special curing agents, i.e. by compounds containing active groups, capable of reaction with functional groups of the main chain or the end groups. The classic example is the curing of epoxy resins: CH3 ...

O

C

O CH2

CH CH2 O

CH3 CH3 ...

O

O CH2

C

NH2 +

R NH2

CH CH2 O

CH3 CH3 ...

O

C

O CH2

CH3

CH CH2 OH

CH3 ...

O

C

O CH2

CH CH2

NH R NH

CH3 OH Further on, the second hydrogen atom is substituted, and a network is formed. According to the classification described in ref. [202], there exist several main methods of obtaining network polymers: 1) Realization of a chemical reaction between two (or more) different functional end groups, attached to a chain of low molecular mass. As a result, a dense network with short chains between cross-link points is formed. 2) Chemical linking of high-molecular compounds by the end groups with the help of a low-molecular cross-linking agent. Consequently, a network with long linear fragments between the cross-linked points is formed. 3) Formation of a network by copolymerization of two- and polyfunctional monomers. The example of such a network is the styrene–divinylbenzene system: ...

CH2

CH CH2

CH CH2

...

...

CH2

CH CH2

CH CH2

...

4) Vulcanization of polymeric chains by involving, in the reaction, functional groups disposed along the main chain. The reaction is performed either by the application of a low-molecular cross-linking agent or by means of radiation and other types of influence on the functional groups.

15

Other possible (and already realized in practice) ways of producing the network systems should also be added. 5) Formation of networks with by means of a reaction of two (or more) heterogeneous polymers by functional groups disposed along the chain of each polymers (i.e. in the repeating units, but not at the ends). 6) Synthesis of polymeric networks with the help of the polycyclotrimerization reaction. For this purpose, oligomers with end groups capable of forming cycles during the reaction [56, 79, 101, 152] are formed. The example of such a reaction is the trimerization of two-functional oligomers (or monomers) containing cyanate end groups. Clearly, other ways of obtaining the polymeric networks are also possible. Recently, a new type of polymer, called ‘interpolymers’ was produced [16, 215]. The interpolymer is a system composed of two (or more) macromolecules, heterogeneous in the chemical structure, chemically bonded to each other through the functional groups disposed in the repeating units of the each macromolecule. A schematic representation of the interpolymer is displayed in Figure 2.

Figure 2. Schematic representation of interpolymer.

A specific example of this system is, for example, a product of interaction between polystyrene and polytrichlorobutadiene: ...

CH2

CH

... + ...

CH2

CCl2

...

CH CCl

CH2

...

AlCl3

...

CH CCl

CCl

CH2

CH

...

...

The formation of interpolymers gives new possibilities of modifying the structure and properties of polymers. Another type of ‘two-cord’ system is the ladder polymer, the example of which is polyphenylsylsesquioxane [113]:

...

Si

O

O ...

Si

Si

O

...

O

...

O O

Si

Chapter II. Packing of macromolecules and polymer density II.1. Increments method and basic physical assumptions After discussing briefly the chemical structure of polymers, let us pass to the volumetric representation of macromolecules, which is necessary for understanding the features of structure formation in polymers. These considerations will be based on the assumptions developed by A.I. Kitaigorodsky in organic crystal chemistry [75]. According to these assumptions, every atom is presented as a sphere with intermolecular radius R. Values of these radii are determined from the data of X-ray structural analysis of ideal crystals of organic substances. In this case, it is assumed that valency-unbonded atoms, entering into an intermolecular (but not chemical) interaction, contact each other along the borders of the spheres. This is schematically represented in Figure 3. Then, if two identical atoms are in contact, the intermolecular radius will be determined from the relation: R = l/2,

(II.1)

where l is the distance between mass centers of two identical valency-unbonded atoms, which, however, are capable of intermolecular physical interaction.

Figure 3. Schematic representation of intermolecular (Van-der-Waals) interaction of two atoms

According to the same assumptions, chemical interaction between two atoms always causes their compression, because the length of the chemical bond di is always shorter than the sum of two intermolecular radii: d i < R1 + R2 .

(II.2)

This is clear from Figure 4, which schematically depicts two chemically bonded atoms. If the intermolecular radii Ri for all atoms participating in the repeat unit, and all lengths of chemical bonds between these atoms are known, their own (Van-der-Waals) volume of the repeat unit could be easily calculated, and a model of this unit (or greater fragment of the macromolecule), in which the volume of each atom is bordered by a sphere with intermolecular radius Ri, could be composed.

17

Figure 4. Schematic representation of two chemically bonded atoms.

Figure 5. Model of polyethylene chain fragment.

Table 1 shows intermolecular radii of some widespread atoms, which compose the majority of polymers. Table 1 Atom C H O N F Cl Br I

Van-der-Waals radii R of different atoms R, nm Atom 0.180 Si 0.117 Sn 0.136 As 0.157 S 0.150 P 0.178 Pb 0.195 B 0.221 Ti

R, nm 0.210 0.210 0.200 0.180 0.190 0.220 0.165 0.200

Table 2 displays bond lengths of various combinations of atoms, also characteristic for most of existing polymers. If these values are known, the volume of the repeat unit of any polymer may be calculated. To conduct this, the own volume of each atom participating in the repeat unit should be preliminarily determined. It is calculated from the formula 3 1 ∆Vi = πR 3 − ∑ πhi2 (3R − hi ), 4 i 3

(II.3)

where ∆Vi is the increment of the own (Van-der-Waals) volume of the present atom; R is the intermolecular radius of this atom; hi is the height of the sphere segment, cut off from the present atom by a neighbor one, chemically bonded to it. The value hi is calculated from relation

18

hi = R −

R 2 + d i2 − Ri2 , 2d i

(II.4)

where Ri is the intermolecular radius of a neighbor valency-bonded atom; di is the length of the chemical bond (see Figure 4). Table 2 Chemical bond length di for same pairs of atoms Bond* di, nm Bond* di, nm Bond* di, nm C–C 0.154 C–F 0.134 O–F 0.161 C–C 0.148 C–F 0.131 O=N 0.120 C=C 0.140 C–Cl 0.177 O=S 0.144 C=C 0.134 C–Cl 0.164 O=P 0.145 C=C 0.119 C–Br 0.194 N–P 0.165 C–H 0.108 C–Br 0.185 N–P 0.163 C–O 0.150 C–I 0.221 N–P 0.158 C–O 0.137 C–I 0.205 S–S 0.210 C–N 0.140 C–B 0.173 S–As 0.221 C–N 0.137 C–Sn 0.215 S=As 0.208 C=N 0.131 C–As 0.196 Si–Si 0.232 C=N 0.127 C–Pb 0.220 P–F 0.155 0.134 H–O 0.108 P–Cl 0.201 C N C≡N 0.116 H–S 0.133 P–S 0.181 C–S 0.176 H–N 0.108 B–B 0.177 C–S 0.156 H–B 0.108 Sn–Cl 0.235 C–Si 0.188 O–S 0.176 As–Cl 0.216 C–Si 0.168 O–Si 0.164 As–As 0.242 * If the same pair of atoms is linked by a single bond, the longer bond corresponds to attachment of this atom to an aliphatic carbon atom; the shorter bond corresponds to attachment of the same atom to an aromatic carbon atom.

Increments of the volumes of various atoms and atomic groups are shown in Table 3. Obviously, the volume of the given atom depends on its surrounding, i.e. on the type of atoms chemically bonded to it. The greater the volume of the neighbor, chemically bonded atom and the shorter the length of the chemical bond, the greater is the compression of the given atom. When increments of the volumes, ∆Vi, of all the atoms entering into the repeat unit of polymers are determined, the relative part of the occupied volume in the total volume of the polymeric substance may be calculated. In the case of polymer, calculations would be appropriate to conduct basing on molar volumes of the repeat unit, because polymers are always polydispersional (i.e. they contain macromolecules of various length), and also because at long lengths of the macromolecule the influence of end groups may be neglected. Then, the own molar volume will equal Vown = N A ∑ ∆Vi , and the total molar volume Vtotal = M/ρ, ρ is density of the i

polymeric substance; M is the molecular mass of the repeat unit; NA is the Avogadro number. Numerous experiments and calculations show that in all cases the condition Vown < Vtotal is fulfilled. Hence, in the first approximation, the volume of the polymeric substance could be divided into two parts: the own (Van-der-Waals) volume of atoms, which they occupy in a solid, and the volume of spaces determined as the difference of Vtotal and Vown. Of interest is determination of the part

19

Table 3 Van-der-Waals volumes of atoms

20

21

22

23

24

25

26

27

28

29

of the occupied volume or, according to the terminology used in organic crystal chemistry, the molecular packing coefficient k:

V k = own = Vtotal

N A ∑ ∆Vi i

M /ρ

.

(II.5)

Clearly, the value of k for the same polymer will depend on temperature and the physical state of the polymer, because the value of ρ depends on them. Calculations performed for many amorphous bulky polymers existing in the glassy state have indicated that the first approximation of k gives its value constant and practically independent of the chemical structure of the polymer [41]. Passing on to polymers with a complicated chemical structure from those with a simple one causes no significant change of the part of the occupied volume (e.g. the value of k). Table 4 indicates the chemical structure and numerical values of coefficients of the molecular packing of some glassy polymers. It also shows that first approximations of the values of k for each of them are equal, indeed. To demonstrate this experimental fact more clearly, Figure 6 displays the dependence of density ρ of various polymers on the relation M N A ∑ ∆Vi . In Figure 6 it is clearly seen that all i

30

Table 4 Values of the coefficients of molecular packing for some glassy and semi-crystalline polymers Structural formula of the repeat unit of polymer Van-der-Waals Packing volume of the coefficient k 3 unit, cm /mol

CH3 CH2

C

41.6

0.678

32.6

0.682

58.5

0.684

69.1

0.680

144.3

0.679

234.7

0.679

263.1

0.680

277.5

0.688

56.4

0.685

CH3 H CH2

C C N CH3

CH2

C C O CH3 O CH3

CH2

C C O C2H5 O

CH3 C O

C

O

CH3

C

O

C O

O

O C

O

O C O

C O

C O

C O O

(CH2)8

C

O C

HN

NH C

O

NH C O

CH2

CH CH CH2

31

—CH2—CH=CH—CH2— CH2

CH C

CH2

59.1 74.3

0.654 0.659

100

0.699

97.8

0.708

110.3

0.693

269.0

0.692

43.9 72.4

0.753 0.663

33.8 54.9

0.700 0.666

58.7 36.0 123.1

0.654 0.744 0.641

134.3

0.664

CH3 (CH2)5

NH C O

CH2

CH CH

H2C HC

CH2 CH2

N CH3 CH2

N

O

O

C

C

C

C

O

O

C

N

O

—CF2—CF2— CH2

CH O C

CH3

O —CH2—CHF— CH2

CH O CH3

—CH2—CCl2— —CH2—CF2— CH3 CH2

C

CH3

C O CH O

CH3

CH3 CH2

C C O C4H9 O

32

CH3 CH2

168.3

0.651

120.0

0.607

85.9

0.696

163.0

0.687

88.8

0.705

C C O C6H13 O

CH3 Si

O

CH2 CH2 CF3 CH2 CH2 CH2 CH3 C N O CH2

CH N

CH2 H2C

CH N

C O CH2

H2C CH2

CH

111.6

0.669

CH2

Cl CH

115.5

0.657

65.6

0.638

89.3

0.650

40.0

0.681

CH3 CH2

CH

S

CH3 CH2

CH C O C2H5

O O CH2

C O

33

O CH CH2 CH3

69.9

0.684

172.5

0.740

70.6

0.677

21.3 126.1

0.752 0.616

118.5

0.667

53.0

0.733

150.8

0.679

103.0

0.620

76.2

0.568

46.4 144.4

0.680 0.692

227.7

0.693

O

C

C NH

O

O CH2

C

NH

CH C O CH3

O —CH2—O— CH3 Si

CH2

O

CH C O C4H9 O F C

CF2

Cl CH3 CH2

C

CH2 C O CH CH2 O

CH2 CH2 CH2

C2H5 Si

O

C2H5 CH3 Si

O

CH3 —CH2—CH2—S— (CH2)2

O C

C O

O O

O C O

O

34

O

154.1

0.696

157.0

0.721

30.2 46.3

0.682 0.666

99.6

0.665

262.1

0.726

C O

O

SO 2

—CH2—CH2— CH2

CH CH3

CH2

CH2

CH

NH C O

(CH2)4

O C NH

O

O

Figure 6. Dependence of density ρ on

  ∑ ∆'

L

L

the values of ρ determined experimentally fit well the same linear dependence on the relation of atoms mass on their volume. In accordance with Equation (II.5), the tangent of this straight line represents the molecular packing coefficient which, in the case of amorphous bulky systems, serves as an universal constant. If it is true, the polymer density ρ may be calculated from the equation

ρ=

kM , N A ∑ ∆Vi i

(II.6)

35

that yields directly from Equation (II.5) under the condition kavg = const. In the case of amorphous bulky polymers, kavg = 0.681. For silicon-containing polymers, the average coefficient of molecular packing is 0.603. Hence, a change of the polymer chemical structure is unable to cause a significant effect on the part of the occupied volume in amorphous polymeric substance, and the value of density, ρ, itself depends on the relation of mass and the Van-der-Walls volume of the repeat unit only. Obviously, here we are dealing with true bulky substances of the amorphous structure. In reality, a polymeric substance with any porosity may be formed, and the coefficient k will have extremely different values. However, in this case, the notion of the packing density, quantitatively estimated by the value of k, loses its usual meaning and must be calculated for pore walls material only. We return to this problem below when discuss parameters of the porous structure of polymers, determined by the sorption method. For copolymers, equation (II.6) has the form

ρ=

kavg (α1M 1 + α 2 M 2 + ... + α n M n )         N A α1 ∑ ∆Vi  + α 2  ∑ ∆Vi  + ... + α n  ∑ ∆Vi           i 1  i 2  i n  

,

(II.7)

where α1, α2, …, αn are molar parts of the components 1, 2, …, n; M1, M2, …, Mn are     molecular masses of the repeat units of the same components;  ∑ ∆Vi  ,  ∑ ∆Vi  ,      i 1  i 2   …,  ∑ ∆Vi  are their Van-der-Waals volumes.    i n In the reduced form, expression (II.7) is: k =n

k avg

ρ=

k =n

∑α k M k

k =1

  N A ∑ α k  ∑ ∆Vi    k =1  i k

,

(II.8)

  where αk, Mk,  ∑ ∆Vi  are the molar part, the molecular mass, and the Van-der   i k Waals volume of the k-th component, respectively. If we want to express the density of copolymer via densities ρ1, ρ2, …, ρn of homopolymers based on the components 1, 2, …, n, expression (II.7) changes to the following form:

ρ=

α1M 1 + α 2 M 2 + ... + α n M n , M M M α1 1 + α 2 2 + ... + α n n ρ1 ρ2 ρn

(II.9)

36

(in this case, it should be taken into account that α1 + α2 + … + αn = 1). In the reduced form, the expression (II.9) is the following: k =n

ρ

∑α k M k

= k =1 k =n

,

(II.10)

M ∑α k ρ k k k =1

Expressions (II.7)–(II.10) may also be used for calculating the density of miscible blends of polymers. Let us now examine the temperature dependences of the molecular packing coefficients of glassy polymers. Calculation of values of k at different temperatures are performed by formulae yielding from the expression (II.5):

k (T ) =

k (T ) =

N A ∑ ∆Vi

[

i

[

i

(

)] ,

(T < Tg);

(II.11)

(

)] ,

(T > Tg);

(II.12)

MVg 1 + α G T − Tg N A ∑ ∆Vi

MVg 1 + α L T − Tg

where Vg is the specific volume of the polymer at the glass transition temperature Tg; αG and αL are the volume expansion coefficients of polymers below and above the glass transition temperature, respectively.

Figure 7. Temperature dependences of the coefficients of molecular packing k for a series of polymers: 1 – poly(n-butyl methacrylate), 2 – poly(n-propyl methacrylate), 3 – poly(ethyl methacrylate), 4 – polystyrene, 5 – poly(methyl methacrylate), 6 – polycarbonate based on bisphenol A.

Calculations by equations (II.11) and (II.12) indicate that temperature dependences of the molecular packing coefficients are of the form depicted in Figure 7. A remarkable property of these temperature dependences in the real equality of the molecular packing coefficient in the first approximation for all bulky polymers at any temperature below the glass transition point. In the second, more accurate approximation, the molecular packing coefficient is the same for every polymer at the glass transition temperature. This value is kg ≈ 0.667.

Table 5 Name 1 Polyethylene

Coefficients of molecular packing k for a series of crystalline polymers Type of elementary cell Chemical formula 2 3 Rhombic Pseudo-monoclinic CH2CH2 Triclinic

Polypropylene: - isotactic - syndiotactic

Monoclinic Monoclinic

1,2-poly(butadiene): - isotactic - syndiotactic

Rhombic Rhombohedral

1,4-trans-poly(butadiene)

Pseudo-hexagonal

1,4-cis-poly(butadiene)

Monoclinic

1,4-cis-polyisoprene

Monoclinic

Polychloroprene

Rhombic

Poly(ethylene terephthalate)

Triclinic

CH2

CH3 CH2

CH CH

CH2 CH2CH=CHCH2 CH2CH=CHCH2 CH2 CH C CH2 CH3 CH2

O CH2 Poly(hexamethylene terephthalate)

CH

CH2

CH C Cl

CH2

O C

C

O

O

Triclinic

O

(CH2)6

O C

C

O

O

ρ, g/cm3 4 1.000 1.014 0.965 1.013

k 5 0.736 0.746 0.710 0.745

0.936 0.910

0.693 0.674

0.963 0.960

0.692 0.690

1.020

0.733

1.010

0.726

1.000

0.725

1.657

0.893

1.455

0.776

1.131

0.652

37

38

2

3

Triclinic

O CH2

CH2

4 1.358

5 0.724

1.274

0.782

1.240 1.248

0.764 0.769

1.157

0.740

1.230

0.758

1.192

0.789

0.813

0.598

O C O C O

Poly(ethylene adipate)

Triclinic

O

(CH2)2

O C

(CH2)4

O Polyamide 6,6: α-isomer β-isomer

Triclinic Triclinic

O

Polyamide 6,10

Triclinic

C

Polyamide 6

Monoclinic

C

O

(CH2)4

C

C O

HN

(CH2)6

NH

HN

(CH2)6

NH

O (CH2)8

C

O C HN

(CH2)5

O Polyamide 11

Triclinic

C HN

(CH2)10

O Poly-4-methylpentene-1

Tetragonal

CH2

CH CH2 CH CH3 CH3

38

1 Poly(ethylene isophthalate)

39

1 Polyvinylchloride

2 Rhombic Monoclinic

3

CH2

CH

4 1.440 1.455

5 0.680 0.687

2.400 2.360

0.794 0.781

1.440

0.742

1.350

0.770

1.110

0.677

1.230

0.719

1.120

0.711

1.506 1.205

0.808 0.723

Cl Polytetrafluoroethylene

Pseudo-hexagonal Hexagonal

Polyvinylfluoride

Hexagonal

Poly(vinyl alcohol)

Monoclinic

–CF2–CF2– CH2

CH F

CH2

CH OH

Polyacrylonitrile

Rhombic

Poly(methyl methacrylate) isotactic

Pseudo-rhombic

CH2

CH C N

CH3 CH2

C C

O CH3

O Rhombohedral

Polyoxymethylene Polyethylene oxide

Hexagonal Hexagonal

CH2 CH

–CH2–O– –CH2–CH2–O–

39

Polystyrene

40

2

3

Rhombic

CH2 CH O CH3

4 1.102 1.154

5 0.663 0.694

40

1 Polypropylene oxide

41

Taking into account that the specific volume at the glass transition temperature Tg equals

Vg = 1

ρg

=

N A ∑ ∆Vi i

,

kg M

(II.13)

where ρg is the polymer density at Tg; and substituting (13) into (11) and (12), we get k (T ) = k (T ) =

kg

[1 + α G (T − Tg )] , kg

[1 + α L (T − Tg )] ,

(T < Tg);

(II.14)

(T > Tg);

(II.15)

Equations (II.14) and (II.15) can be used for obtaining relations, which describe temperature dependences of the density of polymers ρ in the glassy and rubbery states. For this purpose, we substitute (II.14) and (II.15) into equation (II.6):

ρ (T ) =

kg M

[1 + α G (T − Tg )]N A ∑ ∆Vi ,

(T < Tg);

(II.16)

(T > Tg);

(II.17)

i

k (T ) =

kg M

[1 + α L (T − Tg )]N A ∑ ∆Vi , i

Because, as it is seen from the further considerations, values of expansion coefficients αG and αL, as well as the glass transition temperature Tg, can be calculated from the chemical structure of the repeating polymer unit, temperature dependences of density ρ (T) can also be calculated from relations (II.16) and (II.17). In conclusion, let us note that the constancy of the coefficient of molecular packing k is true only for amorphous bulky substances composed of polymers. In the case of crystalline polymeric substances, the situation is significantly changed. If the coefficients of molecular packing for ideal polymeric crystals are calculated with the help of the X-ray analysis data, one can assure himself that, in spite of amorphous ones, the coefficients of molecular packing of crystalline polymers are extremely different. The smallest values of k are typical of aliphatic systems with volumetric side groups, for example, for poly-4-methylpentene-1 and poly-n-butyraldehyde. The highest coefficients of packing are typical of 1,4-trans-β-polyisoprene and polychloroprene. As an example, Table 5 shows the crystallographic values of densities and molecular packing coefficients for a series of typical crystalline polymers. It is clear that the values of k for them vary in a wide range. Hence, crystalline polymers display a rather wide distribution curve of the coefficients of molecular packing (Figure 8).

42

Figure 8. Curve of distribution of the coefficients of molecular packing k for crystalline polymers.

II.2. Relationship between free volume of polymers, coefficient of molecular packing and porous structure Before we start discussing the relationship between the above-mentioned physical characteristics, the term of the ‘free volume’ must be discussed in brief. There are three definitions of the free volume: 1) The free volume represents the difference between the true molar volume of the substance, VM, and its Van-der-Waals molar volume N A ∑ ∆Vi : ∆V = VM − N A ∑ ∆Vi = M / ρ − N A ∑ ∆Vi . i

i

(II.18)

i

The value of ∆V obtained in this way is often called ‘the empty volume’. Clearly, the empty volume depends on temperature, because the molar volume also depends on it: VM = M/ρ. Substituting this relation into equations (II.16) and (II.17), we obtain: 1 + α G T − Tg  ∆V (T ) = N A ∑ ∆Vi  − 1 , (T < Tg); (II.19) kg i  

(

)

(

)

1 + α L T − Tg  ∆V (T ) = N A ∑ ∆Vi  − 1 , (T > Tg); (II.20) kg i   Relations (II.19) and (II.20) describe the temperature dependences of the empty volume. 2) The free volume represents the difference between the volumes of the substance at the absolute zero and at the assigned temperature; to put it differently, the free volume represents an excessive volume occurring as a result of thermal expansion of the substance. This definition of the free volume is most valuable. Moreover, the present free volume is subdivided into the free volume of fluctuation and the expansion volume. 3) The free volume represents the difference between the volume of polymeric substance at the assigned temperature and the volume of the ideal crystal

43

composed of a polymer of the same chemical structure. This definition of the free volume is used extremely seldom. Let us now pass to analysis of the relationship between the free volume of polymers, the coefficient of molecular packing and the porous structure. The porous structure mostly defines their properties. That is why the methods of estimation of the porous structure of polymers and its connection with such characteristics as the coefficient of molecular packing and the free volume of polymer must be discussed in detail. The case is that the size of micropores depends on the method of its estimation. Clearly, interpretation of their nature and the relationship of the characteristics of the microporous structure with the properties of polymers significantly depends on the method of their determination. The properties of many bulky and film polymers significantly depend on the density of packing of macromolecules, and for such systems as sorbents, ionites, etc., used in gel-chromatography and production of ion exchangers, the volume of pores is very important, together with their size distribution, specific surface. Let us present the definition, given in ref. [68]: “Pores are emptinesses or cavities in solids usually connected with each other. They possess various and different form and size, determined significantly by nature and the way of obtaining absorbents”. Usually, the characteristics of a microporous structure are judged by experimental data on equilibrium adsorption, capillary condensation of vapor and mercury pressing in (mercury porosimetry) [121]. Recently, the positron annihilation method has been used [3, 48, 110, 123, 134, 140, 155, 164, 187, 211]. This method helps in determining the characteristics of the microporous structure, when the size of pores is commensurable with the molecule size. Such micropores are inaccessible for sorbate molecules and especially for mercury when mercury porosimetry is used. Polymers and materials prepared from them possess the feature (in contrast to mineral sorbents) that they swell during sorption of vapors of organic liquids. Consequently, their structure changes and usual methods of calculation give no possibility of estimating the true porous structure of the initial material. It stands to reason that vapors of organic liquids, in which polymer does not swell, can be used in sorption experiments. Then the parameters of the porous structure of the initial material can be determined, but these cases are quite rare [107]. Before passing to comparison of parameters of the porous structure with the free volume of the polymer, it should be noted that parameters of the porous structure for the same polymer could be significantly different due to conditions of its synthesis and further processing. For example, a film or fibers may be obtained from various solvents [81], as well as from a solvent–precipitant mixture [97], and will display a different microporous structure and properties. The same can be said about materials obtained by pressing and injection molding and with the help of hydrostatic extrusion as well. Therewith, macropores may also be formed and their total volume may be quite high. If special synthesis methods are used, materials based on polymer networks may be obtained, which possess a large specific surface and extremely large pore radii [115]. Clearly, such macropores are not defined by the packing density of macromolecules. They may be formed by loose packing of formations larger than macromolecules or may be caused by conduction of a chemical process of the network formation under special conditions [167]. Several more general comments should be made. Besides macropores, as mentioned above, micropores are present in a polymeric substance, the size of which is commensurable with the size of sorbate molecules. Clearly, in this case, sorbate

44

molecules cannot penetrate into these micropores (it is assumed that for sorbate molecules to penetrate into pores, the volume of the latter must be several times greater than that of penetrating molecules). Since sorbate molecules may be different, i.e. may possess different sizes, parameters of the porous structure determined from the sorption data will depend on types and sizes of molecules of sorbed substances. That is why such terms as ‘porosity to nitrogen’, ‘porosity to benzene’, etc. have been introduced. Of interest is that the sorption method of determination of the porous structure of polymeric substances cannot be used in the case when a substance contains quite large macropores. This is associated with the fact that under conditions of polymolecular adsorption, when many molecular layers are formed on walls of macropores, their fusion becomes difficult, i.e. capillary condensation is absent. Then, the total volume of pores calculated by the amount of sorbate penetrated into the polymeric substance will be smaller than the true volume of macropores. Starting the analysis of relationship between the physical characteristics of the polymeric substance and its microporous structure, let us introduce some definitions and designations: Ssp is the specific surface of micropores, W0 is the total volume of pores, W0max is the maximal volume of pores accessible for sorbate molecules of any size (per gram of the substance), VF is the free volume (in the present case, the volume of expansion), VE is the ‘empty volume’ (see above), VT is the specific volume of the polymeric substance at given temperature, VW is the Van-der-Waals volume (per gram of the substance), Vid.cr. is the specific volume of the ideal crystal or bulky amorphous polymer (a bulky amorphous polymer is the one in which no sorbate molecule can penetrate into its pores). Let us write down some relations connecting these characteristics: VF = VT – V0 ; (II.21) VE = VT – VW. (II.22) Next, let connect these characteristics with the coefficient of molecular packing k (see above): k = VW/VT; 1 – k = VE/VT. (II.23) As mentioned above, there are so-called non-porous sorbents (for example, crystalline substances), into which no molecules of sorbate can penetrate without swelling. Clearly, that for such substances W0max = 0. At the same time, as seen from the data in Table 5, coefficients of molecular packing of crystals fall within the range from 0.64 to 0.89. Taking into account that the coefficient of molecular packing, by definition, represents a part of the occupied (Van-der-Waals) volume, it can be said that the part of empty (but inaccessible) volume is 1 – k = 0.11–0.36. This empty volume is inaccessible for even small sorbate molecules to penetrate in; let mark it as Vinacc.. Then the volume of the ideal crystal (or bulky amorphous polymer, Vblk) can be written down as Vid.cr. = VW + Vinacc.; Vblk = VW + Vinacc.. (II.24) The volume of the real polymeric substance (which contains micropores accessible for a sorbate) will be summed up from three parts: VT = VW + Vinacc. + W0max. (II.25) Then W0max = VT – Vid.cr.; W0max = VT – Vblk. (II.26)

45

The coefficient of molecular packing in the bulky part of the polymer will be determined from the relation VW k= . (II.27) VT − W0max In the case of estimation of the density of macromolecule packing for the real polymeric substance containing micropores accessible for sorbate molecules, the coefficient of molecular packing, k, should be calculated by the relation VW k= , (II.28) VT − W0 where W0 is the total volume of micropores (per gram of the substance), determined on the basis of sorption measurements. The value of W0max that represents the difference between the specific volume of the substance at the given temperature and volume of the true bulky substance is conceptually identical to the porosity factor P = 1/ρs – 1/ρt, where ρs is the apparent density; ρt is the true density. Therewith, ρs represents the density of the substance at the current temperature, affected by the pores existing in it. It is best to measure the apparent density of substances with the proper geometrical shape, because when using no solvents ρs can be found by dividing the substance weight by its volume. If the apparent density of substances with the improper shape is measured, the pycnometric or dilatometric method can be used. The difficulty is in selection of a liquid that does not wet the surface of the substance and does not penetrate deep into it. The true density ρt represents density of the bulky part of the substance containing no pores. It is best to measure the density of the ideal crystal, because it can be calculated on the basis of crystalline lattice parameters. In the case of amorphous and partly crystalline substances, the method of gradient tubes may be used applying liquids penetrating well into pores. However, it should be taken into account that a mixture of two liquids is used for creation of the density gradient in the tube, each of which may possess different wettability and penetrability into pores. The picture is then distorted, and the determined density is not true. The relations shown above can be estimated unambiguously if a polymer swells in the sorbate, used for estimation of the porous structure of the polymer. If the experiment indicates that W0 is greater than W0max, this indicates that the volume of vapors absorbed by the polymer is greater than the volume of pores existing in it, i.e. the polymer swells during sorption. Let us now consider the experimental and calculated data on determination of the parameters of the polymer structure and coefficients of their molecular packing. These data are shown in Table 6. For ideal polyethylene crystallites, VE = Vinacc. and W0max = 0. The coefficient of molecular packing is quite high. For semi-crystalline polyethylene, the empty volume, VE, is greater than in the case of the ideal crystal and, therewith, a part of it is accessible for penetration of small sorbate molecules. However, the total volume of pores determined by methanol sorption equals 0.01 cm3/g. The molecular packing coefficient for the bulky part of such polyethylene is significantly lower than for the ideal crystal. Polymers in the rubbery state (polyisobutylene, for example) also possess comparatively low values of free volumes and are practically non-porous sorbents (VE = Vinacc.). Contrary to this, polymers produced by polycondensation or polymerization in solution display immensely high values of W0max. In this synthesis method, pores are formed due to elimination of the solvent, distributed in the volume of the synthesized

46

polymer. This is observed from the fact that the same polymers produced by polymerization in the melt are practically non-porous, and values of VE for them are very small, and W0max = 0. Table 6 Parameters of porous structure and coefficients of molecular packing of a series of polymers VE, W0max, W0, Vinacc., K Polymer cm3/g cm3/g cm3/g cm3/g Polyethylene (100% crystallinity) 0.26 ~0 ~0 0.26 0.736

–CH2–CH2– Polyethylene (crystallinity < 100%)

0.35

0.08

0.01

0.27

0.675

0.36

~0

~0

0.36

0.678

1.28 0.22

1.06 ~0

 

0.22 0.22

0.687 0.687

0.82 0.24

0.58 ~0

0.31 ~0

0.24 0.24

0.688 0.688

–CH2–CH2– Polyisobutylene

–CH2–C(CH3)2– Polymethylidenphthalide

CH2 C O C O Polymerization in dimethylformamide solution Polymerization in melt Polyarylate F-1 C

C O

O

O

O C O C O

Polycondensation in chlorinated bisphenol solution pressed at 360°C and under 312.5 MPa pressure

Pores formed during synthesis may be closed in polymer pressing under high pressure, and the porous polymer then becomes non-porous. Therewith, in all cases, W0 is smaller than W0max that indicates the absence of swelling. For all polymers, values of Vinacc. are close to these characteristics for the density of crystallized samples. Of special attention is the fact that independently of the production method, the molecular packing coefficient for amorphous and semicrystalline polymers in their bulky part is the same and close to the average value kavg = 0.681, which was discussed above. For a crystalline sample, the value of k is significantly higher. There is one more interesting point to discuss, associated with molecular packing, namely, the change of the system volume during polymerization, i.e. at transition from monomer to polymer. It is well known that transition from a monomeric liquid to a solid glassy polymer is accompanied by a significant contraction, i.e. volume decrease [76]. The specific volume of the polymer Vp is always smaller than that of monomer Vm, and their difference ∆V = Vp – Vm < 0. One of the reasons for contraction is substitution of longer intermolecular bonds existing in liquid monomers by shorter chemical bonds

47

formed between monomer molecules in the polymer. Therewith, the own Van-derWaals volumes of atoms decrease owing to their ‘compressing’ (see above). Nevertheless, this is not the only reason of contraction. It follows from consideration of the experimentally determined specific volumes that there is another reason for contraction, which is more dense packing of polymeric chains compared with the packing of monomeric molecules. This is indicated by the fact that the packing coefficients of polymers are always greater than those of their monomers (kp > km). Let the total contraction, ∆Vtotal, be presented as a sum of two values: ∆V1, which is the contraction stipulated by substitution of intermolecular bonds by chemical ones, and ∆V2, which is the contraction involved by more dense packing of chains, ∆Vtotal = ∆V1 + ∆V2,

(II.29)

and each of the summands estimated. To do this, values of the specific volume of a polymer should be calculated on the assumption that it displays the packing coefficient, the same as the monomer km, i.e.    ∑ ∆Vi     i p N Vp′ = A ⋅ , (II.30) M km   where  ∑ ∆Vi  is the Van-der-Waals volume of atoms in the repeat unit of the    i p polymer; M is the molecular mass of the unit. Values of Vp′ for some polymers, calculated in this way, are shown in Table 7. They are always greater than experimentally measured values of specific volumes of the polymer, Vp. The difference between Vp′ and Vm is ∆V1 = Vp′ – Vm,

(II.31)

and the remaining part of the contraction is calculated by the formula ∆V2 = ∆Vtotal – V1.

(II.32)

Relative parts of contraction are determined from the relations:

α1 = ∆V1/∆Vtotal; α2 = ∆V2/∆Vtotal.

(II.33) (II.34)

The data shown in Table 7 indicate that in all the cases the smaller part of contraction depends upon opening of double bonds, and the greater part — on dense packing of polymer chains. Therewith, the chemical structure of a monomer and an appropriate polymer significantly affects the values of α1 and α2.

49

Polymer (monomer) 1

CH3 CH2

∆V1, cm3/g 6 0.080

∆V2, cm3/g 7 0.133

α 1, % 8 37.6

α2, % 9 62.4

C C O CH3 O CH3

CH2

1.102

0.890

1.031

0.212

0.071

0.141

33.5

66.5

1.109

0.928

1.045

0.181

0.064

0.117

35.4

64.6

1.046

0.815

0.951

0.231

0.095

0.136

41.1

58.9

1.082

0.873

1.000

0.209

0.082

0.127

39.2

60.8

C C O C2H5 O CH3

CH2

C C O C3H7 O

CH2

CH C O CH3 O

CH2

CH C O C2H5 O

48

Table 7 Changes in volume of the system as a result of polymerization Vm, cm3/g Vn, cm3/g Vtotal, cm3/g V′n, cm3/g 2 3 4 5 1.068 0.855 0.968 0.213

50

1

CH2

CH

2 1.098

3 0.952

4 1.036

5 0.146

6 0.062

7 0.084

8 42.5

9 57.5

1.073

0.841

0.976

0.232

0.097

0.135

41.8

58.2

1.104

0.942

1.028

0.162

0.076

0.086

46.9

53.1

C O C4H9 CH2

O CH O

CH2

C CH3 O CH

49

50

In the set of polyacrylates and polymethacrylates α2 grows first with the volume of the side substituent and then decreases. Decrease of the intensity of the effect of the dense packing of chains, apparently, depends upon steric hindrances. Hence, it follows from the above-said that the notions of porosity and packing density are inadequate. Porosity reflects almost always cavities greater than the molecular size, i.e. quite large ones. As for the packing density of macromolecules themselves, it may be judged by considering the non-porous part of the sample only. As noted above, application of positron annihilation methods is preferable for analyzing the microporous structure of polymers [3, 48, 110, 123, 134, 140, 155, 164, 187, 211]. With the help of these methods, qualitative and quantitative information about the characteristics of submicropores (2–15 Å) in polymers may be obtained. Let us discuss the results of studying annihilation of positrons in two polymers, which are good models of the limiting characteristics of the packing density of macromolecular chains. One of them is polyimide characterized by a highly regular, quasi-crystalline structure, and the second is poly(1-trimethylsilyl-1-propyne) (PTMSP) which, on the contrary, is characterized by a low coefficient of molecular packing. Consider structural changes in PTMSP, which appear during its long exposure at room temperature after synthesis. For comparison, we also display the data on annihilation of positrons for a series of other model polymers. The chemical structures of all above-mentioned systems are shown below. Poly(1-trimethylsilyl-1-propyne) CH3 C

C

Si

H3C

CH3

CH3 n

Polyisoprene CH2

C CH CH2 CH3

n

Polydimethylsiloxane CH3 O

Si CH3

n

Polystyrene CH2

CH

n Polytetraflouroethylene [—CF2—CF2—]n

51

Polyimide O

O

C

C

C

C

O

O

N

N

O n

Observation of the annihilation of positrons in PTMSP was performed with the help of a method of detection of the lifetime spectra of positrons (measurements were made by S.A. Tishin; data not published). Measurements were performed by a thermostabilized spectrometer, which realizes the traditional fast–slow scheme of detection, with a temporal photomultiplier selected and optimized due to an original method [111]. Processing of experimental spectra was performed with the help of wellknown software ‘Resolution’ and ‘Positron FIT’. Table 8 shows the results of separation of parameters of a long-living component at three-component decomposition of positron lifetime spectra for PTMSP, polyimide, polystyrene, polydimethylsiloxane and polytetrafluoroethylene. Clearly, PTMSP possesses an anomalous long lifetime for an ortho-positronium atom, to annihilation of which by a pick-off–decay the origin of a long-living component of the lifetime spectrum in polymers is bound [3, 48, 110, 123, 134, 140, 155, 164, 187, 211]. Hitherto, the maximal lifetime of the long-living component, τD, was observed in polydimethylsiloxane and teflon in solid polymers [123, 164]. Comparison with the results of measurements in model polymers (see Table 8) indicates that neither the presence of an unsaturated bond, nor the presence of a side group or silicon atom separately is the explanation of so high τD for PTMSP. Table 8 Parameters of the longest component of positron lifetime spectrum for a series of polymers and rated values of radius R and volume V of micropores R, Å V, Å3 E, eV Sample τD + 0.03, ns ID ± 0.25, % R0, Å PTMSP 5.78 38.4 6.76 5.10 416.5 0.41 Polytetrafluoroethylene 4.27 21.6 6.05 4.39 265.8 0.51 Polydimethylsiloxane 3.23 41.3 5.45 3.79 170.9 0.63 Polyimide 2.77 38.1 5.14 3.48 132.1 0.71 Polystyrene (atactic) 2.05 40.5 4.56 2.90 76.9 0.90

Two suggestions about the reasons of anomalous long average lifetime of positrons in PTMSP can be made. First, molecular structure of the repeat unit allows a supposition that a high concentration of bulky, low-mobile side groups creates a porous structure with the pore size of about Van-der-Waals volume of –Si≡C3H9 side fragment. Secondly, the size of pores may be associated with a long relaxation time of synthesized PTMSP at room temperature. It may be suggested that the formation and evolution of microcavities of a large size must depend on the motion of large segments of macromolecules or even structural fragments with a long period of regrouping. The lifetime of an ortho-positronium atom regarding the pick-off–annihilation allows estimation of the size of the microcavity in which it was localized before annihilation [140]. The calculation results are also shown in Table 8.

52

In line with the model [140], positronium is considered in a spherical pit surrounded by a layer of electrons, ∆R thick. For wave functions in spherical coordinates: (2ŒR0 )−1 / 2 ⋅ 1 ⋅ sin (Œr / R0 ) in the pit; r %( r ) =  (II.35)  0 outside the pit. The probability of positronium existence outside the limits of density will be:  2 ŒR  R 1 , W ( R) = 1 − + sin  (II.36) R0 2Œ  R0 

where R = R0 – ∆R. Suggesting that the rate of ortho-positronium annihilation inside the electron layer equals 0.5 ns–1, the decomposition rate averaged over spins will be:

λD = 1/τD = 2W(R)

(II.37)

with the constant ∆R = 1.66 Å, selected empirically for solids. Let us consider the results of measurements of PTMSP films porous structure because of their aging. Long-term relaxation of PTMSP films was investigated with the help of measuring positron lifetime spectra. As Table 9 and Figure 9 indicate displaying a series of characteristics of time spectrum decomposition into three components and the calculated radius of micropores R, and durability of samples aging, lifetime of the long-living component decreases with growth of PTMSP exposure time at room temperature. In practice, the intensity of the long-living component does not depend on the relaxation time. Table 9 Long-term relaxation of PTMSP from the data of measurement of the longest component parameters of positron lifetime spectrum (τn is lifetime of intermediate component) Aging time, days RD ± 0.25, % τD ± 0.03, ns τn ± 0.080, ns 13 5.78 38.40 0.687 17 5.68 37.53 0.607 24 5.72 38.09 0.678 83 5.40 38.08 0.507 210 5.09 37.91 0.453

Figure 9. Dependence of sizes R of the positron-sensitive microcavity on time of exposure tc at 25°C for PTMSP

53

The result observed is connected with slow structural relaxation but not the ‘aging’ (if by the ‘aging’ occurrence of the main chain fission is meant), because the latter process is usually accompanied by changes in intensity ID (results of observing long-term aging of polyethylene by the method of positron lifetime variation may be displayed as an example, although ‘aging’ in polymers is a very specific process). Taking into account the relation between τD and the radius of micropores in polymers [140], it must be concluded that in long-term relaxation of PTMSP sizes of pores decrease (see Figure 9) and, probably, the mobility of macromolecular chains reduces due to free volume decrease. As follows from the constancy of ID, the concentration of positronium traps is independent of the exposure time in the studied time interval. Let us now discuss the results of investigation of positron annihilation in polyimide. As the measurements have shown [48], annihilation of positrons in polyimide is significantly different from the one usually observed in most polymers. The annihilation spectrum in polymers is usually characterized by the presence of three or four components with average lifetimes from 100 ps to 4 ns [54, 164, 187]. However, the different structure of the spectrum is observed for polyimide. It displays a single, short-term, component with τ0 = 0.388 ns (Figure 10). Time distribution is approximated well by a single decay line, the tangent of which determines the average lifetime.

Figure 10. Positron lifetime spectrum τ of the starting polyimide film (here N is the number of readings in a channel)

The value of lifetime and the spectrum structure allow a supposition that annihilation in polyimide proceeds from the positron state without forming a positronium atom as it is typical of metals and semiconductors with high mobility of electrons and a regular crystalline structure. In this meaning, polyimide forms an electron structure unique for polymers, characterized by high values and high homogeneity degree of the density function for electrons.

54

Figure 11. Lifetimes τ and intensities of components (%) in the spectra of the original sample (I) and deformed samples of polyimide after recovery lasting for 1 (II) and 24 (III) hrs. Table 10 Sample Initial Deformed Deformed

Annihilation characteristics of polyimide film Recovery τ0, ps τ1, ps τ2, ps lasting, hr 385±5    1 294±30 440±17  24 361±10 531±30 

I2, %  59±5 9±2

Count rate, k⋅10–9, s  0.60±0.15 0.12±0.05

In relation to interaction with positrons, the microstructure of the initial (undistorted) polyimide film possesses no defects. However, time spectra change after deformation (Figure 11 and Table 10). Two components instead of a single one are observed in the deformed sample: with shorter and longer lifetimes. After recovery (resting) during 24 hours at room temperature, an increase of lifetimes of both components and reduction of intensity of longer-term ones are observed. The character of changes taking place allows a supposition that the submolecular structure of polyimide is rebuilt during deformation; intermolecular bonds break, and microdefect free volumes enough for positron localization – are formed. In this case, the value of the long-term component τ2 must reflect changes in the average size, and intensity I2 – concentration of these defects. Analogous changes in the spectra were also observed in annealing defects in metals and semiconductors. These changes are usually analyzed with the help of a positron entrapment model. This model is qualitatively good in reflecting changes in the time spectra observed in polyimide deformation. Reduction of the lifetime of the short component, bound to annihilation in the undistorted part of the polymer, depends on the high rate of capture in the deformed sample. After partial contraction during recovery, the concentration of defects decreases and lifetime τ2 approaches the characteristic one of the original polymer. Therewith, the intensity of the long-term component, I2, formed due to positron annihilation on defects, decreases, too. Growth of the lifetime τ2 may be explained by coagulation (consolidation of small defects into larger ones) during recovery or fast relaxation of small pores and, consequently, by growth of the average capture radius. As indicated in estimations, the concentration of microdefects after partial relaxation decreases more than 7-fold. Therewith, the free volume induced by deformation decreases by a factor of 4 [48]. The values obtained indicate that two processes proceed – fusion of microdefects and relaxation of the smallest ones, though, apparently, the intensity of the latter process is higher.

55

Hence the one-component spectrum is typical of the original polyimide film. In deformed samples, at least two components are observed in time spectra, which are bound to the positron annihilation from the free state and the one localized in micropores, formed at stretching. The lifetime increases and the intensity of the defect component decreases during relaxation. The results obtained with the help of the model of positron capture describe clearly the changes of time distributions observed and allow a supposition that the structure of the free volume during relaxation changes not only as a result of fast recombination of the smallest pores, but also because of their consolidation with the formation of long-term large-size microcavities. Basing on the analysis performed in ref. [48], the following model of positron annihilation and relaxation mechanism bound to it are suggested: before deformation all positrons, captured in small traps with the bond energy slightly higher that the heat energy, annihilate; after deformation, rather long (compared with the positron diffusion length) areas occur, in which the concentration of small traps (of the size ~10 nm) decreases significantly, loosened up areas with deep centers of positron capture are formed simultaneously in which the lifetime of positrons is longer; relaxation happens in the way that pores formed during deformation recombine and, moreover, increase when consolidate. Hence, measuring the lifetime of positrons, the data on changes in structure of the free volume occurring after polymeric film deformation may be obtained. However, interpretation of the information obtained requires a detailed study of the nature of components of a complex time spectrum of annihilation typical for a nonequilibrium state of polymer. No solution of this problem with the help of one of the positron methods was obtained [3, 110, 156]. That is why a complex study of positron annihilation was performed [49] in deformed polyimide with the help of measuring the lifetime of positrons and angular correlation of annihilation radiation. Two series of experiments are described in ref. [49]. In the first series, a polyimide film was stretched by 20%. Then, the film was set free and relaxed freely. Lifetime spectra for the freely relaxed film were measured every 1.5 hours. Parameters of angular distribution were determined every hour during the day. Table 11 Change of annihilation characteristics of polyimide film depending on duration of relaxation after deforming by 20% Lifetime Angular correlation Relaxation lasting I2±1.5, after deforming, h % FWMH± τavg±1, τ1±10, Γ1±0.07, Θρ±0.07, Iρ±1.5, ps ps mrad mrad % 0.05, mrad 0 365 201 74.3 10.44 10.49 7.14 28.2 1 360 176 73.6 10.77    5 368 208 77.2 10.60    24 362 205 73.0 10.48 10.64 7.14 34.7 240 364 200 74.1 10.43 10.72 6.95 32.3 Separated 368 220 76.3     Note. τavg, τ1 and I2 are characteristics of positron lifetime spectra; FWMH is the full width on the middle height of the full spectrum; Γ1 is FWMH of the first Gaussian; Θρ and Iρ are characteristics of the parabolic component of the angular correlation spectrum.

In the second series of experiments, stress relaxation at deformation ε 0 = 20% was studied. The characteristics of angular distributions were determined for films with fixed ends. Measurements were performed with the help of a device that performs deformation of samples directly in the measurement chamber. Stress

56

relaxation curves (dependences of stress σ on time τ) and recovery curves (dependences of deformation ε on time τ) were taken simultaneously. The values of the positron lifetime obtained from spectra are shown in Table 11 and Figure 12. Similar to the above-described results of two-component analysis, changes of annihilation characteristics, which then relaxed gradually to those typical of the initial polyimide sample, were observed in the structure of the time spectrum, approximated by three components, after deformation.

Figure 12. Positron lifetime spectrum as a function of relaxation time for freely relaxing polyimide films (for designation see Table 11).

Three components were separated: the lifetime of the first short-term components (170–220 ps) significantly depend on relaxation time; as displayed by investigations [49], the lifetime of the second one (388±10 ps) is independent of or weakly depends on the sample state. However, significant changes in the intensity of this component are observed. The characteristics of the third component have not changed during the experiment. In the work cited, experiments on measuring the angular correlation were performed (alongside the measurement of the positron lifetime). Making no detailed analysis of the results of these measurements, note that in experiments with fixed ends (under stress relaxation conditions) the free volume significantly increases after deformation, and its further slow relaxation is displayed well, happened at the sacrifice of a decrease of micropore concentration. In most cases, changes of macro- and microparameters of the polyimide film during stress relaxation and recovery after deformation were indicated by the method of positron diagnostics. Non-monotonous changes in the characteristics of positron lifetime spectra and angular distributions of annihilation photons during recovery were observed. Two ranges of changes in positron-sensitive properties of polyimide,

57

associated with ‘fast’ and ‘slow’ relaxation processes, were separated, and differences in the type of relaxation of the polymer microporous structure depending upon the condition of deformation and ‘rest’ were observed. The effects observed are stipulated by formation of areas of the local ‘defrosting’ of molecular mobility. All these experimental facts indicate that the microporous structure of the polymer is rearranged during stress relaxation; this is expressed by the redistribution of the sizes of micropores and their merging. Hence the method of positron annihilation allows not only estimation of the microporous structure of polymers, but also following its change under mechanical loading.

Chapter III. Temperature coefficient of volumetric expansion

The thermal expansion of solids is a consequence of anharmonicity of thermal oscillations of the substance particles. The thermal expansion of polymers has a number of peculiarities connected with various physical transitions occurring in the polymer as temperature is increased. To estimate experimentally the temperature coefficient of volumetric expansion, the temperature dependence of the specific volume of the polymer is determined. Schematically, this dependence is depicted in Figure 13.

Figure 13. Schematic representation of the dependence of specific volume V on temperature T (dilatometric curve) (rate of heating q1 > q2 > q3 > q4).

This dependence as a broken line is typical of many polymers near the glass transition temperature, Tg. At temperatures below the glass transition temperature this dependence is flatter than in the range of temperatures above it. Hence if T < Tg, the temperature coefficient of volumetric expansion (which represents a tangent of dilatometric dependence) is smaller than when T > Tg. In the first case, the temperature coefficient of volumetric expansion is designated as αG, and in the second one – αL. In this connection, the specific volume of the polymeric substance may be calculated by equations V = Vg[1 + αG(T – Tg)], V = Vg[1 + αL(T – Tg)],

(T < Tg); (T > Tg),

(III.1) (III.2)

where Vg is the specific volume of the polymer at the glass transition temperature; T is temperature. The dilatometric dependence shown in Figure 13 is rather simplified. In fact, we are dealing not with a broken line, but with a curve called dilatometric. The curvature of dilatometric dependences may be ambiguous. First of all, transition from the glassy state into the rubbery one is characterized not by an abrupt fracture on the dilatometric curve but by a smooth transition of one branch of the dilatometric curve into another. This is clearly seen from Figure 14 which displays an experimental dilatometric curve for polystyrene, determined near the glass transition temperature (105°C) of this polymer. The glass transition temperature itself is determined by intersection of tangents of two branches of the dilatometric curve. Secondly, if the

59

dilatometric curve is determined in a wide temperature range, we may ensure that in the area below the glass transition temperature it is not linear all the way, but indicates a clear curvature.

Figure 14. Dependence of specific volume V on temperature T for polystyrene.

Figure 15. Dependence of the temperature coefficient of volumetric expansion αG on temperature T for poly(methyl methacrylate).

According to this curve, with decreasing temperature the coefficient of volumetric (or linear) expansion is not the constant of the polymeric substance. Figure 15 displays the experimental temperature dependence of the coefficient of linear expansion for poly(methyl methacrylate), determined in a wide range of temperatures [154]. It is easy to verify that the value of αG decreases with temperature, i.e. the dilatometric dependence at T < Tg is not linear. In this case, to calculate the volume of the polymeric substance, it is not enough to use equation (37), but it is necessary to turn to a more general relation 1 ∂V αG = ⋅ , (III.3) V0 ∂T where αG is the thermal coefficient of volumetric expansion depending on temperature; V0 is the specific volume of the polymer near the absolute zero. Knowing this dependence, the specific volume of a polymeric substance at any temperature T may be calculated by equation (III.3). Dilatometric dependences are not only of practical meaning as the ones allowing searching of the glass transition temperature of polymers. They are also theoretically valuable. First of all, slope changes not associated with the polymeric substance transition from the glassy into the rubbery state are observed at some temperatures below the glass transition temperature. These transitions occur at temperatures below the glass transition temperature (e.g. inside the glassy state area)

60

and are of a somewhat different nature as compared with the main transition. Hence, temperatures of these transitions may be determined by the dilatometric curve. Secondly, according to the concept developed by Boyer and Simha, and formulated by Flory, the transition from the glassy state into the rubbery one takes place at the temperature at which part of the free volume in the polymer becomes the same and equal to fc = 0.025. Due to this concept, the following relation holds [205]: (αL – αG)Tg = 0.113.

(III.4)

This relation is a rough approximation, because it takes into account no curvature of the dilatometric dependence. Taking into account this curvature enables Simha [154] to refine the free volume concept and to determine it with higher accuracy. However, even in this case, the concept is just a rough approximation, although it allows a description of the glassy state–rubbery state transition. Table 12 displays experimental values of αG for a series of glassy polymers. The lower the glass transition temperature, the higher is the coefficient of thermal expansion. This correlates with the Simha–Boyer concept and equation (III.4). Hence, heat-resistant polymers displaying high glass transition temperatures possess lower αG, and traditional polymers softening at low temperature display higher coefficients of thermal expansion which, as a consequence, depend on the chemical structure of the polymer. Table 12 Calculated αG,calc and experimental αG,exp values of thermal coefficients of volumetric expansion and the glass transition temperature Tg for a series of glassy polymers Tg, K Polymer αG,exp⋅104, K–1 αG,calc⋅104, K–1 Poly(methyl methacrylate) 378 2.69 2.55 Poly(ethylene methacrylate) 338 2.99 2.84 Poly-n-propyl methacrylate 308 3.19 3.05 Poly-n-butyl methacrylate 293 3.34 3.22 Poly(methyl acrylate) 293 3.03 2.80 Polystyrene 378 2.50; 2.83 2.50 Polycarbonate based on bisphenol A 423 — 2.27

In ref. [35], the problems discussed above had been studied in detail for polymer networks based on epoxy resins. Cured bulky samples were obtained using epoxy resin ED-20, methyltetrahydrophthalic anhydride as a curing agent, and azelaic acid (to elongate linear fragments between network cross-linked points), and oleic acid (to obtain ‘suspended’ chains). It was found that the coefficients of molecular packing for cured networks based on epoxy resins are higher than those for linear polymers. This is typical of systems containing no ‘dangled’ chains (branches), i.e. when azelaic acid is used as a co-curing agent. At room temperature, the coefficient of molecular packing for them is almost independent of the network composition, and its average value is kavg = 0.694, which is somewhat higher than the average value kavg = 0.681 for linear glassy polymers. Therewith, the average coefficient of molecular packing of cured networks at their glass transition temperature is kg = 0.681, which is also greater than kg = 0.667, typical of linear polymers. According to the data of these measurements (αL – αG)Tg = 0.106.

(III.5)

61

Calculations and measurements have also indicated that for cured epoxy resins the fraction of the free volume, formed due to thermal expansion, is 0.078. This value is calculated from the formula f =

Vg − V0 Vg

= α G Tg ,

(III.6)

where Vg and V0 are specific volumes of the polymer at the glass transition temperature Tg and near the absolute zero, respectively. The same value is determined from the relation k f = 0 − 1, kg

(III.7)

where kg and k0 are the coefficients of molecular packing at the glass transition temperature Tg and near the absolute zero, respectively. For linear polymers, the value of f equals 0.096. As mentioned above, the thermal expansion of substances is a consequence of anharmonicity of thermal oscillations of the substance particles. On this basis, it can be suggested that the coefficient of thermal expansion consists of contributions of various oscillations of these particles. Above all, the role of a weak dispersion interaction must be taken into account. It should be noted that every atom is characterized by self-dispersion interaction, which depends on both the type of the atom and its surrounding atoms, i.e. on the atoms chemically bonded with it. In the calculation scheme [28, 43], to calculate the coefficient of thermal expansion, it was suggested that contributions of each atom are proportional to the part of the Van-der-Waals volume ∆Vi of it in the total Van-der-Waals volume ∑ ∆Vi of the repeat unit of the polymer. i

Moreover, it is essential to take into account the influence of strong intermolecular interactions, which appear in the presence of various polar groups in C O, the repeat unit of polymer. To them are corresponded, first of all, ester O

nitrile –C≡N groups, and various halogens which substitute hydrogen atoms (–CHCl–, –CHF–, –CF3), etc. These groups cause dipole–dipole interactions of various types. The most significant influence is also caused by polar groups, which lead to occurrence of hydrogen bonds. They are, for example, amide NH C , urethane

NH C O

, hydroxylic –OH, acidic

O C OH groups.

O O Clearly, the energy of hydrogen bonds, similar to the dipole–dipole interaction, will depend on the chemical structure of polar groups. Seemingly, their contribution to the coefficient of thermal expansion must be different. However, if different parameters to characterize the energy of strong intermolecular interaction are introduced for each type of the dipole–dipole interaction and hydrogen bonds, this will not only make the calculation scheme more complicated, but will also make impossible calculations of the coefficient of thermal expansion for polymers

62

containing new polar groups. That is why discussion in works [28, 43] was limited by the first approximation, according to which contribution of any dipole–dipole interaction is defined by the same parameter βd, independent of the chemical structure of the polar group. However, since the Van-der-Waals volume of each polar group is different, it will be seen in discussion below that the contribution of each polar group to the coefficient of thermal expansion is also different. Concerning hydrogen bonds, we may also confine to a single parameter βh, which characterizes the energy of hydrogen bonds. The exception is only the class of polyamides which have a specific behavior and require several parameters βh characterizing the energy of hydrogen bonds. Consequently, a relation to calculate the thermal coefficient of volumetric expansion was obtained for polymers existing in the glassy state as follows:

∑α i ∆Vi + ∑ β j αG =

i

∑ ∆Vi

j

,

(III.8)

i

where αi are partial coefficients of thermal volumetric expansion, stipulated by weak dispersion interaction of the i-th atom with the neighbor atoms; ∆Vi is the Van-derWaals volume of the i-th atom; βj are parameters characterizing contribution of each type of specific intermolecular interaction (dipole–dipole, hydrogen bonds) to the coefficient of thermal expansion. Let us consider the physical meaning of parameters αi which characterize the weak dispersion interaction. It is well known that the coefficient of volumetric expansion is described by the relation

αi =

3 Rδ i

γ i2 r0, i

,

(III.9)

where R is the universal gas constant; δi is the anharmonicity coefficient,

δi = 1

2

∂ 3ϕ ∂r 3 r 0,i

; ϕ is the potential of the i-th atom interaction with the adjacent ones; γi

is the harmonic force constant, γ i =

∂ 2ϕ ∂r 2 r 0,i

; r0,i is the distance between the

considered i-th atom and adjacent atoms. To estimate the coefficient δi and the harmonic force constant γi the Lennard– Jones potential can be used

ϕ(r) = D[(r0/r)12 – 2(r0/r)6].

(III.10)

In equation (III.10), the value D characterizes bond energy, and r0 is the equilibrium distance between atoms, unbonded chemically, but participating in the intermolecular interaction. Then, it may be written that

63

γi =

72 Di r02,i

; δi =

756 Di r03,i

; αi =

7 R 7 R . ; Di = 16 Di 16 α i

(III.11)

Table 13 Values of constants αi, βj, ai and bj for various atoms and types of intermolecular interaction Atom or type of intermolecular ai⋅103, bj⋅103, Symbol Symbol αi interaction K–1 Å3K–1 Carbon 0.00 aC 0.02 — αC Hydrogen 19.98 — 1.92 aH αH Oxygen in the backbone 22.95 — 2.21 aO,m αO,m Oxygen in the side group aO,s — –O– 16.00 1.54 αO,s — =O 0.77 8.00 aO,s′ αO,s′ Nitrogen in the backbone 8.62 — 0.83 aN,m αN,m Nitrogen in the side group 6.35 — 0.61 aN,s αN,s Chlorine 4.01 — 0.39 aCl αCl Fluorine 6.90 — 0.66 aF αF Sulfur in the backbone 7.50 — 0.72 aS,m αS,m Sulfur in the side group 2.04 — 0.20 aS,s αS,s Silicon in the backbone 8.30 — 0.80 aSi,m αSi,m Silicon in the side group 0.20 — 0.00 aSi,s αSi,s Boron in carboranes –10.00 — –0.96 aB αB Dipole–dipole interaction* — –55.4 –5.31 bd βd Hydrogen bond** — –139.6 –13.44 bh βh Type of substitution of benzene rings*** bp –25.6 –2.41 — paraβp 16.0 1.54 — methabm βm 16.0 1.54 — orthobo βo Coefficient for polydienes 12.96 — 135.0 b≠ β≠ Aliphatic cycle — –120.0 –11.52 bcycle βcycle * Parameters βd and bd are introduced for each branching in the main or side chain; they are also introduced in the presence of a polar group of any type; if aliphatic polymers possess two CH3-groups or two atoms of F or Cl at carbon atom, then constant bd is neglected. For fragments - CH – (bd = 51) - CH – (bd = 32) - CH – (bd = 51) - CH - (bd = 32) | | | | C–OO–CC–SS–C|| || || || O O O O the additional constant are introduced shown in brackets. ** Constant bh is introduced in the presence of a hydrogen bond of any type for all polymers, except polyamides; for the latter, constants βh and bh are shown in Table 18. *** Constants βp, βm, βo and bp, bm, bo are introduced at substitution of aromatic rings in para-, methaand ortho-positions, respectively; the number of these constants equals to the number of substituted rings. In the case of

structure, 2βp and 2bp are introduced.

The expressions (III.11) allow estimation of the energy of the dispersion interaction for each atom. These values are shown in Table 13. They indicate that values Di really correspond to the energies of the intermolecular interaction, but not to energies of the chemical bond. However, if these values are estimated by other methods [66] (designate them as Di0), it is found that values Di differ several times from Di0. This happens because every atom in any low-molecular substance or

64

polymer is in intermolecular interaction not with a single atom, but with several atoms, with which it coordinates. Then, it should be written: Di = zDi0,

(III.12)

where z is the coordination number. In this case, the total average energy of the weak dispersion interaction is determined from the relation z, where

< D0 > =

∑ ∆Vi i

(

∑1 D0,i ∆Vi

).

(III.13)

i

For copolymers, equation (III.8) gives:     α1 ∑ α i ∆Vi + ∑ β j  + α 2  ∑ α i ∆Vi + ∑ β j  + ... +     j j  i 1  i 2 → αG =     α1 ∑ ∆Vi  + α 2  ∑ ∆Vi  + ... +      i 1  i 2   + α n  ∑ α i ∆Vi + ∑ β j    j  i n → ,   + α n  ∑ ∆Vi     i n

(III.14)

where α1, α2, …, αn are molar parts of the components 1, 2, …, n (do not mix these values up with the values αi present in brackets in the expressions);        α ∆V + β  ,  α ∆V + β  , …,  α ∆V + β  represent the ∑ ∑ ∑ ∑ ∑ ∑ i i j i i j i i j       j j j  i 1  i 2  i n   selection of constants αi, βj for the components 1, 2, …, n of copolymer;  ∑ ∆Vi  ,    i 1      ∑ ∆Vi  , …,  ∑ ∆Vi  are Van-der-Waals volumes of the repeat units of these      i n  i 2 components. In reduced form, the relation (III.14) can be written as follows:

65

k =n



k =1

 i



∑ α k  ∑α i ∆Vi + ∑ β j 

αG =

k

j

k =n

  ∑ α k  ∑ ∆Vi  k =1  i k

.

(III.15)

If it is advisable to express the thermal coefficient of volumetric expansion αG of the copolymer via analogous coefficients αG,1, αG,2, …, αG,n for appropriate homopolymers, the substitution of (III.8) into (III.14) gives:       α1α G,1 ∑ ∆Vi  + α 2α G,2  ∑ ∆Vi  + ... + α nα G, n  ∑ ∆Vi         i 1  i 2  i n , αG =             α1 ∑ ∆Vi + α 2 ∑ ∆Vi + ... + α n ∑ ∆Vi        i 1  i 2  i n

(III.16)

where α1, α2, …, αn are the molar parts of the components 1, 2, …, n of the copolymer. In reduced form, the relation (III.16) becomes

αG =

k =n





k =1 k =n

 i

k

∑ α kα G, k  ∑ ∆Vi    ∑ α k  ∑ ∆Vi  k =1  i k

.

(III.17)

It should be noted that the dependence of αG on the copolymer composition α is not linear but has a curvature dependent on the energy of intermolecular interaction of components and their Van-der-Waals volume (Figure 16).

Figure 16. Schematic representation of the dependence of the thermal coefficient of volumetric expansion αG on the composition of copolymer α .

The calculated values of the coefficients of thermal expansion of a series of polymers in the glassy state are shown in Table 12. Generally, it should be taken into

66

account that, despite the seeming simplicity of this physical characteristic, its experimental determination is rather difficult. That is the reason why greatly different values of αG for the same polymer can be found in the literature. All calculated and experimental values of αG shown in Table 12 characterize a part of the dilatometric straight line which directly adjoins the glass transition temperature. With regard to the thermal coefficient of volumetric expansion αL in the rubbery state, as mentioned above, it can be determined with the help of relation (III.4), although significant errors are possible in this case. The monograph [214] by Van-Krevelen indicates another relation for estimating αL:

αL = εl/VM = εlρ/M,

(III.18)

where εl = 10–3VM, VM is the molar volume (per repeat unit of a polymer); VM = N A ∑ ∆Vi , where NA is the Avogadro number, ∑ ∆Vi is the Van-der-Waals i

i

volume of the repeat unit; M is the molecular mass of the repeat unit; ρ is the polymer density. Taking into account that according to the data by Van-Krevelen VM = 1.60VW in the rubbery state, it follows from the relation (III.18) that the coefficient of thermal volumetric expansion for polymers in the rubbery state is the same and equals

αL ≈ 6.3⋅10–4 K–1.

Chapter IV. Glass transition temperature of polymers

IV.1. Thermomechanical and other methods of evaluation of the glass transition temperature of polymers The thermomechanical method of studying polymers is one of the most widespread methods of experimental determination of glass transition temperature Tg. This method has been developed by V.A. Kargin and T.I. Sogolova. The essence of the method is the following. A polymeric substance is subjected to a permanent or variable load. Therewith, its deformation at each temperature and selected time of the load effect is recorded. It is known that if a polymeric sample is subjected to a constant stress, creep develops in it. Graphically, this looks as depicted in Figure 17. To perform these experiments under comparable conditions, deformation must be measured during a strictly constant time of observation, which may be any but advisable so that reading of deformation would performed on the second, flattened part of the creep curve. Having performed such an experiment at different temperature, a temperature dependence of deformation may be plotted which, in the general case, will be the same as depicted in Figure 18. Before we proceed to consideration of features of the graphic in Figure 18, note that an analogous curve may also be obtained in the case, when the sample is subjected to effect of a low constant force, and temperature rises permanently with time. Such a view of thermomechanical tests is most widespread at present, with temperature usually increasing in accordance with a linear law T = T0 + αt,

(IV.1)

where T0 is the temperature of the onset of the experiment; t is time; α is the rate of temperature growth with time.

Figure 17. Schematic representation of creep (dependence of deformation ε on time t at constant stress).

Usually, the standard rate of temperature growth is from 1.5 to 4 deg/min. This method of thermomechanical investigation is more suitable for the apparatus, although it gives less information than the first one. This concerns especially the cases when fine structural transformations occurring in the sample at heating are studied by the thermomechanical method. We will return to this question below, and let us now consider the thermomechanical curve of the ideal polymer displaying no structural

68

transformations at heating. Such a curve is shown in Figure 18, and it is obtained in the case when an experiment starts from rather low temperatures.

Figure 18. Schematic representation of the thermomechanical curve (dependence of deformation ε on temperature T) on which I is the glassy state of polymer, II is the rubbery state of polymer and III is the viscous flow state of polymer.

In this temperature range, deformation of the sample is small and increases insignificantly with temperature. When the temperature definite for each polymer is reached, deformation begins growing rapidly. If the load is low, increase of deformation proceeds not infinitely but ends quite fast. A plateau occurs on the curve, which is called ‘the rubber-like plateau’. The extent of this plateau on the temperatre axis may be quite long and, as indicated below, it depends on the molecular mass of the polymer. Further heating induces again a sharp increase of deformation, and the polymer sample spreads. The thermomechanical curve depicted in Figure 18 clearly divides the temperature area of the test into three intervals. In the first of them, the polymer is in the solid, glassy state (if it is amorphous). In this state, deformation is rather low and grows only slowly with temperature: the polymer behaves itself almost like lowmolecular substances. The first sharp increase on the thermomechanical curve is connected with the polymer transition from the solid glassy state into the rubbery state. This state is characterized by acquiring rubbery properties by the polymer, i.e. it possesses the ability to develop high reversible deformations under low stress. This state is typical only for polymers, and it is not displayed in the case of low-molecular weight solids. At further heating, a sharp increase of deformation is connected with occurrence of the viscous flow state, for which viscous flowing of the polymeric substance is typical. Appropriate temperatures of transitions from the glassy state to the rubbery state and from the rubbery state into the viscous flow state have received the names ‘the glass transition temperature’ and ‘the flow temperature’. Before we discuss the nature of each of the physical states of polymers, note that depending on the chemical structure of the polymer, i.e. on flexibility or rigidity of its macromolecules, the glass transition temperature may obtain different values. At the present time, there are polymers which display the glass transition temperature from –123 to 600°C. An example of the first of them is polydimethylsiloxane with the following formula: CH3 Si CH3

O n

69

An example of the second one is polynaphthoylenebenzimidazole of the following chemical structure: N

...

C

C N

N C

C

... N O Synthesis of polymers with the lowest possible glass transition temperatures is of great importance for obtaining polymers resistant to cold. Synthesis of polymers with the highest possible glass transition temperature is of the same importance for obtaining solid heat-resistant polymers, capable of service at increased temperatures and stresses. Many polymers, applied at present, possess glass transition temperatures lying inside this broad interval. Shown in Table 14 are some of them and, considering these data, the influence of the chemical structure of polymers on their glass transition temperature could be easily imagined. It should be noted that transition from the glassy state into the rubbery state appears not at a definitely stated temperature but in some temperature range which may be several tens of degrees wide. This is quite clear because the transition considered is not the phase transition (such as melting, for example) but represents a physical transition from one phase to another, keeping one of the phases. In the case of amorphous glassy substances, this phase is liquid with no regard to the aggregate state of the polymeric substance – glassy (solid), rubbery or viscous flow state. Fundamentally, every polymer, if its molecular mass is quite high, may exist in one of the three physical states. Rubbers differ from plastics by the only thing that the glass transition temperature of polymers, on which they are based, is below room temperature, and for plastics – above room temperature. However, if a rubber is cooled down to temperatures below the glass transition point (for example, to –80°C), then a rubber-like substance becomes solid and behaves itself similar to usual glassy polymers. Vice versa, if a solid glassy polymer (for example, poly(methyl-methacrylate)) is heated up to a temperature above its glass transition point (above 100°C), it becomes a rubber-like material and behaves itself as usual rubber at room temperature. Hence, the subdivision of polymeric substances into rubber-like and solid glassy ones is rather conditional and has been formed historically. Table 14 Glass transition temperature Tg of representatives of various series of polymers Name Structural formula Tg, K 1 2 3 1,4-polybutadiene 170 –CH2–CH=CH–CH2– 150 Polydimethylsiloxane

CH3 Si

O

CH3 202

1,4-polyisoprene

CH2

CH C

CH2

CH3

70

1 Polyisobutylene

2

CH3 CH2

3 199

C CH3

Polyvinylmethyl ether

CH2

258

CH O CH3

Polyvinylethyl ether

CH2

248

CH O C2H5

298

Polyvinylacetate

CH2

CH O C

CH3

O Polyvinylchloride

CH2

CH

355

Cl Poly(methyl methacrylate)

CH3 CH2

378

C C O CH3 O

Poly(ethyl methacrylate)

CH3 CH2

338

C C O C2H5 O

Poly(hexyl methacrylate)

CH3 CH2

268

C C O C6H13 O

Poly(methyl acrylate)

CH2

CH

293

C O CH3 Poly(butyl acrylate)

CH2

O CH

233

C O C4H9 Polystyrene

CH2

O CH

378

71

1 Poly(vinyl alcohol)

2

CH2

3 358

CH OH

353

Poly(ethylene terephthalate)

Polyacrylonitrile

C

C O

O

O

CH2

(CH2)2

O 418

CH C N

Polyacrylamide

CH2

426

CH C NH2 O

Polycarbonate based on bisphenol A

422

CH3 O

C

O C

CH3 Polyester based on sebacic acid and phenolphthalein

O 368

C

(CH2)8

O

O

C O C

O

O C Polyester based on isophthalic acid and phenolphthalein

O 543

C

O

C O

O

C

O

O C Polyamide based on sebacic acid and aniline-phthalein

O 451

C

(CH2)8

O

C NH

NH C

O

O C Polyimide based on pyromellitic dianhydride and aniline-phthalein

CO

783

CO N

N CO

O

C

CO O C

O

72

A polymer cannot exist in the gaseous state, because it is much easier to disrupt a chains of macromolecules than to overcome total forces of intermolecular interaction: polymer chains are extremely long and possess numerous intermolecular bonds between neighboring chains; that is why it is impossible to transfer the polymer to the gaseous state without decomposing it. It was mentioned above that in each of three physical states the polymer may exist only if its molecular mass is quite large. Then, at heating up a solid plastic, it transforms consequently from the solid glassy state to the rubbery state, and then — to the viscous flow state (Figure 18). If the molecular mass of the polymer is low, it is transferred from the glassy state directly to the viscous flow state, and the thermomechanical curve has the form schematically represented in Figure 19. Such a curve is also typical for low-molecular weight substances. Hence, the rubbery state is typical for polymers only and is not displayed by low-molecular weight substances.

Figure 19. Schematic representation of thermomechanical curve for a polymer with small molecular mass on which I is the glassy state of polymer and II is the viscous flow state of polymer.

Figure 20. Schematic representation of a series of thermomechanical curves for polymers having different molecular mass M1 < M2 < M3 < M4 < M5 < M6 .

Now it is necessary to determine the molecular mass at which the polymer begins displaying the rubbery state. To accomplish this, let us consider a series of thermomechanical curves of polymer-homologues with various molecular masses. Figure 20 represents schematically such a series; it is clearly seen that with growth of the molecular mass the glass transition temperature increases at first and then practically stops changing. The molecular mass, at which this change stops, is the segment molecular mass. From this molecular mass, the polymer begins indicating the

73

rubbery state. This is quite clear, because the segment is that part of a macromolecule the ends of which are able to slide independently. Consequently, if a macromolecule is longer than the segment length, thermal motion may be performed in its separate places only, not touching the macromolecule as a whole. This causes typical rubbery properties, whereas high reversible deformations without significant mechanical stresses are typical for the polymer. This is the rubbery state. Clearly, if the molecular mass of the polymer exceeds the segment value, its further increase cannot cause growth of the glass transition temperature, because motion of segments as separate parts of macromolecules is not displayed yet. On the contrary, the temperature of polymer transition to the viscous flow state grows permanently with molecular mass, because flow is nothing but slide of separate molecules relative to each other on the whole. It is clear that the longer macromolecule is, the greater is the heat energy required for moving macromolecules relative to each other, i.e. to induce flow. That is why the temperature of transition into the viscous flow state increases permanently with molecular mass. Therefore, it is absolutely clear that the value of the mechanical segment can be determined with the help of the thermomechanical method. This value will be correspond to the molecular mass at which the rubbery state occurs, and exceeding of which may not increase of the glass transition temperature. Figure 21 represents the dependence of the glass transition temperature on molecular mass Mp for polystyrene. This dependence confirms the validity of the above discussion and is described well by some relations, the most widespread among which is the Flory equation Tg = Tg,∞ – α/M,

(IV.2)

where Tg,∞ is the glass transition temperature at molecular mass tending to infinity; α is a parameter; and the equation

β 1 1 = − , Tg Tg, ∞ M

(IV.3)

where β is a parameter.

Figure 21. Dependence of the glass transition temperature Tg on molecular mass Mp for polystyrene.

Let us now discuss experimental and calculation methods of determination of the mechanical segment value, i.e. molecular mass Ms, starting from which the

74

rubbery state occurs. As mentioned above, one of suitable methods of experimental determination of the mechanical segment value is the thermomechanical method. Considering again Figure 20, it may be said that the molecular mass of the segment determined by the thermomechanical method equals M3. The value of the segment depends on the chemical structure of the macromolecule, and the more rigid macromolecule is, the higher the value of its segment is. The lowest value of the segment is displayed by flexible macromolecules, which display quite free rotation of separate units relative to each other. Table 15 shows molecular masses of segments for macromolecules of various polymers. It is clearly seen from this Table that in dependence on the chemical structure of polymers, their macromolecules display absolutely different sizes of segments. The shortest segment is typical for polyisobutylene macromolecules, and the longest for polyarylate macromolecules1. The most rigid-chain polymers possess extremely large mechanical segments and, in many cases, the value of this segment equals the length of a macromolecule. In other words, no separate parts could be moved in such macromolecules, not touching the whole macromolecule. Since when heating such a polymer macromolecules of the polymer move relative to each other as a whole, the temperature of transition to the viscous flow state and the glass transition temperature coincide for it, because the fluidity of polymers by definition is sliding of separate molecules relative to each other. One more interesting feature is noteworthy, clearly seen in Table 15. Different polymers possessing, however, absolutely the same backbone and differing by side substituents only, display different segment lengths. At first glance, this contradicts the fact that the flexibility of macromolecules is not a consequence of their deflection, but the result of rotation of separate units relative to ordinary bonds, i.e. the bonds which link the backbone atoms (in the present case, they are carbon atoms in polyisobutylene, polyethylene, polystyrene, etc.). However, this contradiction is just apparent. Freedom (or restriction) of rotation of separate groups relative to each other depends not only on the macromolecule backbone, but also on type of its side substituents. The more bulky and polar the substituent is, all other factors being the same, rotation of separate units is restricted. Hence, we must consider not only backbone rigidity, but also kinetic rigidity associated with the restriction of rotation, occurring due to the presence of bulky side substituents. In this respect, polystyrene is typical; it has a phenyl ring as a side substituent in the repeat unit. Consequently, this polymer possesses a larger mechanical segment compared with the other polymers shown in Table 15. Now, it is necessary to relate the molecular mass of the segment with the parameters of the chemical structure of the polymer and with its typical temperatures – the glass transition temperature and the temperature of transition into the viscous flow state. However, first of all, we must agree on the method of determining these temperatures. If such a determination is performed thermomechanically, it is very important to choose a method of obtaining these temperatures from the thermomechanical curve. The theory of the thermomechanical method, developed by V.A. Kargin and G.L. Slonymskii, states that the glass transition temperature and the temperature of transition to the viscous flow state must be determined in the manner depicted in Figure 22. A value of deformation ε0 is chosen, plotted on the x-coordinate and in 1

Mechanical segment of macromolecules, the point of our discussion, must not be identified with the Kuhn segment, which characterizes flexibility of isolated chain and is determined from behavior of macromolecules in solution.

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relation to the height of the rubbery plateau. A line parallel to the abscissa axis is then drawn and points of intersection of these lines with the thermomechanical curve give the required glass transition temperature to the viscous flow states. Hence, according to this definition, the glass transition temperature is the temperature at which deformation under the effect of a specific load develops by the value ε0. The temperature of transition to the viscous flow state is the temperature at which irreversible deformation (flow) changes by the same value ε0. The value ε0 may be chosen arbitrarily, but it must not be too high in order not to exceed the rubbery plateau height. In practice, a specific fraction of the rubbery plateau height is taken for the value ε0. Table 15 Values of molecular mass of a segment Ms for different polymers determined by the thermomechanical method Polymer Ms 1000–1200 Polyisobutylene

CH3 CH2

C CH3 12000

Polyvinylchloride

CH2

C Cl 30000–40000

Polystyrene

CH2

C

200 000

Poly(ether ketone) CH3 C

O

C

O

O

CH3

300 000

Polyarylate F-1

C O O

O C

C

O

O C

O

The Kargin–Slonymskii theory based on the consideration of deformation of a polymeric body model with the calculation of temperature change leads to the following relation:

76

lg M = lg M s +

(

)

B Tf − Tg , C + Tf − Tg

(

)

(IV.4)

where M is the molecular mass of the polymer; Ms is the molecular mass of the segment; Tf is the temperature of transition to the viscous flow state; Tg is the glass transition temperature; B and C are parameters of the polymer.

Figure 22. Evaluation of temperatures Tg and Tf according to the value ε0 chosen on the thermomechanical curve in accordance with Kargin–Slonymskii’s theory .

Equation (IV.4) links the molecular mass of the polymer with the molecular mass of the segment and the difference of the temperature of transition into the viscous flow state and the glass transition temperature (Tf – Tg). Not dwelling on the practical meaning of this equation yet, note that its parameters B and C have a clear physical meaning. The parameter B is determined from the relation: B = lg

σt 0 , η 0ε 0

(IV.5)

where σ is the constant stress applied to the sample during the thermomechanical test; t0 is loading time; ε0 is the deformation at which Tg and Tf values are determined (its meaning is clearly seen from Figure 22); η0 is the pre-exponential multiplicand in the Fulcher–Tammann equation which links the viscosity of the system with temperature. This equation has the form: A

η = η 0 e T −T0 ,

(IV.6)

where η is the viscosity of the system at temperature T expressed in Kelvin degrees; η0 is the pre-exponential multiplicand; A and T0 are parameters of the material. Hence, all the values composing the parameter B are determined. The constant C in equation (IV.4) is found from the relation C = 0.434

A . B

(IV.7)

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It should be noted that the Fulcher–Tammann equation describes well the temperature region of the glassy state near the glass transition temperature. We are now interested in this very region. Let us now consider briefly the practical meaning of the Kargin–Slonymskii equation. If parameters Ms, B and C are known, this equation allows determination of the molecular mass of the polymer without its dissolving. This requires determination of the thermomechanical curve for this polymer, determination of the temperature of transition to the viscous flow state Tf and the glass transition temperature Tg and their substitution into the relation (IV.4). Therewith, parameters B and C need not be determined from equations (IV.5) and (IV.7). This only requires determination of three thermomechanical curves of the polymer with the known molecular masses M and solution of the system of three equations composed on the basis of the equation (IV.4) in which the unknown quantities are Ms, B and C. The molecular mass of any amount of polymer-homologues of the given series may then be determined. Table 16 Determination of polymerization degree Ns Ms [174]

M0

Ns [174]

Tg, K

∑ ∆Vi , cm /mol

Ns

Ns*

3460 15625 38073 5625 10000 24287 30246

28 56 104 54 68 86 100

124 279 366 104 147 282 302

195 199 378 171 200 298 378

20.60 41.30 66.00 36.48 48.90 47.73 58.05

128 165 366 136 175 259 351

112 144 320 119 153 227 307

3

Polymer Polyethylene Polyisobutylene Polystyrene Polybutadiene Polyisoprene Poly(vinyl acetate) Poly(methyl methacrylate)

i

However, it should be noted that basing only on the chemical structure of the polymer, the value Ms of it could not be calculated by equation (IV.4). The value of mechanical segment Ms of a macromolecule may be calculated on the basis of the chemical structure of the polymer. The following expression for calculation of Ms (or corresponding polymerization degree Ns = Ms/M0, where M0 is the molecular mass of the repeat unit) has been obtained in the ref. [96]: 1/ 3

  N s = const ⋅ Tg  N A ∑ ∆Vi    i  

.

(IV.8)

The value of Ns may be calculated on the basis of the chemical structure of the repeat unit with the help of atomic approaches, because the glass transition temperature Tg and the Van-der-Waals volume of the unit ∑ ∆Vi (NA is the i

Avogadro number) are estimated with their help. The constant in the formula (IV.8) may be expressed via parameters of the polymeric system and also from the experimental data for standard systems. The estimation gives const = 0.21, if calibration is performed by polystyrene according to data obtained in ref. [177], and const = 0.24 for polystyrene calibration according to refs. [174, 214]. Table 16 shows values of Ns obtained from the data of the ref. [174] and the formula (IV.8), with various constants (Ns with const = 0.24, and Ns* with const =

78

0.21). If const = 0.21 the difference in the values obtained from the ref. [174] does not exceed 10%. So far, we have discussed such physical characteristics of polymers as the glass transition temperature, the temperature of transition to the viscous flow state, the value of the macromolecule segment, which were determined experimentally with the help of the thermomechanical method of polymer investigation. Definite difficulties are met when determining temperature ranges of the solid (glassy), rubbery and viscous flow states of polymers by this method. This especially concerns new polymers. Let us consider generally the possible deformation behavior of polymers in thermomechanical tests. Recall that under these conditions the sample is loaded at increasing temperature. In most cases, the stress acts permanently during the experiment and temperature grows linearly. Fundamentally, the thermomechanical method of investigation allows immediate determination of temperature ranges of all three physical states of the polymer. However, the existence of one or another physical state and appropriate temperature range may be determined reliably only if it is known that the polymer studied behaves itself as a ‘classic’ one, i.e. gives the classic thermomechanical curve depicted in Figure 18. As it is observed in the considerations below, even if the form of the thermomechanical curve coincides with the classic one, in estimation of the properties of a new polymer it is not yet possible to determine unambiguously the temperature ranges of physical states and even of the states themselves. Before we consider this point, let us discuss some procedural questions. A question which appears most often is about the method of determination of transition points from the thermomechanical curve. As mentioned above, the following method is suitable: a definite strain ε0 is chosen, plotted from the temperature axis and from the rubbery plateau. The glass transition temperature and the temperature of transition to the viscous flow state will correspond to temperatures, at which one and the same value ε0 of rubbery and plastic strain occur, respectively. This method is most correct but suitable only when the thermodynamic curve is of the classic form with abrupt bends of the curves in transition temperature ranges. Then, the change of ε0 will not cause large shifts in determination of Tg and Tf. If deformation develops more smoothly, then the adjusted transition points Tg and Tf will be quite undefined. They will be sufficiently dependent on the value of ε0 (Figure 22). That is why another method is used in practice: values of Tg and Tf are determined by cross-points of tangents to two correspondent branches of the thermomechanical curve (Figure 23). In this case, values of Tg and Tf are less dependent on the shape of the thermodynamic curve, and this method is warranted for comparative estimation of polymers. Comparing thermomechanical curves of a series of polymers, the glass transition point may be defined as the temperature at which deformation is developed by the value of a specific percentage of the rubbery plateau height. Then, for each polymer this typical deformation will display different values, because heights of the rubbery plateau are also different. Selection of the determination method of Tg and Tf depends on the shape of the thermomechanical curve of polymers, and any of these methods may be chosen under comparative estimation of their properties. Of importance is just the comparison of several polymers to be performed always by the same method.

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Figure 23. Evaluation of temperatures Tg and Tf by the points of intersection of tangents to two branches of thermomechanical curve.

Let us now describe deformation of polymers during thermomechanical tests. Very often, especially in the initial estimation, a polymer is tested as a powder. The powder may only be compressed, and as it is placed in a special cell, compression is close to the hydrostatic one. First of all, it should be taken into account that a powderlike polymer is not yet a polymeric material, but its basis only. As a result of the thermomechanical test, the substance disposed beyond the punch is characterized, which transmits the load to the sample. That is why we obtain the thermodynamic curve of not a polymeric material, but of a polymer powder. Since the powder may be not densely packed in the cell (not due to the fault of a scientist, but because of its properties), and in connection with probable existence of small amounts of solvent and by-products of the reaction, the shape of the thermomechanical curve may be significantly distorted. That is why estimation of physical states of the polymer on powder samples is of the most ‘rough’ type, especially in the case of new, unknown polymers. Even if the thermomechanical curve is of the classic shape (see Figure 18) and consists of three parts, the statement that the polymer displays all three physical states transiting from one into another at heating should be neglected. It should be taken into account that growth of deformation in a powder sample may be caused by secondary reasons. When the thermomechanical curve is determined, it is best to take a look on the last branch of it. If it exists in the temperature range in which thermal and thermooxidative degradation does not penetrate deep enough, flowing of polymers may be discussed. To be certain that development of a large deformation (up to 100% at compression) is caused by flow and not by deep degradation of the polymer, thermogravimetric analysis (TGA) must be performed (thermogravimetric curve must be obtained) simultaneously. This is most important in the case of heat-resistant polymers for which high strains start to develop in the temperature range from 600 to 800°C. So, this deformation, caused by deep thermal degradation of the polymer, may be erroneously assumed to be the flow. It should also be taken into account that besides degradation, network formation may also occur during thermomechanical tests. These two processes always coexist at polymer heating, but one of them proceeds with much higher rate and defines direction of the whole process. Structuring may be displayed in formation of cross-links between chains of the

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polymer, cycle formation, etc. Consequently, the started flowing of polymer will be stopped, and a plateau appears on the thermomechanical curve, analogous by its shape to the plateau of the rubbery state for linear polymers. That is why the existence of the plateau (more correct, retarded deformation) is not the point to state that the polymer in the present temperature range is in the rubbery state. Ensuring that large deformation in the last region of the thermomechanical curve is cause by polymer flow, it should be attempted to transfer it into a bulky sample by any of existing methods (for example, by hot pressing). After that the sample is again thermomechanically tested. If the shape of the repeated thermomechanical curve is close to the initial one (obtained for the powder sample), chemical changes, possibly occurring during processing, are insufficient. Now we should only ensure that the plateau on the thermomechanical curve reflects the rubbery state, and is not caused by secondary reasons. One of the typical features of the rubbery deformation is its complete reversibility, as well as the ability to be expressed at very low loads (remind that the rubbery modulus is usually three-four decimal degrees lower than the modulus of elasticity for a glassy polymer). When a plateau is found on the thermomechanical curve, it must be checked if deformation is reversible in this temperature range. This is performed by regular loading of the sample, with this load exceeding the initial one, sometimes several times. If at application of additional load deformation increases sharply and recovers fast after its removal, a reversibility may be considered. This is reflected in Figure 24. If values of additional load and additional reversible deformation caused by it are known, the rubbery modulus may be easily calculated dividing the additional load by the deformation increment (Eel = ∆σ/∆ε). Ensuring that the elasticity modulus is of the smallest value (below ~1 MPa), it may be stated that the polymer in the plateau area is really in the rubbery state. For rigid-chain polymers, the elasticity modulus will be significantly higher (~10 MPa), and it is most important here to check reversibility of deformation in the plateau area. Therewith, it is often observed that a part of additional deformation is reversible (rubbery), and the rest of it is irreversible (plastic). In this case, the elasticity modulus is determined by division of ∆σ by the increment of the reversible part of deformation, and it is hardly possible to speak about the true rubbery state.

Figure 24. Checking of reversibility of deformation in the zone of the supposed rubbery-like plateau (see text)

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Figure 25. A lever mechanism with a variable arm for ensuring constant tension in the course of creep process (scheme).

All the above-said is true when thermomechanical tests are also performed under the effect of elongating tension. Secondary effects at elongation may not distort the picture so significantly, as in the case of compression. As thermomechanical investigations are performed at low elongation and the classic thermomechanical curve is obtained, it must only be assured that the plateau reflects the true rubbery. Checking is performed in an analogous way: an additional load is applied, and reversibility of deformation is checked after its removal. If deformation is reversible, it is rubbery. In thermomechanical tests on elongation, the tension should be constant. Contrary to tests on compression, some difficulties are faced here. The cross-section of the sample decreases at deformation and, consequently, stress in it grows under the effect of constant force. To compensate the stress increment caused by developed deformation, the load is transmitted to the sample via a lever mechanism with a variable arm (Figure 25). The arm decreases as deformation is developed, and the load decreases consequently so that the stress in the sample remains constant. After making sure that the plateau on the thermomechanical curve reflects the rubbery state, it can be said with confidence transition to the rubbery state has taken place. The transition ‘point’ is determined by the above-described methods. Until now, we have discussed the thermomechanical curve of semi-crystalline polymers, which is of the classic form and displays three physical states and two transitional temperature areas between them. It should be taken into account that the thermomechanical curve of crystalline polymers may be of the analogous shape (see Figure 18). Besides thermomechanical investigations, X-ray analysis must be performed in order to confirm semi-crystallinity of the sample. Neglecting so far transitions in crystalline polymers, let us discuss separate cases of behavior of semicrystalline polymers under thermomechanical conditions of tests. It may be immediately noticed that classic thermomechanical curves are not always observed. Deviations of the thermomechanical curve shape from the classic one are caused by different reasons. For example, the thermomechanical curve of a semi-crystalline polymer is often of the shape shown in Figure 19. It is absolutely obvious that this polymer displays no viscous flow state. It is also obvious that

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deformation of the sample (under compression) does not reach 100% and, consequently, it is burnt off at high temperatures, and the remaining part is distributed below the punch and prevents it from reaching the bottom. We should also carefully treat the temperature range, in which deformation grows sharply. If deformation near the onset of the horizontal plateau is completely reversible, it is of the rubbery type. However, it may not be stated without additional experiments that the polymer transits to the rubbery state, and the curve uplift points out the glass transition temperature. It is entirely probable that at this temperature chains of the polymer are intensively cross-linked, which therewith deforms, but subsequently loses meltability and solubility. That is why besides thermomechanical tests, thermogravimetric analysis is obligatory. Sometimes a polymer gives the classic thermomechanical curve, the last branch of which is in the area of very high temperatures (700–800°C). Clearly, development of large deformations in this area is caused not by viscous flow, but by intensive thermal degradation. In this case, it must not be said about the temperature of transition into the viscous flow state and the transition into the viscous flow state. It is specially obvious in the cases, when the thermomechanical curve is of the shape depicted in Figure 26. Distortion of the rubbery plateau (if it exists for the polymer tested) is caused by thermal degradation proceeding, excretion of gas products, etc.

Figure 26. Thermomechanical curve of polymer undergoing thermal degradation during the course of measurement (scheme).

Figure 27. Appearance of ‘negative’ deformation on the thermomechanical curve (see text).

Under compression, the thermomechanical curve may also be of the shape depicted in Figure 27. In this case, decrease and even appearance of ‘negative’ deformation is caused by increase of the sample height, as a result of which the punch transmitting the load to the sample is slightly lifted. This event may be caused by

83

elimination of residual solvent at heating up powder-like or bulky samples, excretion of gas products of degradation, etc. In such a case, determination of the true glass transition temperature and the temperature of transition into the viscous flow state is quite difficult. One of the reasons for the appearance of negative deformation may be the following. Preparing for thermomechanical tests, a polymer powder is pressed at room temperature, which is usually well below the glass transition area. Consequently, the polymer is subjected to the induced elastic deformation. Temperature grows during thermomechanical tests and when it reaches the glass transition area, the induced elastic deformation rapidly becomes reversible. The sample expands and, if the test is performed under compression at low loading, ‘negative’ deformation appears on the thermomechanical curve. The higher the pressure applied to the polymer powder at tableting, the greater the negative deformation is and the greater is the distortion of the shape of thermomechanical curve. In this case, the glass transition temperature corresponds to the lowering and not uplifting branch of the thermomechanical curve. In thermomechanical tests of polymers with elongation, deformation also decreases sometimes as temperature grows (Figure 28). The sample contraction may occur as a result of cross-linking, ring formation, etc., proceeding during the test.

Figure 28. Decreasing of deformation on the thermomechanical curve determining under conditions of tension (see text).

In most cases, the thermomechanical curve consists of two (see Figure 19) and not three parts. Usually, this is associated with the insufficient molecular mass of the polymer and with the fact that the polymer transits from the glassy state to the viscous flow one without reaching the rubbery state. Rising of the appropriate branch of the thermomechanical curve defines the temperature of transition into the viscous flow state, and it may be said that the glass transition (softening) and temperatures of transition into the viscous flow state coincide. When the second branch of the thermomechanical curve lies in the area of high temperatures, it should be confirmed that no degradation of the polymer happens in this area, and sharp increase of deformation is really associated with the polymer flowing and not with its degradation. In the latter case, softening and flowing are not yet reached, and the polymer exists in the glassy state in the whole possible temperature range. When transitions from the glassy state to the rubbery state and from the rubbery state to the viscous flow state are expressed clearly on the thermomechanical curve (in temperature ranges of transitions the deformation sharply increases), transition temperatures Tg and Tf can be easily measured. However, the thermomechanical curve is often of the shape depicted in Figure 29. This thermomechanical curve cannot help in determination of the glass transition

84

temperature: the transition is seriously ‘blurred’. That is why other methods should be used. Transitions from the glassy state to the rubbery state and from the rubbery state to the viscous flow state, and the melting point of polymers as well can be determined by dilatometric, calorimetric, and optical methods; it is appropriate to use also dynamic mechanical methods of investigation, etc. (see below).

Figure 29. Thermomechanical curve with ‘blurred’ transition (scheme).

Let us now turn to crystalline polymers. In some cases, the shape of thermomechanical curves of crystalline polymers differs from that of the same curves for semi-crystalline polymers and, in other cases, they nearly coincide. It is impossible to answer the question as to whether a polymer is semicrystalline or crystalline with the help of only the thermomechanical method of investigation. An X-ray pattern should be obtained first, and then data of the X-ray structural analysis must be compared with the results of the thermomechanical study. Crystalline polymers may be obtained directly during synthesis and further processing. For crystalline polymers, the thermomechanical curve may be of the shape depicted in Figure 30. If a polymer is quite extensively crystallized, the rubbery state is nearly completely suppressed, and the polymer displays no significant deformations in a wide temperature range. When transited to the melting point, deformation grows sharply and reaches the maximal value.

Figure 30. Thermomechanical curve of well-crystallized polymer (scheme).

The thermomechanical curve in Figure 30 corresponds to the case in which the rubbery deformation is completely suppressed by crystallization. Under real conditions, semi-crystalline areas exist in the crystalline polymer, which behave themselves at heating as a semi-crystalline polymer. During thermomechanical investigation of semi-crystalline polymers in the glass transition range, deformation will increase with further formation of a plateau (Figure 31). However, the rubbery plateau will not be of the same height as for pure semi-crystalline polymers of the

85

same structure. It will be significantly lower depending on the crystallinity degree. In any case, the glass transition temperature and the temperature of transition to the viscous flow state can be determined for the semi-crystalline polymer.

Figure 31. Thermomechanical curve of semi-crystalline polymer (scheme).

Particular emphasis should be placed on polymers, which easily crystallize during the thermomechanical test. It is then best to perform the experiment with a regular load2.

Figure 32. Schematic representation of the thermomechanical curve for a semi-crystalline polymer determined at periodic variation of load (see text).

Rapid crystallization of a semi-crystalline polymer occurs at temperatures above Tg. That is why the semi-crystalline polymer capable of rapid crystallization in the glassy state behaves itself similar to the normal semi-crystalline polymer only slightly crystallized. In the thermomechanical test, it gives a low deformation up to the glass transition temperature. Passing through this temperature, the polymer develops a large rubber deformation (Figure 32, curve 2). However, crystallization at T > Tg makes the polymer rigid, and if the load is applied regularly, the deformation decreases abruptly (see Figure 32, curve 2). Once the polymer has been deeply crystallized, its deformation is not yet different from the deformation of the preliminarily crystallized polymer (see Figure 32, curve 1). The temperature of transition to the viscous flow states will also coincide (see Figure 32). Let us now discuss briefly other methods of experimental estimation of the glass transition temperature of polymers. One of them is the dilatometric method in which the dependence of the specific volume of the polymer on temperature is 2

In all the cases, when more fine investigations are performed by the thermomechanical method, the load is best to be applied regularly, giving rest to the sample, i.e. a possibility to recover deformation after load removal. Then all effects of the glass transition, crystallization, etc. will be displayed more clearly.

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measured. Schematically, this dependence is depicted in Figure 13. This dependence as a broken line is typical of many polymers near the glass transition temperature. At temperatures below the glass transition temperature, this dependence is more flattened, than at temperatures above it. This is the main reason observed at the glass transition temperature (see Figure 13). Obviously, dilatometric dependences may be obtained at different rates of heating (or cooling) of the sample. Consequently, if the heating rate is changed in a quite wide range, the dilatometric dependences change in the manner depicted in Figure 13. The higher the rate of the sample heating, the higher is temperature at which the bending is observed and, vice versa, the lower the heating rate, the lower is temperature of the bending and the specific volume of the sample in the bending point decreases. Hence, the glass transition temperature depends on the rate of heating or cooling of the sample, which is associated with the relaxation mechanism of the glass transition (see below). It should be noted that changing the heating or cooling rate causes has a smaller effect on the type of dilatometric curve and on the glass transition temperature than the effect of mechanical tension on the type of the thermomechanical curve and the glass transition temperature. To cause a significant change of the glass transition temperature in the first case, the heating or cooling rate must be changed by many decimal degrees, which is impossible to do in an experiment. At the same time, ten- or hundred-fold change of the heating rate changes the glass transition temperature by several degrees only. Meanwhile, the change of the mechanical tension by several percents only may cause a sharp increase or abrupt decrease of the glass transition temperature. A special role in this case is played by the rate of mechanical influence. Analysis of these experimental facts enabled G.M. Bartenev [53] to apply the viewpoint that mechanical and structural glass transitions must differ. By the structural glass transition one means transition from the glassy state to the rubbery state (and the reverse process) in the absence of mechanical tension. This transition and its temperature are associated only with the heating and cooling rate. Mechanical glass transition is associated with the rapid mechanical influence on the polymer and its behavior as a solid under the rapid action of the mechanical force. As temperature changes, the mechanical glass transition may occur at rather lower temperatures than the structural glass transition.

Figure 33. Temperature dependence of heat capacity Cp (scheme).

In addition to the dilatometric method of determination of the glass transition temperature, other methods are also widely used, for example, calorimetric one. In this case, heat capacity Cp of the polymeric substance is measured, the dependence of Cp on T is of the typical shape (Figure 33). In the area of the glassy state, heat

87

capacity grows slowly with temperature, but when Tg is reached, a sharp increase of the heat capacity is observed. When the transitional zone is passed, the heat capacity again increases only slowly with temperature. Apparently, dynamic mechanical analysis is used most widely. In this method, the temperature dependences of the storage modulus E′ and the loss modulus E′′ of the complex elasticity modulus E* = E′ + iE′′ change, as well as the loss-factor tgδE = E′′/E′ (Figure 34). The temperature dependence of tgδE displays several maxima, with the most intensive (and high-temperature) one associated with transition from the glassy state to the rubbery state.

Figure 34. Temperature dependence of the mechanical loss-factor tgδE (scheme).

The temperature dependence of the dielectric loss-factor tgδε = ε′′⁄ε′, where ε′′ and ε′ are the true and the imaginary parts of the complex dielectric constant ε* = ε′ + iε′′, looks the same (Figure 35).

Figure 35. Temperature dependence of the dielectric loss factor tgδε (scheme).

Finally, the glass transition temperature may be determined from a sharp increase of the diffusion coefficient. Neglecting discussion of these methods of experimental determination of Tg, let us guide the reader to special literature [5, 51, 124].

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IV.2. Mechanism of glass transition The mechanism of glass transition has been studied for many years, but there is no general viewpoint on the glass transition mechanism. That is why we shall consider main common viewpoints on this mechanism. Note that these viewpoints do not contradict to each other. First of all, let us consider the concept of the relaxation mechanism of glass transition, put forward by A.P. Alexandrov, Yu.S. Lazurkin, P.P. Kobeko, and G.I. Gurevich. Let us preliminarily note once again that transition to the glassy state and the opposite transition to the rubbery state are not phase transitions of the first and the second kind (remember that according to the Erenfest definition, the phase transition of the first kind is the phase transition at which thermodynamic functions display a sharp jump at the transition temperature, and the derivative of the temperature dependence of the thermodynamic function is ruptured). The phase transition of the second kind is the transition at which temperature dependences of thermodynamic functions at the transition temperature change smoothly, and a sharp jump at the transition temperature is displayed only by first derivatives of these functions. Externally, in the case of polymers, some temperature dependences of thermodynamic functions look as if they were typical of transition of the second kind. For example, the temperature dependence of the specific volume (dilatometric curve) possesses a bend at the glass transition temperature by which this temperature is identified (see Figure 13). However, this resemblance is only superficial, because the transition considered is not associated with appearance of a new phase, as in the case of melting or crystallization from the melt, but is the transition from one physical state to another keeping the unique liquid phase. Hence, by the phase state solid glass represents a liquid, like elastomers and flowing polymers. They differ only by the aggregate state, representing the unique liquid phase.

Figure 36. Location of kinetic unit in the minimum of potential energy (scheme).

Let us now turn to consideration of the relaxation mechanism of glass transition. Let us begin description of this mechanism from low-molecular substances. Each molecule of such substance occupies a definite location in space and oscillates due to thermal fluctuations, i.e. as a result of exceeding the average temperature by the temperature of the present point. To transit to another location in space, a definite amount of energy should be consumed. Let the kinetic unit (atom or molecule) exist in the location, when the potential energy of it is minimal. This case, displayed in Figure 36, corresponds to the stable equilibrium. To transit the kinetic unit to a new state of equilibrium, it is necessary to overcome some energetic barrier, which is the real transition energy and is called the potential barrier. Hence, the potential barrier is the energy, which must be spent by the kinetic unit in order to transit to a new location. Let us consider the composition of this barrier.

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Every thermodynamic system is characterized by the value of the free energy or isobaric–isothermal potential F: F = U – TS,

(IV.9)

where U is the internal energy of the system; T is absolute temperature; S is entropy. The value of the free energy U is composed of the energy of translation of particles, the energy of their rotation, and the oscillation energy of particles in relation to their equilibrium. Concerning entropy, according to the Boltzman definition, it equals S = kBlnW,

(IV.10)

where kB is the Boltzman constant; W is the thermodynamic probability of existence of the system. The notion of entropy as a statistic characteristic is best demonstrated on the example of various conformations of macromolecules. Let us consider one of the border cases, a completely stretched chain (Figure 37,a). Clearly, such chain may possess only a single conformation, and the distance between its ends h will be equal to the full length of the macromolecule. Probability of existence of this conformation is extremely low, because it just the single one. But if we take a globular chain structure (see Figure 37,b), it will obtain a multiplicity of conformation at the same distance between its ends, i.e. its elements may locate differently in space keeping this distance constant. The shorter this distance h (down to some limit), the more conformations of the macromolecule may be composed and greater is the probability of existence of the macromolecule in this (globular) form. Since according to the Boltzman definition entropy is proportional to the logarithm of thermodynamic probability, it will increase with decrease of the distance between macromolecule ends. According to the second law of thermodynamics, a spontaneous process always proceeds with entropy increase. That is why thermal motion in the case of flexible macromolecules causes their coiling.

Figure 37. Schematic representation of straightened (a) and coiled (b) polymer chain (h is the distance between the chain ends).

Turning back to the system composed of small molecules, note that transiting of kinetic unit from one state into another both the internal energy and entropy will change. Change of the free energy will be determined from the relation ∆F = ∆U – T∆S,

(IV.11)

90

where ∆U is the change of the free energy; ∆S is the change of entropy. The probability of the given kinetic unit being in a specific position in space depends on its energy and absolute temperature. This probability is determined by the relation W = ν0e–E/RT,

(IV.12)

where ν0 is the self frequency of oscillation of atoms or molecules; E is the energy of the particle; R is the universal gas constant; T is the absolute temperature. Equation (IV.12) is used in cases when calculation is performed per mole of the substance. If the calculation is performed per molecule, kB (the Boltzman constant) should be introduced instead of R. These values are connected by the relation kB = R/NA, where NA = 6.023⋅1023 mol–1 (the Avogadro number). It is common knowledge that the fundamental frequency of oscillation of atoms in solids is ~1013 Hz, and the oscillation period is determined from the relation τ0 = 1/ν0. It is of interest that the time of settled life of an atom in this location is 10–10–10–11 s. Considering transition of a polymer from the glassy state to the rubbery state, let us introduce such characteristic as the time of this transition. This time τtr is determined from the relation

τ tr = τ tr,0eU tr RT ,

(IV.13)

where τtr,0 is a pre-exponential multiplicand which represents the value reversible to the fundamental frequency of oscillations of atoms; Utr is the energy, which must be spent to perform the transition. The smaller τtr is, the faster the transition is, i.e. the higher is temperature and the lower is transition energy Utr, the easier the transition conduction is. Let us consider a rubbery polymeric body, cooled down at a definite rate of temperature change. If the time of transition of structural elements (kinetic units) from one state to another is short at a definite temperature, such transition is fast and change of the structure will proceed immediately after the temperature change. It may be said that if the transition time τtr is much shorter than the observation time τobs, then the transition will proceed very fast and the equilibrium state of structure of the polymeric body is reached very rapidly immediately after the temperature change. The heat energy, characterized by the RT product, decreases with temperature, and the relaxation time (see equation (IV.13)) grows significantly. However, if it remains shorter than the time of system observation, the transition proceeds quite rapidly. Further decrease of temperature will lead to a sharp increase of τtr and to fulfillment of the condition τobs ≈ τtr, and then τobs < τtr. This corresponds to the case when the observation time becomes much shorter than the transition time. Clearly, this transition cannot be observed, although it proceeds. The transition proceeds so slowly that observation of it becomes difficult. That is why the polymeric substance becomes solid, i.e. transits to the glassy state. It should be noted once again that the structure of the polymeric substance at this transition is in fact not equilibrium. It is somehow frozen in the state preliminary to the transition. If we could manage to observe the transition for a very long time, it would be noted that this transition proceeds, although very slowly. Hence, according to these notions, the transition from the rubbery state into the glassy state is the relaxation process and is associated with the fact that at temperatures definite for every polymer the relaxation

91

time (time of transition) becomes extremely long, and the polymeric substance begins to behave as a glassy solid. This transition theory was subsequently mathematically justified by M.V. Volkenstein and O.B. Ptitsyn, who had performed mathematical analysis of the relaxation transition from the rubbery state to the glassy state and back, and had adjusted conditions under which such transitions may be performed. According to this theory ln

τ tr,0U tr RTg2

+

U tr 1 = lg , RTg q

(IV.14)

where q is the rate of system cooling. Equation (IV.14) shows that the glass transition temperature depends on the rate of polymer cooling. In its initial form, the Volkenstein–Ptitsyn theory does not consider the cooperative character of thermal motion of kinetic units. As shown subsequently, such calculation is absolutely necessary, because due to large length and flexibility of macromolecules the local motion of any kinetic unit touches upon its neighbors. For example, motion in the frames of a single repeat unit of a macromolecule causes motion of the neighbor units and this is the reason for segmental mobility. Calculation of the cooperative ability of motion of kinetic units of various sizes, performed by Ptitsyn and Sharonov, caused refinement of the Volkenstein–Ptitsyn theory and allowed reasonable values of the activation energy of the relaxation process to be determined.

Figure 38. Schematic representation of thermomechanical (a) and dilatometric (b) curves in the zone of transition from the glassy state to the rubbery state (see text) .

Analysis of a polymeric substance with the help of a non-linear model [87, 89] shows [90] that transition from the glassy state to the rubbery state must proceed not in a point but in some interval of temperatures which depends on the rate of heating or cooling q. Schematically, this is shown in Figure 38, which depicts thermomechanical and dilatometric curves in the transition area. Here Tg,1 is the temperature, at which transition from the rubbery state to the glassy state begins at system cooling; Tg,2 is the temperature at which this transition ends. The results obtained in work [90] indicate that at low rates of cooling the glass transition temperature range degenerates and the glass transition temperature only remains. In the general case of analysis of polymer behavior in the area of T < Tg,1, a significant role is played by the cooling rate. If the cooling rate is high, temperatures

92

Tg,1 and Tg,2 differ significantly, i.e. transition to the glassy state proceeds in a wide temperature range. In practice, the glass transition temperature range is really broadened as the rate of cooling (or heating) increases, not in a jump-like manner, but quite smoothly. Therewith, the temperature dependence of deformation in the transition zone becomes more abrupt as the rate of cooling (or heating) decreases. This is clear from Figure 39, which indicates thermomechanical curves for a series of cured compounds based on epoxy oligomers. In accordance with the theoretical results obtained, the glass transition temperature range windes as the heating rate increases (Figure 40).

Figure 39. Thermomechanical curves of cured epoxy compound measured under conditions of compression. Rate of heating 0.62 (1), 1.07 (2), 2.39 (3), 4.61 (4), 6.39 (5), 9.44 (6), and 15.47 (7) deg/min. Composition of specimen: ED-20 – 1 mole, MTHPA – 0.6 mole, azelaic acid – 0.7 mole.

Figure 40. Dependence of temperature differences ∆T = Tg,1 – Tg,2 on heating rate q: (a) correlates with the same composition as in Figure 39; (b) corresponds to the composition of specimen: ED20 – 1 mole, MTHPA – 1.3 mole, oleic acid – 0.7 mole.

The nonlinear model of polymeric substances [87,89] was also used for estimating the activation energy of γ-transition in polymers. Estimation of many properties of polymers requires the values of the activation energies associated with various transitions in polymeric systems. If a polymer is considered as a linear system, then it will be a selection of activation energies, usually associated with the motion of a definite group of atoms in a chain. But if it is considered as a non-linear system, described by a model suggested in ref. [87] (Figure A-6-8), we must know the activation energy of elementary acts only, associated, for example, with a jump of atoms from one equilibrium state to another in a rotary isomer, and which is usually associated with the activation energy of the γtransition. All other relaxation times stipulated by other transitions are expressed via the desired activation energy and temperature coefficients determined by non-linear

93

model parameters [269]. Hence, a temperature dependence of activation energy in the area of α-transition may be obtained, and the Williams–Landel–Ferry formula may be deduced, etc. To use actively this model for calculations of particular polymeric systems, the relation of the parameters of the model with the chemical structure of the repeat unit must be known. Let us demonstrate how expressions for determining activation energy which participate in the temperature dependences of relaxation times (viscosity) may be deduced from correlations of the ref. [269]. It is shown below that the activation energy is linked by simple dependences with the glass transition temperature and, therefore, it may be calculated on the basis of the chemical structure of the repeat unit. Let us consider expressions for relaxation times τ3 and τ5 of a non-linear model, shown in ref. [269], in the range of temperature of the onset Tg,2 and termination Tg,1 of glass transition. With this aim in view, let us perform series expansion of τ2/τ5 and τ2/τ3 in relation to Tg,1 and Tg,2, respectively, and confine to terms of the smallness order one by T – Tg,1 and T – Tg,2. Then,

τ2  d τ2  ≈ ⋅  ⋅ (T − Tg,1 ) , τ 5  dT τ 5  T =T g ,1 because

τ2 τ5

= 0 and T =Tg ,1

τ2 τ3

= 0, T =Tg , 2

where τ2 is the relaxation time of one of elements of the model, which may be expressed by the Arrhenius–Eiring formula. The expressions τ2/τ5 and τ2/τ3 may be expressed via variable δ0, respectively, as follows:

τ2  d τ2   dδ 0  ≈ ⋅  ⋅ (T − Tg ,1 ),   τ 5  dδ 0 τ 5  T =T  dT  T =T g , 1 g ,1 τ2  d τ2   dδ 0  ≈  ⋅  ⋅ (T − Tg, 2 ) .   τ 3  dT τ 3  T =T  dT  T =T g,2 g,2  d τ2   d τ2  ⋅  ⋅  and  are of As according to ref. [269], values   dδ 0 τ 5  T =Tg ,1  dδ 0 τ 3  T =Tg , 2 the same order of magnitude, future consideration will take into account only the d (δ 0 ) at Tg,1. As expressions τ2/τ5, and Tg will be taken for Tg,1. Let determine dT according to ref. [269],

94

−1

 16 K  δ 0 = 1 − ⋅ 1  , K1 = f1e −Y1 , K 3 = f 3e −Y3 ,  21 K 3  Y1,3 = 1 − 2 1 − T Tg1,3 ,

(

)

then dδ 0 16 2 K1  1 1 1 1  = δ0 ⋅ − ⋅ . dT 21 K 3  Tg3 1 − Y3 Tg1 1 − Y1  The relation

K1 K3

= 57/32, δ0 = –2.8 [269]. K1 value corresponds to the T =Tg ,1

elasticity of a chemical bond, and K3 – to the elasticity modulus of an intermolecular bond, that is why Tg,1 = Td (to the temperature of onset of intense thermal degradation), Tg,3 = Tm (to the melting point). Then,

dδ 0 dT

T =Tg ,1

  T 7.6  1 = − m  Tm  Tg,1 Td  1− T m 

   1 . Tg,1  1− Td 

 d τ2  ⋅  ≈ −0.3 may be found. Taking into account the From ref. [269]   dδ 0 τ 5  T =Tg ,1 expression

dδ 0 and the value dT

 d τ2   ⋅  , the dependence τ2/τ5 on temperature  dδ 0 τ 5  T =Tg ,1

in the area of Tg (suggesting Tg = Tg,1) may be presented as follows:     T τ2 2.3  1 1  ≈− − m⋅  T − Tg .  Tm  Td τ5 Tg Tg  1−  1− T Td  m 

(

)

(IV.15)

The expression obtained, according to ref. [269], describes the transition to the glassy state. It may be displayed that the Bachinski formula is also reduced to an analogous form. Actually, to describe the viscosity of liquids η, Bachinski [768] has suggested an empirical formula:

η=

C . v −ω

(IV.16)

95

Here v is the specific volume of the liquid; ω is the molecular ‘limited’ volume of the liquid (according to Bachinski, Mω = ∑ ε i , where εi are constants of atoms forming i

the molecule; M is the molecular mass), C is the constant. The expression (IV.16) describes well the liquid viscosity in a wide range of temperatures and pressures, the constant C may be found from the Camerling–Ownnes theory of similarity of molecular systems. Bachinski has shown [768] that for many liquids the following correlation is fulfilled: CM 1/ 6 = 0.531⋅10–4, 1 / 2 1/ 3 Tg ω

(IV.17)

Thus, the accuracy of determination of C depending on the chemical structure is ±6.2%. Let us transform the expression (IV.16). For this purpose, let us consider the temperature dependence of the specific volume, which may be presented in the following form: v = ω[1 + αv(T – Tg)],

(IV.18)

where αv is the coefficient of volumetric expansion which, according to ref. [269] in the case of the potential ‘6 – 12’, is equal to 1/42Tg. With regard to equation (IV.16), correlation (IV.18) has the following form:

η0 T − Tg = , η 42Tg

(IV.19)

where η0 = C/ω. Expression (IV.19) will be analogous to equation (IV.15) obtained from the non-linear model, therewith, η0 = η2 = K1τ2, and η = K1τ5. But contrary to the Bachinski formula deduced for transition from liquid to glass, the desired expression describes the transition from the side of low temperatures (transition from the glassy state to the rubbery state). That is why     Tg  Tm 1 1  − ⋅ η 2 = 2.3  42η0 . Tm  Tg Td Tg  1−  1− T Td  m 

(IV.20)

From equation (IV.20) the expression for the activation energy may be deduced. Let us estimate the following value for the real polymeric systems:   Tg  T 1 1 − m⋅ A = 2.3  Tm  T Tg Tg d 1−  1− T Td m 

   42 .   

(IV.21)

96

Shown in the Table are initial data (glass transition temperatures Tg, melting points Tm, temperatures of the onset of intense thermal degradation Td), and results of calculation of values A for a series of polymers. Values of Tg, Tm and Td are usually taken from experimental data but, in some cases, when, for example, a polymer is not crystallized or when its melting point lies above the temperature of onset of intense thermal degradation and thus may not be measured experimentally, they are calculated by the expressions shown below. Calculation results indicate (Table 17′) that the value A falls within the range from 24 to 82. In some cases, it becomes negative. It is typical of polymers, for which temperature of the onset of intense thermal degradation lies near the glass transition temperature, or the melting point and the glass transition temperature are close, i.e. for heat-resistant polymers. As obvious, coefficients A calculated for the most of polymers are positive that, according to equation (IV.20), correspond to positive values of viscosity. Similar to a non-linear model of polymeric body, in which viscosity is associated with relaxation mechanisms, according to the Mandelshtamm– Leontovich theory, the viscosity in our expressions will be the second one, because the non-linear model suggested [269] describes behavior of a polymeric substance by a selection of kinetic equations. However, as the Table indicates, for some polymers A < 0. The latter means that the second viscosity of these polymers is negative. The result obtained has a physical meaning and, as follows from ref. [768], is associated with nonequilibrium of the medium in the range of the glass transition temperature. This result must be analyzed in more detail.  ∆E *  h  expression (IV.20) may With regard to η0 = C/ω and η 2 = 2 ⋅ exp  kTg  Nl q   be presented in the form:  Nl 2 q AC  ∆E * , = ln ⋅ kTg ω   h

(IV.22)

where N is the number units in the macromolecule; l is the unit size; q = exp(∆E/kT); ∆E* is the difference in the energies of rotary isomers.

97

Table 17′ Polymer 1

CH2

Initial data and calculated results of value A for a series of polymers Tm, K Td, K Tg, K (exper./calc.) 2 3 4 263 449/451 673

Tg/Tm

Tg/Tm

Tm/Td

5 0.583

6 0.391

7 0.667

8 39.1

378

513/509

618

0.743

0.612

0.824

46.6

249

405/413

664

0.603

0.375

0.622

46.5

302

508/500

678

0.605

0.445

0.737

35.2

453

633/625

588

0.724

0.770

1.06

-21.5

205

332/314

663

0.654

0.309

0.474

71.4

A

CH CH3

CH2

CH

CH2

CH C2H5

CH2

CH CH2

CH CH3 CH3

CH2

O

(CH2)4

CH

O C

C O

97

O

(CH2)4

98

O

(CH2)3

O C

(CH2)4

O CH2

C

2 214

3 311/315

4 659

5 0.679

6 0.325

7 0.478

8 77.6

254

359/387

704

0.655

0.361

0.550

64.1

260

417/398

704

0.653

0.369

0.565

62.2

150

234/234

771

0.640

0.194

0.303

82.2

O

CH O CH2 CH2

CH3

CH O CH3

CH3 Si O CH3 CH2CH2 CH2

CH

CH3

O CH2

CH

213

410/410

713

0.519

0.575

0.299

37.8

246

438/397

651

0.620

0.379

0.609

50.9

CH3 CH2CH2O CH2

206

339/348

794

0.592

0.259

0.438

60.4

198

348/356

742

0.556

0.267

0.480

50.5

194

309/311

721

0.622

0.269

0.431

67.4

CH O CH3

(CH2)4O

98

1

99

1

2 318

3 433/465

4 629

5 0.682

6 0.506

7 0.739

8 47.6

223

323/334

655

0.667

0.340

0.510

71.2

351

530/539

621

0.651

0.565

0.868

23.7

323

499/494

622

0.654

0.519

0.794

35.1

323; 348

499/496

621

0.651

0.520

0.799

34.0

319

467/481

653

0.662

0.488

0.737

42.5

298

/458

640

0.651

0.716

0.466

44.8

CH3 C

CH2

C O CH3 O (CH2)2

O C

(CH2)4 C O

O C

(C H 2 ) 4

C

O

O NH

(C H 2 ) 6

NH

O

C

(CH2)8

O

C NH

(CH2)6

NH

O NH

(CH2)5

C O

NH

(CH2)10

C O

CH2

CH O C

99

O

CH3

100

2 199

3 /300

4 713

5 0.663

6 0.420

7 0.279

8 78.6

620

/801

667

0.774

0.930

1.200

–180.9

CH3 CH2

C CH3

C O

C O O

O C O C O

100

1

101

Considering a polymer in the glassy state as a frozen liquid and using data of the Camerling–Ownnes theory of molecular similarity, as Bachinski has done, the expression for C may be presented in the following form in analog to formula (IV.17): CM 1/ 6 = 10 −b . Tg1/ 2ω 1/ 3 Here b is the exponent value which, in the case of polymeric systems, is somewhat dependent on chemical structure and will be found during comparison with particular     characteristics of polymers. Because ω ~ N  ∑ ∆Vi  , where  ∑ ∆Vi  is the  i  r.u.  i  r.u. 1/ 3

Van-der-Walls

volume

of

the

repeating

unit,

  l ~  ∑ ∆Vi   i 

and

C = 10 − b Tg1/ 2ω 1/ 3 M 1/ 6 ,  ATg1/ 2 M 1/ 6  ∆E * . = 2.3 27 − b + ln 1/ 3   kTg M r.u.  

(IV.23)

If for real polymer systems logarithm is taken from Tg, M, and Mr.u., average by polymeric systems, it is observed that fluctuations stipulated by the difference of parameters Tg, M, and Mr.u. for particular polymeric system from average ones, give a low error (< 10%) for determination of the activation energy ∆E* of this system. In turn, the logarithm from average values of parameters by polymeric systems is taken into account via constant b. That is why, expression (IV.23) may be reduced to the form: ∆E* = 2.3(27 – b)kTg. In accordance with the experimental data from refs. [214], the value of b may be estimated, which is equal to 15.7. Respectively, ∆E* ≅ 26kTg.

(

(IV.24)

)

(

)

* * Shown below are some calculated ∆Ecalc values and experimental ∆Eexper of the activation energy for a series of polymers according to the data from refs. [214].

Polymer

PEO

PCTFE

PE

PVA

Tg, K * ∆Ecalc , kJ/mol * ∆Eexper , kJ/mol

206 44.8

323 70.3

213 46.0

303 65.8

Butyl rubber 200 45.6

37.7

72.7

50.3

60.8

49.9

102

* * Deviations between ∆Ecalc and ∆Eexper observed do not exceed ~15%. Considering jumps of atoms or groups of atoms along the chain in rotary isomers as diffusion of these atoms in polymer, the ratio ∆E*/d2 may be determined by data from ref. [214] (p. 298, Figure XVIII.7), where d is the size of the groups of atoms diffusing. But ∆E* is deduced from expression (IV.23), from which the size d of the jumping group of atoms may be determined immediately. For example, in the case of PS, ∆E*/d2 = 600, Tg = 378 K, ∆E* = 82.2 kJ/mol, and d = 5.7 Å that corresponds to the group C6H5. However, jumps of a group of hydrogen atoms are possible in PS. If sizes of these atoms are known, the activation energy may be deduced from the expression ∆E*/d2. It is equal ∆EH* = 19.7 kJ/mol. From expression (IV.24) the temperature of transition of this group of atoms Tg,1 may be estimated. If calculated, Cg,1 = 91 K, which by the order of magnitude corresponds to the temperature of δ-transition in PS (accurate coincidence is out of the question, because the constant in expression (IV.24) is found for the γ-transition). Estimations performed allows a supposition that if the repeating unit of polymer possesses side (asymmetric) groups of atoms different by size, a δ-transition appears in such a polymer. Discussed above is one of the concepts of the polymeric substance transition from the glassy state to the rubbery state and back. This concept is based on assumption of the relaxation mechanism of transition, the time of this transition (the relaxation time) depending on temperature and being determined by the energetic barrier height to be overcome in order to provide with the transition. Since different macromolecules possess different flexibility or rigidity, they are characterize by different heights of the energetic barrier and, consequently, by different glass transition temperatures. The more flexible the macromolecule is, the lower the height of the energetic barrier is which characterizes the system composed of such macromolecules. The concept of the polymeric substance transition from the glassy state to the rubbery state and back, based on the relaxation mechanism of transitions, is not unique. Another concept that explains these transitions is the concept by S.N. Zhurkov, based on acceptance of a significant role of intermolecular bonds in polymers. Let us discuss this concept. Every polymeric macromolecule possesses polar groups, which display various energies of interaction with each other. They form bonds which are not chemical, but are of the physical nature and are called intermolecular bonds. Such bonds may be rather weak as, for example, in polyisobutylene, polypropylene, polyethylene, and in other analogous polymers, but they may also be very strong as for example, in polyamides, polyesters, etc. In the former case, it is spoken about weak Van-der-Waals bonds, the energy of which is very low. In the latter case, strong NH C hydrogen bonds which appear due to existence of amide groups (polyO amides), and strong dipole–dipole interaction which appear due to existence of ester O groups are considered. Intermolecular bonds may be stipulated by an C O existence of not only these polar groups, but also of other groups capable of strong
 
 
     

 ≡N groups (for example,
  

  -groups (for example, in poly(vinyl alcohol), and many others. Polar groups in neighboring macromolecules under definite conditions may interact with each other forming cross-linked bridges between them, which are

Š

Š

103

quite strong. Recall that we are dealing not with chemical bonds, but with physical ones. Schematically, it is represented in Figure 41.

Figure 41. Schematic representation of a network of physical (intermolecular) bonds.

It is necessary to pay attention to one more very important circumstance. Contrary to chemical bonds, intermolecular bonds are rather weak. Energy necessary for dissociation them is by a decimal degree lower than that for chemical bonds. Recall that the energy of chemical bonds dissociation is of several tens of kcal/mol, and that of intermolecular bonds is several kcal/mol. The strongest bonds are displayed by amide, ester, hydroxyl and some other groups. However, the energy of dissociation of intermolecular bonds formed by these groups does not exceed ten kcal/mol (in fact, this energy is lower). Hence, intermolecular bonds are not stable, but decompose permanently under the effect of thermal fluctuations and are formed in new places. They somewhat migrate along the polymeric chain and in doing so link not strictly specified chains of macromolecules, but the chains, in which they appear at the current moment. However, at each definite temperature the amount of intermolecular bonds existing simultaneously is much the same for polymers of a definite chemical structure. It may be said that under equivalent consideration, intermolecular bonds act in definite places of macromolecules forming a peculiar network and making impossible their motion in relation to each other. Taking into account that, in fact, macromolecular bonds are formed and dissociate permanently under the effect of thermal fluctuations, an equilibrium may be shifted to one or another side, i.e. to the side of formation or dissociation of intermolecular bonds. As with usual chemical reaction, the equilibrium shifts to the side of decomposition of intermolecular bonds as temperature increases, and to the side of their formation as temperature decreases. When polymeric substance existing in the rubbery state is cooled, the amount of forming intermolecular bonds grows permanently. When a definite temperature is reached, the amount of bridge bonds formed reaches the value enough for fixing separate macromolecules relative to each other and allowing not translation mobility of them. Therewith, vibration mobility remains. Formation of a quite frequent and strong network of intermolecular bonds with polymer cooling down to a definite temperature promotes a definite loss of mobility of macromolecules at this temperature, and the polymeric substance as a whole becomes solid, i.e. transits to the glassy state. Temperature, at which this phenomenon is observed, is the glass transition temperature. When polymeric substance in the glassy state is heated, the equilibrium shifts to the side of intermolecular bond decomposition. When the process of their dissociation is carried far enough, macromolecules obtain translation mobility, and the polymeric substance softens, i.e. transits to the rubbery state. According to the concept considered, this is the nature of transition of polymeric substance from the rubbery state to the glassy state and back.

104

To perform this transition, a definite number of intermolecular bonds must be formed in each polymeric substance with no dependence on their chemical structure. Designate this number as x. Concentration of active groups in the polymer, capable of formation of intermolecular bonds, are designated as N. In this case, taking into account that intermolecular bonds dissociate and are formed permanently in other places, and that an equilibrium between the number of formed and dissociated bonds exists, to describe this process, a common equation of chemical kinetics describing equilibrium process may be used: x ↔ 2(N – 2x).

(IV.25)

Left part of the equation includes the number of intermolecular bonds formed at a definite temperature; right part of it in brackets represents the number of active groups remained, capable of forming these bonds. Digit two appears in brackets, because formation of each bond consumes two active groups. This is also the reason of appearance of two before brackets. Since the reaction considered is equilibrium, the law of mass action may be applied to it, according to which the relation of concentration of the substance formed to concentration of the initial substance is a constant value, equal to the equilibrium constant. Therewith, stoichiometric coefficients of the reaction equation are included in indices of degrees of corresponding concentrations. Hence, in the present case

(N − 2 x )2 x

= k eq ,

(IV.26)

where keq is the equilibrium constant. According to Arrhenius, temperature dependence of the equilibrium constant is described by the following relation k eq = A * e

E − RT

,

(IV.27)

where A* is a steric factor; E is the bond energy (in this case, the intermolecular bonds energy); R is the universal gas constant; T is absolute temperature. Substituting (IV.27) into (IV.26), we get E

x = (N − 2 x )2 A * e RT .

(IV.28)

Equation (IV.28) associates the number of intermolecular bonds formed with the number of active groups capable of formation of these bonds and with temperature. To clear up the mechanism of glass transition, it is necessary to interlock a part of polar groups one way or another in order to exclude them from the interaction. Interlocking may be performed by different methods, for example, by an insignificant change of the chemical structure of the macromolecule or by introducing small molecules of low-molecular substances acting as plasticizers. They may be introduced by sorption of gaseous small molecules of a sorbate by polymer. Assume that n of such small molecules were introduced into a polymeric substance, so that the same number of polar groups were interlocked. In this case, (IV.28) will be written as:

105

E

x = (N − 2 x − n )2 A * e RT1 ,

(IV.29)

where n is the number of solvent molecules. To fulfill the equilibrium condition and the number of intermolecular bonds formed to become x again, temperature must be changes, i.e. it must be decreased to decrease the value in brackets in the right part of equation (IV.29). That is why, T is substituted by T1 in this equation. Solving jointly the system of equations (IV.28) and (IV.29), we obtain the following relation n  E T1 − T  . 2 ln1 − = ⋅  N − 2 x  R T1T

(IV.30)

Š

In equation (IV.30) T correlates with the glass transition temperature of the ZLWK WKH JODVV WUDQVLWLRQ WHPSHUDWXUH RI original non-plasticized polymer, and T1 plasticized polymer, because by the condition x is the number of intermolecular bonds necessary for formation of a firm spatial network. When this network is formed, the polymer transits to the glassy state. Equation (IV.30) may be used for description of transition of a polymer from the solid state to the rubbery state, when a definite amount of plasticizer is introduced into it. However, this equation may be simplified significantly. First, assume that the number of intermolecular bonds is significantly smaller than the total number of polar groups capable of intermolecular interaction (as experiments have indicated, such happens indeed in practice: to obtain a firm network of intermolecular bonds and to ‘cure’ the polymer, it is necessary to fix no more than 10% of polar groups). In other words, the condition is fulfilled: N >> 2x.

(IV.31)

Let us introduce the second assumption, namely, operation with a small number of plasticizer molecules (clearly, such assumption might be not introduced). Then it will be written that n/N << 1.

(IV.32)

Taking the logarithm of the equation (IV.30) and taking into account that under the condition (IV.32) ln(1 – n/N) = –n/N, we get 2

n E T1 − T = ⋅ . N R T1T

(IV.33)

If a low amount of plasticizer molecules is introduced into the polymer, the glass transition temperatures of the initial and plasticized polymers will not be T − T1 ∆T = . Under significantly different. So it may be assumed that T ≈ T1 and T1T T2 these conditions, the relation (IV.33) will obtain the form

106

2

n E ∆T = ⋅ . N R T2

(IV.34)

Therefrom, ∆T = or

2nRT 2 EN

(IV.35)

∆T = αn,

(IV.36)

2 RT 2 . EN

(IV.37)

where

α=

Equation (IV.36) reflects the Zhurkov rule of molar parts. According to this rule, decrease of the glass transition temperature of plasticized polymer compared with the glass transition temperature of the initial polymer is proportional to the molar part of introduced plasticizer. This rule means that independent of chemical structure, depression of the glass transition temperature is the same, if equal molar parts of plasticizer are introduced. This is quite clear, because according to the present concept, solidification of polymer (transition from the rubbery state to the glassy state) proceeds at formation of equal amount of intermolecular bonds. If a part of polar groups is eliminated from the process in one way or another, then decrease of the glass transition temperature will depend only on the amount of these groups, but not on chemical structure of interlocking molecules. Table 17 Values of energies of intermolecular interaction for various polar groups E, kcal/mol Polymer Group 2.25 Poly(methyl methacrylate)

O

C Polyacrylonitrile Polyisobutylene Polyethylene

O CH3 –C≡N –CH3 –CH2–

2.22 0.67 0.32

Experiments and calculations performed by S.N. Zhurkov et al, lead to values of energies of the intermolecular interaction for various groups, shown in Table 17. This Table indicates clearly that energies of the intermolecular interaction are low, by a decimal degree lower than energies of chemical bonds. The highest energy of intermolecular interaction is displayed by hydrogen bonds and by bonds appeared at the sacrifice of interaction of carbonyl groups with nitrile groups, as well. The lowest energy of intermolecular interaction (Van-der-Waals forces) is typical of

Š&+ Š JURXSV 2

The higher the energy of intermolecular interaction is, the greater amount of thermal energy is required for overcoming these bonds and the higher the glass transition temperature of polymers possessing strong intermolecular bonds is. The lower the energy of intermolecular interaction is, the lower the amount of thermal

107

energy is required to failure these bonds and the lower the temperature of polymer transition from the glassy state to the rubbery state is. Polymers such as polyethylene, polyisobutylene, polypropylene, etc. display low glass transition temperatures. In turn, polymers such as polyesters, polyamides, polyvinyl alcohol, polyacrylonitrile, etc., which possess strong intermolecular interaction, display higher glass transition temperatures. Clearly, this consideration neglects skeleton rigidity of the macromolecule, i.e. possibility or impossibility for separate groups in the main chain to rotate around the ordinary bonds. Besides, this factor may be decisive and cause a very significant influence on the glass transition temperature of the polymer. One more important circumstance is noteworthy. In the case of glass transition of low-molecular liquids, molecules of which contain polar groups, all of them may interact with each other, because no steric hindrances for this exist. In the case of the glass transition of polymers, macromolecules are denuded of this possibility due to their great length. Only in an ideal polymer crystal, free from any distortions and dislocations, polar groups in the repeating units are capable of complete interaction with each other. At the glass transition of a polymeric liquid (i.e. at transition from the viscous flow state or the rubbery state to the glassy state), structure of the polymer is far from the ideal crystal. That is why, due to steric hindrances, only a part of polar groups may enter the intermolecular interaction. According to tests on measurements of heat capacity of polymeric substances at temperature change, it was found that only a small part of polar groups enters the intermolecular interaction during polymer cooling down to the glass transition temperature. The amount of these groups is ~10% of the total number of groups capable of intermolecular interaction. And even this small amount of interacting groups is enough for the polymeric substance to transit from the rubbery state to the glassy state due to formation of a spatial network of intermolecular bonds. Formation of these bonds at polymer cooling and their dissociation at its heating may be detected by direct methods of physical study. The first among them is the IR-spectroscopy method, with the help of which it was clearly displayed that at polymer cooling down to the glass transition temperature or polymer heating above this temperature an intensive formation or dissociation of intermolecular bonds proceeds. So detailed consideration of this concept is associated with the circumstance that in the subsequent discussion describing calculation methods of estimation of the glass transition temperature of polymers Tg various variants of appearance of the intermolecular interaction will be considered, which affect on calculated values of Tg. Actually, intermolecular interaction between polar groups located in neighboring chains causes formation of a physical network of intermolecular bonds. However, such interaction may also be performed between polar groups, located in the same repeating unit of the macromolecule. Then these groups are excluded from the interaction between the neighboring chains, and the glass transition temperature decreases. The same effect will also be observed at an anomalous addition of monomers during polymerization or copolymerization (‘head-to-head’ or ‘tail-to-tail’ attachment). In this case, intermolecular interaction is performed between polar groups located in neighboring repeating units of the same chain; clearly, therewith they are excluded from the interaction between neighboring chains, which causes a decrease of the glass transition temperature. All these points will analyzed in detail below.

108

Besides the Zhurkov rule of plasticization, discussed above in detail, there are many other rules. For example, because viscosity of low-molecular fluids–plasticizers in by many decimal degrees smaller than viscosity of polymer, it is assumed that equal volumes of the fluid may equally decrease the glass transition temperature of the polymer: ∆T = æ

v , V +v

(IV.38)

where V is the polymer volume; v is the plasticizer volume; æ is the coefficient equal for all fluids. Equation (IV.38) reflects the Kargin–Malinskii rule of volumetric parts. It operates better for weak polar polymers, whereas the Zhurkov rule is more valuable for strong polar polymers. The cases are often, when both rules are unable to describe behavior of real polymeric systems. Let us dwell in more detail on this point below.

IV.3. Calculation of the glass transition temperature of linear polymers This section discusses the calculation scheme for estimation of the glass transition temperature Tg developed in works [6, 128]. According to this scheme, the coefficients of molecular packing for polymers of various chemical structures are approximately equal at the glass transition temperature of each of polymers, this value kg being estimated by the value kg ≈ 0.667 for linear polymers. The coefficient of molecular packing k0 near the absolute zero is also approximately equal for all polymers and is 0.731. These measurements indicate that the transition from the glassy state to the rubbery state proceeds, when a definite part of empty volume is reached, equal fE = 1 – kg = 0.333, and expression for the glass transition temperature may be presented in the following form: Tg =

k 0 k g − 1 0.0906 = . αG αG

(IV.39)

It should be noted that the material discussed in the present Chapter is tightly connected with the material discussed in Chapter III. Using expressions for αG deduced in Chapter III, it must be written down that R 1 1 = , Tg 0.455 ZD0 1

where

∑Z D

1 = i ZD0

i 0i

∆Vi

∑ ∆Vi i

(IV.40)

; Z is the coordination number of the i-th atom; D0i is

the bond energy of the i-th atom with neighboring ones; ∆Vi is the Van-der-Waals volume increment of the i-th atom.

109

Numerical values of 1/ZiD0i, typical of every atom and every type of intermolecular interaction, were determined with the help of statistical treatment of experimental data by the method of least squares. As mentioned above, the values of bond energies obtained as a result of processing of the experimental data correspond to energies of the intermolecular interaction. Presence of polar groups in polymers, which causes strong intermolecular interaction, is taken into account by introduction of special constants. For practical calculations, it is advisable to rewrite the formula (IV.40) in the following form:

Tg =

∑ ∆Vi i

∑ ai ∆Vi + ∑ b j i

,

(IV.41)

j

where ai is a selection of constants, which characterizes energy of a weak dispersion interaction as the averaged contribution of each atom into this interaction; bj is a selection of constants, which characterizes energy of a strong specific intermolecular interaction (dipole–dipole, hydrogen bonds, etc.). Values ai and bj are associated with parameters αi by the following relations: ai = bj =

αi αi = ; k 0 / kg − 1 0.0906 βj k0 / kg − 1

=

βj 0.0906

(IV.42) .

(IV.43)

Using the relation (IV.41), the glass transition temperature of a tremendous number of polymers may be calculated. This is associated with the circumstance that the approach described is ‘atomistic’, i.e. each atom is characterized by its constant ai (their values are shown in Table 13). As for specific intermolecular interactions (dipole–dipole, hydrogen bonds), they are characterized by their constants bj, independent of chemical structure of the polar group. For example, dipole–dipole interactions of various types are characterized by the same constant bd = –55 ⋅ 10–3 Å3 K–1. The situation with hydrogen bonds in polyamides is somewhat more complicated. This is connected with specificity of their influence on Tg in the frames of the present class of polymers (Table 18)*. As an example, Table 19 indicates experimental and calculated values of glass transition temperatures for a series of model polymers, starting with polymers of a simple chemical structure to quite complicated systems. Selections of parameters of the relation (IV.41) are also present there. The approach developed [6, 128] and the relation (IV.41) obtained on its basis allow taking into account the influence of the type of unit attachment to the polymers chain on the glass transition temperature of it. If attachment is usual (‘head-to-tail), as * Clearly, such approach displays advantages before popular calculation method, based on group contributions [214]. If a polymer possesses a group, which contribution into the glass transition temperature is unknown, the calculation becomes impossible. In the same approach discussed, contribution of any group is simply composed of the contributions of atoms composing it plus the contribution of specific interactions, if these groups are polar.

110

schematically represented in Figure 42a, the intermolecular interaction may be performed between neighboring polymer chains with formation of a physical network of intermolecular bonds (see Figure 41). But if a part of chains displays an anomalous attachment (for example, ‘head-to-head’), as schematically represented in Figure 42b, the intermolecular interaction will proceed between two polar groups, located in neighboring units in the same chain. Obviously, then these polar groups are excluded from formation of the physical network between neighboring chains (see Figure 42b). Table 18 Values of constants βj and bj IRU DPLGH 1+&2 JURXSV Polyamide type bj⋅103, Å3K–1 βj⋅103, Å3K–1 Aromatic polyamides –11.5 –119 Aliphatic polyamides of all types* –20.4 –212 Aliphatic – aromatic polyamides containing CH2-groups** –14.0 – 145 Aliphatic – aromatic polyamides based on iso-, terephthalic acid –18.1 –188 and metha-, para-phenylene diamine, containing an even number CH2-groups** Aliphatic – aromatic polyamides based on iso-, terephthalic acid 16.3 –169 and metha-, para-phenylene diamine containing an odd number of CH2-groups** * When the number (n RI &+2 JURXSV SHU 1+&2 JURXS n > 5, additional constants bCH =

Š

Š Š

Š

Š

Š

Š

Š

2

–23.0 are introduced in amount m(n – 5), where m LV WKH DPRXQW RI 1+&2 JURXSV SDUWLFLSDWLQJ LQ the repeating unit. ** When the number (n RI &+2 JURXSV LQ WKe repeating unit n > 6, additional constants bCH = –

Š Š

2

23.0 are introduced in amount (n – 6).

Figure 42. Nature of intermolecular interaction in case of molecules ‘head-to-tail’ (a) and ‘head-tohead’ (b) joined .

Since strong intermolecular interactions are taken into calculation in the relation (IV.41) by parameters bj, clearly, it should be presented that ∑ ∆Vi Tg =

i

∑ ai ∆Vi + (1 − γ )∑ b j i

,

(IV.44)

j

where γ is the part of anomalously joined chains. According to the relation (IV.44), the glass transition temperature of polymer with anomalous attachment of chains will be decreased. It will be displayed on a particular example below.

111

Van-der-Waals volumes of the repeating units

∑ ∆Vi , values i

Polymer

∑ ai ∆Vi , and i

Table 19

∑ b j and the glass transition temperatures Tg for a series of polymers j

∑ ∆Vi , Å

3

i

∑ ai ∆Vi ⋅10 , 3

i

3

1

CH2

CH C

−1

∑ b j ⋅10 ,

Tg

3

j

3

Calc.

Exp.

−1

2 81.3

ÅK 3 319

ÅK 4 81.5

5 203

6 200

79.0 71.8

279 478

81.5 0

219 150

225 150

51.2

239

–55.4

279

258; 293

68.3

319

0

214

199

68.3

319

–45.7

250

248

CH2

CH3

Š&+ Š&+ &&OŠ&+ Š 2

2

CH3 Si

O

CH3 CH2

CH CH3 CH3

CH2

C CH3

CH2

CH

111

C2H5

112

CH

2 60.7

3 295

4 –55.4

5 253

6 258

77.8

375

–55.4

244

248

O CH3

CH2

CH O C2H5

CH2

CH

111.9

534

–36.0

225

221

CH2

O C4H9 CH

48.9

199

–55.4

341

355

63.7 96.4

238 422

0 –166.2

268 377

256 378

113.5

502.0

–166.2

338

338

147.6

661.0

–146.7

287

293

Š&+ Š&&O Š Cl

2

2

CH3 CH2

C C O CH3 O

CH3 CH2

C C O C 2H5 O

CH3 CH2

C C O C4H9 O

112

1

CH2

113

1

CH3 CH2

2 181.8

3 820.0

4 –127

5 262

6 268

41.5

309.0

–195.0

365

358

166

578.4

–136.4

375

353

54.0

183

–55.4

422

418

231.8

1121

–428

335

325

116

560

–214

335

335

64.3

363

–210

421

426

C C O C6H13 O

CH2 CH OH C

C O

O

O

(CH2)2

O

CH2 CH

KN C

(CH2)4

O C

C NH

(CH2)6

NH

O (CH2)5

NH

O CH2

CH C NH2 113

O

114

CH2

CH

CH3 CH2

C

CH2

CH

2 109.8

3 320

4 –27.7

5 376

6 378

127

399

–83.1

401

435

60.2

356

–195.0

374

379; 439

238.8

771

–162.0

431

422

451.1

1480

–300.5

382

363

C OH O CH3 O

C

O C

CH3 C O

(CH2)8

O

C O O

O C O C

O

114

1

115

1

C

C O

O

2 465.4

3 1163

4 –351.3

5 573

6 593

437.3

1060

–242

535

543

395.5

1066

–453.3

646

653

519.9

1331

–298.4

503

500

O C

O

O C

C

C O

O

O

O

O C

NH

C HN C

O C

O

O C O C N O

O

CH3 O

C CH3

O

C N C O

115

C

O

116

O

S

O

2 368.1

3 1133

4 –468

5 553

6 560

262.3

692

–76.8

416

420

389.7

1003

–326.1

576

583

S

O

O

O

C

O

O C O O

O C

C

O

O C

O

116

1

O

Table 19′ Experimental and calculated values of the glass transition temperature, Tg, for the broaden list of polymers, presented in the alphabetic order ( ∑ a i ∆Vi + i ∆ V , ∑ i Tg, K Tg, K No. Polymer References + ∑ b j )⋅103 i (exp.) (calc) 3 j Å , Å3K–1 1. Cyanoethyl cellulose 132(0 gr) 483 (1 gr); 453 389 (1 326 CH2 OH 188(1 gr) 244 (0 gr) gr); 541 CH O (0 gr)

O CH CH CH CH OH O CH2 2.

Ethyl cellulose

CH2 O CH2

CH2

C N

CH3

CH O O CH CH CH CH CH3 3.

CH2

Methyl cellulose

O

O CH2

204(2 gr) 654 (2 gr); 168(1 gr) 449 (1 gr)

316

312 (2 gr); 374 (1 gr)

275, 327

151(1 gr) 370 (1 gr); 170(2 gr) 496 (2 gr)

423

343 (2 gr); 408 (1 gr)

326

318

328

472

CH3

CH2 O CH3

CH O O CH CH CH CH CH3 4.

O

O CH3

Nylon 8,6 [Poly(iminoadipoyliminooctamethylene)]

O

(CH2)4

C NH O

(CH2)8

811

117

NH C

266

2 Nylon 10 [Poly(imino-1-oxodecamethylene)]

NH C

3 184

4 580

5 315

6 317

7 480

368

1161

333, 319

317

472, 473

402

1280

322

314

477

300

929

313

323

472

201

640

315, 316, 365, 333 343

314

480, 668, 681, 716

218

699

313 - 316, 314

312

284, 475, 480, 679

539

1750

323

308

477

235

756

314

311

480

(CH2)9

O 6.

Nylon 10,10 [Poly(iminosebacoyliminodecamethylene)]

NH C

(CH2)8

C NH

O 7.

(CH2)10

O

Nylon 10,12 [Poly(iminodecamethyleneiminododecanedioyl)]

NH

(CH2)10

NH C

(CH2)10

O 8.

O

Nylon 10,6 [Poly(iminoadipoyliminodecamethylene)]

NH C

(CH2)4

O 9.

C

C NH

(CH2)10

O

Nylon 11 [Poly(imino-1-oxoundecamethylene)]

NH C

(CH2)10

O 10.

Nylon 12 [Poly(imino-1-oxododecamethylene)]

NH C

(CH2)11

O 11.

Nylon 12,18 [Poly(iminododecamethyleneiminooctadecanedioyl)]

NH

(CH2)12

NH C

(CH2)16

O 12.

Nylon 13 [Poly(imino-1-oxotridecamethylene)]

NH C O

(CH2)12

C O

118

1 5.

1 13.

2 Nylon 14,18 [Poly(iminotetradecamethyleneiminooctadecanedioyl)]

NH

(CH2)14

NH C

(CH2)16

(CH2)18

NH C

(CH2)16

7 477

641

2102

323

305

477, 478

64.7

170

384

380

469

198

577

316

343

470

215

634

318

339

471, 472

300

929

323

323

284, 317, 320, 325

334

1044

319

320

473, 477, 478

249

752

331

331

464, 473

O

Nylon 3 [Poly(imino-1-oxotrimethylene)]

NH C

6 307

C

O 15.

5 321

O

Nylon 18,18 [Poly(iminooctadecanedioyliminooctadecamethylene)]

NH

4 1866

C

O 14.

3 573

(CH2)2

O 16.

Nylon 4,6 [Poly(iminotetramethyleneiminoadipoyl)]

NH

(CH2)4

NH C

(CH2)4

O 17.

O

Nylon 5,6 [Poly(iminopentamethyleneiminoadipoyl)]

NH

(CH2)5

NH C

(CH2)4

O 18.

C O

Nylon 6,10 [Poly(iminohexamethyleneiminocebacoyl)]

NH

(CH2)6

NH C

(CH2)8

O 19.

C

C O

Nylon 6,12 [Poly(iminohexamethyleneiminododecanedioyl)]

NH

(CH2)6

NH C

(CH2)10

C

O 20.

O

Nylon 6,7 [Poly(iminohexamethyleneiminopimeloyl)]

(CH2)6

NH C O

(CH2)5

C O

119

NH

2 Nylon 6,8 [Poly(iminohexamethyleneiminosuberoyl)]

NH

(CH2)6

NH C

(CH2)6

O 22.

(CH2)6

NH C

(CH2)7

6 328

7 473

283

868

331

326

476

133

405

325

328

325, 473, 479, 480

249

752

318, 333

331

365, 464, 472, 473

266

811

328

328

473

150

464

323

323

480, 680

334

1044

333

320

472

368

1161

323

317

477

C O

Nylon 7 [Poly(imino-1-oxoheptamethylene)]

NH C

5 330

O

O 23.

4 811

C

Nylon 6,9 [Poly(iminohexamethyleneiminoazelaoyl)]

NH

3 266

(CH2)6

O 24.

Nylon 7,6 [Poly(iminoadipoyliminoheptamethylene)]

NH C

(CH2)4

Nylon 7,7 [Poly(iminopimeloyliminoheptamethylene)]

NH C

(CH2)5

C NH

(CH2)7

O

O 26.

(CH2)7

O

O 25.

C NH

Nylon 8 [Poly(imino-1-oxooctamethylene)]

NH C

(CH2)7

O 27.

Nylon 8,10 [Poly(iminooctamethyleneiminodecanedioyl)]

NH

(CH2)8

NH C

(CH2)8

O 28.

C O

Nylon 8,12 [Poly(iminooctamethyleneiminododecanedioyl)]

NH

(CH2)8

NH C O

(CH2)10

C O

120

1 21.

1 29.

2 Nylon 8,22 [Poly(iminooctamethyleneiminodocosanedioyl)]

NH

(CH2)8

NH C

(CH2)20

O 30.

4 1750

5 321

6 308

7 477

167

522

319

320

473, 479, 480

283

868

318

326

471

146

394

354 Vicat softening temperature

371

589

209

641

353

326

733

264

892

330

296

733

C O

Nylon 9 [Poly(imino-1-oxononamethylene)]

NH C

3 539

(CH2)8

O 31.

Nylon 9,6 [Poly(iminoadipoyliminononamethylene)]

NH C

(CH2)4

O 32.

C NH

(CH2)9

O

Poly(1,1,1-trifluoro-isopropyl methacrylate)

CH3 CH2

C C O CH CF3 O

33.

CH3

Poly(1,1,5-trihydrooctafluoroallyl-2-cyanoacrylate)

C N CH2

C CO O CH2 (CF2)3

34.

CF2H

Poly(1,1,7-trihydrododecafluoroheptyl-2-cyanoacrylate)

C N CH2

C CF2H

121

CO O CH2 (CF2)5

2 Poly(1,1-dichloro-2-fluoroethylene)

CCl2

3 68.9

4 205

5 ~320 Low molecular weight

6 336

7 646

86.6

318

263–273

272

445

132

314

373

420

733

76.1

409

191

186

491, 492

102

366

253

279

539

85.3

288

263

296

539-542

CH F

36.

Poly(1,1-difluoro-2-trifluoromethoxyethylene)

CF2

CH O CF3

37.

Poly(1,1-dihydrotrifluoroethyl-2-cyanoacrylate)

C N CH2

C CO O CH2 CF3

38.

Poly(1,1-dimethylsilazane)

CH3 Si

NH

CH3 39.

Poly(1,1-dimethyltetramethylene)

CH3 C

(CH2)3

CH3 40.

Poly(1,1-dimethyltrimethylene)

CH3 C

(CH2)2

CH3

122

1 35.

1 41.

2 Poly(1,2-butadiene)

CH2

3 64.2

4 240

5 266, 269

6 268

7 734

98.3

426

262

231

498

306

494

653

620

296

418

931

453

449

297

165

579

258 Brittle point

285

582

CH CH CH2

42.

Poly(1,2-dimethyl-1-butenylene)

CH3 C

C CH2

CH2

CH3 43.

Poly(1,3,4-oxadiazolediyl-1,4-phenylene-3,3-phthalidylidene-1,4-phenylene)

N N C

C O

C O CO

45.

Poly(1,3,4-oxadiazolediyl-1,4-phenyleneoxy-1,4-phenyleneisopropylidene-1,4-phenyleneoxy-1,4phenylene)

N N C

CH3

C

O

C

O 46.

CH3

Poly(1,3-dimethylbutyl acrylate)

CH2

O

CH3

C O CH CH2

CH CH3

O

CH3

123

CH

2 Poly(1,3-dioxa-2-propyl-4,6-cyclohexylenemethylene)

CH2

CH2

HC O

48.

CH

CF2

CH2

CH2

CF2

O CF2

CH2

O C

C

O

O (CH2)4

O CF2

CH2

O C

(CF2)3

O 50.

51.

CF2

7 273, 324

324

1232

287

263

328

387

1363

303

284

328

230

816

301

282

328

64.2

375

166, 171

171

735, 736

98.3

489

197

201

507, 507

CH3

O CF2

Poly(1,4-butadiene)

CH2

C O

Poly(1,4,7-trioxy-3,3,5,5-tetrafluoroheptamethyleneisophthaloyl)

O CH2

6 289

CH3

Poly(1,4,7-trioxy-3,3,5,5-tetrafluoroheptamethylenecarbonyl-1,3-phenylene-hexafluorotrimethylene1,3-phenylenecarbonyl)

O CH2

5 322

O

O 49.

4 498

CH

Poly(1,4,7-trioxy-3,3,5,5-tetrafluoroheptamethylene-5-pentyloxyisophthaloyl)

O CH2

3 144

O C

C

O

O

–CH2–CH=CH–CH2– 52.

Poly(1-ethyl-1-butenylene)

C

CH CH2

CH2

CH3

CH2

124

1 47.

1 53.

2 Poly(17-fluoroalkyl methacrylate)

3 321

4 1239

5 310

6 259

7 686

86.0

368

241

234

662

132

667

192

198

504

235

1205

220

195

273

119

454

~250

262

525

162

623

251

260

589

CH3 CH2

C C O CH2

(CF2)7

CF3

O 54.

Poly(1-bromo-1-butenylene)

CH2

C CH CH2 Br

55.

Poly(1-butyl-1-butenylene)

CH2

C CH CH2 (CH2)3

56.

CH2

C

CH CH2

(CH2)9 57.

CH3

Poly(1-decyl-1-butenylene)

CH3

Poly(1-ethyl-1-methyltetramethylene)

CH3 C

(CH2)3

CH2 58.

CH3

Poly(1H,1H,3H-hexafluorobutyl acrylate)

CH2

CH C O CH2

CHF

CF3 125

O

CF2

2 Poly(1H,1H,5H-octafluoropentyl acrylate)

CH2

3 189

4 724

5 238

6 261

7 589

206

696

309

296

603

261

946

286

276

603

316

1197

258

264

603

277

1145

256 Crystalline

242

589

CH C O

CH2

(CF2)3

CF2H

O 60.

Poly(1H,1H,5H-octafluoropentyl methacrylate)

CH3 CH2

C C O CH2

(CF2)3

CF2H

O 61.

Poly(1H,1H,7H-dodecafluoroheptyl methacrylate)

CH3 CH2

C C O CH2

(CF2)5

CF2H

O 62.

Poly(1H,1H,9H-hexadecafluorononyl methacrylate)

CH3 CH2

C C O CH2

(CF2)7

CF2H

O 63.

Poly(1H,1H,-pentadecafluorooctyl acrylate)

CH2

CH C O CH2 O

(CF2)6

CF3

126

1 59.

1 64.

2 Poly(1H,1H-heptafluorobutyl acrylate)

CH2

3 167

4 645

5 243

6 259

7 315, 589, 590

184

523

~330

352 (syndio)

602

139

519

247

268

589

119

538

226

221

509

184

939

190

196

286, 505

115

502

221

229

504, 505, 507

CH C O CH2

(CF2)2

CF3

O 65.

Poly(1H,1H-heptafluorobutyl methacrylate) syndiotactic

CH3 CH2

C C O CH2

(CF2)2

CF3

O 66.

Poly(1H,1H-pentafluoropropyl acrylate)

CH2

CH C O CH2

CF2

CF3

O 67.

Poly(1-heptene)

CH2

CH (CH2)4 CH3

68.

Poly(1-heptyl-1-butenylene)

C

CH CH2

(CH2)6 69.

CH2

CH3

Poly(1-isopropyl-1-butenylene)

C CH CH2 CH CH3

127

H3C

CH2

2 Poly(1-methyloctamethylene)

CH

3 154

4 664

5 215

6 232

7 538

81.3

457

159, 183

178

502, 508

140

509

283 low molecular wieght

275

499

115

578

196

199

504

132

528

293

250

286, 396, 505, 506

146

537

282

272

624

(CH2)7

CH3 71.

Poly(1-pentenylene)

–CH=CH–(CH2)3– 72.

Poly(1-phenyl-1-butenylene)

C

73.

CH CH2

CH2

Poly(1-propyl-1-butenylene)

C

CH CH2

CH2

CH2 CH2 74.

CH3

Poly(1-tert-butyl-1-butenylene)

H3C

C

CH CH2

C

CH3

CH2

CH3 75.

Poly(2,2-dimethylbutoxyethylene)

CH2

CH O

CH3

CH2 C CH2 CH3

CH3

128

1 70.

1 76.

2 Poly(2,3,3,3-tetrafluoropropylene)

CH2

3 71.9

4 240

5 315

6 300

7 291

135

371

378

364

291

160

366

435 softening point

437

287

137

338

406

405

548, 557

CF CF3

77.

Poly(2,3,4,5,6-pentafluorostyrene)

CH2

CH

F

F

F

F F

78.

Poly(2,4,6-trimethylstyrene)

CH2

CH

H3C

CH3

CH3 79.

Poly(2,4-dichlorostyrene)

CH2

CH Cl

129

Cl

2 Poly(2,4-diisopropylstyrene)

CH2

3 211

4 548

5 ~435

6 385

7 287

143

340

385

420

300, 548, 555, 563, 574

137

338

379

405

280, 548, 558

120

307

374

391

287

CH CH3 CH CH3

H3C 81.

CH CH3

Poly(2,4-dimethylstyrene)

CH2

CH CH3

CH3 82.

Poly(2,5-dichlorostyrene)

CH2

CH Cl

Cl 83.

Poly(2,5-fluorostyrene)

CH2

CH F

F

130

1 80.

1 84.

2 Poly(2,5-diisopropylstyrene)

CH2

3 211

4 548

5 441

6 385

7 287

143

340

416

420

548

137

338

440

405

287, 494, 559

188

431

440

437

701

CH CH3 CH CH3

H3C CH H3C 85.

Poly(2,5-dimethylstyrene)

CH2

CH CH3

H3C 86.

Poly(2,6-dichlorostyrene)

CH2

CH

Cl

87.

Cl

Poly(2,6-xylenyl methacrylate)

CH3 CH2

CH3

C C O O 131

CH3

2 Poly(2-bromoethyl methacrylate)

3 135

4 380

5 325

6 355

7 560

206

624.6

339

330

551

204

711

340

287

545

135

349

450

388

556

94.5

372

256

254

500

CH3 CH2

C C O (CH2)2

Br

O 89.

Poly(2-butoxycarbonylstyrene)

CH2

CH C O (CH2)3

CH3

O 90.

Poly(2-butoxymethylstyrene)

CH2

CH CH2 O (CH2)3 CH3

91.

Poly(2-carboxystyrene)

CH2

CH C OH O

92.

Poly(2-chloro-1,4,4-trifluoro-1-butenylene)

F C

C CH2 Cl

CF2

132

1 88.

1 93.

2 Poly(2-chlorobenzoyloxyethylene)

CH2

CH

3 152

4 425

5 335

6 355

7 278, 628, 653

128

374

>365

342

361

124

302

392

410

548

159

562

313

283

514

226

518

418

436

706

Cl

O O C 94.

Poly(2-chloroethyl methacrylate)

CH3 C

CH2

C O (CH2)2

Cl

O 95.

Poly(2-chlorostyrene)

CH2

CH Cl

96.

Poly(2-cyclohexylethylethylene)

CH2

CH CH2

97.

CH2

CH2 CH CH2

Poly(2-decanehydronaphthyl methacrylate) CH3 CH2

CH2 CH2 CH2

C

CH2 CH2 C O CH CH CH2 O

133

CH2 CH CH2 CH2 CH2

2 Poly(2-ethoxycarbonylphenyl acrylate)

CH2

3 200

4 617

5 303

6 325

7 560

172

444.6

391

387

551

170

531

347 low viscosity

320

545

126

406

345

310

273

234

659

340, 328338

355

723

CH C

O

O O 99.

C

O

C 2H 5

Poly(2-ethoxycarbonylstyrene) CH2 CH

C O CH2

CH3

O 100.

Poly(2-ethoxymethylstyrene)

CH2

CH CH2

101.

O CH2

CH3

Poly(2-ethyl-1,3-dioxa-4,6-cyclohexylenemethylene)

CH2 CH2

CH CH O

O CH CH2

102.

CH3

Poly(2-ethyl-2-methylpropylene terephthalate)

CH3 CH2

C CH2

H3C CH2

O C O

C O O

134

1 98.

1 103.

2 Poly(2-ethylhexyl methacrylate)

3 216

4 797

5 263 Brittle point

6 271

7 581

143

404

376

354

555

131

337

384

389

562

240

801.6

318

299

551

121

346

358

350

690

CH3 CH2

104.

C C O CH2

CH

(CH2)3

O

CH2

CH3

Poly(2-ethylstyrene)

CH2

CH CH2

105.

CH3

CH3

Poly(2-fluoro-5-methylstyrene) CH2 CH F H3C

106.

Poly(2-hexyloxycarbonylstyrene)

CH2

CH C O

(CH2)5

CH3

O 107.

Poly(2-hydroxyethyl methacrylate)

CH3 CH2

C O CH2

CH2 OH

135

O

C

2 Poly(2-hydroxymethylstyrene)

CH2

4 327

5 433

6 410

7 273

138

433 (80% isotactic), 369 (atactic), 357 (58% syndio)

328, 359, 311 (80%

560, 604606, 663

isotactic),

319(80% isotactic), 374

393 (58%

(atactic),

CH CH2

109.

3 134

OH

Poly(2-hydroxypropyl methacrylate)

CH3 CH2

110.

C O CH2

CH

O

OH

syndiotactic)

CH3 206

548.6

400

376

551

223

637.6

341

350

551

CH C O CH2

CH CH3

O

CH3

Poly(2-isopentyloxycarbonylstyrene)

CH2

387 (58% syndiotact.)

Poly(2-isobutoxycarbonylstyrene)

CH2

111.

C

CH

O C O CH2

CH2 CH CH3

CH3

136

1 108.

1 112.

2 Poly(2-isopentyloxymethylstyrene)

CH2

3 221

4 725

5 351 (low viscosity)

6 305

7 545

204

636

~330

321

553

189

469.6

419

402

551

187

555

361 (low viscosity)

337

545

CH CH2 O

(CH2)2

CH CH3 CH3

113.

Poly(4-isopentyloxystyrene)

CH2

CH

CH3 O 114.

(CH2)2

CH CH3

Poly(2-isopropoxycarbonylstyrene)

CH2

CH CH3 C O CH CH3 O

115.

Poly(2-isopropoxymethylstyrene)

CH2

CH CH2

137

CH3 O CH CH3

2 Poly(2-isopropyl-1,3-dioxa-4,6-cyclohexylenemethylene)

CH2

4 424

5 329

6 340

7 273

182

534

319

341

560

155

421

403 (low viscosity)

368

556

123

496

223

248

587

153

451

362 (low viscosity)

339

545

CH2 CH CH O

CH3 117.

3 144

O CH CH

CH3

Poly(2-methoxycarbonylphenyl acrylate)

O CH3 C O CH2

CH

O C O 118.

Poly(2-methoxycarbonylstyrene)

CH2

CH C O CH3 O

119.

Poly(2-methoxyethyl acrylate)

CH2

CH C O CH2

CH2

O CH3

O 120.

Poly(2-methoxymethylstyrene)

CH2

CH CH2

O CH3

138

1 116.

1 121.

2 Poly(2-methoxystyrene)

CH2

3 136

4 363

5 ~348

6 375

7 568

109

318

355

343

273

148

556

241 Brittle point

266

582, 706

165

645

235 Brittle point

256

582

126

314

293, 371375, 453, 443, 455

401

670, 671, 689, 697, 709, 726

CH O CH3

122.

Poly(2-methyl-1,3-dioxa-4,6-cyclohexylenemethylene) CH2 CH2 CH CH O

O CH CH3

123.

Poly(2-methylbutyl acrylate)

CH2

124.

C O CH2

CH CH2

O

CH3

CH3

Poly(2-methylpentyl acrylate)

CH2

125.

CH

CH C O CH2

CH CH2

O

CH3

Poly(α-methylstyrene)

CH3 CH2

C

CH2

CH3

139

2 Poly(2-methylstyrene)

CH2

3 126

4 315

5 409

6 400

7 282, 548, 555, 563, 568

183

487

358

376

585

199

833

228 Brittle point

239

582

255

977

286

261

575

223

712

365

313

551

CH CH3

127.

Poly(2-naphthyl acrylate)

CH2

CH C O O

128.

Poly(2-octyl acrylate)

CH2

CH

CH3

C O

CH

(CH2)5

CH3

O 129.

Poly(2-octyloxystyrene)

CH2

CH O

130.

(CH2)7

CH3

Poly(2-pentyloxycarbonylstyrene)

CH2

CH C O O

(CH2)4

CH3

140

1 126.

1 131.

2 Poly(2-pentyloxymethylstyrene)

CH2

(CH2)4

6 277

7 545

246

730

336 low viscosity

337

545

213

462

397 (low viscosity)

461

556

189

533.6

381

354

551

187

621

370 low viscosity

301

545

CH CH2 O

133.

5 320

CH3

Poly(2-phenethyloxymethylstyrene)

CH2

4 798

CH CH2 O

132.

3 221

CH2

CH2

Poly(2-phenoxycarbonylstyrene) CH2 CH C O O

134.

Poly(2-propoxycarbonylstyrene)

CH2

CH C O CH2

CH2 CH3

O 135.

Poly(2-propoxymethylstyrene)

CH2

CH CH2 O CH2

CH2 CH3 141

2 Poly(2-tert-butylaminoethyl methacrylate) CH3 CH2

C

4 647

5 306

6 300

7 560, 597

206

563

345

367

585

66.7

217

300

307

291, 620, 647, 648

206

609

293

338

650

182

553

318

329

599, 706

CH3

C O (CH2)2 137.

3 194

NH C CH3

O Poly(2-tert-butylphenyl acrylate) CH2 CH

CH3

C O O H3C C CH3

138.

139.

CH3 Poly(3,3,3-trifluoropropylene) CH2 CH

CF3 Poly(3,3-dimethyl-3-phenylpropionyloxyethylene)

CH2

CH O C O CH3 CH2

C CH3

140.

Poly(3,3-dimethylbutyl methacrylate)

CH3 CH2

C C O CH2 O

CH3 CH2

C

CH3

CH3

142

1 136.

1 141.

2 Poly(3,3-phthalidylidene-1,4-phenyleneoxy-5-tert-butylisophthaloyloxy-1,4-phenylene)

O C

C

H3C

CO

4 870

5 552

6 525

7 313

296

550

513, 538

538

313, 323

390

720

591, 543

542

6, 313

137

338

401

405

548, 558

C O O

O

O

3 457

C

CH3

CH3 142.

Poly(3,3-phthalidylidene-1,4-phenyleneoxycarbonyloxy-1,4-phenylene)

O C O

C

O

O CO 143.

Poly(3,3-phthalidylidene-1,4-phenyleneoxyisophthaloyloxy-1,4-phenylene)

C

O C O

O

C O O

CO 144.

Poly(3,4-dichlorostyrene)

CH2

CH

Cl 143

Cl

2 Poly(3,4-dimethylbenzoylethylene)

CH2

3 162

4 411

5 315

6 346

7 267

143

340

384

420

548

233

747

274

312

601

143

340

377

420

563

CH C O

CH3 CH3 146.

Poly(3,4-dimethylstyrene)

CH2

CH

CH3 CH3 147.

Poly(3,5,5-trimethylhexyl methacrylate)

CH3 CH2

C

CH3

C O CH2 O 148.

Poly(3,5-dimethylstyrene)

CH2

H3C

CH

CH3

CH2

CH CH2

C CH3

CH3

CH3

144

1 145.

1 149.

2 Poly(3-bromobenzoyloxyethylene)

CH2

CH

3 159

4 436

5 331

6 365

7 278, 628

151

428

338

352

278, 628, 653

124

302

363

410

548

176

652

248

270

511

184

505

320

364

560

Br

O O C 150.

Poly(3-chlorobenzoyloxyethylene)

CH2

CH

Cl

O O 151.

C

Poly(3-chlorostyrene)

CH2

CH

Cl 152.

Poly(3-cyclohexylpropylethylene)

CH2

CH CH2

153.

CH2

CH2

CH2 CH CH2

Poly(3-dimethylaminophenyl acrylate)

CH2

CH2 CH2 CH2

CH C O O 145

H3C N CH3

2 Poly(3-ethoxycarbonylphenyl acrylate)

CH2

3 200

4 617

5 297

6 324

7 560

143

404

~303

354

555

117

311

398

376

686

129

369

355

349

686

121

451

277, 278, 223

268

685, 718, 724

CH C O O O C O C2H5

155.

Poly(3-ethylstyrene)

CH2

CH

CH2 CH3 156.

Poly(3-fluoroalkyl α-fluoroacrylate)

F CH2

C

O C O CH2 157.

CF3

Poly(3-fluoroalkyl methacrylate)

CH3 CH2

C

O C O CH2 158.

CF3

Poly(3-hydroxybutyrate)

CH2

CH O

O C

(CH2)3

OH

146

1 154.

1 159.

2 Poly(3-hydroxymethylstyrene)

CH2

(CH2)2

7 273

157

611

217

257

594

182

534

311

341

560

140

586

198

239

587

148

556

228 Brittle point

266

582

CH CH3 O CH3

O Poly(3-methoxycarbonylphenyl acrylate)

CH2

6 410

CH C O

161.

5 398

OH

Poly(3-methoxybutyl acrylate)

CH2

4 327

CH

CH2 160.

3 134

CH C O O C O O CH3

162.

Poly(3-methoxypropyl acrylate)

CH2

CH C O (CH2)3 O CH3 O

163.

Poly(3-methylbutyl acrylate)

CH2

CH C O CH2

CH3

147

O

CH2 CH CH3

2 Poly(3-methylstyrene)

CH2

3 126

4 315

5 370

6 400

7 548, 555, 563, 572

157

561

289

280

596, 597

148

565

267, 257

262

560, 582

145

363.4

389

399

544

208

632

318

329

651

CH

CH3 165.

Poly(3-oxabutyl methacrylate)

CH3 CH2 O 166.

167.

C C O (CH2)3

Poly(3-pentyl acrylate) CH2 CH

CH3

CH3

CH2 O C O CH CH2 Poly(4-acetylstyrene)

CH2

O

CH3

CH

O C CH3 168.

Poly(4-benzoylbutyryloxyethylene)

CH2

CH O

O C

(CH2)3 C O

148

1 164.

1 169.

2 Poly(4-benzoylstyrene)

CH2

3 204

4 528

5 371

6 386

7 544

213

539

383

395

578

175

476

317

329

627

159

436

365

365

278, 628, 652, 653

CH

O C 170.

Poly(4-biphenyl acrylate)

C H2 O 171.

CH C

O

Poly(4-bromo-3-methoxybenzoylethylene)

CH2

CH C O

O CH3 Br 172.

Poly(4-bromobenzoyloxyethylene)

CH2

CH O

O

C

Br 149

Poly(4-bromostyrene)

CH

2

3 131

4 310

5 391, 414430

6 423

7 548-550, 668

174.

Br Poly(4-butoxycarbonylphenyl acrylate)

234

909

286

257

560

254

669

401 softening point

380

611

206

625

349

330

544

204

711

< 283

287

552

CH2

175.

177.

CH C O

C O

O

O

(CH2)3

CH3

Poly(4-butoxycarbonylphenylmethacrylamide) CH3

CH2

176.

CH2

C C NH

C O

O

O

Poly(4-butoxycarbonylstyrene) CH2 CH

O C O (CH2)3 Poly(4-butoxymethylstyrene) CH2

(CH2)3

CH3

CH3

CH

CH2 O (CH2)3 CH3

150

1 173.

1 178.

2 Poly(4-butoxystyrene)

CH2

(CH2)3

7 552, 553

178

586

279

304

552, 553, 555

217

715

334

303

278, 628

183

284

CH3

Poly(4-butylyloxybenzoyloxyethylene)

CH2

6 301

CH

(CH2)3 180.

5 ~320

CH3

Poly(4-butylstyrene)

CH2

4 621

CH

O 179.

3 187

CH O C

O

(CH2)2

CH3

O 181.

Poly(4-carboxyphenylmethacrylamide)

CH3 CH2

527 >473 (Softening (decompoint) position)

611

C C OH

O

O

151

C NH

2 Poly(4-carboxystyrene)

CH2

3 135

4 293

5 386

6 388

7 544

140

354

418

396

548

129

324

395

398

548

140

354

387

396

548

CH

COOH 183.

Poly(4-chloro-2-methylstyrene)

CH2

CH CH3

Cl 184.

Poly(4-chloro-3-fluorostyrene)

CH2

CH

F Cl 186.

Poly(4-chloro-3-methylstyrene)

CH2

CH

CH3 Cl

152

1 182.

1 187.

2 Poly(4-chlorobenzoylethylene)

CH2 O 188.

3 142

4 372

5 310, 362

6 333

7 627, 628

151

428

357

352

278, 628, 652, 653

124

302

383, 388401

410

287, 548550, 667

150

624

233-238

240

754

204

731

~263

279

650

CH Cl

C

Poly(4-chlorobenzoyloxyethylene) CH2 CH

O O

C

Cl

189.

Poly(4-chlorostyrene) CH2 CH

190.

Poly(4-cyanobutyl acrylate)

Cl

CH2

CH C O

C N

(CH2)4

O 191.

Poly(4-cyclohexylbutyryloxyethylene)

CH2

CH O C O CH2

(CH2)2

CH2 CH2 CH2

153

CH2 CH CH2

193.

2 Poly(4-decylstyrene) CH2 CH

(CH2)9 CH3 Poly(4-dimethylaminocarbonylstyrene) CH2 CH

3 280

4 1120

5 208

6 250

7 554

174

437

398

398

544

314

1138

221

276

554

181

568

343

319

278, 628

CH3 O C N CH3 194.

Poly(4-dodecylstyrene)

CH2

CH

(CH2)11 195.

CH3

Poly(4-ethoxybenzoyloxyethylene)

CH2

CH O

O

C

O CH2

CH3

154

1 192.

1 196.

2 Poly(4-ethoxycarbonylphenyl acrylate)

CH2

197.

3 200

4 617

5 310

6 324

7 766

220

491

441 (Softening point)

448

611

172

445

367

387

544

153

442

~359

346

553

162

476

325

305

627, 628

CH C O

C O C2H5

O

O

Poly(4-ethoxycarbonylphenylmethacrylamide)

CH3 CH2

C C NH

C O CH2

O

O

198.

Poly(4-ethoxycarbonylstyrene) CH2 CH

199.

Poly(4-ethoxystyrene)

O C O CH2

CH2

CH3

CH

O CH2 200.

CH3

Poly(4-ethylbenzoylethylene)

CH2 O

CH3

CH C

CH2

CH3 155

2 Poly(4-ethylstyrene)

CH2

4 404

5 300, 351

6 354

7 552, 554, 555

139

390

368

356

686

151

448

353

337

686

115

286

368

402

548-550, 563-565

CH

CH2 202.

3 143

CH3

Poly(4-fluoroalkyl α-fluoroacrylate)

F

CH2

C C O CH2

CF2

CF2H

O 203.

Poly(4-fluoroalkyl methacrylate)

CH3 CH2

C C O CH2

CF2

CF2H

O 204.

Poly(4-fluorostyrene)

CH2

CH

F

156

1 201.

1 205.

2 Poly(4-hexadecylstyrene)

CH2

O 208.

(C H 2 ) 4

213

721.4

339

295

544

240

801.6

339

299

554

238

888

253

268

552

CH

C

O

(C H 2 ) 5 C H 3

Poly(4-hexyloxymethylstyrene)

C H2

7 554

CH3

Poly(4-hexyloxycarbonylstyrene)

CH2

6 325

CH

O C 207.

5 278

CH3

Poly(4-hexanoylstyrene)

C H2

4 1175

CH

(CH2)15 206.

3 382

CH

O

(C H 2 ) 5

C H3

157

C H2

2 Poly(4-hexylstyrene)

CH2

5 246

6 277

7 554

206

549

363

376

544

189

470

368

402

544

189

520

342

363

278, 628

CH3

Poly(4-isobutoxycarbonylstyrene)

CH2

4 765

CH

(CH2)5 210.

3 212

CH

CH3 O C O CH2 211.

CH CH3

Poly(4-isopropoxycarbonylstyrene)

CH2

CH

CH3 O C O CH CH3 212.

Poly(4-isopropylbenzoyloxyethylene)

CH2

CH O C O

CH3 CH CH3

158

1 209.

1 213.

2 Poly(4-methoxy-2-methylstyrene)

CH2

3 152

4 386

5 ~358, 363

6 394

7 568

200

482

379

415

560

202

410

453 (Softening point)

493

611

155

421

386

368

544

CH CH3

O CH3 214.

Poly(4-methoxycarbonylphenyl methacrylate)

CH3 CH2

215.

C C O

C O CH3

O

O

Poly(4-methoxycarbonylphenylmethacrylamide)

CH3 CH2

216.

C C NH

C O CH3

O

O

Poly(4-methoxycarbonylstyrene)

CH2

CH

159

O C O CH3

2 Poly(4-methoxymethylstyrene) CH2 CH

CH2

3 153

4 451

5 350

6 339

7 552

136

363

386

375

553, 568570, 672

126

315

366, 374, 382

400

548, 555, 569, 572, 573, 708

206

509

400

405

544

O CH3

218.

Poly(4-methoxystyrene) CH2 CH

219.

Poly(4-methylstyrene) CH2 CH

220.

Poly(4-morpholinocabonylstyrene) CH2 CH

O CH3

CH3

CH2 O

C N CH2

CH2 O CH2

160

1 217.

1 221.

2 Poly(4-nonadecyl styrene) CH2 CH

(CH2)18 222.

263

1031

220

255

554

417

1198

305

348

554

247

843

323

293

544

CH3

Poly(4-octanoylstyrene)

CH2

7 273

CH

(CH2)17 224.

6 360

CH3

Poly(4-octadecylstyrene)

CH2

5 305

CH

(CH2)8 223.

4 1206

CH3

Poly(4-nonylstyrene)

CH2

3 434

CH

(CH2)6

CH3

161

O C

226.

2 Poly(4-octylstyrene) CH2 CH

(CH2)7 CH3 Poly(4-p-anisoylstyrene) CH2 CH

3 246

4 943

5 228

6 261

7 554

211

521

376

405

544

194

515

~373 (softening point)

377

287

185

423

434

437

571, 576

O CH3 227.

Poly(4-phenoxystyrene) CH2 CH

O 228.

Poly(4-phenyl styrene)

CH2

CH

162

1 225.

1 229.

2 Poly(4-phenylacetylstyrene)

CH2

Poly(4-piperidinocarbonylstyrene) CH2 CH

CH2 CH2

232.

6 393

7 544

214

534

387

401

544

170

453

375

375

544

189

534

365

354

544

CH2 CH2 CH2

Poly(4-propionylstyrene) C H2 C H

O C C H2 C H3 Poly(4-propoxycarbonylstyrene) C H2 C H

C

O

CH2

CH2

C H3

163

O

5 351

CH2

CO N 231.

4 561

CH

O C 230.

3 221

234.

2 Poly(4-propoxymethylstyrene) C H2 C H

C H2 O Poly(4-propoxystyrene)

C H2

C H2

C H2

Poly(4-p-toluoylstyrene)

CH2

5 295

6 301

7 552

170

531

343

320

553

220

524

372

420

544

178

511

359

348

555

CH3

C

O

Poly(4-sec-butylstyrene)

C H2

CH

C H3

CH

CH3

CH

H3C 236.

C H2

4 621

CH

O 235.

C H2

3 187

C H2

C H3

164

1 233.

1 237.

2 Poly(4-tert-butylbenzoylethylene) C H2 C H

H 3C 238.

C

O

C

C H3

C H3 Poly(4-tert-butylbenzoyloxyethylene) CH2 CH

3 196

4 515

5 377

6 344

7 628

206

572

374

360

278, 628, 655

238

607

356, 403451

392

758, 760

206

563

344

366

585

C O

O

H3C

C CH3 CH3

239.

Poly(4-tert-butylcyclohexyl methacrylate) C H3

C H2

C C

240.

C H2 CH C H2

O

O Poly(4-tert-butylphenyl acrylate)

C H2

CH C

C H3 C

C H3

C H3

C H3 O

C C H3

CH3 165

O

C H2 CH C H2

2 Poly(4-tert-butylstyrene) CH2 CH

CH3

C

3 178

4 446

5 399–404

6 399

7 666

348

1156

237

301

554

196

631

343

311

544

127

236

429

494

714

C H3

C H3

242.

Poly(4-tetradecylstyrene)

CH2

CH

(CH2)13 243.

Poly(4-valerylstyrene)

CH2

O 244.

CH3

CH

C (CH2)3

Poly(4-vinyl phenol) CH2 CH

OH

CH3

166

1 241.

1 245.

2 Poly(5,7-dihydro-1,3,5,7-tetraoxobenzo[1,2-c:4,5-c']dipyrrole-2,6[1H,3H]-diyl-1,4-phenyleneoxy1,4phenylene)

N 246.

N

CO

CO

CO

CO

CO

CO

CO

CO

N

N

CO

CO

CO

CO

CO

CO

CO

CO

Poly(5-bromo-2-butoxystyrene) CH2

5 523, 773, 653

6 649

7 298, 300-309

319

519

618

615

303

286

709

408

403

299

303

734

408

413

299

303

844

383

359

299

208

667

320

312

547

S

(CH2)2 CH (CH2)4 CH3

CH3

N

(CH2)3

C

(CH2)3

CH3

N

(CH2)9

CH O

(CH2)3

CH3

167

Br

4 476

O

Poly(5,7-dihydro-1,3,5,7-tetraoxobenzo[1,2-c:4,5-c']dipyrrole-2,6[1H,3H]-diylnonamethylene)

N 250.

CO

Poly(5,7-dihydro-1,3,5,7-tetraoxobenzo[1,2-c:4,5-c']dipyrrole-2,6[1H,3H]-diyl-4,4dimethylheptamethylene)

N 249.

CO

Poly(5,7-dihydro-1,3,5,7-tetraoxobenzo[1,2-c:4,5-c']dipyrrole-2,6[1H,3H]-diyl-3methylheptamethylene)

N 248.

CO

Poly(5,7-dihydro-1,3,5,7-tetraoxobenzo[1,2-c:4,5-c']dipyrrole-2,6[1H,3H]-diyl-1,4-phenylenethio-1,4phenylene)

N 247.

CO

3 309

2 Poly(5-bromo-2-ethoxystyrene)

CH2

3 174

4 487

5 353

6 357

7 547

225

680

310 (low viscosity)

331

547

191

511

308

374

547

157

408

359

385

547

225

755

322 (low viscosity)

298

547

CH O CH2

CH3

Br 252.

Poly(5-bromo-2-isopentyloxystyrene)

CH2

CH

O (CH2)2

CH3

Br 253.

CH CH3

Poly(5-bromo-2-isopropoxystyrene)

CH2

CH CH3 O CH CH3

Br 254.

Poly(5-bromo-2-methoxystyrene)

CH2

CH O CH3

Br 255.

Poly(5-bromo-2-pentyloxystyrene)

CH2

CH O

Br

(CH2)4

CH3

168

1 251.

1 256.

2 Poly(5-bromo-2-propoxystyrene) CH2 CH

O

257.

5 327 (low viscosity)

6 331

7 547

177

644

250

275

580

186

677

223

274

755

156

492

350

317

686

194

469

360

414

548

CH

CH2 CH2 C N C O CH O CH2 CH3 O

Poly(5-cyano-3-thiapentyl acrylate)

CH2

4 577

CH3

Br Poly(5-cyano-3-oxapentyl acrylate)

CH2

258.

(CH2)2

3 191

CH C O

(CH2)3

S

(CH2)2

C N

O 259.

Poly(5-fluoroalkyl methacrylate)

CH3 CH2

C C O CH2

CF2

CF3

O 260.

Poly(5-tert-butyl-2-methylstyrene) CH2 CH

H3C

CH3

H3C C 169

CH3

2 Poly(6-cyano-3-thiahexyl acrylate)

CH2

3 203

4 769

5 215

6 264

7 755

203

769

215

264

755

204

823

233

248

587

237

944

214

251

755

206

696

320

296

686

CH C O

(CH2)3

S

(CH2)3

C N

O 262.

Poly(6-cyano-4-thiahexyl acrylate)

CH2

CH C O

S

(CH2)4

(CH2)2

C N

O 263.

Poly(7,7,8,8-tetrafluoro-3,6-dioxaoctyl acrylate)

CH2

CH C O (CH2)2

(CH2)2

O

O CF2

CF2H

O 264.

Poly(8-cyano-7-thiaoctyl acrylate)

CH2

CH C O

(CH2)7

S

CH2

C N

O 265.

Poly(8-fluoroalkyl methacrylate)

CH3 CH2

C C O CH2 O

(CF2)3

CF2H

170

1 261.

1 266.

2 Poly(9-iodononylethylene)

CH2

5 267

6 260

7 649

125

303

513

413

287, 291, 295, 566

143

210

487-618; 628-649

682

287, 494, 751-753, 759

129

335

395

385

728

173

551

330

314

729

190

576

304

330

732

CH2I

Poly(α,β,β-trifluorostyrene)

CF2

4 858

CH (CH2)8

267.

3 223

CF

268.

Poly(acenaphthylene)

269.

Poly(allyl-2-cyanoacrylate)

CH

CH

C N CH2

C CH CH2

CO O CH2 270.

Poly(allyloxyethyl-2-cyanoacrylate)

C N CH2

C CO O (CH2)2

271.

O CH2

CH CH2

Poly(allyloxyisopropyl-2-cyanoacrylate)

C N CH2

C CO O CH2 CH O CH2

171

CH3

CH CH2

2 Poly(aniline)

3 87.7

4 247

5 373

6 355

7 761

128

390

314, 347

329

627, 628

138

391

344

353

278, 398, 568, 580, 628, 652, 653

127

381

333

333

511, 514, 661

112

498

218, 217

225

588, 600, 620-624, 709

145

472

330 Vicat softening point

307

599

NH 273.

Poly(benzoylethylene)

CH2

CH

O C 274.

Poly(benzoyloxyethylene)

CH2

CH O

O C 275.

Poly(benzylethylene)

CH2

CH CH2

276.

Poly(butoxyethylene)

CH2

CH O

277.

(CH2)3

CH3

Poly(butyl chloroacrylate)

Cl CH2

C C O O

(CH2)3

CH3

172

1 272.

1 278.

2 Poly(butyl cyanoacrylate)

3 150

4 457

5 358

6 328

7 613

245

706

321

347

725

193

785

223

246

699

200

643

291

311

699

200

602

304

332

699

102

449

223

227

509-513

C N CH2

C C O

(CH2)3

CH3

O 279.

Poly(butylene 2,6-naphthalate)

C O C

(CH2)4

O

O

O 280.

Poly(butylene adipate)

C

(CH2)4

C O

O 281.

282.

283.

(CH2)4

O

O

Poly(butylene isophthalate)

C

C O

O

O

(CH2)4

Poly(butylene terephthalate)

C

C O

O

O

Poly(butylethylene)

CH2

(CH2)4

O

O

CH (CH2)3 173

CH3

2 Poly(butylimino-2,2,3,3,4,4-hexafluoropentamethylenebutyliminoadipoyl)

N

285.

CH2

(CF2)3

CH2 N

(CH2)3

(CH2)3 O

CH3

CH3 C

(CF2)3

(CH2)3 O

C N

6 254

7 481

383

1508

~293 Brittle point

252

481

121

476

253

254

600

116

358

~303

324

750

206

500

405

412

6

282

594

480

475

6

O

(CH2)6

CH3 CH S

(CH2)3

CH3

Poly(butyryliminoethylene)

CH2

5 283–288

C

Poly(butylthioethylene)

CH2

4 1508

O (CH2)3

CH3

287.

(CH2)4

Poly(butyliminohexafluoroglutarylbutyliminohexamethylene)

N

286.

C

3 383

CH NH C

(CH2)2

CH3

O 288.

289.

Poly(carbonyl-1,3-phenylenecarbonyloxy-1,3-phenyleneoxy)

C

C O

O

O

O

Poly(carbonyl-1,3-phenylenecarbonyloxy-1,4-phenylene-1,4-phenyleneoxy)

C

C O

O

O

O

174

1 284.

1 290.

2 Poly(carbonyl-1,4-phenylenecarbonyloxy-1,4-phenylene-(methyl)phenylmethylene-1,4-phenyleneoxy)

3 391

4 788

5 483-493

6 496

7 6

282

552

540

511

6

494

834

598

592

6

554

1053

563

526

6

CH3

291.

292.

C

C

O

O

O

C

O

Poly(carbonyl-1,4-phenylenecarbonyloxy-1,4-phenylene-1,4-phenyleneoxy)

C

C O

O

O

O

Poly(carbonyl-1,4-phenylenesulfonyl-1,4-phenylenecarbonyloxy-1,4-phenylene-3,3-phthalidylidene1,4-phenylene)

C

SO2

O

C O O

O C O CO

293.

Poly(carbonyl-1,4-phenylenesulfonyl-1,4-phenylenecarbonyloxy-1,4-phenylenediphenylmethylene1,4-phenyleneoxy)

C O

SO 2

C O

C

O

O 175

2 Poly(carbonyl-1,4-phenylenesulfonyl-1,4-phenylenecarbonyloxy-1,4-phenyleneisopropylidene-1,4phenyleneoxy)

3 437

4 837

5 518

6 522

7 6

64.5

210

~325, 373

307

238

773

331

308

283, 286, 325, 360, 397, 399, 413, 415, 476, 523, 608, 629-645 706

272

932

329

292

706

153

464

349

330

650

CH3

C

SO2

O 295.

C O

C

O

CH3

Poly(chlorotrifluoroethylene)

CF

O

CF2

Cl 296.

Poly(cyclodecyl methacrylate)

CH3 CH2

297.

C

CH2 C O CH CH2 O

CH2

CH2

CH2

CH2

CH2 CH2 CH2

Poly(cyclododecyl methacrylate)

CH3 CH2

298.

C

CH2 C O CH CH2 O

CH2

CH2

CH2

CH2

CH2

CH2

Poly(cyclohexanoyloxyethylene)

CH2

CH O

CH2 C CH CH2 O

CH2 CH2 CH2

CH2 CH2 CH2

176

1 294.

1 299.

2 Poly(cyclohexyl acrylate)

CH2

300.

CH

CH2 C O CH CH2 O

3 153

4 532

5 292

6 288

7 583

167

413

387 Vicat softening point

404

599

170

456

384

373

742

170

552

298

308

650

124

326

393, 406

380

514-520

CH2 CH2 CH2

Poly(cyclohexyl chloroacrylate)

Cl CH2

C

CH2 C O CH

CH2

CH2

CH2

O 301.

CH2

Poly(cyclohexyl methacrylate)

CH3 CH2

302.

C

CH2 C O CH CH2 O

CH2 CH2 CH2

Poly(cyclohexylacetoxyethylene)

CH2

CH O C O CH2

303.

Poly(cyclohexylethylene) CH2 CH H2C

CH2 CH CH2

CH2 CH2 CH2

CH CH2 177

H2C CH2 H2C

2 Poly(cyclohexyloxyethylene)

CH2

CH CH2

306.

5 354

6 306

7 624

204

614

346

332

706

221

695

326

318

706

107

301

348

355

514

250

1050

203

238

277, 581, 596, 756

CH2 CH2 CH2

Poly(cyclooctyl methacrylate)

CH3

CH2

4 438

CH O CH2

305.

3 134

C

CH2 C O CH CH2 O

CH2

CH2 CH2 CH2

CH2

Poly(cyclooctylmethyl methacrylate)

CH3 CH2

C C O CH2 O

307.

CH2 CH2

Poly(cyclopentylethylene)

CH2

308.

CH2 CH CH2

CH CH2 CH CH2

CH2 CH2

Poly(decyl methacrylate)

CH3 CH2

C

O

C

O

CH2 CH2 CH2

(C H 2 ) 9

C H3

178

1 304.

1 309.

2 Poly(decyl-2-cyanoacrylate) C N

CH2 310.

4 996

5 192

6 254

7 731

214

1034

~183, 211

207

600, 620, 622, 623

194

649

289-297

299

597

106

757

134; 138140

140

740, 741

160

489

292

327

560

80.7

504

173, 186

160

493, 711

C

CO O Poly(decyloxyethylene)

CH2

(CH2)9 CH3

CH O

311.

3 253

(CH2)9

CH3

Poly(diethylaminoethyl methacrylate)

CH3 CH2

C C O (CH2)2

CH3

CH2

CH3

N

O 312.

CH2

Poly(diethylsiloxane)

C2H5 Si

O

C2H5 313.

Poly(dimethylaminoethyl methacrylate)

CH3 CH2

C C O CH2 O

314.

Poly(dimethylsilylenemethylene)

CH3 CH2

N CH3

CH3 Si

179

CH3

CH2

2 Poly(dimethylsilylenetrimethylene)

3 115

4 665

5 203

6 173

7 490

263

809

311

325

489

208

967

208

215

461

71.3

330

246

216

351, 352

140

649

199

216

351

147

465

296

316

352, 461

123

572

201

215

351

CH3 Si

(CH2)3

CH3 316.

Poly(di-p-tolylsilylenetrimethylene)

CH3

Si

(CH2)3

CH3 317.

Poly(dithiodecamethylene)

–S–S–(CH2)10– 318.

Poly(dithioethylene)

–S–S–CH2–CH2– 319.

Poly(dithiohexamethylene)

–S–S–(CH2)6– 320.

Poly(dithiomethylene-1,4-phenylenemethylene)

S 321.

S

CH2

Poly(dithiopentamethylene)

–S–S–(CH2)5–

CH2

180

1 315.

1 322.

2 Poly(dodecafluorobutoxyethylene)

CF2

CF

3 174

4 713

5 263–273

6 244

7 445

267

1094

270 Brittle point

244

581, 586

284

1229

208 conflicting data

231

273, 581, 598, 600, 601

262

630

420, 425, 418, 430, 405, 431

416

684, 700, 702, 709, 715

604

1250

483

483

738, 739

478

988

515

484

738, 739

O CF2 CF2 323.

CF2

CF3

Poly(dodecyl acrylate)

CH2

CH C O

(CH2)11

CH3

O 324.

Poly(dodecyl methacrylate)

CH3

CH2

C C O (CH2)11

CH3

O 325.

Poly(ehter ether ketone)

O

O

C O

326.

Poly(ether imide) 1 O N

C

CH3 O

O

C

C

O

C C

CH3

O

327.

C

O

O

Poly(ether imide) 10 O

N

N

O O

O

C C

O

O

N

O 181

C

2 Poly(ether imide) 11 O N

329.

O

O

N

O

O

C

O

O

C

C

C

O

O

N

O

C

C

C

O

O

O

S

O

500

465

738, 739

646

1462

487

442

738, 739

572

1246

485

459

738, 739

488

1030

499

474

738, 739

O

C

C

O

O

O N

Poly(ether imide) 15

N

1028

O

O

C

O

478

N

C

O

7 738, 739

O

O

C

Poly(ether imide) 14 O

6 467

N

Poly(ether imide) 13 O

5 488

O

Poly(ether imide) 12

N

332.

C C

O

331.

O

C

N

330.

O

O

4 1203

O

O

C

3 562

S

O

O

C

C

C

C

O

O

N

182

1 328.

1 333.

2 Poly(ether imide) 16

O N

334.

335.

C

C

O

O S

C

C

C

O

O

C

C

C

O

O

O

SO2

O

C

C

484

738, 739

656

1505

451

436

738, 739

582

1143

533

509

738, 739

520

1074

488

484

738, 739

O

O N

O

O

CH3 O

C CH3

O O

C C O

N 183

O

530

O

O

C

C

1165

N

C

C

564

N

O Poly(ether imide) 2

N

7 738, 739

O

O

C

Poly(ether imide) 19 O

N

S

6 455

O

O

C

O

5 482

N

Poly(ether imide) 18 N

337.

O

4 1073

O

O

C

O

336.

S

C

Poly(ether imide) 17 O

N

O

3 488

2 Poly(ether imide) 20

O N

339.

O

SO2

C

C

C

C

O

O

O

340.

341.

N

342.

O

O

C

C

C

O

O SO 2

7 738, 739

394

812

528

485

738, 739

666

1402

402

475

738, 739

572

1184

483

483

738, 739

487

1008

512

483

738, 739

N

O

O

C

C

C

O

O

O

O

C

O

O

C

C

O

O

O

O

C

O

N

C

C

O Poly(ether imide) 24 O N

O

C

Poly(ether imide) 23 O

6 514

N

Poly(ether imide) 22 N

5 538

O

C

O

4 969

O

O

Poly(ether imide) 21

N

3 498

O N

O C

C

C

C

O

O

O

N

184

1 338.

1 343.

2 Poly(ether imide) 25 O

N

344.

O

C

C

C

C

O

O

CO

CO

347.

N

S

CO

CO

CO

N

401

379

738, 739

572

1246

507

459

738, 739

488

1073

504

455

738, 739

656

1505

475

436

738, 739

O N

S

O

CO

CO

CO

CO

N

S O

1330

O

CO

Poly(ether imide) 29

504

(CH2)6

Poly(ether imide) 28

N

7 738, 739

O CO

O

6 455

O

Poly(ether imide) 27

N 346.

O

CO

O

5 467

N

Poly(ether imide) 26

N 345.

C

O

4 1442

O

O

O

O

3 656

O CO

CO

CO

O

O

N

185

CO

2 Poly(ether imide) 3 O N

349.

O

C

O

C

O

O

CO

CO

CO

CO

CO

CO

N

1143

540

509

738, 739

498

969

539

514

738, 739

666

1402

503

475

738, 739

487

1008

521

483

738, 739

O

N

O

CO

CO

CO

CO

O

O

CO

O

N

Poly(ether imide) 33

N

582

N

SO2 O

7 738, 739

O

SO 2

CO

Poly(ether imide) 32

6 451

O

Poly(ether imide) 31

N

5 472

O

SO2

CO

O

352.

O

4 1246

N

Poly(ether imide) 30

N

351.

C

C

O

350.

O

3 562

O

CO

CO

CO

CO

N

186

1 348.

1 353.

2 Poly(ether imide) 34

N 354.

O

CO

CO

CO

CO

Poly(ether imide) 35

N 355.

O

O

CO

CO

CO

N 356.

562

1286

466

437

738, 739

553

1122

550

493

738, 739

CO

CO

469

862

548

544

738, 739

638

1381

497

462

738, 739

O N

O

CO

CO

CO

CO

N

Poly(ether imide) 38

N

7 738, 739

O CO

O

6 461

O

Poly(ether imide) 37

N

357.

O

CO

O

5 514

N

Poly(ether imide) 36

O

4 855

N

O

CO

3 394

O

CO

CO

CO

CO

O

O

N

187

2 Poly(ether imide) 39

O N 359.

O

CO

CO

CO

CO

C

O

O

C

O

O

N

478

1028

482

465

738, 739

478

1028

505

465

738, 739

646

1462

471

442

738, 739

597

1164

549

513

737

O

O

O CO

CO

CO

N

Poly(ether imide) 41

N

7 738, 739

O

CO

O

362.

C

C

O

361.

6 467

N

Poly(ether imide) 40

N

5 512

O

O

360.

4 1203

O

Poly(ether imide) 4

N

3 562

O

O

CO

CO

CO

CO

O

O

N

Poly(ether imide) 42

CF3 C CF3

N

CO

O CO

CO O

CO

N

188

1 358.

1 363.

2 Poly(ether imide) 43

CF3 C

N

CF3

364.

CO

O

O

CO

CO

CO

C

N

CF3

CO

CO

O

366.

Poly(ether imide) 46

N

CF3

367.

CO

CO

CO

CO

CO

CO

Poly(ether imide) 47

N

1164

491

513

737

597

1164

529

513

737

478

988

536

484

738, 739

394

812

532

485

738, 739

O

CO

CO

O

597

N

O

O N

7 737

N

Poly(ether imide) 45

C

6 513

CO

O

CF3

5 529

O

CO 365.

4 1164

N

Poly(ether imide) 44

CF3

3 597

O N

O

CO

CO

CO

CO

N 189

2 Poly(ether imide) 48

O N 369.

N 370.

371.

CO

CO

CO O

O

5 487

6 451

7 738, 739

478

1047

499

465

738, 739

394

855

497

461

738, 739

683

1444

503

473

737

598

1189

554

503

737

O

N

O

CO

CO

CO

CO

O N

Poly(ether imide) 5 O N

4 1246

O

CO

Poly(ether imide) 49

3 562

O

C

O

O

C C

C

N

O

O Poly(ether imide) 50 O

O

O N

CO CO CO CO

N

O

372.

Poly(ether imide) 51 H3C

CH3 CH2

H3C

N CH3

O CO CO

O CO CO

N

190

1 368.

1 373.

2 Poly(ether imide) 52

CH3

H3C CH2

N

H3C 374.

Poly(ether imide) 53 H3C

5 538

6 503

7 737

598

1189

545

503

737

581

1107

581

525

737

471

975

523

483

737

O

CO

CO

CO

CO

CH3 N

H3C

N

O

O CO

CO

CO

CO

N

CH3

Poly(ether imide) 54

H3C

CH3 N

H3C 376.

4 1189

CH3

CH2

375.

O

3 598

O

O

CO

CO

CO

CO

N

CH3

Poly(ether imide) 55

O N

CO CO

CO

CO O

191

CF3

N

2 Poly(ether imide) 56

O N

O

CO

CO

CO

CO

3 471

4 975

5 523

6 483

7 737

471

975

508

483

737

692

1401

526

494

737

692

1401

519

494

737

562

1286

403

437

738, 739

N

CF3

378.

Poly(ether imide) 57

O

O

N

CO

CO

CO

CO

N

CF3

379.

Poly(ether imide) 58 CH3

CH3

C

C

CH3

CH3

O CO

N

CO

CO

CO

N

O

380.

Poly(ether imide) 59 CH3

CH3

C

C

CH3

CH3

O N

CO CO CO CO

N

O

381.

Poly(ether imide) 6 O

N

C

O O

O

C

C

C

O

O

O N

O

192

1 377.

1 382.

383.

2 Poly(ether imide) 60 CH3

CH3

C

C

CH3

CH3

O

CO

N

CO

CO

CO

O

N

CO

CO

CO

O O

N

O CO

CO

CO

Poly(ether imide) 64

O O

Poly(ether imide) 65 N

O

CO

N

6 494

7 737

608

1310

502

464

737

608

1310

501

464

737

608

1351

481

450

737

683

1444

500

473

737

439

916

533

479

737

N

N

O

CO

CO

CO

CO

O

5 519

N

CO

Poly(ether imide) 63

4 1401

N

O

CO

O

O

387.

CO

CO

Poly(ether imide) 62

O

386.

O

O

O

385.

N

Poly(ether imide) 61 O

384.

O CO

3 692

N

CO CO

CO O

N

193

CO

2 Poly(ether imide) 66

O N

389.

O

CO

CO

CO

CO

Poly(ether imide) 67

4 916

5 528

6 479

7 737

439

916

503

479

737

439

916

527

479

737

523

1092

513

479

737

553

1122

502

493

738, 739

N

O N

3 439

CO CO CO CO

N

O 390.

Poly(ether imide) 68

O N

391.

O

CO

CO

CO

CO

N

O

Poly(ether imide) 69 O

N

CO CO

CO

N

CO O 392.

Poly(ether imide) 7 O N

C

O O

O

C

C

C

O

O

N

O

194

1 388.

1 393.

2 Poly(ether imide) 70

O

O N

394.

Poly(ether imide) 71 O

O

CO

CO

CO

CO

3 523

4 1092

5 508

6 479

7 737

523

1092

522

479

737

523

1092

518

479

737

523

1134

499

461

737

523

1134

498

461

737

N

O CO

N

CO CO CO

N

O 395.

Poly(ether imide) 72

O

O N

396.

Poly(ether imide) 73 O

O

CO

CO

CO

CO O

N

N

CO CO

CO

CO

N

O 397.

Poly(ether imide) 74 O

N

CO

O

CO CO CO

N

195

O

2 Poly(ether imide) 75

O

O N

400.

Poly(ether imide) 76

O

CO

CO

CO

CO

C

N

CH3

4 1134

5 494

6 461

7 737

565

1137

538

497

737

565

1137

509

497

737

565

1137

520

497

737

656

1442

489

455

738, 739

N

O

CH3

3 523

CO CO

CO

CO

N

O 401.

Poly(ether imide) 77

O

CH3 C

N

CH3

CO CO CO CO

N

O 402.

Poly(ether imide) 78

O

CH3 C

N

CH3 403.

Poly(ether imide) 79

CO

CO

CO

CO

CO O

N

O

O

CO

CO

CO

CO

O N

O

N

196

1 399.

1 404.

2 Poly(ether imide) 8

O N

C

O

O

C C

C

C

CO

CO

CO

CO

C

483

738, 739

520

1075

509

484

738, 739

612

1246

503

491

738, 739

577

1378

408

397

738, 739

N

O

CH3

CO CO

CO

CH2 N

CH3 O

C CH3

O CO CO

N

(CH2)6

197

CO

508

N

CH3

CO

CO

1251

O

CH3

Poly(ether imide) 83

N

604

O

CH3

O

408.

CO CO

Poly(ether imide) 82

N

7 738, 739

O

CO O

407.

C CH3

CO

Poly(ether imide) 81

N

6 496

O O

406.

5 520

N

CH3

Poly(ether imide) 80

N

4 946

O

O 405.

3 469

2 Poly(ether imide) 84 O

O

O N

410.

CO

CO

CO

CO

CO

N

C

C C

C

O

O

Poly(ethyl chloroacrylate)

5 500

6 473

7 738, 739

683

1444

504

473

766

638

1381

478

462

738, 739

111

294

366 Vicat softening point

377

588, 599, 614, 615

131

427

300

307

560

176

553

325

318

613

N

O

O

N

Cl CH2

4 1444

O

O

O

3 683 N

O

Poly(ether imide) 9 O

412.

CO

CO

O

O N

411.

CO

Poly(ether imide) 85 O

O

C C O CH2

CH3

O

413.

Poly(ethyl ethacrylate)

CH2 CH2

CH3

C C O CH2

CH3

O

414.

Poly(ethyl ethoxycarbonyl methacrylate) CH2

CH2

C O CH2

C

O

C O CH2 O

CH3

CH3

198

1 409.

1 415.

2 Poly(ethyl-2-cyanoacrylate)

3 116

4 279

5 422

6 416

7 730

85.3

297

268

287

526

211

548

398

385

725

211

548

398

385

699, 727

315

1150

293

274

481

315

1150

~278

274

481

C N CH2

C CO O CH2 CH3

416.

Poly(ethyl-2-propylene) CH2 CH3

CH2 417.

C

CH3 Poly(ethylene 2,6-naphthalate)

O

(CH2)2

O C O

C O

418.

Poly(ethylene 2,6-naphthalenedicarboxylate)

(CH2)2

O C C O

O

O 419.

Poly(ethylimino-2,2,3,3,4,4-hexafluoropentamethyleneethyliminoadipoyl)

N

420.

CH2

(CF2)3

CH2 N

CH2

CH2

CH3

CH3

C

(CH2)4

O

Poly(ethyliminohexafluoroglutarylethyliminohexamethylene)

N

C

CH2 O

C N

O

(CH2)6

O CH2 CH3

199

CH3

(CF2)3

C

2 Poly(ethylthioethylene)

CH2

3 86.5

4 297

5 266

6 291

7 601

62.3

207

310 (~60% syndio)

301

501

128

391

347, 336

327

687, 705

91.8

340

268

270

445

312

1268

269

246

612

137

634

208 - 228

216

496, 509, 512

CH S CH3

CH2 422.

Poly(formyloxyethylene)

CH2

CH O

O C 423.

H

Poly(glycidyl methacrylate)

CH3 CH2

C C O CH2

CH O

O 424.

Poly(hexafluoromethoxyethylene)

CF2

CF O

425.

CF3

Poly(hexyl hexyloxycarbonyl methacrylate)

CH2

CH2

C O

C

O

O C O (CH2)5 426.

CH2

Poly(hexylethylene)

CH2

CH (CH2)5 CH3

(CH2)5

CH3

CH3

200

1 421.

1 427.

2 Poly(hexyloxyethylene)

CH2

CH

3 146

4 676

5 199

6 216

7 600, 620624, 709

212

436

553, 513

486

378, 474, 486

273

713

383

383

485

212

394

< 500

538

378

212

394

< 500

538

378

212

353

618, 580 600

601

378, 379

258

459

578

562

379

O (CH2)5

428.

NH

429.

CH3

Poly(imino-1,3-phenyleneiminoisophthaloyl)

NH C

C

O

O

Poly(imino-1,3-phenyleneiminosebacoyl)

NH

NH C

(CH2)8

O 430.

C

O

O

NH C

C

O

O

Poly(imino-1,4-phenyleneiminoterephthaloyl)

NH 433.

NH C

Poly(imino-1,4-phenyleneiminoisophthaloyl)

NH 432.

O

Poly(imino-1,3-phenyleneiminoterephthaloyl)

NH 431.

C

NH C

C

O

O

Poly(imino-1,5-naphthyleneiminoterephthaloyl) NH C

O

O

201

NH C

1 434.

435.

2 Poly(imino-1-methyl-3-oxotrimethylene)

NH CH CH2

C

CH3

O

Poly(imino-1-oxoethylene-1,4-phenylene-2-oxoethyleneiminooctadecamethylene) NH C CH2

436.

C NH

CH2

CH2

CH2

CH2

C NH

CH2

CH2

CH2

CH2

O 438.

7 469

478

1354

351

353

478

392

1107

358

354

478

512

1467

338

349

478

543

1278

438

425

285

266

826

338

322

406

300

992

336

302

483

(CH2)18

O

Poly(imino-2,2,4-trimethylpentamethyleneiminoadipoyl)

CH3

CH3

C CH2

CH CH2

NH C

CH3 440.

C NH

Poly(imino-1-oxotrimethylenefluoren-9-ylidene-3-oxotrimethyleneiminotrimethylenefluoren-9ylidenetrimethylene) NH C (CH2)2 C (CH2)2 C NH (CH2)3 C (CH2)3

NH CH2

6 407

O

O

439.

5 369

(CH2)11

O O Poly(imino-1-oxotrimethylene-1,4-phenylene-3-oxotrimethyleneiminooctadecamethylene)

NH C

4 201

(CH2)18

O O Poly(imino-1-oxotrimethylene-1,4-phenylene-3-oxotrimethyleneiminododecamethylene) NH C

437.

CH2

3 81.8

O

Poly(imino-2,2-dimethylpentamethyleneiminoazelaoyl)

(CH2)4

C O

CH3 C CH3

(CH2)3

NH C O

(CH2)7

C O

202

NH CH2

1 441.

2 Poly(imino-2,2-dimethylpentamethyleneiminodipoyl)

3 249

4 801

5 350

6 311

7 483

266

878

344

303

483

256

531

430

481

375

255

627

393

407

334

313

749

465

418

444

CH3 NH CH2

C

(CH2)3

NH C

CH3 442.

(CH2)4

O

C O

Poly(imino-2,2-dimethylpentamethyleneiminopimeloyl)

CH3 NH CH2

C

(CH2)3

NH C

CH3 443.

(CH2)5

O

C O

Poly(imino-2,2-dimethylpentamethyleneiminoterephthaloyl)

CH3 NH CH2

C

(CH2)3

NH C

C

O

O

CH3 444.

Poly(imino-5-methylisophthaloyliminohexamethylene)

NH C

C NH

O

(CH2)6

O CH3

445.

Poly(imino-5-tert-butylisophthaloyliminomethylene-1,3-phenylenemethylene)

NH C

C NH

O

CH2

CH2

O

H3C C CH3 203

CH3

2 Poly(imino-5-tert-butylisophthaloyliminomethylene-1,4-phenylenemethylene)

NH C

C NH

O H3C

3 313

4 708

5 477

6 442

7 444

272

696

343

391

373

388

1057

393

367

285, 317

261

906

278

288

464

256

656

377

390

482

CH2

CH2

O C CH3 CH3

447.

Poly(iminoadipoyliminomethylene-2,5-dimethyl-1,4-phenylenemethylene)

H3C NH

C

(CH2)4

O

C

NH CH2

CH2

O H3C

448.

Poly(iminoadipoyliminotrimethylenefluoren-9-ylidenetrimethylene)

NH

C

(CH2)4

O

449.

(CH2)3

C

(CH2)3

O

Poly(iminoadipoyliminotrimethylenemethyliminotrimethylene)

NH C O 450.

C NH

(CH2)4

C NH

(CH2)3

O

N

(CH2)3

CH3

Poly(iminocarbonyl-1,4-phenylene-2-oxoethyleneiminohexamethylene)

NH C O

CH2

C NH O

(CH2)6

204

1 446.

1 451.

2 Poly(iminoethylene-1,4-phenyleneethyleneimino-1,11-dioxoundecamethylene)

NH (CH2)2

(CH2)2

NH C

(CH2)9

O 452.

(CH2)2

NH C

(CH2)12

(CH2)2

NH C

(CH2)14

(CH2)2

NH C

(CH2)16

(CH2)2

NH C

363

478

443

1241

358

357

478

478

1354

348

353

478

341

900

378

379

478

232

721

355

322

483

O

(CH2)8

O 456.

366

C

Poly(iminoethylene-1,4-phenyleneethyleneiminosebacoyl)

NH (CH2)2

1127

O

O 455.

409

C

Poly(iminoethylene-1,4-phenyleneethyleneimino-1,18-dioxooctadecamethylene)

NH (CH2)2

7 478

O

O 454.

6 360

C

Poly(iminoethylene-1,4-phenyleneethyleneimino-1,16-dioxohexadecamethylene)

NH (CH2)2

5 369

O

O 453.

4 994

C

Poly(iminoethylene-1,4-phenyleneethyleneimino-1,14-dioxotetradecamethylene)

NH (CH2)2

3 358

Poly(iminoglutarylimino-2,2-dimethylpentamethylene)

C O

CH3 NH C

C O

NH

CH2

C CH3

(CH2)3 205

O

(CH2)3

2 Poly(iminohexamethyleneimino-1-oxotrimethylenefluoren-9-ylidene-3-oxotrimethylene)

NH

(CH2)6

NH C

(CH2)2

C

(CH2)2

O

458.

5 395

6 367

7 285, 317

249

752

323

331

476

382

1000

427-437

382

318

304

608

<500

500

378

256

572

426

447

375

273

652

398

419

444

O

NH C

(CH2)5

C

O 459.

4 1057

C

Poly(iminohexamethyleneimino-4-methylpimeloyl)

NH (CH2)6

3 388

O

Poly(iminohexamethyleneiminocarbonyl-1,4-phenylene-2,2-butylidene-1,4-phenylenecarbonyl)

CH3 NH

(CH2)6 NH C O

C

C

CH2

O

CH3 460.

Poly(iminoisophthaloylimino-1,4-phenylenemethylene-1,4-phenylene)

NH C

C NH

O 461.

CH2

O

Poly(iminoisophthaloylimino-2,2-dimethylpentamethylene)

CH3 NH C

C NH

O 462.

O

CH2

C (CH2)3 CH3

Poly(iminoisophthaloylimino-3,4-dimethylhexamethylene)

NH C O

C NH O

(CH2)2

CH3 CH CH CH3

(CH2)2

206

1 457.

1 463.

2 Poly(iminoisophthaloylimino-4,4'-biphenylylene)

NH C

Poly(iminoisophthaloyliminohexamethylene)

C NH

O

C NH

604

390

396

285, 317, 357, 484

246

599

438

411

285, 317, 444

273

717

388

381

485

395

828

448

477

285, 317

329

715

473

460

444

CH2

CH2

Poly(iminoisophthaloyliminooctamethylene)

C NH

O

(CH2)8

O

Poly(iminoisophthaloyliminotrimethylenefluoren-9-ylidenetrimethylene)

NH C

C NH

O

(CH2)3

C

(CH2)3

O

Poly(iminomethylene-1,3-cyclohexylenemethyleneimino-5-tert-butylisophthaloyl)

NH CH2

239

O

NH C

468.

7 378, 379

(CH2)6

Poly(iminoisophthaloyliminomethylene-1,3-phenylenemethylene)

O

467.

6 544

O

NH C 466.

5 558

O

NH C

465.

4 529

C NH

O 464.

3 288

CH CH2

CH2 CH2

CH CH2 CH2

NH C

C

O

O

H3C C

CH3 207

CH3

2 Poly(iminomethylene-1,3-phenylenemethyleneimino-1-oxotrimethylenefluoren-9-ylidene-3oxotrimethylene)

NH CH2

CH2

NH C

(CH2)2

C

(CH2)2

O 470.

CH2

NH C

(CH2)4

CH2

CH2

NH C

(CH2)10

CH2

NH C

(CH2)16

CH2

285

321

757

432

424

285

341

897

378

380

463, 478

443

1234

348

359

478

392

1104

363

355

478

O

Poly(iminomethylene-1,4-phenylenemethyleneiminopentadecanedioyl)

NH CH2

347

C

O 474.

346

O

Poly(iminomethylene-1,4-phenylenemethyleneiminooctadecanedioyl)

NH CH2

689

C

O 473.

239

C

O

Poly(iminomethylene-1,4-phenylenemethyleneiminododecanedioyl)

NH CH2

7 285, 317

C

NH C O

472.

6 432

O

Poly(iminomethylene-1,3-phenylenemethyleneiminocarbonyl-2,2'-biphenylenecarbonyl)

NH CH2

5 423

C

O 471.

4 914

O

Poly(iminomethylene-1,3-phenylenemethyleneiminoadipoyl)

NH CH2

3 395

NH C O

(CH2)13

C O

208

1 469.

1 475.

2 Poly(iminomethylene-1,4-phenylenemethyleneiminotridecanedioyl)

NH CH2

CH2

NH C

(CH2)11

CH2

NH C

(CH2)9

CH2

NH C

(CH2)4

O H3C

C

7 478

324

878

380

369

463, 478

306

841

382

364

444

313

749

461

418

444

307

783

383

392

478

O

Poly(iminomethylene-5-tert-butyl-1,3-phenylenemethyleneiminoadipoyl)

NH CH2

6 361

C

O 477.

5 373

O

Poly(iminomethylene-1,4-phenylenemethyleneiminoundecanedioyl)

NH CH2

4 992

C

O 476.

3 358

C O

CH3

CH3 478.

Poly(iminomethylene-5-tert-butyl-1,3-phenylenemethyleneiminoisophthaloyl)

CH2

NH CH2

H3C

NH C

C

O

O

C CH3 CH3

479.

Poly(iminooctamethyleneimino-1-oxoethylene-1,4-phenylene-2-oxoethylene)

NH

(CH2)8

NH C

CH2

C O

209

O

CH2

2 Poly(iminooxalylimino-2,2-dimethylpentamethylene)

3 181

4 482

5 382

6 376

7 483

239

648

376

369

482

307

751

423

409

488

304

566

< 500

537

378

307

751

418

409

361, 406

307

751

421, 432

409

406

CH3 NH C

C NH CH2

O O 481.

C

CH2

CH2

CH3

Poly(iminopentamethyleneiminocarbonyl-1,4-phenylene-2-oxoethylene)

NH

(CH2)5

NH C

CH2

O 482.

CH2

C O

Poly(iminoterephthaloylimino-1,4,4-trimethylheptamethylene)

CH3

CH3 NH C

C NH CH

O 483.

O

C

O

CH3

C NH

CH2

O

Poly(iminoterephthaloylimino-2,2,4-trimethylheptamethylene)

CH3 NH C O 485.

(CH2)3

Poly(iminoterephthaloylimino-1,4-phenylenemethylene-1,4-phenylene)

NH C 484.

(CH2)2

C NH CH2 C O

CH3 CH2

O

C NH CH2 CH CH2 O

(CH2)3

CH3

Poly(iminoterephthaloylimino-2,4,4-trimethylheptamethylene)

NH C

CH

CH3

CH3 C CH3

(CH2)3

210

1 480.

1 486.

2 Poly(iminoterephthaloylimino-3-ethylhexamethylene)

NH C O 487.

C NH

(CH2)2

O

O

C NH

(CH2)2

CH2

CH3

CH

O

O

5 403

6 383

7 406

290

727

416

399

406

288

488

613

590

378, 379

290

690

425

420

406

341

900

393

379

484, 487

239

561

413

426

406

(CH2)3

Poly(iminoterephthaloylimino-4,4'-biphenylylene)

NH C

4 713

CH CH3

H3C

489.

(CH2)3

Poly(iminoterephthaloylimino-3-isopropylhexamethylene)

NH C

488.

CH

3 273

C NH O

Poly(iminoterephthaloylimino-4,4-dimethylheptamethylene)

CH3 NH C O 490.

(CH2)3

O

O

C NH O

C NH O

(CH2)6 211

O

(CH2)3

(CH2)12

Poly(iminoterephthaloyliminohexamethylene)

NH C

C CH3

Poly(iminoterephthaloyliminododecamethylene)

NH C 491.

C NH

2 Poly(iminoterephthaloyliminomethylene-2,5-dimethyl-1,4-phenylenemethylene)

3 279

4 564

5 498

6 495

7 373

290

769

388

377

406

348

841

446-455

414

318

222

568

357

391

482

456

1373

358

332

285, 317

H3C NH C

C NH

O

CH2

CH2

O H3C

493.

Poly(iminoterephthaloyliminononamethylene)

NH C

C NH

O 494.

(CH2)9

O

Poly(iminotetramethyleneiminocarbonyl-1,4-phenylene-2,2-butylidene-1,4-phenylenecarbonyl)

CH3 NH

(CH2)4 NH C O

C

C

CH2

O

CH3 495.

Poly(iminotetramethyleneiminocarbonyl-1,4-phenylene-2-oxoethylene)

NH

496.

(CH2)4 NH C

CH2 C

O

O

Poly(iminotrimethylenefluoren-9-ylidenetrimethyleneiminosebacoyl)

NH

(CH2)3

C

(CH2)3

NH C O

(CH2)8

C O

212

1 492.

1 497.

2 Poly(isobutoxyethylene)

CH2

CH O

498.

CH2

CH C O CH2 O

499.

4 424

5 254

6 264

7 273, 333, 495, 588, 600, 620, 622-626

131

466

249 Brittle point, 230

281

582, 706

145

398

363

364

588, 617

CH3 CH CH3

Poly(isobutyl acrylate)

CH2

3 112

CH3 CH CH3

Poly(isobutyl chloroacrylate)

Cl CH2

C

CH3

C O CH2 O 500.

Poly(isobutyl methacrylate)

CH CH3 148

CH3 CH2

C

CH3

C O CH2 CH O 501.

Poly(isobutylethylene)

CH2

CH

440 326, 337 336 (random), (random), random), 530 (iso) 281 279 (iso) (isotactic)

506, 581, 583, 596, 607, 706, 710

CH3 102

375

302, 297 333

272

281, 524, 527-537

CH2 CH CH3 213

CH3

2 Poly(isohexylethylene)

CH2

CH2 CH2

5 239

6 246

7 511

135

364

372

371

278, 370, 658

119

465

259

256

511

94.9

344

270, 261

276

588, 600, 620, 624, 709

114

388

267-270

294

506, 582, 583

CH3 CH CH3

Poly(isonicotinoyloxyethylene)

CH2

4 557

CH CH2

503.

3 137

CH O C O

504.

N

Poly(isopentylethylene)

CH2

CH CH2

CH3 CH CH3

CH2 505.

Poly(isopropoxyethylene)

CH2

CH O CH CH3 CH3

506.

Poly(isopropyl acrylate)

CH2 O

CH C O

CH3 CH CH3

214

1 502.

1 507.

2

3 128

Poly(isopropyl chloroacrylate)

4 318

Cl CH2

CH3

C C O O

508.

Poly(isopropyl methacrylate)

131

CH3 C

CH3 C O CH CH3 O

Poly(isopropylethylene)

CH2

6 402

7 588, 599

CH CH3 CH2

509.

5 363 Vicat softening point

306 (iso), 428 (iso), 300 (isotactic), 314 417 (syndio) (syndio) 358 (syndiotactic)

706

85.3

287

323

297

514, 521, 528, 534

349

1195

303-308

292

481

349

1195

~293

292

481

60.7

240

242, 246, 228, 245, 248, 251

253

273, 333, 577, 600, 691, 694, 696, 704, 709, 719

CH CH CH3 CH3

510.

Poly(isopropylimino-2,2,3,3,4,4-hexafluoropentamethyleneisopropyliminoadipoyl)

N CH2

(CF2)3

CH CH3

CH2

N

H3C CH

CH3 511.

O

(CH2)4

C O

CH3

Poly(isopropyliminohexafluoroglutarylisopropyliminohexamethylene)

N

C

(CF2)3

H3C CH O

(CH2)6

CH3

Poly(methoxyethylene)

CH2

C N

O CH CH3

CH3 512.

C

CH CH3

215

O

2 Poly(methyl chloroacrylate)

3 94.2

4 216

5 413 Vicat softening point

6 437

7 599

84.5

198

404

426

579, 618

112

289

357

387

579, 616

Cl CH2

C C O CH3 O

514.

Poly(methyl fluoroacrylate)

F CH2

C C O CH3 O

515.

Poly(methyl fluoromethacrylate)

CF3 CH2

C C O CH3 O

516.

Poly(methyl methacrylate)

96.4

CH3 CH2

C C O CH3

301 (iso), 311, 323 320 (iso), 224 (isotactic), 431 (syndio) (syndio) 378 (syndiotactic)

766

O 517.

Poly(methyl phenylacrylate)

155

CH2 O

C C O CH3

364

391, 397

426

619

216

1 513.

1 517.

2 Poly(methylimino-2,2,3,3,4,4-hexafluoropentamethylenemethyliminoadipoyl)

N

CH2 (CF2)3

CH2 N

CH3 518.

CH3

(CH2)4

O

Poly(methyliminohexafluoroglutarylmethyliminohexamethylene)

N

C

(CF2)3

CH3 O 519.

C

C N

4 989

5 298-303 (Brittle point)

6 284

7 481

281

989

298-303

284

481

138

496

301

278

707

133

437

319

304

757

186

407

500, 357, 423, 481

457

273, 362, 495, 595, 669, 698

270

1154

198-320

234

580, 757

C O

(CH2)6

O CH3

Poly(methylphenylsilylenemethylene)

Si

3 281

CH2

CH3 520.

Poly(N-butylacrylamide)

CH2

CH C NH

(CH2)3

CH3

O 521.

Poly(N-carbazolylethylene) CH2 CH

N

522.

Poly(N-dodecylacrylamide)

CH2

CH O

(CH2)11

CH3

217

C NH

2 Poly(neopentyl methacrylate)

3 165

4 465

5 299-312

6 355

7 664, 706

119

399

332 softening point, crystalline sample

298

511

182

630

213 Brittle point

225

580, 706

135

364

360

371

278, 658

216

986

215 Britte point, 184

219

580, 706

CH3 CH2

524.

C

CH3

C O CH2

C

O

CH3

CH3

Poly(neopentylethylene)

CH2

CH CH2

H3C

C

CH3

CH3 525.

Poly(heptyl acrylate)

CH2

CH

O C O C7H15 526.

Poly(nicotinoyloxyethylene)

CH2

CH O C O

527.

N

Poly(nonyl acrylate)

CH2

CH C O O

(CH2)8

CH3

218

1 523.

1 528.

2 Poly(nonylethylene)

CH2

3 188

4 900

5 236

6 209

7 512

148

630

216

235

706

133

373

390

357

580

133

307

401

433

580

150

331

433

453

579

123

436

270

282

291

CH (CH2)8 CH3

529.

Poly(n-pentyl acrylate)

CH2

CH CH3

O C O (CH2)4 530.

Poly(N-sec-butylacrylamide)

CH2

531.

C NH

CH

O

CH3

CH2

Poly(N-tert-butylacrylamide)

CH2

532.

CH

CH

CH3

C NH

C

O

CH3

CH3

CH3

Poly(N-tert-butylmethacrylamide)

CH3 CH2

533.

C

CH3 C NH C CH3 CH3 O

Poly(octafluoro-4-methyl-1-butenylene)

CF

CF

CF CF3

219

CF2

535.

2 Poly(octamethyene p,p'-dibenzoate)

C

C O

O

O

Poly(octyl acrylate)

CH2

3 344

4 1055

5 315

6 326

7 722

199

896

208 Brittle point

222

580, 592, 706

180

853

194

211

600, 620-623

112

394

308

284

333-336

77.8

401

204

194

333

77.8

290

277

268

333

(CH2)8 O

CH C O

(CH2)7

CH3

O 536.

Poly(octyloxyethylene)

CH2

CH O (CH2)7

537.

CH3

Poly(oxy-tert-butylethylene)

O CH2 CH3

CH C

CH3

CH3 538.

Poly(oxy-1,1-dimethylethylene)

CH3 O C

CH2

CH3 539.

Poly(oxy-1,2-dimethylethylene)

CH3 O CH CH3

CH

220

1 534.

1 540.

2 Poly(oxy-1,3-phenylenecarbonyloxycarbonyl-1,3-phenyleneoxytetramethylene)

O

C O C O

541.

O

(CF2)3

O C

(CF2)3

543.

544.

O C

C

O

O

C

O

521

1598

345

326

314

206

500

411

412

380, 409, 426, 432-434

458

962

476

473

6

456

851

543-553

536

6

SO 2

Poly(oxy-1,4-phenylene-3,3-phthalidylidene-1,4-phenyleneoxy-1,4-phenylenesulfonyl-1,4-phenylene)

O

7 350

O

Poly(oxy-1,4-phenylene(methyl)phenylmethylene-1,4-phenyleneoxy-1,4-phenylenesulfonyl-1,4phenylene) CH3

O

6 297

C

Poly(oxy-1,3-phenyleneoxyisophthaloyl)

O

5 < 293

O

O 542.

4 956

(CH2)4

Poly(oxy-1,3-phenylenehexafluorotrimethylene-1,3-phenyleneoxycarbonyl-1,3phenylenehexafluorotrimethylene-1,3-phenylenecarbonyl)

O

3 284

O

SO2

C

CO

221

O

2 Poly(oxy-1,4-phenylene-9,9-anthronylidene-1,4-phenyleneoxyterephthaloyl)

O

O C

C

O

O

C

3 455

4 753

5 627

6 604

7 6

389

894

428

435

297

366

787

478

465

6

400

726

571

551

376

340

833

428

408

375

CO 546.

Poly(oxy-1,4-phenylenecarbonyl-1,4-phenyleneoxy-1,4-phenyleneisopropylidene-1,4-phenylene)

CH3 O

C

O

C

O 547.

Poly(oxy-1,4-phenylenecarbonyl-1,4-phenyleneoxy-1,4-phenylenesulfonyl-1,4-phenylene)

O 548.

CH3

CO

O

SO 2

Poly(oxy-1,4-phenylenecarbonylimino-1,4-phenylenesulfonyl-1,4-phenyleneiminocarbonyl-1,4phenylene)

O

C NH

SO2

NH C

O 549.

O

Poly(oxy-1,4-phenylenecarbonylimino-2,2-dimethylpentamethyleneiminocarbonyl-1,4-phenylene)

CH3 O

C NH CH2

C

(CH2)3

O

CH3

NH C O

222

1 545.

1 550.

2 Poly(oxy-1,4-phenylenediphenylmethylene-1,4-phenyleneoxy-1,4-phenylenesulfonyl-1,4-phenylene)

O

551.

C

O

O C C

552.

4 1071

5 503

6 482

7 6

499

1279

424

390

395

420

917

448

458

297

296

610

463, 554

485

308, 378, 379

SO 2

Poly(oxy-1,4-phenylenefluoren-9-ylidene-1,4-phenyleneoxysebacoyl)

O

3 516

(CH2)8

O

C O

Poly(oxy-1,4-phenylenehexafluoro-2,2-propylidene-1,4-phenyleneoxy-1,4-phenylenecarbonyl-1,4phenylene] CF 3

O

C

O

CF3 553.

O

Poly(oxy-1,4-phenyleneiminoisophthaloylimino-1,4-phenylene)

O

NH C

C NH O

223

O

C

2 Poly(oxy-1,4-phenyleneisopropylidene-1,4-phenyleneoxy-1,4-phenyleneazo-1,4-phenylene) CH3

O 555.

C

O

3 389

4 960

5 448

6 405

7 297

389

894

433

435

6

482

1090

438

442

6

399

854

458, 463473

467

6

555

1394

393

398

297

482

1090

438

442

297

N N

CH3 Poly(oxy-1,4-phenyleneisopropylidene-1,4-phenyleneoxy-1,4-phenylenecarbonyl-1,4-phenylene)

CH3 O

C

O

CO

CH3 556.

Poly(oxy-1,4-phenyleneisopropylidene-1,4-phenyleneoxy-1,4-phenylenecarbonyl-1,4phenylenecarbonyl-1,4-phenylene)

CH3

O

C

O

CH3 557.

C

C

O

O

Poly(oxy-1,4-phenyleneisopropylidene-1,4-phenyleneoxy-1,4-phenylenesulfonyl-1,4-phenylene)

CH3 O

C

O

SO2

CH3 558.

Poly(oxy-1,4-phenyleneisopropylidene-1,4-phenyleneoxy-1,4phenylenesulfonylmethyliminotetramethylenemethyliminosulfonyl-1,4-phenylene) CH3 CH3 CH3 O

C

O

SO 2

N

(CH2)4

N

SO 2

CH3

559.

Poly(oxy-1,4-phenyleneisopropylidene-1,4-phenyleneoxy-1,4-phenyleneterephthaloyl-1,4-phenylene)

CH3 O

C CH3

O

C

C

O

O

224

1 554.

1 560.

2 Poly(oxy-1,4-phenyleneisopropylidene-1,4-phenyleneoxy-1,4-phenylenethionyl-1,4-phenylene)

3 394

4 940

5 438

6 419

7 6

365

806

453

453

6

357

808

453

442

6

290

633

423

458

6

188

373

487

504

297, 322, 381–383

366

787

478

465

297

CH3 O

C

O

S

CH3 561.

Poly(oxy-1,4-phenylenemethylene-1,4-phenyleneoxy-1,4-phenylenesulfonyl-1,4-phenylene)

O 562.

SO 2

O

O

SO 2

O

O C

C

O

O

Poly(oxy-1,4-phenylenesulfonyl-1,4-phenylene)

O 565.

O

Poly(oxy-1,4-phenyleneoxy-1,4-phenyleneoxyterephthaloyl)

O

564.

CH2

Poly(oxy-1,4-phenyleneoxy-1,4-phenyleneoxy-1,4-phenylenesulfonyl-1,4-phenylene)

O 563.

O

SO2

Poly(oxy-1,4-phenylenesulfonyl-1,4-phenyleneoxy-1,4-phenylenecarbonyl-1,4-phenylene)

O

SO 2

O

C 225

O

2 Poly(oxy-1,4-phenylenesulfonyl-1,4-phenyleneoxy-1,4-phenylenecyclohexylidene-1,4-phenylene)

O

SO 2

O

3 438

4 948

5 478

6 462

7 297

516

1081

503

482

297

399

854

449

467

292, 297, 361, 369

394

940

438

419

297

C H2C CH2 H2C CH2 CH2

567.

Poly(oxy-1,4-phenylenesulfonyl-1,4-phenyleneoxy-1,4-phenylenediphenylmethylene-1,4-phenylene)

O

568.

SO 2

O

C

Poly(oxy-1,4-phenylenesulfonyl-1,4-phenyleneoxy-1,4-phenyleneisopropylidene-1,4-phenylene)

CH3 O

SO 2

O

C CH3

569.

Poly(oxy-1,4-phenylenesulfinyl-1,4-phenyleneoxy-1,4-phenyleneisopropylidene-1,4-phenylene)

CH3 O

S O

O

C CH3

226

1 566.

1 570.

2 Poly(oxy-1,4-phenylenesulfonyl-1,4-phenyleneoxy-1,4-phenylenemethylene-1,4-phenylene)

O 571.

SO 2

O

3 365

4 806

5 453

6 453

7 319

458

962

473

476

297

377

748

518

504

6

367

850

448

432

297

465

951

508

489

297

CH2

Poly(oxy-1,4-phenylenesulfonyl-1,4-phenyleneoxy-1,4-phenylenemethylphenylmethylene-1,4phenylene)

CH3

O

572.

C

SO2

O

SO 2

Poly(oxy-1,4-phenylenesulfonyl-1,4-phenyleneoxy-1,4-phenylenethio-1,4-phenylene)

O 574.

O

Poly(oxy-1,4-phenylenesulfonyl-1,4-phenyleneoxy-1,4-phenylenesulfonyl-1,4-phenylene)

O 573.

SO 2

SO 2

O

S

Poly(oxy-1,4-phenylenesulfonyl-1,4-phenyleneoxy-2,6-dimethyl-1,4-phenyleneisopropylidene-3,5dimethyl-1,4-phenylene) CH HC 3

3

CH3 O

SO 2

O

C

H3C

CH3

227

CH3

2 Poly(oxy-1,4-phenylenesulfonyl-1,4-phenyleneoxyterephthaloyl)

O 576.

SO 2

SO 2

O

Poly(oxy-1-methyltrimethylene)

O CH CH2

5 522

6 540

7 380

338

637

523

531

322

368

665

533

553

322

367

850

448

432

6

77.8

346

223

225

364

122

473

271

258

338

SO 2

Poly(oxy-1,4-phenylenethio-1,4-phenyleneoxy-1,4-phenylenesulfonyl-1,4-phenylene)

S

4 574

SO2

SO 2

O 579.

O

Poly(oxy-1,4-phenylenesulfonyl-4,4'-biphenylylenesulfonyl-1,4-phenylene)

O 578.

C

Poly(oxy-1,4-phenylenesulfonyl-2,7-naphthylenesulfonyl-1,4-phenylene)

O

577.

O C

3 310

O

SO 2

CH2

CH3 580.

Poly(oxy-2,2,2-trichloroethylethylene)

O CH2

CH CH2 CCl3

228

1 575.

1 581.

2 Poly(oxy-2,2,3,3,4,4,5,5-octafluorohexamethyleneoxycarbonyliminohexamethylene-iminocarbonyl)

O CH2

(CF2)4

CH2

O C NH

(CH2)6

(CF2)4

CH2

O C

(CH2)2

S

(CH2)2

S

(CH2)2

O 583.

(CF2)3

CH2

O C

(CH2)2

S

(CH2)2

S

(CH2)2

585.

(CF2)3

CH2

(CF2)3

CH2

F

F

F

F

F

F

F

F

O C

(CH2)4

(CF2)3

CH2

221

421

312

1412

~233

221

421

326

1025

314

318

357

241

1000

216

241

309-422

460

1655

301

278

314

C O

Poly(oxy-2,2,3,3,4,4-hexafluoropentamethyleneoxycarbonyl-1,3-phenylenedecafluoropentamethylene1,3-phenylenecarbonyl)

O CH2

~235 Brittle point

O

O 586.

1534

O

Poly(oxy-2,2,3,3,4,4-hexafluoropentamethyleneoxyadipoyl)

O CH2

339

C

Poly(oxy-2,2,3,3,4,4-hexafluoropentamethyleneoxy-4,4'-octafluorobiphenylylene)

O CH2

7 745, 746

O

O 584.

6 272

C

Poly(oxy-2,2,3,3,4,4-hexafluoropentamethyleneoxy3,6-dithiaoctanedioyl)

O CH2

5 271

O

Poly(oxy-2,2,3,3,4,4,5,5-octafluorohexamethyleneoxy-3,6-dithiaoctanedioyl)

O CH2

4 1199

NH C

O 582.

3 326

O C

C O

229

O

(CF2)5

2 Poly(oxy-2,2,3,3,4,4-hexafluoropentamethyleneoxycarbonyl-1,3-phenylenehexafluorotrimethylene1,3-phenylenecarbonyl)

O CH2

(CF2)3

CH2

O C

(CF2)3

589.

(CF2)3

(CF2)4

C

O

O

CH2

O CH2

CH3

CH2 CH CH2 O

C O CH H3C

CH3

CH2

CH3 C

O C

CH3 H3C

1035

318

313

314

272

1058

247

257

358

271

642

442

422

279

460

1281

318

359

404

O

CH3 O

324

CH C

Poly(oxy-2,6-dimethyl-1,4-phenyleneisopropylidene-3,5-dimethyl-1,4-phenyleneoxysebacoyl)

H3C

7 314

CH2

CH O C C

6 288

CH2

Poly(oxy-2,2,4,4-tetramethyl-1,3-cyclobutyleneoxycarbonyl-trans-1,4-cyclohexylenecarbonyl)

H3C

591.

O C

Poly(oxy-2,2,3,3,4,4-hexafluoropentamethyleneoxymethylene-1,4-phenylenemethylene)

O CH2 590.

CH2

5 290

O

Poly(oxy-2,2,3,3,4,4-hexafluoropentamethyleneoxycarbonyl-3,3'-biphenylylenecarbonyl)

O CH2

4 1406

C

O 588.

3 405

O CH3

(CH2)8

C O

230

1 587.

1 592.

2 Poly(oxy-2,6-diphenyl-1,4-phenylene)

3 236

4 483

5 493, 498

6 489

7 447, 450, 546, 673

663

1842

365

360

404

190

642

322

296

329

390

1114

380

350

329

O

593.

Poly(oxy-2,6-diphenyl-1,4-phenylenemethylene-3,5-diphenyl-1,4-phenyleneoxysebacoyl) O

CH2

O C

(CH2)8

O

594.

Poly(oxy-2-acetoxytrimethyleneoxy-1,4-phenylene)

O CH2

CH CH2

C O

O

O O C CH3 595.

Poly(oxy-2-acetoxytrimethyleneoxy-1,4-phenylene-1-ethyl-1,3-cyclohexylene-1,4-phenylene)

CH2 O CH2

CH CH2 O

C

CH

H2C CH2 CH2 CH3 CH2

231

O C CH3

O

2 Poly(oxy-2-acetoxytrimethyleneoxy-1,4-phenyleneisopropylidene-1,4-phenylene)

3 317

4 906

5 333

6 350

7 329

294

799

403

368

329

372

944

373

394

276

375

1036

338

362

276

CH3 O CH2

CH CH2

O

C CH3

O O C CH3 597.

Poly(oxy-2-acetoxytrimethyleneoxy-1,4-phenylenesulfonyl-1,4-phenylene)

O CH2

CH CH2

O

SO 2

O O C 598.

CH3

Poly(oxy-2-acetoxytrimethyleneoxy-2,6-dichloro-1,4-phenyleneisopropylidene-3,5-dichloro-1,4Cl Cl phenylene) CH3

O CH2

CH CH2

O

C CH3

O C CH3

Cl

Cl

O 599.

Poly(oxy-2-benzoyloxytrimethyleneoxy-1,4-phenyleneisopropylidene-1,4-phenylene)

CH3 O CH2

CH CH2 O

O C

O

C CH3

232

1 596.

1 600.

2 Poly(oxy-2-butenylene oxycarbonyliminohexamethyleneiminocarbonyl) cis-trans

O CH2

CH CH CH2

O C NH

(CH2)6

HC

CH CH2

O C

(CH2)8

O 602.

O CH2

257

1163

232

221

408

127

472

~293

269

363

152

492

333

309

359

330

940

348

351

359

352

964

413

365

359

O

CH3 Cl

Poly(oxy-2-hydroxytrimethyleneoxy-1,4-phenylene)

O CH2

7 408, 427

C CH2 CH2

603.

6 269

C

Poly(oxy-2-ethyl-2-chloromethyltrimethylene)

CH2

5 234, 228

O

Poly(oxy-2-bytenyleneoxysebacoyl) cis, trans

O CH2

4 918

NH C

O 601.

3 247

CH CH2 O OH

604.

Poly(oxy-2-hydroxytrimethyleneoxy-1,4-phenylene-1,3,3-trimethyltrimethylene-1,4-phenylene)

CH3 O CH2 605.

CH CH2 O

CH CH2

C

OH

CH3

CH3

Poly(oxy-2-hydroxytrimethyleneoxy-1,4-phenylene-1-ethyl-1,4-cyclohexylene-1,4-phenylene)

O CH2

CH CH2 O OH

CH2

CH2

CH2

CH2

C H2C

CH

233

CH3

2 Poly(oxy-2-hydroxytrimethyleneoxy-1,4-phenylene-1-methyl-1,4-cyclohexyleneisopropylidene-1,4phenylene)

H3C

O CH2

CH CH2 O

CH2

607.

5 408

6 385

7 359

296

836

368

354

359

279

756

373

369

359

337

862

388

391

329

245

708

353

346

359

CH C CH2

OH

4 1003

CH3

CH2

C

3 386

CH2

CH3

Poly(oxy-2-hydroxytrimethyleneoxy-1,4-phenyleneisobutylidene-1,4-phenylene)

CH3 O CH2

CH CH2 O

CH2

OH 608.

C CH3

Poly(oxy-2-hydroxytrimethyleneoxy-1,4-phenyleneisopropylidene-1,4-phenylene)

CH3 O CH2 609.

CH CH2 O

C

OH

CH3

Poly(oxy-2-hydroxytrimethyleneoxy-1,4-phenylenemethyl(phenyl)methylene-1,4-phenylene)

CH3 O CH2

CH CH2 O

C

OH

610.

Poly(oxy-2-hydroxytrimethyleneoxy-1,4-phenylenemethylene-1,4-phenylene)

O CH2

CH CH2 O OH

CH2

234

1 606.

1 611.

2 Poly(oxy-2-hydroxytrimethyleneoxy-1,4-phenylenesulfonyl-1,4-phenylene)

O CH2

CH CH2

O

3 256

4 648

5 428

6 395

7 329

334

793

388

421

359

306

939

358

326

359

342

1276

290

268

424

SO2

OH 612.

Poly(oxy-2-hydroxytrimethyleneoxy-2,6-dichloro-1,4-phenyleneisopropylidene-3,5-dichloro-1,4phenylene)

Cl

Cl CH3 O CH2

CH CH2 O

C

OH

CH3 Cl

613.

Cl

Poly(oxy-2-hydroxytrimethyleneoxy-2-chloro-1,4-phenyleneisopropylidene-3-chloro-1,4-phenylene)

Cl

Cl CH3

O CH2 614.

CH CH2 O

C

OH

CH3

Poly(oxy-2-pentyloxyisophthaloyloxy-2,2,3,3,4,4-hexafluoropentamethylene)

(CH2)4

CH3

O O

C

C O CH2

O

O

CF2

CF2

CF2

CH2 235

2 Poly(oxy-2-propionyloxytrimethyleneoxy-1,4-phenyleneisopropylidene-1,4-phenylene)

3 334

4 997

5 331

6 335

7 329

410

1155

330

355

404

646

1587

371

407

404

311

1239

~248

251

421

CH3 O CH2

CH CH2

O

C

O O C 616.

CH3 CH2

CH3

Poly(oxy-3,3',5,5'-tetramethyl-4,4'-biphenylylenesebacoyl)

H3C

CH3

O

O C

(CH2)8

O H3C 617.

CH3

Poly(oxy-3,3',5,5'-tetraphenyl-4,4'-biphenylylenesebacoyl)

O C

O

(CH2)8

O

618.

C O

C O

Poly(oxy-3-heptafluoropropylglutaryloxy-2,2,3,3,4,4,5,5-octafluorohexamethylene)

O C O

CH2 CH CH2

C

CF2

O

CF2

CF3

O CH2

(CF2)4

CH2

236

1 615.

1 619.

2 Poly(oxy-3-heptafluoropropylglutaryloxy-2,2,3,3,4,4-hexafluoropentamethylene)

O C

CH2

O

CH CF2 CF2

620.

CH2

C

O CH2

(CF2)3

(CH2)2

4 1224

5 243 - 248

6 254

7 421

281

749

353

375

373

348

725

503

480

297

342

1276

282

268

424

274

794

359

345

409

CH2

O CF3

Poly(oxy-3-oxotrimethyleneiminomethylene-2,5-dimethyl-1,4-phenylenemethyleneimino-1oxotrimethylene) H3C

O

3 311

C NH CH2

CH2

NH C

O

(CH2)2

O H3C

621.

Poly(oxy-4,4'-biphenylyleneoxy-1,4-phenylenesulfonyl-1,4-phenylene)

O 622.

O

SO 2

Poly(oxy-4-pentyloxyisophthaloyloxy-2,2,3,3,4,4-hexafluoropentamethylene)

O

C

C O CH2

O

O

CF2

CF2

CF2

CH2

O (CH2)4 623.

CH3

Poly(oxy-5-butyl-1,3-phenyleneoxyisophthaloyl)

O

CH3

C

O

O 237

(CH2)3

O C

2 Poly(oxy-5-ethyl-1,3-phenyleneoxyisophthaloyl)

O

O C

C

O

O

3 240

4 615

5 395

6 390

7 409

308

972

335

317

409

223

526

426

424

409

364

1260

307

289

430

CH2 CH3 625.

Poly(oxy-5-hexyl-1,3-phenyleneoxyisophthaloyl)

O

O C

C

O

O

(CH2)5 CH3 626.

Poly(oxy-5-methyl-1,3-phenyleneoxyisophthaloyl)

O

O C

C O

O

O

CH3 627.

Poly(oxy-5-nonyl-1,3-phenyleneoxy-2-fluoroisophthaloyl)

O

(CH2)8 CH3

F

O C

C

O

O

238

1 624.

1 628.

2 Poly(oxy-5-nonyl-1,3-phenyleneoxy-5-fluoroisophthaloyl)

O

O C

C

O

O

(CH2)8

3 364

4 1260

5 293

6 289

7 334

359

1238

304

290

409

342

1152

314

297

409

316

972

343

325

374

F

CH3 629.

Poly(oxy-5-nonyl-1,3-phenyleneoxyisophthaloyl)

O

O C

C

O

O

(CH2)8 CH3 630.

Poly(oxy-5-octyl-1,3-phenyleneoxyisophthaloyl)

O

O C

C

O

O

(CH2)7 CH3 631.

Poly(oxy-5-oxopentamethyleneiminomethylene-1,4-phenylenemethyleneimino-1-oxopentamethylene)

O

(CH2)4

C NH CH2

NH C O

(CH2)4 239

O

CH2

2 Poly(oxy-5-pentyloxyisophthaloyloxy-1,4-phenylenemethylene-1,4-phenylene)

O C

C O

O

(CH2)4

7 380

404

1033

443

391

380

342

1276

243

268

424

376

1011

411

372

380

SO2

O O

(CH2)4

CH3

Poly(oxy-5-pentyloxyisophthaloyloxy-2,2,3,3,4,4-hexafluoropentamethylene)

O C

C O CH2

O

(CF2)3

CH2

O O

(CH2)4

CH3

Poly(oxy-5-pentyloxyisophthaloyloxy-4,4'-biphenylene)

O

6 360

CH3

C O

O

635.

5 383

CH2

Poly(oxy-5-pentyloxyisophthaloyloxy-1,4-phenylenesulfonyl-1,4-phenylene)

O C

634.

4 1092

O O

633.

3 393

C

C O

O

O O

(CH2)4

CH3

240

1 632.

1 636.

2 Poly(oxy-5-propyl-1,3-phenyleneoxyisophthaloyl)

O

O C

C

O

O

3 257

4 704

5 394

6 365

7 409

428

1597

291

268

409

393

1419

295

277

409

326

865

341

377

407

CH2 CH2 637.

CH3

Poly(oxy-5-tridecyl-1,3-phenyleneoxyisophthaloyl)

O

O C

C

O

O

(CH2)12 CH3 638.

Poly(oxy-5-undecyl-1,3-phenyleneoxyisophthaloyl)

O

O C

C

O

O

(CH2)10 CH3 639.

Poly(oxyadipoyloxy-1,4-phenyleneisopropylidene-1,4-phenylene)

CH3 O C

C O

C

O

CH3

241

O

(CH2)4

2 Poly(oxyadipoyloxy-2,6-dimethyl-1,4-phenyleneisopropylidene-3,5-dimethyl-1,4-phenylene)

H3C

3 392

4 961

5 366

6 408

7 404

341

832

381

410

404

295

1261

217

234

290, 406

77.8

374

185

208

709

112

533

203

210

330

332

1114

293

298

314

CH3 CH3

O C

(CH2)4

O

C O

C

O

CH3 H3C

641.

CH3

Poly(oxyadipoyloxy-3,3',5,5'-tetramethyl-4,4-biphenylylene)

H3C O C

(CH2)4

O

CH3

C O O H3C

642.

Poly(oxyadipoyloxydecamethylene)

O C

(CH2)4

C O

(CH2)10

O

O 643.

CH3

Poly(oxybutylene)

–O–(CH2)4– 644.

Poly(oxybutylethylene)

O CH2

CH (CH2)3

645.

CH3

Poly(oxycarbonyl-1,3-phenyleneoxy-1,3-phenylenecarbonyloxy-2,2,3,3,4,4hexafluoropentamethylene)

O C O

O

C O CH2 O

(CF2)3

CH2

242

1 640.

1 646.

2 Poly(oxycarbonyl-1,4-cyclohexylenecarbonyloxy-1,4-phenyleneisopropylidene-1,4-phenylene) trans

CH2 O C 647.

CH C O CH2

O

4 825

5 423

6 422

7 279

148

523

240

283

676

348

1188

287

293

411

131

498

220

263

675

521

1541

313

338

441

326

970

335

336

442

CH3

CH2

CH

3 348

CH2

C CH3

O

Poly(oxycarbonyl-1,5-dimethylpentamethylene)

CH3 O C CH

(CH2)3

CH3

O 648.

CH

Poly(oxycarbonyl-2,6-naphthylenecarbonyloxydecamethylene)

O C C O

O

(CH2)10

O 649.

Poly(oxycarbonyl-3-methylpentamethylene)

O C

(CH2)2

O 650.

CH

(CH2)2

CH3

Poly(oxycarbonylimino-1,4-phenylenemethylene-1,4-phenyleneiminocarbonyloxyhexadecamethylene)

O C NH

CH2

NH C O

O 651.

O

Poly(oxycarbonylimino-4-methyl-1,3-phenyleneiminocarbonyloxynonamethylene)

O C NH

NH C O CH3

O

(CH2)9 243

O

(CH2)16

2 Poly(oxycarbonylimino-4-methyl-1,3-phenyleneiminocarbonyloxyoctamethylene)

O C NH

NH C O

O 653.

CH3

Poly(oxycarbonyliminomethylene-1,4-phenylenemethyleneiminocarbonyloxydecamethylene)

CH2

NH C O

CH2

NH C O

O

1094

322

329

441

394

1220

318

323

441

463

1475

320

314

441

227

672

323

338

316

270

882

319

306

314

(CH2)12

Poly(oxycarbonyliminomethylene-1,4-phenylenemethyleneiminocarbonyloxy-hexadecamethylene)

CH2

NH C O

O

(CH2)16

O

Poly(oxycarbonylneopentylenesulfonylneopentylene)

CH3

CH3 CH2

O

C

CH2

SO 2

CH2

C

CH2

CH3

CH3

Poly(oxycarbonyloxy-1,3-phenylenehexafluorotrimethylene-1,3-phenylene)

O C O O

360

O

O C NH CH2

657.

7 442

(CH2)10

Poly(oxycarbonyliminomethylene-1,4-phenylenemethyleneiminocarbonyloxydodecamethylene)

O C

6 340

O

O C NH CH2

656.

5 337

(CH2)8

O

655.

4 909

O

O C NH CH2 654.

3 309

(CF2)3

244

1 652.

1 658.

2 Poly(oxycarbonyloxy-1,4-phenylene-2,2-butylidene-1,4-phenylene)

3 256

4 643

5 407

6 398

7 385, 394

273

732

410

373

384

280

636

394

440

385, 388

278

650

448

428

384, 385, 388

CH3 O C O

C

O

CH2 CH3

659.

Poly(oxycarbonyloxy-1,4-phenylene-2,2-pentylidene-1,4-phenylene)

CH3 O C O

C CH2

O

CH2 CH3 660.

Poly(oxycarbonyloxy-1,4-phenylenebenzylidene-1,4-phenylene)

O C O

CH

O

661.

Poly(oxycarbonyloxy-1,4-phenylenecyclohexylidene-1,4-phenylene)

O C O O

C CH2

H3C

CH2 CH2

245

H2C

663.

2 Poly(oxycarbonyloxy-1,4-phenyleneethylidene-1,4-phenylene)

O C O

CH

O

CH3

Poly(oxycarbonyloxy-1,4-phenylenehexafluoro-2,2-propylidene-1,4-phenylene)

3 222

4 530

5 403

6 419

7 388

270

577

449

468

387

256

643

422

398

388

365

859

393

425

388

205

506

420

405

388, 402

207

551

~383

376

402

CF3 O C O

C

O 664.

CF3

Poly(oxycarbonyloxy-1,4-phenyleneisobutylidene-1,4-phenylene)

CH3 O C O

C CH2

O

CH3 665.

Poly(oxycarbonyloxy-1,4-phenyleneisopropylidene-1,3-phenyleneisopropylidene-1,4-phenylene)

O C O O 666.

CH3

CH3

C

C

CH3

CH3

Poly(oxycarbonyloxy-1,4-phenylenemethylene-1,4-phenylene)

O C O

CH2

O 667.

Poly(oxycarbonyloxy-1,4-phenylenethio-1,4-phenylene)

O C O O

S

246

1 662.

1 668.

2 Poly(oxycarbonyloxy-2,2,3,3,4,4,5,5-octafluorohexamethylene)

O C O CH2

CF2

CF2

CF2

CF2

3 182

4 816

5 232

6 223

7 389

294

593

453, 493, 504

496

384, 386, 387

305

723

443 - 452

422

385

299

676

427

442

385

289

682

385

424

388

CH2

O 669.

Poly(oxycarbonyloxy-2,6-dichloro-1,4-phenyleneisopropylidene-3,5-dichloro-1,4-phenylene)

Cl

Cl

CH3 O C O

C CH3

O Cl 670.

Cl

Poly(oxycarbonyloxy-2-chloro-1,4-phenylenecyclohexylidene-3-chloro-1,4-phenylene) Cl

Cl

C

O C O O

671.

H2C

CH2

H2C

CH2 CH2

Poly(oxycarbonyloxy-2-chloro-6-methyl-1,4-phenyleneisopropylidene-3-chloro-5-methyl-1,4phenylene) Cl Cl

CH3 O C O

C

O

CH3 H3C

672.

CH3

Poly(oxycarbonyloxy-2-isopropyl-1,4-phenyleneisopropylidene-1,4-phenylene)

CH3 CH3

CH CH3

O

C CH3

247

O C O

2 Poly(oxycarbonyloxy-2-methoxy-1,4-phenyleneisopropylidene-1,4-phenylene)

H3C

4 625

5 418

6 424

7 388

311

697

408

446

385, 393

255

577

413

442

388

272

714

368, 373, 418, 363 383

381

384-386, 393

O

O C O

C

O 674.

3 265

CH3

Poly(oxycarbonyloxy-2-methyl-1,4-phenylenecyclohexylidene-3-methyl-1,4-phenylene)

H3C

CH3

O C O O H2C H2C 675.

C

CH2

CH2 CH2

Poly(oxycarbonyloxy-2-methyl-1,4-phenyleneisopropylidene-1,4-phenylene)

H3C CH3 O C O O 676.

C CH3

Poly(oxycarbonyloxy-2-methyl-1,4-phenyleneisopropylidene-3-methyl-1,4-phenylene)

H3C

CH3 CH3

O C O O

C CH3

248

1 673.

1 677.

2 Poly(oxycarbonyloxy-3-methyl-1,4-phenylenebenzylidene-2-methyl-1,4-phenylene)

4 685

5 455

6 457

7 385

140

633

230 low molecular wieght sample

221

389

419

1167

321

359

441, 442

214

1070

232

200

330, 333

365

1393

265

262

314

H3C

CH3 O C O

3 313

CH

O

678.

Poly(oxycarbonyloxyhexamethylene)

O C O

(CH2)6

O 679.

Poly(oxydecamethyleneoxycarbonylimino-1,4-phenylenemethylene-1,4-phenyleneiminocarbonyl)

O

(CH2)10

O C NH

CH2

O 700.

O

Poly(oxydecylethylene)

O CH2

CH (CH2)9

701.

NH C

CH3

Poly(oxydimethylsilylene-1,3-phenylenehexafluorotrimethylene-1,3-phenylenedimethylsilylene)

CH3 O

Si

(CF2)3

Si CH3

249

CH3

CH3

2 Poly(oxydimethylsilylene-1,3-phenylenetetrafluoroethylene-1,3-phenylenedimethylsilylene)

Si

(CF2)2

CH3 Si

O

CH3

CH3

Si

Si

Poly(oxydimethylsilyleneoxydimethylsilylene-2,4,5,6-tetrafluorophenylenedimethylsilylene)

CH3 O

CH3

228

946

245

241

446

300

1429

212

210

446

446

2084

221

214

428

F

CH3

CH3

Si

Si CH3

CH3 F

F

Poly(oxydimethylsilyleneoxypentyleneoxyisophthaloyloxyneopentylenedimethylsilylene)

CH3

CH3

C O CH2

C CH2

Si

O

CH3

CH3

CH3 O

371

F F

706.

286

CH3 F

Si

293

F

CH3

O

1021

CH3

F

705.

292

Si

Poly(oxydimethylsilylene-2,4,5,6-tetrafluorophenylenedimethylsilylene)

O

7 450

CH3

CH3 704.

6 266

CH3

Poly(oxydimethylsilylene-1,4-phenyleneoxy-1,4-phenylenedimethylsilylene)

O

5 271

Si

CH3 703.

4 1271

CH3

CH3 O

3 338

Si CH3

O

(CH2)5

O C O

250

1 702.

1 707.

2 Poly(oxydimethylsilyleneoxypentyleneoxyterephthaloyloxyneopentylenedimethylsilylene)

CH3

CH3

C O CH2

C CH2

Si

O

CH3

CH3

CH3 O

Si

O

(CH2)5

O C

CH3 708.

O

Poly(oxydiphenoxymethyleneoxy-1,4-phenyleneisopropylidene-1,4-phenylene)

3 446

4 2046

5 238

6 218

7 429

407

998

365

408

405

390

920

398

424

405

356

868

385

410

405

O CH3 O C O

C CH3

O 709.

Poly(oxydiphenoxymethyleneoxy-3,3'-dimethyl-4,4'-biphenylene)

O

CH3

O C O O 710.

CH3

Poly(oxydiphenoxymethyleneoxy-4,4'-biphenylene)

O O C O O 251

2 Poly(oxydiphenylsilylene-1,3-phenylene)

O

712.

713.

Si

O

CH3

CH3

Si

Si

CH3

CH3

Poly(oxydodecamethyleneoxycarbonylimino-1,4-phenylenemethylene-1,4-phenyleneiminocarbonyl)

O

(CH2)12

O C NH

CH2

5 ~331

6 341

7 295

354

1378

~273

286

451

453

1291

316

351

441

115

542

220

212

351, 352

181

586

291

309

279

NH C

O 714.

4 765

Si

Poly(oxydiphenylsilyleneoxydimethylsilylene-1,4-phenylenedimethylsilylene)

O

3 261

O

Poly(oxyethylenedithioethylene)

–O–(CH2)2–S–S–(CH2)2– 715.

Poly(oxyethyleneoxycarbonyl-1,4-cyclohexylenecarbonyl) trans

O CH2

CH2

CH2 O C CH CH2 O

CH2 CH C CH2

O

252

1 711.

1 716.

2 Poly(oxyethyleneoxycarbonyl-1,4-phenylene-sec-butylidene-1,4-phenylenecarbonyl)

O CH2

CH2

O C O

717.

CH CH2

CH2

CH2

4 840

5 380

6 368

7 318

211

548

386

385

411, 412

211

548

392

385

412

299

726

390

412

443

282

664

412, 366

425

442, 443

333

852

379

391

443

C

CH3

O

Poly(oxyethyleneoxycarbonyl-2,6-naphthylenecarbonyl)

O CH2

3 309

O C O

C O

718.

Poly(oxyethyleneoxycarbonyl-2,7-naphthylenecarbonyl)

O CH2

719.

CH2

O C

C

O

O

Poly(oxyethyleneoxycarbonylimino-1,4-phenylene ethylene-1,4-phenyleneiminocarbonyl)

O

(CH2)2

O C NH

(CH2)2

NH C

O 720.

O

Poly(oxyethyleneoxycarbonylimino-1,4-phenylene methylene-1,4-phenyleneiminocarbonyl)

O

(CH2)2

O C NH

CH2

O 721.

NH C O

Poly(oxyethyleneoxycarbonylimino-1,4-phenylene tetramethylene-1,4-phenyleneiminocarbonyl)

O

(CH2)2

O C NH

NH C O

253

O

(CH2)4

2 Poly(oxyethyleneoxyisophthaloyl)

O

723.

(CH2)2

O C

C

O

O

Poly(oxyethylenetetrathioethylene)

3 166

4 485

5 324

6 342

7 412

153

733

233

209

351

77.8

355

203

219

330

43.7

186

243

235

353, 354

251

1041

218 - 223

241

421

224

918

218 – 223

244

421

368

979

357

376

442

279

866

317

322

411

–O–(CH2)2–S–S–S–S–(CH2)2– 724.

Poly(oxyethylethylene)

O CH2

CH CH2

725.

CH3

Poly(oxyethylidene)

O CH CH3 726.

Poly(oxyglutaryloxy-2,2,3,3,4,4,5,5-octafluorohexamethylene)

O C

C O CH2

(CH2)3

Poly(oxyglutaryloxy-2,2,3,3,4,4-hexafluoropentamethylene)

O C

C O CH2

(CH2)3

(CF2)3

Poly(oxyheptamethyleneoxycarbonylimino-1,4-phenylenemethylene-1,4-phenyleneiminocarbonyl)

O

(CH2)7

O C NH

CH2

O 729.

CH2

O

O 728.

CH2

O

O 727.

(CF2)4

O

Poly(oxyhexamethyleneoxycarbonyl-2,6-naphthylenecarbonyl)

O

(CH2)6

O C O

NH C

C O

254

1 722.

1 730.

2 Poly(oxyhexamethyleneoxycarbonyliminomethylene-1,4-phenylenemethyleneiminocarbonyl)

O

(CH2)6

O C NH

CH2

CH2

O 731.

4 846

5 329

6 345

7 441

146

712

206

205

330

122

298

403

410

414, 440

374

806

433, 513

464

288, 291, 425

333

722

462

461

313, 403, 404, 426

298

673

423

443

380

NH C O

Poly(oxyhexylethylene)

O CH2

3 292

CH (CH2)5 CH3

732.

733.

Poly(oxyisophthaloyl)

C

O

O

Poly(oxyisophthaloyloxy-1,4-phenylenebenzylidene-1,4-phenylene)

O C O

734.

O C

C O

CH

O

Poly(oxyisophthaloyloxy-1,4-phenyleneisopropylidene-1,4-phenylene)

CH3 O C O 735.

C O

C

O

CH3

Poly(oxyisophthaloyloxy-1,4-phenylenemethylene-1,4-phenylene)

O C

O

CH2 255

O

C O

2 Poly(oxyisophthaloyloxy-2,2,3,3,4,4-hexafluoropentamethylene)

O C O 737.

C O CH2

(CF2)3

O

C O

C

O

CH3

H3C

O

7 410, 424

399

819

498

487

404

365

772

461

473

403

C O

366

882

438

415

401

332

723

418

459

403

CH3 CH2

O H3C

739.

6 289

CH3

Poly(oxyisophthaloyloxy-2,6-dimethyl-1,4-phenylenemethylene-3,5-dimethyl-1,4-phenylene)

O C

5 298

O

H3C 738.

4 858

CH2

Poly(oxyisophthaloyloxy-2,6-dimethyl-1,4-phenyleneisopropylidene-3,5-dimethyl-1,4-phenylene) H3C CH3 CH3

O C

3 248

CH3

Poly(oxyisophthaloyloxy-2-methyl-1,4-phenyleneisopropylidene-3-methyl-1,4-phenylene)

H3C

CH3 CH3

O C O 740.

C O

C

O

CH3

Poly(oxyisophthaloyloxy-2-methyl-1,4-phenylenemethylene-3-methyl-1,4-phenylene)

H3C O C O

C O O

CH3 CH2

256

1 736.

1 741.

2 Poly(oxyisophthaloyloxy-4,4'-biphenylylene)

O C

4 594

5 437, 583, 480

6 475

7 309, 423

121

644

<193

188

420, 422, 447, 747

245

980

238

250

358

357

838

379

426

444

271

863

325

314

279

329

1154

279

285

358

C O

O 742.

3 282

O

Poly(oxymethyl-3,3,3-trifluoropropylsilylene)

CH3 O

Si CH2

743.

CH2

O CH2

744.

CF3

Poly(oxymethylene-1,3-phenylenemethyleneoxy-2,2,3,3,4,4-hexafluoropentamethylene)

CH2

O CH2

(CF2)3

CH2

Poly(oxymethylene-1,3-phenylenemethyleneoxycarbonylimino-1,4-phenylenemethylene-1,4phenyleneiminocarbonyl)

O CH2

CH2

O C NH

CH2

NH C

O 745.

Poly(oxymethylene-1,4-cyclohexylenemethyleneoxycarbonyl-trans-1,4-cyclohexylenecarbonyl)

O CH2 746.

O

CH2 CH CH2

CH2 CH CH2 CH2

CH2 O C CH CH2 O

CH2 CH C CH2 O

Poly(oxymethylene-1,4-phenyleneoxy-1,4-phenylenemethyleneoxy-2,2,3,3,4,4hexafluoropentamethylene)

O CH2

O CH2

(CF2)3

CH2

257

O CH2

2 Poly(oxymethylene-5-tert-butyl-1,3-phenylenemethyleneoxycarbonylimino-1,4-phenylenemethylene1,4-phenyleneiminocarbonyl)

O CH2

CH2

O C NH

CH2

O

3 425

4 993

5 387

6 428

7 444

233

635

357

367

350

180

874

220

206

367

153

750

220

204

367

142

707

214

201

210

1123

197

187

351, 352, 366 351

154

550

248

280

446

NH C O

H3C C CH3 CH3 748.

Poly(oxymethyleneoxy-1,4-phenylenecarbonyloxycarbonyl-1,4-phenylene)

O CH2

O

C O C O

749.

O

Poly(oxymethyleneoxy-2,2,3,3,4,4,5,5-octafluorohexamethylene)

–O–CH2–O–CH2–(CF2)4–CH2– 750.

Poly(oxymethyleneoxy-2,2,3,3,4,4-hexafluoropentamethylene)

–O–CH2–O–CH2–(CF2)3–CH2– 751.

Poly(oxymethyleneoxyethylenedithioethylene)

–O–CH2–O–(CH2)2–S–S–(CH2)2– 752.

Poly(oxymethyleneoxytetramethylenedithiotetramethylene)

–O–CH2–O–(CH2)4–S–S–(CH2)4– 753.

Poly(oxymethylpentafluorophenylsilylene)

CH3 O

Si

F

F

F

F F

258

1 747.

1 754.

2 Poly(oxymethylpentafluorophenylsilyleneoxydimethylsilylene)

CH3 O

Si

3 226

4 1027

5 208

6 220

7 446

232

826

303

281

279

217

682

341

318

279

381

1140

303

334

316

402

1104

345

364

442

CH3 O

Si

F

F CH3

F

F F

755.

Poly(oxyneopentyleneoxycarbonyl-1,4-cyclohexylenecarbonyl) trans

CH3 O CH2

C CH2 CH3

756.

CH2 O C CH CH2 O

CH2 CH C CH2 O

Poly(oxyneopentyleneoxyterephthaloyl)

CH3 O CH2

C CH2

O C

C

O

O

CH3 757.

Poly(oxyneopentylenesulfonylneopentyleneoxycarbonyliminohexamethyleneiminocarbonyl)

CH3 O CH2

CH3

C CH2

SO2

CH3 758.

CH2

C CH2

O C NH

CH3

(CH2)6

NH C

O

O

Poly(oxynonamethyleneoxycarbonyimino-1,4-phenylenemethylene-1,4-phenyleneiminocarbonyl)

O

(CH2)9

O C NH

NH C O

259

O

CH2

2 Poly(oxyoctamethyleneoxycarbonylimino-1,4-phenylenemethylene-1,4-phenyleneiminocarbonyl)

O

(CH2)8

O C NH

CH2

O

(CH2)5

O

C

(CH2)4

O 762.

(CH2)5

O C NH

CH2

(CH2)5

O C NH O

764.

(CH2)5

O

316

350

210

864

204

243

290

333

852

368

391

442

258

721

325

358

442

262

787

311

333

411

O NH C CH3

Poly(oxypentamethyleneoxycarbonyl-2,6-naphthylenecarbonyl)

O

326

NH C

Poly(oxypentamethyleneoxycarbonylimino-4-methyl-1,3-phenyleneiminocarbonyl)

O

953

O

O 763.

301

C

Poly(oxypentamethyleneoxycarbonylimino-1,4-phenylenemethylene-1,4-phenyleneiminocarbonyl)

O

7 442

O

Poly(oxypentamethyleneoxyadipoyl)

O

6 370

C O C O

761.

5 352

O

Poly(oxypentamethyleneoxy-1,4-phenylenecarbonyloxycarbonyl-1,4-phenylene)

O (CH2)5

4 1041

NH C

O 760.

3 385

C O

C O

O

260

1 759.

1 765.

2 Poly(oxypentamethyleneoxyterephthaloyl)

O 766.

(CH2)5

O

C

C

O

O

Poly(oxypimeloyloxy-2,6-dimethyl-1,4-phenyleneisopropylidene-3,5-dimethyl-1,4-phenylene)

H3C

3 217

4 682

5 283, < 318

6 318

7 306, 310, 311, 312

409

1041

357

393

404

358

913

369

392

404

60.7

266

198, 201

228

330, 333, 688

CH3 CH3

(CH2)5

O C O

C O

C

O

CH3 H3C

767.

Poly(oxypimeloyloxy-3,3',5,5'-tetramethyl-4,4'-biphenylylene)

H3C (CH2)5

O C O

O

Poly(oxypropylene)

CH2

CH3

C O H3C

768.

CH3

CH O

CH3

CH3

261

2 Poly(oxyterephthaloyloxy-1,4-phenylene-9,9-anthronylidene-1,4-phenylene)

O C O

3 455

4 753

5 570, 627

6 604

7 395

374

763

473

490

425

437

776

654, 590

563

395

364

704

534

517

439

C O O

C C O

770.

Poly(oxyterephthaloyloxy-1,4-phenylenebenzylidene-1,4-phenylene)

O C O

771.

CH

O

Poly(oxyterephthaloyloxy-1,4-phenylenefluoren-9-ylidene-1,4-phenylene)

O C O

772.

C O

C O O

C

Poly(oxyterephthaloyloxy-1,4-phenylenehexafluoroisopropylidene-1,4-phenylene)

CF3 O C O

C O

C

O

CF3

262

1 769.

1 773.

2 Poly(oxyterephthaloyloxy-1,4-phenyleneisopropylidene-1,4-phenylene)

3 333

4 681

5 478

6 489

7 400, 403, 404, 407, 426, 435, 438, 439

399

778

498

513

404

360

865

463, 413

416

436

434

1048

403, 413

414

289, 293, 294, 436

366

839

~428, 461

436

401, 417, 436

CH3 O C O 774.

C O

C

O

CH3

Poly(oxyterephthaloyloxy-2,6-dimethyl-1,4-phenyleneisopropylidene-3,5-dimethyl-1,4-phenylene)

H3C

CH3

CH3 O C O

C O

C

O

CH3 H3C

775.

CH3

Poly(oxyterephthaloyloxy-2-chloro-1,4-phenyleneisopropylidene-3-chloro-1,4-phenylene)

Cl

Cl CH3

O C O 776.

C O

C

O

CH3

Poly(oxyterephthaloyloxy-2-isopropyl-1,4-phenyleneisopropylidene-3-isopropyl-1,4-phenylene) CH3 CH3

CH3

CH

CH CH3 CH3

O C O 777.

C O

C

O

CH3

Poly(oxyterephthaloyloxy-2-methyl-1,4-phenyleneisopropylidene-3-methyl-1,4-phenylene)

H3C

CH3 CH3

O C

C

O

CH3

263

O

C O

2 Poly(oxyterephthaloyloxy-2-sec-butyl-1,4-phenylene-isopropylidene-3-sec-butyl-1,4-phenylene)

3 468

4 1225

5 373

6 382

7 436

302

1079

298, 268

280

274, 416, 417

251

839

276

299

417

234

762

318, 264

307

274, 416, 417

256

721

358

355

279, 437

CH3 O C O

779.

O

O

782.

CH3

O CH2

CH2

CH CH3

CH CH3

CH3

CH3

C O

(CH2)10

O

Poly(oxyterephthaloyloxyheptamethylene)

O C 781.

C

Poly(oxyterephthaloyloxydecamethylene)

O C 780.

C O

C O

(CH2)7

O

Poly(oxyterephthaloyloxyhexamethylene)

O C

C

O

O

O

(CH2)6

Poly(oxyterephthaloyloxymethylene-1,4-cyclohexylenemethylene) 70% trans

O C O

C O CH2 O

CH2 CH CH2

CH2 CH CH2 CH2

264

1 778.

1 783.

2 Poly(oxyterephthaloyloxyneopentylenesulfonylneopentylene)

CH3 O C O 784.

785.

786.

C O CH2

C

CH2

O

CH3

C

O

O

O

SO 2

CH2

C

O

O

O

C

O

O

O

6 376

7 316

C

285

1000

308, 270

285

274, 416, 417

268

921

318

291

416

217

682

318

318

274, 416

CH2

CH3

(CH2)8

Poly(oxyterephthaloyloxypentamethylene)

O C

5 378

(CH2)9

Poly(oxyterephthaloyloxyoctamethylene)

O C

4 878

CH3

Poly(oxyterephthaloyloxynonamethylene)

O C

3 330

(CH2)5

787.

Poly(oxytetrafluoroethylene)

64.4

330

225

195

356, 357

788.

Poly(oxytetramethylene)

77.8

374

189, 178

208

346, 347, 348, 695

789.

Poly(oxytetramethylenedithiotetramethylene)

183

863

197

212

351, 352

790.

Poly(oxytetramethyleneoxy-1,4-phenylenecarbonyloxycarbonyl-1,4-phenylene)

284

874

348

325

350

−O−CF2−CF2− −O−(CH2)4−

−O−(CH2)4−S−S−(CH2)4− O

(CH2)4

O

C O C O

265

O

2 Poly(oxytetramethyleneoxy-1,4-phenyleneisopropylidene-1,4-phenyleneoxy-1,4-phenylenesulfonyl1,4-phenyleneoxy-1,4-phenyleneisopropylidene-1,4-phenylene) CH3 O

(CH2)4

O

O

O

(CH2)4

SO2

O C NH

CH2

O 793.

794.

(CH2)4

CH2 O C CH CH2 O

(CH2)4

7 297

316

788

382

401

442

215

824

263

261

279

245

706

349

347

411

261

1106

216

236

273, 408, 427

200

602

290, 353

332

274, 279, 416, 417

CH3

NH C

CH2 CH C CH2 O

Poly(oxytetramethyleneoxycarbonyl-2,6-napthylenecarbonyl)

O

6 396

O

Poly(oxytetramethyleneoxycarbonyl-1,4-cyclohexylenecarbonyl)

O

5 413

CH3

Poly(oxytetramethyleneoxycarbonylimino-1,4-phenylenemethylene-1,4-phenyleneiminocarbonyl)

O

4 1735

C

C CH3

792.

3 687

O C O

C O

795.

Poly(oxytetramethyleneoxysebacoyl)

O 796.

(CH2)4

O C

(CH2)8 C

O

O

Poly(oxytetramethyleneoxyterephthaloyl)

O

(CH2)4

O C

C

O

O

266

1 791.

1 797.

Poly(oxytrimethylene)

2

3 60.7

4 295

5 195

6 206

7 347, 362, 363

798.

Poly(oxytrimethyleneoxy-1,3-phenylenecarbonyl-oxycarbonyl-1,3-phenylene)

267

878

326

304

350

267

795

368

336

350

176

707

214

249

290

299

726

392

412

442

207

570

345, 213

363

442, 744

198

744

267

266

279

−O−(CH2)3− O

(CH2)3

O

C O C O

799.

O

Poly(oxytrimethyleneoxy-1,4-phenylenecarbonyloxycarbonyl-1,4-phenylene)

O

(CH2)3

O

C O C O

800.

O

(CH2)3

O C

(CH2)4

O 701.

O

Poly(oxytrimethyleneoxyadipoyl)

C O

Poly(oxytrimethyleneoxycarbonylimino-1,4-phenylenemethylene-1,4-phenyleneiminocarbonyl)

O

(CH2)3

O C NH

CH2

NH C

O 802.

O

(CH2)3

O C NH O

803.

O

Poly(oxytrimethyleneoxycarbonylimino-4-methyl-1,3-phenyleneiminocarbonyl)

NH C CH3

O

Poly(oxytrimethyleneoxycarbonyl-1,4-cyclohexylenecarbonyl) trans

(CH2)3

CH2 CH C CH2 O

267

O

CH2 O C CH CH2 O

2 Poly(oxytrimethyleneoxycarbonyl-2,6-napthylenecarbonyl)

O

(CH2)3

3 228

4 626

5 346

6 364

7 411

183

523

368, 308

350

279, 321, 416, 417

260

585

453

445

712

207

456

420

454

560

165

604

268 Brittle point

273

581, 706

O C O

C O

805.

Poly(oxytrimethyleneoxyterephthaloyl)

O 806.

(CH2)3

O C

C

O

O

Poly(pentabromobenzyl acrylate)

CH2

Br

Br

CH C O

Br

CH2

O Br 807.

Poly(pentachlorophenyl acrylate)

CH2

Cl

Br Cl

CH C O

Cl

O Cl 808.

Poly(pentyl methacrylate)

CH3 CH2

C

O C O (CH2)4

CH3

Cl

268

1 804.

1 809.

2 Poly(pentylethylene)

CH2

3 137

4 620

5 242

6 221

7 509

150

326

467

460

291

138

380

330

363

560, 585

172

558

270

308

560

94.4

260

359, 383, 353, 365

363

692, 709, 713, 720

129

586

207

220

600, 621

CH (CH2)4 CH3

810.

Poly(perfluorostyrene)

CF2

CF

F

F

F

F F

811.

Poly(phenyl acrylate)

CH2

CH C O O

812.

Poly(phenyl ethyl acrylate)

CH2

CH C O

(CH2)2

O 813.

Poly(phenylene sulfide)

S 814.

Poly(pentyloxyethylene)

CH2

CH (CH2)4

CH3

269

O

2 Poly(p-phenylene terephthalamide) Kevlar

NH C

CH2

CH2

7 709

94.9

409

224

232

588, 621

96.4

353

283

273

650

114

452

236 Brittle point, 228

252

589, 706

128

383

344 Vicat softening point

334

588, 599

CH3

Poly(propionyloxyethylene)

CH2

6 601

CH O

817.

5 508, 580 600

O

Poly(propoxyethylene)

CH2

4 353

C NH

O 816.

3 212

CH O C

CH2

CH3

O 818.

Poly(propyl acrylate)

CH2

CH C O (CH2)2

CH3

O 819.

Poly(propyl chloroacrylate)

Cl CH2

C C O CH2 O

CH2

CH3

270

1 815.

1 820.

2 Poly(propylethylene)

CH2

3 85.3

4 363

5 233

6 235

160

366

409 softening point

437

7 362, 496, 503, 509, 510, 512, 522, 528, 543 287

112

434

253

258

600, 620

131

476

251, 253, 256

275

582, 583

145

407

347 Vicat softening point

356

599

CH CH2 CH2

821.

CH3

Poly(2,4,5-trimethylstyrene)

CH2

CH CH3

H3C CH3 822.

Poly(sec-butoxyethylene)

CH2

CH O CH CH3 CH2

823.

CH3

Poly(sec-butyl acrylate) conventional

CH2

CH C O CH CH2 O

824.

CH3

CH3

Poly(sec-butyl chloroacrylate)

Cl CH2

C C O CH CH2 CH3

271

O

CH3

2 Poly(sulfonyl-1,2-cyclohexylene)

4 300

5 401

6 393

7 316

118

300

381

393

316

309

724

413

427

467, 468

309

681

467

453

467, 468

360

940

398

383

467, 468

362

674

< 573 softening point

537

376

CH CH

SO2

CH2

H2C H2C 826.

3 118

CH2

Poly(sulfonyl-1,3-cyclohexylene)

CH2 SO 2

CH

CH

CH2

CH2 CH2 827.

Poly(sulfonyl-1,3-phenyleneiminoadipoylimino-1,3-phenylene)

SO 2

NH C

(CH2)4

O 828.

O

Poly(sulfonyl-1,3-phenyleneiminoadipoylimino-1,4-phenylene)

SO 2

NH C

(CH2)4

O 829.

C NH O

Poly(sulfonyl-1,3-phenyleneiminoazelaoylimino-1,3-phenylene)

SO 2

NH C

(CH2)7

O 830.

C NH

C NH O

Poly(sulfonyl-1,3-phenyleneiminocarbonyl-1,4-naphthylenecarbonylimino-1,3-phenylene)

SO 2

NH C O

C NH O

272

1 825.

1 831.

2 Poly(sulfonyl-1,3-phenyleneiminododecanedioylimino-1,3-phenylene)

SO 2

NH C

(CH2)10

O 832.

NH C

(CH2)8

NH C

7 467, 468

377

997

385

378

467, 468

343

884

398

388

467, 468

342

621

568 - 583

551

376

360

857

451

420

467, 468

362

591

< 573 softening point

613

376

O

(CH2)6

C NH

O 834.

6 371

C NH

Poly(sulfonyl-1,3-phenyleneiminosuberoylimino-1,3-phenylene)

SO 2

5 380

O

O 833.

4 1111

C NH

Poly(sulfonyl-1,3-phenyleneiminosebacoylimino-1,3-phenylene)

SO 2

3 412

O

Poly(sulfonyl-1,4-phenyleneimino-2-methoxyisophthaloylimino-1,4-phenylene)

O CH3 SO2

NH C

C NH

O 835.

O

Poly(sulfonyl-1,4-phenyleneiminoazelaoylimino-1,4-phenylene)

SO 2

NH C

(CH2)7

O 836.

C NH O

Poly(sulfonyl-1,4-phenyleneiminocarbonyl-1,4-naphthylenecarbonylimino-1,4-phenylene)

SO2

NH C O

C NH O

273

2 Poly(sulfonyl-1,4-phenyleneiminocarbonyl-1,4-phenylenemethylene-1,4-phenylenecarbonylimino-1,4phenylene)

SO 2

NH C

CH2

O 838.

NH C

(CH2)10

NH C

(CH2)5

NH C

(CH2)8

NH C O

842.

401

467, 468

326

721

436

452

467, 468

377

913

444

413

467, 468

343

800

453

429

467, 468

410

707

590

580

376

O

(CH2)6

C NH O

Poly(sulfonyl-1,4-phenyleneiminoterephthaloyl-1,4-phenylenecarbonylimino-1,4-phenylene)

SO2

433

C NH

Poly(sulfonyl-1,4-phenyleneiminosuberoylimino-1,4-phenylene)

SO 2

1027

O

O 841.

412

C NH

Poly(sulfonyl-1,4-phenyleneiminosebacloylimino-1,4-phenylene)

SO 2

7 376

O

O 840.

6 564

C NH

Poly(sulfonyl-1,4-phenyleneiminopimeloylimino-1,4-phenylene)

SO 2

5 < 573 softening point

O

O 839.

4 723

C NH

Poly(sulfonyl-1,4-phenyleneiminododecanedioylimino-1,4-phenylene)

SO 2

3 408

NH C

C

C NH

O

O

O

274

1 837.

1 843.

2 Poly(sulfonyl-1,4-phenylenemethylene-1,4-phenylene)

SO 2 844.

3 196

4 371

5 497

6 529

7 383

366

839

436

423

6

131

411

380, 316, 346, 313, 304

319

560, 583, 584, 706, 709

102

311

337 Highly crystalline sample softening point

328

511

301

1115

297 Brittle point

270

277, 581

318

1325

201-264

240

593

CH2

Poly(terephthaloyloxy-2-methyl-1,4-phenyleneisopropylidene-3-methyl-1,4-phenyleneoxy)

H3C

CH3 CH3

845.

C

C O

C

O

O

CH3

Poly(tert-butyl acrylate)

CH2

CH

CH3

C O C

CH3 CH3

O 846.

Poly(tert-butylethylene)

CH2

CH

H3C

C

O

CH3

CH3 847.

Poly(tetradecyl acrylate)

CH2

CH

O C O (CH2)13 848.

CH3

Poly(tetradecyl methacrylate)

CH3 CH2

C CH3

275

O C O (CH2)13

2 Poly(tetradecylethylene)

CH2

3 273

4 1026

5 246

6 266

7 509, 512, 521

CH (CH2)13 CH3

850.

Poly(tetrathiodecamethylene)

246

1159

197

212

461

851.

Poly(tetrathioethylene)

109

717

249

212

351, 352

852.

Poly(tetrathiomethylene-1,4-phenylenemethylene)

184

848

276

284

461

109

405

256, 228

269

452, 453

109

405

221

269

453

564

1790

331

315

463

−S−S−S−S−(CH2)10− −S−S−S−S−(CH2)2− S

853.

S

S

CH2

Poly(thio-1,2-cyclohexylene)

S

CH CH

H2C

CH2

H2C 854.

CH2

S

CH2

Poly(thio-1,3-cyclohexylene)

S

CH2 CH CH CH2

CH2 CH2

855.

Poly(thio-11-oxoundecamethyleneiminoethylene-1,4-phenyleneethyleneimino-1-oxoundecamethylene)

S

(CH2)10

C NH O

(CH2)2

(CH2)2

NH C O

(CH2)10

276

1 849.

1 856.

2 Poly(thio-1-ethylethylene)

S

CH2

858.

CH CH2

C

CH3

O

Poly(thio-1-methyltrimethylene)

S

CH2

5 218

6 217

7 330

88.1

301

285

293

466

86.5

388

214

223

462

86.5

333.2

259

260

677

121

504

~223

240

316

139

539

293

258

465

CH3

Poly(thio-1-methyl-3-oxotrimethylene)

S

4 399

CH CH2

857.

3 86.5

CH2 CH CH3

859.

Poly(thio-2,2-dimethylene) Poly(thioisobutylene)

CH3 S

CH2

C CH3

860.

Poly(thio-2-ethyl-2-methyltrimethylene)

CH3 CH2 S

CH2

C CH2 CH3

861.

Poly(thio-3-methyl-6-oxohexamethylene)

S

(CH2)2

CH CH3

(CH2)2

C O

862.

Poly(thiodifluoromethylene)

45.7

250

155

183

456-458

863.

Poly(thiomethylene)

35.3

205

218

172

454

−S−CF2−

277

−S−CH2−

2 Poly(thioneopentylene)

3 104

4 414

5 233

6 251

7 316, 453

69.5

309

226

225

330, 454, 455

CH3 S

CH2

C CH2 CH3

865.

Poly(thiopropylene)

S

CH2

CH CH3

866.

Poly(thiotrimethylene)

69.5

364

~228

191

453, 459, 460

867.

Poly(tridecanolactone)

233

1031

237

226

763

329

1006

293

327

721

49.7

171

304

291

445

159

416 (random), 489 (iso), 365 (sindio)

−S−(CH2)3−

(CH2)12

C O O

868.

Poly(triethylene glycol p,p'-dibenzoate) C

869.

C O CH2

O Poly(trifluoroethylene)

CF2

CH2

O CH2

CH2

O CH2

CH2

O

O

CH F

870.

Poly(trimethylsilyl methacrylate)

CH3 CH2

C C O O

CH3 Si CH3 CH3

341 (iso), 382 (random), 400 (sindio) 325 (iso), 436 (sindio)

609

278

1 864.

1 871.

Poly(trithiodecamethylene)

2

3 227

4 1064

5 203

6 213

7 461

872.

Poly(trithiomethylene-1,4-phenylenemethylene)

165

555

291

297

461

424

1317

319

322

441

458

1427

324

321

441

560

1767

321

317

441

355

1086

328

327

441

389

1201

323

324

441

−S−S−S−(CH2)10− S

873.

S

S

CH2

CH2

Poly(ureylene-1,4-phenylenemethylene-1,4-phenyleneureylenedecamethylene)

NH C NH

CH2

O 874.

Poly(ureylene-1,4-phenylenemethylene-1,4-phenyleneureylenedodecamethylene)

CH2

O

(CH2)12

O CH2

O

NH C NH

(CH2)6

O

NH C NH

CH2

O

Poly(ureyleneoctamethyleneureylene-1,4-phenylenemethylene-1,4-phenylene)

NH C NH

(CH2)8

NH C NH O

CH2 279

O

(CH2)18

O

Poly(ureylenehexamethyleneureylene-1,4-phenylenemethylene-1,4-phenylene)

NH C NH 877.

NH C NH

Poly(ureylene-1,4-phenylenemethylene-1,4-phenyleneureyleneoctadecamethylene)

NH C NH

876.

(CH2)10

O

NH C NH 875.

NH C NH

2 Poly(vinyl trimethyl silane)

3 115

4 314

5 378-417

6 366

7 693

159

564

286

282

674

182

534

~349

341

278, 628

346

1442

188, 250

240

665

142

393

372

361

665

CH

CH2 H3C

Si

CH3

CH3 879.

Poly[(1,2-diethoxycarbonyl)ethylene]

CH CH O C

C O

O H3C 880.

O

CH2 CH2

CH3

Poly(4-acetoxybenzoyloxyethylene)

CH2

CH O C CH3

O C O 881.

O

Poly[(1-heptoxycarbonyl-1-heptoxycarbonylmethylene)ethylene] O C O CH2

(CH2)6

CH3

C O (CH2)6

CH3

C CH2 O

882.

Poly[(1-methoxycarbonyl-1-methoxycarbonylmethylene)ethylene] O C O CH3 CH2

C CH2 C O CH3 O

280

1 878.

1 883.

2 Poly[(1-methylcyclohexanoyloxy)ethylene]

CH2

O

CH3

C

C

CH2

5 359

6 353

7 650

165

571

283

289

650

182

534

333

341

278, 628

230

552

345

417

545

CH2 CH2 CH2

CH2

Poly[(2,2-dimethylvaleryloxy)ethylene]

CH2

4 482

CH

O 884.

3 170

CH O C O

H3C C

(CH2)2

CH3

CH3 885.

Poly[(2-acetoxybenzoyloxy)ethylene]

CH2

CH O C O O C

CH3

O 886.

Poly[(2-benzoyloxymethyl)styrene]

CH2

CH CH2

O C 281

O

2 Poly[(2-butoxyethoxy)methylstyrene]

CH2

O (CH2)3

Poly[(2-ethylhexyloxy)ethylene]

CH2

4 925

5 < 235

6 268

7 552

180

779

207

231

600, 621

164

488

338

336

278, 628

155

443

321

349

278, 580, 628, 653

CH CH2 O (CH2)2

888.

3 248

CH3

CH O CH2 CH CH2 CH2

889.

CH2 CH3

CH3

Poly[(2-methoxybenzoyloxy)ethylene]

CH2

CH2

CH O C O O CH3

890.

Poly[(2-methylbenzoyloxy)ethylene]

CH2

CH O C O CH3

282

1 887.

1 891.

2 Poly[(2-nitrosoethyl) methacrylate]

3 125

4 411

5 328

6 304

7 610

164

488

~317

336

278, 628

155

443

324

349

278, 628, 653

155

439

366

353

278, 628

CH3 CH2

C C O CH2

CH2

N O

O 892.

Poly[(3-methoxybenzoyloxy)ethylene]

CH2

CH O C O O CH3

893.

Poly[(3-methylbenzoyloxy)ethylene]

CH2

CH O C O CH3

894.

Poly[(3-nitrobenzoyloxy)ethylene]

CH2

CH O C NO 2

283

O

2 Poly[(3-trimethylsilylbenzoyloxy)ethylene]

CH2

3 217

4 573

5 353

6 378

7 659

218

752

267

290

489

276

857

325

322

489

CH O C O H3C

Si

CH3

CH3 896.

Poly[(4-dimethylaminophenyl)methylsilylenetrimethylene]

H3C N CH3

Si

(CH2)3

CH3 897.

Poly[(4-dimethylaminophenyl)phenylsilylenetrimethylene]

H3C N CH3

Si

(CH2)3

284

1 895.

1 898.

2 Poly[(4-methoxybenzoyloxy)ethylene]

CH2

3 164

4 488

5 360

6 336

7 278, 570, 628, 652, 653

155

443

343

350

278, 628, 652, 653

155

439

395

353

278, 628

306

1093

320

280

561

CH O C

O CH3

O 899.

Poly[(4-methylbenzoyloxy)ethylene]

CH2

CH O C

CH3

O 900.

Poly[(4-nitrobenzoyloxy)ethylene]

CH2

CH O C

NO 2

O 901.

Poly[(4-pentadecafluoroheptyl)styrene]

CH2

CH

CF3

285

(CF2)6

2 Poly[(4-phenylbenzoyloxy)ethylene]

CH2

3 214

4 552

5 358

6 388

7 278, 628

200

571

346

350

278, 628

206

691

313

298

651

217

573

408

379

659, 660

142

473

348

300

514

CH O C O

903.

Poly[(4-propionyloxybenzoyloxy)ethylene]

CH2

CH O

904.

C

O C

O

O

CH2

CH3

Poly[(4-p-toluoylbutyryloxy)ethylene]

CH2

CH O C

(CH2)3

CH3

O 905.

Poly[(4-trimethylsilylbenzoyloxy)ethylene]

CH2

906.

CH O

CH3

C

Si

O

CH3

CH3

Poly[(cyclohexylmethyl)ethylene]

CH2

CH CH2

CH2 CH CH2

CH2 CH2 CH2

286

1 902.

1 907.

2 Poly[(cyclopentylmethyl)ethylene]

CH2

908.

CH

CH2 CH2 CH CH2

4 391

5 333

6 317

7 514

131

434

~328, 338

302

591

172

656

243

262

490

314

1292

253–255

243

656

177

670

288–293

264

656

CH2 CH2

Poly[(heptafluoro-2-propoxy)ethylene]

CH2

3 124

CH O

F 3C 909.

CF

CF3

Poly[(methyl)phenylsilylenetrimethylene]

CH3 Si

910.

(CH2)3

Poly[(nonadecafluorodecanoyloxy)ethylene]

CH2

CH O C

(CF2)8

CF3

O 911.

Poly[(nonafluorovaleryloxy)ethylene]

CH2

CH O C

CF3 287

O

(CF2)3

2 Poly[(pentadecafluorooctanyloxy)ethylene]

CH2

3 259

4 1040

5 258–263

6 249

7 656

122

421

315

290

656

142

362

393

392

654

94.9

297

~319, < 348

319

656, 657

210

652

327

321

656

CH O C

(CF2)6

CF3

O 913.

Poly[(pentafluoropropionyloxy)ethylene]

CH2

CH O C

CF2

CF3

O 914.

Poly[(tert-butoxycarbonylamino)ethylene]

CH2

CH NH

CH3

O C O C CH3 CH3 915.

Poly[(trifluoroacetoxy)ethylene] CH2 CH

O C 916.

CF3

O Poly[(undecafluorocyclohexylcarbonyloxy)ethylene] CH2 CH

O

CF2 C CF CF2 O

CF2 CF2 CF2

288

1 912.

1 917.

2 Poly[2-(2-dimethylaminoethoxycarbonyl)styrene]

CH2

3 218

4 597.6

5 342

6 365

7 551

199

610

363

326

650

199

599

323

332

650

199

610

328

326

650

CH CH3 C O

(CH2)2

N CH3

O 918.

Poly[2,2,3,3-tetramethylvaleryloxy)ethylene]

CH2

CH O CH3 CH3 C

C

C

CH2

CH3

O CH3 CH3 919.

Poly[2,2,3,4-tetramethylvaleryloxy)ethylene]

CH2

CH O CH3 C

C

CH

CH

CH3

O CH3 CH3 CH3 920.

Poly[2,2,4,4-tetramethylvaleryloxy)ethylene]

CH2

CH O CH3 C

C

CH2

C CH3

CH3 289

O CH3

CH3

2 Poly[2,2-difluoro-2-(2-heptafluorotetrahydrofuranyl)ethyl acrylate]

CH2

CH

F

C O CH2

923.

5 275 Brittle point

6 283

7 587

261

529

~471 (softening point)

493

546

192

608

319

316

560

272

1000

250

272

552

F2

F2

O

Poly[3-(4-biphenyl)styrene]

CH2

4 739

CF2

O 922.

3 209

F2

CH

Poly[3-chloro-2,2-bis(chloromethyl)propyl acrylate]

Cl CH2

CH

CH2

C O CH2

C

O

CH2

CH2Cl

Cl 924.

Poly[4-(1-ethylhexyloxymethyl)styrene]

CH2

CH CH3 CH2 CH2

O CH (CH2)4

CH3

290

1 921.

1 925.

2 Poly[4-(1-hydroxy-1-methylbutyl)styrene]

CH2

5 ~403 softening point

6 365

7 567

168

375

~438 softening point

448

567

236

731

~364 softening point

323

567

CH3

CH2

CH2

CH3

Poly[4-(1-hydroxy-1-methylethyl)styrene]

CH2

4 553

CH

OH C 926.

3 202

CH

OH C

CH3

CH3 927.

Poly[4-(1-hydroxy-1-methylhexyl)styrene]

CH2

CH

OH C

CH3 CH3

291

(CH2)4

2 Poly[4-(1-hydroxy-1-methylpentyl)styrene]

CH2

7 567

185

464

459

399

567

246

665

323 low viscosity

370

567

CH

CH3

CH2

CH3

Poly[4-(1-hydroxy-3-morpholinopropyl)styrene]

CH2

6 341

CH3

Poly[4-(1-hydroxy-1-methylpropyl)styrene]

HO C 930.

5 356 softening point

CH3

(CH2)3 CH2

4 642

CH

HO C 929.

3 219

CH

CH2 HO CH

(CH2)2

N CH2

CH2 O CH2

292

1 928.

1 931.

2 Poly[4-(1-hydroxy-3-piperidinopropyl)styrene]

CH2

HO CH

(CH2)2

N CH2

5 327 low viscosity

6 369

7 567

212

656

319

323

552

240

677.4

314

354

567

CH2 CH2 CH2

Poly[4-(2-hydroxybutoxymethyl)styrene]

CH2

4 688

CH

CH2

932.

3 254

CH

OH CH2 933.

O

CH2

CH CH2

Poly[4-(3-morpholinopropionyl)styrene]

CH2

CH

CH2 O

CH3

C

(CH2)2

N

293

CH2

CH2 O CH2

2 Poly[4-(3-piperidinopropionyl)styrene]

CH2

CO

(CH2)2

N CH2

Poly[4-(4-biphenyl)styrene]

CH2

936.

937.

4 701

5 311

6 353

7 567

261

529

593 (softening point)

493

546

212

721

293

294

552

272

1067

231

255

552

CH

CH2

935.

3 248

CH2 CH2 CH2

CH

Poly[4-(4-hydroxybutoxymethyl)styrene] CH2 CH

CH2 O (CH2)4 Poly[4-(octyloxymethyl)styrene] CH2 CH

CH2

O

(CH2)7

OH

CH3

294

1 934.

1 938.

2 Poly[4-(sec-butoxymethyl)styrene]

CH2

3 204

4 646

5 313

6 316

7 552

262

630

433

416

297

290

674

446

430

380

273

591

483

462

297

280

1284

210

218

451

CH

CH3 CH2 O CH CH2 939.

CH3

Poly[di(oxy-1,4-phenylene)carbonyl-1,4-phenylene]

O

O

C O

940.

Poly[di(oxy-1,4-phenylene)oxyisophthaloyl]

O

941.

O

C

O

O

Poly[di(oxy-1,4-phenylene)sulfonyl-1,4-phenylene]

O 942.

O C

O

SO 2

Poly[di(oxydimethylsilylene)-1,4-phenylenedimethylsilylene]

CH3 O

Si

CH3

Si

Si

CH3

CH3

295

CH3

O

CH3

2 Poly[di(oxyethylene)oxy-1,4-phenylenecarbonyl-oxycarbonyl-1,4-phenylene]

O

(CH2)2

O

(CH2)2

O

(CH2)4

C O CH2

O 945.

O CH2

CH2

(CH2)7

C O CH2

CH2

O CH2

CH2

(CH2)2

O

(CH2)12

O 948.

(CH2)2

O C NH

(CH2)6

O

290

254

1081

205

235

290

260

916

272

284

349

339

1480

202

229

290

185

758

226

244

290

151

602

244

251

290

O

C O CH2

CH2

O CH2

CH2

(CH2)3

C O CH2

CH2

O CH2

CH2

O

O

O

Poly[di(oxyethylene)oxymalonyl]

C

241

O

O 949.

227

NH C

Poly[di(oxyethylene)oxyglutaryl]

C

838

O

Poly[di(oxyethylene)oxydodecanedioyl]

C

202

O

O 947.

7 350

O

Poly[di(oxyethylene)oxycarbonyliminohexamethyleneiminocarbonyl)]

O

6 316

O

O 946.

CH2

Poly[di(oxyethylene)oxyazelaoyl]

C

5 314

O

Poly[di(oxyethylene)oxyadipoyl]

C

4 930

C O C O

944.

3 294

CH2

C O CH2 O

CH2

O CH2

CH2

O

296

1 943.

1 950.

2 Poly[di(oxyethylene)oxymethylmalonyl]

O CH2

CH2

O CH2

CH2

O C O

951.

CH

CH2

O CH2

CH2

O C

4 625

5 244

6 269

7 290

305

1344

214

227

290

441

1951

205

226

290

134

521

265

257

290

237

987

226

240

290

219

920

213

238

290

C

CH3 O

Poly[di(oxyethylene)oxynonylmalonyl]

O CH2

3 168

CH

O (CH2)8

C O

CH3 952.

Poly[di(oxyethylene)oxyoctadecanedioyl]

O CH2

CH2

O CH2

CH2

O C

(CH2)18

O 953.

O

Poly[di(oxyethylene)oxyoxalyl]

O CH2 954.

C

CH2

O CH2

CH2

O C

C

O

O

Poly[di(oxyethylene)oxypentylmalonyl]

O CH2

CH2

O CH2

CH2

O C

CH

O (CH2)4

C O

CH3 955.

Poly[di(oxyethylene)oxypimeloyl]

O CH2

CH2

O CH2

CH2

O C

C O

297

O

(CH2)5

2 Poly[di(oxyethylene)oxypropylmalonyl]

O CH2

CH2

O CH2

CH2

O C O

CH

C

CH2

O

3 185

4 714

5 235

6 259

7 290

271

1163

199

233

290

237

1000

212

237

290

168

680

244

247

290

259

794

325

326

350

179

909

218

197

368

341

804

422

424

444

CH3 957.

Poly[di(oxyethylene)oxysebacoyl]

O CH2

CH2

O CH2

CH2

O C

(CH2)8

O 958.

O

Poly[di(oxyethylene)oxysuberoyl]

O CH2

CH2

O CH2

CH2

O C

(CH2)6

O 959.

O

CH2

O CH2

CH2

O C

(CH2)2

O

C O

Poly[di(oxymethylene)oxy-1,4-phenylenecarbonyl-oxycarbonyl-1,4-phenylene]

O CH2

O CH2

O

C O C O

961.

C

Poly[di(oxyethylene)oxysuccinyl]

O CH2 960.

C

O

Poly[di(oxymethylene)oxy-2,2,3,3,4,4-hexafluoropentamethylene]

–O–CH2–O–CH2–O–CH2–(CF2)3–CH2– 962.

Poly[imino-5-tert-butylisophthaloylimino-(2,5-dimethylhexamethylene)]

NH C

C NH

O H3C

O C CH3 CH3

CH2

CH CH3

(CH2)2

CH CH2 CH3

298

1 956.

1 963.

2 Poly[imino-5-tert-butylisophthaloylimino-(3,4-dimethylhexamethylene)]

3 341

4 804

5 446

6 424

7 444

306

754

436

406

444

306

855

351

358

373

290

770

398

377

406

CH3 NH C

C NH

O

(CH2)2

C

CH

(CH2)2

CH3

O

H3C

CH

CH3

CH3 964.

Poly[imino-5-tert-butylisophthaloyliminohexamethylene]

NH C

C NH

O

(CH2)6

O

H3C

C

CH3

CH3 965.

Poly[iminomethylene(2,5-dimethyl-1,4-phenylene)methyleneiminosuberoyl]

H3C NH CH2

CH2

NH C

(CH2)6

O

C O

H3C 966.

Poly[iminomethylene-1,4-phenylenemethyleneiminocarbonyl(1,3,3-trimethyl-5-oxopentamethylene)]

CH3 NH CH2

CH2

NH C CH CH2

C CH2

C

CH3

O

299

O

CH3

2 Poly[oxy(allyloxymethyl)ethylene]

O CH2

O CH2

5 195

6 215

7 330

82.5

311

259

265

332

139

662

194

210

330

75.5

306

251, 248

247

332, 703

104

479

212

217

330

173

840

190

206

330

87.3

400

211

218

330

CH CH2

Poly[oxy(bromomethyl)ethylene]

O CH2

4 544

CH CH2

968.

3 117

CH CH2Br

969.

Poly[oxy(butoxymethyl)ethylene]

O CH2

CH CH2 O

970.

(CH2)3

CH3

Poly[oxy(chloromethyl)ethylene]

O CH2

CH CH2Cl

971.

Poly[oxy(ethoxymethyl)ethylene]

O CH2

CH CH2

972.

O CH2

Poly[oxy(hexyloxymethyl)ethylene]

O CH2

CH CH2 O

973.

CH3

(CH2)5

Poly[oxy(methoxymethyl)ethylene]

O CH2

CH CH2 O CH3

CH3

300

1 967.

1 974.

2 Poly[oxy(methyl)phenylsilylene]

3 129

4 471

5 187, 273

6 274

7 748

340

960

331

354

448

293

807

394

363

431

306

691

416 Proposed structure

443

377

CH3 O

975.

Si

Poly[oxy(methyl)phenylsilyleneoxy-1,4-phenyleneisopropylidene-1,4-phenylene]

CH3 O

Si

CH3 O

C CH3

976.

Poly[oxy-1,3-phenyleneoxy-5-(heptafluoropropyl)isophthaloyl]

O

O C

C

O

O CF2

977.

CF2

CF3

Poly[oxy-1,4-phenylene-(2-cyano)-2-phenyltrimethylene-1,4-phenylene]

C N O

CH2

C CH2

301

2 Poly[oxy-1,4-phenylene-2,2-di(ethoxycarbonyl)trimethylene-1,4-phenylene]

CH2

3 335

4 1003

5 327

6 334

7 377

281

562

507

500

331

311

641

484

485

331

CH3

O C O O

CH2

C CH2 C O O CH2

979.

CH3

Poly[oxy-2-(1-naphthyl)-6-phenyl-1,4-phenylene]

O

980.

Poly[oxy-2-(2-biphenylyl)-6-phenyl-1,4-phenylene]

O

302

1 978.

1 981.

2 Poly[oxy-2-(2-chlorobenzoyloxy)trimethyleneoxy-1,4-phenyleneisopropylidene-1,4-phenylene] CH3

O CH2

CH CH2

O

3 406

4 1140

5 339

6 356

7 329

303

634

513

478

331

331

943

338

351

329

C

O

CH3

O C CH2 Cl

982.

Poly[oxy-2-(4-tert-butylphenyl)-6-phenyl-1,4-phenylene]

O

CH3

C

CH3

CH3 983.

Poly[oxy-2-(chloroacetoxy)trimethyleneoxy-1,4-phenyleneisopropylidene-1,4-phenylene]

CH3 O CH2

CH CH2 O

C CH3 303

O C CH2Cl

O

2 Poly[oxy-2,2-bis(chloromethyl)trimethylene]

3 124

4 420

5 265

6 295

7 273, 300, 339-345

375

1014

383

370

329

451

1212

399

372

329

445

1660

300

268

410

CH2Cl O CH2

C CH2 CH2Cl

985.

Poly[oxy-2-acetoxytrimethyleneoxy-1,4-phenylenemethyl(phenyl)methylene-1,4-phenylene]

CH3 O CH2

CH CH2

O

C

O C CH3 O 986.

Poly[oxy-2-benzoyloxytrimethyleneoxy-1,4-phenylenemethyl(phenyl)methylene-1,4-phenylene]

CH3 O CH2

CH CH2

O

C

O C CH2 O 987.

Poly[oxy-5-(pentadecafluoroheptyl)isophthaloyloxy-2,2,3,3,4,4-hexafluoropentamethylene]

O

C

C O CH2

O

O CF2

CF2

CF2

CF2

CF2

CF2

CF2

CF2

CF2

CH2

CF3

304

1 984.

1 989.

2 Poly[oxy-6-phenyl-2-(m-tolyl)-1,4-phenylene]

3 252

4 506

5 492

6 498

7 331

252

506

491

498

331

385

881

~478

437

337

162

372

500, < 433, 418

435

390–392

H3C

O

990.

Poly[oxy-6-phenyl-2-(p-tolyl)-1,4-phenylene]

H3C O

991.

Poly[oxycarbonyldi(oxy-1,4-phenylene)sulfonyl-1,4-phenyleneoxy-1,4-phenylene]

O

O C O

SO2

O

O 992.

Poly[oxycarbonyloxy-1,3-(2,2,4,4-tetramethylcyclobutylene)

H3C

305

CH3 C O C O CH CH C O H3C CH3

2 Poly[oxycarbonyloxy-1,4-phenylene(methyl)phenylmethylene-1,4-phenylene]

3 297

4 661

5 449

6 449

7 384, 386

271

679

410

399

388

369

1525

231

242

446

CH3 O C O

C

O

994.

Poly[oxycarbonyloxy-4,6-dimethyl-1,2-phenylenemethylene-3,5-dimethyl-1,2-phenylene]

CH2 O C O O 995.

H3C

CH3 H3C

Poly[oxydi(pentafluorophenyl)silylenedi(oxydimethylsilylene)]

F F

F

F

F CH3 O

Si

O

Si

F

F CH3

F

F F

CH3 O

Si CH3

CH3

306

1 993.

1 996.

2 Poly[oxymethylpentafluorophenylsilylenedi(oxydimethylsilylene)] CH3 CH3 CH3

O

997.

Si

O

Si

F

F CH3

F

F

O

Si CH3

999.

CH3

Si

O

CH3

Si

O

CH3

CH3

Si

Si

CH3

CH3

CH3

CH3

CH3

CH2

CH2

CH2

CH2

CH2

CH3

Si

Si CH3

Si

O

Si

O

Si

O

Si

O

CH2

CH2

CH2

CH2

CH2

CH3

CH3

CH3

CH3

CH3

Poly[penta(oxydiethylsilylene)-1,4-phenyleneoxy-1,4-phenylenedimethylsilylene]

O

CH3

CH3

CH3

CH3

CH3

CH2

CH2

CH2

CH2

CH2

Si

O

Si

6 213

7 446

538

2491

231

216

372

666

4215

193

158

372

750

4438

208

169

372

CH3

Poly[penta(oxydiethylsilylene)-1,4-phenylenedimethylsilylene]

O

5 190

CH3

CH3 O

4 1343

Si

F Poly[oxytri(dimethylsilyleneoxy)(methyl)phenylsilylene-1,3-phenylene(methyl)phenylsilylene)]

CH3

998.

O

3 286

O

Si

O

Si

O

Si

CH2

CH2

CH2

CH2

CH3

CH3

CH3

CH3

CH3

O

Si CH3 307

CH2

CH3

CH3 O

Si

CH3 O

CH3

Si

CH3 O

CH3

Si

O

CH3

CH3

CH3

Si

Si

CH3

CH3

1001. Poly[tetra(oxydimethylsilylene)-1,4-phenylenedimethylsilylene]

CH3 O

Si

CH3

CH3 O

CH3

Si

O

CH3

Si

O

CH3

CH3

CH3

Si

Si

CH3

CH3

1002. Poly[tetra(oxydimethylsilylene)-1,4-phenyleneoxy-1,4-phenylenedimethylsilylene]

CH3 O

Si CH3

CH3 O

Si

CH3 O

CH3

Si CH3

Si

O

6 185

7 372

423

2238

201

189

372

507

2449

221

207

372

126

581

213

217

330

347

848

423

409

297

357

808

453

442

297

Si

CH CH2 CH2

O CH2

CH CH2

1004. Poly[tri(oxy-1,4-phenylene)carbonyl-1,4-phenylene]

O

5 198

CH3

CH3

1003. Poly[thio-1-(allyloxymethyl)ethylene]

S

4 2286

CH3

CH3 O

3 423

O

O

C O

1005. Poly[tri(oxy-1,4-phenylene)sulfonyl-1,4-phenylene]

O

O

O

SO2

308

1 2 1000. Poly[tetra(oxydimethylsilylene)-1,3-phenylenedimethylsilylene]

1 2 1006. Poly[tri(oxydimethylsilylene)-1,4-phenylenedimethylsilylene]

CH3 O

Si

CH3 O

CH3

Si

O

CH3

CH3

CH3

Si

Si

CH3

CH3

1007. Poly[tri(oxydimethylsilylene)oxy(methyl)-2-phenylethylsilylene]

3 351

4 1764

5 211

6 199

7 372

380

2159

171

176

449

344

1901

201

181

449

359

1962

148

183

449

CH2 CH3 O

Si

CH3 O

CH3

Si

CH3 O

CH3

Si

CH2 O

CH3

Si CH3

1008. Poly[tri(oxydimethylsilylene)oxy(methyl)phenylsilylene]

CH3

CH3 O

Si

O

CH3

Si

CH3 O

CH3

Si

CH3 O

Si

CH3

1009. Poly[tri(oxydimethylsilylene)oxy(methyl)trimethylsiloxysilylene]

CH3 H3C CH3 O

Si

O

Si CH3

CH3 O

Si CH3

O O

Si CH3

309

CH3

CH3

Si CH3

CH3 O

Si

CH3 O

CH3

Si

Si

O

(CH2)2

O

(CH2)2

O

(CH2)2

O

C NH (CH2)6 O

7 372

304

1131

260

269

765

204

826

224

247

587

194

767

236

253

589

390

677

583

576

6

NH C O

1012. Poly1H,1H-nonafluoro-4-oxahexyl acrylate)

CH2

6 220

CH3

1011. Poly[tri(oxyethylene)oxycarbonyliminohexamethyleneiminocarbonyl]

O

5 236

Si

CH3

CH3

4 1982

CH3

CH3 O

3 436

CH C O CH2

(CF2)2

O CF2

CF3

O 1013. Poly1H,1H-nonafluoropentyl acrylate)

CH2

CH C O CH2

(CF2)3

CF3

O 1014. Polyarylate of phenolphthalein and terephthalic acid

O C O CO

O C

C

O

O

310

1 2 1010. Poly[tri(oxydimethylsilylene-1,4-phenyleneoxy)-1,4-phenylenedimethylsilylene]

1 2 1015. Polyamide of 1,7-bis(aminophenoxy)naphthalene and dicarboxylic acid

NH C

5 412–536

6 537

7 717

389

745

522

483

6

443

711

643

623

6

396

655

588

605

6

O

1016. Polyamide of adipic acid and anilinephthalein

O

4 564

C NH

O

C (CH2)4

3 303

C NH O

NH C O CO

1017. Polyamide of anilinefluorene and terephthalic acid

NH C

NH C

C

O

O

1018. Polyamide of anilinephthalein and isophthalic acid

C

C NH

O

O

NH C O 311

CO

NH

NH C

C

O

O

C

3 396

4 613

5 633

6 646

7 6

396

613

643

646

6

579

872

646

664

6

471

746

643-653

631

6

O CO 1020. Polyamide of diphenylphthalidedicarboxylic acid and 1,4-phenylenediamine

C O

C NH C

NH

O O

CO 1021. Polyamide of diphenylphthalidedicarboxylic acid and anilinephthalein

C O

C NH C

O

NH C

O

O

CO

CO

1022. Polyamide of diphenylphthalidedicarboxylic acid and benzidine

C O

C NH C

O O

CO

NH

312

1 2 1019. Polyamide of anilinphthalein and terephthalic acid

1 2 1023. Polyamide of phenolanthrone and terephthalic acid

NH

NH C

C

O

O

C

3 461

4 689

5 683

6 669

7 6

457

1018

435

449

6

480

829

600

579

6

C O 1024. Polyamide of sebacic acid and anilinephthalein

C (CH2)8

C NH

O

O

NH C O CO

1025. Polyamide of 4,4'-diphenyloxidedicarboxylic acid and anilinephthalein

C O

O

C NH O

NH C O 313

CO

C

O

C O

O

3 474

4 894

5 573–593

6 530

7 6

333

722

457

461

683

474

894

585

530

6

516

940

543

549

6

O C

O

O CO 1027. Polyarylate of Bisphenol A and isophthalic acid

CH3 O

C

O C

C

O

O

CH3

1028. Polyarylate of diphenylphthalidedicarboxylic acid and 4,4'-diphenyleneoxide

C

C O C

O

O

O

O O

CO 1029. Polyarylate of diphenylphthalidedicarboxylic acid and bisphenol A

C O

C O CO

CH3

C O

C

O

CH3

314

1 2 1026. Polyarylate of 4,4'-diphenyloxidedicarboxylic acid and phenolphthalein

1 2 1030. Polyarylate of diphenylphthalidedicarboxylic acid and resorcin O

O C O

3 390

4 720

5 543

6 542

7 6

455

754

627

604

6

513

911

587

563

6

437

817

543

535

6

C C

O O

CO

1031. Polyarylate of phenolanthrone and terephthalic acid

O C

O C

C

O

O

C O 1032. Polyarylate of phenolfluorene and 4,4'-diphenyldicarboxylic acid

O C

O C

C

O

O

1033. Polyarylate of phenolfluorene and isophthalic acid

O C

O C

C

O

O 315

O C

O C

C

O

O

1035. Polyarylate of phenolphthalein and 4,4-diphenyldicarboxylic acid

O C

O C

C

O

O

3 437

4 776

5 590

6 563

7 6

465

812

603

573

6

573

936

583–598, 613

612

6

390

720

543

542

6

O CO 1036. Polyarylate of phenolphthalein and diphenylphthalidedicarboxylic acid

O

O C C

C C

O O

O

CO

CO

1037. Polyarylate of phenolphthalein and isophthalic acid

O C O CO

O

O C

C

O

O

316

1 2 1034. Polyarylate of phenolfruorene and terephthalic acid

1 2 1038. Poly(carbonyl-1,4-phehylenesulfonyl-1,4-phenylenecarbonyloxy-1,4-SKHQ\OHQHIOXRUHQ_-9-ylidene-1,4phenyleneoxy)

C

SO2

C O

O

O C

5 623

6 580

7 6

383

861

433

445

6

452

1180

373

383

6

532

830

687-703

641

6

O

1039. Polyester of phenolphthalein and adipic acid

C

4 933

C

O

O

3 541

(CH2)4

O

C O

O CO 1040. Polyester of sebacic acid and phenolphthalein

C (CH2)8

C O

O

O

O C O CO

1041. Polyimide of anilinefluorene and 3,3',4,4'-tetracarboxydiphenyl

N C

CO

CO

CO

CO

N

317

3 467

4 840

5 586–592

6 556

7 6

558

889

623–673

628

6

550

893

623–673

616

6

493

814

623–664

606

6

CF3 N

CO

C

CO

CO

CF3

CO

N

O

1043. Polyimide of anilineanthrone and 3,3',4,4'-tetracarboxydiphenyloxide

N

CO

O

CO

CO CO

N C

CO 1044. Polyimide of anilinefluorene and 3,3',4,4'-benzophenonetetracarboxylic acid

N

CO

CO

CO

CO CO

N C

1045. Polyimide of anilinephthalein and 3,3',4,4'-tetracarboxydiphenyloxide

O N

C

O O

C

C

C

O

O

N C O CO

318

1 2 1042. Polyimide of 4,4'-diaminodiphenyloxide and 3,3',4,4'-tetracarboxydiphenylhexafluoroisopropylene

2

3 229

4 398

5 563–573

6 576

7 766

237

395

563–573

600

766

456

619

723–788

737

6

409

520

789

787

6

399

854

459

467

683

CO O

N

N

CO

CO

1047. Polyparabanic acid 2

CO CH2

N

N

CO

CO

1048. Polypyromelliteimide of anilinefluorene

N

CO

CO

CO

CO

N C

1049. Polypyromelliteimide of anilinephthalein

N

CO

CO

CO

CO

N C O CO

1050. Polysulfone of Bisphenol A and diphenylene sulfone

CH3 O

O

C CH3

SO 2

319

1 1046. Polyparabanic acid 1

320

The above-described effect of influence of chain attachment on Tg of the polymer is also spread upon polymeric systems, composed of isomers. For example, polyamidophenyl quinoxalines, analyzed in the study in [83], display the following structure: O

N

N

N

NH C O

N

C NH O

O

n

PAPQ-1 O

N

O

N

C NH O

N

C O NH

N

n

PAPQ-2 In the case of PAPQ-2, polar groups representing phenolquinoxaline cycles are in tight contact with each other, and the intermolecular interaction occurring at the sacrifice of these groups is displayed in the frames of the unit; then, the dipole–dipole interaction between units of neighboring chains is not realized. PAPQ-1 and PAPQ-2 are isomers with the same gross-formula. Work [83] represent experiments and calculations on the determination of glass transition temperatures of two pairs of polymers, the structure of which is shown above. Since strong intermolecular interaction occurring at the sacrifice of separate polar groups significantly affects Tg, it is obvious that when this interaction takes place inside the unit, Tg of such polymers must be lower than for polymers with a completely analogous structure, but with such disposition of polar groups in which intermolecular interaction manifests itself between units of neighboring chains. Calculations and experiments have confirmed that this indeed happens in this way. When the glass transition temperature is calculated for PAPQ-1, ∑ b j = 4bp + 2bh + 2bd + 2 ⋅ 1 bd , and for PAPQ-2 ∑ b j = 4bp + 2bh . Consequently, 2

j

j

the calculated value of Tg for PAPQ-1 is 265°C, and for PAPQ-2 it is 208°C. Appropriate experimental values are 275 and 210°C, respectively. At the same gross-formula of the repeat unit, the glass transition temperature also depends on the location of separate aromatic fragments. For example, polyimides PI-1and PI-2 possess identical gross-formulae, but display different locations of aromatic cycles. Calculating the glass transition temperature for PI-1, it is obtained that ∑ b j = 2bd + 2bp , and in the case of PI- ∑ b j = 2bd + 5bp . In accordance j

Š

j

with this, Tg for polyimide PI-1 is 238°C, and for PI- approximately the same values are obtained.

Š 295°C. Experimentally,

321

O

O C

C

O

C

C

O

O PI-1

O

O

C

C

C

C

O

O

N

N

O

N

N

O

O

PI-2 It should be noted that special attention must be paid to the cases when a significant deviation in experimental and calculated data on the glass transition temperature is observed. This may be caused by both the occurrence of a polar group possessing a special influence on the energy of intermolecular interaction and, vice versa, exclusion of any polar group from formation of the network of physical bonds between neighboring chains of the polymer. In the first case, introduction of a new constant bj is required, but it should always be remembered that the greater is the number of constants introduced to the calculated scheme, the lower becomes its predictive power and, in the border case, when every new polymer requires introduction of a new constant bj, the predictive power of the calculation scheme becomes equal to zero. It should be also taken into account that anomalous addition of units to each other may not weaken, but strengthen intermolecular interaction between neighboring chains, and then the glass transition temperature will increase. Let us now turn to the problem of plasticization rules. Let us use two poly(ether ketones)∗ as polymers: C

O

C

O

PAEK 1

O

PAEK 2

O CH3

H3C C

O O

C O

C O As plasticizers, let us choose phthalic acid esters – dimethylphthalate (DMP), dimethylterephthalate (DMTP) and dibutylphthalate (DBP):

∗

These poly(ether ketones) are produced in works [769 – 771].

322

O C O CH3 O

O C O CH3 ,

,

C O CH3 O

C O CH3

C O

(CH2)3

CH3

C O

(CH2)3

CH3

O

O DMP

DMTP

DBP

Dependence of the glass transition temperature for PAEK 1 on the plasticizer is shown below: Glass transition temperature, K Spl, % (mass) DMP DMTP DBP 0 380 380 380 2.5 365 364 369 5.0 355 347 365 10.0 345 341 364 15.0 339 * 364 20.0 330 * 364 Dependence of the glass transition temperature for PAEK 2 on the plasticizer concentration is the following: Glass transition temperature, K DMP DMTP 0 470 470 2.5 425 417 5.0 408 394 10.0 387 380 15.0 362 * 20.0 353 * * - dimethylterephthalate is crystallized out of the polymer. Spl, % (mass)

DBP 470 441 426 415 406 403

IV.4. Influence of plasticization on the glass transition temperature of polymers Let us analyze the changing effect of the part of the plasticizer introduced into polymer on the glass transition temperature of the plasticized system. As mentioned above, there are two fundamental concepts for description of these dependences and two regularities corresponded to them: the Zhurkov rule of molar parts and the Kargin–Malinskii rule of volumetric parts. According to the Zhurkov rule, decrease of the glass transition temperature is proportional to the molar part of the plasticizer introduced, i.e. when it is fulfilled, the dependence of the glass transition temperature on the molar part of the plasticizer must be linear. The experiments performed indicated∗ that dependence Tg(α) is not linear for all studied systems. ∗

Author’s data, not published.

323

According to the Kargin–Malinskii rule, decrease of the glass transition temperature during plasticization is proportional to the volumetric part of the plasticizer in the polymer (æ), i.e. Tg(æ) must be linear. The experimental data for systems considered give no linear dependence. Hence, both rules do not describe the real behavior of the given polymer– plasticizer systems. Let us consider a possibility to describe the dependence of the glass transition temperature on the part of plasticizer, basing on the notions and calculation schemes, discussed above. According to the given ideas, the glass transition temperature of a linear polymer is described by correlation (IV.41). If the plasticizer molecule interacts with polar groups of the polymer and screens them, then the specific intermolecular interaction between polymer chains cannot be displayed. Then, because these groups are excluded from the intermolecular interaction, constants βj must be excluded from equation (IV.41), and this exclusion must be made in proportion to the molar part of the plasticizer introduced. The quantity of parameters βj excluded depends on the quantity of dipole–dipole interactions displayed by the given polar group of the polymer, as well as on the number of polar groups contained in the plasticizer molecule. In the general form, we obtain the following equation:

Tg =

∑ ∆Vi i

∑ ∆Vi ai + ∑ β j − mbd nα i

,

(IV.45)

j

where m is the quantity of parameters bd related to a single polar group of the polymer; n is the quantity of polar groups in the plasticizer molecule; α is the molar part of the plasticizer. The value of m is determined according to the rules discussed above. In the case of poly(arylether ketone) based on phenolphthalein (PAEK 2), m = 2.5; for diane poly(arylether ketone) (PAEK 1), m = 2. With regard to the plasticizers applied, dimethylphthalate and dibutylphthalate, for them n = 2. Then, equation (IV.45) is reduced to the form: for PAEK 2,

Tg =

∑ ∆Vi i

∑ ∆Vi ai + ∑ β j − 2.5bd ⋅ 2α i

,

(IV.46)

.

(IV.47)

j

and for PAEK 1,

Tg =

∑ ∆Vi i

∑ ∆Vi ai + ∑ β j − 2bd ⋅ 2α i

j

Calculations performed by equations (IV.46) and (IV.47) lead to a decrease of the glass transition temperature of the plasticized sample compared with the original

324

polymer. However, the above-considered reason for decrease of the glass transition temperature is not unique. The second reason may be trivial and associated with the so-called effect of dilution. It is meant that a polymer with a relatively high softening temperature is diluted by the plasticizer with a much lower softening temperature. If the polymer and the plasticizer are miscible, the system may be considered analogous to a two-component mixture of polymers, fully miscible with each other (the mixture possesses general glass transition temperature at any composition). Then the phase diagram (Tg–composition) for the mixture of polymer and plasticizer may be calculated by correlation (IV.59) applied to estimation of the glass transition temperature of mixtures of miscible polymers. In this case, the correlation is reduced to the form:     α p  ∑ ∆Vi  + α pl  ∑ ∆Vi       i p  i  pl

Tg =

αp

   ∑ ∆Vi     i p

(Tg )p

+ α pl

   ∑ ∆Vi     i  pl

(Tg )pl

,

(IV.48)

+ 0.06α pα pl

    where  ∑ ∆Vi  and  ∑ ∆Vi  are Van-der-Waals volumes of the repeat unit of      i p  i  pl polymer and the plasticizer molecule, respectively; (Tg)p and (Tg)pl are the glass transition temperature of the polymer and the softening temperature of plasticizer, respectively. Let us consider results of calculation (Figure 42′) performed by equations (IV.46), (IV.47) and (IV.48) for plasticized systems PAEK 1 and PAEK 2. Clearly, the dependence of the glass transition obtained by equation (IV.48) with regard to the dilution influence is much deeper than the one calculated by equations (IV.46) and (IV.47) with regard to screening of polar groups of the polymer by plasticizer molecules. However, both dependences give no values of depression of the glass transition temperature, coincident to experimental ones. Hence, taking into account each mechanism of plasticization separately does not enable adequate description of the real behavior of the plasticized system. However, if the effects of ‘dilution’ and ‘screening’ are summed up, the obtained dependence of the glass transition temperature on the plasticizer concentration is close to the experimental one in the area of plasticizer concentration, when it is miscible with the polymer. Calculated data are shown below. Calculated glass transition temperatures for plasticized PAEK 1 Glass transition temperature Tg, K Cpf, % (mass) DMP DBP Exp. Calc. (dil. + scr.) Exp. Calc. (dil. + scr.) 2.5 365 367 368 366 5.0 355 355 365 352 10.0 345 333 364 327 15.0 339 313 364 305 20.0 330 296 364 285

325

Calculated values of glass transition temperature for plasticized PAEK 2 Glass transition temperature Tg, K Cpf, % (mass) DMP DBP Exp. Calc. (dil. + scr.) Exp. Calc. (dil. + scr.) 2.5 425 448 441 447 5.0 408 428 426 425 10.0 384 393 415 387 15.0 362 363 406 353 20.0 253 336 403 323

Figure 42′. Calculated dependence of glass transition of PAEK-11 (a, b) and PAEK-21 (c, d) on the content of dimethylphthalate (a, c) and dibutylphthalate (b, d) .

Discussing the PAEK 2–DBP system, it may be noted that starting from 7– 10% (mass) of dibutylphthalate concentration in the system, clear separation, displayed in appearance of film dimness, begins. From this point, decrease of the glass transition temperature is decelerated and, consequently, an abrupt deviation from the calculated dependence appears (Figure 42′d). Using dimethylphthalate as a plasticizer for PAEK 2, the plasticizer is much better joined with polymer. No signs of separation are observed in the range of concentrations considered (0–20% (mass)). In this case, the calculated dependence of the glass transition temperature on the plasticizer concentration, obtained by summation of temperature depressions of dilution and screening effects of polar groups, is close to the experimental one throughout the length (Figure 42′c).

326

Note that the present analysis requires no introduction of any fitting parameters and is based only on the chemical structure of the plasticizer and the polymer. It may be mentioned in passing that comparing the calculated and experimental data, the system composition (the part of plasticizer), at which miscibility fails (PAEK 2–DBP, Figure 42′e), may be determined,. Concerning the case of PAEK 1 and PAEK 2 plasticization by dimethylterephthalate, in the present case, such analysis cannot be performed completely. This is associated with the circumstance that dimethylterephthalate is a crystalline substance, and only the melting point is known for it. The dilution effect may not be calculated, because, on the one hand, a mixture of crystalline and amorphous substance exists and, on the other hand, although crystallinity of DMTP is lost at low concentrations and its glass transition temperature is unknown. In this case, the only depression of the glass transition temperature associated with screening of polar groups in the polymer may be estimated. Clearly, the values of depressions calculated by equations (IV.46), (IV.47) coincide with the analogous values calculated for dimethylphthalate. Hence, description of the plasticizer influence on the glass transition temperature is quite complicated and, for adequate description, both mechanisms of plasticization must be taken into account. Let us now turn to description of copolymers. For copolymers, the relation (IV.41) is transformed to the form:     α1  ∑ ∆Vi  + α 2  ∑ ∆Vi  + ... +      i 1  i 2 → Tg =     α1 ∑ ai ∆Vi + ∑ b j  + α 2  ∑ ai ∆Vi + ∑ b j  + ... +     j j  i 1  i 2   + α n  ∑ ∆Vi     i n →   + α n  ∑ ai ∆Vi + ∑ b j    j  i n

,

(IV.49)

  where α1, α2, …, αn are the molar parts of the components 1, 2, …, n;  ∑ ∆Vi  ,    i 1      ∑ ∆Vi  , …,  ∑ ∆Vi  are the Van-der-Waals volumes of the repeat units of the      i n  i 2 components

1,

2,

…,

n;

   a ∆V + b  , ∑ ∑ i i j   j  i 1

   a ∆V + b  , ∑ ∑ i i j   j  i 2

…,

   a ∆V + b  are selections of constants for the components 1, 2, …, n ∑ ∑ i i j   j  i n (remember that α1 + α2 + … + αn = 1).

327

Correlation (IV.49) always leads to non-linear dependences of Tg on the polymer composition. If we want to express Tg of the copolymer via the glass transition of appropriate homopolymers, then using correlations (IV.41) and (IV.49), it may be written       α1 ∑ ∆Vi  + α 2  ∑ ∆Vi  + ... + α n  ∑ ∆Vi         i 1  i 2  i n Tg = ,        ∑ ∆Vi   ∑ ∆Vi   ∑ ∆Vi         i n  i 1  i 2 + α2 + ... + α n + α1 Tg,1 Tg,2 Tg, n

(IV.50)

where Tg,1, Tg,2, …, Tg,n are glass transition temperatures of homopolymers, prepared from the components 1, 2, …, n. In another form, the relation (IV.50) may be presented as follows: 1 1 1 1 = β1 + β 2 + ... + βn , Tg Tg,1 Tg,2 Tg, n

(IV.51)

where   α1 ∑ ∆Vi     i 1 ; β1 =             α1 ∑ ∆Vi + α 2 ∑ ∆Vi + ... + α n ∑ ∆Vi        i 1  i 2  i n   α1 ∑ ∆Vi     i 2 ; β2 =       α1 ∑ ∆Vi  + α 2  ∑ ∆Vi  + ... + α n  ∑ ∆Vi         i 1  i 2  i n ………………………………………………………   α1 ∑ ∆Vi     i n . βn =       α1 ∑ ∆Vi  + α 2  ∑ ∆Vi  + ... + α n  ∑ ∆Vi         i 1  i 2  i n Hence, coefficients β1, β2, …, βn by their meaning are parts of the Van-derWaals volumes of the units 1, 2, …, n. In the reduced form, relation (IV.49) is presented in the form:

328





k =1

 i

k

∑ α k  ∑ ∆Vi 

Tg =

where

k =n

  ∑ α k  ∑ ai ∆Vi + ∑ b j  k =1 j  i k

k =n

,

(IV.52)

αk is the molar part of the k-th component;

   ∑ ∆Vi     i k

and

   a ∆V + b  are the Van-der-Waals volume and the selection of constants for ∑ ∑ i i j   j  i k the k-th component, respectively; n is the total number of components in copolymer. In the reduced form, correlations (IV.50) and (IV.51) are the following: k =n

Tg =





∑ α k  ∑ ∆Vi 

 i k ,    ∑ ∆Vi    k =n  i  α ∑ k T k g, k k =1 k =1

(IV.53)

and 1 k =n 1 = ∑ βk . Tg k =1 Tg, k

(IV.54)

Dependences (IV.49)–(IV.54) of the glass transition temperature Tg of the copolymer on its composition are non-linear, and in the case of two-component copolymers they look as it is schematically represented in Figure 43. These dependences do not take into account weakening of the intermolecular interaction in copolymer units, although this weakening must cause a decrease of the glass transition temperature compared with the additive value. In practice, ‘dips’ on the dependences of Tg on composition are deeper than the ones calculated from equations (IV.49)–(IV.54) and equal ∆T = 20–30°C [78]. That is why a modified relation was suggested in the work [39], which takes this circumstance into account:

%J =

    α  ∑ ∆'  + α   ∑ ∆'  +  +       ∆' + ∑ β  + α   ∑ , ∆' + ∑ β     L

L

 α  ∑ ,   L

L

L

L

L

M

M

L

L

L

M

M

  +  +  

→ , (IV.55)

329

→

  + α Q  ∑ ∆'L   L Q

  α Q  ∑ , L ∆'L + ∑ β M  + [α ( − α ) + α  ( − α  ) +  + α Q ( − α Q )]⋅     L M  Q

where all designations are the same as in the formula (IV.51); 0.03 is an empirical coefficient which takes into account decrease of strong intermolecular interactions at copolymerization.

Figure 43. Dependences of the glass transition temperature Tg on composition α for two-component copolymers calculated from equations (IV.49) (curve 1) and (IV.54) (curve 2)

In the reduced form, equation (IV.54) may be presented in the following form:

Tg =

k =n





k =1

 i

k

∑ α k  ∑ ∆Vi 

k =n   ∑ α k  ∑ ai ∆Vi + ∑ b j  + 0.03 ∑α k (1 − α k ) k =1 k =1 j  i k

k =n

,

(IV.56)

    where αk,  ∑ ∆Vi  and  ∑ ai ∆Vi + ∑ b j  are the molar part, the Van-der-Waals     j  i k  i k volume and the selection of constants for the k-th component of the copolymer, respectively.

330

To calculate the glass transition temperatures of copolymers, equations (IV.55) and (IV.56) require no knowledge of the experimental glass transition temperatures of homopolymers. Another variant of the formula for calculation of the glass transition temperature of copolymers may be deduced by combining equations (IV.41) and (IV.55):     α1 ∑ ∆Vi  + α 2  ∑ ∆Vi  + ... +      i 1  i 2 → Tg =        ∑ ∆Vi   ∑ ∆Vi   ∑ ∆Vi         i n  i 1  i 2 +α2 + ... + α n + α1 Tg,1 Tg,2 Tg, n

(IV.57)

  + α n  ∑ ∆Vi     i n → , + [α1 (1 − α1 ) + α 2 (1 − α 2 ) + ... + α n (1 − α n )]⋅ 0.03 where Tg,1, Tg,2, …, Tg,n are the glass transition temperatures of homopolymers, composed of the components 1, 2, …, n. In the reduced form, equation (IV.57) can be written as: k =n

Tg =





∑ α k  ∑ ∆Vi 

 i k .    ∑ ∆Vi    k =n k =n  i k + 0 . 03 α ∑ k T ∑α k (1 − α k ) g, k k =1 k =1 k =1

(IV.58)

Let us now discuss the influence of sequences distribution in polymers on the glass transition temperature of them. Imagine that the following sequences exist: 1) Diades AA and BB; 2) Triades AAA and BBB; where A and B are the components of a copolymer. Let the molar parts of components A and B be 0.5:0.5. Now Tg values may be calculated for random copolymers consisting of the sequences mentioned above. For the random copolymer consisting of A and B components, we obtain:     α A  ∑ ∆Vi  + α B  ∑ ∆Vi       i A  i B Tg = , (IV.59)      ∑ ∆Vi   ∑ ∆Vi       i A  i B +αB + 0.03[α A (1 − α A ) + α B (1 − α B )] αA Tg Tg

( )A

( )B

331

where αA and αB are the molar parts of the components A and B, respectively;      ∑ ∆V  and  ∑ ∆V  are the Van-der-Waals volumes of the components A and i i    i A  i B B, respectively; Tg,A and Tg,B are the calculated glass transition temperatures for homopolymers based on the components A and B, respectively. For instance, for copolymers of ethylene (component A) and vinyl acetate   (component B), it is obtained: Tg,A = 213 K, Tg,B = 301 K;  ∑ ∆Vi  = 34.2 Å3,    i A    ∑ ∆V  = 79.65 Å3: i    i B (CH2CH2)n

(

CH2

CH

)m

O

A

C O CH3 B Then, Tg value for the random copolymer in accordance with equation (IV.59) is equal to 250 K, when αA:αB = 0.5:0.5. Let us now examine the case, when the copolymer composed of the same components is the random copolymer consisting of diades AA and BB. Since Tg,AA = Tg,A and Tg,BB = Tg,B, equation (IV.59) is reduced to the following form:     α A ⋅ 2 ∑ ∆Vi  + α B ⋅ 2 ∑ ∆Vi       i A  i B Tg = . (IV.60)     2 ∑ ∆Vi  2 ∑ ∆Vi       i A  i B + αB + 0.03[α A (1 − α A ) + α B (1 − α B )] αA Tg Tg

( )A

( )B

(It should be remembered since the molar ratio of the components A and B is 0.5:0.5, the molar ratio of the components AA and BB is the same). Substituting all the values into equation (IV.60), we obtain Tg = 259 K. Let us now discuss the case when the random copolymer consists of the triades AAA and BBB. Then, in accordance with equation (IV.59), we get:     α A ⋅ 3 ∑ ∆Vi  + α B ⋅ 3 ∑ ∆Vi       i A  i B Tg = . (IV.61)     3 ∑ ∆Vi  3 ∑ ∆Vi       i A  i B +αB + 0.03[α A (1 − α A ) + α B (1 − α B )] αA Tg Tg

( )A

( )B

332

Substituting all the parameters into equation (IV.61), we obtain that Tg = 262 K. Of course, in principle, any copolymer may be ‘constructed’ of arbitrary sequences. For example, they may be the following: AAA, ABBA, BBB, ABAB, BBBA, etc. In this case, the equation for calculating the glass transition temperature is reduced to the form:       + α ABBA  ∑ ∆Vi  + α BBB  ∑ ∆Vi  + α AAA  ∑ ∆Vi         i  AAA  i  ABBA  i  BBB → Tg =        ∑ ∆Vi   ∑ ∆Vi   ∑ ∆Vi         i  AAA  i  ABBA  i  BBB + + α ABBA + α BBB α AAA Tg Tg Tg (IV.62) AAA ABBA BBB       + α ABAB  ∑ ∆Vi  + α AAAB  ∑ ∆Vi  + α BBBA  ∑ ∆Vi         i  ABAB  i  AAAB  i  BBBA → ,        ∑ ∆Vi   ∑ ∆Vi   ∑ ∆Vi         i  ABAB  i  AAAB  i  BBBA + α AAAB + α BBBA α ABAB Tg Tg Tg

( )

( )

( )ABAB

( )AAAB

( )

( )BBBA

where αAAA, αABBA, αBBB, αABAB, αAAAB, and αBBBA are the molar parts of the   , sequences AAA, ABBA, BBB, ABAB, AAAB, and BBBA;  ∑ ∆Vi     i  AAA          ∑ ∆Vi   ∑ ∆Vi   ∑ ∆Vi   ∑ ∆Vi  , , , , and          i  BBB  i  ABBA  i  ABAB  i  AAAB    ∑ ∆Vi  are the Van-der-Waals volumes of the same sequences, respectively;    i  BBBA Tg,AAA, Tg,ABBA, Tg,BBB, Tg,ABAB, Tg,AAAB, and Tg,BBBA are the glass transition temperatures of homopolymers based on the components AAA, ABBA, BBB, ABAB, AAAB, and BBBA. Obviously, the molar parts of all the sequences cannot be arbitrary, because the molar parts of the original components A and B are given. Consequently, the way of looking at the problem of influence of the sequences distribution is the following: 1) Using the above approach, all the sequences are drawn as homopolymers and Tg values are calculated for them. 2) The properties of the copolymer containing the given parts of the sequences are calculated; it should be only remembered that original components are taken in the given proportions, and the molar parts of the sequences must correspond to the molar parts of original components. Let us now consider the influence of chain branchings on the glass transition temperature. Branchings of chains are observed quite often and are either a consequence of side processes proceeding in the course of synthesis of linear polymers, or are created purposefully for modification of properties. For example, branches may already be contained in the structure of monomers of the following type:

333

R' CH2

CH; R

CH2

C R

where R is the hydrocarbon radical: —CnH2n+1; —O—CnH2n+1; C O CnH2n +1 , etc. O In the general case, the number of possible variants of chain branchings may be quite high. Consider the main among them, schematically represented in Figure 44, which depicts the backbone and possible types of branches.

Figure 44. Different types of branchings in polymers (see text).

In the simplest (but of the lowest probability) case, branches may possess equal length and be located in each repeat unit (see Figure 44a). Another variant corresponds to branches of the same length, regularly distributed along the backbone in a definite number of units (see Figure 44b). The third type corresponds to the case, when branches of various length exist, but they are regularly distributed along the backbone in a definite number of units (see Figure 44c). Then a variant is possible when branches of different lengths are randomly distributed along the backbone (see Figure 44d). Two more types are to be considered. According to one of them, branches of different lengths are randomly distributed along the backbone, and each branch possesses one more branch of different length on it (see Figure 44e). Clearly, this general case may include all particular cases, when branches are identical and regularly distributed along the backbone, and secondary branches are identical or different, etc. One more principal case consists of a tree-like branching (see Figure 44f). In this case, occurrence of every new branch causes occurrence of one more branch. Note that at the present time, such systems have been synthesized in practice [82], and they are named ‘dendric polymers’ or ‘dendrimers’. In all cases considered, branches may be of the same chemical origin in relation to the backbone, or of a different one. The case, when branches are of a different chemical origin in relation to the backbone corresponds to grafted polymers.

334

In these systems, in most cases, a microphase separation occurs due to thermodynamic inconsistency of the backbone and the grafted chain. Therefore, each phase may possess, for example, its own glass transition temperature which, however, in many cases differs from the glass transition temperature of individual components. That is why Tg for grafted copolymers may be calculated only in the case of complete miscibility of the initial and the grafted polymer. However, the reverse task may also be solved – estimation of the composition of each phase by temperatures of relaxation transitions in it, assuming that each of the phases includes a definite number of foreign units. Work [24] analyzes the influence of branchings of different types on the physical characteristics of polymers. In homopolymers, branches may be joined directly to the backbone by substitution of one of the atoms or via a spacer of another chemical origin. Let us consider all these variants on particular examples. Let us choose polyethylene and its derivatives as one of the model polymers for this consideration: Structure I CH2

CH

(CH2

CH2)m

(CH2)n CH3 Let us consider the influence of branchings on the glass transition temperature Tg. Note that if m = 0, we get the case, depicted in Figure 44a, when branches are of the same length and distributed on each repeat unit. Let us discuss in detail calculation of this structure. The glass transition temperature is calculated from equation (IV.41). For the structure I, depicted above,

∑ ∆Vi = (51.3 + 17.1n + 34.2m)Å3; i

∑ ai ∆Vi + ∑ b j = (185.34 + 80.25n + 160m)⋅10–3 Å3K–1. i

j

Calculating values of

∑ b j , it should be taken into account that every branch j

requires introduction of a single increment bd. As a consequence, we get Tg (K ) =

51.3 + 17.1n + 34.2m ⋅103 . 185.34 + 80.25n + 160.5m

(IV.63)

The results of calculation performed in equation (IV.63), are shown in Figure 45. Clearly, if branches locate in every repeat unit (m = 0), then Tg is strongly dependent on the number of CH2–groups in the branch. Tg sharply increases as n decreases, when n < 5. For the case m = 0, a series of branched polymers exists, the properties of which have been studied experimentally. For example, when m = 0 and n = 0, polypropylene is obtained, for which the calculated value of Tg is 277 K, and the experimental value is 263 K. If m = 0 and n = 1, we get polybutene-1 with the calculated value of Tg of 258 K and the experimental one of 248 K. The same

335

convergence typical of the present method is also observed for other polymers at m = 0 and n = 2 and 6.

Figure 45. Dependences of the glass transition temperature Tg on n for structure I. Numbers at curves denote m values (see text).

If branches are less frequent (m > 0), influence of the number of units in the branching on Tg decreases and, when branches become rather rare (m = 10), the branching length does not practically affect Tg (note, by the way, that when m → ∞, this correlates with a star-like polymer). Let us now consider the case when branches in polyethylene are randomly distributed along the backbone, these branches being of different lengths. Assume that the value of m adheres to the random distribution F ( m) = 1 − e

m mavg

( m ≥ 0) ,

(IV.64)

where mavg is the average value of m. The function of density distribution is of the following form f ( m) =

1 mavg

e

− m mavg .

(IV.65)

Assume also that the polymerization degree of branches is distributed by the Flory law q(n) = γ2ne–γn,

(IV.66)

1 ; nw is the weighted mean value of n. nw For future calculations assume that the border value n = 10. Then the number of units in a branch will be determined from the correlation

where γ =

nbr =

i = nk

∑

i =1

i 2γ 2 e − iγ .

(IV.67)

336

The formula for calculation of the glass transition temperature Tg deduced on the basis of equation (IV.41) and with regard to all mentioned above, obtains the following form for structure I i = nk

51.3 + 17.1 ∑ i 2γ 2 e − iγ + 34.2 i =1 i = nk

Tg (K ) =

185.3 + 80.25

1 mavg

∑

j 2 β 2 e − jβ

j =1 j = mk

∑ i 2γ 2e −iγ + 160.5 ∑

i =1

where β =

j = mk

⋅103 ,

(IV.68)

jβe − jβ

j =1

.

Results of the calculation performed by formula (IV.68) are indicated in Figure 46. The character of Tg dependence on nw is analogous to Tg dependence on n with the only difference that the former are more smooth. Generally, results of the calculations performed [24] for branched polyethylene indicate that the greatest influence on the glass transition temperature is cause by short-chain branchings, frequently located along the backbone.

Figure 46. Dependences of Tg on nw for structure I. Numbers at curves denote mavg values (see text).

Let us now turn to the next type of branchings when new branches appear on the original branch. In this case, the structural formula of branched polyethylene is the following Structure II CH2

CH

(CH2

CH2)n

(CH2)x CH

(CH2)y CH3

(CH2)z CH3 This correlates with the scheme in Figure 44e. In this case, the formula for calculation of Tg, deduced from equation (IV.41), obtains the following form:

337

7J . =

 +  [ + \ + ] + P ⋅  .  +  [ + \ + ] + P − 

(IV.69)

The meaning of designations x, y, z, and m is seen from the structural formula of branched polyethylene. Note that if x + y + z = 0 and m = 0, this corresponds to a polymer of the following structure ... CH2 CH ... CH CH3 CH3 For this polymer, the calculated glass transition temperature is 294 K, and experimental 302 K. Analysis of equation (IV.69) indicates that Tg depends only on the sum (x + y + z) and the value m. Figure 47a shows dependences of Tg on (x + y + z) at various values of m, i.e. at various frequencies of branches, determined by the value 1/m. Here the influence of branches is approximately the same as in the previous cases. However, the appearance of secondary branches causes a sharper increase of the glass transition temperature, which is most clearly displayed at short and frequent (m = 0) branchings. Let us consider tree-like branchings. In the case of polyethylene with a trifunctional cross-linked point, the chemical structure of such system is shown below: Structure III ... ... CH2 CH ... (CH2 )n (CH2)n

CH

(CH2 )n ...

CH .. .

(CH2)n

CH2 CH

(CH2)n

(CH2 )n

CH .. .

CH

(CH2)n ... (CH2)n CH ... .. . CH

(CH2 )n ...

When determining

∑b j ,

it should be taken into account that every new

j

branch causes a necessity of introducing the increment bd, and its influence becomes most noticeable when the shortest branches exist. If these branches are of the same length (n =const), the formula for calculation of the glass transition temperature, deduced from equation (IV.31), obtains the following form: Tg (K ) =

17.1n + 11.0 ⋅103 . 80.25n − 15.25

(IV.70)

The dependence of Tg on n, calculated by equation (IV.70), is indicated in Figure 47b (curve 1). It is clearly observed that if tree-like branchings exist, Tg of the system changes must more significantly than in all previous cases. When branches become shortest (n = 1), Tg reaches 160°C. This is much higher than Tg of the original polyethylene. The reason for such a sharp increase of Tg is that at short branchings the

338

chemical structure of the polymer significantly differs from the chemical structure of polyethylene, the role of dipole–dipole interaction increasing. Consider another variant, when branches are distributed by lengths according to the Flory law. In this case, the formula for calculation of Tg deduced from (IV.41) obtains the following form: i = nk

Tg (K ) =

17.1 ∑ i 2γ 2e − iγ + 110 i =1 i = nk

80.25

∑i

2 2 − iγ

γ e

⋅ 103 ,

(IV.71)

− 15.25

i =1

and the results of calculations performed by equation (IV.71) are shown in Figure 47b (curve 2). It may be noted that when branches are distributed by lengths, dependence of Tg on nw is weaker than with branches of the same length. Clearly, comparison is set at n = nw . Let us now discuss branchings of the same chemical origin as the backbone, but joined to it by untying of another chemical structure. As an example, let us consider the structure of the branched polymer, displayed below: Structure IV ... CH2 CH (CH2 CH2)m ... O CH2 (CH2 )n CH3 A particular case of this system (m = 0) represents a selection of vinyl ethers with different values of n. For this case, the formula for calculation of Tg is of the following form Tg (K ) =

78.1 + 17.1n ⋅103 289.4 + 80.25n

(IV.72)

Figure 47. Dependences of Tg on (x + y + z) for structure II (a) and dependences of Tg on n (1) and nw (2) for structure III (b). Numbers at curves denote m values (see text) .

339

and calculations by equation (IV.72) lead to the dependence of Tg on n depicted in Figure 48. This dependence is analogous to the above-described ones. Therewith, concordance of the calculated and experimental values of Tg usual for the present method is also observed in this case.

Figure 48. Dependence of Tg on n for structure IV.

If branches are distributed by lengths and randomly located along the backbone, the formula for calculation of Tg transforms to: i = nk

78.1 + 17.1 ∑ i 2γ 2 e − iγ + 34.2 Tg (K ) =

i =1 i = nk

289.4 + 80.25

∑

i =1

2 2 − iγ

i γ e

j = mk

∑

jβe − jβ

j =1 j = mk

+ 160.5

∑

⋅ 103 . jβe

(IV.73)

− jβ

j =1

In the general case, the dependence of Tg on the average number of CH2– groups in branches at various values of m looks as depicted in Figure 49. The character of these dependences is also analogous to that considered above for branched polyethylene.

Figure 49. Dependences of Tg on nw for structure IV. Numbers at curves denote m values (see text).

340

In conclusion, let us consider a series of organic glasses based on polymethacrylates. The general formula of these systems is of the following form: Structure V CH3 ...

CH2

C

...

C O CH2

(CH2)n CH3

O The formula for calculation of Tg for polymethacrylates, deduced from equation (IV.41), is of the form Tg (K ) =

113.85 + 17.1n ⋅103 . 355.0 + 80.25n

The dependence of Tg on the length of side branch n is shown in Figure 50. As the branch length increases, the glass transition temperature abruptly decreases and transition from the glassy state to the rubbery state is observed.

Figure 50. Dependences of Tg on n for structure V.

In summary, several conclusions on the influence of branchings on the glass transition temperature Tg can be made. The first conclusion is that as the length of branches increases, the properties of polymers change insignificantly and approach the properties of homopolymers of the chemical structure identical to branches. This is true in the case when branches located in each repeat unit. If the branches are less frequent and distributed along the backbone of the polymer is a random way, their influence on properties is more significant, if their chemical nature differs from that of the backbone. The highest influence on the physical properties is exerted by short branchings. For the glass transition temperature, the influence of short branchings is formally taken into account by introduction of constants bd to the formula (IV.41) in the case of every new branching. This influence is associated with increase of chain rigidity and occurrence of an additional intermolecular interaction, when a branch containing a polar group is joined. It should also be noted that existence of a great number of short-chain branches brings about a qualitative change of the chemical structure of the polymer. If short-chain branchings are of the tree-like form, the

341

chemical structure of the system obtained is just roughly similar to that of the initial polymer. In conclusion of this Section, let us consider a possibility of estimating energies of the intermolecular interaction and its components, stipulated by various types of intermolecular interaction. Basing on the approach developed in refs. [6, 128], we obtain that the total energy of intermolecular interaction is described by the following correlation:

D =

R ∑ ∆Vi i

  0.455 ∑ ai ∆Vi + ∑ b j    j  i 

m,

(IV.74)

where m is the number of atoms in the polymer repeat unit; R is the universal gas constant; the meaning of the rest of parameters is the same as in equation (IV.41). For copolymers, we obtain: k =n   R ∑ α k  ∑ ∆Vi    k =n k =1  i k D = ∑ α k mk , k =n   k =1 0.455 ∑ α k  ∑ ai ∆Vi + ∑ b j    k =1 j  i k

(IV.75)

where mk is the number of atoms in the k-th repeat unit. Substituting equation (IV.41) into formula (IV.75), we obtain: D =

RTg 0.455

m,

(IV.76)

and for copolymers, D =

RTg, cop k = n ∑ α k mk , 0.455 k =1

(IV.77)

where Tg,cop is the glass transition temperature of the copolymer. Energy of dispersion interaction d is determined from the following expression:

D d=

R ∑ ∆Vi i

  0.455 ∑ ai ∆Vi     i 

For a statistic copolymer, we obtain:

m.

(IV.78)

342

k =n   R ∑ α k  ∑ ∆Vi    k =n k =1  i k Dd= ∑ α k mk . k =n   k =1   0.455 ∑ α k ∑ ai ∆Vi   k =1  i k

(IV.79)

The energy of the dipole–dipole and hydrogen bonds is determined from the formula:  mR∑ ∆Vi    1 1 i − D dd + h = D − D d = .  0.455  ∑ ai ∆Vi + ∑ b j ∑ ai ∆Vi  j i   i

(IV.80)

The ratio of the dispersion interaction in the total intermolecular interaction equals:

∑b j D d j = 1+ . D a ∑ i ∆Vi

(IV.81)

i

For statistic copolymers, we obtain: k =n





∑α k  ∑ b j 

D d k =1  j k = 1+ . k =n  D  ∑α k  ∑ ai ∆Vi  k =1  i k

(IV.82)

The ratio of dipole–dipole and H-bonding interaction in total intermolecular interaction may be determined from the following expression:

∑ ai ∆Vi + ∑ b j

D dd + h Dd i = 1− =1− D D

j

∑ ai ∆Vi i

∑b j =−

j

∑ ai ∆Vi

,

(IV.83)

i

which for statistic polymers is reduced to the form: k =n

( )

∑α k b j k D dd + h k = − =1 . k =n D ∑α k (ai ∆V )k k =1

(IV.84)

343

Hence, using the above-described approach, not only the glass transition temperature of polymers and copolymers may be estimated, but also such important characteristics as total energy of intermolecular interaction and its components, stipulated by various types of the intermolecular interaction (weak dispersion interaction, strong dipole– dipole interaction, and hydrogen bonds), may be determined. Running ahead, let us note that such estimation is also correct for polymer networks, which are analyzed below.

IV.5. Calculation of the glass transition temperature of polymer networks It is experimentally determined that the glass transition temperature, as well as many other properties of network systems, depends on the number of the repeat units between cross-linked points m in the way as schematically represented in Figure 51. If the network is not dense, its glass transition temperature depends only slightly on m, but when the number of the repeating units in chains between neighboring crosslinked points decreases significantly, the glass transition temperature increases sharply and reaches extremely high values. Tests and calculations indicate that the glass transition temperature begins increasing, when the number of units in linear fragments binding cross-linked points becomes smaller than 4–5. This is the vary value of m, at which it may be said about the transition from rarefied to frequent networks.

Figure 51. Schematic representation of dependence of the glass transition temperature Tg of networks on the number of repeat units m in linear fragments between cross-linked points.

Let us consider two border cases: 1) Network is extremely sparse; this correlates with the case of m → 0. In this case, the properties of networks do not practically differ from the properties of a linear polymer of the same chemical structure. Consequently, the glass transition temperature Tg may be calculated from equation (IV.41). 2) Network is extremely frequent and consists of cross-linked points only (the notion of the ‘network cross-linked point’ will be discussed in detail below). In this border case, the network may soften at heating only at the sacrifice of its thermal degradation by chemical bonds in cross-linked points. The softening temperature of this network, or more precisely the temperature of onset of intensive thermal degradation, may be calculated from equation (VII.5).

344

In most cases, real networks respond to the intermediate variant, because they contain quite long but finite linear fragments between cross-linked points. Calculation of the influence of both these linear fragments and network cross-linked points on the glass transition temperature has led to the following equation, which represents the generalized form of equations (IV.41) and (VII.5).

7J =

   ∑ ∆9    UI ,    + ∑ E  +  ∑ . ∆9    FUOS  OFK  L

L

  ∑ D ∆9   L

L

L

M

M

L

(IV.85)

L

L

  where  ∑ ∆9  is the Van-der-Waals volume of the repeating fragment of the   UI   is a selection of constants for linear chains network;  ∑ ai ∆Vi + ∑ b j    i j   l.ch. L

L

  participating in the repeating fragment of the network;  ∑ . ∆9  is a selection   FUOS of constants for the network cross-linked point. This equation takes into account both border cases of extremely rarefied and extremely frequent networks, and intermediate cases. Actually, if a network consists of linear fragments only, influence of cross-linked points may be neglected and    ∑ . ∆9  → 0; equation (IV.85) transforms into (IV.41). On the contrary, in a   FUOS network is the most frequent, i.e. consists of cross-linked points only, equation (IV.75) transits into (VII.5). In all intermediate cases, both summands from the denominator of equation (IV.85) act. L

L

L

L

L

L

Figure 52. Schematic representation of networks of different types: (a) network consisting of fragments of polymer chains of the same chemical structure and possessing tetra-functional cross-linked point; (b) network consisting of fragments of polymer chains of the same chemical structure but possessing tri-functional cross-linked point; (c) network consisting of linear chains of the same chemical structure and cross-links of different chemical structure possessing tri-functional cross-linked point.

Let us consider now two the most important questions. Question one is connected with estimation of the Van-der-Waals volume of the repeating fragment of

345

the network. Let us analyze several variants of networks, schematically represented in Figure 52. The network depicted in Figure 52a possesses a tetra-functional crosslinked point (four chains yield from each cross-linked point, all chains possessing the same chemical structure). For this network, the Van-der-Waals volume of the repeating fragment of the network is calculated as *

         ∑ ∆Vi  = 2(m − 2) ∑ ∆Vi  + 4 ∑ ∆Vi  +  ∑ ∆Vi  ,          i  r.f.  i l.ch.  i l.ch.  i  crl.p.

(IV.86)

  where  ∑ ∆Vi  is the Van-der-Waals volume of the repeat unit in linear chains    i  l.ch.   is the Van-der-Waals volume of the connecting cross-linked points;  ∑ ∆Vi     i  crl.p. *

  network cross-linked point;  ∑ ∆Vi  is the Van-der-Waals volume of the unit of    i l.ch. the linear fragment, linked to the cross-linked point (this unit is of somewhat different chemical structure compared with ‘normal’ units). Another variant reflects the case when the network possesses a tri-functional cross-linked point which links linear chains of the same chemical structure (see Figure 52b). For this network, the formula for calculation of the Van-der-Waals volume of the repeating fragment obtains the form: *

         ∑ ∆Vi  = (m − 2) 3  ∑ ∆Vi  + 3 ∑ ∆Vi  +  ∑ ∆Vi  ,         2 i  i  r.f. l.ch.  i  l.ch.  i crl.p.

(IV.87)

where designations are the same as in equation (IV.86). One more type of the network is depicted in Figure 52c. This network is composed of cross-linked chains of the same chemical nature and cross-linking bridges of different chemical nature. In this case, the formula for calculation of the Van-der-Waals volume of the repeating fragment of the network obtains the form: *

       ∑ ∆Vi  = (m1 − 2) ∑ ∆Vi  + 2 ∑ ∆Vi  +→        i  r.f.  i  l.ch.,1  i  l.ch.,1 *

     m − 2  →+ 2 ∆Vi  +  ∑ ∆Vi  +  ∑ ∆Vi  ∑      2  i l.ch.,2  i  l.ch.,2  i  crl.p.

,

(IV.88)

346

  where  ∑ ∆Vi  is the Van-der-Waals volume of the repeat unit of linear    i  l.ch.,1 fragments of cross-linked chains, m1 being the average amount of these units, located *

  is the same for the unit between cross-linked points of the network;  ∑ ∆Vi     i  l.ch.,1 linked to cross-linked point (possessing different chemical structure compared with   is the Van-der-Waals volume of the repeat unit of the ‘normal’ units);  ∑ ∆Vi     i  l.ch.,2 linear fragment of cross-linking bridges, m2 being the amount of these units; *

   ∑ ∆Vi  is the same for the unit linked to the cross-linked point.    i  l.ch.,2 Let us now discuss the notion of ‘cross-linked point of the network’. To formulate this notion, equation (IV.85) was calibrated in work [30] basing on the experimental data on the glass transition temperatures for numerous and wellcharacterized network systems. Consequently, it was obtained that the following definition of the cross-linked point must be assumed for better coincidence of calculated and experimental data: the cross-linked point of a network is a group of atoms which includes the atom, from which branching of chains takes place, and neighboring atoms, chemically bonded with it, with their closest substituents. Shown below are networks of different chemical nature, in which cross-linked points, according to this definition, are marked by the dotted line. Basing on the present definition of the cross-linked point and the examples displayed, it is quite easy to identify the cross-linked point in a network of any chemical nature. .. . CH2 CH2 ...

CH2

CH2

CH CH2

CH2

... ...

CH2

CH2

C CH2

CH2

CH2

CH2

CH2

...

...

...

CH2

CH CH2

CH CH2

CH CH2

CH

...

...

CH2

CH CH2

CH CH2

CH CH2

CH

...

CH2

...

347

OH CH2

... N

CH2

...

OH

CH CH2

O

O CH2

CH CH2

...

CH CH2

O

O CH2

CH CH2

...

N CH2

OH

OH

... CH3 Si

CH3

O

CH3 ...

Si

...

O

CH3 CH3

Si O Si

O

CH3 O

Si

...

CH2

CH3

CH2

...

CH3 CH3

O .. .

.. .

Analyzed below are various variants of networks and examples of calculations of their glass transition temperatures Tg are given. Let us now dwell on estimation of value of the molecular mass of the averaged fragment between neighboring crosslinked points. This estimation may be performed on the basis of the chemical structure of the network and the experimentally determined glass transition temperature. For this purpose, the experimental value of Tg must be substituted to equation (IV.85), and the equation must be solved by m. Let us perform this analysis in general for various types of networks depicted in Figure 52. Let consider first the network depicted in Figure 52a; four chains yield from each cross-linked points of this network. Therewith, all linear fragments are of the same chemical structure of the repeat unit. Then, considering the repeating fragment of the network, it should be written on the basis of equations (IV.85) and (IV.86): *

Tg =

      + 4 ∑ ∆Vi  +  ∑ ∆Vi  2(m − 2) ∑ ∆Vi         i  l.ch.  i  l.ch.  i  crl.p.

, (IV.89)

*

      + 4 ∑ ai ∆Vi + ∑ b j  +  ∑ K i ∆Vi  2(m − 2) ∑ ai ∆Vi + ∑ b j        j j  crl.p.  i  l.ch.  i  l.ch.  i

  is a selection of constants for the repeat unit of linear where  ∑ ai ∆Vi + ∑ b j    j  i  l.ch. *

  intercross-linked chains;  ∑ ai ∆Vi + ∑ b j  is the same for border chains linked   j  i  l.ch.

348

  to cross-linked points;  ∑ K i ∆Vi  is a selection of constants for the network    i  crl.p. cross-linked point; other designations are the same as in formula (IV.86). The molecular mass of linear chains between cross-linked points Mc may be determined as M c = ( m − 2) M + 2M * + 1 M crl.p. ,

(IV.90)

2

where M is the molecular mass of the ‘normal’ unit; M* is the molecular mass of the unit linked to the cross-linked point; Mcrl.p. is the molecular mass of the cross-linked point. Denote:    ∑ ∆Vi  = A;    i  l.ch.

  + 2 ∑ ∆Vi     i  l.ch.



1 2

  =C; = B ;  ∑ ai ∆Vi + ∑ b j    i j  crl.p.   l.ch. 

∑ ∆Vi 

 i *

  + 2 ∑ ai ∆Vi + ∑ b j    i j   l.ch.



1 2



∑ K i ∆Vi 

 i

 crl.p.

= D.

(IV.91)

Then, after some transformations, equations (IV.89) and (IV.90) obtain the form: m−2= Mc =

B − Tg D Tg C − A

B − Tg D

;

Tg C − A

(IV.92)

M + 2M * + 1 M crl.p. .

(IV.93)

2

Let us now analyze a network with tri-functional cross-linked points with the same chemical structure of chain units yielding from the cross-linked point (see Figure 52b). For the repeating fragment of this network, it should be written:  P −   ∑ ∆9 

L

7J =

L

 P −   ∑ D ∆9 + ∑ E   L

L

L

M

M

   +  ∑ ∆9  OFK 

L

L

   +  ∑ ∆9  OFK 

L

L

   +  ∑ D ∆9 + ∑ E    OFK  L

L

L

M

   FKOS

, (IV.94)

M

   +  ∑ . ∆9   OFK  L

L

L

   FKOS

where all designations are the same as in equation (IV.89). Then

 − %J  ;  %J − 

2−  = 

(IV.95)

349

where *

*           ; D1 =  ∑ ai ∆Vi + ∑ b j  . (IV.96) B1 = ∑ ∆Vi +  ∑ ∆Vi  +  ∑ K i ∆Vi          i j i  i  l.ch.  i  crl.p.     l.ch. crl.p.

Therefrom, B1 − Tg D1 M + 2 M * + 2 M crl.p. . M c = ( m − 2) M + 2M * + 2 M crl.p. = 2 3 3 T C−A 3 g

(IV.97)

The third variant of the network depicted in Figure 52c consists of linear chains, cross-linked by chains of a different chemical structure; the cross-linked point of the network is tri-functional. For this network, the dependence of Tg on the composition is of the following form: *

Tg =

      (m1 − 2) ∑ ∆Vi  + + 2 ∑ ∆Vi  + 2  ∑ ∆Vi  3       i  l.ch.,1  i  l.ch.,1  i  crl.p. *

      (m2 − 2) ∑ ai ∆Vi + ∑ b j  + 2 ∑ ai ∆Vi + ∑ b j  +  ∑ K i ∆Vi  +       j j  crl.p.  i  l.ch.,1  i  l.ch.,1  i

→

 m − 2  + 2 ∆Vi  ∑   2  i 

→

, (IV.98)

*

    +  ∑ ∆Vi  + 1  ∑ ∆Vi  3     l.ch.,2  i  crl.p. l.ch.,2  i *

   m − 2  ai ∆Vi + ∑ b j  + 2 +  ∑ ai ∆Vi + ∑ b j  ∑    2  i j j   l.ch.,2  i  l.ch.,2

where m1 and m2 are the numbers of ‘normal’ units of cross-linked chains and crosslinking bridges between neighboring cross-linked points. All parameters with index * relate to chains linked to the cross-linked points. The relation between m1 and m2 is detected from the molar ratio of components 1 and 2: m1/m2 = q1/q2, where q1 and q2 are numbers of moles of components 1 and 2, respectively. Substituting this relation into equation (IV.98), we get

m1 − 2 =

B2 − 2 A1 − A2 − Tg ( D2 − 2C − C2 ) −   q Tg  C1 + 1 C2  − q 2   2

    q q −2Tg  C1 − 1 C2  + 2  A1 + 1 A2  2 q2  2 q2    →   q1 A2  −  A1 + 2q2  

→

,

(IV.99)

350

where     A1 =  ∑ ∆Vi  ; A2 =  ∑ ∆Vi  ;      i  l.ch.,1  i l.ch.,2





        =  ∑ ∆'  +  ∑ ∆'  +  ∑ ∆'  +  ∑ ∆'  ;   OFK    FUOS   OFK    FUOS L

L

L

L

L

L

L

L

    C1 =  ∑ ai ∆Vi + ∑ b j  ; C2 =  ∑ ai ∆Vi + ∑ b j  ;     i j i j   l.ch.,1   l.ch.,2 *

*

      +  ∑ K i ∆Vi  +  ∑ ai ∆Vi + ∑ b j  D2 = 2 ∑ ai ∆Vi + ∑ b j  .       j j  crl.p.  i  i  l.ch.,1  i  l.ch.,2 Let us introduce designations:   q B* = B2 − 2 A1 − A2 + 2 A1 + 1 A2  ; q2   q q D* = D 2 −C 2 + 1 C2 ; C* = C1 + 1 C 2 . q2 q2 Then we obtain B * −Tg D * . m1 − 2 = Tg C * − A * q A* = C1 + 1 A2 ; q2

(IV.100)

(IV.101)

In a network composed of heterogeneous chain fragments, located between cross-linked points, it is advisable to determine two values of Mc (Mc,1 which is the molecular mass of the chain fragment between cross-linked points for cross-linked chains, and Mc,2 which is the same for cross-linking bridges). Therewith, Mc,1 = (m1 – 2)M1 + 2M1* + Mcrl.p.,1; q  Mc,2 = (m2 – 2)M2 + 2M2* + Mcrl.p.,2 =  2 m1 − 2  + 2M2* + Mcrl.p.,2,  q1 

(IV.102) (IV.103)

where Mcrl.p.,1 is the molecular mass of the part of the cross-linked point adhered to cross-linked chains (black circles in Figure 52c); Mcrl.p.,2 is the molecular mass of the part of the cross-linked point adhered to cross-linking bridges (white circles in Figure 52c). Therewith, the value Mcrl.p.,1 + Mcrl.p.,2 is equal to the molecular mass of the whole cross-linked point, Mcrl.p.. Then, considering (IV.101), expressions (IV.102) and (IV.103) become of the following form M c,1 = q M c,2 = 1 q2

B * −Tg D * Tg C * − A *

M 1 + 2 M 1* + M crl.p.,1;

(

)

 B * −Tg D *    M + 2 M 2* − M 2 + M crl.p.,2 .  Tg C * − A *  2  

(IV.104)

(IV.105)

351

The average value of molecular mass Mavg of the chain between network cross-linked points may be calculated from the equation Mavg = αMc,1 + (1 – α)Mc,2,

(IV.106)

where α = q1/(q1 + q2). Let us now discuss in detail the procedure of calculating Tg and Mc for all three variants. The first of them correlated with a network with a tetra-functional cross-linked point and chains of the same chemical structure yielding from it. As the simplest example, let us analyze the hypothetical network based on polyethylene

Structure VI ... ...

(CH2)m- 2

... ...

C

CH2

(CH2)m -2

CH2

...

C

CH2

(CH2)m-2

CH2

(CH2 )m- 2 (CH2)m -2

CH2

C

CH2

...

.. .

CH2 CH2

C ...

CH2

... C

...

CH2

.. .

...

...

C

(CH2)m -2

CH2

C

...

.. .

CH2 .. .

For this network, the Van-der-Waals volume of the repeating fragment will be determined from the relation        ∑ ∆Vi  =  ∑ ∆Vi  + 2 ∑ ∆Vi  ( m − 2) ,        i  r.f.  i  crl.p.  i  v.

(IV.107)

  where  ∑ ∆Vi  is the Van-der-Waals volume of the cross-linked point (marked    i  crl.p.   by dotted lines);  ∑ ∆Vi  is the Van-der   i  v. Therewith∗,


   Š -group.

∗

Numbers of atoms correspond to those given in Table 3.

2

352

   ∑ ∆Vi  = ∆VC,1 + 4∆VC,10 + 8∆VH,124 = 5.0 + 4⋅13.1 + 8⋅2.0 = 73.4 Å3;    i  crl.p.    ∑ ∆Vi  = ∆VC,10 + 2∆VH,124 = 13.1 + 2⋅2.0 = 17.1 Å3.    i  v. Then    ∑ ∆Vi  = 73.4 + 2⋅17.1(m – 2) = 50 + 34.2m.    i  crl.p.   Value  ∑ ai ∆Vi + ∑ b j  = 2(aC∆VC,10 + aH2∆VH,124)(m – 2) = 2(0.021⋅13.1 +   i j   l.ch. 19.98⋅2⋅2.0)(m – 2)⋅10–3 = (160.39m – 320.78)⋅10–3 Å3K–1.   = KC(∆VC,1 + 4∆VC,10) + KH⋅8⋅∆VH,124 = [1.15(5.0 + 4⋅13.1) + Value  ∑ K i ∆Vi     i  crl.p. 2.307⋅8⋅2.0]⋅10–3 = 102.92⋅10–3 Å3K–1. Substituting these values to equation (IV.89), we get Tg (K ) =

73.4 ⋅103 . 160.39m − 217.86

(IV.108)

Relation (IV.108) is true, when m ≥ 2. The dependence of Tg on m calculated from this relation is depicted in Figure 53.

Figure 53. Dependence of Tg on m for the model polyethylene network with tetra-functional crosslinked point

353

If m = 1 or m = 0, the situation changes. When m = 1, the structure of the network VI looks as follows .. . ... ...

...

C

.. .

.. .

CH2 C

C CH2 ...

CH2

...

...

CH2

...

C

...

C .. . This network consists of cross-linked pointV RQO\ HDFK &+2-group being adhered to two neighboring cross-linked points. Then      ∑ ∆Vi  =  ∑ ∆Vi  = ∆VC,1 + 4⁄2∆VC,10 + 8⁄2∆VH,124 = 5.0 + 2⋅13.1 + 4⋅2.0 =      i  r.f.  i  crl.p.

Š

= 39.2 Å3;    ∑ K i ∆Vi  = KC⋅(∆VC,1 + 4⁄2∆VC,10) + KH⋅8⁄2∆VH,124 = [1.15(5.10 + 2⋅13.1) +    i  crl.p. 2.307⋅4⋅2.0]⋅10–3 = 54.35⋅10–3 Å3K–1. Since the present network contains no linear fragments, and for it    a ∆V + b  = 0, softening of it may occur only as a result of its thermal i i j ∑ ∑   j  i  l.ch. degradation, i.e. degradation by chemical bonds. Substitution of all parameters into equation (IV.89) gives Tg = Td =

39.2 ⋅103 = 721 K . 54.34

If m = 0, the structure of the network VI obtains the form .. .

... ...

...

C

...

... ...

C

...

C C ... ...

C

...

.. .

C

C

.. .

.. .

...

C

...

...

C

...

.. . i.e. possesses the ideal diamond structure, for which

...

354

   ∑ ∆Vi  = ∆VC,1 = 5.0 Å3;    i  r.f.

   a ∆V + b  = 0; ∑ ∑ i i j   i j   l.ch.

   ∑ . ∆9  = KC⋅∆VC,1 = 1.15⋅10−3 Å3K–1.   FUOS L

L

L

Substitution of these values into equation (IV.89) gives Tg = Td =

5.0 ⋅103 = 870 K . 5.75

Hence, the calculated temperature of onset of intense thermal degradation of diamond is ~600°C. It is known from literature that diamond is burnt off completely when heated up to 850–1000°C. Let us now consider behavior of the network Structure VII ...

(CH2)m -2

CH2

CH CH2

.. .

(CH2)m -2

CH2

CH

...

(CH2)m -2

CH2

CH

...

CH2 (CH2 )m- 2 CH2 ...

(CH2)m -2

CH2

CH CH2

... which possesses a tri-functional cross-linked point, marked by dotted lines. For this network,    ∑ ∆9  = ∆VC,6 + ∆VC,10 + 7∆VH,124 = 9.0 + 3⋅13.1 + 7⋅2.0 = 62.3 Å3;   FUOS L

L

   ∑ ∆Vi  = (∆VC,10 + 2∆VH,124)(m – 2) = (13.1 + 2⋅2.0)(m – 2) =    i  l.ch. = (17.1m – 34.2) Å3;        ∑ ∆Vi  =  ∑ ∆Vi  + 1.5  ∑ ∆Vi  = 62.3 + 1.5(17.1m – 34.2) =        i  r.f.  i  crl.p.  i  l.ch. = (26.65m + 11) Å3;      a ∆V + b  +  ∑ K i ∆Vi  = 1.5[(aC∆VC,10 + 2aH∆VH,124)(m – 2)l.ch. + ∑ ∑ i i j     i j i     l.ch. crl.p. + [KC(∆VC,6 + ∆VC,10) + 7KH∆VH,124]crl.p. = {[1.5(0.021⋅13.1 + 2⋅19.98⋅2.0)×

355

×(m – 2)] + 1.15(9.0 + 3⋅13.1) + 7⋅2.307⋅2.0}⋅10−3 = (120.3m – 152.76) 10−3 Å3K−1. Substituting all obtained above values into equation (IV.94), we get Tg (K ) =

25.65m ⋅103 . 120.3m − 152.76

(IV.109)

The dependence of Tg on m is shown in Figure 54. It is clearly seen that Tg abruptly decreases as m increases, i.e. as the size of linear fragments between neighboring cross-linked points increases.

Figure 54. Dependence of Tg on m for the model polyethylene network with tri-functional cross-linked point.

Equation (IV.109) is true at m ≥ 2. If m = 1, we obtain the network of the following structure: .. .. . . ...

CH

CH2

CH CH2 CH

...

CH2 CH CH2 CH CH2 CH ... .. .. . . This network consists only of cross-linked points, which are its repeating fragments; these fragments−cross-linked points are marked by dotted lines. For this network ...

     ∑ ∆Vi  =  ∑ ∆Vi  = ∆VC,6 +1.5∆VC,10 + 4∆VH,124 =      i  r.f.  i  crl.p. = 9.0 + 1.5⋅13.1 + 4⋅2.0 = 36.65 Å3;

356

   a ∆V + b  = 0; ∑ ∑ i i j   i j   l.ch.    ∑ K i ∆Vi  = KC(∆VC,6 +1.5∆VC,10) + KH⋅4⋅∆VH,124 =    i  crl.p. = [1.15(9.0 + 1.5⋅13.1) + 2.307⋅4⋅2.0]⋅10−3 = 53.41⋅10−3 Å3K. Substituting the values obtained into equation (IV.94), we get Tg = Td =

36.65 3 ⋅ 10 = 686 K . 53.41

Finally, let us consider structure VII of the network under the condition m = 0. This network displays the form: ... ... ...

CH CH CH

...

...

CH CH ...

...

CH ...

Here the branching atom is every carbon atom, so it should be written down that    ∑ ∆Vi  = ∆VC,6 + ∆VH,124 = 9.0 + 2.0 = 11.0 Å3;    i  crl.p.    a ∆V + b  = 0; ∑ ∑ i i j   i j   l.ch.      ∑ K i ∆Vi  =  ∑ K i ∆Vi  = KC⋅∆VC,6 + KH⋅∆VH,124 =      i  r.f.  i  crl.p. = (1.15⋅9.0 + 2.307⋅3⋅2.0]⋅10−3 = 14.964⋅10−3 Å3K−1. Substituting these values of network parameters into equation (IV.94), we obtain Tg = Td =

11.0 ⋅103 = 735 K . 14.964

Let us now discuss the network of the third type, in which cross-liked chains are of the same chemical structure, and cross-linking bridges are of a different one. As an example, let us analyze the real network based on epoxy resin of the following chemical structure [30]

357

Structure VIII OH ...

CH2

OH

CH CH2

O

O

CH2

CH CH2

O

O m

OH

OH ...

CH2

CH CH2 OH

O

O

CH2

CH CH2 OH

O

O CH2

CH CH2

CH2

N CH2

CH CH2

m

CH2

...

N

...

The stoichiometric composition of the network conforms to 2 moles of epoxy oligomer per mole of curing agent, diamine. The cross-linked point of the network is marked by dotted lines. For the repeating fragment of the network, we get:    ∑ ∆Vi  = (∆VN,144 + 2∆VC,56 + ∆VC,21 + 4∆VH,124) + [2∆VC,39 + 2∆VC,40 + 2∆VC,130    i  r.f. + 2∆VC,132 + 2∆VH,125 + 6∆VH,124 + 2∆VC,20 + 4∆VC,18 + ∆VH,124 + (2∆VO,130 + 2∆VC,40 + ∆VC,39 + ∆VC,132 + 5∆VH,124 + ∆VH,125 + 2∆VC,20 + 4∆VC,18 +4∆VH,124)m]l.ch.,1 + (4∆VC,18 + ∆VC,19 + 4∆VH,124 + 1/2∆VC,12 + 1/2⋅2⋅∆VH,124)l.ch.,2 = (0.9 + 2⋅14.6 + 10.2 + 4⋅2.0) + [2⋅12.2 + 2⋅16.2 + 2⋅2.7 + 2⋅5.6 + 2⋅4.7 + 6⋅2.0 + 2⋅11.6 + 4⋅12.7 + 4⋅2.0) + (2⋅2.7 + 2⋅16.2 + 12.2 + 5.6 + 5⋅2.0 + 4.7 + 2⋅11.6 + 4⋅12.7 + 4⋅2.0)m] + (4⋅12.7 + 8.4 + 4⋅2.0 + 1/2⋅12.6 + 1/2⋅2⋅2.0) = (300.6 + 152.3)m Å3;    a ∆V + b  = aC(2∆VC,39 + 2∆VC,40 + 2∆VC,20 + 4∆VC,18) + aH(2∆VH,125 + ∑ i i ∑ j  i j   l.ch.,1 ∆VH,124 + 4∆VH,124) + aO,0(2∆VO,130) + aO,δ(2∆VO,132) + [aC(2∆VC,40 + ∆VC,39 + 2∆VC,20 + 4∆VC,18) + aH(5∆VH,124 + ∆VH,125 + 4∆VH,124) + aO,0(2∆VO,130) + aO,s( ∆VO,132)]m + bd⋅2 + bh⋅2 + bm + (bd + bh + bm)m = {0.021(2⋅12.2 + 2⋅16.2 + 2⋅11.6 + 4⋅12.7) + 19.98(2⋅4.7 + 6⋅2.0 + 4⋅2.0) + 22.95⋅2⋅2.7 + 16.0⋅2⋅5.6 + [0.021(2⋅16.2 + 12.2 + 2⋅11.6 + 4⋅12.7) + 19.98(5⋅2.0 + 4.7 + 4⋅2.0) + 22.95⋅2⋅2.7 + 16.0⋅5.6]m + 2(–55.4) + 2(– 139.6) + 16.0 + (–55.4 – 139.6 + 16.0)m}⋅10–3 = (519.3 + 490.6m)⋅10–3 Å3K–1;    a ∆V + b  = aC(4∆VC,18 + ∆VC,19 + 1/2∆VC,12) + aH(4∆VH,124 + ∑ i i ∑ j  i j   l.ch.,2 1/2⋅2⋅∆VH,124) + bp = [0.021(4⋅12.7 + 8.4 + 1/2⋅12.6) + 19.98(4⋅0.2 + 2.0) + + (–25.6)]⋅10–3 = 175.6⋅10–3 Å3K–1;    ∑ K i ∆Vi  = KN∆VN,144 + KC(2∆VC,56 + ∆VC,21) + KH⋅4∆VH,124 = [2.52⋅0.9 +    i  crl.p. 1.15(2⋅14.6 + 10.2) + 2.307⋅4⋅2.0]⋅10–3 = 66.1⋅10–3 Å3K–1. Substituting all calculated values of network parameters into equation (IV.98), we get:

358

Tg (K ) =

300.6 + 152.3m 10 − 3 . 519.3 + 490.6m + 175.6 + 66.1

(IV.110)

If m = 0, we obtain the structure of cured epoxy resin based on diglycidyl ether of corresponding bisphenol HO

OH

For this resin, the calculated value of Tg,calc = 395 K; the experimental value is Tg,exp = 396 K.

Figure 55. Dependence of Tg,calc on m for the cured epoxy resin (see text).

Figure 55 depicts the dependence of Tg on m, calculated by formula (IV.110). When the linear fragment of the network decreases, the glass transition temperature increases, especially intensive, when m → 0. Let us now consider the influence of structure features of polymeric networks and their defects on the glass transition temperature Tg. a) Influence of cross-links distribution in the network on the glass transition temperature Let us analyze first the influence of cross-links distribution along the chains upon the glass transition temperature of the network. As a well-studied model network, a system based on polystyrene cross-linked by divinylbenzene is chosen. This network displays the structure:

359

.

Structure IX

...

(CH2

CH)m -1

CH2

CH CH2

CH

(CH2

CH)m -1

CH2

CH

...

...

(CH2

CH)m -1

CH2

CH CH2

CH

(CH2

CH)m -1

CH2

CH

...

.. This network is a tri-functional one, i.e. three chains yield from every crosslinked point of it (marked by dotted lines). It possesses    ∑ ∆'  = ∆VC,7 + 2∆VC,10 + ∆VC,19 + 5∆VH,124 = 8.7 + 2⋅13.1 + 8.4 + 5⋅2.0 =   FUOS = 53.3 Å3;    ∑ ∆Vi  = ∆VC,7 + ∆VC,19 + 5∆VC,18 + 6∆VH,124 + (∆VC,10 + ∆VC,7 + ∆VC,19 +    i  l.ch.,1 L

L

5∆VC,18 + 8∆VH,124)(m – 1) = 8.7 + 8.4 + 5⋅12.7 + 6⋅2.0 + (13.1 + 8.7 + 8.4 + 5⋅12.7 + 8⋅2.0)(m – 1) = (109.7m – 17.1) Å3;    ∑ ∆Vi  = 2∆VC,18 + 2∆VH,124 = 2⋅12.7 + 2⋅2.0= 29.4 Å3;    i  l.ch.,2    ∑ K i ∆Vi  = KC(2∆VC,7 + 2∆VC,10 + ∆VC,19) + KH⋅5∆VH,124 = [1.15(8.7 + 2⋅13.1    i  crl.p. + 8.4) + 2.307⋅5⋅2.0]⋅10–3 = 72.9⋅10–3 Å3K–1;    a ∆V + b  = aC(∆VC,7 + ∆VC,19 + 5∆VC,18) + aH6∆VH,124 + [aC(∆VC,10 + ∑ ∑ i i j   j  i  l.ch.,1 ∆VC,7 + ∆VC,19 + 5∆VC,18) + aH8∆VH,124](m – 1) + bd/2 + bd/2⋅(m – 1) = {0.021(8.7 + 8.4 + 5⋅12.7) + 19.98⋅6⋅2.0 + [0.021(13.1 + 8.7 + 8.4 + 5⋅12.7) + 19.98⋅8⋅2.0](m – 1) + (–55.4)/2 + (–55.4)/2⋅(m – 1)}⋅10–3 = (–80.2 + 294.0m)⋅10–3 Å3K–1;    a ∆V + b  = aC2∆VC,18 + aH2∆VH,124 + bp/2 = (0.021⋅2⋅12.7 + 19.98⋅2.0 i i j ∑ ∑   i j   l.ch.,2 – 25.6/2)⋅10–3 = 67.6⋅10–3 Å3K–1.

360

Substituting all calculated values of network parameters into equation (IV.98), we get Tg (K ) =

64.6 + 109.7m − 3 10 . 60.3 + 294.0m

(IV.111)

Formula (IV.111) is true under the condition m ≥ 1. The dependence of Tg on m calculated from formula (IV.111) is depicted in Figure 56 (note that when m → ∞, we get the calculated value of Tg for linear polystyrene, which equal 373 K). It is seen that the glass transition temperature sharply increases as the number of polystyrene units between neighboring cross-links decreases, when the number of these units (m – 1) → 0. When m = 0, the network structure IX is of the following composition .. ..

...

CH2

CH CH2

CH CH2

CH

...

...

CH2

CH CH2

CH CH2

CH

...

.. . that corresponds to 100% cross-linking. Cross-linked point of this network is marked by dotted lines. It possesses    ∑ ∆'  = 2⋅1/2⋅∆VC,10 + ∆VC,7 + 2⋅2/2⋅∆VH,124 + ∆VH,124 + ∆VC,19 = 2⋅1/2⋅13.1 +   FUOS L

L

8.7 + 2⋅2/2⋅2.0 + 2.0 + 8.4 = 36.2 Å3;   = 0, because all atoms in cross-linked The value of  ∑ ai ∆Vi + ∑ b j    i j   l.ch.,1 chains are included in cross-linked points composition.    ∑ ∆Vi  = 2∆VC,18 + 2∆VH,124 = 2⋅12.7 + 2⋅2.0= 29.4 Å3;    i  l.ch.,2

361

   a ∆V + b  = aC2∆VC,18 + aH2∆VH,124 + bp/2 = (0.021⋅2⋅12.7 + 19.98⋅2.0 ∑ ∑ i i j   i j   l.ch.,2 – 25.6/2)⋅10–3 = 67.6⋅10–3 Å3K–1;    ∑ K i ∆Vi  = KC(2⋅1/2⋅∆VC,10 + ∆VC,7 + ∆VC,19) + KH(2⋅1/2⋅∆VH,124 + ∆VH,124) =    i  crl.p. [1.15(2⋅1/2⋅13.1 + 8.7 + 8.4) + 2.307(2⋅1/2⋅2.0 + 2.0)]⋅10–3 = 48.57⋅10–3 Å3K–1. Substituting these values into equation (IV.98), we get Tg (K ) =

36.2 + 29.4 − 3 10 = 565 . 48.57 + 67.6

Let us now consider the influence of cross-linked points distribution along the chains on the glass transition temperature Tg. The analysis will be performed in terms of the cross-linking degree α. For the fragment of the network ... CH2 CH ...

.. let us determine all values of parameters from equation (IV.98).    ∑ ∆Vi  = 2∆VC,18 + 2∆VH,124 + ∆VC,10 + ∆VC,7 + ∆VC,19 + 3∆VH,124 = 2⋅12.7 + 2⋅2.0    i 1 + 13.1 + 8.7 + 8.4 + 3⋅2.0 = 65.6 Å3;    a ∆V + b + K ∆V  = a 2∆V C C,18 + aH2∆VH,124 + bp/2 + KC(∆VC,10 + ∆VC,7 ∑ ∑ ∑ i i j i i   j i  i 1 + ∆VC,19) + KH∆VH,124 = [0.021⋅2⋅12.7 + 19.98⋅2⋅2.0 – 1/2⋅25.6 + 1.15(13.1 + 8.7 + 8.4) + 2.307⋅3⋅2.0]⋅10–3 = 116.2⋅10–3 Å3K–1. Now it should be taken into account that in the repeat unit of polystyrene ...

CH2

...

CH

.. 2*

Š Š

which yields from the network cross-
    2     

 lines) is included into the cross-linked point composition. Then, for this polystyrene unit, we get:

362

*

     ∑ ∆Vi  =  ∑ ∆Vi  = ∆VC,10 + ∆VC,7 + ∆VC,19 + 5∆VC,18 + 8∆VH,124 = 13.1 + 8.7      ps  i 2  i + 8.4 + 5⋅12.7 + 8⋅2.0 = 109.7 Å3; *

   a ∆V + b + K ∆V  = a (∆V + ∆V C C,7 C,19 + 5∆VC,18) + aH6∆VH,124 + bd/2 + ∑ i i ∑ j ∑ i i  j i  i 2 KC∆VC,10 + KH2∆VH,124 = [0.021(8.7 + 8.4 + 5⋅12.7) + 19.98⋅6⋅2.0 – 55.4/2 + 1.15⋅13.1 + 2.307⋅2⋅2.0]⋅10–3 = 238.1⋅10–3 Å3K–1. For a ‘normal’ polystyrene unit, ...

CH2

...

CH

  we obtain  ∑ ∆Vi  = 109.7 Å3;    i  ps    a ∆V + b  = a (∆V C C,10 + ∆VC,7 + ∆VC,19 + 5∆VC,18) + aH8∆VH,124 + bd/2 = ∑ ∑ i i j   j  i  ps [0.021(13.1 + 8.7 + 8.4 + 5⋅12.7) + 19.98⋅8⋅2.0 – 55.4/2]⋅10–3 = 294.1⋅10–3 Å3K–1. When all parameters of equation (IV.98) are calculated, let us analyze three variants. 1) Uniform distribution of cross-links For this case, equation (IV.98) obtains the following form:

Tg (K ) =

  α  ∑ ∆Vi  +    i 1 *

    α  ∑ ai ∆Vi + ∑ b j + ∑ K i ∆Vi  + α  ∑ ai ∆Vi + ∑ b j + ∑ K i ∆Vi  +     j i j i  i 1  i 2

→

  + (1 − α ) ∑ ∆Vi     i  ps   + (1 − 2α ) ∑ ai ∆Vi + ∑ b j    j  i  ps

→

(IV.112)

,

where α is the cross-linking degree equal to 1/m, which characterizes the part of linked phenyl groups. Substituting values of all above-determined parameters of the network into (IV.112), we get

363

Tg ( K ) =

α ⋅ 65.6 + (1 − α ) ⋅109.7 ⋅ 10 3 . α ⋅116.2 + α ⋅ 238.1 + (1 − 2α ) ⋅ 294.0

(IV.113)

Formula (IV.113) is true under the condition 0 < α ≤ 0.5. If α > 0.5, no free units of polystyrene remains, and formula (IV.113) reduces to the form Tg (K ) =

α ⋅ 65.6 + (1 − α ) ⋅109.7 ⋅10 3 . α ⋅116.2 + (1 − α ) ⋅ 238.1

(IV.114)

When α = 0, we obtain the calculated glass transition temperature of linear polystyrene. The dependence of Tg on the cross-linking degree α determined with the help of equations (IV.113) and (IV.114) is shown in Figure 57, curve 1.

Figure 57. Dependence of Tg on the cross-linking degree α for the polystyrene and divinylbenzene network: 1 – uniform distribution of cross-linked points; 2 – irregular distribution of crosslinked points; 3 – the most ununiform distribution of cross-linked points. The points denote Tg values detected experimentally.

Figure 57 indicates that the dependence of Tg on the cross-linking degree α consists of two parts – with fast and slow increase of Tg. This is caused by a specific influence of network cross-linked points upon Tg, which is indicated by the above analysis. 2) A random distribution of cross-links It is common knowledge that the function of distribution by distances for a random field of points fitting a straight line is the following F (l ) = 1 − e

−l / l avg

,

(IV.115)

where l is the distance between two neighboring points, and in the case under consideration – the distance between cross-linked points; lavg is the average distance. Turning to the polymerization degree m, we get F (m) = 1 − e

− m / mavg

.

(IV.116)

364

If cross-linked points are located alongside each other ...

CH2

CH CH2

CH CH2

CH CH2

CH CH2

CH CH2

CH

...

...

CH2

CH CH2

CH CH2

CH CH2

CH CH2

CH CH2

CH

...

WKH DPRXQW RI

Š Š &+2

JURXSV EHORQJLQJ WR VW\UHQH UHVLGXHV PDUNHG E\ GRWWHG OLQHV

in the scheme) and simultaneously included into the cross-linked point composition will be lower than in the case, when cross-linked points are separated by linear fragments: ...

CH2

CH CH2

CH CH2

CH CH2

CH CH2

CH CH2

CH

...

...

CH2

CH CH2

CH CH2

CH CH2

CH CH2

CH CH2

CH

...





*

Consequently, the value of α  ∑ ai ∆Vi + ∑ b j + ∑ K i ∆Vi  decreases by the    i

j

i

2

value proportional to the amount of cross-links located alongside each other. Then   (1 − α ) ∑ ai ∆Vi + ∑ b j  increases by the same value.   j  i  ps Since we are interested in cross-linked points located alongside each other, F(m) at m = 1 is F (1) = 1 − e

−1 / mavg

. Then, taking into account that mavg = 1/α, we get

F(1) = 1 – e–α.

(IV.117)

Then equation (IV.103) in the second summand in the denominator will obtain α – α(1 – e–α) = α/eα, and in the third summand, 1 – 2α + α(1 – e–α) = 1 – α – α/eα instead of (1 – 2α). When substitutions are made, we get

365

Tg (K ) =

  α  ∑ ∆Vi  +    i 1   α α  ∑ ai ∆Vi + ∑ b j + ∑ K i ∆Vi  +  eα  j i  i 1

  + (1 − α ) ∑ ∆Vi     i  ps

→ + (1 − α



*

   a ∆V + b + K ∆V  + ∑ ∑ ∑ i i j i i   j i  i 2

 a ∆V + b  ∑ i i ∑ j  j  i  ps

→

(IV.118)

,

− α / eα )

Substituting all calculated values of the network parameters into equation (IV.118), we get Tg (K ) =

α ⋅ 65.6 + (1 − α ) ⋅109.7 ⋅10 3 . α α ⋅ 238.1 + (1 − α − α / e ) ⋅ 294.0 α ⋅116.2 + eα

(IV.119)

Formula (IV.119) is valuable under the condition (α + α/eα) ≤ 1; this conditionally correlates with α ≤ 0.66. If (α + α/eα) ≥ 1, no free units of polystyrene remain, and formula (IV.119) transits into equation (IV.114). The dependence of Tg on α obtained with the help of (IV.119) and (IV.114) is depicted in Figure 57, curve 2. 3) The most ununiform distribution of cross-linked points. This distribution corresponds to the case when a large number of cross-linked points locates alongside each other and form significant sequences of cross-linked points linked by linear fragments of polystyrene chains (see scheme in Figure 58).

Figure 58. Hypothetical case of the most ununiform distribution of cross-linked points of network (scheme).

At this distribution of cross-linked points, the amount of polystyrene units linked to the cross-linked point becomes small, and it may be written that

366

Tg (K ) =

    α  ∑ ∆Vi  + (1 − α ) ∑ ∆Vi       i 1  i  ps     α  ∑ ai ∆Vi + ∑ b j + ∑ K i ∆Vi  + (1 − α ) ∑ ai ∆Vi + ∑ b j      j i j  i 1  i  ps

.

(IV.120)

Substituting all values of the network parameters into (IV.120), we obtain α ⋅ 65.6 + (1 − α ) ⋅109.7 ⋅10 3 . α ⋅116.2 + (1 − α ) ⋅ 294.0

Tg (K ) =

(IV.121)

Calculations performed by equation (IV.121) lead to the dependence of Tg on α displayed in Figure 57, curve 3. This figure also displays experimental data on the dependence of Tg on the cross-linking degree α, taken from ref. [160]. Experimental points fit well curve 2; this testifies that cross-linked points are distributed in the real network in a random way. b) Influence of the cross-linked point structure on the glass transition temperature of network Let us discuss one more question on the influence of the chemical structure of the network upon its glass transition temperature. In this case, networks of equal chemical structure of linear fragments but of different structure of cross-linked points will be discussed. For this analysis, polydimethylsiloxane networks of the following chemical structure are suitable [27]: H3C ...

CH2

CH2

Si O

H2C ...

...

Si

Si O

CH3 CH2

CH3 CH2

Si

(O

CH3

CH3

CH3

CH3 Si)n -2 CH3

O

Si

CH2

...

CH3

H2C

(A) H3C

...

O

Si O Si

CH3

CH3

CH2

CH2

CH2

CH2

Si

CH2

CH2 Si O Si CH2 H3C CH3

CH2

CH3 ...

O

O

CH3

CH3 (O

Si)n -2 CH3

O

Si

CH2

...

CH3

(B)

At low values of n (high cross-linked network) synthesis is based on application of individual compounds with accurately assigned value of n. That is why at conversion close to 100%, the value of Mc in these networks is determined with high accuracy. Prior to analyzing the influence of the structure of cross-linked points on Tg, let us repeat the definition of the network cross-linked point from the point of view of its chemical structure: cross-linked point of a network is a group of atoms, including the atom from which branching begins, plus neighboring atoms chemically bonded with the former one with their closest substituents (therewith, another end of the

367

branch may also be included into the corresponding cross-linked point, otherwise this will be a simple branching). Then for considered networks, cross-linked points are of the structure limited by dotted lines. Let us determine all values of constants for these systems, included into equation (IV.85):    ∑ ∆Vi  = 3∆VC,106 + 3∆VC,109 + 3∆VC,109 + 3/2[4∆VC,106 + 2(n – 1)3∆VC,106] +    i  r.f. 21∆VH,124 + 3/3[12∆VH,124 + 6(n – 2) ∆VH,124] + 3∆VO,135 + 3/2[∆VO,135 + (n – 2)∆VO,135] + 3∆VSi,172 + 3/2[2∆VSi,173 + (n – 2)∆VSi,172] = (239.85 + 108.15n) Å3;    a ∆V + b  = a {3∆V C C,109 + 3/2[4∆VC,106 + 2(n – 2)∆VC,106]} + aH{6∆VH,124 + ∑ ∑ i i j   j  i  3/2[12∆VH,124 + 6(n – 2)∆VH,124] + aO,03/2[∆VO,135 + (n – 2)∆VO,135] + aSi3/2[2∆VSi,173 + (n – 2)∆VSi,172]. Substitution of all values to the present relation gives the value of    a ∆V + b  = (176.0 + 721.5n)⋅10–3 Å3K–1 (therewith, ∑ b j = 0 , because ∑ ∑ i i j   j j  i  polymer contains no separate polar groups which induce strong intermolecular interaction). For the cross-linked point of the network (A), we get    ∑ K i ∆Vi  = KC(3∆VC,106 + 3∆VC,109) + KH15∆VH,124 + KO3∆VO,135 +    i  crl.p.,A KSi3∆VSi,172 = 252.3 Å3K−1.    a ∆V + b  , and  K ∆V  into ∑ ∑ i i j   ∑ i i  j  i  crl.p.,A  i  equation (IV.85) gives the following dependence of the glass transition temperature Tg of the network on the polymerization degree:   Substitution of  ∑ ∆Vi  ,    i  r.f.

Tg (K ) =

239.8 + 108.1n ⋅ 10 3 . 428.3 + 721.5n

(IV.122)

Graphically, this dependence is shown in Figure 59. For networks with tetrafunctional cross-linked pints (scheme B), it can be easily shown that the calculated dependence of Tg on the polymerization degree n is identical to relation (IV.122), because each value in the numerator and the denominator of this relation is multiplied by the same number 4/3. Let us now discuss the influence of cross-linked points structure of networks from polydimethylsiloxane on their glass transition temperature. Reasoning from chemical ideas, possible cross-linked points may be depicted as shown below.

368

Figure 59. Dependence of Tg on 1/n for A and B structures (see text).

To calculate Tg of networks containing various cross-linked points, it is     advisable to determine first the values of  ∑ ai ∆Vi + ∑ b j  and  ∑ ∆Vi  for a     j  i  l.f.  i  l.f. single linear fragment of the network between two neighboring cross-linked points:    ∑ ∆Vi  = 2∆VSi,172 + 4∆VC,106 + ∆VSi,172(n – 2) + 2∆VC,106(n – 2)+ ∆VO,135 +    i  l.f. ∆VO,135(n – 2) + 12∆VH,124 + 6∆VH,124(n – 2);    ∑ ∆Vi  = (72.1n – 0.5) Å3;    i  l.f.    a ∆V + b  = a [4∆V C C,106 + 2∆VC,106(n – 2)] + aO,0[∆VO,135 + (n – 2)∆VO,135] ∑ ∑ i i j   j  i  l.f. + aH[12∆VH,124 + 6(n – 2)∆VH,124] + aSi[2∆VSi,172 + (n – 2)∆VSi,172];    a ∆V + b  = (481n – 11.5)⋅10−3 Å3K−1. ∑ i i ∑ j  j  i l.f.     and  ∑ ai ∆Vi + ∑ b j  for all The resultant values of  ∑ ∆Vi      i j  i  l.f.  l.f. considered networks are equal. Displayed below are possible cross-linked points of networks from polydimethylsiloxane and relations obtained from equation (IV.85) which allow calculation of Tg in accordance with the number of the repeat units n in linear fragments connecting these cross-linked points.

369

... H3C CH3 ...

Si

O

CH3 H3C

Si

CH3

O

CH3

Si O

Si

O

CH3

Si

...

(1)

...

(2)

Tg (K ) =

144.2n + 33.7 ⋅10 3 962n + 11.3

CH3

... CH3 ...

Si

O

CH3 H3C

CH3

CH3

Si O

Si

O

CH3

Si

Tg (K ) =

108.15n + 52.9 ⋅10 3 721.5n + 46.5

CH3

.. . CH3 ...

...

...

CH3

CH3

Si O

Si

CH3

CH2

CH3

CH3

CH2

CH3

Si O

Si ...

CH3

CH3

CH3

CH3

CH3

CH3

Si O

Si

Si

Si

Si

O

O

O

CH3

CH2

CH3 ...

...

...

...

CH3

CH3

CH3

CH3

CH3

Si O

Si

O

O

CH3

CH3

CH3

CH3

CH3

(4)

Tg (K ) =

72.1n + 58.1 3 ⋅10 481n + 61.2

Tg ( K ) =

72.1n + 49.3 3 ⋅10 481n + 48.8

CH3

CH3

Si

72.1n + 66.1 3 ⋅10 481n + 72.6

CH3 Si ...

Si

Tg (K ) =

...

Si O

Si

(3)

O

...

Si O

Si ...

CH3

CH3

(5)

According to the relation described above, dependences of Tg of networks on the number n of polydimethylsiloxane units are shown in Figure 60. It is obvious that the structure of a cross-linked point affects significantly Tg only if n = 1÷4 (α = 0.25 – 1). As the distance between cross-linked points increases, the role of the latter decreases abruptly, and already at n = 10 the glass transition temperature for all networks approaches Tg of polydimethylsiloxane, which equals 150 K.

370

Figure 60. Dependences of Tg of networked polydimethylsiloxanes on 1/n for different types of crosslinked points. Designations 1–5 of cross-linked points in the Figure correspond to their designations in the text.

c) Influence of dangled chains and other defects of network on the glass transition temperature of polymer networks Among the multitude of various variants of dangled chains, let us consider two of them in detail. The first variant is a modification of network I, when a part of crosslink bridges is ‘broken’ in halves: ... ...

...

C CH2

(CH2)m -2

... ...

C

...

CH2 CH2

(CH2)m -2

CH2

.. .

C

CH2

(CH2)m-2

CH2

C

...

.. .

CH2 _ -2 (CH2 )m 2

CH3 CH3 _ -2 (CH2 )m 2

... ...

C

...

CH2 CH2

(CH2)m -2

CH2

C

CH2

(CH2)m-2

CH2

C

...

CH2 ... ... In this case, the cross-linked point of the network is kept, but it becomes trifunctional (in the present example, as it is obvious from the scheme, chemical structure of a cross-linked point also remains unchanged). The repeating fragment of the present structure consists of the repeating fragment of tri-functional network and dangled chain. Then for the Van-der-Waals volume, it may be written down that ...

371

         ∑ ∆Vi  =  ∑ ∆Vi  + 1.5 ∑ ∆Vi  (m − 2) +  ∑ ∆Vi  ,          i  r.f.  i  crl.p.  i  v.  i  d.ch.

(IV.123)

  where  ∑ ∆Vi  is the Van-der-Waals volume of the network cross-linked point,    i  crl.p.   marked by dotted lines in the scheme;  ∑ ∆Vi  is the Van-der-Waals volume of    i  v.

Š&+ Š JURXS  ∑ ∆Vi  2

 i Therewith,

d.ch.

is the Van-der-Waals of the dangled chain.

    m     ∑ ∆Vi  =  ∑ ∆Vi   − 2  +  ∑ ∆Vi  ,   2       i  i  d.ch.  i  v.  CH 3   where  ∑ ∆Vi  is the Van-der-Waals volume of the end CH3    i  CH 3   Values of  ∑ ∆Vi  and    i  crl.p.

(IV.124)

Š JURXS

   ∑ ∆Vi  were calculated above;    i  v.

   ∑ ∆Vi  = ∆VC,13 + 3∆VH,124 = (17.2 + 3⋅2.0) = 23.2 Å3;    i  CH 3 Prior to further calculations, one more important note should be made. Principally, there may not appear a situation, when a network possessing no dangled chains, and a network containing them are of the identical structure. To put it differently, appearance of dangled chains always implies a change in chemical structure of the network, which affects the glass transition temperature and other properties. So,    ∑ ∆Vi  = 73.4 + 1.5⋅17.1(m – 2) + 17.1(m/2 – 2) + 23.2 = (11.1 + 34.2m) Å3;    i  r.f.    a ∆V + b  = 1.5(aC∆VC,10 + aH2∆VH,124)(m – 2) + (aC∆VC,10 + ∑ i i ∑ j  i j   l.ch. aH2∆VH,124)(m/2 – 2) + (aC∆VC,13 + aH3∆VH,124) + bd = [1.5(0.021⋅13.1 + 19.98⋅2⋅2.0)(m – 2) + (0.021⋅13.1 + 19.98⋅2⋅2.0)(m/2 – 2) + 0.021⋅17.2 + 19.98⋅3⋅2.0 + (–55.4)]⋅10–3 = (160.4m – 336.2)⋅10–3 Å3K–1.

372

Substituting all values calculated above into equation (IV.85), we get Tg (K ) =

α (11.1 + 34.2m) + (1 − α )(5.0 + 34.2m) ⋅10 3 α (160.4m − 336.2 + 102.9) + (1 − α )(160.4m − 217.86)

(IV.125)

5.0 + 34.2m + 6.1α ⋅10 3 . 160.4m − 15.44α − 217.86

(IV.126)

or Tg (K ) =

Equation (IV.126) is valuable up to α ≤ 5, because at m ≥ 2 break of every linear chain forms two cross-linked points of the network, from which dangled chains yield. The dependence of the glass transition temperature Tg on α determined by formula (IV.126) is displayed in Figure 61. It is obvious that Tg grows weakly with α, the slope of this dependence decreasing as the number of units between cross-links m increases. At first glance, it may seem that appearance of network defects as dangled chains must cause a decrease but not an increase of Tg. However, it should be taken into account that in this particular case, appearance of dangled chains does not decrease the number of cross-linked points and even change their chemical structure. Moreover, appearance of every branching requires introduction of the constant bd that promotes increase of Tg.

Figure 61. Dependences of Tg on α (with different m) for tetra-functional polyethylene network with dangled chains and constant numbers of cross-linked points.

In the case when a cross-linked point loses is original meaning due to chains ‘break’, i.e. it is no more the cross-linked point of the network, the glass transition temperature obviously decreases. This type of the network is shown below: CH3 _ -2 (CH2 )m 2 .. .. . . CH2 ... C CH2 (CH2)m -2 CH2 C CH2 (CH2)m-2 CH2 C ... ...

CH2 .. .

...

373

In the case of this ‘break’ of network linear fragments a linear chain is formed, and constants ai and bj instead of Ki should be used in calculation of the former crosslinked point, i.e.    a ∆V + b  = aC(∆VC,1 + 4∆VC,10) + aH8∆VH,124 = [0.021(5.0 + 4⋅13.1) + ∑ i i ∑ j  i j   crl.p. 19.98⋅8⋅2.0]⋅10–3 = 320.9⋅10–3 Å3K–1. Then for a network fragment between two vertical dotted lines, it should be given:    ∑ ∆Vi  = 73.4 + 17.1(m – 2) + 17.1(m/2 – 2)⋅2 + 23.2⋅2 = (17.2 + 34.2m) Å3;    i  f.    a ∆V + b  = {320.0 + 0.021[13.1(m – 2) + 13.1(m/2 – 2)2 + 17.2⋅2] + ∑ i i ∑ j  j  i  f. 19.98[2⋅2.0(m – 2) + 2⋅2.0(m/2 – 2)⋅2 + 6⋅2.0]}⋅10–3 = (80.2 + 160.4m)⋅10–3 Å3K–1. Substituting all these values and previously calculated parameters of the network into equation (IV.85), we obtain Tg ( K ) =

α (17.2 + 34.2m) + (1 − α )(5.0 + 34.2m) ⋅10 3 α (80.2 + 160.4m) + (1 − α )(160.4m − 217.86)

or Tg ( K ) =

5.0 + 34.2m + 12.2α ⋅10 3 , 160.4m + 298.1α − 217.86

(IV.127)

(IV.128)

Figure 62. Dependences of Tg on α (with different m) for tetra-functional polyethylene network with dangled chains when the formation of them reduce a quantity of cross-linked points.

374

where α is a part of the former cross-linked points in the network, from which 2 broken chains yield. Relation (IV.128) it valuable at α ≤ 0.25. The dependence of Tg on α calculated by equation (IV.128) is shown in Figure 62. Clearly, the glass transition temperature decreases abruptly as the part of crosslinked points with yielding two broken chains increases. The reason is that the place of branching loses its role of the cross-linked point, and linear fragment between two neighbor cross-linked points two-fold increases. Let us now consider a tri-functional network with dangled chains. The scheme of this hypothetical network based on polyethylene is shown below: ...

...

(CH2)m -2

CH2

CH CH2

(CH2)m -2

CH2

CH2

CH

...

(CH2)m -2

CH2

CH2

CH

...

CH2 _ -2 (CH2 )m 2

CH3 CH3 _ -2 (CH2 )m 2

CH2 ...

(CH2)m -2

CH2

CH CH2

..

. In this case, when linear chains break, the former cross-linked point with yielding a dangled chain loses its significance as a cross-linked point, and is included to composition of a linear fragment with a branching. For the repeating fragment of the network bordered by two vertical dotted lines, we obtain:

   ∑ ∆Vi  = 62.3 + 17.1m – 34.2 + 17.1(m/2 – 2) + 23.2 = (25.65m +17.1) Å3;    i  f.    a ∆V + b  = a [∆V + 3∆V C C,6 C,10 + ∆VC,10(m – 2) + ∆VC,10(m/2 – 2) + ∆VC,13] ∑ ∑ i i j   i j   f. + aH[7∆VH,124 + 2∆VH,124(m – 2) + 2∆VH,124(m/2 – 2) + 3∆VH,124] + bd = {0.021[9.0 + 3⋅13.1 + 13.1(m – 2) + 13.1(m/2 – 2) + 17.2] + 19.98[7⋅2.0 + 2⋅2.0(m – 2) + 2⋅2.0(m/2 – 2) + 3⋅2.0] – 55.4}⋅10–3 = (24.8 + 120.3m)⋅10–3 Å3K–1. Substituting all these and previously calculated parameters of the network into equation (IV.85), we get: Tg (K ) =

α (17.1 + 25.65m) + (1 − α )(11.0 + 25.65m) ⋅10 3 α (24.8 + 120.3m) + (1 − α )(120.3m − 152.76)

or Tg ( K ) =

11.0 + 25.65m + 6.1α ⋅10 3 , 120.3m + 177.56α − 152.76

where α is a part of structures with dangled chains.

(IV.129)

(IV.130)

375

Figure 63. Dependences of Tg on α (with different m) for tri-functional polyethylene network with dangled chains . Figure 64 (right). Schematic representation of different defects of network (see text).

Dependences of Tg on α with different m are shown in Figure 63. As the part of dangled chains grows, Tg abruptly decreases, especially with small m. Considered above were hypothetical network based on polyethylene with fragments between cross-linked points and dangled chains of the same chemical structure. Let us now consider a system with dangled chains of a different structure compared with backbones of the network. Such a system based on epoxy resin was synthesized and investigated in ref. [35]. Epoxy resin ED-20 was cured by methyltetrahydrophthalic anhydride, some part of which was substituted by oleic acid in order to obtain a network with dangled chains. Results of calculations and experiments performed for such networks are discussed in detail in refs. [35, 125]. Besides dangled chains, the network may also contain such defects as isolated loops. Figure 64 displays defects of various types as branchings or isolated loops. Cross-linked points are the only branching points, in which chains yielding from them are joined to neighboring chains, i.e. form a cross-link. These structures are marked by ‘+’ sign. If a chain forming no cross-link but representing a simple branch yields from the branching point or forms an isolated loop at the sacrifice of attachment by its end to the yielding point or another place of the same chain, these places are not also cross-linked points (marked by ‘–’ sign). This must be taken into account when equation (IV.85) is used for calculation of the glass transition temperature. If a branching point is a cross-linked point (in the sense discussed above), constants Ki must be used for calculation by equation (IV.85). If a branching point is not the cross-linked point, for the same atoms constants ai and bj must be substituted into equation (IV.85). Using equation (IV.85), the amount of defects in the real network structure may be estimated by comparing calculated and experimental values of Tg. Let us perform such detailed analysis on the example of network based on methyl methacrylate and 1-methoxy-1-vinyl-1-sila-2,7-dioxa-4,5-(1,2-carborano) cycloheptane copolymer. Chemical structure of this polymer is shown below [67]: CH3 CH2

C

CH2

C O CH3 O

O x

Si

CH2 C

I

CH

II

O CH3 O CH2

C B10H10

y

376

Under the thermal effect upon the copolymer, the cycle in structure II transforms into a linear structure, and a network of the following chemical structure is formed: CH3 CH2

C

CH2

CH

C O CH3

Si

O

C C CH2

O CH2

O

x

I

O

B10H10

y

CH3

II According to the definition of the cross-linked point given above, its chemical structure in the present network is the following: ...

CH2

...

O

...

CH Si

O

...

O . Let us calculate values of

∑ ∆Vi , ∑ ai ∆Vi + ∑ b j i

i

j

and

∑ K i ∆Vi

for the

i

present network. For structure I,    ∑ ∆Vi  = ∆VC,10 + ∆VC,13 + ∆VC,1 + ∆VC,48 + ∆VC,41 + 8∆VH,124 + ∆VO,129 + ∆VO,139    i I = 96.8 Å3. For structure II,    ∑ ∆Vi  = ∆VC,10 + ∆VC,41 + ∆VC* + 2∆VC,40 + 2∆VC,117 + 10∆VH,124 + 10∆VH,128 +    i  II 3∆VO* + ∆VSi,174 + 6∆VB,185 + 4∆VB,184 = 208.8 Å3. Increments of the Van-der-Waals volume for atoms marked by ‘*’ sign are absent in Table 3, so they are displayed below: C Si

1.54 1.88

C

1.54

1.08

H ∆' ∗ = 8.0 Å3; &

Si

1.64

O

1.50

C ∆' ∗ = 1.95 Å3; 2

C O O

1.64 1.64

Si

1.64

O

1.88

3 C ∆'6L = 30.1 Å .

377

Let us now calculate values of

∑ ai ∆Vi + ∑ b j i

for the ideal network with the

j

structure shown above. For structure I,    a ∆V + b  = a (∆V C C,10 + ∆VC,13 + ∆VC,1 + ∆VC,48 + ∆VC,41) + aH(8∆VH,124) + ∑ i i ∑ j  i j  I aO,s∆VO,129 + aO′′,s ∆VO,139 + 3bd = 255.8⋅10–3 Å3K–1. For structure II,    a ∆V + b  = a (2∆V C C,40 + 2∆VC,117 + ∆VC,41) + aH(7∆VH,124 + 10∆VH,128) + ∑ ∑ i i j   j  i  II aB(6∆VB,185 + 4∆VB,184) + bd = 401.0⋅10–3 Å3K–1.   Let us now calculate  ∑ K i ∆Vi  for cross-linked point of the network    i  crl.p. with structure displayed above.   d  ∑ K i ∆Vi  = KC(∆VC,10 + ∆VC*) + KH(3∆VH,124) + K O (3∆VO*) + KSi(∆VSi*) =    i  crl.p. 69.9⋅10–3 Å3K–1. Let us consider in the general form dependence of the glass transition temperature on chemical structure of the network taking into account that it possesses defects in the structure and branchings displayed in Figure 64. As noted above, if a branch forms no cross-link, then, though chemical structure of the cross-linked point in the branching point is the same as in the case of the cross-link formation, this crosslinked point loses its role, and constants ai and bj instead of Ki must be used for it in calculation by equation (IV.85). The equation (IV.85) is rewritten in the form:

Tg =

∑ ∆Vi i

       a ∆V + b  + y ∑ K i ∆Vi  + (1 − y ) ∑ ai ∆Vi + ∑ b j  ∑ ∑ i i j       j j  i  crl.p.  i  l.ch.  i  crl.p.

, (IV.131)

where y is a part of branching points acting as cross-linked points. Comparing calculated and experimental glass transition temperatures, the value of y may be found from the relation, deduced from equation (IV.131) .

378



y=



  −  ∑ ai ∆Vi + ∑ b j    j  l.ch.  i  crl.p.

∑ ∆Vi −  ∑ ai ∆Vi + ∑ b j 

1

Tg, exp i

 i

j

     ∑ K i ∆Vi  +  ∑ ai ∆Vi + ∑ b j      j  i  crl.p.  i  l.ch.

,

where Tg,exp is the experimental glass transition temperature. If to form a network copolymers are used, as it takes place in this case, the situation becomes more complicated. Assume that one of components of copolymer does not form a network. Then in the ideal case (100% of cross-linking)

7

J

Q  ∑ ∆9  + Q ∑ ∆9 

 − 

=



  − Q  ∑ D ∆9 + ∑ E   L

L

L

M

M

L

L

,

L



L

   + Q ∑ D ∆9 + ∑ E   , L

L

L

M

M

 ,,      +  ∑ . ∆9       FUOS ,,   L

(IV.132)

L

L

(n is the part of structure II). If a part of branching points are not cross-linked points, then   − Q  ∑ ∆9  +  , 7J =      − Q  ∑ D ∆9 + ∑ E  + Q ∑ D ∆9 + ∑ E  +      ,  OFK   + Q ∑ ∆9    ,,  →     + \ ∑ . ∆9  +  − \  ∑ D ∆9 + ∑ E      FUOS   ,, 

L

L

L

L

L

L

M

M

L

M

L

M

(IV.133)

L

L

L

L

L

L

L

L

M

M

where y is the part of active cross-linked points. To calculate the part of active y and inactive (1 – y) cross-linked points,   equation (IV.133) will be used. First, let us calculate the value of  ∑ ai ∆Vi + ∑ b j  :   j  i     a ∆V + b  = aC(∆VC,10 + ∆VC*) + aH(3∆VH,124) + aO,0(2∆VO*) + ∑ ∑ i i j   j  i  crl.p.,II aO,s∆VO* + aSi∆VSi* = 490.4⋅10–3 Å3K–1.

379

  Substituting found values of  ∑ ∆Vi  ,    i I    a ∆V + b  , ∑ ∑ i i j   j  i  II (IV.133), we get

   ∑ K i ∆Vi  ,    i  crl.p.

Tg (K ) =

   ∑ ∆Vi  ,    i  II

   a ∆V + b  , ∑ ∑ i i j   j  i I

   a ∆V + b  ∑ ∑ i i j   j  i  l.ch.

96.8 + 112.1n ⋅10 3 . 255.8 − 420.5ny + 635.6n

into

equation

(IV.134)

Experimental dependences of Tg on composition of component II, taken from ref. [20] are shown in Figure 65. It also displays calculated dependences of Tg on concentration of the second component with different parts of acting cross-linked points y. Calculation was performed by equation (IV.134). The Figure displays that, for example, if the part of component II is 20%, the amount of active cross-linked points is ~58% and of inactive – 42%. At the amount of this component of 10%, the part of active cross-linked points is ~55%, etc. Hence, although networks obtained in ref. [20] are high cross-linked ones, but are significantly defective. If 20% of structure II is contained by the copolymer, the part of the sol fraction is 18.5%. Consequently, a part of inactive cross-linked points is contained in the branched copolymer, being washed off the system. The remaining part of inactive cross-linked points (42 – 18.5 = 23.5%) is in the network composition and forms a structure of branchings or isolated loops type.

Figure 65. Dependences of Tg on proportion n of 1-methoxy-1-vinyl-1-sila-2,7-dioxa-4,5-(1,2carborano)cycloheptane; the proportion of active cross-linked points: 1 (1), 0.9 (2), 0.8 (3), 0.7 (4), 0.6 (5), 0.5 (6), 0.4 (7). The points denote the experimental values.

Let us give one more example of estimation of the network structure based on comparison of the calculated and experimental glass transition temperatures Tg [105]. Consider ebonites as an example of this network, because various types of networks may be realized in them, and because essential but conflicting experimental data on their structure and mechanical properties have also been obtained. Let us calculate Tg of polyisoprene-based ebonite. Let us assume that the amount of sulfur participating in the reaction corresponds to a single atom of sulfur per repeat unit of isoprene.

380

Several variants of formation of a network, in the composition of which structures of four types may participate, will be considered. .. .. . . 1)

...

CH2

CH C CH2 S .. .

3) ...

CH2

...

;

2)

...

CH2

CH C CH2 S

CH3

... ;

CH3

S .. .

CH C CH2

.. . 4) ...

... ;

CH2

CH C CH2

... .

CH3

(S)n CH3 .. .

Structure 1 contains monosulfide bridges, structure 2 – disulfide bridges, structure 3 – polysulfide ones, and structure 4 represents original units of polyisoprene. For structure 1, all atoms are included into cross-linked points of the   network. That is why for it  ∑ ai ∆Vi + ∑ b j  = 0. Then, we obtain:   j  i     ∑ ∆Vi  = 2∆VC,10 + ∆VC* + ∆VC** + ∆VC,13 + 8∆VH,124 + ∆VS,157 = 92.2 Å3;    i 1    ∑ K i ∆Vi  = KC(2∆VC,10 + ∆VC* + ∆VC** + ∆VC,13) + KH8∆VH,124 + KS∆VS,157 =    i 1 136.9⋅10−3 Å3K−1. For constants of the Van-der-Waals volume for several atoms included in the structure of networks 1–4 are absent in Table 3, they are shown below: S C

1.76 1.54

C

S 1.54

1.54

C ∆' ∗ = 10.2 Å3; C 

H S

2.10

S

1.76 1.54

C

1.54

1.54

C ∆' ∗∗ = 6.1 Å3; 

C 2.10

S ∆'$∗ = 18.9 Å3.

For structure 1, the glass transition temperature Tg calculated from equation (IV.85) coincides with temperature of the onset of intense thermal degradation Td (because all atoms are included into the cross-linked point of the network): Tg = Td = 673 K. Consequently, if sulfur is completely consumed for polyisoprene vulcanization in a manner that monosulfide bonds are formed only, then Tg must coincide with the temperature of onset of intense thermal degradation, and it equals to 400°C. In practice, ebonites possess the glass transition temperature of 60–120°C [84]. That is

381

why it must be assumed that besides monosulfide bridges, di- and, probably, polysulfide ones must occur. Let us calculate Tg for a network system possessing various types of the bridges between polyisoprene chains. For structure 2,    ∑ ∆Vi  = 2∆VC,10 + ∆VC* + ∆VC** + ∆VC,13 + 8∆VH,124 + ∆VS,164 = 111.1 Å3;    i 2    ∑ K i ∆Vi  = KC(2∆VC,10 + ∆VC* + ∆VC** + ∆VC,13) + KH8∆VH,124 + KS∆VS,164 =    i 2 172.8⋅10−3 Å3K−1. For structure 3,    ∑ ∆Vi  = 2∆VC,10 + ∆VC* + ∆VC** + ∆VC,13 + 8∆VH,124 + ∆VS,164 + (n – 2)∆VS*;    i 3    ∑ ∆Vi  = (73.3 + 18.9n) Å3;    i 3    ∑ K i ∆Vi  = 172.8⋅10−3 Å3K−1;    i 3    a ∆V + b  = (n – 2)a ∆V ; S S* i i j ∑ ∑   j  i 3    ∑ ai ∆Vi  = (141.7n – 283.5)⋅10−3 Å3K−1;    i 3 For structure 4,    ∑ ∆Vi  = 2∆VC,10 + ∆VC,16 + ∆VC,15 + ∆VC,13 + 8∆VH,124 = 81.5 Å3;    i 4    a ∆V + b  = a (2∆V C C,10 + ∆VC,16 + ∆VC,15 + ∆VC,13) + aH8∆VH,124 + bd + b≠ ∑ i i ∑ j  j  i 4 = 400.7⋅10−3 Å3K−1. Let us deduce a relation binding Tg of ebonite to molar parts of bridges containing mono-, di- and polysulfide groups. Let β1, β2, …, βn be molar parts of cross-linked points (bridges) containing mono-, di- and n-sulfide groups. At assigned stoiciometric relation between sulfur S and polyisoprene, occurrence of a single disulfide bond leads to unlinking of one polyisoprene unit. Occurrence of a trisulfide bond causes appearance of two unlinked polyisoprene units, etc. Consequently, the part β1 of unlinked polyisoprene units will be equal to

β1 = β 2 + 2 β 3 + ... + (n − 1) β n =

i=n

∑ (i − 1) β i .

i =2

(IV.135)

382

Then, on the basis of the above considerations and equation (IV.135), we get       β  ∑ ∆9L  + β   ∑ ∆9L  +  + β Q  ∑ ∆9L  +  L   L   L  → 7J =        β  ∑ . L ∆9L  + β   ∑ . L ∆9L  +  + β Q  ∑ . L ∆9L  +  L  L   L     L =Q +  ∑ ∆9L  ∑ β L L −   L   L = →   L =Q     +  ∑ D L ∆9L   +  ∑ D L ∆9L + ∑ E M  ∑ β L L −    L  M  L     L  = i=n

Therewith, it is taken into account that

= 1 – β1 – β2 – … – βn =

i =2

i=n

1–

∑ (i − 1) β i

(IV.136)

∑ βi .

i =1

Let us consider some particular cases. Assume that the network is formed only of mono- and disulfide bonds. Then       β1 ∑ ∆Vi  + β 2  ∑ ∆Vi  + (1 − β1 − β 2 ) ∑ ∆Vi         i 1  i 2  i 4 Tg = . (IV.137)       β1 ∑ K i ∆Vi  + β 2  ∑ K i ∆Vi  + (1 − β1 − β 2 ) ∑ ai ∆Vi + ∑ b j        j  i 1  i 2  i 4

Taking into account β2 = 1 – β1 – β2, finally, we get       (1 − 2 β 2 ) ∑ ∆Vi  + β 2  ∑ ∆Vi  + β 2  ∑ ∆Vi         i 1  i 2  i 4 Tg = . (IV.138)       (1 − β 2 ) ∑ K i ∆Vi  + β 2  ∑ K i ∆Vi  + β 2  ∑ ai ∆Vi + ∑ b j        j  i 1  i 2  i 4 Equation (IV.138) binds Tg of the ebonite network with the molar part β2 of disulfide bridges. Substituting numerical values of all increments into this equation, we obtain 92.2 + 8.2 β 2 ⋅103. (IV.139) 136.9 + 299.7 β 2 The dependence of Tg on β2 is shown in Figure 66. As the part of disulfide bonds increases, Tg smoothly decreases and at β2 = 0.5 (disulfide bridges exist only) becomes equal 60°C. Tg =

383

Figure 66. Calculated dependence of Tg for ebonites containing only mono- and disulfide cross-links on the part β2 of disulfide links.

Taking into account that literature data exist on the formation of bridges only with a limited number of sulfur atoms (the average value is assumed to be 2.6 [84]), let us consider the question of the influence of the part of bridges of different length (up to three sulfur atoms) upon Tg. Designate the number of linked units as mlnk = m1 + m2 + m3, where m1, m2 and m3 are the numbers of units linked by mono-, di- and trisulfide bridges, respectively. The total number of units both linked and free equals mlnk + m2 + 2m3 = m1 + m2 + m3 + m2 + 2m3 = m1 + 2m2 + 3m3. Then m1 ; m1 + 2m2 + 3m3 m3 β3 = ; m1 + 2m2 + 3m3

β1 =

m2 ; m1 + 2m2 + 3m3 m2 + 2m3 β4 = . m1 + 2m2 + 3m3

β2 =

Divide numerators and denominators of these expressions by m3

β =

2 2 2 2 + 2 2 +  ;

 β = 2 2 + 2 2 +  ;

β =

2 2 2 2 + 2 2 +  ;

2 2 +  β = 2 2 + 2 2 +  .

(IV.140)

Substituting equations (IV.140) into equation (IV.136), we get

%J . =

2 2 + 2 2 +  +  2 2 +  ⋅ . 2 2 + 2 2 +  +  2 2 + 

(IV.141)

Let us now consider a triangle (Figure 67), analogous to the Gibbs triangle, on the sides of which parts of bridges with various bonds are plotted, and isolines represent levels of equal Tg. Clearly, Tg of ebonites may change in an extremely wide range at the existence of mono-, di- and trisulfide bridges. The maximum softening temperature (monosulfide bonds only exist), coincident with the temperature of

384

Figure 67. Triangular diagram characterizing the influence of composition of network of ebonites on Tg; the numbers at isolines denote Tg.

intensive degradation in this case, equals 400°C. The minimal value Tg = −10°C is typical of the network containing trisulfide bridges only. A case may also be imagined when sulfuric bridges may be of different lengths, and a definite distribution by length exists in the real system. Such variants were discussed in ref. [105], where the glass transition temperature of ebonites was calculated as the value dependent on the function of distribution by lengths of cross bridges composed of sulfur atoms. Consequently, it has been concluded that the bridges may not contain a large number of sulfur atoms. So the statement set in literature that the bridges are mostly formed of disulfide chains is true.

Chapter V. Temperature of transition into the viscous flow state for amorphous polymers

V.1 Estimation of temperature of transition into the viscous flow state of polymers Let us note once again that, in dependence on temperature, amorphous polymer may exist in three physical states: glassy, rubbery and viscous flow. Methods of estimation of the glass transition temperature, Tg, of polymers based on chemical structure of the repeat unit are discussed above. Estimation of not only glass transition temperature Tg, but also temperature of transition into the viscous flow state Tf, which determine the temperature range of the rubbery state, by the chemical structure of polymer is extremely desirable. Therewith, the value of Tf is necessary for processing of polymers. The position of Tf in relation to Tg of the polymer is closely linked with the molecular mass of the chain M. An attempt to estimate Tf was made in ref. [96]. It is based on application of an expression that sets the dependence of zero (Newtonian) shear viscosity η0 on the molecular mass M [177]:

η0 =

0.5 1.5 ρRTπ 2 M 3   M e   1  M e   1 −  +  K    , 15 M e2   M   3  M  

(V.1)

where K = ζb 2 N 02 k BTπ 2 M 2 (b is the segment length; N0 is the number of segments composing the present macromolecule; ζ is the friction coefficient of the segment, including external and internal friction of the initial macromolecule; kB is the Boltzmann constant); Me is the mass of the macromolecule, starting from which entanglements of chains affect significantly the melt viscosity; R is the universal gas constant. According to refs. [175, 176], equation (V.2) holds in the range of Me < M < 5Me. It may be shown that the value of K does not depend upon M. Actually, b2 = 2 n0a , where n0 is the number of repeat units in a segment; a is the typical size of the repeat unit; ζ = ζ0n0 (ζ0 is the friction coefficient for a single repeat unit). Then K = ζb 2 N 02 k BTπ 2 M 2 = ζ 0 n0 n0 a 2 N 02 k BTπ 2 M 2 =

(V.2)

= ζ 0 (n0 aN 0 )2 k BTπ 2 (M a n0 N 0 )2 = ζ 0 a 2 k BTπ 2 M a2 .

Hence, the value of K depends upon characteristics of the repeat unit only and   is independent of the molecular mass; therewith, a ~  ∑ ∆Vi     i  the Van-der-Waals volume of the repeat unit.

23

, where

∑ ∆Vi i

is

386

Equation (V.2) is used in ref. [84] for determination of the temperature of transition into the rubbery state Tf. Displaying no details of analysis, let us write down the final expression for estimation of Tf: Tg Tf

=1+

1 ln C

B 3 1 0.5  X 1.5  ( AX ) − 1 +    3 

,

(V.3)

where C is a constant (C = 26); X = M/Ms, M is the molecular mass of the polymer, Ms is the molecular mass of the mechanical segment of the macromolecule, i.e. the molecular mass starting from which the glass transition temperature becomes independent of M; A = Ms/Me; B = (A0.5 – 1)3 + 1/3. As the data in ref. [177] indicate, for most polymers (polystyrene, poly-αmethylstyrene, polyvinylacetate, polyisobutylene, polyisoprene) A ≈ 2.4. For polyethylene only, A ≈ 5.2–6.8, and for polybutadiene, A ≈ 4.3. Calculations for different values of A give the following constants B: A = 2.4 → B = 0.5; A = 4.3 → B = 1.57; A = 5.2 → B = 2.43; A = 6.8 → B = 4.48. Substituting values of parameters A, B and C into equation (V.3), for the general case we obtain: Tg Tf

=1+

1 ln 26

B . 3   1.5  0.5   M    M  1    2.4  − 1 +  M M 3   s  s     

(V.4)

To use equation (V.4), it is necessary first to estimate the value of Ms. It may be calculated with the help of equation (V.1) with regard to the condition that Ms = Ns⋅M0, where Ns is the polymerization degree of a mechanical segment of macromolecule; M0 is the molecular mass of the repeat unit of the polymer. The value of Ns (or Ms) determines the transition temperature to the rubbery state of the polymer, when Tg is yet independent of the molecular mass. However, when N < Ns (or M < Ms), the polymer also transits into the glassy state, but then Tg will depend on the molecular mass. In this case, Tg = Tf (but therewith, it should be taken into account that Tg is lower than the true glass transition temperature). Expression (V.4) enables Tf to be also determined for this case. It is obvious that the border transition temperature into the viscous flow state is limited by the temperature of onset of intense thermal degradation Td (see Chapter VII for details). From this point of view, not at every molecular mass a polymer may transit into the viscous flow state. Using the expression (V.4) and assuming that Tf = Td, the maximal polymerization degree (or the maximal molecular mass) may be determined, at which the polymer may be transited into the viscous flow state (Table 20). However, temperature ranges of the viscous flow state, obtained in this case, are by 15–20% higher than really observed ones. This is due to the fact that due to

387

polydispersity of synthetic polymers, a part of chains transits into the viscous flow state (actually, a superposition of the viscous flow and the rubbery states is observed), which causes overload of longer chains. Table 20 Maximum degree of polymerization Nmax at which the polymer can be transferred from the glassy state to the viscous flow state without degradation Polymer Tg/Td [84] Xmax Ns Nmax Polyethylene 0.575 112 3136 ∼28 Polyisobutylene 0.420 72 144 10368 Polystyrene 0.612 18 320 5760 Polyvinylacetate 0.716 227 2043 ∼9 Poly(methyl methacrylate) 0.506 40 307 12280 Poly(vinyl chloride) 0.630 15 230 3450

To illustrate the above-said, let us analyze some experimental data displayed in ref. [109], in particular, by polyisobutylene and polyvinyl chloride. Analysis of the data on polyisobutylene with the help of expression (V.4) indicates that at N > 10 400 the polymer is unable to be transited into the viscous flow state. All the rest molecules with higher polymerization degree would inevitably degrade (under the influence of either mechanical stresses or temperature) when tried to be transited into the viscous flow state. The latter causes the fact that long molecules disrupt due to their overload. Broken chains may be found after thermomechanical analysis by determination of the molecular-mass distribution. The value of Tfcalc at N = 10 400 found with the help of expression (V.4) equals to 181°C, whereas Tfexp = 120°C. Reduction of Tf ( Tfexp − Tfcalc = 61°C) is stipulated by the fact that, according to the data from [109], longer molecules are ~50-fold overloaded because of shorter molecules transition into the viscous flow state. If the experiment is conducted at the load of 0.07 MPa, long molecules in the range of Tf are affected by the load of ~3.5 MPa. Hence, if we want to obtain Tf independent of more or less disregarded attendant factors as polydispersity with the help of thermomechanical analysis, it is necessary to introduce additional limits for the polymerization degree of images, therewith, setting the condition that the value of ∆T = Tfexp − Tfcalc is smaller than the limits of the experimental error. The polymerization degree N, up to which polydispersity does not affect Tf, may be estimated from the empirical expression: NTg N sTf

≈ 2.3.

(V.5)

The influence of polydispersity must be displayed in dynamic mechanical tests. In this case, asymmetry of the temperature dependence of the mechanical loss tangent in the area of the α-transition (at T < Tα) is also associated with transition of shorter chains with N < Ns into the viscous flow state. A polymer possessing no lowmolecular component (N > Ns) must display a symmetric maximum of losses.

388

V.2 Dependence of Newtonian viscosity on molecular mass of polymer in a wide range of its change It is common knowledge that the reptation Doi–Edwards model has allowed substantial developments in the rheology of concentrated solutions and melts of polymers. Already in initial works by Doi–Edwards [225], the following law for Newtonian viscosity η0 was obtained:

η0 ~ (M/Me)3 at M ≥ Me,

(V.6)

where M is the molecular mass of the polymeric chain; Me is the molecular mass starting from which physical entanglements between chains affect molecular motion of the chain. This formula was deduced only with regard to reptational motions in the chain. Further on, with regard to fluctuation of the tube contour, in which reptational motion proceeds, Doi [226] succeeded in refining dependence (V.6) for M/Me >> 1. The form of the expression deduced by Doi  M   η0 ~   Me 

3

M  1 − 1.47 e   M  

0.5  3

 

(V.7)

enabled the dependence of η0 on M/Me to be described for M/Me >> 1. However, expression (V.7) in the range of values M/Me < 20 indicates an abrupt deviation with experimental data [226]. More accurate calculation of fluctuations of the tube contour has enabled Lin [174, 175] to deduce an expression: 3 1.5  0.5 3  M   1M    Me    1 − 1.47 η0 ~    +  e  , 3 M    M    M e    

(V.8)

which describes the behavior of η0 down to M/Me ~ 5 or even 1, if the value of Me is corrected in each of these areas. As indicated in the work by Lin [174], Doi had insufficiently utilized the possibilities of the theory. Moreover, as it will become clear below, if summation in appropriate expressions deduced in ref. [227] is performed completely, then the dependence η0 = f(M/Me) may be obtained within the framework of the Doi–Edwards theory. This dependence describes efficiently the experimental data in the total area of M/Me variation, i.e. from unit to as large as one likes value (in the area preceding the chain rupture at shear). As indicated in ref. [96], calculated estimation of the glass transition temperature of polymers in the range Me < M < Ms requires taking into account not only the chemical structure of the repeat unit of the polymer, but also its viscosity depending on the molecular mass. Obtaining reliable analytical dependences of viscosity on molecular mass is of the utmost importance for proteins when calculating the glass transition temperature

389

(specificity of proteins is so that, in contrast to synthetic polymers, many of them have the molecular mass in the range Me < M < Ms). In contrast to protein macromolecules, synthetic polymers usually possess a wide molecular-mass distribution. The presence of low-molecular fractions causes a change of the glass transition temperature of the system and spectra of relaxation times of it. Hence, for both proteins and synthetic polymers, a significant role is played by entanglements between chains (the notion of ‘entanglement’ is interpreted below). By virtue of the fact that the molecular mass of protein macromolecules is not large enough, entanglements affect the glass transition temperature. In this respect, proteins occupy a special place among high-molecular compounds. Taking into account the polymerization degree is also important for many heat-resistant polymers, which possess no rubbery state due to low polymerization degree N < Ncr, where Ncr is the critical polymerization degree, and at heating transit directly to the viscous flow state. An attempt was made in ref. [772] to deduce an analytical expression, which allows description of the dependence of Newtonian viscosity in the whole range of changes of the molecular mass Me ≤ M ≤ Mcr. Following this work and according to refs. [226, 227], let us write down an expression for the relaxation shear modulus: G (t ) =

4nkTL µ A (t ) µ B (t ) µ C (t ) . 5a

(V.9)

Here

µA = 1+

N cr

 tp 2  ,  A  

∑ exp − τ

p =1

    95    , µ  = ∑    + [α  − ]H[S −  5= π 5   τ   µC =

N0

8

(V.10)

 tp 2  ,   C

∑ π 2 n 2 exp − τ

n =1

where τA, τB, τC are relaxation times for motions of different types: τA is relaxation time of the linear chain between neighboring entanglements; τB is maximal relaxation time of the whole chain; τC is time of reptational motion of chain. In accordance with ref. [227] expression (V.10) may be rewritten in following form:

   95    µ  =  + [α  − ]∑   H[S −  5= π 5  τ 

the the the the

(V.11)

(α(E) is the function of transformation of a chain length element (affinor), which depends on the stretch ratio). Then, transferring from summation to integration, under the condition

390

N0

8

∑ π 2 p2 = 1

p =1

it may be written down N0

8

∑ π 2 p2

p =1

=

N0

∫

dp

2 1 p

.

(V.12)

Let us consider the range of such times t when µA(t) = 1. With regard to expression (V.11), let us write down that ′ , µBµC = µC + [α(E) – 1] µ B where ′ = µB

  p2 n2   exp ∑ π 4 p 2 n 2  − t  τ + τ   . C  p, n   B 64

(V.13)

In future discussion, we will be interested in the maximal Newtonian, or ‘zero’ shear viscosity η0 which, according to ref. [226], is determined as ∞

η0 = ∫ G (t )dt , 0

and in this case equal   QN7/      (α  − )τ % ∑    +τ & ∑    . η = D  5  3 π S Q S  + Q τ % 3 π Q    τ&

(V.14)

Because according to ref. [227], 2

3

N  N  τ B = 2 0  τ A , τ C = 6 0  τ A ,  Ne   Ne  then

τ B 1 Ne = . τ C 3 N0

(V.15)

Let us now calculate sums included in expression (V.14) with regard to condition (V.15). In accordance with transition (V.12), let us write down

391

N N

0 0 1 1 dpdn ∑ π 4 p 2n 2 2 1 N e 2 → ∫ ∫ p 2n 2 2 1 N e 2 . n n p + p + 1 1 3 N0 3 N0

64

Transiting to new variables x = 1/p and y = 1/n, let us calculate the integral 0.5

1/ N0

∫

1

 1 Ne    xN 0 0 .5 2 3 N 0   1 Ne  y dy   x ⋅ arctg . = −1 +  1 Ne 2 1 Ne 2  3 N0  x + N0 x y2 + 3 N0 3 N0

Taking into account that

(V.16)

1 Ne 2 x << 1, integral (V.16) may be presented 3N 0 N 0

in the following form: 1/ N0

∫

1

 1 Ne  y 2 dy  = −1 +  1 Ne 2  3 N0  x y2 + 3 N0

0 .5

 1 Ne   x ⋅ arctg  3 N0 

0.5

x.

(V.17)

With regard to expression (V.17), let us calculate the integral 1/ N0 1/ N0

∫

1

∫

1

1.5 0 .5 0.5   1 Ne    1 Ne  7 1  3N 0  x 2 y 2 dy  . (V.18)   arctg  −  = +  1 N e 2 12 4  N e   3 N 0    3 N0  x y2 +  3 N0

Analogously, the sum at τC may be calculated, which equals 8

N0

1 = . 3 1 n

∑ π 2n4 → ∫ n

dn

(V.19)

4

Substituting values of sums (V.18) and (V.19) into expression (V.14), we obtain: η =

        1    1    QN7/  

 − 0   DUFWJ  10  τ +  τ  . (V.20) (α  − ) +             D      10    1     1      

As indicated in ref. [227], α(E) may be presented as

α  =  +

 ∑ εαα +  α

(ε ), 

where εαα are components of the strain tensor of the chain.

392

  If a polymer is considered incompressible  ∑ ε αα = 0  , then the relaxation   α  process τB is displayed in the non-linear area only. With regard to volumetric compressibility (which, in fact, was made by Lin) in δL 1 B (B is a constant, which may be found from the condition the form of ∑ ε αα = L0 3α of equality of the viscosity calculated by expression (V.20) and the Rous viscosity at M = Me); fluctuations of the chain contour will contribute to the linear viscosity. But taking into account fluctuations of the chain contour length, according to ref. [226], causes a change of coefficient at τC, which will be taken into account by constant C. Then expression (V.20) may be presented in the following form: η0 =

1 .5 0 .5 0 .5    1 Ne   1 Ne   Ne 4nkTL0  δL 7 1  3 N 0      −   arctg + C  . (V.21) τ C  B +  15a  3 N0   3 N 0   N 0  L0 12 4  N e  

~ Considered in ref. [226] are fluctuations of the contour length L (t ) , according

(

)

2 1/ 2 ~ to which values of fluctuations are δL = L (t ) − L and are determined from the

correlation:

δL ≈ L(Me/M0)1/2. The same work has indicated that the average contour length of the chain is determined as follows: ~ L (t ) ≡ L . The analogous meaning of L(t) is indicated in ref. [174]. According to this work, the contour length of the chain is determined from the same correlations. One more important point of the work [174] is that at time, t equal to the correlation time of fluctuations δL(t) and designated as τB, a part of tube, which is still in the stress state, shortens to length L0. Further on, expression for G(t) in ref. [174] is presented in the form: G (t ) = 0.5

TL  t t   Bµ B , + Cµ C 5a  τB τ C 

δL  M e  =  . L  M  Because according to refs. [226] and [174]

where B =

393

δL  M e  =  L0  M 

0.5

0.5  M   , C ~ 1 − 1.47 e    M   

3

and taking into account that Ne/N0 = Me/M, expression (V.21), finally, may be presented in the following form:

η0 =

 M  ρRTπ 2  KM e  15  Me 

3

  M   M   M   ,  + f P   + f D  1.28 f L   Me   Me   M e  

(V.22)

where fL is the Lin function, fL(M/Me) = 1/3 (Me/M)1.5; fD is the Doi function, fD(M/Me) = [1–1.47(Me/M)0.5]3; fP is an additional function obtained in work [767],

1 (  !

 

 



     0 ) =  − 0 DUFWJ 0  , ρ is density.      The numerical coefficient at fL(M/Me) is calculated under the condition that at

M/Me = 1, η0 from expression (V.22) coincides with the value of η0R from the Rouse formula, i.e.

η0R =

ρRTπ 2 KM . 36

Correlation (V.22) is true at M/Me ≥ 1. In the case when M/Me < 1, in equation (V.22) it should be taken M/Me = 1 in order to transit to the Rouse expression. Let us consider two functions ϕ(M/Me) which take into account influence of the molecular mass (polymerization degree) on Newtonian viscosity η0:  M   M   =   ϕ P   Me   Me  and

3

  M   M   M    + f P   + f D  1.28 f L   Me   Me   M e  

(V.23)

ϕL(M/Me) = [fL(M/Me) + [1 – (Me/M)0.5]3](M/Me)3,

where ϕP(M/Me) is the function deduced in ref. [767]; ϕL(M/Me) is the function deduced in refs. [174, 175]. Figure 67′ represents calculation results for the ratio ϕP(M/Me)/ϕL(M/Me) depending on X = M/Mc in the range of variation of X: 1 ≤ X ≤ 40. A difference between the results suggested in ref. [767] and the results by Lin is observed in the area of high values X ~ 10. But if results obtained by the Lin formula for ϕL and by the Doi formula for ϕD are compared, it is found that this deviation is much higher. At X ~ 10 ϕD/ϕL = 0.676, and ϕP/ϕL = 0.935, i.e. the Lin formula overestimates ϕ in the area of large values of X. Equation (V.23) enables data to be improved, obtained by the Lin formula in the range of high X, and approximate them to the experimental data.

394

Figure 67′. Dependence of ϕP/ϕL on X.

Basing on expression (V.22), the calculation scheme for determination of the transition temperature to the viscous flow state Tf, suggested in ref. [96], may be improved. Using the same expressions presented in ref. [96], the expression for Tg/Tf will have the following form: Tg Tf

= 1+

[

]

3   1 1   ln B ′ X 1.5  (AX )0.5 − 1.47 + 0.427 + 1.08(AX )1.5 1 − × C  9 AX   

 1  × arctg   3 AX 

0.5 

−1

(V.24)

  , 

where A = Mc/Me, X = M/Mc, C = 26 [96], B′ = (A0.5 – 1.47)3 + 0.427 + 1.08A1.5(1 – 1/9A)arctg(3A)-0.5. When A = 2.4 [230], B′ =1.805. If the ratio Tg/Tf is determined at 1/A = X = 2.4-1, then Tg/Tf = 1.08 and the glass transition temperature Tg* of polymer, which possesses M < Mc, is Tg* = Tg/1.08 = 419 K. For example, if Tg = 180°C, then Tg* = 146°C. The maximal shift of Tg at N = Ne will represent ∆Tg = 34°C. Calculations by the Lin formula indicate Tg/Tf = 1.066, Tg* = 152°C and ∆Tg = 28°C. Expressions obtained for η0 = f(N/Nc) allow a significant refinement of sought for dependence of η0 in the area of Ne ≤ N ≤ Nc. However, when they are used for determination of Tg* in the area of Ne ≤ N ≤ Nc, they refine Tg* insignificantly compared with the Lin expression [174, 175]. That is why, in future consideration, determining Tg* or Tf at N > Nc, more simple Lin expression may be used. Basing on the approach considered, irregular polymeric systems may be subdivided into three classes according to the type of transition from the glassy state into the viscous flow state. 1. Substances of the polymeric type (polymers in their classic meaning), which display Tg independent of the polymerization degree.

395

2. Polymeric systems of the Edwards–Doi type, in which a significant role of the reptation motion is played and Tg depends on the polymerization degree N, therewith Ne ≤ N ≤ Nc. 3. Substances of the Rouse type – oligomeric systems with the polymerization degree N < Ne. In the case of the Rouse substance, the Newtonian viscosity of a melt may be presented in the following form:

η0R =

ρN Aζ a a 2 ρRTπ 2 N. KM = 36 36M a

In accordance with the principle of determination of typical temperatures which was used by us in ref. [96], the Rouse glass transition temperature will be determined from the condition of equality of the coefficients of viscosity at a temperature change

ρ Rζ a (TR ) N R = 1, ρ Eζ E (TE ) N e

(V.25)

where ρR and ρE are densities of polymer melts, included in the Rouse and the Lin formulae, respectively; ζa and ζE are friction coefficients of chains from the same formulae, respectively. Suggesting that ρR = ρE, correlation (V.25) transforms to the following equation:

ζ %# = ζ %

0 # .

(V.26)

Taking into account that ζ(T) ~ exp(∆E*/RT) and that, according to ref. [96], ∆E* = CTg, expression (V.26) may be presented in the form: N TR 1 = 1 + ln e . TE C′ N Here C ′ = C

Tg

(V.27)

Tg

= 1.08 and C′ ≅ 28, N < Ne. TE TE Let polymer possess Nc = 360, Ne = 150, and NR = 75. Then ,

TR =

Tg TE = . 1.025 1.067

Let Tg = 453 K, then TR = 409 K = 136°C, Tg = 180°C, TE = 146°C. The value of B′ (and of A, as well) may be determined as the cross point of the tangent appropriate to the Rouse law and the tangent appropriate to the Edwards–Doi law for low-molecular fractions of the current polymer.

396

As indicated above, a significant role in determination of Tg in the range of N < Nc is played by the parameter A = Nc/Ne. An additive scheme of determination of Nc basing on their chemical structure of the repeat unit of the polymeric chain was suggested in ref. [96]. Ref. [767] gives the additive scheme for determination of Ne. Attempts to determine Ne were already made long ago – since the reptation model had appeared. In a series of works [230, 231] Ne was estimated on the basis of topological ideas considering cross-linked points as a consequence of chain entanglements. Analytical expressions obtained are complicated for analysis and do not solve the final task – setting of a correlation between Ne and the chemical structure of the chain. Moreover, it has been already indicated in works [233, 234] that topological cross-linked points are unable to make a significant contribution to the thermodynamic properties of polymers because of low probability of their appearance. The latter experimental data [229] indicate that Ne significantly depends on temperature. Basing on the results of works [229, 233, 234], let us suggest the following scheme of Ne determination. First, all cross-linked points in polymer are of the physical type and appear due to capture of one chain by a potential pit of another chain at thermal motion of them relative to each other. Hence, a network with physical cross-linked points appears in the polymer. Secondly, a stress σ = Eelε appears in between cross-linked points of the chain, i.e. any fixation of parts of the chain causes occurrence of stress between fixation points due to thermal motion (this fact is known in the theory of the rubbery state). And, thirdly, the reptation motion inside a tube of diameter d appears when a cross-linked point–oscillator formed by captured unit of a neighboring chain loses its stability under the effect of applied stress. The points formulated above may be presented in the following form: Eeε max

πd 2 = f max , 4

(V.28)

where Ee is the elasticity modulus of the network, Ee = 3ρRT/Me; Me is the molecular mass of the chain between physical cross-linked points, Me = µNe; µ is the molecular mass of the repeat unit; d is the diameter of the tube in which the reptation proceeds; fmax is the maximal force at which the cross-linked point–oscillator loses its stability, and reptation motion starts. Monograph [28] indicates solution of the problem of stability loss by an oscillator under the effect of high intensities and, according to these data,  3 Tm f max = ak BT 1 − 1 − e T 4  εmax = const1/ar0.

 ,  

(V.29) (V.30)

Substituting expressions (V.29) and (V.30) into expression (V.28), we obtain: Ne = d2/(C∞b2), where

(V.31)

397

 3 Tm  C∞ = 1 − 1 − const 2 . (V.32) 4e T   Expression (V.31) is analogous to that obtained in ref. [96]. Moreover, it enables the meaning of parameter C∞ to be defined concretely. The dependence of C∞ on T completely indicates experimentally measured dependences of Ne on temperature [96]. Using the Beaman rule Tg/Tm = 2/3, expression (V.32) may be presented in the form:  9 Tg C ∞ = 1 − 1 −  8e T 

  const . 2  

(V.32)

In fact, Tg/Tm may change in a wide range depending on the chemical structure of the repeat unit, and this fact may be taken into account with the help of an appropriate additive scheme [96]. Using formulae (V.31) and (V.33), and the expression for Ne, as well, deduced in work [96], Nc/Ne may be presented in the form: 1F O   7J  = FRQVW7J   −  − . (V.34) H 7  1H G    The value of const is constant for all polymers, l03 =  ∑ ∆Vi  , d2 is the    i  r.u. cross-section square of the repeat unit. This analysis makes it possibility to describe all three states of the polymer, the temperature of transition from one state to another, as well as the area of N < Nc via typical temperatures. This allows approaching from new positions the interpretation of results of acoustic spectroscopy for polydispersity polymers, influence of short chains (N < Nc) at T < Tα (temperature of α-transition), for example, β-transition.

Chapter VI. Melting point of polymers

The melting point is determined as the temperature at which a polymer transits from the crystalline state into the viscous flow state. In contrast to low-molecular substances, in which this process proceeds in a jump-like manner, in the case of polymers melting is observed in a temperature range. This happens due to polydispersity of polymeric chains, their branching and imperfection of crystallites formed. The equilibrium and experimental melting points are distinguished. The equilibrium melting point Tm0 = ∆H m ∆S m , where ∆Hm is the melting enthalpy, ∆Sm is the melting entropy. The equilibrium melting point is determined by the point of the phase equilibrium between a monocrystal of the polymer and its melt. Since perfect monocrystals are difficult to obtain from the polymer, the equilibrium melting point is determined by methods of extrapolation, for example, by extrapolation of the dependence of the experimental melting point on the size of crystallites or on the molecular mass of the polymer. The melting point Tm is a physical characteristic and is most difficult to calculate. The case in point is calculation based on the chemical structure of the repeat unit of the polymer. Let us discuss two approaches to solution of this problem. One of them is based on estimation of the relation between the glass transition temperature Tg and the melting point Tm. It should be noted that, according to the Beaman rule [132], Tg/Tm ≈ 2/3. However, the detailed analysis of a large range of polymers of extremely various structures has indicated [172] that this relation varies in a wide range, although for a broad group of polymeric systems it is ~2/3. The equation expressing the relation between glass transition temperature Tg and melting point Tm is deduced in ref. [42] basing on the experimental data, according to which the coefficient of molecular packing of a crystalline polymer at the melting point is approximately equal to the coefficient of molecular packing of a amorphous polymer of the same structure at the glass transition temperature, i.e. melting of a crystalline polymer and transition of an amorphous polymer from the glassy state into the rubbery state occurs when the same part of the empty volume is reached. The relation for Tg/Tm is presented in the following form [42]:   ∑ ∆Vi  Tg  i = − A Tm  ∑ (δ i ∆Vi ) + ∑ γ j  j   i

−1

.

(VI.1)

Here δi = (k0 – kg)/ki (ki is the partial coefficient of packing of the i-th atom); γj are the constants taking into calculation the contribution of strong intermolecular interactions; A = kg/(k0 – kg) = 10.418; the meaning of the rest of designations is the same as in equation (IV.39). Values of δi and γj are shown in Table 21.

399

Table 21 Values of parameters δi and γi of various atoms and types of intermolecular interaction Atom or type of intermolecular interaction Designation δi γi, Å3 Silicon 0.0840  δSi Carbon 0.08685  δC Hydrogen 0.0740 δH  Oxygen in the backbone 0.0621  δO,b Oxygen in a branch 0.0963 δO,s  Nitrogen in the backbone – 0.0212  δΝ,b Dipole–dipole interaction -0.0727  γd Hydrogen bond -0.0188  γh p-Substitution -0.100   Hydrogen bonds in polyamides:   Aromatic polyamides 0.422   Even aliphatic polyamides 0.392   Even–odd aliphatic polyamides 0.392   Odd aliphatic polyamides 0.467   Aliphatic–aromatic polyamides containing an even 0.445   number of –CH2-groups Aliphatic–aromatic polyamides containing an odd 0.544   number of –CH2-groups Aliphatic–aromatic polyamides based on iso-, 0.445   terephthalic acids and p, m-phenylenediamines containing –CH2-groups only

In many cases, calculations performed by equation (VI.1) indicate good coincidence with experiment. To increase the accuracy of calculations, it is advisable to know group contributions ∑ (δ i ∆Vi ) + ∑ γ j of polymeric groups, from which i

j

many polymers are composed. These group contributions are obtained by summing up values of δ i ∆Vi for atoms participating in the current group. Then values of γj typical of each polar group and type of substitution in aromatic cycles, and small readjustments, which increase accuracy of the calculation, as well, are added to the sum. The readjustments are deduced by comparing experimental and calculated values of Tm for a series of polymeric standards. Values of ∑ (δ i ∆Vi ) + ∑ γ j for a i

j

series of atomic groups are shown in Table 22. Table 22

Corrected values of quantities

∑ (δ i ∆Vi ) + ∑γ j for a series of basic fragments (See text) i

j

Chemical structure of a fragment and neighboring atoms bonded to it

&««Š&+2Š««& 1

C......

CH

.......C

∑ (δ i ∆Vi ) + ∑γ j i

j

2 1.434 2.795

CH3 C H3 C ......

C C H3

.......C

4.302

400

1

C......

CH

.......C

2 4.228

C2H5 C......

CH

.......C

3.652

O CH3 C......

CH

.......C

2.702

&««Š&&O2Š««&

3.770

Cl

C......

CH

.......C 7.889

CH3 C......

C

.......C 9.177

C......

C

.......C

1.576

O C......

CH

.......O

3.064

CH3 CH3 C ......

C

.......O

4.623

CH3

C......

CH

.......O 8.175

C......

.......O 6.745

C......

.......O

2««Š&+2Š««2

6.645

1.972

401

1

2

.......O

O......

7.028

O......

.......O

6.928

.......O

9.826

CH3 O...... CH3

&««Š2Š««& &««Š2Š««& &««Š1+Š««& S......

.......C

S......

.......C

S......

.......S

0.211 0.168 0.153 6.567

6.467

6.671

6.571

S......

S......

.......S

.......N

6.724

6.624

S......

.......N

S......

.......O

S......

.......O

6.849

6.749

402

&««Š6Š««& 1

O

O

C

C N

N

C......

2 1.338

C

12.000

C

O C......

.......C

O C

.......C O

9.455

C O

C......

C

.......C 13.290

CF3 C......

.......C

C

5.997

CF3 .......C

C......

.......C

C......

6.463

6.263

NH

C......

C

.......C

8.630

N

&««Š&+2Š««2 &««Š2Š««& &««Š&+2Š««1 1.54 C......

C O

1.50

.......C

1.703 0.130 1.573 2.083

O 1.54 C......

C O

1.37

.......C

2.039

O C......

C

.......N

1.856

O C......

.......N

6.619

403

1

2

O......

6.519

.......N

1.37

C......

1.54

NHC

.......C

2.403

O

O......

6.802

.......N

6.776

.......N

N......

N......

C......

6.902

.......N

O......

6.676

.......N O

O

C

C

N

N C

C

O

O

.......C

16.646

O C O......

9.882

N

.......C

C O

O C C......

9.600

N

.......C

C

.......C

C O O......

NH N

8.912

404

1

2

O C

9.704

S......

N

.......C

C O NH

S......

C

.......C

8.734

N C H3 O ......

Si

.......O

5.986

6L«« 2 ««6L

0.042

C H3

ŠŠ

O......

C

.......O

2.541

O C......

N

.......C

2.049

CH3 N......

.......N 10.685

C......

.......C 10.172

Š Š

&«« &)2 ««&

2.048

Application of these values enables the melting point to be calculated with high accuracy. This is indicated by Table 23, which displays experimental values of Tm for a series of polymers, as well as the calculated values obtained with the application of atomic and group contributions.

Table 23 Experimental and calculated values of melting point Tm for a series of polymers Tm, K Tm, K Tm, K calculated by calculated by Polymer experimental equation (VI.1) equation (VI.5) 1 2 3 4 Polyethylene 410 410  Polypropylene 449 451 421 Polybutene 405 413 418 Poly-4-methylpentene-1 508 500 504 Poly(ethylene adipate) 323 334 321

405

1 Poly(ethylene oxide) Poly(propylene oxide) Poly(tetramethylene oxide) Poly(tetramethylene adipate) Poly(trimethylene adipate) Polystyrene Poly-α-vinylnaphthalene Poly(hexamethylene adipamide) Poly(hexamethylene sebacamide) Polyamide-6 Polyamide-11 Polyvinylisobutyl ether Poly(methyl methacrylate) Polyvinylethyl ether Poly(dimethyl siloxane)

2 339 340; 348 338; 309 332 311 513; 523 633 539 499; 488 488; 499 467; 455 438 433 417 234

3 348 356 311 314 315 509 625 539 494 496 481 497 465 398 234

4 314 342 349 334 311          

Melting points of copolymers cannot be described by a simple relation, deduced from equation (VI.1) using the rule of additivity. Another approach [29] is based on consideration of the repeat unit of the polymer as a set of anharmonic oscillators. According to ref [114], the free energy of an anharmonic oscillator is: F ≤ F0 +

mω 2 2 β 3 a − a − βa y 2 , 0 2 3

(VI.2)

where

=ω  = α   =ω  F0 = kT ln 2sh ; y2 = cth ; ω = ; 0 2mω m 2kT    2kT  (m is the mass of the atom; α is the elasticity coefficient of the oscillator; β is the coefficient of anharmonicity of the oscillator; a is the value which characterizes the displacement of the equilibrium point of the harmonic oscillator relative to zero). The value of a in expression (VI.2) is deduced from the condition of the minimum of the free energy by a, i.e. when mω2a – βa2 – β〈y2〉0 = 0.

(VI.3)

The condition of stability loss of this system is considered, and the critical temperature Tcr, at which stability is lost, is determined from the expression: Tcr =

where

=ω , A +1 k B ln A −1

(VI.4)

406

2

2mω  mω 2  A= . η  2 β  The critical temperature Tcr is a phase transition temperature, i.e. of melting in this case. If the both Lennard–Jones potential, with the help of which parameters of the equation (VI.2) are determined, and the expression for the coefficient of volumetric expansion are used, the following correlation for estimation of the melting point may be obtained [29]:

∑ K i ∆Vi 1 = i , Tm ∑ ∆Vi

(VI.5)

i

where Ki = 18.52R/(zDi) (R in the universal gas constant; z is the coordination number; Di is the energy of intermolecular interaction occurred at the expense of the i-th atom). The number of values of Ki is determined by the number of atoms composing the repeat unit. But because some of atoms participate in the composition of polar groups with strong dipole–dipole interaction, hydrogen bonds, etc., the latter may be taken into account by addition of a part of energy strong intermolecular interaction, stipulated by contribution of the i-th atom, to the energy of dispersion interactions Di. Then K i,1 ~

1 1 ; ; K i,2 ~ Di Di + ∆Did

.  = L

'

L

 ; etc., + ∆' K

(VI.6)

L

where ∆Did is the contribution of the i-th atom into the dipole–dipole interaction;

∆'K is the contribution of the i-th atom into hydrogen bonding; etc. Computerized calculations performed according to equation (VI.5) by the method of least squares have indicated [29] that for polymers, containing atoms of carbon, hydrogen and oxygen in the repeat unit, for satisfactory calculation of the melting point of a series of polymers basing on chemical structure of the repeating L

d unit, it is enough to know DH, DH , D0, and D0d parameters (Table 24); values of Di were calculated at z = 4.

Element Hydrogen Oxygen

Table 24 Numerical values Ki, Di and ∆Di for hydrogen and oxygen atoms Di, kcal/mol Designation Ki,x⋅103, deg–1 ∆Di, kcal/mol KH 10.42 0.88  10.03 0.92 0.04 d KH 16.5 0.56 KO  13.3 0.70 0.14 d KO

407

Table 23 displays calculated values of Tm for a series of polymers containing atoms of carbon, hydrogen and oxygen in the repeat unit. In spite of quite good convergence of the calculated and experimental data, it should be noted that the present method is of limited application.

Chapter VII. Temperature of onset of intense thermal degradation of polymers

Let us consider the characteristic of thermal stability of polymers, determined with the help of thermogravimetric analysis. The temperature dependences of the substance mass at continuously increasing temperature are determined with the help of this method (thermogravimetric curves). It is common knowledge that for most of polymers, thermogravimetric curves possess the shape schematically represented in Figure 68. Estimating the thermal stability of a polymer, let us use the temperature of onset of intense thermal degradation Td determined by the point of intersection of tangents to two branches of the thermogravimetric curve (see Figure 68).

Figure 68. Schematic representation of the thermogravimetric curve and the method of evaluation of Td value.

The correlation, indicated in ref. [88], which estimates the temperature of onset of thermal degradation Td is deduced on the basis of consideration of valency bound atoms as a set of anharmonic oscillators which form the repeat unit of the polymer. At this point, approach is the same as for estimation of the glass transition temperature Tg and the melting point Tm. But in this case, the energy of chemical bonds but not the energy of intermolecular interaction is taken into calculations, although the latter also displays a significant influence on the energy of dissociation of chemical bonds. For example, it is common knowledge that the energy of dissociation of C–C (carbon–carbon) bonds changes in the range from 30 to 90 kcal/mol [64] depending on which group contains carbon atoms, i.e. according to their valence surrounding. The same also relates to other pairs of valence bonded atoms (C–O, C–S, C–N, etc.). When a polymer is heated up, its volume changes, this change being composed of two parts: increase of the free volume and change of chemical bonds lengths. Analysis of these changes has led to the following dependence of the temperature of onset of intense thermal degradation Td on the parameters of the chemical structure of the polymer [88]:

409

∑ α xi ∆Vi 1 = 42 i , Td ∑ ∆Vi

(VII.1)

i

where α xi is the partial coefficient of volumetric expansion of the i-th atom, appeared at the sacrifice of the change chemical bonds lengths; ∆Vi is the Van-der-Waals volume of the i-th atom; ∑ ∆Vi is the Van-der-Waals volume of the repeat unit of i

the polymer. Therewith  3Rβ   , α xi =   γ 2d  0 i 

(VII.2)

where β is the coefficient of anharmonicity, β = −

γ =

1 ∂ 3ζ 2 ∂d 3

; γ is a force constant, d0

∂ 2ζ

; ζ is the potential of chemical interaction. All values are given for the i-th ∂d 2 atom, chemically bonded with other atoms. To estimate the energy of dissociation of chemical bonds, let us use the Morse potential

(

)

−2   ζ ( d ) = E  e − a ( d − d 0 ) − 1 − 1 ,  

(VII.3)

where d is the distance between chemically bonded atoms; d0 is the equilibrium distance; E is the energy of dissociation of chemical bonds. Taking into account expression (VII.3),

α xi = Then Td =

3 R    . 4  ad 0 E i

(VII.4)

∑ ∆Vi i

∑ Ki ∆Vi

,

(VII.5)

i

where K i = 31.5

R . (ad 0 E )i

(VII.6)

In the case of dissociation by C–H bonds, the values of constants are d0 = 0.108, Eavg = 396 J/mol, a = 0.0266 nm–1. Values of Ki are shown in Table 25.

410

Table 25 Conventional symbols and numeric values of constants Ki Designation of Atom or group Note Ki⋅103, Å3K–1 constant 1 2 3 4 Carbon KC 1.150 — 1.920 Suitable for carbon atom participating Carbon K Cd in a polar group Hydrogen KH 2.307 — d 0.556 Suitable for hydrogen atom Hydrogen KH participating in a polar group Oxygen KO 0.058 — d 1.572 Suitable for oxygen atom participating Oxygen KO in a polar group d 2.520 Suitable for nitrogen atom Nitrogen KN participating in a polar group d Nitrogen 0.411 Suitable for nitrogen atom K N,c participating in a heterocycle Sulfur KS 1.900 — 6.300 Suitable for sulfur atom participating Sulfur K Sd in a polar group Fluorine KF 1.360 — d Chlorine 2.500 Suitable for chlorine atom K Cl participating in a polar group h 3.450 Suitable in the presence of hydrogen Hydrogen bond at the K OH bond at the sacrifice of OH-group sacrifice of OH-group h 2.200 Suitable in the presence of hydrogen Hydrogen bond at the K NHCO bond at the sacrifice of NHCO-group sacrifice of NHCO-group

Because atoms may participate in the composition of polar groups possessing a specific intermolecular interaction, their contribution to thermal stability will be different from the contribution of the same atoms possessing only weak Van-derWaals interaction. For example, polymers may contain the following polar groups: CH3;

C

;

O NH C O O

C O

;

C OH;

O ;

OH;

C N;

NH C

O N

N

C

C O

;

O O ;

S

;

Cl;

O CH2

.

O

When Td is calculated by equation (VII.5), taking into account of the intermolecular interaction is performed in the following way. If the atom considered is not a participant of the polar group, its parameter Ki is multiplied by the appropriate Van-der-Waals volume. But if the atom is included into the polar group, its parameter Ki is designated as K ih or K id (h is the hydrogen bond, d is the dipole–dipole interaction), and K ih or K id is multiplied only by the appropriate Van-der-Waals volume, i.e. contribution of the atom into the Van-derWaals interaction is not taken into account as the significantly weaker value. For representatives of various classes of polymers, examples of calculation of Td values are shown in Table 26.

411

Table 26 Calculated and experimental values of temperatures of onset of intense thermal degradation Td in inert medium for a series of polymers ∑ K i ∆Vi ⋅10 3 , ∑ ∆Vi , Td, K Td, K Chemical structure of the repeat unit i i Exper. Calc. Å3 Å3/K 1 2 3 4 5 48.6 34.2 713 704 CH2CH2 81.4

51.5

673

673

186.1

115.6

623; 633; 643

621

372.6

231.2

593; 653

621

241.0

166.1

663

688

CH

177.6

109.9

633

618

CH3

164.3

96.4

600– 623

588

213.1

126.15

563

591

CH3 CH2

C

CH

(CH2)5

NH

O C

(CH2)4

O

O

C NH

(CH2)6

NH

O

(CH2)2

O C

C

O

O

CH2

CH2

C C O CH3 O

CH2

C O C O

CF2CF2 CH2

CH

70.8

54.9

773

775

95.0

54.1

570

569

606.0

396

683

653

2

3

4

5

C N C

C HN

O

O

NH C O C O

1

412

C

C O

O

O

570.0

389.5

728

683

777.0

513.0

733; 728

660

527.0

381.0

753

723

668.0

437.4

663; 653; 658

655

O C O C O

N

N

C

C

O

N

N

C

C

O

C

O

O C O

N

O O

N

C

C

C

C

O

O

N

CH3 O

C CH3

N

O O C

S

C

O

O

O

Equation (VII.5) allows not only estimation of the temperature of onset of intense thermal degradation of the polymer at its heating in an inert medium, but also makes it possible to solve a series of other problems. For example, for some time it was not clear whether intense thermal degradation may be started at temperatures below the glass transition temperature Tg. To put it differently, must the polymer at heating first transit from the glassy state into the rubbery state or into the viscous flow state in order to make possible thermal degradation, or may the latter also proceed in the glassy state? For many polymers, the temperature of onset of intense degradation is indeed above the glass transition temperature or the melting point. However, in a number of cases, degradation starts at temperatures below the glass transition temperature or the melting point. The latter is stipulated by the fact that stability of a chemical bond depends crucially on dipole–dipole interactions and hydrogen bonds. Let us discuss this in more detail. If at the same backbone a polymer possesses polar groups in its composition which occupy a significant Van-der-Waals volume compared with other atoms, not participating in the composition of these groups, this polymer possesses a lower temperature of thermal degradation. Let us show it on the example of polystyrene CH2

CH

In the case of polystyrene, existence of a bulky polar phenyl group leads to the following expression for calculation of Td by expression (VII.5):

413

Td (K ) =

109.9 d K C ⋅ 22.2 + K H ⋅ 6.0 + K Cd ⋅17.5 + K H ⋅10

103 = 618

(deduction of this expression requires the data from Table 3 to be used, which indicate Van-der-Waals volumes of atoms). Formally, if polystyrene is considered as a substituted polyethylene, in which one atom of hydrogen in every unit is substituted by phenyl, it may be concluded that such substitution causes a decrease of Td. This reduction may start before the glass transition temperature is reached. Polymers with bulky side substituents containing polar groups behave In this manner. Polymethylidenephthalide is one of these polymers: CH2 C O C O The formula for calculation of the temperature of the onset of intense degradation of polymethylidenephthalide, deduced from (VII.5), is of the following form: Td ( K ) =

126.15 d d K C ⋅ 21.6 + K Cd ⋅ 83.3 + K H ⋅ 4 + K H ⋅8 + KO ⋅ 9.25

103 = 592 .

The presence of a bulky polar grouping in a side chain induces high glass transition temperature, equal to 390°C for this polymer. This value of Tg appears as the result of both calculations and experiments [55]. The latter should be discussed in more detail, because in the cases in which the intense thermal degradation starts before the glass transition temperature is reached, direct determination of Tg becomes impossible. In this connection the following method of determination of Tg for polymethylidenephthalide was used in refs. [55]: a series of methylidenphtalide copolymers with styrene, methyl methacrylate and methyl acrylate of different compositions was synthesized. The glass transition temperature was determined for these copolymers. Figure 69 represents dependences of Tg on the composition of these copolymers. As the concentration of methylidenephthalide increases, Tg grows first, but when Tg reaches values at which the intense thermal degradation starts, this growth terminates. In this area of high concentrations of methylidenephthalide, softening of copolymers proceeds at the sacrifice of their decomposition. At the same time, extrapolation of dependences of Tg on composition to the 100% concentration of methylidenephthalide results in Tg = 390°C for polymethylidenephthalide. Hence, the onset of intense degradation of this polymer is at 300°C and, consequently, for polymethylidenephthalide, the onset of this process is at much lower temperatures, than the glass transition temperature. On the contrary, in the case of weakly polar groups of polymers (polyethylene, polydimethylsiloxane, etc.), the glass transition temperature and the melting point are much lower than the temperature of thermal degradation. A case may be selected (on the example of polyheteroarylenes), when the glass transition temperature and the temperature of thermal degradation will be practically coincident.

414

Hence, when analyzing the influence of the chemical structure of polymers on their thermal characteristics and when predicting the polymer properties, it should be taken into account that one and the same groupings may cause the opposite influence on various thermal characteristics.

Figure 69. Dependences of Tg on the molar part α of styrene (1), methyl methacrylate (2) and methyl acrylate (3) for the copolymers of these components with methylidenephthalide.

Of special attention is the case when degradation of a polymer starts from dissociation by end groups. If this dissociation does not lead to formation of the same end groups, as mentioned above, calculation of the value must be performed on the basis of the chemical structure of the repeat unit of the polymer. In this case, if dissociation of end groups, which proceeds easier than decomposition in the polymer backbone, leads to formation of the same end groups, the calculation must be performed on the basis of the chemical structure of end groups only. In this case, polyformaldehyde decomposition is typical: HO CH2

O

(CH2

O)n ...

Calculation of Td by the structure of end groups (marked by dotted lines) leads to the following value: Td,1 ( K ) =

39.4 d h ⋅ 4 + K OH ⋅10.3 K Cd ⋅ 25.1 + K H

103 = 447 .

This value coincides well with the experimental value of Td, determined by the first change of the mass of polyformaldehyde under conditions of thermogravimetric analysis (Figure 70). This analysis was performed for a sample stabilized by

Š&+ Š&+ Š2Š JURXSV LQWURGXFHG LQWR SRO\PHU LQ  FRQFHQWUDWLRQ ,W LV 2

2

common knowledge [69] that introduction of the units mentioned prevents premature decomposition of polyformaldehyde proceeding by the end groups. That is why, calculation of temperature of the second decrease of the sample mass is conducted by structure of the repeat unit. It should be noted preliminarily that polyformaldehyde is a polar polymer, chains of which possess a strong intermolecular interaction. Then

415

Td,2 (K ) =

Figure 70.

32.5 d h K Cd ⋅ 25.1 + K H ⋅ 4 + KO ⋅ 3.4

103 = 583 .

7KHUPRJUDYLPHWULF FXUYH IRU SRO\IRUPDOGHK\GH VWDELOL]HG E\  RI

Š&+ Š&+ Š2Š 2

2

groups. (The experiment was performed in the medium of argon at the heating rate of 5 deg/min).

This value coincides also well with the experimental value, determined from the thermogravimetric curve (see Figure 70). Hence, analysis of the influence of the chemical structure on thermal degradation of a polymer makes it possible to perform scanning by various polar groups existing both on the ends of macromolecules and in the repeat units. Therewith, it may occur that the temperature of decomposition of these groups is lower than the onset temperature of intense thermal degradation of the whole polymer. It is advisable to perform further scanning with regard to chemical transformations of these groups at polymer heating. Clearly, such calculation analysis is unable to substitute completely experimental studies of thermal degradation of polymers, which is extremely complicated. In the case of copolymers, equation (VII.5) obtains the form:       α1 ∑ ∆Vi  + α 2  ∑ ∆Vi  + ... + α n  ∑ ∆Vi         i 1  i 2  i n Td = ,       α1  ∑ K i ∆Vi  + α 2  ∑ K i ∆Vi  + ... + α n  ∑ K i ∆Vi         i 1  i 2  i n

(VII.7)

  where α1, α2, …, αn are the molar parts of components 1, 2, …, n;  ∑ ∆Vi  ,    i 1      ∑ ∆Vi  , …,  ∑ ∆Vi  are the Van-der-Waals volumes of these components;      i n  i 2        ∑ K i ∆Vi  ,  ∑ K i ∆Vi  , …,  ∑ K i ∆Vi  are the series of constants for        i n  i 1  i 2 components 1, 2, …, n. In the reduced form, correlation (VII.7) becomes

416

k =n

Td =





∑ α k  ∑ ∆Vi 

k =1  i k . k =n     K i ∆Vi αk   k =1  i k

∑

(VII.8)

∑

If it is desirable to express the temperature of onset of intense thermal degradation via analogous values for homopolymers composed of components 1, 2, …, n, then, according to equation (VII.5) and expression (VII.7), it may be written that:       α1  ∑ ∆Vi  + α 2  ∑ ∆Vi  + ... + α n  ∑ ∆Vi         i 1  i 2  i n Td = ,        ∑ ∆Vi   ∑ ∆Vi   ∑ ∆Vi         i n  i 1  i 2 +α2 + ... + α n α1 Td,1 Td,2 Td, n

(VII.9)

where Td,1, Td,2, …, Td,n are temperatures of onset of intense thermal degradation of homopolymers synthesized from components 1, 2, …, n. In the reduced form, expression (VII.9) is the following: k =n

Td =





∑ α k  ∑ ∆Vi 

 i k .    ∑ ∆Vi    k =n  i  α ∑ k T k d, k k =1 k =1

(VII.10)

In a different form, correlation (VII.7) looks as follows: 1 1 1 1 = β1 + β2 + ... + β n , Td Td,1 Td,2 Td, n where   α1  ∑ ∆Vi     i 1 ; β1 = k =n   ∑ α k  ∑ ∆Vi  k =1  i k

(VII.11)

417

  α 2  ∑ ∆Vi     i 2 ; β2 = k =n     ∑ α k  ∑ ∆Vi  k =1  i k …………………………   α n  ∑ ∆Vi     i n . βn = k =n     ∑ α k  ∑ ∆Vi  k =1  i k

Chapter VIII. Optical and opto-mechanical properties of polymers

VIII.1 Refractive index The refractive index is the most important optical property of polymers. This characteristic is directly associated with the dielectric constant of the substance. In the general case, three molecular processes contribute to the statistic dielectric constant: orientation of constant momenta in a field, the relative displacement of positive and negative ions inside the molecule and the displacement of electrons in relation to nuclei. These three processes describe orientational, atomic and electron polarization, respectively. In the area of sound frequencies, the dielectric constant is usually independent of frequency. As frequency increases, orientational polarization falls behind the field and finally has no effect on the dielectric constant. Usually, this transition proceeds in the area of radio frequencies. At much higher frequencies, usually in the infrared range, another type of transition is observed, when the frequency of influence approaches self frequencies of ion oscillations or exceeds them. In the optical range, the contribution of orientational polarization is negligible; the contribution of atomic polarization may also be neglected. In this range, not the dielectric constant ε but refractive index n is measured, the square of which equals ε under the condition that both these values are determined at one and the same frequency. The refractive index somewhat changes with frequency in the optical range which is associated with a gradual approximation to self frequencies of oscillation of electrons existing in the ultraviolet range. The simplest method, in which an attempt is made to take into account electrostatic interactions between molecules, is the method of the local Lorentz field. The Lorentz calculation of the local field intensity is also valid at optical frequencies under the same conditions, as in the statistic case, if the wavelength of the alternative field is long compared with the lattice spacing. As applied to polymers, this means that the wavelength must be long compared with the size of the elementary cell (or approximately comparable with the average distance between atoms of neighboring macromolecules). A change of the refractive index n in dependence on density at the current frequency with the Lorentz correction adheres to the following law: n2 −1 M ⋅ = R. n2 + 2 ρ

(VIII.1)

This expression represents the Lorenz–Lorentz formula, which is the optical analogue to the Clausius–Mossotti formula (see below). In equation (VIII.1), M is the molecular mass (of the repeat unit, in the case of polymers); ρ is density; R is molecular refraction. In the case of glassy polymers, the value of ρ is calculated by the formula (II.6), substitution of which into equation (VIII.1) gives

419

n2 − 1 n2 + 2

⋅

N A ∑ ∆Vi i

kavg

= R = ∑ mi Ai ri ,

(VIII.2)

i

where NA is the Avogadro number; kavg is the average coefficient of molecular packing (for bulky monolith bodies, kavg = 0.681; for films, kavg = 0.695); mi is the number of atoms of the i-th type in the repeat unit; ri is the specific refraction of atoms of the i-th type; Ai is the atomic mass; ∆Vi are increments of Van-der-Waals volumes of atoms participating in the repeat unit of polymer. Molecular refraction R is the additive value and is composed of refractions Ri of separate atoms and constants for types of chemical bonds (double, triple). Some values of Ri, required for calculations, are shown in Table 27 composed according to the data by Eisenlore for the wavelength of 0.5893 µm (D). To calculate the refractive index n of copolymers, equation (VIII.2) has the form: n2 − 1 2

n +2

=

kavg (α1R1 + α 2 R2 + ... + α n Rn )         N A α1 ∑ ∆Vi  + α 2  ∑ ∆Vi  + ... + α n  ∑ ∆Vi           i 1  i 2  i n  

,

(VIII.3)

where α1, α2, …, αn are the molar parts of the components 1, 2, …, n; R1, R2, …, Rn are the molar refractions of homopolymers based on components 1, 2, …, n;        ∑ ∆Vi  ,  ∑ ∆Vi  , …,  ∑ ∆Vi  are their Van-der-Waals volumes.        i n  i 1  i 2 Table 27 Atomic refractions of series of atoms in organic compounds according to the data by Eisenlore Atoms, atomic groups and features of structure Symbol RD, cm3/mol Carbon RC 2.418 Hydrogen RH 1.100 Oxygen: RO– In OH 1.525 In ethers * 1.643 RO< In CO 2.211 RO= Chlorine RCl 5.967 Bromine RBr 8.865 Iodine RI 13.900 Double C=C bond R= 1.733 2.398 Triple C≡C bond R≡ Nitrogen: In primary amines 2.322 R H 2 NC

In secondary amines

RHN(C)2 R

2.502

N(C)3 In tertiary amines 2.840 RC–N=C In imides (tertiary) 3.776 RN≡C In nitriles ** 3.118 * As well as in α-oxides, no constant being introduced for the tertiary oxide cycle. ** Values of nitrogen atomic refraction in imides and nitriles include constants for double and triple carbon–nitrogen bonds.

420

In the reduced form, correlation (VIII.3) may be written down as follows: k =n

n2 −1 n2 + 2

k avg =

k =n

∑ α k Rk

k =1

  N A ∑ α k  ∑ ∆Vi    k =1  i k

.

(VIII.4)

Equations (VIII.1) and (VIII.2) allow sufficiently accurate estimation of the refractive index of polymers and copolymers on the basis of their chemical structure. However, in the case of polymers and copolymers with low glass transition temperatures, the value of n is often somewhat underestimated. This is associated with the fact that the coefficient of molecular packing k for such systems is somewhat lower than the average value kavg. That is why for more accurate estimation of the refractive index, it is desirable to take into account the temperature dependence of k, described by equations (II.14) and (II.15). Substitution of these dependences into expression (VIII.2) gives n2 −1 n2 + 2

=

Rkg

[1 + α G (T − Tg )]N A ∑ ∆Vi ,

(T < Tg);

(VIII.5)

(T > Tg),

(VIII.6)

i

2

n −1 2

n +2

=

Rkg

[1 + α L (T − Tg )]N A ∑ ∆Vi , i

where kg = 0.667. Consequently, using equation (VIII.4) for copolymers, we obtain: k =n

n2 −1 n2 + 2

kg =

∑α k Rk

k =1

k =n

  1 + α G T − Tg N A ∑ α k  ∑ ∆Vi    k =1  i k

[

(

)]

, (T < Tg);

(VIII.7)

, (T > Tg).

(VIII.8)

k =n

n2 −1 n2 + 2

kg =

∑α k Rk

k =1

k =n

  1 + α L T − Tg N A ∑ α k  ∑ ∆Vi    k =1  i k

[

(

)]

Table 28 displays refractive indices for a series of amorphous polymers. Clearly, the value of n depends on the chemical structure of the polymer and increases at transition from aliphatic polymers to aromatic ones. It is also clear that for 20 polymers existing in the rubbery state at room temperature (T > Tg), values of nD calculated with regard to the temperature dependence of the coefficient of molecular packing, i.e. with application of expression (VIII.5), correlate much better with

421

20 experimental values of nD . Therewith, values of the glass transition temperature Tg determined by equation (IV.41) and values of the coefficients of volumetric expansion αG and αL determined by correlations (III.8) and (III.5) were used for calculation. Table 28 20 20 Molar refractions R, experimental n D, exp and calculated n D, calc values of the refractive indices for a series of amorphous polymers 20 20 Ri , n D nD , exp Polymer , calc * Tg,calc, K i cm3/mol 1 2 3 4 5 282 20.126 1.479 1.49 / 1.47 CH2 CH

∑

C O CH3 O CH

CH2

24.744

1.469

1.49 / 1.47

267

33.980

1.466

1.50 / 1.48

242

24.744

1.490

1.490

377

29.362

1.485

1.50 / 1.49

338

38.598

1.483

1.50 / 1.49

287

44.233

1.568

1.56

378

33.343

1.591

1.60

376

37.961

1.587

1.59

401

C O C2H5 O CH2

CH C O C4H9 O CH3

CH2

C C O CH3 O CH3

CH2

C C O C2H5 O

CH3 CH2

C C O C4H9 O CH3

CH2

C C O O

CH2

CH

CH3 CH2

C

422

1

CH2

CH

2 38.210

3 1.610

4 1.62

5 356

15.497

1.467

1.49 / 1.47

253

20.126

1.467

1.49

301

34.986

1.578

1.58

353

69.983

1.585

1.58

431

117.662

1.610

1.60

582

Cl

CH2

CH O CH3

CH2

CH O C

CH3

O

CH2

CH O

CH3 O

C

O C

CH3 C

C O

O

O

O O C O C O

* Column 4 indicates two values of ncalc: the first of them is determined by equation (VIII.2), and the second – by equation (VIII.6), i.e. with regard to the temperature dependence of the coefficient of molecular packing k.

More comprehensive data on polymers, for which refractive indices were measured, are shown in Table 28′, the experimental data in which are borrowed from Polymer Handbook [453]. Table 28′ Average refractive indices of polymers Polymer 1 Cellulose Cellulose nitrate Cellulose triacetate Cellulose tripropionate Ethyl cellulose Methyl cellulose (low viscosity) Natural rubber Poly(acrolein) Poly(acrylic acid)

n (exp) 2 1.54 1.5 – 1.514 1.47 – 1.48 1.48 – 1.49 1.479 1.497 1.519 – 1.52 1.529 1.527

n1 (calc) 3 1.49 (am); 1.54 (cryst) 1.52 1.48 1.49 1.47 1.48 1.51 1.51 1.49

423

1 Poly(acrylonitrile) Poly(allyl mathacrylate) Poly(N-allyl methacrylamide) Poly(N-benzyl methacrylamide) Poly(n-butyl methacrylate) Poly(benzyl methacrylate) Poly(bornyl methacrylate) Poly(2-bromoethyl methacrylate) Poly(p-bromophenyl methacrylate) Poly(2-bromo-4-trifluoromethylstyrene) Poly(1,2-butadiene) Poly(1,3-butadiene) Poly(butene) (isotactic) Poly(butyl acrylate) Poly(sec-butyl α-bromoacrylate) Poly(sec-butyl α-chloroacrylate) Poly(2-tert-butyl-1,3-butadiene) Poly(tert-butyl methacrylate) Poly(butylmercaptyl methacrylate) Poly(N-butyl methacrylamide) Poly(o-chlorobenzyl mathacrylate) Poly(2-chloro-1-(chloromethyl)ethyl methacrylate) Poly(2-chlorocyclohexyl methacrylate) Poly(o-chlorodiphenylmethyl methacrylate) Poly(2-chloroethyl α-chloroacrylate) Poly(2-chloroethyl methacrylate) Poly(1-(o-chlorophenyl)ethyl methacrylate) Poly(chloroprene) Poly(o-chlorostyrene) Poly(m-cresyl methacrylate) Poly(o-cresyl methacrylate) Poly(cyclohexyl α-bromoacrylate) Poly(cyclohexyl α-chloroacrylate) Poly(cyclohexyl α-ethoxyacrylate) Poly(cyclohexyl methacrylate) Poly(p-cyclohexylphenyl methacrylate) Poly(1-decene) Poly(2-decyl-1,3-butadiene) Poly(2,3-dibromopropyl methacrylate) Poly(1,3-dibromopropyl methacrylate) Poly(2,6-dichlorostyrene) Poly(2-diethylaminoethyl methacrylate) Poly(1,1-diethylpropyl methacrylate) Poly(2,3-dimethylbutadiene) (methyl rubber) Poly(1,2-diphenylethyl methacrylate) Poly(diphenylmethyl methacrylate) Poly(p-divinylbenzene) Poly(dodecyl methacrylate) Poly(2-ethoxyethyl acrylate) Poly(2-ethoxyethyl methacrylate) Poly(3-ethoxypropyl acrylate) Poly(ethyl acrylate) Poly(ethyl α-chloroacrylate)

2 1.52 (1.5187) 1.5196 1.5476 1.5965 1.483 1.5680 1.5059 1.5426 1.5964 1.5 1.5000 1.5154 1.5125 1.4631 – 1.466 1.542 1.500 1.5060 1.4638 1.5390 1.5135 1.5823 1.5270 1.5179 1.6040 1.533 1.517 1.5624 1.5541.558 1.6098 1.5683 1.5707 1.542 1.532 1.4969 1.5066 1.5575 1.4730 1.4899 1.5739 1.5270 1.6248 1.5174 1.4889 1.525 1.5816 1.5933 1.6150 1.4740 1.471 1.4833 1.465 1.4685 1.502

3 1.49 1.51 1.53 1.57 1.49 1.56 1.53 1.53 1.58 1.55 1.53 1.51 1.50 1.48 1.53 1.52 1.51 1.50 1.56 1.52 1.57 1.53 1.53 1.59 1.53 1.52 1.56 1.54 1.62 1.55 1.55 1.55 1.53 1.51 1.52 1.56 1.49 1.50 1.56 1.53 1.63 1.52 1.51 1.51 1.58 1.58 1.61 1.49 1.46 1.48 1.47 1.47 1.51

424

1 Poly(ethyl methacrylate) Poly(ethylene) (density 0.914 g/cm3) (density 0.94 – 0.945 g/cm3) (density 0.965 g/cm3) Poly(ethylmercaptyl methacrylate) Poly(2-fluoroethyl methacrylate) Poly(4-fluoro-2-trifluoromethylstyrene) Poly(2-(heptafluorobutoxy)ethyl acrylate) Poly(heptafluorobutyl acrylate) Poly(2-heptyl-1,3-butadiene) Poly(hexadecyl methacrylate) Poly(2,2,3,3,4,4-hexafluorobutyl acrylate) Poly(n-hexyl methacrylate) Poly(2-hydroxyethyl methacrylate) Poly(iminoadipoyliminohexamethylene) (Nylon 6,6) Poly(iminoadipoyliminotetramethylene) Poly(imino(1-oxohexamethylene)) (Nylon 6) Polyisobutene Poly(isobutyl methacrylate) Polyisoprene Poly(2-isopropyl-1,3-butadiene) Poly(isopropyl methacrylate) Poly(p-isopropylstyrene) Poly(methacrylonitrile) Poly(p-methoxybenzyl methacrylate) Poly(2-methoxyethyl acrylate) Poly((N-2-methoxyethyl)methacrylamide) Poly(4-methoxy-2-methylstyrene) Poly(o-methoxyphenyl methacrylate) Poly(methoxypropyl acrylate) Poly(o-methoxystyrene) Poly(p-methoxystyrene) Poly(methyl acrylate) Poly(methyl α-bromoacrylate) Poly(methyl α-chloroacrylate) Poly(methyl methacrylate) Poly(N-methyl-methacrylamide) Poly(4-methyl-1-pentene) Poly(1-methylcyclohexyl methacrylate) Poly(2-methylcyclohexyl methacrylate) Poly(3-methylcyclohexyl methacrylate) Poly(4-methylcyclohexyl methacrylate) Poly(o-methylstyrene) Poly(α-naphthyl methacrylate) Poly(β-naphthyl methacrylate) Poly(m-nitrobenzyl methacrylate) Poly(2-nitro-2-methylpropyl methacrylate) Poly(nonafluoropentyl acrylate) Poly(octafluoropentyl acrylate) Poly(oxycarbonyloxybis(1,4-(3,5-dichlorophenylene))) Poly(oxycarbonyloxy1,4-(2,6-dichloro)phenylene-isopropylidene-1,4(2,6-dichloro)phenylene) Poly(oxycarbonyloxy1,4-phenylene-1,3-dimethylbutylidene-1,4phenylene) Poly(oxycarbonyloxy1,4-phenylene-1-methyl-butylidene-1,4-phenylene)

2 1.485 1.51 1.52–1.53 1.545 1.547 1.4768 1.46 1.390 1.367 1.5000 1.4750 1.392 1.4813 1.5119 1.53 1.53 1.53 1.505–1.51 1.477 1.521 1.5028 1.4728 1.554 1.52 1.552 1.463 1.5246 1.5868 1.5705 1.471 1.5932 1.5967 1.472 – 1.480 1.5672 1.517 1.4893 1.5398 1.459 – 1.465 1.5111 1.5028 1.4947 1.4975 1.5874 1.6410 1.6298 1.5845 1.4868 1.360 1.380 1.6056 1.6056

3 1.50 1.49 1.52 1.54 1.54 1.47 1.51 1.40 1.39 1.50 1.50 1.41 1.49 1.49 1.52 1.52 1.52 1.49 1.50 1.51 1.51 1.50 1.58 1.49 1.54 1.46 1.51 1.57 1.54 1.46 1.57 1.57 1.46

1.5671

1.57

1.5745

1.57

1.54 1.51 1.49 1.52 1.51 1.52 1.52 1.52 1.52 1.60 1.59 1.59 1.57 1.53 1.37 1.38 1.61 1.60

425

1 Poly(oxycarbonyloxy1,4-phenylene-1-propylbutylidene-1,4-phenylene) Poly(oxycarbonyloxy1,4-phenylene-sec-butylidene-1,4-phenylene) Poly(oxycarbonyloxy1,4-phenylenebutylidene-1,4-phenylene) Poly(oxycarbonyloxy1,4-phenylenecyclohexylidene-1,4-phenylene) Poly(oxycarbonyloxy1,4-phenylenediphenyl-methylene-1,4-phenylene) Poly(oxycarbonyloxy1,4-phenyleneethylidene-1,4-phenylene) Poly(oxycarbonyloxy1,4-phenyleneisobutylidene-1,4-phenylene) Poly(oxycarbonyloxy1,4-phenyleneisopropylidene-1,4-phenylene) Poly(oxy-2,6-dimethylphenylene) Poly(oxydimethylsilylene) (poly(dimethyl siloxane)) Poly(oxyethylene) (high molecular weight) Poly(oxyethyleneoxymaleoyl) (poly(ethylene maleate) Poly(oxyethyleneoxysuccinoyl) (poly(ethylene succinate)) Poly(oxyethyleneoxyterephthaloyl) (amorphous) (poly(ethylene terephthalate)) Poly(oxymethylene) Poly(oxy-1-oxopentamethylene) Poly(oxypropylene) Poly(pentachlorophenyl methacrylate) Poly(pentadecafuorooctyl acrylate) Poly(pentafluoropropyl acrylate) Poly(pentafluorovinyl propionate) Poly(1-phenyl-n-amyl methacrylate) Poly(phenyl α-bromoacrylate) Poly(phenyl methacrylate) Poly(1-phenylallyl methacrylate) Poly(1-phenylcyclohexyl methacrylate) Poly(1-phenylethyl methacrylate) Poly(2-phenylethyl methacrylate) Poly(N-(2-phenylethyl)methacrylate) Poly(2-(phenylsulfonyl)ethyl methacrylate) Poly(n-propyl methacrylate) Poly(propylene) (density 0.9075 g/cm3) Poly(styrene) Poly(tetradecyl methacrylate) Poly(tetrafluoro-3-(heptafluoropropoxy)propyl acrylate) Poly(tetrafluoro-3-(pentafluoroethoxy)propyl acrylate) Poly(tetrafluoro-3-(trifluoromethoxy)propyl acrylate) Poly(2-(1,1,2,2-tetrafluoroethoxy)ethyl acrylate) Poly(tetrafluoroethylene) Poly(2,2,2-trifluoro-1-methylethyl methacrylate) Poly(trifluorochloroethylene) Poly(2-trifluoroethoxy)ethyl acrylate) Poly(trifluoroethyl acrylate) Poly(trifluoroethyl methacrylate) Poly(trifluoroisopropyl methacrylate) Poly(trifluorovinyl acetate) Poly(3,3,5-trimethylcyclohexyl methacrylate) Poly(undecafluorohexyl acrylate) Poly(vinyl acetate) Poly(vinyl alcohol) Poly(vinyl benzoate) Poly(vinyl butyl ether) Poly(viny sec-butyl ether) (isotactic) Poly(vinyl butyral) Poly(vinyl chloride)

2 1.5602 1.5827 1.5792 1.5900 1.6539 1.5937 1.5702 1.5850 1.575 1.43 1.4563 1.4840 1.4744 1.5750

3 1.57 1.57 1.57 1.58 1.61 1.58 1.57 1.58 1.59 1.41 1.45 1.46 1.46 1.54

1.48 1.465 1.4495 1.608 1.339 1.385 1.364 1.5396 1.612 1.5706 1.5573 1.5645 1.5487 1.5592 1.5857 1.5682 1.484 1.5030 1.59 – 1.592 1.4746 1.346 1.348 1.360 1.412 1.35 – 1.38 1.4185 1.42 – 1.43 1.419 1.407 1.437 1.4177 1.375 1.485 1.356 1.4665 1.49 – 1.53 1.5775 1.4563 1.4740 1.48 – 1.49 1.54 – 1.55

1.42 1.47 1.46 1.61 1.35 1.39 1.37 1.55 1.59 1.56 1.56 1.56 1.55 1.55 1.57 1.56 1.50 1.51 1.60 1.49 1.36 1.36 1.37 1.41 1.31 1.45 1.42 1.42 1.41 1.44 1.45 1.40 1.52 1.36 1.49 1.50 1.58 1.48 1.48 1.50 1.57

426

1 Poly(vinyl chloroacetate) Poly(vinyl decyl ether) Poly(vinyl dodecyl ether) Poly(vinyl ethyl ether) Poly(vinyl-2-ethylhexyl ether) Poly(vinyl formal) Poly(vinyl hexyl ether) Poly(vinyl isobutyl ether) Poly(vinyl methyl ether) Poly(vinyl methyl ether) (isotactic) Poly(vinyl octyl ether) Poly(vinyl pentyl ether) Poly(vinyl phenyl sulfide) Poly(vinyl propionate) Poly(vinylcarbazole) Poly(vinylfuran) Poly(vinylidene chloride) Poly(vinylidene fluoride) Poly(vinylnaphthalene) Poly(1-vinyl-2-pyrrolidone) Poly(N-vinylphthalimide) Poly(2-vinyltetrahydrofuran) Poly(2-vinylthiophene)

2 1.512 1.4628 1.4640 1.4540 1.4626 1.50 1.4591 1.4507 1.467 1.4700 1.4613 1.4581 1.6568 1.4665 1.683 1.55 1.60 – 1.63 1.42 1.6818 1.53 1.6200 1.55 1.6376

3 1.54 1.48 1.48 1.47 1.49 1.50 1.48 1.48 1.47 1.47 1.48 1.48 1.67 1.48 1.64 1.56 1.57 1.38 1.63 1.55 1.60 1.53 1.65

VIII.2 Stress-optical coefficient Polymeric materials have been used widely in the photoelasticity method of investigation of stresses. This method is based on the phenomenon of birefringence, found by D. Brüster, which appears in transparent optically isotropic materials under deformation and is stipulated by anisotropy of the refraction index in the mutually perpendicular directions. In the case of glassy polymers, the first consequence of applied load is the variation of interatomic distances and valence angles in a polymeric chain. These changes determine the instaneous elastic deformation. Elastic deformation is associated with the mobility of atoms composing chains of macromolecules within a statistical segment of a macrochain. Occurrence of the birefringence and its value at deformation of polymers existing in the glassy state are generally stipulated by the dislocation of electronic shells of atoms and electronic clouds forming chemical bonds, as well as by distortion of the valence angles that leads to anisotropy of polarizability of elementary units of macromolecules. In the glassy state, birefringence may also be associated sometimes with the elastic orientation of optically anisotropic macromolecules or their parts (for example, mobile side methyl groups in polyacrylates and fluoride groups in poly(methacrylate) esters near their equilibrium state. Therewith, the so-called elastic component of birefringence occurs, which reaches its maximum almost immediately after the load application. In the case of the ideal elastic body, total birefringence would be controlled by elastic deformation, because under these conditions elastically deformed polymer would exist in the equilibrium state. However, it should be noted that the behavior of real polymeric substances differs from the elastic one. The change of deformation and values of birefringence with time is typical of them even in the glassy state.

427

For optically sensitive polymeric materials that are in the glassy state, the experimental Wertheim law is applicable up to certain stress levels. This law connects the optical retardation δ at any point of the polymeric model existing in the planestress state with the difference of the main principal stresses σ1 and σ2 acting in the plane model in the same point, and thickness of the model d:

or

δ = Cσ(σ1 – σ2)d

(VIII.9)

∆n = n1 – n2 = Cσ(σ1 – σ2)d,

(VIII.10)

where Cσ is the stress-optical coefficient; ∆n is the birefringence; n1 and n2 are refractive indices along and perpendicular to the optical axis. Introducing the light wavelength λ into equation (VIII.10), a correlation is obtained which connects birefringence with the fringe order: C d m = σ (σ 1 − σ 2 ) , λ

(VIII.11)

where m is the fringe order for a given wavelength λ. Coefficient Cσ is usually considered to be independent of the wavelength, but as more accurate investigation has indicated, it is not true. In practice of the photoelasticity method, to characterize he optical sensitivity of polymeric materials, another value – the stress-related value of material strip σ 10.0 – is used. This value equals

σ 10.0 =

λ . Cσ d

(VIII.12)

In the case of a linear dependence between stress and strain, birefringence may be expressed via strain. Then, the dependence is reduced to the following form:

δ = Cε(ε1 – ε2)d,

(VIII.13)

where Cε is the strain-optical coefficient of the material; ε1 and ε2 are the principal strains. Both optical coefficients are connected with each other by the correlation: C E λ Cε = σ = , 1 1 + µ ε 0.0

(VIII.14)

where µ is the Poisson ratio; E is the elasticity modulus; ε 10.0 is the strain-related value of material strip. Since the value of birefringence of a polymer in the glassy state, in a definite range, is proportional to the anisotropy of polarizability of the elementary unit bonds, the increase of polarizability anisotropy of macromolecules and, consequently, the optical sensitivity of the polymer may be achieved by introduction into the starting monomer or oligomer of groups with high polarizability anisotropy (such as aromatic

428

cycles of types of benzene, naphthalene, anthracene, carbonyl groups and any molecular groupings containing double or triple bonds, i.e. groupings containing mobile π-electrons) to the molecule. At the present time, there exist two approaches to numerical estimation of the stress-optical coefficient Cσ for polymers based on the chemical structure of the repeating unit of them. The first approach suggested in ref. [36] is empirical. On the basis of numerous experimental data, treated in the cited reference, a correlation has been suggested that connects Cσ with the parameters of the chemical structure of the repeat unit of the polymer:

Cσ =

∑ Ci i

N A ∑ ∆Vi

+∏,

(VIII.15)

i

where Ci are constants which characterize contributions of every atom and type of intermolecular interaction into the stress-optical coefficient (Table 29); ∑ ∆Vi is the i

Van-der-Waals volume of the repeat unit composed of volumes of atoms participating in this unit; NA is the Avogadro number; ∏ = 0.3544⋅10−4 cm2/kG is the universal parameter. Table 29 Values Ci characterizing contributions of each atom and type of intermolecular interaction to the stress-optical coefficient Atom or type of intermolecular interaction Ci⋅103, Symbol MPa–1 cm3/mol CC Carbon –2.0492 CH Hydrogen –0.5227 CO,b Oxygen in the backbone 3.1980 CO,s Oxygen in the side group –0.7568 Nitrogen in the backbone 7.1750 CΝ,b Nitrogen in the side group 1.3030 CΝ,s CCl Chlorine –3.4760 CS Sulfur –0.7900 Cd Dipole–dipole interaction * –1.6000 Ch Hydrogen bond –6.2100 p-Substitution of aromatic cycles ** Cp 1.7000 * Coefficient Cd is applied to every group of any chemical nature; if two identical groups locate at the same atom, a single coefficient Cd must be introduced. For phenyl group, Cd = –2.15⋅10–3. ** Coefficient Cp is introduced in the case of p-substitution of aromatic cycles; the number of coefficients Cp introduced equals to the number of aromatic cycles substituted in the p-position.

Table 30 shows values of stress-optical coefficient Cσ for a series of polymers in the glassy state. The value of Cσ changes is extremely wide range in relation to chemical structure of polymer – from low negative values for poly(methyl methacrylate) and poly-α-methylstyrene to extremely high positive values typical of aromatic polymers. High optical sensitivity of these polymers is indicated by their structure: the presence of a large number of condensed cycles characterized by a significant anisotropy of polarizability. As follows from Table 30, compounds saturated by nitrogen and sulfur atoms and aromatic cycles possess the highest values

429

of Cσ. A significant influence on the optical sensitivity is also caused by the type of substitution of aromatic cycles. Para-substitution promotes increase of Cσ. The presence of a large number of polar C=O-groups leads to a decrease of optical sensitivity. Possessing high negative anisotropy of polarizability, this group decreases the total positive effect. The fact that the C=O-group is the carrier of the negative effect has been observed in ref. [206] and confirmed in ref. [100] in the study of the opto-mechanical properties of polymers displayed in Table 30. Table 30 Experimental and calculated values of stress-optical coefficients Cσ for a series of glassy polymers Polymer Cσ⋅106, Cσ⋅106, Mpa-1 exper. MPa-1 calc. 1 2 3 10.7 10.65

CH2

CH

CH3 CH2

–3.3

–2.80

–2.0

–4.57

15.7

22.30

7.3

14.80

22.9

16.70

C C

CH3

O

CH3 CH2

CH2

C

CH

CH3

CH2

CH Cl

Cl CH2

CH

Cl

430

1

CH3 CH2

2 24.5

3 10.10

57.5

43.9

111.0

105.0

161.0

160.0

90

77.3

150.0

143.0

93

83.0

C C O CH2 O

CH2

CH N

CH3 O

C

O C

CH3

C C

N

O

N

O

N

C

N

C C

C HN

O

NH C

O

O C O

C C

C O

O

N

N

S O

N

N

C C

C O O

O C O C O

431

1

C

C NH

O

O

2 77.4

3 66.4

62.0

60.1

NH C

C

C NH

O

O

NH C

Using equation (VIII.15), the contribution of each group to the stress-optical coefficient may be estimated quantitatively. This contribution is characterized by the value of ∑ Ci N A ∑ ∆Vi , with the higher value of Cσ corresponding to the lower i

i

absolute value of ∑ Ci i

N A ∑ ∆Vi , because this relation is negative. A sequence of i

separate groups is formed due to the contribution

∑ Ci i

N A ∑ ∆Vi ⋅107 to the stressi

optical coefficient:

;

C O O

O

O

O

O

C

C

C

C

N

N

;

;

;

O;

C

.

O

O

C

C

C

C

O

O

O

O

–145.2

–157.5

–166.8

C ;

–223.4 C

;

;

–323.3

–336.0

C ;

C

–315.9

N

N

O –345.7

C

–348.3

–486.4

The calculation scheme considered for determination the stress-optical coefficient enables the expected optical sensitivity of the polymer to be estimated and is of a definite meaning for synthesis of polymers applicable in the photoelasticity method of investigation of stresses. For copolymers, equation (VIII.15) obtains the following form:

432

      α1  ∑ Ci  + α 2  ∑ Ci  + ... + α n  ∑ Ci         i 1  i 2  i n + ∏, Cσ =         N A α1 ∑ ∆Vi  + α 2  ∑ ∆Vi  + ... + α n  ∑ ∆Vi          i 1  i 2  i  n  

(VIII.16)

  where α1, α2, …, αn are the molar parts of the components 1, 2, …, n;  ∑ Ci  ,    i 1      ∑ Ci  , …,  ∑ Ci  are series of constants for the components 1, 2, …, n;      i 2  i n        ∑ ∆Vi  ,  ∑ ∆Vi  , …,  ∑ ∆Vi  are Van-der-Waals volumes of the        i n  i 1  i 2 components 1, 2, …, n; NA is the Avogadro number. In the reduced form, equation (VIII.16) obtains the form: k =n

Cσ =





∑ α k  ∑ Ci 

k =1  i k + ∏, k = n    N A  ∑ α k  ∑ ∆Vi      k =1  i k  

(VIII.17)

    where αk,  ∑ Ci  and  ∑ ∆Vi  are the molar part, the series of constants and the      i k  i k Van-der-Waals volume of the k-th element, respectively. If it is desired to express the stress-optical coefficient for copolymers via stress-optical coefficients of the components, equation (VIII.16) changes to:     α1 (Cσ ,1 − ∏ ) ∑ ∆Vi  + α 2 (Cσ ,2 − ∏ ) ∑ ∆Vi  + ... +      i 1  i 2 → Cσ =     α1 ∑ ∆Vi  + α 2  ∑ ∆Vi  + ... +      i 1  i 2   + α n Cσ , n − ∏  ∑ ∆Vi     i n → ,   + α n  ∑ ∆Vi     i n

(

)

(VIII.18)

where Cσ,1, Cσ,2, …, Cσ,n are stress-optical coefficient of the components 1, 2, …, n. In the reduced form, equation (VIII.18) obtains the following form:

433

k =n

Cσ =





 i

k

∑ α k (Cσ , k − ∏ ) ∑ ∆Vi 

k =1

k =n

  ∑ α k  ∑ ∆Vi  k =1  i k

,

(VIII.19)

where Cσ,k is the stress-optical coefficient of the k-th component. Let us now consider a semi-empirical method of estimation of the stressoptical coefficient Cσ suggested in ref. [91]. The matter is that the calculation scheme for determination of Cσ, described above, which enables Cσ of polymer to be determined with high accuracy based on chemical structure of the repeating unit, does not link Cσ with other opto-mechanical thermal parameters (elasticity modulus, glass transition temperature, etc.). Let us first perform general analysis. Let us consider a thin plate lying in the XOY plane. Because the light wavelength is much less than transverse sizes of the sample, it may be considered as an infinite plate. The light beam falls along the Z axis. Since the plate is thin, then according to [86] U ZZ =

µ (U XX + U YY ) , 1− µ

(VIII.20)

where UZZ, UXX and UYY are displacements along the corresponding axes; µ is the Poisson ratio. When the plate is undistorted, its material represents an isotropic dielectric with the dielectric constant ε0. When the plate is deformed, the optical symmetry of the medium changes and, consequently, the substance becomes optically anisotropic and may be described by introducing dielectric constant tensor εik. According to [85]

εik = ε0δik + a1Uik + a2Ullδik,

(VIII.21)

where Uik is the strain tensor; a1 and a2 are opto-elastic constants. We shall be interested in the value of εZZ which, in accordance with (VIII.21), may be presented in the following form:

ε ZZ = ε 0 − a1

  µ (U XX + U YY ) + a2 1 − 2µ U XX + 1 − 2 µ U YY  = 1− µ 1− µ  1− µ 

 1 − 2µ µ  (U XX + U YY ). = ε 0 +  a2 − a1 − − 1 1 µ µ  

(VIII.22)

Let σYY = 0 (uniaxial deformation). Then µUXX = –UYY and

εZZ = ε0 + [a2(1 – 2µ) – a1µ]UXX; E U XX − µ 2U XX = EU XX , σ XX = 1− µ 2

(

)

(VIII.23) (VIII.24)

434

where E is the elasticity modulus. From the system of equations (VIII.23) and (VIII.24), the dependence of εZZ on σYY may be obtained:

ε ZZ = ε 0 +

1 [(1 − 2µ )a2 − µa1 ]σ XX . E

(VIII.25)

2 , and ε0 by n02 (where nZZ and n0 are refractive indices Substituting εZZ by nZZ in the Z direction and of isotropic substance, respectively), the stress-optical coefficient Cσ may be found from expression (VIII.25) which, according to the definition, will be equal:

Cσ =

∂nZZ 1 = [(1 − 2 µ )a2 − µa1 ]. ∂σ XX σ = 0 2n0 E XX

(VIII.26)

Using for a1 and a2 the expressions deduced in ref. [50], according to which  ∂ε  1 a1 = 6 ρ   − (ε 0 − 1)(ε 0 + 2 ) ; ∂ ρ  T 3  ∂ε  2 a2 = −3ρ   + (ε 0 − 1)(ε 0 + 2 ), ∂ ρ  T 3 where ρ is the polymer density, we obtain the final expression: Cσ =

(

)(

)

 ∂ε   1 2 2 2  (µ + 1) n0 + 2 n0 − 1 − 3ρ    . 2n0 E  3  ∂ρ T 

(VIII.27) (VIII.28)

(VIII.29)

 ∂ε  Let us estimate the term   by equation (VIII.29). Let us rewrite  ∂ρ T

 ∂ε     ∂ρ  T

as  ∂ε   ∂ε ∂T    =   . ⋅ ∂ ρ  T  ∂T ∂ρ T

(VIII.30)

When T << Tg, it may be assumed that in a definite temperature range the ∂ε = const < 0. Consequently, coefficient of volumetric expansion αG = const, and ∂ρ expression (VIII.29) may be reduced to the following form: Cσ =

 ∂ε  1   + ∏* , 2n0α G E  ∂ρ T

which is an analogue of equation (VIII.15). In expression (VIII.31)

(VIII.31)

435

∏* =

(1 + µ ) ( n02 − 1) ( n02 + 2 ) 3n0 E

,

where µ is the Poisson ratio; n0 is the refractive index; E is the elasticity modulus. Because according to the above-said, in the case of amorphous polymers, 0.0962 (see Section IV.3), and for isotropic dielectric, according to ref. [85], αG = Tg

∑ ε i ∆Vi

ε0 = i

∑ ∆Vi

, expression (VIII.31) obtains the final form:

i

Cσ =

15.6Tg n0 E

∑ Ci ∆Vi

⋅ i

∑ ∆Vi

+ ∏* ,

(VIII.32)

i

where Ci = ∂ε/∂T; ∆Vi are the increments of Van-der-Waals volumes of atoms. Values of Ci are practically independent of temperature and stress, as well as change weakly with temperature and stress in the optical range of frequencies [57, 106]. To determine Cσ, we will base on the following reasons. Because ∏* is the order zero value, and Cσ is the first order value by smallness, Ci may be presented as Ci = Ci,0 + δCi, where Ci,0 is the order zero component, and δCi is the component of the first order by smallness. Then determination of Cσ from expression (VIII.32) will be reduced to solution of the system of equations

∑ Ci ∆Vi i

∑ ∆Vi

+ ∏′ = 0 ;

∑ δCi ∆Vi i

∑ ∆Vi

= Cσ′ ,

(VIII.33)

i

i

∏* n0 E C n E where Cσ′ = σ 0 ; ∏′ = . 15.6Tg 15.6Tg The second equation in the system (VIII.33) is used in ref. [91] for calculation of constants δCi for various atoms and types of intermolecular interaction. Calibration of the method was performed on the basis of experimental values of Cσ, n0, E and Tg for a series of amorphous polymers studied well by solving the excessive system of equations, composed on the basis of correlation (VIII.32). Consequently, a selection of constants δCi shown in Table 31 was obtained. Knowing these constants, the value of Cσ′ may be determined first from expression (VIII.33), and then Cσ may be calculated from the correlation Cσ =

Cσ′ ⋅15.6Tg n0 E

.

(VIII.34)

436

Table 31

Values of constants δCi for different atoms and types of intermolecular interaction Element or type of molecular interaction Designations δCi⋅106, MPa-1 Carbon – 0.005 δCC Hydrogen – 0.118 δCH Oxygen in the backbone 2.660 δCO,b Oxygen in the side group –0.700 δCO,s Nitrogen in the backbone 16.620 δCN,b Nitrogen in the side group 0.640 δCΝ,s Sulfur 0.740 δCS Dipole–dipole interaction 6.470 δCd Hydrogen bond –66.040 δCh Para-substitution –0.730 δCp Metha-substitution –3.410 δCm

Let us perform this analysis in detail for poly(phenyl quinoxoline)

C C

N N

O

N N

C C

which possesses the maximal stress-optical coefficient. For this polymer,

∑ δCi ∆Vi

= δCC(4∆VC* + 4∆VC,21 + 4∆VC,19 + 2∆VC,20 + 20∆VC,18) + δCH,20∆VH,124 +

i

δCO,b∆VO,131 + 4δCN,s∆VN,151 + 2δCd = –0.005(4⋅11.1 + 4⋅10.2 + 4⋅8.4 + 2⋅11.6 + 20⋅12.7) + (–0.118)⋅20⋅2.0 + 2.66⋅2.1 + 4⋅0.64⋅6.1 + 2⋅6.47 = 27.4 MPa–1 cm3/mol; ∑ ∆Vi = 462.5 Å3. i

27.4 = 0.0592. 462.5 Substituting the value of Cσ′ , Tg = 563 K, n0 = 1.62 and E = 1900 MPa into expression (VIII.34), we obtain Then Cσ′ =

Cσ =

0.0592 ⋅15.6 ⋅ 563 3 ⋅10 = 169 B. 1.62 ⋅1900

The semi-empirical approach to estimation of the stress-optical coefficient Cσ of polymers, developed in ref. [91], also enables the elasticity modulus of the linear glassy polymer to be calculated. Using correlation (VIII.34) for the determination of Cσ′ , it may be indicated that

437

Table 32 Opto-mechanical properties of a series of heat-resistant aromatic polymers Structure of the repeating unit Tg, K Cσ⋅106, MPa-1 Cσ′ ∑ ∆Vi , Calc. Exper. i MPa-1⋅cm3/mol 1 2 3 4 5 563 27.4 169.0 161.1

N

E, MPa Calc. Exper.

n Calc.

6 1810

7 1900

8 1.62

3120

3100

1.62

N

O

N

N

O

O

C

C

2017.0

120.7

111.2

N

N C

788

C

C

O

O

O C 437

O

438

C

C O

O

O

2 593

3 12.940

4 88.1

5 92.7

6 2200

7 2190

8 1.60

543

10.600

72.8

72.8

1640

1630

1.63

630

7.813

80.2

77.4

1260

1260

1.64

C O C O O

O

O

S

C

O O

C O

C O

C O

NH

NH C

438

1

439

   ∑ δCi ∆Vi  ⋅15.6T g   Cσ′ ⋅15.6Tg  i  = E= . Cσ n0    ∑ ∆Vi C n   σ 0  i 

(VIII.35)

Substituting equation (IV.41) into correlation (VIII.35) for calculation of Tg of linear polymer, finally, we obtain:    ∑ δCi ∆Vi  ⋅ 15.6    i  E= .   Cσ n0  ∑ ai ∆Vi + ∑ b j    j  i 

(VIII.36)

The results of calculation of E for representatives of different classes of heatresistant polymers are shown in Table 32. It should be noted that the elasticity moduli of glassy polymers at temperatures below Tg differ insignificantly from each other (for example, two-fold difference must not be considered large, because it may be indicated as a result of tests at different deformation rates, for samples of different shape, for samples of the same polymer with different prehistory of production, etc.). That is why we must proceed with caution when estimating the elasticity modulus of linear glassy polymers. The stress-optical coefficient Cσ is a fundamental characteristic of materials applied to the photoelasticity method of stress investigation. According to this method, a model of the full-scale construction is prepared from transparent optically sensitive polymers to which appropriate loads are the applied. Consequently, a birefringence appears in the material of the model, and one may evaluate the stress– strain state on the basis of the construction of the pattern of fringes. In the case of bulky models, the method of deformation freezing is quite efficient. The essence of this method is that the model is heated up to a temperature at which the model material transits into the rubbery state. The model is then loaded and cooled under load down to room temperature, which for usual stress-optical materials is approximately 80–100°C below the glass transition temperature. Strains occurring at loading of the model and optical anisotropy are frozen. Further on, the model is sawn into thin plates which are then studied. Paying no attention to other varieties of the photoelasticity method of stress investigation, let us note that successful development of these methods is possible only by creation of new polymeric materials with the required opto-mechanical properties. Solution of various problems requires materials of two types, elastic and viscoelastic. Investigation of stress fields in stratified elastic media by the method of dynamic photoelasticity requires polymeric materials with different moduli possessing the ratio of elasticity moduli from 2 to 10 and elastic properties at both static and dynamic loading. Such materials have appeared recently in the practice of dynamic photoelasticity [45].

440

Figure 71. Stress relaxation for specimens ED-20 + MTHPA (1), polyisocyanurate network (2) and viscoelastic material (3) the composition of which is indicated in Figure 72

A difficulty in obtaining such materials is that all polymeric glasses independently of the chemical structure possess nearly identical elasticity moduli of ~103 MPa. Seemingly, to obtain materials with different moduli, polymers may be synthesized which at the experimental temperature (room, for example) would exist in the zone of transition from the glassy state to the rubbery state. Because the elasticity modulus decreases abruptly in this zone, a material can always be selected possessing an acceptable elasticity modulus. However, materials in the transition zone possess extremely viscoelastic behavior, whereas the photoelasticity method of stress investigation require elastic materials for solving this task. These two incompatible requirements may be fulfilled in polymer networks of a definite chemical structure by synthesizing high cross-linked networks which, possessing extremely high rubbery modulus, would possess low glass transition temperatures (significantly below the room temperature). The current state of the problem in the sphere of prognosis of properties of such systems enabled the structure of high cross-linked networks to be predicted, which would fulfill these two conflicting requirements [46, 47]. When the above-considered approach is used, the glass transition temperature and the elasticity modulus are predicted, and network carbofunctional organosilicon polyisocyanurates with various but short lengths of the linear fragment were chosen as a particular object of investigation [45, 46]: R CH3 O C

CH3

N

N C

C O

CH3

N

O R

R

where CH3 R:

NH C O CH2 O

CH2

O CH2

Si

CH3 O

CH3 n

Si CH2 CH3

O CH2

CH2

O C NH O

441

A significant property of these materials is that despite the values of moduli typical of the zone of transition from the glassy state to the rubbery state, they display rubbery, as glass or rubber, but not viscoelastic behavior as usual materials existing in the transition zone. Let us compare the relaxation behavior of the given materials and materials (elastic and viscoelastic), which are usually applied to the photoelasticity method [47]: epoxy oligomer ED-20 cured by polysebacic acid anhydride (viscoelastic material) and oligomer ED-20 cured by methyltetrahydrophthalic anhydride (Tg = 115°C, elastic material). Existence of the transition zone (from the glassy state into the rubbery state) of a viscoelastic material in the temperature range from –5 to 34°C enables, changing the test temperature, comparison of the relaxation properties of these polymers to be performed at identical values of the initial elasticity modulus. Analyzing the data shown in Figure 71, it may be concluded that the curve of stress relaxation for a polyisocyanurate network is similar to the relaxation curve of a glassy polymer. The absolute value of stress decrease, when the part of low relaxation rate for the sample of network polyisocyanurate is reached, is approximately the same as for the glassy material (ED-20 + MTHPA) and significantly lower than for a viscoelastic polymer. Figure 72 indicates dependences of mechanical I(t) = ε(t)/σ and optical D(t) = m(t)/σd creep on time: m(t) is the order interference fringes at the moment of measurement; σ is the stress in the sample; d is the sample thickness in the translucence direction. Figure 72b indicates that the compliance of the viscoelastic material smoothly increase after loading. However, in contrast to a typical viscoelastic polymer, for silicon-containing polyisocyanurate networks (see Figure 72b) the creep process rapidly attenuates.

Figure 72. Mechanical I(t) and optical D(t) creep curves of network polyisocyanurates at n = 6.2 (see the structural formula in the text), σ = 1.4 MPa (a) and viscoelastic polymer based on epoxy oligomer ED-20 cured with the mixture of polysebacic anhydride (27.3 %) in the presence of ozelaic acid (13.8%) (b)

442

Hence it may be noted that the elastic behavior, complicated only by weak signs of viscoelasticity, is typical of carbofunctional organosilicon polyisocyanurates synthesized in Refs. [45, 46]. Table 33 indicates the values of equilibrium elasticity moduli, obtained by approximation of relaxation curves of stresses attracting new relaxation memory functions (see below) which, besides high correlation coefficients, yield a series of the physical characteristics of the material (number of microdefects, initial entropy of the system, etc.). This Table also indicates values of the stress-related value of material strip σ 10.0 . As observed from the Table, stress-optical sensitivity n increases with the value of the linear fragment in the sequence of polyisocyanurates differing by the amount of dimethylsiloxane units between network cross-linked points. Obviously, despite a decrease of the concentration of groups with high polarizability anisotropy (isocyanurate cycle, aromatic rings), optical sensitivity increases due to an increase of macrochain mobility, which depends on both the concentration of cross-linked points in the network and the amount of dimethylsiloxane units possessing a low potential energy of Si–C–Si bond rotation [52]. Table 33 Opto-mechanical properties of macrodiisocyanates networks Static loading Dynamic loading N 1.2 2.5 3.8 5.4 6.2

E, MPa

σ 1.0 O , MPa (at 22°C)

Cp, m/s

Ed, MPa

σ 1.0 O , MPa

866 283 149  43

1.26 1.14 1.08 0.80 0.69

1980 1700 1270 1000 800

4822 3454 1897 1140 727

2.6 2.3 2.1 1.8 1.6

The dynamic characteristics of the opto-mechanical properties of polymers may differ significantly from statistic ones due to the influence of the time factor. For example, under the effect of short-term pulse loads the processes associated with recording of the optical pattern of fringes in the model last from several microseconds to hundreds of microseconds. In this case, usual quasi-static creep and stress relaxation tests may not reflect the essence of phenomena proceeding in the polymeric material under dynamic influence. Analysis of photograms indicates that longitudinal and transverse waves spread in a rod from network polyisocyanurates at a constant rate. Values of the rate of longitudinal wave spreading Cspr and dynamic elasticity modulus Ed calculated by 2 , where ρ is the material density, are indicated in Table 33. It the equation Ed = ρCspr is obvious that these materials have a wide range of values of the longitudinal wave rate Cspr (500 … 2000 m/s) and dynamic elasticity modulus Ed (300 … 5000MPa). Hence, optically sensitive materials based on network polyisocyanurates with short chains between neighboring cross-linked points of the network differ significantly from the traditionally used materials which have practically equal values of the dynamic modulus different static elasticity modulus. This creates problems in their application to solution of some dynamic tasks. To estimate the viscoelastic properties of network polyisocyanurates under the conditions of dynamic influence, let us consider pulses of fringe patterns m(t) in different cross-sections l of rods. For comparison, values of pulses m(t) are depicted on a single graphic (Figure 73) with a time displacement which takes into account the

443

rate of wave spreading in materials. In contrast to a typical viscoelastic polymer (see Figure 73b), an insignificant change of the shape of pulses and their duration is typical of samples of network polyisocyanurates with n = 1, …, 9. This change is observed at an increase of the distance passed in the rods and testifies their low viscoelasticity under pulsed loading. Consequently, optically sensitive network polyisocyanurates are suitable for the study of the stress–strain state of stratified media by the dynamic photoelasticity method.

Figure 73. Change of impulses of pattern of fringes m(t) in different sections l of network polyisocyanurates rods at n = 2.5 and l = 50 (1); 150 mm (3) (a) and viscoelastic polymer at l = 60 (1); 100 (2); 160 mm (3) (b), the composition of which is indicated in Figure 72.

Considered above were elastic polymeric materials. However, to solve a series of tasks, optically sensitive materials with viscoelasticity are required. Clearly, the viscoelastic behavior is typical of the transition zone from the glassy to the rubbery state. The currently available photoelasticity methods of creep modeling (the photocreep method) are based on a special selection of materials which, besides high optical sensitivity, would possess a clearly expressed viscoelastic behavior. To solve this problem, it is necessary to obtain polymer networks existing at test temperature in the transition area from the glassy to the rubbery state. Therewith, the viscoelasticity of materials increases as the test temperature approaches glass transition temperature Tg. Therefrom, it is required to obtain a series of polymeric materials with assigned Tg, differently spaced in relation to the test temperature. To predict the thermal and physical characteristics of newly synthesized epoxy anhydridoacidic compounds used for solving the present task, the above-described calculation scheme of determination of Tg was used for polymer networks in ref. [35]. To obtain optically sensitive polymers differing by their thermal and optomechanical indices, to synthesize model materials, epoxy oligomer ED-20, curing agent – methyltetrahydrophthalic anhydride, cocuring agents – dicarboxylic azelaic acid and monocarboxylic oleic acid in various stoichiometric ratios were used in order to regulate the cross-linking frequency and the molecular mass of linear fragment Ms.

444

Figure 74. Dependence of the stress-related values of material strip reduced to the specific stress m/σ on loading time t. Formula of compositions is: 1) ED-20 – 1 mole, methyltetrahydrophthalic anhydride (MTHPA) – 1.4 mole, oleic acid – 0.6 mole; 2) ED-20 – 1 mole, MTHPA – 0.2 mole, azelaic acid – 0.9 mole; 3) ED-20 – 1 mole, MTHPA – 1.6 mole, oleic acid – 0.4 mole

Consequently, a set of polymer networks was obtained, the glass transition temperatures of which are close to room temperature. Obviously, materials based on these networks possessed clearly expressed viscoelasticity. Such polymers possess the ability to creep in an extremely wide range of absolute values of compliance and process rates. This also causes the clearly indicated time dependence of the stressrelated values of material strip reduced to specific stress (Figure 74).

Chapter IX. Dielectric constant of polymers and organic solvents

Calculations of the dielectric constant of polymers from their chemical structure is an important problem both from the point of view of directed synthesis of polymers with the given dielectric constant and for estimation of the polarity (magnetic momentum) of the repeat unit of the polymer that is also of a significant meaning for predicting polymer solubility in organic solvents. That is why it is also worthwhile to perform quantitative estimation of the dielectric constant for organic liquids which are solvents for polymers. It must be noted immediately that the problem of calculation of the dielectric constant of organic liquids is more complicated than of polymers. This is associated with the circumstance that the coefficient of molecular packing for amorphous polymers is approximately identical and depends only slightly on the chemical structure of the polymer. As mentioned above, in the first approximation, the coefficient of molecular packing for glassy amorphous polymers at room temperature is estimated by the value 0.681. In more accurate approximation, the coefficient of molecular packing is approximately identical for all polymers at their glass transition temperatures Tg; this value is kg = 0.667. As shown below, this enables accurate calculations of dielectric constant ε to be performed for polymers at room temperature. For organic liquids, their coefficient of molecular packing significantly depends on the chemical structure; the coefficient of molecular packing for organic liquids varies in the range from 0.45 to 0.705. This circumstance makes it difficult to calculate the density (or molar volume) of a liquid, which is necessary for further calculation of the dielectric constant. It is common knowledge that the relationship between polarization vector P and the vector of electric field intensity E in vacuum and in a dielectric is of the following form: D = E + 4πP = εE, where D is the vector of electric field density. The theory leads to the following expression for the dielectric constant in the case of nonpolar dielectrics:

ε = 1+

∑ niα i i

1 − ∑ niα i β i

,

i

where ni is the concentration of atoms, ions or molecules of the i-th type; α is the polarizability of these structural elements; βi is the factor taking into account the dipole–dipole interaction. It is common knowledge that the dielectric constant of a homogeneous dielectric ε indicates how many times the value of electric field intensity E decreases inside the dielectric compared with the value of external field E0: E E= 0. ε

446

Macroscopic field E is the vector sum, E = E0 + Ei, of the external field E0 and the field Ei stipulated by the substance polarization P in the external field: Ei = –4πP. In weak fields, polarization is proportional to the macroscopic field P = χE, where χ is the macroscopic susceptibility of the substance. Therefrom, with regard to the above-considered equations, we obtain the wellknown connection of macroscopic characteristics:

ε = 1 + 4πχ.

(IX.1)

At microscopic description, the value of polarization is determined by microscopic polarizability χ 0 = ∑ niα i (ni is the concentration and αi is the i

polarizability of structural elements of the substance) and the value of local field Eloc: P = χ0Eloc, where the local field equals Eloc = E + βP. The coefficient of depolarization β for isotropic dielectric equals to 4π/3. A connection between macroscopic χ and microscopic χ0 polarizabilities may be deduced from the above-considered correlations:

χ=

χ0 . 1 − βχ 0

The formula connecting the dielectric constant with microscopic polarizability (the Clausius–Mossotti formula) may be also deduced:

ε − 1 4π 4π = χ0 = ∑ niα i . ε +2 3 3 i

(IX.2)

Electron polarizability αel in the range of optical frequencies equals:

α el =

f 0k e2 , ∑ 2 2 m k ω 0k − ω D

where ωD = 3.2⋅1015 rad/s is the frequency corresponding to D – the sodium bend; ω0,k and f0,k are the frequency and strength of oscillators for the electron spectrum of the substance associated with the 0 → k transition. By the order of magnitude, electron polarizability αel is 10–24 cm3. The appropriate contribution in equation (IX.2) is of the following form:

447

ρ 4π ni (α el )i = ∑ ∑ Ri , 3 i M i where ρ is density; M is molecular mass; Ri is the molar refraction of the i-th structural element (in the case of polymers, calculation is performed per a repeat unit). Orientational polarizability αdip in weak fields associated with the rotation of constant dipoles d, equals:

α dip =

d2 . 3k BT

At room temperature, for dipoles d ≅ 1D the value of αdip also equals approximately 10–24 cm3. The appropriate contribution to the Clausius–Mossotti formula, associated with the presence of polar groups in the structure, will be presented in the following form:

(

)

4π ρ ∑ ni α dip i = M ∑ ∆P j , 3 i j where the sum by j is summed up by all polar groups of structural elements. Finally, equation (IX.2) will have the following form:

ε −1 M ⋅ = ∑ Ri + ∑ ∆Pj . ε +2 ρ i j

(IX.3)

In the case of polymers, calculation is performed per a repeat unit. Taking into account that the density of polymers may be calculated by equation (II.6), we reduce it to the following expression, suitable for calculation of the dielectric constant: k avg P ε −1 = . ε + 2 N A ∑ ∆Vi

(IX.4)

i

The value of molar polarizability P is additive and is composed of polarizabilities of atoms, as well as polarizability constants associated with the presence of various types of chemical bonds (double, triple) and other features of the structure of molecules. In this case, the situation is the same as with estimation of the molar refraction. For non-polar dielectrics, the dielectric constant is stipulated by deformation polarizability only and, according to the Maxwell correlation, is practically coincident with the square of the refractive index in the area of high frequencies ε ≈ n2. For these polymers (polyethylene, polytetrafluoroethylene, polybutadiene, etc.), the molar refraction R is practically coincident with the molar polarization P. For polar dielectrics, the situation is more complicated. Under the effect of an electric field the orientation of constant dipoles proceeds in them. These dipoles occur at the expense of the presence of polar groups in the polymer, for example, such as

448

Š2+ Š&2Š Š&22Š Š&O Š1+&2Š Š1+&22Š Š&1 HWF 7KLV FDXVHV

exceeding of refraction R by polarization P for these groups. Let us analyze the correction, necessary to be introduced into refraction in order to calculate polarizability of the molecule. For polar groups of various chemical nature, polarizability is given as Pi = Ri + ∆Ri,

(IX.5)

where Ri is the molar refraction of this group; ∆Ri is a correction associated with the orientation of dipoles. Values of these corrections were calculated with the help of linear regression analysis based on comparison of refractive indices and dielectric constants of a great number of polar polymers. The calculation itself was performed using formulae (IX.3′) and (IX.3′′), the combination of which gave the following expression:   kavg  ∑ Ri + ∑ ∆R j    i j ε −1  . = N A ∑ ∆Vi ε +2

(IX.6)

i

Calculation results are indicated in Table 34, which gives numerical values of ∆Ri for various polar groups most often met in polymers. If these values are known, dielectric constants for a wide range of organic polymers of various classes may be calculated. The results of the calculations performed indicate good coincidence with the experimental data [133, 214] (see Table 34a). Table 34

Š 2Š Group

Values ∆Ri for calculation of polarizability Group ∆Ri, cm3/mol 3.557 C NH

∆Ri, cm3/mol 21.000

O

C

5.371

Š)

0.845

8.728

Š&O

3.900

17.085

Š&) Š

1.352

O

10.300

O O C O O C O

2

O

Š2+ Š&≡N

3.500 5.464

N O

To obtain more accurate values of the dielectric constant for polymers at room temperature, it is desirable to take into account the temperature dependence of the

449

coefficient of molecular packing. This relates, first of all, to polymers existing in the rubbery state at room temperature. According to ref. [128], the temperature dependence k(T) for these polymers is described by the correlation: Table 34a Calculated and experimental values of the dielectric constant for a series of polymers Polymer Error, % εcalc εexper 1 2 3 4 Polytetrafluoroethylene 1.98 2.00; 1.96; – 1.0 2.01; 2.10 Poly(4-methyl-1-pentene) 2.27 2.13 6.6 Polypropylene 2.27 2.15; 2.20 3.2 Polyisobutylene 2.23 2.23 0.0 Poly(vinyl cyclohexane) 2.38 2.25 5.8 Poly(1-butene) 2.25 2.27 – 0.9 Polyethylene 2.23 2.20; 2.30 1.4 2.35 2.1 2.40 Poly(α,α,α′,α′-tetrafluoro-p-xylylene) Polyisoprene 2.28 2.37 – 3.8 Poly(o-methyl styrene) 2.54 2.49 2.0 Poly(1,4-butadiene) 2.27 2.51 – 9.6 2.51 5.6 2.65 Poly(β-vinyl naphthalene) Polystyrene 2.57 2.55; 2.60; 2.50 0.8 2.57 – 1.2 2.54 Poly(α-methyl styrene) Poly(cyclohexyl methacrylate) 2.70 2.58 4.6 Polychlorotrifluoroethylene 2.70 2.80; 2.60 3.8 2.60 1.9 2.65 Poly(α-vinyl naphthalene) Poly[oxy(2,6-dimethyl-1,4-phenylene)] 2.77 2.75; 2.65; 2.60 0.7 Poly[1,1-cyclohexane bis(4-phenyl)carbonate] 2.97 2.60 14.2 Poly(p-xylylene) 2.58 2.65 – 2.6 Poly(p-chlorostyrene) 2.88 2.82; 2.63; 2.65 2.1 Polyvinylbutyral 2.70 2.69 0.4 Ethylcellulose 2.71 2.70 0.4 Poly(isobutyl methacrylate) 2.71 2.70 0.4 Poly(dimethyl siloxane) 2.75 2.75 0.0 Poly[oxy(2,6-diphenyl-1,4-phenylene)] 2.78 2.80 – 0.7 Poly(m-chlorostyrene) 2.88 2.80 2.8 Poly(n-butyl methacrylate) 2.64 2.82 – 6.4 Poly(vinylidene chloride) 2.87 2.90; 2.92; 2.85 0.7 Bisphenol-A-polycarbonate 3.11 3.00; 3.05; 2.90 2.0 Poly(N-vinyl carbazole) 2.69 2.90 – 7.2 Poly[1,1-ethane bis(4-phenyl)carbonate] 3.18 2.90 9.6 Poly(3,4-dichlorostyrene) 3.16 2.94 7.5 Poly(chloro-p-xylylene) 2.89 2.95 – 2.0 Poly(vinyl chloride) 3.14 3.15; 3.05; 2.95 – 0.3 Poly(1,4-cyclohexylidene dimethylene terephthalate) 2.94 3.00 – 2.0 Poly(ethyl methacrylate) 2.84 2.80; 3.00; 2.90 1.4 Poly(oxy-2,2-dichloromethyltrimethylene) 3.06 3.00 2.0 Poly(p-methoxy-o-chlorostyrene) 2.94 3.08 – 4.5 Poly(methyl methacrylate) 2.94 2.94; 3.15; 3.10 0.0 Poly[thio(p-phenylene)] 2.99 3.10 – 3.5 Polyoxymethylene 2.96 2.95; 2.85; 3.10 0.3 Poly(tetramethylene terephthalate) 3.09 3.10 – 0.3 3.20; 3.16; 3.10 1.9 3.26 Poly(ethyl α-chloroacrylate) Poly[4,4’-isopropylidene diphenoxydi(4-phenylene)sulfone] 2.93 3.18 – 7.9 Poly(ether etherketone) 3.00 3.20 – 6.2 Poly(hexamethylene sebacamide) 3.60 3.80; 3.20; –5.3

450

Poly(vinyl acetate)

1

2 3.10

Poly(ethylene terephthalate)

3.29

Poly(p-hydroxybenzoate) Poly[2,2’-(m-phenylene-5,5’-bibenzimidazole)] Poly(methyl α-chloroacrylate) Poly[4,4’-diphenoxydi(4-phenylene)sulfone] Poly(hexamethylene adipamide)

3.25 2.72 3.47 3.54 4.13

Poly[N,N’-(p,p’-oxydiphenylene)pyromellitimide] Poly[4,4’-sulfondiphenoxydi(4-phenylene)sulfone] Polyacrylonitrile

3.39 3.72 3.01

k (T ) =

kg

(

1 + 3.56 ⋅10 − 4 T − Tg

)

3 3.02; 3.30; 3.20; 3.22; 3.25 3.40; 3.50; 3.10; 3.25 3.28 3.30 3.45; 3.32; 3.40 3.44 4.14; 4.10; 4.00; 3.50 3.50 3.80 3.26; 3.15; 3.10; 4.00

4 2.6 1.2 –0.9 -–17.6 0.6 2.9 – 0.2 – 3.1 – 2.1 – 2.9

; Tg < T.

At room temperature we obtain k (298) =

kg

(

1 + 3.56 ⋅ 10 − 4 298 − Tg

)

.

For copolymers, the equation to calculate the dielectric constant is presented in the following form:

ε −1 = ε +2

k avg (α1P1 + α 2 P2 + ... + α n Pn )         N A α1  ∑ ∆Vi  + α 2  ∑ ∆Vi  + ... + α n  ∑ ∆Vi           i 1  i 2  i n  

,

(IX.5)

where α1, α2, …, αn are the molar parts of the components 1, 2, …, n of copolymer;        ∑ ∆V  ,  ∑ ∆V  , …,  ∑ ∆V  are the Van-der-Waals volumes of the same i i i        i 1  i 2  i n components; P1, P2, …, Pn are polarizabilities of the components 1, 2, …, n. In the reduced form, equation (IX.5) looks as follows: k =n

ε −1 = ε +2

kavg k =n

∑ α k Pk

k =1

  N A ∑ α k  ∑ ∆Vi    k =1  i k

,

(IX.6)

451

  where αk and  ∑ ∆Vi  are the molar part and the Van-der-Waals volume of the k-th    i k component, respectively; Pk is its polarizability. It is desirable to express the dielectric constant of a copolymer via dielectric constants of the components, then equation (IX.5) may be rewritten in the following form:

ε −1 = ε +2

  ε − 1  ε − 1  ∆Vi  + α 2 2 ∆Vi  + ... + α1 1 ∑ ∑     ε1 + 2  i ε2 + 2  i 1 2     α1 ∑ ∆Vi  + α 2  ∑ ∆Vi  + ... +      i 1  i 2

→

 ε − 1  + αn n ∆Vi  ∑  ε n + 2  i    + α n  ∑ ∆Vi     i n

→

(IX.7)

n,

where ε1, ε2, …, εn are dielectric constants of the components 1, 2, …, n. In the reduced form, equation (IX.7) becomes the following: k =n

ε −1 



∑ α k ε k + 2  ∑ ∆Vi 

k ε − 1 k =1  i k = , k =n  ε +2  ∑ α k  ∑ ∆Vi  k =1  i k

(IX.8)

  where αk is the molar part of the k-th element;  ∑ ∆Vi  is the Van-der-Waals    i k volume of the k-th element. To estimate the dielectric constant of organic liquids which are solvents of polymers, let us now turn to the calculation scheme. As mentioned above, the coefficient of molecular packing for organic liquids depends significantly on the chemical structure and is not a constant value. That is why calculation of the dielectric constant by formula (IX.3) is difficult, because the density of the liquid can not be calculated with acceptable accuracy. However, this is not the main reason why the dielectric constant of liquids may not be estimated with the help of the Clausius–Mossotti equation with acceptable accuracy. For example, if the value ε is calculated for such a solvent as n-propyl alcohol and the value ∆Ri is assumed the same as for polymers, we will get the following numerical values: ∆Ri = 3.3 cm3/mol, ρexper = 0.799 g/cm3, ∑ ∆Vi = 70.65 Å3, ∑ ∆Ri = 17.579 cm3/mol, P = i

i

21.079 cm3/mol. Substituting these values into equation (IX.3), we obtain ε = 2.17, which is by the order of magnitude lower than the experimental value, which equals

452

20.1. The same calculations have also been performed for a series of other liquids and, therewith, the result is analogous to the former one: in all cases, the calculated dielectric constant is significantly lower than the experimental one. Hence, even if it is possible to calculate the density of a liquid with high accuracy, this would not lead to proper values of the dielectric constant calculated by equation (IX.3). It might be suggested that the value ∆Ri for the same polar group contained in polymers and low-molecular liquids must be different. The calculations performed indicate that this is true not only for comparison of the behavior of organic liquids and polymers, but also for comparison of liquids themselves related to the same class. For example, contribution of OH-group to the value of ∆Ri is different in the sequence of alcohols and depends on the chemical structure of alcohol. In all cases, for liquids related to the same class, the contribution of a polar group to the value ∆Ri increases with the Van-der-Waals volume of the liquid. Such analysis has been performed on the basis of equation (IX.6), into which the average value of the coefficient of molecular packing kavg for liquids of various classes is substituted, and values ∑ ∆Ri i

are calculated from Tables shown in refs. [28] and [128]. Table 34b Dependence of ∆Ri on the Van-der-Waals volume of liquids   ∆Ri = f  ∑ ∆Vi  Class of liquids  i  Chlorinated compounds containing 1 carbon atom ∆R 1 = −0.49 ∆V + 43.8 Cl

∑

i

i

Chlorinated compounds containing more than 1 carbon atom Chlorinated compounds with a double bond at chlorine atom Alcohols

2 ∆RCl = −0.332∑ ∆Vi + 45.52

Acids

Esters

  ∆RCOOH = 3.75 ⋅10 −3  ∑ ∆Vi  − 0.825∑ ∆Vi + i  i  + 57.0 ∆RCOO = 0.18∑ ∆Vi + 20.2

Ketones

∆RCO = 0.525∑ ∆Vi + 8.25

Ethers Aldehydes

∆RO = 29.0

i

= ∆RCl = −0.325∑ ∆Vi + 33.5 i 2

  ∆ROH = −1.708 ⋅ 10 −3  ∑ ∆Vi  + 0.761∑ ∆Vi + i  i  + 0.343 2

i i 2

Nitriles

  ∆RCOH = −1.67 ⋅ 10 −3  ∑ ∆Vi  + 0.751∑ ∆Vi + i  i  + 2.648 ∆RCN = 0.525∑ ∆Vi + 9.75

Nitrocompounds

∆R NO2 = 0.667∑ ∆Vi + 3.64

i i

Therewith, experimental values of the dielectric constant ε were used, and values ∆Ri for each polar group were calculated by equation (IX.4). A total of eleven

453

classes of organic liquids was analyzed, indicated in Table 34b. Therewith, for chlorinated compounds, the influence of the chlorine atom on polarization, OH-group – for alcohols, COOH-group – for acids, etc. was taken into account. Figure 74′ indicates the dependence of ∆ROH on the Van-der-Waals volume of an alcohol molecule. Clearly, all points fit well the generalized curve that may be approximated with the help of a correlation 2

  ∆ROH = −1.708 ⋅ 10 − 3  ∑ ∆Vi  + 0.761∑ ∆Vi + 0.343 .   i  i 

Figure 74′. Dependence of correction ∆R to polarizability on the Van-der-Waals volume

∑ ∆Vi

for

i

OH-group

Correlations for calculation of ∆Ri values for all remaining classes of liquids were obtained in an analogous manner. These correlations are shown in Table 34b. If they are known, contribution of each polar group to the value ∆Ri may be calculated easily. These calculations were performed for a multiplicity of organic liquids (Table 35). The calculations were performed with the help of correlations indicated in Table 34b; Van-der-Waals volumes and molar refractions were determined according to the common procedure [28, 128]. The calculations performed have indicated quite good coincidence with experimental values of ε, which could not be obtained by other methods. Hence, it is possible to calculate the dielectric constant of polymers and their solvents; this may be performed on the basis of the chemical structure of the repeat unit of a polymer or a molecule of an organic liquid. Table 35 Values of the Van-der-Waals volume, polarizability, calculated and experimental values of dielectric constant for a series of organic liquids P, ∑ ∆Vi , Å3 Liquids Error, % εcalc εexper i cm3/mol 1 2 3 4 5 6 n-Pentane 97.40 25.29 1.932 1.844 4.80 Isopentane 97.40 25.29 1.932 1.843 4.80

454

1 n-Hexane Cyclohexane n-Heptane n-Octane n-Nonane n-Decane Benzene Toluene o-Xylene m-Xylene p-Xylene Ethylbenzene Naphthalene Isopropylbenzene Styrene Chlorobenzene Bromobenzene Ethylbromide 1,2-Dibromoethane Methyliodide Ethyliodide Cyclohexanol Methanol Ethanol n-Propanol n-Butanol Isobutanol n-Pentanol n-Hexanol n-Octanol Acetic acid Butyric acid Isovaleric acid n-Valeric acid Methyl formate Ethyl formate Methyl acetate Ethyl acetate n-Propyl acetate Ethyl propyonate n-Butyl acetate Acetone Methyl ethylketone Diethylketone Cyclohexanone Diethyl ether Dipropyl ether Diisopropyl ether Dibutyl ether Diamyl ether Diisoamyl ether Butylethyl ether Ethanale Propyonale Butynale Nitromethane

2 114.00 102.00 132.00 149.00 166.00 183.00 88.24 105.00 121.00 121.00 121.00 122.00 134.00 139.00 118.00 102.00 109.00 68.02 89.83 64.68 81.75 110.00 36.51 53.58 70.65 87.72 87.72 105.00 122.00 150.00 55.16 89.29 106.00 106.00 57.29 74.36 74.36 91.43 109.00 109.00 126.00 64.84 81.91 98.98 104.00 89.86 124.00 124.00 158.00 192.00 192.00 124.00 47.77 64.84 81.91 46.19

3 29.31 27.71 34.53 39.14 43.76 48.38 26.31 30.93 35.54 35.54 35.54 35.54 41.65 40.16 35.08 35.07 34.07 19.20 26.97 19.62 24.24 32.79 11.90 16.52 21.14 25.75 25.75 30.37 34.99 44.23 17.24 26.47 31.09 31.09 21.82 26.44 26.44 31.05 35.67 35.67 40.29 21.44 26.05 30.67 33.09 25.87  35.11        

4 1.945 1.990 1.941 1.947 1.951 1.950 2.261 2.240 2.568 2.374 2.270 2.220 2.339 2.207 2.256 4.160 5.400 9.380 4.780 7.000 7.820 14.790 32.660 26.610 21.440 17.580 17.580 14.630 12.500 10.450 6.240 2.890 2.650 2.650 9.640 7.170 7.170 6.090 5.450 5.450 5.080 22.550 18.920 17.070 17.930 4.400 3.410 4.880 3.030 2.810 2.810 3.410 23.410 18.000 14.720 39.050

5 1.890 2.023 1.924 1.948 1.972 1.991 2.284 2.379 (25°) 2.568 2.374 2.270 2.412 2.540 (85°) 2.380 2.430 (25°) 5.621 5.400 9.390 4.780 7.000 7.820 15.000 32.630 24.300 20.100 17.100 17.700 13.900 13.300 10.340 6.150 2.970 2.640 2.660 8.500 7.160 6.680 6.020 5.690 5.650 5.010 20.700 18.510 17.000 18.300 4.335 3.390 (26°) 3.880 3.060 (25°) 2.770 (25°) 2.820 3.060 (25°) 21.100 18.500 13.400 35.870 (30°)

6 2.90 – 1.60 0.90 – 0.05 – 1.20 – 2.10 – 1.00 – 5.80 0.00 0.00 0.00 – 8.00 – 7.90 – 7.30 – 7.20 – 26.00 0.00 – 0.10 0.00 0.00 0.00 – 1.40 – 0.70 8.80 6.01 2.28 – 1.10 4.80 – 6.40 0.80 0.70 – 2.60 0.20 – 0.60 13.10 – 0.10 7.10 1.00 – 4.30 – 3.60 1.20 8.90 2.20 0.40 – 2.00 1.50 0.60 25.80 – 1.00 1.40 0.30 11.40 10.90 2.70 9.80 8.90

455

1 Nitroethane 1-Nitropropane 2-Nitropropane Nitrile acetate Nitrile propyonate Nitrile butyrate Nitrile pentanate Nitrile isoheptanate

2 63.26 80.33 80.33 48.96 66.03 83.09 100.00 117.00

3    16.72 21.34 25.95 30.57 35.19

4 30.540 27.080 26.760 40.780 25.240 20.450 18.110 16.730

5 28.060 (30°) 23.240 (30°) 25.520 (30°) 37.500 27.200 20.300 17.400 15.500

6 8.80 16.50 4.90 8.70 – 7.20 0.70 4.10 7.90

Chapter X. Equilibrium rubbery modulus for polymer networks

X.1 Calculations of the equilibrium modulus To estimate the equilibrium rubbery modulus E∞ and molecular mass of an linear fragment Mc in the case of elastomer networks in the case of fairly sparse cross-links, the equation of the classic rubber elasticity theory is used: E∞ = 3

ρRT , Mc

(X.1)

where ρ is the density of a cross-linked elastomer; R is the universal gas constant; T is absolute temperature. Application of equation (X.1) to high-crosslinked networks, the linear fragment of which contains an extremely small number of units, down to 1 and even lower, causes a substantial divergence between the experimental and calculated values of E∞. For equation (X.1) to be true for description of the properties of highcrosslinked networks, the so-called front-factor Φ is introduced into it: E∞ = 3

ρRT Φ. Mc

(X.2)

However, introduction of an unpredictable front-factor into equation (X.1) does not improve the situation, because, comparing the calculated and experimental values of E∞, we may only estimate this front-factor. In this connection, ref. [31] indicates an attempt to obtain a generalized correlation for estimation of E∞ and Mc which is true both for sparse and high-crosslinked networks. Let us perform a detailed analysis of the influence of a great number of network cross-linked points on the equilibrium rubbery modulus. Preliminarily, it should be noted that for sparse networks, the Van-der-Waals volume of cross-linked points is extremely lower than the Van-der-Waals volume of linear fragments. That is why it may be neglected when the compressibility of a network system is estimated. In the case of high-crosslinked networks, it is impossible to do this, because the total Van-der-Waals volume of cross-linked points is approximately identical to the total Van-der-Waals volume of linear fragments and may even exceed it. Considering an elastomer network as a system consisted of two subsystems – elastic and rotational-isomeric, let us analyze first the latter one. Ref. [28] indicates that determination of the elasticity coefficient of the rotational-isomeric subsystem requires knowing of the difference between the energies of rotational isomers, which depends on the size of ‘molecular defects’ lid in the polymer in the following manner:

()

2 ∆Ei = 4 Di a 2 lid ,

(X.3)

457

where Di is the average energy of intermolecular interaction, typical of the atom of this type; a is the Morse potential constant:

(

)

2   ϕ ( R ) = D  e − a ( R − r0 ) − 1 − 1 ,  

(X.4)

where D is the depth of the potential trough; r0 is the equilibrium distance between atoms in harmonic approximation.

Figure 75. ‘Cylinder of interaction’ of two hydrogen atoms (see text).

To determine the size of the molecular defect lid , let us consider a ‘cylinder of interaction’ (Figure 75), the notion of which was introduced in work [91]. Its volume is composed from volumes of interacting atoms ∆Vi and the volume of the defect itself, characterized by δi∆Vi: cryst = 2∆Vi + δ icryst ∆Vi , Vcyl amph = 2∆Vi + δ iamph ∆Vi , Vcyl

(X.5) (X.6)

amph where ∆Vi is the Van-der-Waals volume of the i-th atom; values δ icryst and δ i participating in correlations (X.5) and (X.6), are discussed in detail in work [31]; cryst is the volume of the cylinder of interaction for a crystalline therewith, Vcyl amph polymer, and Vcyl is the same for an amorphous polymer.

Therefrom, the defect value equals lid =

amph cryst = Vcyl Vcyl δ amph − δ icryst ∆Vi = i , S cyl 2S cyl 2

(X.7)

where ∆Vi is a part of the Van-der-Waals volume of the i-th atom which is overlapped by the Van-der-Waals volume of the atom, chemically bonded to it; digit 2 in the denominator appears due to the equality of the amplitude of the i-th atom jump-over to a half of the defect size; Scyl is the cross-section of the cylinder of interaction. Let us estimate the value of lid . For this purpose, let us consider the limiting case when the base radius of the cylinder of interaction equals the Van-der-Waals radius of the i-th atom. Then

458

(

)

1 amph − δ icryst Ri . lid = δ i 3

(X.8)

For a hydrocarbon polymer, Ri = RH; the defect size is the constant value which equal lid = 0.053RH, where RH is the Van-der-Waals radius of the hydrogen atom, equal to 1.17 Å. Substituting the value of lid obtained into expression (X.3), we obtain the value of hydrogen atom contribution to the difference of the energies of rotational isomers: ∆EH = 4.56 kJ/mol. The value found correlates by the order of magnitude with spectroscopic data on the difference of energies of rotational isomers. For further analysis, let us rewrite expression (X.3) in the following form: 2

 ld  ∆Ei = 4 Di a 2 Ri2  i  ,  Ri   

(X.9)

where Ri is the Van-der-Waals radius of the i-th atom; lid is the size of the defect formed by this atom. Ref. [28] indicates estimation of the value aRi; therewith, it was found that aRi ≈ 6. To estimate the contribution of the rotational-isomeric subsystem to the temperature dependence of the elasticity modulus of the polymer in the transition region and in the range of the rubbery state, values of lid for various atoms and types of the intermolecular interaction should be found. As the calculations performed have indicated [28], melting points of polymers Tm and the energy of intermolecular interaction Di, included in expression (X.9), for the atom of the present type depend on the fact whether it participates in the composition of a group of atoms performing the hydrogen bond or dipole–dipole interaction. That is why when ∆Ei is calculated by expression (X.9), the influence of the types of specific effect mentioned will be generally displayed via the energy of intermolecular interaction Di of the atom of this type. Therewith, as calculation amph

indicates, the constants δ i and δ icryst are independent of the influence of hydrogen bonds and dipole–dipole interaction. For the case when the repeat unit consists of a selection of atoms of different types, let us introduce the notion of efficient difference in the energies ∆Eeff of rotational isomers, which may be found as follows. Apply a correlation deduced in ref. [28] to estimate the elasticity modulus of amorphous polymers:

E=

∑ ∆Vi i

∆V S ∑ κ il i ii i

,

(X.10)

where ∆Vi is the Van-der-Waals volume of the i-th atom of a repeat unit; Si is the Van-der-Waals surface of the i-th atom, through which the intermolecular interaction

459

occurs; κi is the elasticity coefficient of the i-th atom bond; li is the characteristic size of the bond (see Figure 75). For the rubbery state, the denominator in expression (X.10) is equal to ∆Vi Si ∑ κ l , where κri is the elasticity coefficient of the rotational–isomeric subsystem. ri i i According to ref. [28], the value of κri is

κ ri =

RT

( )

2 qi lid

,

(X.11)

 ∆E  where qi = exp i  ; ∆Ei is the difference in the energies of rotational isomers.  RT  With regard to correlation (X.11), the denominator of equation (X.10) obtains the form: 2 ( lid ) qi Si ∆Vi . ∑ i

(X.12)

RTli

Let us introduce a value qeff so that it is determined from the condition: 2 d 2 ( ) qeff Seff ∑ ∆V . leff ( lid ) qi Si ∆Vi = ∑ i

li

i

leff

(X.13)

i

Since leff, according to correlation (X.3), is calculated from the formula

(leffd )2 = 4∆DEeffa 2 , eff

d = 〈l〉, then condition (X.13) may be reduced where Deff = αRTm, and Seff = 〈S〉 and leff to the following form: 2 ( lid ) qi Si ∆Vi ∑ i

li

= T =Tm

∆Eeff qeff S 4αRTm a 2 l

Vsurf .

(X.14)

The left part of expression (X.14) is known, and 〈S〉 and 〈l〉 in the right part are known, too. That is why the problem is reduced to obtaining a solution of equation of the type b = xex, where x = ∆Eeff/RTm, and b includes all the known components of equation (X.14). On this basis, the influence of atoms of various types on ∆Eeff may be determined. If the hydrogen atom is assumed to be the most typical one included in polymers, and the value ∆EH appropriate to it is taken, appearance in the structure of a repeat unit from atoms of another type with ∆Ei will cause a change of ∆Eeff.

460

Therewith, this change depends on the weight coefficient of this atom in the repeat unit. The value ∆Eeff will either increase or decrease: ∆Eeff > ∆EH, if ∆Ei > ∆EH, and ∆Eeff < ∆EH, if ∆Ei < ∆EH. Let us now consider the influence of the cross-linking degree on the elasticity modulus of a polymer network existing in the rubbery state. Representing the polymer network as a mixture of linear fragments and cross-linked points, let us write down that 1/Enet = γ/El.f. + (1 – γ)/Ecrl.p., or E net =

E l.f.

γ + (1 − γ )

El.f. Ecrl.p.

,

(X.15)

where El.f. and Ecrl.p. are the moduli for linear fragments and cross-linked points, respectively; γ is the concentration of linear fragments. Because equation (X.15) was deduced by summing up compressibilities of linear fragments of the polymeric chain and cross-linked points, consequently,     γ =  ∑ ∆Vi  ∑ ∆Vi , where  ∑ ∆Vi  is the Van-der-Waals volume of the    i  l.f. i  i  l.f. linear fragment;

∑ ∆Vi

is the Van-der-Waals volume of the repeating fragment of

i

the polymer network. As regards the notion of the cross-linked point from the present viewpoint, it was formulated above: the cross-linked point of a network is a group of atoms consisted of the atom, from which branching starts, plus neighboring atoms, chemically bonded to the former one, and their nearest substituents.   The value  ∑ ∆Vi  ∑ ∆Vi may be reduced to the following form:   i  l.f. i 0

     ∑ ∆Vi  n ∑ ∆Vi       i  l.f.  i  l.f. = , 0 ∑ ∆Vi     i n ∑ ∆Vi  +  ∑ ∆Vi       i  l.f.  i  crl.p.

(X.16)

0

  where  ∑ ∆Vi  is the Van-der-Waals volume of the repeat unit of the linear    i  l.f. fragment; n is the number of repeat units per single cross-linked point. Introducing designations ξ = El.f./Ecrl.p.

  and β =  ∑ ∆Vi     i  crl.p.

0

   ∑ ∆Vi  ,    i  l.f.

where El.f. is the elasticity modulus of the linear polymer; Ecrl.p. is the elasticity

461

  modulus of a polymer network consisted of cross-linked points only;  ∑ ∆Vi  is    i  crl.p. the Van-der-Waals volume of the network cross-linked point, we reduce to the following correlation: γ = n/(n + β), and expression (X.15) is transformed to the form: El.f. n βξ = γ + (1 − γ )ξ = + . E net n+β n+β

(X.17)

Since the polymer composed of cross-linked points only has the modulus many orders of magnitude higher than the modulus of the linear elastomer, then ξ << 1, and for high-crosslinked networks ξ << β. That is why the second term in expression (X.17) may be neglected. Therefrom, we obtain that El.f. n = . E net n + β

(X.18)

d Let us turn back to efficient values leff, qeff, leff and Seff. Taking into account that the elasticity modulus of the rotational-isomeric subsystem is proportional to the mean-statistic number of chains n concluded in linear fragments between cross-linked points, correlation (X.10) may be reduced to the form:

El.f. =

leff RT , d leff qeff S eff n

( )

(X.19)

and for the case of n0 = 1 and temperature T0, the rotational-isomeric modulus of such a hypothetical polymer equals:

(El.f. )0 =

leff RT0

(leffd )2 qeff0 Seff

.

(X.20)

Taking the ratio of these moduli, we obtain 0 El.f. T qeff = , (El.f. )0 T0 nqeff

(X.21)

 ∆E   ∆E  0 = exp eff  . where qeff = exp eff  and qeff  RT   RT0  Substituting this expression into correlation (X.18), we obtain: 0 E net n + β T qeff = . (El.f. )0 n 2 T0 qeff

(X.22)

462

For practical calculations, it should be taken into account that the repeating fragment of the network consists of nΦ/2 of repeat units. That is why expression (X.22) with regard to functionality Φ must be reduced to the form: E net 2(n + β ) T = (El.f. )0 Φn 2 T0

0 qeff

qeff

.

(X.23)

Let us analyze in more detail equation (X.23) comparing it with equation (X.1) of the classical rubber elasticity theory. Clearly, these equations are analogous, equation (X.23) describing the modulus of both high-crosslinked and sparse networks. Actually, writing down n = Mmix/M0, from equation (X.23) we obtain: 0 2 (M c / M 0 + β ) T qeff , E net = (El.f. )0 Φ (M c / M 0 )2 T0 qeff

(X.24)

where M0 is the molecular mass of the repeat unit. For sparse networks, β << Mc/M0. Then 0 2 M 0 T qeff E net = (El.f. )0 . Φ M c T0 qeff

(X.25)

Comparing expressions (X.1) and (X.25), we obtain that

(El.f. )0 = 3RΦT0 q0eff

.

(X.26)

2M 0 qeff

0 , and If measurements of the modulus are performed at T = T0, then qeff = qeff equation (X.23) is reduced to the form:

Emix = (El.f. )0

2( n + β ) Φn 2

.

(X.27)

In the presence of large chain fragments between cross-linked points (sparse networks) the condition β << n is fulfilled, and equation (X.27) is reduced to the correlation: E net = (El.f. )0

2 Φn

(X.28)

0 qeff and M0n = Mc, we obtain: and, with regard to correlations T0 T = qeff

E net =

3ρRT , Mc

(X.29)

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i.e. correlation (X.29) is entirely in agreement with equation (X.1). Correlation (X,27) is applicable to practical calculations of the modulus of high-crosslinked networks, the value (El.f.)0 being calculated either by equation (X.26) or determined from experimental data on the equilibrium modulus for sparse networks, when n >> 1. 0 Let us perform a series of transformations. When T = T0 and qeff = qeff , expression (X.29) obtains the form

(El.f. )0 = 3ρRΦT 2M 0

.

(X.30)

Then from correlations (X.30) and (X.27) we obtain: E net =

3ρRT (n + β )

. (X.31) M 0n2 Taking into account that Mc = M0n, we obtain from expression (X.31) that E net =

3ρRT (n + β ) . M cn

(X.32)

Dividing the numerator and the denominator of expression (X.32) by n, we reduce it to the form: E net =

3ρRT ( n + β n) . Mc

(X.33)

Mc =

3ρRT ( n + β n) . E net

(X.34)

or

For a sparse network, β << 1 and n >> 1, that is why β/n → 0, and generalized equation (X.33) is transformed into usual equation (X.1). To estimate the value Mc for the case of high-crosslinked networks, expression (X.34) may be used. Preliminarily, for convenience, it may be transformed with regard to n = Mc/M0 to the following form:  M  3 ρRT 1 + 0  β  Mc  E net = . Mc

(X.35)

Then from equation (X.35) we get: Mc =

3 ρRT + 3ρRT (3ρRT + 4 E net M 0 β ) 2 E net

.

(X.36)

For the sparse networks, β → 0 and expression (X.36) transforms to equation (X.1).

464

As an example, let us discuss model networks based on polydimethylsiloxane used in ref. [188]. The network is of the following structure: ... O (H3C CH3 ...

(O

Si)m +1 CH3

O

(H3C

Si CH3)m +1 O CH3 Si

O

(Si

O

CH3

Si

CH3)m +1

...

O)m +1

O .. . The cross-linked point of the network is marked by dotted lines. For this network,    ∑ ∆Vi  = ∆VSi,175 + 4∆VO,135 = 34.7 Å3;    i  crl.p.    ∑ ∆Vi  = m(∆VSi,172 + 2∆VC,106 + 4∆VO,135 + 6∆VH,124) = 72.1 Å3;    i  l.f. β = 34.7/72.1 = 0.481. Calculation of the equilibrium rubber modulus of high cross-linked networks by equation (X.27) and the glass transition temperature Tg by equation (IV.85) requires taking into account all details of chemical structure of the network. The point is that when m = 0, the structure of this network obtains the form ... O H3C CH3 ...

O

Si

CH3

O O

CH3 H3C

...

Si CH3 Si

O

O Si

Si CH3

CH3

O O

Si

O

...

O ...

O ...

CH3 In this case, only a part of the unit

Si CH3

O

remains as a linear fragment

465

between neighboring cross-linked points, because the oxygen atom in this unit is a component of the neighboring cross-linked point. Determining n as *

0

   ∑ ∆Vi     i  l.f.

   ∑ ∆Vi  , where    i  l.f.

*

   ∑ ∆Vi  is the Van-der-Waals volume of    i  l.f.

CH3 Si

group, we get n = 0.99. If m = 1, n = 1.99, etc. Of prime importance is

CH3 taking into account of this fact in calculation of the glass transition temperature Tg, which is calculated by equation (IV.85). In this case, equation (IV.85) transformed to the formula 0

Tg =

*

       ∑ ∆Vi  + m ∑ ∆Vi  +  ∑ ∆Vi         i  crl.p.  i  l.f.  i  l.f.. 0

*

       ∑ K i ∆Vi  + m ∑ ai ∆Vi  +  ∑ ai ∆Vi         i  crl.p.  i  l.f.  i  l.f. 0

,

(X.37)

*

    where  ∑ ai ∆Vi  and  ∑ ai ∆Vi  are series of increments for structures      i  l.f.  i  l.f. CH3 Si

CH3 O

and

CH3 considered,

Si

, respectively (note that in the case of the network

CH3

∑b j

= 0).

j

Calculation of the equilibrium rubbery modulus Enet by equation (X.27) requires the value (El.f.)0 to be estimated first. As mentioned above, this may be performed by two methods. The first method concludes in application of expression (X.30), i.e. the value (El.f.)0 is determined empirically. Substituting ρ = 1.169 g/cm3, Φ = 4, M0 = 74.15, T = 293 K into this equation, we obtain that (El.f.)0 = 230 MPa. The second method concludes in application of expression (X.27) and experimental value of Enet for a sparse network. If the experimental value of Enet determined in ref. [188] is used then, basing on expression (X.27), we obtain that (El.f.)0 = 197 MPa. Calculations of Enet by expression (X.27) using this value of (El.f.)0 give equilibrium rubbery moduli shown in Table 36. Clearly, Enet increases sharply with shortening of the distance between neighboring cross-linked points of the network, and for high-crosslinked network (n = 1) reaches high values. Therewith, the glass transition temperature remains below room temperature.

466

Table 36 M 1 1 2 3 4 6 9

n 0.99 1.99 2.99 3.99 4.99 6.99 9.99

Values of β, Enet and Tg for polydimethylsiloxane networks Emix, MPa β 0.482 148.0 0.241 61.5 0.160 38.2 0.120 27.6 0.096 21.6 0.069 15.0 0.048 10.3

Tg, K 183 166 161 158 156 154 153

X.2 Heteromodular and gradient-modulus polymers Concluding this Chapter, let us consider application of calculation schemes which allow estimation of the glass transition temperature Tg and the equilibrium rubbery modulus E∞ to create polymeric materials with unusual properties. The case in point is the production of elastic polymeric materials of two types: 1) heteromodular ones, displaying the elasticity modulus constant for every sample, which changes in a very wide range when transiting from sample to sample; 2) gradient-modulus ones displaying the elasticity modulus changing smoothly in the frames of the same sample along the assigned direction, therewith, the material possesses no layers or interfaces. In the latter case, it is necessary to obtain a smooth transition from rubber to plastic or vice versa in the volume of the same material that enables heteromodular constructions to be created without using traditional methods of bonding – gluing, welding, etc. Complexity of realization of this idea is associated with two main features of polymeric material behavior, which must be clarified. Figure 76 schematically shows the temperature dependence of the elasticity modulus for a typical polymer: unfortunately, in the glassy state range, the elasticity modulus weakly depends on the chemical structure of the polymer and varies in the range (2–3)⋅103 MPa. This the first difficulty of obtaining heteromodular and gradient-modulus materials.

Figure 76. Schematic representation of dependence of elasticity modulus E on temperature T.

In the area of transition from the glassy state into the rubbery state, the elasticity modulus decreases abruptly in a narrow range of temperature (∆T = 20–30°)

467

by several orders of magnitude (see Figure 76). In this connection, at first glance, it may seem that the problem of obtaining polymers with different elasticity moduli lying, for example, in the range from 3⋅103 MPa to 3 MPa, is quite simple: to accomplish this, polymers possessing the glass transition temperature Tg close to room temperature must be produced (if heteromodular materials should work at room temperature). However, it is common knowledge that materials in the transition zone display a clearly expressed viscoelastic behavior and, moreover, their mechanical properties change sharply at extremely small changes, both decrease (transition to plastic) and increase (transition to rubber), of temperature. This is the second difficulty of obtaining heteromodular materials, which besides the wide range of change of the elasticity modulus must possess elastic but not viscoelastic properties. Moreover, they must retain the assigned gradient of the properties in a wide temperature range. Theoretically, the wide range of the elasticity modulus without applying any plasticizers or fillers may be obtained by creating high-crosslinked network structures containing bulky cross-linked points bonded by flexible linear chains of controlled length (Figure 77).

Figure 77. Schematic representation of network consisting of bulky cross-linked points and short flexible chains as linear fragments.

This yields from generalized equation (X.33) for estimation of the equilibrium rubbery modulus E∞ for network systems. According to equation (X.33), a high value of E∞ (or Enet) may be reached transiting to high-crosslinked networks with bulky cross-linked points when n = 1 and β > 1. Therewith, to keep glass transition temperature Tg low, linear fragments connecting cross-linked points must be extremely flexible. Polyisocyanurates networks, the chemical structure of which is displayed in Chapter VIII, were synthesized [45, 46] as the structures containing rigid bulky cross-linked points bonded by linear flexible chains (R). The role of a cross-linked point (marked by dotted lines) is played by isocyanurate cycle possessing the functionality equal to three. Short organosilicon chains, the structure of which is indicated in Chapter VIII, were used as linear fragments. In another method, polyisocyanurate copolymer networks were produced [127]. The general principle of synthesis of these materials is that application of olygomeric diols as one of the original components interacting with a diisocyanate, 2,4-toluene diisocyanate, in particular, by the urethane-forming reaction, gives first macrodiisocyanates:

468

H3C

CH3

O C N

N C O O NH C O

O R O C

HN

Industrial oligomeric rubbers with hydroxyl end groups may be used as diols. One of them is PF-OP-15 polyether representing the oligomeric copolymer of tetrahydrofurane and propylene oxide: R:{

[O

(CH2)4]m

(O CH2

CH)n }p CH3

m/n = 19.7/1.7; p = 1.2.

Figure 78. Dependence of calculated glass transition temperature Tg of polyisocyanurate polymers based on diisocyanates with different structure R as dictated by the mole fraction α of rubber PF-OP-15:

1) R:

3) R:

; 2) R:

CH2

H

CH2

H

; 4) R:

;

(CH2)6

.

Macrodiisocyanates obtained from oligomeric rubbers form polymer networks by the reaction of polycyclotrimerization in situ and copolymerize with diisocyanates of any chemical structure (aromatic, alicyclic or aliphatic), capable of forming polyisocyanurate networks at an acceptable rate. In reality, mixed copolymeric structure is formed in this case. The polymer may contain a network composed of products of homopolycyclotrimerization of oligomer and diisocyanate simultaneously

469

with a network formed by interaction of the oligomer with diisocyanate producing structures with the arbitrary disposition of linking chains. As the excess of diisocyanate in the initial reactionary mixture increases compared with oligomeric macrodiisocyanate, the concentration of rigid cyclic structures in the network formed increases. Hence, the ratio of flexible (polyether) and rigid (isocyanurate cycles with joint aromatic cycles) fragments may be changed as desired that regulates mechanical properties in a wide range. To check the real possibility of this regulation of polymer networks properties, calculation of Tg value was performed preliminarily with application of equation (IV.85). Figure 78 indicates a smooth decrease of Tg value as the part of flexible rubber fragment increases. Calculations of the equilibrium elasticity modulus performed by equation (X.27) indicated [127] that at low Tg (below room temperature) the equilibrium modulus E∞ may obtain high values, intermediate between values of the moduli for rubber and plastics. Such preliminary calculations were performed for networks with organosilicon linear fragments [45]. Consequently, an expression was deduced for the structure considered, which connects Tg of the network with the amount of dimethylsiloxane units n in the linear fragment: Tg =

751 + 108.15n . 2,450 + 721.5n

(X.38)

Calculations performed according to expression (X.38) are shown in Table 37, which indicates that the glass transition temperature lies below room temperature and decreases as the value of n increases approaching the glass transition temperature of polydimethylsiloxane at n = 44. Table 37 also indicates the values of equilibrium moduli, which obtain different values in the range from 3 to 870 MPa in dependence on the value of n. Synthesis of polymer networks with the above-discussed structure performed confirmed correctness of the suppositions and calculations made [45, 46]. For example, E∞ of obtained networks with organosilicon linear fragments changes in the range from 3 to 3⋅103 MPa (Figure 79) in dependence on the length of the linear fragment. Table 37 Values of calculated glass transition temperatures Tg and equilibrium modulus of elasticity E∞ of cured macrodiisocyanate depending on the number of dimethylsiloxane repeat units n Eexper, MPa Ecalc, MPa Tg, °C 0 33   1 –2 884 886 2 –5 249 283 3 – 41 123 149 6 – 67 40 43 9 – 80 22 24 19 – 99 8 7 44 – 112 3 3

Of special importance for these materials is the process of microphase separation associated with the substantially different surface energy of organosilicon chains (21 dyn/cm) and isocyanurate cross-linked points (35 dyn/cm). Microphase separation has been confirmed by X-ray photoelectron spectroscopy, electron microscopy and dynamic mechanical analysis.

470

Figure 79. Logarithmic dependence of the elasticity modulus E on a number of dimethylsiloxane units n in linear fragment.

Generally speaking, the microphase separation may proceed both in linear block-copolymers or mixtures of polymers and in networks. In these cases, two clearly expressed maxima are observed on the temperature dependence of mechanical loss factor tgδ. Figure 80 represents the temperature dependence of mechanical loss factor tgδ for bulky samples of considered polyisocyanurates possessing different lengths of linear organosilicon fragments. The low-temperature peak is shifted to the side of low temperatures with increasing length of the linear fragment, approaching Tg of polydimethylsiloxane. The high-temperature maximum associated with devitrification of the whole system does not practically depend on the value n starting from n = 2.

Figure 80. Temperature dependence of the mechanical loss-factor tgδ for bulky specimens of polyisocyanurate networks with different length of linear organosilicon fragments n: 1 – 1.2; 2 – 2.1; 3 – 3.2; 4 – 5.4; 5 – 9.2; 6 – 22 .

Hence, two transition temperatures, one of which may be related to the devitrification temperature of the polydimethylsiloxane microphase in the polymeric network and the second – to the devitrification temperature of a microphase

471

containing isocyanurate cross-linked points with attached branchings, are clearly observed. Thereby, the existence of microphase separation in the system associated with thermodynamic incompatibility of polydimethylsiloxane fragments with isocyanurate cross-linked points and branchings attached to them should be admitted. Taking into account that the transition temperatures of both microphases are shifted towards each other, the composition of microphases may be calculated basing on the condition that the transition temperature must coincide with the glass transition temperature of microphases of the present composition. Because the transition temperature in the microphase generally composed of organosilicon fragments is higher than the glass transition temperature of polydimethylsiloxane, it is obvious that neighboring fragments are contained in the polydimethylsiloxane microphase. To answer the question about the structure of these fragments, the glass transition temperatures of the structures shown below have been calculated in ref. [45]: CH3 NH C O CH2 O

CH2

O CH2

Si CH3

CH3 O

Si n CH3

CH2

O CH2

CH2

O C NH O

The glass transition temperature was calculated from formula (IV.41), the step-by-step calculation of Tg for polydimethylsiloxane fragment with different n and attached parts of chains marked by dotted lines being performed. These parts were ‘lengthened’ until the calculated glass transition temperature coincided with the experimental temperature of the first transition. Consequently, the coincidence of calculated and experimental values of the transition temperature was observed in the case when polydimethylsiloxane domains contained parts of chains marked by double dotted lines. Therewith, the structure of these parts is independent of the value n, i.e. of the length of the organosilicon chain. This method of estimating the composition of microdomains may be extended to any object characterized by microphase separation. If a microphase separation proceeds in the system, then in calculation of the equilibrium rubbery modulus the notion of the ‘network cross-linked point’ from the viewpoint of its chemical structure must be broadened. Actually, the presence of a rigid microphase (the above-considered one, for example, which includes isocyanurate cross-linked points with attached branchings) enables the microphase to play the role of a cross-linked point. Therewith, all one needs to do is to be convinced that the glass transition temperature of this microphase is above room temperature. The presence of this ‘macrocross-linked point’ causes a significant increase of β value in equation (X.27) that promotes an increase of the equilibrium rubbery modulus. Ref. [45] displays this analysis performed for the system discussed above. Consequently, it was obtained that the glass transition temperature of rigid domains was 33°C, and their Van-der-Waals volume equaled 751 Å3. Taking into account that the Van-der-Waals volume of the repeat unit of polydimethylsiloxane equals 180.15 Å3, the value β = 751/180.15 ≈ 4.2. Experimentally, the equilibrium rubbery moduli were determined [46] with the help of measurements of stress relaxation curves approximated with the help of a physically proved relaxation memory function (see below). The mechanical behavior of polyisocyanurate networks containing organosilicon linear fragments was demonstrated above (see Figure 71). Figure 79 indicates the dependence of the elasticity modulus of polyisocyanurate networks on the number of repeat units of polydimethylsiloxane chains linking cross-linked points.

472

The elasticity modulus of such networks overlaps the range of moduli typical of the transition zone from the glassy state to the rubbery state. However, in spite of this, the mechanical behavior of present materials is not viscoelastic as for all polymers in the transition zone, but elastic typical of polymeric glasses. This is proved by the form of stress relaxation curves displayed in Figure 71. Let us call attention once again to the mechanical behavior of polyisocyanurate network with linear organosilicon fragments with n = 6.2. At these sizes of linear chains, the initial stress σ0 is approximately coincident with σ0 for a viscoelastic material (7.5 MPa), but mechanical behavior is significantly different from the former one: stress relaxes fast by a low value at the initial moment of time, and then stress decrease stops, i.e. the material behaves itself as an elastic glassy polymer1. Let us now turn to analysis of the properties of gradient-modulus materials produced on the basis of oligomeric rubber PF-OP-15 and 2,4-toluylene diisocyanate. For producing gradient-modulus materials, initial components are dosed smoothly to ensure directed control of the chemical composition of the network in the same sample.

Figure 81. Dependence of the elasticity modulus lgE or E on concentration of 2,4-toluylene diisocyanate G in its mixture with oligomeric macrodiisocyanate; the concentration of G varies along the specimen length l.

In this manner, the samples were produced in refs. [19, 129] in which the elasticity modulus changed from 4.5 MPa (typical of rubbers) to 2000 MPa (typical of 1 It should be taken into account that there are no absolute elastic polymeric materials (excluding ideal crystals). That is why the terms ‘elastic behavior’ and ‘elastic material’ in relation to polymers are conditional: by elastic behavior we mean extremely slow stress relaxation.

473

plastics). Therewith, this change proceeded smoothly in the same material with no interfaces and intermediate layers. Figure 81 displays an example of the dependence of the elasticity modulus on the concentration of 2,4-toluylene diisocyanate in the initial mixture. It is clearly observed that the elasticity modulus changes linearly along the sample and, hence, as mentioned above, a smooth transition from rubber to plastic is performed with no interfaces. To analyze the mechanical behavior of materials obtained, stress relaxation curves were measured for microspecimens cut off from the initial macrospecimen in different points of the gradient (Figure 82). This Figure indicates for comparison the stress relaxation curve for a viscoelastic material – epoxy resin ED-20 cured by polysebacic acid anhydride in the presence of azelaic acid, for which Tg is located near room temperature, i.e. for the same material, the relaxation behavior of which is shown in Figure 71. Stress relaxation curves were plotted in ‘relative stress’ – time coordinates. Relative stress was calculated as σ/σ0, where σ is the current relaxing stress, σ0 is the initial stress developed at the moment of the end of ‘immediate’ setting of deformation.

Figure 82. Curves of relative stress relaxation σ/σ0 for polyisocyanurate networks prepared from oligomeric macrodiisocyanate and 2,4-toluylene diisocyanate when the concentration of the latter in percent by weight is: 1 – 6, 2 – 46, 3 – 27, respectively; 4 – epoxy oligomer ED-20 cured with methyltetrahydrophthalic anhydride; 5 – epoxy viscoelastic polymer (the composition is indicated in the text).

From this Figure, it is obvious that as for usual viscoelastic material the relative stress relaxes fast down to zero, for polymer networks obtained in refs. [19, 129] a slower decrease of stress is observed, typical of polymeric glasses or rubbers with further transition to extremely low stress relaxation. Hence, in spite of the fact that a definite part of the material in the same specimen possesses values of the modulus typical of the transition zone, mechanical behavior is elastic as for glasses of rubbers but not viscoelastic typical of all polymers in the transition zone. Dynamic mechanical analysis indicates that tgδ for the networks obtained is extremely low which is typical of elastic materials, despite the values of the storage modulus E′ typical of the transition zone. Basing on polyisocyanurate networks, films with a gradient of the elasticity modulus perpendicular to the film surface (in direction of thin) were obtained. These

474

films formed from high-crosslinked polymer networks, possess good mechanical properties (strength of 50 MPa and ultimate elongation of 90%). Hence, obtaining of gradient-modulus materials allows us to avoid the main drawback of high-crosslinked polymer networks – their brittleness.

Chapter XI. Description of relaxation processes in polymers

XI.1 Stress relaxation Various variants of the memory function in the appropriate Boltzmann– Volterra equations have been suggested to date to describe processes of stress relaxation and creep. Combined description of these memory functions and their resolvent is present in monograph [112]. The memory function contains three or four parameters, moreover they usually possess a fractional exponent of time, because only in this case experimental data may be described with a good approximation by stress relaxation and creep. Analysis of proposed memory functions has indicated that at proper selection of parameters they give the course of relaxation processes with acceptable accuracy. However, the physical meaning of these parameters is not always clear, although attempts to disclose the physical meaning of some memory functions have been made [74, 104]. Reference [7] suggested an approach to production of relaxation memory functions based on consideration of thermodynamic functions and their changes during relaxation. Suppose that the stress relaxation proceeds as a result of interaction and diffusion of kinetic units – relaxants. Relaxants may be various groups of atoms, repeat units, larger fragments and the whole fragments of macromolecules. Among relaxants are also separate elements of the free (empty) volume, i.e. microcavities, stress concentrators, etc. Interacting with each other, these microcavities may unite, be rearranged and diffuse in a polymeric material during relaxation, forming a structure which promotes a decrease of relaxing stress. The polymeric material may then be considered as the one consisting of relaxants and non-relaxants, the most part of the sample material after ‘instantaneous’ setting of deformation consisting of relaxants interacting with each other forming a non-relaxing material. Appearance of particles of two types (relaxants and non-relaxants) and their diffusion cause production of entropy in the system, which increases in the course of stress relaxation. The production of entropy (or the rate of appearance of entropy) is determined dS 1 by the expression , where S is entropy, t is time, V is the volume of the system. dt V Relaxation memory functions were obtained in ref. [7], assuming that the driving force of the process is production of the system (sample) entropy, which increases up to the maximum value during stress relaxation. It is common knowledge that if the system consists of two types of particles, the entropy of mixing S in this system determined with the help of the Boltzmann equation is calculated from the expression m*! , S = k B ln m1*! m2*!

(XI.1)

where m* is the total number of particles (in this case, the number of relaxants and non-relaxants per specific volume); m1* and m*2 are numbers of relaxants and non-

476

relaxants in the specific volume, respectively; kB is the Boltzmann constant. Taking into account that two types of particles are present, it may be written down that S = kB ln

m* !

(α m ) !(1 − α ) m  ! *

*

,

(XI.2)

where α is the part of relaxants in the total number of particles. Using an approximate formula for calculating the factorial at large m*, basing on equation (XI.2), we obtain:

S = kB ln

( )

2π m* m*

(

2πα m* α m*

)

α m*

m*

*

e− m

2π (1 − α )m* (1 − α ) m* 

(1−α ) m* − (1−α ) m*

.,

(XI.3)

e

After some transformations and neglecting low components of the expression, we obtain the following form: S = –kBm*[αlnα + (1 – α)ln(1 – α)].

(XI.4)

The value α of changes with time t from 1 to 0.5, because at α = 0.5 the entropy of mixing reaches its maximum. Assume that the memory function in the Boltzmann–Volterra equation is associated with the reverse dependence of the type 1 1 T (τ ) = S 0  −  S S max

∞

 *  ∫ T (τ ) dτ , 0

(XI.5)

where T*(τ) is the variable part of the memory function. Then, substituting expression (XI.4) into equation (XI.5), we obtain T (τ ) =

S0  1 1   , − k B m  α lnα + (1 − α ) ln(1 − α ) ln 0.5 

(XI.6)

∞

where m = m* ∫ T * (τ )dτ . 0

Now it is necessary to indicate the dependence of α on τ (remember that α is the part of relaxants in the total number of particles in the system). As mentioned above, the change of α with time τ may be stipulated by two reasons: interaction of relaxants and their transition into non-relaxants, and diffusion of kinetic units (note that the mechanism associated with diffusion of microdefects is described in refs. [104, 119]). Let us discuss these reasons. Since the interaction of relaxants is complicated, it is reasonable to describe it by the reaction equation of the n-th order. If in a usual chemical reaction, for example, the third order is rarely observed (because this requires active collision of three particles simultaneously), in this case, relaxants are ‘condensed’ in the sample, and the elementary act of their interaction may include several relaxants together (for

477

example, fusion of several microcavities into a single one). Therewith, the reaction order may be fractional. For this case, the following kinetic equation is true: dc = kc n , dτ

(XI.7)

where k is the reaction constant; c is the concentration. Integrating (XI.7) from τ = 0 to t, we obtain that c=

c0

(

)

1

1 + c0n −1 (n − 1) kτ n−1

,

where c0 is the initial concentration of relaxants of any type (assume for ease that these concentrations are equal for different types of relaxants). Then

α=

c = c0

1

(1 + k *τ / β )β

,

(XI.8)

1 where k * = kc0n −1 ; β = ; n is the reaction order. n −1 To obtain the relaxation memory function, it is necessary to substitute expression (XI.8) into equation (XI.6). Therewith, it should be taken into account that at the end of deformation the fraction of relaxants is not 1, but somewhat smaller, equal to 1 – α0. Taking into account all the above-mentioned, we obtain    S0  1 → T1 (τ ) = −  k B m1      1 1     − α 0  ln −α0  +  β β * *      1+ k τ / β   1+ k τ / β  

(

)

(

)

   1 1  → − .     ln 0.5  1 1     + α 0  ln1 − +α0   1 − β β * *      1+ k τ / β   1+ k τ / β  

(

)

(

(XI.9)

)

The function T1(τ) is of the physical sense only under the condition that 1 ≥ 0.5 . Hence, the memory function T1(τ) contains four parameters: β * 1+ k τ / β S 1 and α0. A = 0 ; k * = kc0n −1 ; β = k B m1 n −1

(

)

478

According to requirements on the relaxation memory function, the value α0 must be extremely low; in ref. [7] α0 was assumed to be equal to 10–10 basing on approximation of stress relaxation curves for a series of polymers. At low values of k*τ/β, i.e. in initial stages of relaxation, the function T1(τ) is

(

substantially simplified. Actually, at low k*τ/β, the value 1 + k *τ / β Then

)β

  S0  1 → T1 (τ ) = −  k B m1   1   1   − α 0  ln − α0  +  *   1 + k *τ  + k τ 1    

(

)

(

)

(XI.10)

  1 1  − → .     ln 0.5  1 1 1 − + α 0  ln1 − +α0    1 + k *τ    1 + k *τ     

(

)

(

= 1 + k*τ.

)

After further transformations, we obtain T1 (τ ) = −

  S0  1 1  −  * . kB m1  k τ + α 0 ln k *τ + α 0 − 1 ln 0.5     

(

) (

(XI.11)

)

Neglecting the second summand in the braces due to its smallness, it is obtained that T1 (τ ) = −

S0 1 . * kB m1 k τ + α 0 ln k *τ + α 0 − 1  

(

) (

(XI.12)

)

Let us integrate expression (XI.12) after some preliminary transformations: t

∫ T1 (τ )dτ = −

0

 k *τ + α  0 .  d *∫ * *   e k B m1k 0 k τ + α 0 k τ + α 0   ln e e S0

t

1

(XI.13)

Then, we obtain t

∫ T1 (τ )dτ = −

0

S0 k B m1k

*

ln

(

ln k *τ + α 0 e

)t = − 0

S0 k B m1k

*

ln

(

)

ln k *t + α 0 − 1 . ln α 0 − 1

(XI.14)

479

t

Let us substitute the value of ∫ T1 (τ )dτ into the Boltzmann equation: 0

 t  σ = σ 0 1 − ∫ T1 (τ )dτ  ,  0 

(XI.15)

where σ0 is the initial stress appeared as a result of ‘instantaneous’ deformation. Consequently, we obtain the relation for description of stress relaxation in the initial part (at low values of t):

σ =σ0 +

σ 0 S0 k B m1k

*

ln

(

)

ln k *t + α 0 − 1 . ln α 0 − 1

If expression (XI.16) is true, the dependence of σ on ln

σ 0 S0

(XI.16)

(

)

ln k *t + α 0 − 1 must lnα 0 − 1

, and the section cut-off on k B m1k * the axis of ordinates – σ0. Figure 83 displays such dependence for poly(methyl methacrylate). It is clear that experimental points fit the straight line with high accuracy. represent a straight line, tangent of which equals

(

)

ln k *t + α 0 − 1 (for explanation see text). Stress relaxation for ln α 0 − 1 PMMA-material under conditions of uniaxial compression when the temperature is 295 K and the deformation is 2.2%.

Figure 83. Dependence of σ on ln

Let us now consider the diffusion mechanism of relaxation. When the continuous-time random walks of kinetic units take place, the amount of sites occupied by them at time τ and, consequently, the part of non-relaxants 1 – α is determined from the correlation [146]: (1 – α) = aτb/2,

(XI.17)

480

where 0 < b < 1; a is a constant. In the case, if b = 1, correlation (XI.17) corresponds to the Fick diffusion: 1/ 2

4  Dτ  1−α =   l π 

, 1/ 2

4 D   l π  Substituting correlation (XI.17) into equation (XI.6), we obtain

where l is the sample size; D is the diffusion coefficient; i.e. a =

T2 (τ ) = −

S0 k B m2

 1 1  −  γ , γ γ γ  aτ ln aτ + (1 − aτ ) ln(1 − aτ ) ln 0.5 

.

(XI.18)

where γ = b/2. The function T2(τ) is of the physical meaning only under the condition, if aτγ ≤ 0.5. S The memory function (XI.18) contains three parameters: A = − 0 , a and k B m2 γ. It represents a function with a non-essential singularity at τ = 0. At low values of aτγ the function T2(τ) may be simplified. In this case, correlation (XI.18) will be reduced to the form: T2 (τ ) = −

S0  1 1  −  γ . k B m2  aτ (ln aτ γ − 1) ln 0.5 

(XI.19)

Neglecting the second summand in the square brackets, we obtain     S0  1 . T2 (τ ) = − (XI.20) k B m2  γ ln aτ γ   aτ  e   After a series of preliminary transformations, let us integrate correlation (XI.20): t

∫ T2 (τ )dτ = −

0

S0 k B m2 aγ

 aτ γ d ∫ aτ γ ln aτ γ  e  0 e e t

τ 1−γ

Let us make variable substitution: 1/ γ

aτ γ ex  ex  = x ; τγ = ; τ =   e a a Then

.

    .

481

t

1 / γ −1

 xe    a x ln x

x2 

S0

∫ T2 (τ )dτ = − k Bm2 aγ ∫

0

x1

dx.

(XI.21)

dx.

(XI.22)

In what follows we obtain that t

∫ T2 (τ )dτ = −

0

S 0 e1 / γ −1 k B m2 a

Taking into account the x1 = 0 and x2 = t

∫ T2 (τ )dτ = −

0

1/ γ

x2 1 / γ − 2

∫

γ x 1

x

ln x

at γ , we get e

 a 1 / γ −1 li  t1−γ 1/ γ e   k B m2 a γ  S 0 e1 / γ −1

 , 

(XI.23)

where li is the integral logarithm. t

Substituting the value of

∫ T2 (τ )dτ

into the Boltzmann equation (XI.15), we

0

deduce the correlation describing stress relaxation in the initial section of the relaxation curve under the condition that atγ << 1: 1 / γ −1  σ S e1 / γ −1  a  li   t1−γ , σ =σ0 + 0 0  k B m2 a1 / γ γ  e 

(XI.24)

 a 1 / γ −1  t1−γ  , we obtain the If correlation (XI.24) is fulfilled in σ– li   e   straight line, the tangent of which equals

σ 0 S 0 e1 / γ −1

, and the section cut-off from k B m2 a1 / γ γ the axis of ordinates equal to σ0. If the limiting stage of relaxation is interaction of relaxants and transition of them into a non-relaxing material, memory function (XI.9) or reduced expression (XI.16) on the initial part must act. If the limiting stage of the process is diffusion of non-relaxants formed, memory function (XI.18) or reduced expression (XI.24) in the initial part must act. t

As an example, Figure 84 shows dependences of σ on

∫ T1 (τ )dτ

and on

0 t

∫ T2 (τ )dτ . Clearly, in this case, the memory function T1(τ)represents better the stress 0

relaxation for poly(methyl methacrylate).

482

t

t

0

0

Figure 84. Dependences of σ on ∫ T1* (τ )dτ (1) and ∫ T2* (τ )dτ (2). (Experiment conditions are similar to those shown in Figure 83).

Let us now turn back to the relaxation memory function, based on the analysis of kinetics of relaxants interaction and their transition into a non-relaxing material. Memory function (XI.9) has been deduced from the condition that the interaction of relaxants represents a unilateral irreversible reaction of the n-th order, i.e. it is assumed that the relaxants interacted transit into a non-relaxing material in a way that makes the process irreversible. Ref. [44] discusses the interaction of relaxants by a reversible reaction of the n-th order, i.e. it is assumed that a non-relaxing material may produce relaxants during the process. The kinetic equation of this reaction is of the following form: −

dα = kα n − k (1 − α ) n . dt

(XI.25)

Equation (XI.25) is deduced under the condition that rate constants of direct and reverse reactions are identical and equal to k. This leads to the situation when the system transits to the equilibrium state, parts of relaxants and non-relaxants become the same and equal to 0.5. Equation (XI.25) is integrated up to the end only in some particular cases, for example, at n = 2. In the general case, when n is a fractional value, only numerical integration may be performed. To find the dependence of the transformation degree α on time t, the Runge–Kutta numerical method with automatic selection of the integration step was used in ref. [44]. The resultant values of values α, calculated at various small steps with respect to t, were used for computer calculations t

of the values of the integral of the variable part of the memory function ∫ T3* (τ )dτ , 0

where T3* (τ ) = −

1 1 + . (α − α 0 ) ln(α − α 0 ) + (1 − α + α 0 ) ln(1 − α + α 0 ) ln 0.5

(XI.26)

483

Hence, we get three relaxation memory functions T1(τ), T2(τ) and T3(τ), therewith S T1 (τ ) = − 0 ⋅ T1* (τ ) ; (XI.27) k B m1 S T2 (τ ) = − 0 ⋅ T2* (τ ) ; (XI.28) k B m2 S T3 (τ ) = − 0 ⋅ T3* (τ ) , (XI.29) k B m3 where T1* (τ ) , T2* (τ ) and T3* (τ ) are variable parts of memory function T1(τ),T2(τ) and T3(τ), respectively. These variable parts of the relaxation memory function are described by correlations: T1* (τ ) = −

where α =

1 1 − , (α − α 0 ) ln(α − α 0 ) + (1 − α + α 0 ) ln(1 − α + α 0 ) ln 0.5 1

(1 + k τ / β ) β *

T2* (τ ) = −

(XI.30)

;

1 γ

aτ ln aτ

γ

γ

γ

+ (1 − aτ ) ln(1 − aτ )

−

1 . ln 0.5

(XI.31)

The variable part of the memory function T3(τ) is described by correlation (XI.26). Substituting memory function T1(τ), T2(τ) and T3(τ) into the Boltzmann equation, we obtain: t

σ S σ = σ 0 − 0 0 ∫ T1* (τ ) dτ ; k B m1

(XI.32)

σ S σ = σ 0 − 0 0 ∫ T2* (τ )dτ ; k B m2

(XI.33)

σ S σ = σ 0 − 0 0 ∫ T3* (τ )dτ . k B m3

(XI.34)

0 t

0 t

0

In equations (XI.32)–(XI.34) ∞

m1 = m1* ∫ T1* (τ )dτ ;

(XI.35)

m2 = m2* ∫ T2* (τ ) dτ ;

(XI.36)

0 ∞ 0

484 ∞

m3 = m3* ∫ T3* (τ )dτ ,

(XI.37)

0

where m1* and m3* are the quantities of inhomogeneities (relaxants) in the polymeric material which interact with each other in the course of relaxation; m*2 is the number of diffusing inhomogeneities in the material during stress relaxation. If equations (XI.32)–(XI.34) describe correctly the shape of experimental t

t

0

0

stress relaxation curves, then in coordinates σ– ∫ T1* (τ )dτ , σ– ∫ T2* (τ ) dτ and σ– t

∫ T3 (τ )dτ *

a straight line must be formed, tangents of which are equal to σ0S0/kBmi,

0

and the section cut-off from the axis of ordinates is σ0. To use equations (XI.32), (XI.33) and (XI.34), it is necessary to know values t

t

t

0

0

0

of integrals ∫ T1* (τ )dτ , ∫ T2* (τ ) dτ and ∫ T3* (τ ) dτ . These values depend on two pairs of parameters: k* and β, a and γ, respectively. These integrals have been determined by computerized numerical methods [13, 44] at varied parameters k* and β, a and γ. Tables 38–40 show numerical values of variable parts T1* (τ ) , T2* (τ ) and T3* (τ ) of relaxation memory function T1(τ), T2(τ) and T3(τ), taken from refs. [13, 44]. Basing on these values, the stress relaxation curves may be approximated in a wide range of process time t. According to equations (XI.32), (XI.33) and (XI.34), to process experimental data by the root squares fitting, it is advisable to perform the following procedure. Initially, all values of integrals

t

t

t

0

0

0

* * * ∫ T1 (τ )dτ , ∫ T2 (τ )dτ and ∫ T3 (τ )dτ in the form of

three dimensions are stored in the computer memory. Each dimension contains values t

t

t

0

0

0

of integrals ∫ T1* (τ )dτ , ∫ T2* (τ ) dτ and ∫ T3* (τ ) dτ at different chosen times t for each pair of system parameters k* and β, a and γ (dimension 1 corresponds to Table 38, dimension 2 – to Table 39, dimension 3 – to Table 40). Every experimental dependence σ(t) is approximated by equation (XI.32), (XI.33) or (XI.34), and values of pairs of parameters k* and β, a and γ are automatically selected, for which the sum of deviation squares of the experimental values from the experimental ones is minimal, and the correlation coefficient is maximal. Experimental checking of memory function (XI.9), (XI.18) and (XI.26) and of the whole procedure was performed in a series of works [11, 12, 14, 38] on the example of poly(methyl methacrylate), polyoxadiazole, polyimide, polybenzoxazole, and other polymers.

485

Table 38 t

Values of ∫ T1* (τ ) dτ 0

t 1

β = 0.2 2

β = 0.3 3

β = 0.6 6

β = 0.7 7

β = 0.8 8

17.46 18.80 20.05 20.69 21.06 21.30 21.44 21.62 21.66 21.66 21.66 21.66 21.66 21.66 21.66 21.66 21.66 21.66 21.66 21.66 21.66 21.66 21.66 21.66

17.45 18.78 20.01 20.63 20.98 21.19 21.31 21.45 21.47 21.47 21.47 21.47 21.47 21.47 21.47 21.47 21.47 21.47 21.47 21.47 21.47 21.47 21.47 21.47

17.45 18.76 19.98 20.57 20.91 21.10 21.22 21.33 21.34 21.34 21.34 21.34 21.34 21.34 21.34 21.34 21.34 21.34 21.34 21.34 21.34 21.34 21.34 21.34

133.4 144.5 156.6 164.2 169.7 174.1 177.8 186.0 196.5 210.3 217.5 222.0 227.2 229.7 231.6 231.9 232.0 232.0 232.0 232.0 232.0 232.0 232.0

β = 0.4 β = 0.5 4 5 k∗ = 0.1 17.49 17.47 18.86 18.82 20.20 20.11 20.92 20.78 21.37 21.19 21.67 21.45 21.88 21.62 22.22 21.86 22.39 21.94 22.39 21.94 22.39 21.94 22.39 21.94 22.39 21.94 22.39 21.94 22.39 21.94 22.39 21.94 22.39 21.94 22.39 21.94 22.39 21.94 22.39 21.94 22.39 21.94 22.39 21.94 22.39 21.94 22.39 21.94 k∗ = 0.01 133.4 133.4 144.5 144.4 156.5 156.4 164.0 163.9 169.5 169.3 173.8 173.6 177.4 177.1 185.4 185.0 195.4 194.7 208.1 206.7 214.2 212.1 217.6 215.0 221.2 217.6 222.4 218.2 222.8 218.2 222.8 218.2 222.8 218.2 222.8 218.2 222.8 218.2 222.8 218.2 222.8 218.2 222.8 218.2 222.8 218.2

0.5 1 2 3 4 5 6 9 15 30 45 60 90 120 180 240 300 360 720 1,440 2,880 5,760 10,080 100,000

17.57 19.06 20.62 21.56 22.22 22.73 23.13 23.98 24.88 25.70 25.87 25.88 25.90 25.90 25.90 25.90 25.90 25.90 25.90 25.90 25.90 25.90 25.90 25.90

17.52 18.93 20.34 21.14 21.66 22.03 22.31 22.82 23.21 23.31 23.31 23.31 23.31 23.31 23.31 23.31 23.31 23.31 23.31 23.31 23.31 32.31 23.31 23.31

0.5 1 2 3 4 5 6 9 15 30 45 60 90 120 180 240 300 360 720 1,440 2,880 5,760 10,080

133.5 144.6 156.8 164.5 170.2 174.8 178.6 187.3 198.6 214.7 223.1 230.3 238.9 244.3 250.7 254.0 255.9 257.1 258.0 258.1 258.1 258.1 258.1

133.4 144.4 156.4 163.8 169.3 173.5 177.0 184.8 194.3 205.8 210.7 213.2 215.2 215.4 215.4 215.4 215.4 215.4 215.4 215.4 215.4 215.4 215.4

133.4 144.4 156.3 163.8 169.2 173.4 176.9 184.6 193.9 205.1 209.7 211.9 213.5 213.5 213.5 213.5 213.5 213.5 213.5 213.5 213.5 213.5 213.5

133.4 144.4 156.3 163.7 169.1 173.3 176.8 184.4 193.7 204.6 208.9 210.9 212.2 212.2 212.2 212.2 212.2 212.2 212.2 212.2 212.2 212.2 212.2

100,000

258.1

232.0

222.8

215.4

213.5

212.2

218.2

486

1

2

3

0.5 1 2 3 4 5 6 9 15 30 45 60 90 120 180 240 300 360 720 1,440 2,880 5,760 10,080 100,000

1,027 1,110 1,201 1,258 1,300 1,334 1,362 1,427 1,514 1,644 1,724 1,784 1,872 1,936 2,027 2,093 2,145 2,188 2,341 2,472 2,556 2,578 2,579 2,579

1,027 1,110 1,200 1,257 1,299 1,333 1,361 1,426 1,513 1,640 1,719 1,776 1,860 1,919 2,001 2,059 2,101 2,136 2,244 2,308 2,319 2,319 2,319 2,319

0.5 1 2 3 4 5 6 9 15 30 45 60 90 120 180 240 300 360 720 1,440 2,880 5,760 10,080 100,000

7,900 8,556 9,256 9,692 10,010 10,270 10480 10970 11,620 12,570 13,170 13,610 14,270 14,760 15,470 16,000 16,430 16,800 18,230 19,760 21,360 22,930 24,090 24,090

7,900 8,556 9,256 9,692 10,010 10,270 10,480 10,970 11,620 12,570 13,160 13,600 14,260 14,750 15,450 15,980 16,390 16,760 18,130 19,560 20,940 22,130 22,820 22,820

0.5 1 2 3 4

59,840 65,230 70,910 74,420 76,970

59,840 65,230 70,910 74,420 76,970

4

5 k∗ = 0.001 1,027 1,027 1,110 1,110 1,200 1,200 1,257 1,257 1,299 1,299 1,333 1,333 1,361 1,361 1,426 1,425 1,512 1,511 1,639 1,637 1,716 1,714 1,772 1,769 1,853 1,849 1,910 1,905 1,988 1,980 2,041 2,029 2,079 2,065 2,109 2,092 2,194 2,163 2,227 2,182 2,227 2,182 2,227 2,182 2,227 2,182 2,227 2,182 k∗ = 0.0001 7,900 7,900 8,556 8,556 9,256 9,256 9,692 9,692 10,010 10,010 10,270 10,270 10,480 10,480 10,970 10,970 11,620 11,620 12,570 12,570 13,160 13,160 13,600 13,600 14,260 14,260 14,740 14,740 15,440 15,440 15,900 15,950 16,380 16,360 16,730 16,720 18,080 18,050 19,460 19,390 20,720 20,590 21,170 21,450 22,170 21,780 22,170 21,780 k∗ = 0.00001 59,840 59,840 65,230 65,230 70,910 70,910 74,420 74,420 76,970 76,970

6

7

8

1,027 1,110 1,200 1,257 1,299 1,333 1,361 1,425 1,511 1,637 1,713 1,768 1,847 1,901 1,974 2,022 2,055 2,081 2,142 2,154 2,154 2,154 2,154 2,154

1,027 1,110 1,200 1,257 1,299 1,332 1,360 1,425 1,511 1,636 1,712 1,767 1,845 1,899 1,970 2,017 2,049 2,073 2,127 2,135 2,135 2,135 2,135 2,135

1,027 1,110 1,200 1,257 1,299 1,332 1,360 1,425 1,510 1,636 1,711 1,766 1,844 1,897 1,967 2,013 2,043 2,067 2,116 2,121 2,121 2,121 2,121 2,121

7,900 8,556 9,256 9,692 10,010 10,270 10,480 10,970 11,620 12,570 13,160 13,600 14,260 14,740 15,440 15,950 16,360 16,710 18,030 19,350 20,500 21,280 21,520 21,520

7,900 8,556 9,256 9,692 10,010 10,270 10,480 10,970 11,620 12,570 13,160 13,600 14,260 14,740 15,440 15,950 16,350 16,700 18,020 19,320 20,440 21,150 21,340 21,340

7,900 8,556 9,256 9,692 10,010 10,270 10,480 10,970 11,620 12,570 13,160 13,600 14,260 14,740 15,440 15,950 16,350 16,700 18,010 19,300 20,390 21,060 21,210 21,210

59,840 65,230 70,910 74,420 76,970

59,840 65,230 70,910 74,420 76,970

59,840 65,230 70,910 74,420 76,970

487

1 5 6 9 15 30 45 60 90 120 180 240 300 360 720 1,400 2,880 5,760 10,080 100,000

2 78,990 80,670 84,500 89,610 96,900 101,400 104,700 109,700 113,300 118,600 122,500 125,700 128,400 139,000 150,800 163,600 177,500 189,700 189,700

3 78,990 80,670 84,500 89,610 96,900 101,400 104,700 109,700 113,300 118,600 122,500 125,600 128,400 139,000 150,600 163,200 176,800 188,300 188,300

4 78,990 80,670 84,500 89,610 96,900 101,400 104,700 109,700 113,300 118,600 122,500 125,600 128,300 138,900 150,500 163,100 176,400 187,500 187,500

5 78,990 80,670 84,500 89,610 96,900 101,400 104,700 109,700 113,300 118,600 122,500 125,600 128,300 138,900 150,500 162,900 176,200 187,100 187,100

6 78,990 80,670 84,500 89,610 96,900 101,400 104,700 109,700 113,300 118,600 122,500 125,600 128,300 138,900 150,500 162,900 176,000 186,800 186,800

7 78,990 80,670 84,500 89,610 96,900 101,400 104,700 109,700 113,300 118,600 122,500 125,600 128,300 138,900 150,400 162,800 175,900 186,600 186,600

8 78,990 80,670 84,500 89,610 96,900 101,400 104,700 109,700 113,300 118,600 122,500 125,600 128,300 138,900 150,400 162,800 175,800 186,500 186,500 Table 39

t

Values of ∫ T2* (τ ) dτ 0

t 1

a = 0.315 2

a = 0.284 3

0.5 1 2 3 4 5 6 9 15 30 45 60 90 120 180 240 300 360 720 1,440 2,880 5,760 10,080 100,000 t

0.144 0.236 0.375 0.494 0.576 0.656 0.727 0.900 1.142 1.464 1.609 1.675 1.707

0.189 0.317 0.523 0.693 0.843 0.977 1.099 1.418 1.915 2.754 3.303 3.693 4.198 4.488 4.745 4.808

a = 0.199

a = 0.169

0.5 1 2

0.521 0.837 1.316

0.660 1.081 1.744

a = 0.252 4 γ = 0.1 0.247 0.425 0.722 0.977 1.205 1.417 1.614 2.144 3.029 4.711 5.991 7.036 8.684 9.948 11.780 13.030 13.910 14.540 15.850

a = 0.220 5

a = 0.189 6

a = 0.158 7

0.323 0.567 0.984 1.353 1.691 2.007 2.307 3.132 4.567 7.492 9.902 12.000 15.590 18.640 23.670 27.780 31.250 34.240 46.290 56.880 61.700

0.421 0.750 1.326 1.844 2.327 2.784 3.220 4.443 6.628 11.285 15.300 18.930 25.410 31.180 41.320 50.190 58.160 65.440 100.100 146.400 200.800 252.000 277.400

a = 0.139 γ = 0.2 0.860 1.431 2.353

a = 0.1096

a = 0.0798

0.556 1.003 1.799 2.527 3.213 3.867 4.498 6.283 9.541 16.700 23.070 28.960 39.780 49.680 67.710 84.090 99.290 113.600 187.100 301.100 470.700 707.900 948.100 1694.000 a = 0.050

1.127 1.913 3.221

1.601 2.757 4.721

2.577 4.493 7.811

488

1 3 4 5 6 9 15 30 45 60 90 120 180 240 300 360 720 1,440 2,880 5,760 10,080 100,000 t

2 1.694 2.011 2.286 2.531 3.133 3.979 5.124 5.647 5.885 6.003

3 2.285 2.754 3.173 3.555 4.536 6.047 8.530 10.10 11.17 12.47 13.13 13.56

4 3.130 3.819 4.448 5.029 6.574 9.097 13.740 17.140 19.830 23.910 26.800 30.800 33.160 34.570 35.390

5 4.349 5.368 6.310 7.195 9.596 13.680 21.740 28.180 33.650 42.770 50.270 62.200 71.460 78.930 85.090 106.500 117.800

6 6.447 8.031 9.514 10.920 14.800 21.610 35.740 47.660 58.250 76.860 93.160 121.300 145.300 166.800 186.000 274.300 382.800 493.300 567.100

7 10.780 13.530 16.130 18.620 25.590 38.090 65.000 88.580 110.100 149.300 185.000 249.300 307.400 361.000 411.200 667.700 1063.000 1649.000 2471.000 3310.000

a = 0.126

a = 0.104

a = 0.0819 γ = 0.3 1.913 3.076 4.894 6.381 7.677 8.841 9.908 12.690 17.110 24.900 50.350 34.500 40.450 44.460 49.180 51.430 52.390 52.690

a = 0.0598

a = 0.378

a = 0.0157

2.621 4.265 6.888 9.080 11.020 12.790 14.430 18.800 26.030 39.720 50.220 58.900 72.860 83.880 100.500 112.600 121.700 128.700 147.500

4.064 6.680 10.940 14.560 17.810 20.810 23.620 31.240 44.240 70.200 91.380 109.800 141.400 168.300 213.600 251.300 283.800 312.400 436.300 568.700 669.000

a = 0.0494 γ = 0.4 3.858 5.928 9.030 11.490 13.590 15.450 17.140

a = 0.0346

a = 0.0198

9.064 15.020 24.870 33.380 41.120 48.330 55.140 73.880 106.600 174.900 233.100 285.400 379.000 462.800 611.800 744.400 865.600 978.300 1545.000 2401.000 3650.000 5383.000 7146.000 12981.00 A = 0.005

5.370 8.310 12.790 16.400 19.540 22.340 24.910

8.928 13.890 21.560 27.850 33.370 38.360 42.970

29.87 46.81 73.33 95.35 114.80 132.70 149.30

0.5 1 2 3 4 5 6 9 15 30 45 60 90 120 180 240 300 360 720 1,440 2,880 5,760 10,080 100,000 t

1.219 1.901 2.914 3.701 4.358 4.926 5.429 6.664 8.396 10.750 11.830 12.320 12.560

1.488 2.362 3.697 4.762 5.673 6.478 7.204 9.049 11.830 16.240 30.350 20.650 22.600 23.440 23.780

a = 0.079

a = 0.0642

0.5 1 2 3 4 5 6

2.456 3.712 5.526 6.909 8.053 9.036 9.902

3.007 4.583 6.904 8.710 10.230 11.560 12.740

489

1 9 15 30 45 60 90 120 180 240 300 360 720 1,440 2,880 5,760 10,080 100,000 t 0.5 1 2 3 4 5 6 9 15 30 45 60 90 120 180 240 300 360 720 1,440 2,880 5,760 10,080 100.000

2 12.020 14.970 18.990 20.870 21.750 22.210

3 15.720 20.110 26.900 30.870 33.380 36.050 37.060

4 21.450 28.120 39.410 46.970 52.530 60.110 64.860 69.760 71.490 71.860

5 31.620 42.360 61.830 76.130 87.580 105.300 118.700 137.600 150.300 158.900 164.900 175.000

6 55.220 75.440 114.100 114.400 170.100 212.900 248.400 305.900 352.000 390.500 423.400 553.900 664.600 706.500

7 193.90 269.50 420.50 544.70 654.10 845.60 1014.00 1306.00 1562.00 1792.00 2004.00 3047.00 4571.00 6734.00 9665.00 12595.00

a = 0.05

a = 0.0403

a = 0.0209

a = 0.0113

a = 0.00158

4.738 6.850 9.792 11.980 13.760 15.280 16.600 19.810 24.230 30.180 32.950 34.250 34.920

5.767 8.384 12.080 14.870 17.170 19.160 20.910 25.250 31.510 40.870 46.140 49.370 52.580 53.600

a = 0.0306 γ = 0.5 7.420 10.830 15.710 19.450 22.570 25.290 27.740 33.850 43.030 57.930 67.480 74.220 82.960 87.970 92.330 93.240

10.42 15.28 22.33 27.80 32.42 36.50 40.17 49.57 64.13 89.34 107.00 120.70 141.10 155.60 174.80 186.20 193.00 196.90

17.95 26.42 38.84 28.60 56.93 64.34 71.08 88.57 116.40 167.30 205.50 236.90 287.60 328.10 391.20 439.50 478.00 509.70 620.30 680.20

102.4 150.4 221.1 277.2 325.4 368.6 408.1 511.9 681.0 1003.0 1258.0 1476.0 1850.0 2169.0 2713.0 3178.0 3590.0 3965.0 5757.0 8281.0 11745.0 16307.0 20775.0 34975.0 Table 40

t

Values of

∫ T3 (τ )dτ *

0

t 1

β = 0.2 2

β = 0.3 3

0.5 1 2 3 4

17.570 19.060 20.740 21.700 22.370

17.52 18.94 20.47 21.29 21.82

β = 0.4 β = 0.5 4 5 K∗ = 0.1 17.49 17.47 18.87 18.83 20.32 20.23 21.06 20.93 21.52 21.35

β = 0.6 6

β = 0.7 7

β = 0.8 8

17.460 18.800 20.180 20.840 21.230

17.45 18.78 20.13 20.78 21.15

17.45 18.77 20.11 20.73 21.10

490

1 5 6 9 15 30 45 60 90 120 180 240 300 360 720 1,440 2,880 5,760 10,080 100,000

2 22.880 23.290 24.170 25.150 26.140 26.400 26.470 26.516 26.520 26.521 26.521 26.521 26.521 26.521 26.521 26.521 26.521 26.521 26.521

3 22.20 22.48 23.02 23.50 23.79 23.81 23.81 23.81 23.81 23.81 23.81 23.81 23.81 23.81 23.81 23.81 23.81 23.81 23.81

0.5 1 2 3 4 5 6 9 15 30 45 60 90 120 180 240 300 360 720 1,440 2,880 5,760 10,080 100,000

133.50 144.69 157.73 165.29 171.35 175.93 179.75 188.68 200.54 217.70 227.25 233.75 242.53 248.07 254.85 258.50 260.62 261.90 265.01 265.38 265.39 265.39 265.39 265.39

133.40 144.53 157.44 165.17 170.79 175.23 178.91 187.39 198.35 213.32 220.82 225.43 230.88 233.66 236.25 237.11 237.41 237.52 237.69 237.69 237.69 237.69 237.69 237.69

0.5 1 2 3 4 5 6 9 15 30 45

1,027 1111.5 1209.9 1268.4 1311.2 1345.4 1373.9 1441.2 1532.3 1669.9 1752.6

1,027 1111.6 1209.6 1268.0 1310.7 1344.7 1373.2 1440.0 1530.4 1666.1 1746.7

4 5 21.84 21.62 22.06 21.80 22.45 22.12 22.73 22.31 22.85 22.38 22.85 22.38 22.85 22.38 22.85 22.38 22.85 22.38 22.85 22.38 22.85 22.38 22.85 22.38 22.85 22.38 22.85 22.38 22.85 22.38 22.85 22.38 22.85 22.38 22.85 22.38 22.85 22.38 k∗ = 0.01 133.40 133.40 144.50 144.48 157.36 157.29 165.02 164.91 170.57 170.42 174.94 174.74 178.54 178.30 186.79 186.42 197.29 196.63 211.11 209.76 216.55 215.57 221.22 218.71 225.14 221.83 226.82 223.00 228.08 223.76 228.36 223.88 228.42 223.90 228.43 223.91 228.45 223.91 228.45 223.91 228.45 223.91 228.45 223.91 228.45 223.91 228.45 223.91 k∗ = 0.001 1,027 1,027 1111.5 1111.4 1209.4 1209.3 1267.7 1267.5 1310.3 1310.1 1344.3 1344.0 1372.7 1372.4 1439.3 1438.9 1529.3 1528.6 1664.0 1662.7 1743.5 1741.6

6 21.480 21.640 21.910 22.060 22.104 22.105 22.105 22.105 22.105 22.105 22.105 22.105 22.105 22.105 22.105 22.105 22.105 22.105 22.105

7 21.38 21.53 21.77 21.90 21.93 21.93 21.93 21.93 21.93 21.93 21.93 21.93 21.93 21.93 21.93 21.93 21.93 21.93 21.93

8 21.32 21.46 21.68 21.79 21.82 21.82 21.82 21.82 21.82 21.82 21.82 21.82 21.82 21.82 21.82 21.82 21.82 21.82 21.82

133.40 144.47 157.26 164.85 170.33 174.62 178.15 186.17 196.20 208.87 214.29 217.10 219.76 220.67 221.21 221.28 221.29 221.29 221.29 221.29 221.29 221.29 221.29 221.29

133.40 144.46 157.23 164.80 170.26 174.54 178.04 186.00 195.89 208.26 213.41 216.01 218.38 219.14 219.56 219.61 219.61 219.61 219.61 219.61 219.61 219.61 219.61 219.61

133.40 144.45 157.21 164.77 170.21 174.47 177.96 185.87 195.67 207.81 212.78 215.23 217.42 218.09 218.44 218.48 218.48 218.48 218.48 218.48 218.48 218.48 218.48 218.48

1,027 1111.3 1209.1 1267.3 1309.8 1343.8 1372.1 1438.6 1528.2 1661.8 1740.3

1,027 1111.3 1209.0 1267.2 1309.7 1343.6 1371.9 1438.3 1527.8 1661.2 1739.4

1,027 1111.2 1208.2 1267.0 1309.5 1343.4 1371.7 1438.1 1527.5 1660.7 1738.6

491

1 60 90 120 180 240 300 360 720 1,440 2,880 5,760 10,080 100,000

2 1813.0 1902.3 1966.9 2061.3 2128.4 2180.5 2222.8 2390.4 2536.4 2638.6 2685.1 2692.0 2692.5

3 1805.0 1889.8 1949.9 2035.3 2093.6 2137.0 2170.9 2293.4 2373.2 2405.2 2409.8 2409.9 2409.9

0.5 1 2 3 4 5 6 9 15 30 45 60 90 120 180 240 300 360 720 1,440 2,880 5,760 10,080 100,000

7900.0 8543.0 9257.0 9696.1 10020.0 10279.0 10496.0 11003.0 11682.0 12705.0 13318.0 13768.0 14439.0 14932.0 15668.0 16208.0 16639.0 16999.0 18531.0 20172.0 21886.0 23592.0 24827.0 26080.0

7900.0 8543.0 9257.0 9696.1 10020.0 10279.0 10496.0 11003.0 11682.0 12700.0 13310.0 13758.0 14425.0 14914.0 15642.0 16174.0 16597.0 16949.0 18429.0 19963.0 21465.0 22789.0 23555.0 24104.0

0.5 1 2 3 4 5 6 9 15 30 45 60 90 120 180 240 300

59,840 65,248 71,358 74,932 77,514 79,553 81,247 85,191 90,455 98,281 102,922 106,304 111,320 114,985 120,433 124,422 127,605

59,840 65,248 71,358 74,931 77,514 79,553 81,246 85,190 90,454 98,278 102,918 106,300 111,313 114,975 120,417 124,401 127,577

4 5 1800.6 1798.0 1883.2 1879.2 1940.9 1935.5 2021.6 2013.4 2075.3 2064.3 2114.2 2100.4 2143.7 2127.3 2244.0 2215.0 2296.8 2255.0 2310.7 2262.7 2311.4 2262.9 2311.4 2262.9 2311.4 2262.9 k∗ = 0.0001 7900.0 7900.0 8543.0 8543.0 9257.0 9257.0 9696.1 9696.1 10020.0 10020.0 10279.0 10279.0 10496.0 10496.0 11003.0 11003.0 11682.0 11682.0 12699.0 12699.0 13309.0 13308.0 13756.0 13755.0 14421.0 14419.0 14909.0 14905.0 15633.0 15627.0 16161.0 16153.0 16580.0 16569.0 16928.0 16914.0 18381.0 18351.0 19860.0 19796.0 21252.0 21122.0 22382.0 22139.0 22936.0 22583.0 23243.0 22789.0 K∗ = 0.00001 59,840 59,840 65,248 65,248 71,358 71,358 74,931 74,931 77,514 77,514 79,553 79,553 81,246 81,246 85,190 85,190 90,453 90,453 98,277 98,277 102,916 102,915 106,297 106,295 111,309 111,307 114,970 114,967 120,409 120,404 124,390 124,383 127,563 127,554

6 1796.3 1876.6 1932.0 2007.9 2056.9 2091.3 2116.5 2196.5 2229.6 2234.6 2234.7 2234.7 2234.7

7 1795.1 1874.7 1929.4 2004.0 2051.8 2084.9 2109.0 2184.0 2212.9 2216.7 2216.8 2216.8 2216.8

8 1794.1 1873.2 1927.5 2001.1 2048.0 2080.3 2103.5 2175.1 2201.4 2204.5 2204.5 2204.5 2204.5

7900.0 8543.0 9257.0 9696.1 10020.0 10279.0 10496.0 11003.0 11682.0 12699.0 13308.0 13754.0 14418.0 14903.0 15623.0 16148.0 16562.0 16906.0 18332.0 19755.0 21037.0 21983.0 22363.0 22520.0

7900.0 8543.0 9257.0 9696.1 10020.0 10279.0 10496.0 11003.0 11682.0 12699.0 13308.0 13754.0 14417.0 14902.0 15621.0 16145.0 16558.0 16900.0 18319.0 19726.0 20978.0 21877.0 22218.0 22348.0

7900.0 8543.0 9257.0 9696.1 10020.0 10279.0 10496.0 11003.0 11682.0 12699.0 13308.0 13753.0 14416.0 14901.0 15620.0 16142.0 16554.0 16896.0 18303.0 19704.0 20934.0 21801.0 22117.0 22230.0

59,840 65,248 71,358 74,931 77,514 79,553 81,246 85,189 90,453 98,276 102,915 106,294 111,305 114,965 120,401 124,379 127,549

59,840 65,248 71,358 74,931 77,514 79,553 81,246 85,189 90,452 98,276 102,914 105,294 111,304 114,963 120,398 124,376 127,545

59,840 65,248 71,358 74,931 77,514 79,553 81,2476 85,189 90,452 98,276 102,914 106,293 111303 114,962 120,397 124,373 127,542

492

1 360 720 1,440 2,880 5,760 10,080 100,000

2 130,268 141,656 154,141 167,823 182,744 195,357 211,967

3 130,233 141,581 153,980 167,475 181,990 193,966 209,097

4 130,216 141,544 153,899 167,298 181,608 193,256 207,617

5 130,206 141,522 153,850 167,192 181,377 192,826 206,718

6 130,199 141,507 153,818 167,121 181,223 195,538 206,120

7 130,194 141,496 153,795 167,071 181,113 192,335 205,700

8 130,190 141,488 153,778 167,033 181,032 192,184 205,393

Calculation results are shown in Table 41 and in Figure 85. Clearly, the memory function T1(τ) displays the course of stress relaxation better than T2(τ) does. Therewith, the correlation coefficient is close to 1. Consequently, in this case, the limiting stage of the process in the initial stage of the stress relaxation curve is the interaction of relaxants and their transition into a non-relaxing material. Table 41 also shows that the value k*, proportional to the rate constant of interaction k, is constant at different temperatures. The main contribution to the rate of relaxation is made by the value kBmi/S0, proportional to the number of kinetic units mi which cause relaxation. These values naturally decrease as temperature increases. Naturally, more accurate determination of parameters of memory functions T1(τ) and T2(τ) requires longer experiments.

t

Figure 85. Dependences of σ on

* ∫ T1 (τ )dτ (a) and 0

t

∫ T2 (τ )dτ *

(b) for bulky specimens of PMMA

0

(deformation is 2.2%) at T, K: 294 (1), 313 (2), 323 (3) and 333 (4) Table 41 Values of parameters k*, β, Ai = σ0S0/(kBmi ), kBmi/S0, a,γ, memory functions Tl(τ) and T2(τ), initial stresses σ0, and correlation coefficients r for poly(methyl methacrylate) with deformation equal to ε0 = 2.2% T2(τ); a = 0.05 min-1; γ = 0.5 T1(τ); k*=0,0001 min-1; β = 0.2 4 T, K k B m1 k B m2 A2, Al⋅10 , r r σ0, MPa σ0, MPa S0 MPa MPa S0 294 1.6 49.64 310,250 0.998 2.272 40.39 17.78 0.991 313 2.3 52.29 227,350 0.999 3.197 39.33 12.30 0.995 323 1.8 38.25 212,500 0.999 2.465 28.20 14.44 0.991 333 2.3 39.25 170,650 0.999 3.177 26.31 8.28 0.991

493

In conclusion, one practical problem should be discussed, which is associated with the reliability of determination of relaxation parameters by approximation of stress relaxation curves. t

For this purpose, let us consider dependences of k * ∫ T3* (τ ) dτ on lgt for the

0 * memory function T3 (τ ) dτ , shown in Figure 86. Clearly, for every chosen value the

rate constants of the reaction k* for different values of β in initial stages coincide first, and then diverge. The duration of the relaxation process, at which divergence of these curves depends on k*: the higher this value is, the shorter is duration of the process of t

the onset of divergence of k * ∫ T3* (τ ) dτ dependences on t. This area of t and k* 0

(cross-hatched in Figure 87) characterizes the duration of the relaxation process, which must be realized in the experiment in order to obtain reliable values of β which characterize the reaction order n, because β = 1/(n – 1). The experiment performed at shorter values of t, allows no reliable determination of the reaction order. Such experiment enables us to determine the rate constant only. For example, if the rate constant equals 0.1 min–1, the duration of a relaxation process performed in experiments must exceed 2 min; at k* = 0.01 min–1 this duration is already 60 min, etc.

t

Figure 86. Dependences of k

*

∫ T3 (τ )dτ *

on lgt when k* = 0.1 (1); 0.01 (2); 0.001 (3); 0.0001 (4),

0 and 0.00001 (5).

Hence, the minimal time, during which the stress relaxation experiment must be conducted for reliable determination of the process parameters, may be determined with the help of Figure 87. Actually, a reduction of the rate constant of the process must cause a significant increase of the experiment duration. The above-discussed procedure of approximation of stress relaxation curves σ(t) is true for the case of linear mechanical behavior of polymeric materials, when the parameters of the process are independent of its duration and deformation value. We should dwell on the possibility of description of non-linear relaxation processes, which are most typical of polymeric materials, even at low strains.

494

Figure 87. Logarithmic dependence of the minimum duration of relaxation process t on k* at which reliable determination of β parameter is possible.

At the present time, the most wide-spread method of approximation of stress relaxation curves in the non-linear area of mechanical behavior is the one based on the main cubic theory by Iliyushin [73]. According to ref. [73], the relaxation modulus Er(t) = σ(t)/ε0 is first approximated in the linear area of viscoelasticity and then, by introduction of one more parameter and application of the same relaxation memory function but with different parameters, relaxation curves in the non-linear area are approximated. Another method is based on the application of equations containing a fractional exponent of the time degree of magnitude, this index being assumed to be dependent on the value of deformation [220] which is maintained constant during the relaxation process. In both cases, although good coincidence of the experimental and calculated curves is reached, the physical meaning of the introduced new parameters is not disclosed. In this section, let us consider an approach to description of stress relaxation curves in the non-linear area with the help of physically stated parameters included in the relaxation memory function (XI.9). It should be noted that the memory function (XI.9) has been obtained on the basis of the mixing entropy of only two types of units – relaxants and non-relaxants; if the number of types of kinetic units making a significant contribution to the relaxation process exceeds two, several rate constants of interaction of relaxants of various types appear with respective several values of k*. Finally, this will lead to the appearance of a spectrum of interaction constants, which is analogous to the spectrum of relaxation times. However, numerous tests have indicated that to describe the stress relaxation curves, a single rate constant of relaxants interaction k* is enough even at a significant duration of the process t. This means that although relaxants may be of different types, only one of them makes a substantial contribution to the relaxation process. That is why in further discussion, transiting to the description of the approximation procedure of relaxation dependences in the non-linear area of the mechanical behavior of polymers, we shall use the memory function T1(τ). The meaning of such approximation is the following [15]. Let us write down an expression for the temperature dependence of the rate constant: k * = k0* exp(−∆U / RT ) ,

(XI.38)

where k0* is the pre-exponential multiplicand; ∆U is the activation energy of the interaction process; R is the universal gas constant; T is the absolute temperature. It is

495

common knowledge that the free volume of polymers increases during their deformation (in this case, the free volume means the empty volume, which represents the difference between the real volume of the polymeric substance and the Van-derWaals volume of atoms occupied by them in the polymeric substance). At significant deformation of solid (glassy and crystalline) polymers, the free volume increases up to a very high value, which simplifies significantly the jump-over of kinetic units from one position into another. This very fact leads to induced elasticity, i.e. to induced softening of the material. That is why if we assume that the activation energy of relaxants interaction decreases as mechanical stress increases, at quite high stress values this may cause appearance of an excessive free volume. Based on this phenomenon, the expression for the temperature dependence of stress relaxation time is deduced [1, 65]. Hence, it may be presented  ∆U 0 − δσ r   ∆U 0 − δEr ε 0  * k * = k0* exp − ,  = k0 exp − RT RT    

(XI.39)

where Er is the relaxation modulus; ∆U0 is the initial interaction energy of relaxants; σr is the relaxing stress; ε0 is the constant deformation; δ is the fluctuation volume in which the elementary act of relaxants interaction proceeds. Therewith, for the linear area of mechanical behavior when stress is not so high yet to form an excessive free volume, let us assume the value δ = 0, i.e. the rate

constant k * = k0* exp(− ∆U 0 RT ) and does not depend on mechanical stress. As the given deformation ε0 increases, a moment appears when a large excessive free volume forms that greatly simplifies interaction of relaxants and accelerates the relaxation process. This, from the positions considered, is the transition to the non-linear behavior. In this case, the value k* is not constant, but becomes dependent on the relaxation modulus according to expression (XI.39). Taking this into account, we can approximate the stress relaxation curves in the non-linear area and carry out simultaneous determination of excessive fluctuation volume δ, in which the elementary act of relaxants interaction proceeds. Before describing the procedure of approximation of relaxation curves with the help of the suggested approach, let us rewrite the Boltzmann equation in the following form: t

E (t ) = E0 − E0 ∫ T (τ ) dτ ,

(XI.40)

0

where E0 is the initial modulus appearing after the ‘instantaneous’ assignment of deformation; T(τ) is the relaxation memory function. Experiment indicates that the best approximation of relaxation curves for glassy polymers is reached at the application of the memory function T1(τ), which we will use in future discussion. Substituting the memory function T1(τ) into equation (XI.40), we obtain: t

E S E (t ) = E0 − 0 0 ∫ T1* (τ ) dτ , k B m1 0

(XI.41)

496

where T1* (τ ) is the variable part of the memory function T1(τ), described by equation (XI.30). For the case of stress relaxation in the non-linear area of mechanical behavior,

α=

1   ∆U 0 − δEr ε 0   k 0* exp − τ RT   1 +  β  

     

β

.

(XI.42)

It follows from comparison of equations (XI.8) and (XI.42) that the component k 0* exp(− ∆U RT ) , independent of the relaxation modulus Er, corresponds to k* in equation (XI.8) for the linear part of the mechanical behavior. Hence, it may be indicated that

α=

1   δE ε   k * exp r 0 τ  RT  1 +  β  

     

β

.

(XI.43)

The procedure of approximation is concluded in determination of the value δ, at which the value of function ϕ(δ), which is the sum of the squares of deviations, is minimal: n

(

)

ϕ (δ ) = ∑ Ei, calc − Ei, exper 2 , i =1

where n is the number of experimental points; Ei,calc and Ei,exper are the values of the relaxation modulus calculated by equation (XI.42) and determined experimentally, respectively. The calculation algorithm is the following. The values of relaxation moduli for the experimental stress relaxation curves are stored in the computer sequentially in the ascending order of deformation values ε0. Each introduced curve, except the first one, is compared with the previously introduced curve. If every value of the modulus of the newly introduced curve at one and the same relaxation time is lower than that of the averaged curve, and the mean arithmetic value of relative deviations exceeds 10%, such a curve is assumed to be related to the non-linear area of the mechanical behavior. Then, for the averaged curve, the relaxation parameters for the linear area are calculated by the above-described method and, basing on them, approximation of the case related to the non-linear area is performed. Search for the minimum of the function ϕ(δ) is conducted by the method of the reverse variable step, and integration of the relaxation memory function is performed by the Simpson method with the given accuracy (usually, 0.001%).

497

Function α participating in the integrand (XI.41) contains the relaxation modulus Er which, as the function itself (XI.43), depends on time τ. However, at numerical integration of the relaxation memory function (XI.9), the relaxation modulus is taken in the area between two experimentally determined points ti and ti+1 as a constant value equal to Ei+1,exper, which, as shown below, does not affect the approximation results. It must be noted that it follows from equations (XI.9) and (XI.41) that at the point τ = 0 α = 1, and the difference (α – α0) equals 0.9999999999, which is approximated by the computer as 1 and considered as an error, because the argument of the natural logarithm becomes 0. To eliminate this difficulty, the Boltzmann equation may be rewritten in the following form: t

E (t ) = E1 − E0 ∫ T (τ ) dτ , t1

where t1 and E1 are values of time and relaxation modulus, respectively, for the initial point of the experimental curve. Simultaneously, such presentation significantly reduces the working time of the program, because there is no need for integration of the steepest (starting) part of the relaxation curve.

XI.2 Sorption and swelling processes Let us apply the above-considered apparatus for describing relaxation phenomena in polymers to the processes of sorption and swelling. The point is that these processes proceed not only by filling in of separate pores in a polymeric substance, but also cause conformation rearrangements of macromolecules, i.e. are accompanied by relaxation processes. This idea has been put forward long ago in a series of works [63, 67, 71]. However, the detailed analysis of this process with regard to the relaxation mechanism of sorption and swelling was performed in refs. [10, 72]. This approach makes it possible to identify mechanism of the sorption of vapors by polymers. As mentioned above, the mechanism is associated with the fact that the penetration of vapors of low-molecular liquids into a polymer causes a rearrangement, sometimes significant, of submolecular organization which, in turn, affects the kinetics of further sorption and diffusion. Let us perform analysis on the basis of solving a system of differential equations [120]: f ⋅ν −

∂P ∂ν = 0 ; v ⋅ν − = 0, ∂x ∂x

(XI.44)

where f is the force affecting the diffusing particle; v is the number of diffusing particles; ν is the velocity of movement of diffusing particles; D is the diffusion coefficient; P is the osmotic pressure equal to (v/N)RT; N is the total number of particles in the system; R is the universal gas constant; T is the absolute temperature. From equation (XI.44) with regard to the expression for P, it is obtained that

498

D(t ) =

RT ν (t ) ⋅ . N f

(XI.45)

Since the polymer is a viscoelastic medium, movement of the diffusing particle in it, determined by its velocity v, depends on kinetic relaxation processes of the medium. To put it differently, it is necessary to connect the rate of diffusing particle, which becomes dependent on time, v(τ), as the particle penetrates into the viscoelastic substance, with the force f participating in the system of equations (XI.44). Let us perform general analysis of sorption and swelling processes using the hereditary Boltzmann–Volterra theory and choosing the above-mentioned memory function for describing the creep of polymeric substances. Because resolvents of memory function (XI.9) and (XI.18) have not as yet been found, the same memory function (XI.9) and (XI.18) but with the parameters different from those suitable for description of stress relaxation, may be used in the description of creep of polymers. With the help of memory function (XI.9) and (XI.18), creep is described by the correlations:  t  ε (t ) = ε 0 1 + ∫ T1 (τ )dτ  ;    0  t   ε (t ) = ε 0 1 + ∫ T2 (τ )dτ  ,    0 

(XI.46)

(XI.47)

where ε(t) is the deformation developed up to the moment of time t; ε0 is the elastic (instantaneous) deformation. If the process is limited by the rate of interaction of relaxants, the creep is described by equation (XI.46), and if it is limited by diffusion of inhomogeneities in the material, the creep is described by equation (XI.47). Let us analyze sorption and swelling with the help of a device applied in ref. [72] using memory functions T1(τ) and T2(τ). The value ν will equal

ν (τ ) =

d [ε (τ )l0 ], dτ

(XI.48)

where l0 is the initial length typical of the sample. Using expressions (XI.46) and (XI.47), we obtain:

or

ν(τ) = ε0l0T1(τ)

(XI.49)

ν(τ) = ε0l0T2(τ).

(XI.50)

From correlations (XI.49) and (XI.50), we get

499

l f ν (τ ) = 0 T1 (τ ) SE

(XI.51)

l f ν (τ ) = 0 T2 (τ ) , SE

(XI.52)

or

where S is the cross-section of the sample; E is the instantaneous elasticity modulus; f is the force loading the sample. Substituting correlations (XI.51) and (XI.52) into expression (XI.46), we obtain D(τ ) =

RT l0 T1 (τ ) N SE

(XI.53)

D (τ ) =

RT l0 T2 (τ ) . N SE

(XI.54)

or

Solving the equation of diffusion with the variable diffusion coefficient D(τ), as it was made in ref. [72], for a plate of thickness l, the kinetic dependence of the relative amount of sorbed substance may be calculated: ∞    2n + 1  2 RT   2n + 1  2  M (t ) 1 ⋅ =∑ f * (0)  − exp −  π exp − π  ×   M (∞) n = 0 ( 2n + 1) 2    l  N l      

×

RT *   f (t )   N 

   2n + 1  2 RT  *  − ⋅ − f π exp ( 0 )    ∑ 2 l  N  n = 0 ( 2n + 1)     ∞

1

(XI.55)

  2n + 1  2 RT   − exp −  π f * (∞)  ,  ⋅ l  N     where t  l0   f (t ) = T1 (τ ) dτ + B1  ∫ SE   0

(XI.56)

t  l  f * (t ) = 0  ∫ T2 (τ ) dτ + B2  . SE   0

(XI.57)

*

or

Therewith, l f * (0) = 0 B1 or SE l f * (∞) = 0 (A1 + B1 ) or SE

l f * (0) = 0 B2 , SE l f * (∞) = 0 (A2 + B2 ) , SE

(XI.58) (XI.59)

500

where ∞

A1 = ∫ T1 (τ )dτ 0

Because

∞

or A1 = ∫ T2 (τ )dτ . 0

RT * f (t ) << 1 , then on the basis of equation (XI.55) it may be N

written that M (t ) f * (0) − f * (t ) = . M (∞) f * (0) − f * (∞)

(XI.60)

Substituting expressions (XI.56), (XI.58) or (XI.57), (XI.59) into equation (XI.60), we obtain, respectively: t

∫ T1(τ )dτ M (t ) 0 = M (∞) A1

(XI.61)

or t

∫ T2 (τ )dτ

M (t ) 0 = M (∞ )

A2

.

(XI.62)

In the case of desorption or syneresis, kinetic equations are reduced to the form: t

M (t ) = 1 − ∫ T1 (τ ) dτ M0

(XI.63)

0

or t

M (t ) = 1 − ∫ T2 (τ ) dτ , M0

(XI.64)

0

where M0 is the initial mass of the swelling sample. Taking into account that relaxation memory function (XI.9) and (XI.18) contain constant values, which may be taken out of the integral, we obtain from expressions ((XI.61) and (XI.62): t

∫ T1 (τ )dτ *

M (t ) 0 = M (∞ ) ∞ * ∫ T1 (τ )dτ 0

or

(XI.65)

501

t

∫ T2 (τ )dτ *

M (t ) 0 = , M (∞ ) ∞ * ∫ T2 (τ )dτ

(XI.66)

0

where T1* (τ ) and T2* (τ ) are variable parts of memory function T1(τ) and T2(τ), respectively. To confirm experimentally the possibility of describing sorption with the help of expressions (XI.65) and (XI.66), measurement of kinetics on free films and coating based on cured epoxy resin ED-20 were performed in ref. [10]. Figure 88 indicates the kinetic curves of sorption measured on samples of cured epoxy resins in the form of both free films and coatings on an aluminum substrate with various types of surface treatment. The calculation results are shown in Table 42. Calculations of the kinetic parameters of sorption were performed by equations (XI.65) and (XI.66) using t

tabulated values of integrals

* ∫ T1 (τ )dτ and 0

t

∫ T2 (τ )dτ , shown above. If equations *

0

(XI.65) and (XI.66) describe properly the process of sorption, graphs in coordinates t

t

0

0

M(t)– ∫ T1* (τ ) dτ or M(t)– ∫ T2* (τ ) dτ must represent straight lines starting from the origin of coordinates, and the tangents of them are equal to M (∞) A1* or M (∞) A2* . This requirement is fulfilled by equation (XI.64), which is clearly seen in Figure 89. t

Experimental points fit well the calculated straight line in M(t)– ∫ T2* (τ ) dτ 0

coordinates which yields from the origin of coordinates. The correlation coefficient varies from 0.998 to 0.996. Parameters of the memory function T2(τ) are shown in Table 42. Table 42

Type of sample Free film Cover on degreased aluminum surface Cover on treated aluminum surface

Kinetic parameters of sorbing processes Parameters of memory function T2(τ) –10 D⋅10 , (time in min) 2 cm /s a γ 4.361 0.0346 0.4 3.451 0.0209 0.5 2.707

0.1040

0.3

Rate of change Mt/M0 at t 1 min 0.0297 0.0427

60 min 0.00395 0.00414

0.0653

0.00433

Description of kinetic curves of sorption using the memory function T1(τ) indicated [10] the worst coincidence of the calculated and experimental values of M(t). Remember that the memory function T1(τ) is valid if the course of the relaxation process is limited by the rate of relaxants interaction. When this process is limited by their diffusion in the material (i.e. self-diffusion), the memory function T2(τ) is valid.

502

As the calculations performed have shown [10], in the case of sorption of vapors of low-molecular liquids, the process is described well with the help of the memory function T2(τ), i.e. self-diffusion of relaxants is the limiting stage of the process lying in the basis of sorption. The Table also shows the diffusion coefficients calculated by the usual Fick equation. Table 42 indicates the rate of change of the relative mass overweight {d[M(t)/M(∞)]}/dt, which is calculated on the basis of correlation (XI.62):  M (t )  1 * v = d  dt = * T2 (τ ) , ∞ M ( ) A2  

(XI.67)

where the value A2* is taken from Table 39. If the parameters of the memory function T2* (τ ) are known, it is easy to calculate v at various moments of time t.

Figure 88. Kinetic curves of water vapor sorption at p/ps = 0.08: 1 – free film of ED-20; 2 – ED-20 coating degreased aluminum foil; 3 – ED-20 coating aluminum foil processed with γ-aminopropyltriethoxysilane.

t

Figure 89. Dependence of Mt/M0 on ∫ T2* (τ ) dτ . 0

503

Hence, the method of description of the kinetic curves of sorption, which takes into account the relaxation type of this process, enables kinetic dependences to be approximated with high accuracy using the new relaxation memory function. If the Fick equation with the constant diffusion coefficient is used for this purpose, the adequate description cannot be carried out, and the process may be approximated only over a short initial period of time.

Chapter XII. Solubility of polymers

XII.1 Specific cohesive energy of organic liquids and polymers Hildebrand solubility parameter To predict the solubility of polymers in various organic solvents, as well as for preliminary estimation of the miscibility of polymers with each other or with plasticizers, the characteristic as the solubility parameter δ is often used. This characteristic has been introduced by Hildebrand for description of solutions of nonelectrolytes. The Hildebrand solubility parameters is determined from the correlation

δ=

∆E0 , V

(XII.1)

where ∆E0 = ∆H0 – RT; ∆H0 is the latent heat of liquid evaporation; R is the universal gas constant; T is the absolute temperature; V is the molar volume of the liquid. The square of the solubility parameter represents specific cohesive energy of the liquid, i.e. the value of cohesive energy divided by the molar volume:

δ2 = ∆E0/V.

(XII.2)

These notions are also applied to polymers, therewith, estimations are indicated per repeat unit. In this case, the problem is that the value δ may be experimentally determined only for low-molecular liquids evaporating without decomposition. For polymers which cannot be evaporated without decomposition, values of δ are determined by indirect methods or calculated from the constants of energies for separate atoms and groups of atoms [141, 150, 206]. Taking into account the nature of molecules in liquids and polymers leads to the following equation for calculating the specific cohesive energy [25]:

δ2 =

∆E

*

N A ∑ ∆Vi i

=

∑ ∆Ei* i

N A ∑ ∆Vi

,

(XII.3)

i

where ∆E* = k∆E0 is the cohesive energy of the liquid or repeat unit of the polymer, reduced by the number of times by which the Van-der-Waals volume of the molecule (or unit) is smaller than the molar volume; k is the coefficient of molecular packing of the liquid or polymer. The value of ∆E* is additive and is represented as ∆E* =

∑ ∆Ei* , where ∆Ei* i

is the contribution of each atom and type of intermolecular interaction to ∆E*. Values of ∆Ei* are shown in Table 43, which indicates the appropriate notes. With the help of them, the solubility parameter δ may be calculated for many polymers with different chemical structures.

505

Table 43 * Values ∆Ei for different atoms and types of intermolecular interaction Atom and type of intermolecular interaction Designation ∆Ei* , cal/mol 1 2 3 Carbon 550.7 * ∆E C Hydrogen 47.7 * ∆E H Oxygen 142.6 * ∆EO Nitrogen 1205.0 * ∆E N Fluorine 24.2 ∆E F* Sulfur 1750.0 ∆ES* Chlorine –222.7 * ∆E Cl Bromine 583.0 * ∆E Br Iodine 1700.0 ∆EI* Double bond –323.0 ∆E ≠* Aromatic cycle (skeleton) 713.0 ∆Ε* Dipole–dipole interaction Dipole–dipole interaction in non-polar aprotic solvents of: amide type dimethylsulfoxide type Hydrogen bond Specific interactions in the presence of =CCl2 group In stressed three-five-component cycles in the presence of O atom Hydrocarbon radical isomerism

∆E d* ∆E a*, N ∆E a*, S ∆E h* ∆E =* CCl 2

1623.0

1623.0 2600.0 3929.0 2600.0

* ∆EO, c

2430.0

∆Ei*

– 412.0

Notes: * 1) The constant ∆E ≠ is introduced in the presence of double bonds, not participating in polar groups. * 2) The constant ∆E is introduced in the presence of aromatic cycles in the amount appropriate to these cycles. * 3) The constant ∆E d is introduced in all the cases in the presence of a polar group of any type; in the case of chlorinated compounds in the presence of two and more Cl atoms, added to the same carbon * atom, two ∆E d should be introduced. * 4) The constant ∆E a , N is introduced at the calculation of δ for dipolar aprotic solvents of the amide type; therewith, the usual dipole–dipole interaction at the sacrifice of polar groups is taken into account by introduction of the appropriate amount of constants. 5) The same as for the previous case, but for solvents of the dimethylsulfoxide type.

506

* 6) The constant ∆E h is introduced in the presence of a hydrogen bond of any type. * 7) The constant ∆E = CCl takes into account the specific interaction in the presence of the 2

Cl C

Cl * group; therewith, the constant ∆E d is not introduced. * 8) The constant ∆EO, c takes into account appearance of polarity in stressed 3–5-component cycles

containing the heteroatom O. * 9) The constant ∆Ei is introduced at transition from normal hydrocarbons to isomers.

The experimental methods of determination of solubility parameter δ conclude in the following. The value of intrinsic viscosity η of the polymer is evaluated in a selection of solvents with different values of the solubility parameter. Further on, dependences of η of the polymer on the parameter δ of solubility of the same solvent, in which they were measured, are composed. Figure 90 indicates the schematic representation of this dependence. The maximum of this dependence determines the parameter of polymer solubility which, in this case, equals the parameter of liquid solubility, in which the intrinsic viscosity of this polymer is maximal (see Figure 90).

Figure 90. Schematic representation of dependence of intrinsic viscosity [η] of polymers in different solvents on solubility parameter of solvent δp.

Another experimental method of estimating the value δ is the measurements of the equilibrium swelling degree and plotting the dependence of this value on the solubility parameter of the liquid in vapors of which the swelling degree was measured. This dependence is analogous to the one depicted in Figure 90. It should be noted that experimental methods of determination of δ are difficult and not always reliable. For preliminary estimation of δ, the calculation methods are preferable. To predict solubility of polymers, the calculated value δp for the polymer is sometime compared with experimental values δs for solvents. If values δ for polymers and solvents differ significantly, then no dissolution takes place. However, the coincidence of solubility parameters of the polymer and the solvent does not yet guarantee dissolution of the polymer in the current solvent. In practice, in the case of coincidence of δ values, dissolution is observed in 50% of cases only (see below).

507

Table 44

‹ 1 2 3 4 5 6 7 8 9 10 11 12 13 14

Parameters of solubility of a series of polymers Name ∆Ei* , N A ∆Vi , i i cm3/mol cal/mol Poly(methyl methacrylate) 5043 58.5 Poly(ethyl methacrylate) 5689 69.0 Poly-n-propyl-methacrylate 6335 79.3 Poly-n-butyl-methacrylate 6981 89.6 Poly(methyl acrylate) 4397 48.2 Poly(ethyl acrylate) 5043 58.5 Polystyrene 5500 66.0 Polyisobutylene 2584 41.6 Polyacrylonitrile 4623 32.6 Poly(ethylene terephthalate) 10418 102.4 Nylon 6,6 18210 139.2 Phenolphthalein and isophthalic acid 27189 234.7 polyarylate Phenolphthalein and terephthalic acid 27189 234.7 polyarylate Phenolphthalein and 4,4’31397 279.2 diphenyldicarboxylic acid polyarylate

∑

∑

δ, cal0.5/cm1.5 Calc.

Exper. [54]

9.3 9.1 8.9 8.8 9.55 9.3 9.1 7.9 11.9 10.1 11.4 10.7

9.1; 9.5; 9.4 8.95   10.1 9.4 9.1; 8.6; 8.7 7.95; 7.8; 8.05    10.8

10.7

10.7

10.6

10.4

Table 44 indicates solubility parameters δ for polymers of various chemical structures. The chemical structure significantly affects the value δ. The presence of C O strong polar groups of Cl, , C≡N types, etc. leads to increase of the O cohesive energy and, consequently, of the value δ. The solubility parameter is most strongly increased by hydrogen bonds appearing in the presence of OH and HNC groups, and by aromatic cycles, as well. O For copolymers, the equation for calculating the solubility parameter is presented in the form:       α1  ∑ ∆Ei*  + α 2  ∑ ∆Ei*  + ... + α n  ∑ ∆Ei*         i 1  i 2  i n δ2 = , (XII.4)         N A α1  ∑ ∆Vi  + α 2  ∑ ∆Vi  + ... + α n  ∑ ∆Vi           i 1  i 2  i n     where α1, α2, …, αn are the molar parts of components 1, 2, …, n;  ∑ ∆Vi  ,    i 1      ∑ ∆V  , …,  ∑ ∆Vi  are the Van-der-Waals volumes of components 1, 2, …, i     i n  i 2

508

      n;  ∑ ∆Ei*  ,  ∑ ∆Ei*  , …,  ∑ ∆Ei*  are the cohesive energies of components        i n  i 1  i 2 1, 2, …, n; NA is the Avogadro number. In the reduced form, equation (XII.4) is the following: k =n

δ2 =





∑ α k  ∑ ∆Ei* 

k =1  i k , k =n   NA α k  ∆Vi    k =1  i k

∑

(XII.5)

∑

  where αk and  ∑ ∆Vi  are the molar part and the Van-der-Waals volume of the    i k   k-th component, respectively;  ∑ ∆Ei*  is its cohesive energy.    i k If it is desirable to express the solubility parameter via the solubility parameters of the components of the copolymer, it should be written down:       α1δ 12  ∑ ∆Vi  + α 2δ 22  ∑ ∆Vi  + ... + α nδ n2  ∑ ∆Vi         i 1  i 2  i n , δ2 =       α1  ∑ ∆Vi  + α 2  ∑ ∆Vi  + ... + α n  ∑ ∆Vi         i 1  i 2  i n

(XII.6)

where δ1, δ2, …, δn are solubility parameters of the components 1, 2, …, n. In the reduced form, this equation is presented as

δ2 =

k =n





k =1 k =n

 i

k

∑ α k δ k2  ∑ ∆Vi    ∑ α k  ∑ ∆Vi  k =1  i k

,

(XII.7)

  where αk,  ∑ ∆Vi  , and δk are the molar part, the Van-der-Waals volume and the    i k solubility parameter for the k-th component, respectively.

509

XII.2 Solubility criterion The problem of prediction of the solubility of polymers has been urgent for many years. One of the methods of preliminary estimation of polymer solubility concludes in comparing values of Hildebrand solubility parameters δ for the polymer δp and the solvent δs. Therewith, it is assumed that if the condition δp ≈ δs is fulfilled, polymer dissolution in the given solvent may be expected. However, experience suggests that with the help of such comparison we may only ‘neglect’ the solvents, in which no dissolution of the given polymer proceeds. These are systems for which δp >> δs or δp << δs. With the help of this estimation, it is possible to shorten significantly the list of solvents to be tested, in which the polymer may dissolve. Estimation and experience [128] indicate that this method makes it possible, for example, to neglect 120–130 organic liquids from the list of 160 solvents as obviously invalid for dissolution. Therewith, the polymer is dissolved in a half of the rest of solvents conformed to the condition δp ≈ δs. Consequently, fulfilling the condition δp ≈ δs may not guarantee solubility of the polymer. It is desirable to obtain a more accurate method of preliminary estimation of polymers in relation to the solvents, for which the condition δp ≈ δs is met. Let us consider in detail the solubility criterion suggested in refs. [32, 95], which possesses quite high predictive force. The feature of the problem of solubility lies in the fact that here, in contrast to determination of some simplest properties, not only the chemical structure, but also the particular supermolecular structure of the polymer should be taken into account. Actually, it is common knowledge that a crystalline polymer is dissolved much less readily than an amorphous polymer of the same chemical structure. Oriented samples are also less soluble compared with isotropic samples. Probably, in the case of isotropic samples of amorphous polymers, the supermolecular structure may also be different; however, this problem is still the subject of discussion. Presented at an International Conference in 1979 in London were experimental and theoretical data on the absence of a ‘nodular’ structure in amorphous polymers, the data of electron-microscopic investigations of the surface of films and chips being added to artifacts [142]. However, it can be hardly imagined that if the surface of the film obtained from the solution and the surface of chip of a block sample obtained from the melt give one and the same electron-microscopic picture of globules, this picture is the consequence of artifacts. It is noted that the neutron scattering method is the only one that may give direct information on this problem, although interpretation of data is not uniform. This discussion has been continuing. Basing on results of X-ray scattering, it is stated [189] that a level of packing regularity does also exist in amorphous polymers. Analyzing the problem of solubility in ref. [32], the authors used the model of the supermolecular structure developed in refs. [92, 93], in which the supermolecular structure of amorphous polymers is modeled as globules. Therewith, an attempt was made in these works to justify the absence of a long period at small-angle X-ray scattering. It is also assumed that every globule consists of globules–macromolecules [4, 102]. Supposing that both types of globules are connected with each other by bond belts, let us discuss the most typical elementary event of dissolution, i.e. decomposition of particles down to separate globular macromolecules, which is schematically represented in Figure 91.

510

Figure 91. Schematic representation of polymer surface and elementary event of dissolution.

Here, the case of an uneven surface is shown (for example, a powder or a film with surface roughness), when the load is applied from the side of the solvent to the globule belt by which it is connected with neighboring globules, and detachment of the globule from the polymeric substance and its transition into the solvent proceeds at the sacrifice of bond belt rupture. It should be noted that to estimate the solubility of polymers, it is necessary to consider the change of free energy, i.e. to take into account both energetic and entropic components. In this case, we will consider only amorphous polymers composed of macromolecular globules, which keep their form in solution. In this particular case, conformational selection is maintained, and the change of entropy at transition of macromolecules into the solution will be slightly different from the ideal case only due to the difference in sizes of solvent molecules and macromolecular globules. That is why in this case, in the first approximation, the entropy change at dissolution may be neglected. When analyzing the solubility of crystalline polymers or amorphous non-globular polymers with flexible macromolecules, the entropic component of free energy may be significant, and the energetic criterion considered is changed. Let us dwell on the physical assumptions used in determination of the solubility conditions. When a polymer sample is immersed into a solvent, the globules which exist on the sample surface will be detached first. Let us consider forces affecting the globules. Figure 92 indicates the globule cross-section and the belt of the globule bonding it to other globules of the supermolecular structure of the polymer. At the moment of the polymer dipping into solvent, the solid (globule)–liquid interface is formed. Formation of the unit of this surface includes work WA determined by the process of adhesive wetting: WA = γs-p – (γp + γs),

(XII.8)

where γp and γs are surface tensions of the polymer and the solvent, respectively; γs-p is interfacial tension. Therewith, WA represents the work of adhesion, i.e. the work required for separation of the surfaces (restoration of the initial state). The work of adhesion induces occurrence of forces affecting the globule of the supermolecular structure; these forces depending on the size and sign of curvature of the surfaces which form the globule and the bond belt. These forces lead to detachment of the globule from the rest of the polymer sample. But as soon as this detachment happens, and the globule transits into the solvent, a fresh (new) surface of another globule, which was closed, is formed. It is also wetted by the solvent, and the same forces appear. When this globule is detached, the situation is repeated: new surfaces of the previously screened globules are formed and due to the effect of wetting forces they are subsequently transferred into the solvent.

511

Figure 92. Cross-section of the globule and the belt whereby the globule is linked to the other globules of supermolecular structure (schematic).

Let us consider in more detail the forces affecting the globule and appearing during its adhesive wetting by the solvent. According to the image in Figure 92, two forces affect the initial globule of the supermolecular structure and the bond belt. They are the forces determined by WA, i.e. by the surface tension of the solvent, and the interfacial tension applied to the surface of the supermolecular structure globule, disposing over the bond belt. The latter force strives to detach the globule from the polymer (this happens due to different signs of curvature of these surfaces). Determining the second force, it may be assumed that the surface to which the force detaching the globule is applied is the tore surface (shaded in Figure 92). Therewith, the wetting force intensity will be applied to the half of the tore surface. Because the cross size of the linking belt is significantly smaller than the greater radius of the belt, further on, all calculations are performed on typical sizes of the bond belt, the surface of which is also assumed to be toroidal. Let us determine the conditions of polymer dissolution. Figure 93 indicates a triangle, apexes of which locate in centers of globules of the supermolecular structure. Therewith, [A′D] = [DB′] = r, [AD] = [DB] = R, [DO] = [OE] = Rt = R/2, where r is the smaller radius of the linking tore, R is the radius of globule of the supermolecular structure, Rt is the larger radius of the linking tore.

Figure 93. Characteristic distances between globules (see text).

Globule 1 will be affected by the adhesive wetting force which compresses globule to the polymer. According to the Laplace law, this force is determined from the correlation:

512

f gl = πRt

γ s + γ p − γ s-p γ s + γ p − γ s-p = πR . R 2

(XII.9)

Moreover, the force ft is applied to the bond belt striving to detach the globule from the substance:

θ 1 1   γ s + γ p − γ s-p . f t = 2πRr ⋅ sin  − 2  r Rt 

(

Since Rt = R/2,

)

(XII.10)

θ = π 6 , then 2

(

)

1 1  f t = πRr  −  γ s + γ p − γ s - p . r R

(XII.11)

For the globule to detach from the supermolecular structure, two conditions must be fulfilled. The first condition is that the detaching force applied from the side of the solvent, ft, must be greater than fgl which compresses the globule to the substance, i.e.

(γ s + γ p − γ s - p ) 3 − 8Rr  > 0 . 



(XII.12)

For globules, 8r/3R << 1. That is why condition (XII.12) may be presented in the following form: γs + γp > γs-p.

(XII.13)

The second condition is formed in the following manner: the work performed by the surface tension forces of the solvent A must exceed the energy of rupture of intermolecular bonds in the bond belt, i.e. specific cohesive energy of the polymer δ p2 would be lower or equal to the work of the solvent on rupture of the linking tore, 2

related to the specific volume of the linking tore δ p* . The value A may be presented in the form: A = (ft – fgl)εmaxr, and 2

δ p* =

(

)

A 8r  ε  γ s + γ p − γ s - p  3 −  max . Vt R  4πr 

(XII.14)

Because 8r/3R << 1, expression (XII.14) transforms to: 2

(

δ s* = γ s + γ p − γ s - p

)3ε4max . πr

(XII.15)

513

So, the second condition may be written down as follows: 2

δ p2 ≤ δ s* .

(XII.16)

Let us transform expression (XII.16): 2

δ s* = δ s2 ρ

γ s + γ p − γ s-p γs

,

(XII.17) 2

* * where δ s2 is the cohesive energy density of the solvent; δ s* = 3γε max 4πr * , ε max is the maximal distance between solvent molecules, which are still affected by forces of interaction, r* is the characteristic size of the linkage of the Frenkel cluster of the * r . Making designation µ = δ p2 δ s2 and taking into solvent, ρ = ε max r * ε max

account that, according to refs. [147 – 149],

(

)

γ s - p = γ s + γ p − 2Φ γ sγ p 1 / 2 ,

(XII.18)

the condition of solubility (XII.16) may be presented in the following form: 1/ 2

γ p  µ ≤ 2 ρΦ  γs 

,

(XII.19)

where µ = δ p2 δ s2 ; δp and δs are Hildebrand parameters for the polymer and the solvent, respectively; γp and γs are the surface tensions of the polymer and the solvent, respectively; Φ=

(

)

4 VsVp 1 / 3

(Vs1/3 + Vp1/3 )2

.

(XII.20)

Here Vs and Vp are the molar volumes of the solvent and the polymer (per one unit), respectively. Note that values Φ are of the order of unity. Expression (XII.19) is deduced from the condition that there is no swelling of the polymer. Let us consider another case, i.e. when polymer swells. This means that the solvent penetrates into a cavity between globules A, B, and C (see Figure 92). In this case, condition (XII.15) may be formally presented in the same form, however, the effective value (γp/γs)eff will play the role of the relation γp/γs. This effective value will be smaller than the true value γp/γs, because penetration of the solvent into the cavity between globules causes wedging and decreases force ft necessary for detachment of globule B. This question is discussed in detail in ref. [32]. Consequently, the following expression is deduced for the solubility criterion:

514

µ < 2 ρΦ Φ − Φ 2 − 1 + a  ,  

(XII.21)

where a = γs-p/γs; other parameters are the same as in equation (XII.19). Recall that µ = δ p2 δ s2 ; ρ is a constant; the value Φ is calculated from formula (XII.20). It follows from expression (XII.21) that the value amin = 1 – Φ2 exists, at which dissolution is possible (for example, at Φ ≈ 0.95, amin = 0.1). Because µ is always positive, solubility is possible at amin < a < 1. The maximal value µmax will equal

µmax = 2ρΦ2.

(XII.22)

Hence, according to criterion (XII.21), solubility will be observed in the case when the right part of criterion (XII.21) is greater than the left part, both parts of the criterion being calculated on the basis of the chemical structure of the repeat unit of polymer and solvent molecule. Experimental testing of the solubility criterion (XII. 21) is performed on the example of approximately 300 systems of the polymer–solvent type. For polymeric objects of the study, both traditional amorphous polymers (poly(methyl methacrylate), polystyrene, polyvinylacetate, polyisobutylene, butadiene and isoprene rubbers, etc.) and a series of heat resistant polymers of the cardo-type structure (polyarylates, aromatic polyamides and polyimides, polyphenylquinoxaline) were chosen in ref. [32]; about 50 organic liquids were used as solvents. As mentioned above, solubility must be observed if calculated values  2 ρΦ Φ − Φ 2 − 1 + a  appear greater or equal to the value µ = δ p2 δ s2 ; therewith,   δp and δs values used may be both experimental and theoretical. The value ρ introduced above is not determined directly. However, if the value ρ is nearly constant, then introducing designation

β = Φ Φ − Φ 2 − 1 + a  ,  

(XII.23)

µ < 2ρβ,

(XII.24)

we obtain that

i.e. in (µ, β) coordinates, the dependence µ = 2ρβ represents a straight line yielding from the origin of coordinates. Above it there must be a point which indicates the absence of solubility, and below it – the presence of solubility. Calculations performed for the mentioned number of pairs indicate (Figure 94) that points corresponding to the case of insolubility (dark points) are usually disposed above the mentioned line, and the points corresponded to solubility (light points) are located below it. Hence, the range of values µ is divided into two fields, one of which represents the field of insolubility, and another one – the field of solubility. However, each of these fields contains certain numbers of ‘foreign’ points, the fraction of which

515

is about 15%. Generally, the number of dark and light points is approximately equal, and the value µ = δ p2 δ s2 varies around unity.

Figure 94. Dependence of µ on β for different pairs of the ‘polymer–solvent’ type (for explanation see text).

The dotted line in Figure 94, corresponding to the dependence µ = 2ρβ, is traced in the manner that the number of ‘foreign’ points above and below the straight line would be minimal and approximately equal. If these conditions are fulfilled, the value 2ρ = 1.374. Figure 94 indicates immediately that if the only condition δs ≈ δp is fulfilled, dissolution may be observed in 50% of cases only. Consequently, for systems in which the condition δs ≈ δp is fulfilled, according to criterion (XII.19) or (XII. 21), solubility may be predicted with higher accuracy equal to 85%. Taking into account that real non-solvents are preliminarily neglected, for which δs >> δp and δs << δp, the predictive strength of the criterion may be considered high. Usually, solubility is observed when the surface tension of the solvent is close to the surface tension of the polymer; the interfacial tension is then low and the value a is also low. But if the equality δs ≈ δp is fulfilled, the surface tension of the solvent is significantly lower than the surface tension of the polymer, the relation of the interfacial tension to the surface tension of the solvent is high, a becomes high, and β becomes low. Points in the diagram (see Figure 94) appears in the left upper field. Hence, two factors promote the increase of β value: equality of the molar volumes of the solvent and the repeat unit and the equality of the surface tensions of the polymer and the solvent. In its turn, according to Figure 94, the increased value β promotes the polymer solubility. It has been assumed for a long time that the like must dissolve in the like. However, no acceptable definition of such substances has been presented as yet. From the point of view of the solubility criterion expressed by expression (XII.19) or (XII. 21), these substances may be considered as the ones possessing similar densities of the cohesive energy (the Hildebrand solubility parameter δ), molar volumes and surface tension coefficients. In this case, solubility and miscibility are usually observed.

516

Figure 94 indicates that some exclusions do also exist. For example, in the case of polyarylate, C O C

O C

O

O C

O

according to criterion (XII. 21), there should be no dissolution in chloroform, and cyclohexanol must dissolve the polymer. In practice, the situation is opposite: chloroform dissolves efficiently many heat-resistant polymers, and cyclohexanol – not. These very exclusions form the main part of distortion of criterion (XII. 21). Apparently, in the case of chloroform, dissolution proceeds practically without swelling, which confirms the preliminary experiments. Other possible reasons of deviations from criterion (XII. 21) will be analyzed below. According to the above-discussed ideas, estimation of the solubility of the polymer with the given chemical structure in one or another solvent concludes in the following facts. For this polymer and solvent, values of the solubility parameter δ are calculated from formula (XII.3). Then the value of the surface energy for polymers, γp, is calculated by equation (XIII.18) or equations (XIII.28) and (XIII.29). It is also possible to calculate γp with the help of a parachor using equation (XIII.1). The molar volume of the repeat unit of the polymer, necessary for this purpose, is determined as

Vp =

N A ∑ ∆Vi i

k avg

,

(XII.25)

where kavg = 0.681. If polymer density, dp, is known, then Vp = M/dp, where M is the molecular mass of the repeat unit. The surface tension for the solvent, γs, may be assumed as both experimental and calculated from formula (XIII.11). The value Φ is calculated from formula (XII.20). Values of Vs and Vp required for this purpose are determined from correlations Vp = M/dp and Vs = M/ds, where ds and dp are densities of the solvent and the polymer, respectively. After that, the value of the interfacial tension, γs-p, is calculated from equation (XII.13), and then – the value a = γs-p/γs. Further on, the value 1.347Φ Φ − Φ 2 − 1 + a  is calculated, and the result is compared with   µ = δ p2 δ s2 . If µ < 1.347Φ Φ − Φ 2 − 1 + a  , then dissolution of the polymer in the   current solvent may be expected with 85% probability. Let us indicate an example of calculation for the poly(methyl methacrylate)– benzene system. CH3 CH2

C C O CH3 O

Polymer

Solvent

517

First, it is necessary to determine solubility parameters for the polymer, δp, and the solvent, δs.   * * * + 8 ∆EH + 2 ∆EO + ∆Ed* = 5⋅550.7 + For the polymer,  ∑ ∆Ei*  = 5 ∆EC    i p 8⋅47.7 + 2⋅142.6 + 1623 = 5043.3 cal/mol = 21,081 J/mol (all values ∆Ei* are taken from Table 43).    ∑ ∆Vi  = ∆VC,10 + ∆VC,13 + ∆VC,1 + ∆VC,49 + ∆VC,41 + ∆VO,139 + ∆VO,129 + 8∆VH,124    i p = 13.1 +17.2 + 5.0 + 15.9 + 20.3 + 5.8 + 3.4 + 8⋅2.0 = 96.7 Å3 (all numbers of atoms correspond to the numbers in Table 3).     Substituting calculated values of  ∑ ∆Ei*  and  ∑ ∆Vi  into equation      i p  i p (XII.3), we obtain that

δ p2 =

21,081 = 362 J/cm3; δp = 19.0 (J/cm3)1/2. 0.6023 ⋅ 96.7

For the solvent,   *  ∑ ∆Ei*  = 6 ∆E * + 6 ∆EH + ∆E* = 6⋅550.7 + 6⋅47.7 + 713 = 4303 cal/cm3 = C    i s 17,988 J/ cm3.    ∑ ∆Vi  = 6∆VC,18 + 6∆VH,124 = 6⋅12.7 + 6⋅2.0 = 88.2 Å3.    i s According to these parameters, from equation (XII.3) we obtain that

δ p2 =

17,988 = 338.6 J/cm3; δs = 18.4 (J/cm3)1/2. 0.6023 ⋅ 88.2

It is clear that δs ≈ δp, and it may be suggested that poly(methyl methacrylate) is soluble in benzene. The surface tensions of the polymer and the solvent should then be calculated.     Substituting values of  ∑ ∆Ei*  and  ∑ ∆Vi  , and m = 15 (the number of atoms      i p  i p in the repeat unit of poly(methyl methacrylate)) into equation (XIII.22), we obtain that poly(methyl methacrylate) relates to polar groups of the type 1):

γ p = 0.0751

21,081 (96.7) 2 / 3 ⋅151 / 3

= 30.5 dyn/cm.

518

For benzene, according to expression (XIII.11), we obtain

γ s = 0.0287

17,988 (88.2) 2 / 3

= 26.1 dyn/cm.

The molar volume of benzene Vs = 89 cm3/mol; the molar volume of poly(methyl methacrylate), according to equation (XII.25), equals Vp =

0.6023 ⋅ 96.7 = 85.5 cm3/mol. 0.681

Now it is necessary to calculate the value of Φ. According to equation (XII.20), it indicates Φ=

4 ⋅ (89 ⋅ 85.5)1 / 3

(89

1/ 3

)

1/ 3 2

+ 85.5

= 1.

Further on, interfacial tension is calculated from formula (XII.18):

γs-p = 26.1 + 30.5 – 2⋅1⋅(26.1⋅30.5)1/2 = 0.17 dyn/cm. The value of a = γs-p/γs = 0.17/26.1 = 0.00656. Substituting all values of parameters of the polymer–sovlent system obtained into criterion (XII.21), we obtain that

µ=

δ p2 δ s2

=

362 < 1.374 ⋅1.0 ⋅ 1.0 − 1.0 2 − 1 + 0.00656  ; 338.6   1.069 < 1.263

Because the left part of criterion (XII.21) is smaller than the right part of it, then poly(methyl methacrylate) must dissolve in benzene, which is observed in reality. Let us display one more example of calculation for a more complicated polymer–solvent system. As a polymer, let us choose anilinphthalein polypyromellitimide, and nitrobenzene – as a solvent: O O C

C

N

O

N C

C

O

O

N O

C O C

Polymer

O

Solvent

519

For the polymer,   * * *  ∑ ∆Ei*  = 30 ∆EC + 14 ∆EH + 6 ∆EO + 2 ∆E *N + 3 ∆Ed* + 4 ∆E * = 30⋅550.7 +    i p 14⋅47.7 + 6⋅142.6 + 2⋅1205 + 3⋅1623 + 4⋅713 = 28,175 cal/mol = 117,733 J/mol;    ∑ ∆Vi  = 4∆VC,64 + 8∆VC,19 + 14∆VC,18 + 2∆VC,21 + ∆VC,34 + ∆VC,49 + 14∆VH,124 +    i p 5∆VO,139 + ∆VO,129 + 2∆VH,144 = 4⋅14.1 + 8⋅8.4 + 14⋅12.7 + 2⋅10.2 + 7.9 + 15.7 + 14⋅2.0 + 5⋅5.8 + 3.4 + 2⋅0.9 = 407.6 Å3. In accordance with equation (XII.3), we obtain: 117,773 = 479.7 J/cm3; δp = 21.9 (J/cm3)1/2. δ p2 = 0.6023 ⋅ 407.6 For the solvent,   * * *  ∑ ∆Ei*  = 6 ∆EC + 5 ∆EH + ∆E * + ∆Ed* = 6⋅550.7 + 5⋅47.7 + + ∆E *N + 2 ∆EO    i p 1205 + 2⋅142.6 + 713 + 1623 = 7,369 cal/mol = 30,802 J/mol;    ∑ ∆Vi  = 5∆VC,18 + ∆VC,21 + 5∆VH,124 + ∆VH,150 + 2∆VO,140 = 5⋅12.7 + 10.2 + 5⋅2.0    i p + 7.0 + 2⋅7.2 = 105.1 Å3. According to equation (XII.3), we obtain: 30,802 = 486.6 J/cm3; δs = 22.06 (J/cm3)1/2. δ s2 = 0.6023 ⋅105.1 It is clear that δs ≈ δp, i.e. the present polyimide, principally, may dissolve in nitrobenzene. Then, the surface tension of the polymer and the solvent is calculated. The number of the atoms in the repeat unit of the present polyimide m = 52. Substituting all parameters of polyimide into equation (XIII.18), we obtain that

γ p = 0.0751

117,773 ( 407.6) 2 / 3 ⋅ 521 / 3

= 43.1 dyn/cm.

According to expression (XIII.11), for nitrobenzene we get:

γ s = 0.0287

30,802 (105.1) 2 / 3

= 39.7 dyn/cm.

The molar volume of nitrobenzene Vs = 103 cm3/mol; the molar volume Vp of polyimide, according to equation (XII.25), equals Vp =

0.6023 ⋅ 407.6 = 360.5 cm3/mol. 0.681

Let us calculate the value of Φ by equation (XII.20):

520

Φ=

4 ⋅ (103 ⋅ 360.5)1 / 3

(1031/ 3 + 183.61/ 3 )2

= 0.9576.

The value of the interfacial tension, calculated from formula (XII.18), equals:

γs-p = 43.1 + 39.7 – 2⋅0.9576⋅(43.1⋅39.7)1/2 = 3.58 dyn/cm. The value of a = γs-p/γs = 3.58/26.1 = 0.0901. Substituting all values of parameters of the polymer–solvent system into criterion (XII.21), we obtain that

µ=

δ p2 δ s2

=

479 < 1.374 ⋅ 0.9576 ⋅  0.9576 − 0.9576 2 − 1 + 0.0901  ;   486.6 0.986 < 1.149.

According to criterion (XII.21), the present polyimide will dissolve in nitrobenzene, which is observed in practice [6].

XII.3 Influence of molecular mass and degree of macromolecule orientation on solubility Solubility criterion (XII.21) is true for the case of isotropic amorphous polymers with a globular supermolecular structure. Moreover, this criterion does not take into account the influence of the polymerization degree of the polymer on solubility, although it is common knowledge that it may be significant at transition to greater molecular masses. Ref. [95] indicates the attempt to take into account the type of supermolecular structure and the polymerization degree of polymers on their solubility, as well as to set a connection between the Flory–Huggins theory and the chemical structure of the polymer and the solvent. Solubility criterion (XII.21) contains constant ρ, which is described by the correlation:

εp r ρ = max s , s rp ε max

(XII.26)

p where ε max is the maximal relative deformation of intermolecular bonds in the s polymer at the moment of their rupture and polymer transition into the solvent; ε max is the maximal deformation of the liquid, i.e. the deformation, at which the continuity is distorted; rs is the typical size of the bond of Frenkel clusters in the solvent; rp is the lower radius of the linking globule for the polymer. Imagine that in the area of deformation the polymer and the solvent behave themselves as elastic bodies [77] characterized by elasticity moduli Ep and Es, respectively. Then

521

( )

( )

2 2 p s δ p2 = Ep ε max 2 ; δ s2 = Es ε max 2,

(XII.27)

and the value µ will obtain the following form:

µ=

δ p2 δ s2

=

( ). s )2 Es (ε max

2 p Ep ε max

(XII.28)

At the moment of globule detachment, tensions in the globule and the solvent p s = Esε max . That is why expression (XII.24) is reduced to are equal, i.e. σ = Epε max the form:

rp rs

≤ 2β .

(XII.29)

1/2 Because rp = n1/2 p a p and rs = ns as (where ap and as are the sizes of repeat units of the polymer and the solvent, respectively; np and ns are numbers of units in the connecting globule in the polymer and of solvent molecules, respectively), then taking into account the rotary-isomeric theory [58], we obtain



2/3



2/3

  ∆Ep  ∑ ∆Vi  exp RT  ;    i p

ap2 =  

  ∆E  ∑ ∆Vi  exp RTs  ,  i s

as2 =  

(XII.30)

(XII.31)

where ∆Vi,p and ∆Vi,s are Van-der-Waals volumes of the i-th atoms participating in the repeat unit of the polymer and the molecule of the solvent, respectively; ∆Ep and ∆Es are differences in the energies of rotary isomers of the polymer and the solvent. With this regard, expression (XII.29) may be reduced to the form:      ∑ ∆Vi   1/ 2   ns    i s    ⋅β ,  q≤2  np          ∑ ∆Vi     i  p   ∆Ep − ∆Es  . where q = exp   2 RT 

(XII.32)

522

The value ns will be calculated (with accuracy up to a constant) from the condition that ns equals the number of solvent molecules, which cover the connection globule by a single layer. Then  ∆Vp   ns = 6 np   ∆Vs 

2/3

ns 6  ∆Vp    = 1/3 np ns  ∆Vs 

2/3

;

(XII.33)

.

    Here ∆Vp =  ∑ ∆Vi  ; ∆Vs =  ∑ ∆Vi  . Substituting expression (XII.33)      i p  i s into formula (XII.32), we obtain that 1/ 2

 6   q ≤ 2  n1/3   p 

⋅β ,

(XII.34)

1/ 2

 6   , and q by its meaning equals to µ (q = µ). i.e. ρ =   n1/3   p  Taking into account that not the whole connecting globule is washed over by the solvent (a part of it is a part of the globule–macromolecule), we may present the expression for ρ in the following form: 1/ 2

 6  ρ =  1/3  n   p 

C −1 ,

(XII.35)

where C is the part of the connecting globule surface washed over by the solvent. C is the constant, and its value may be determined from the condition ρ = 0.687 at np = 24 [94] (at the polymerization degree N = 104). Then C = 2.1, and solubility condition (XII.34) will reduce to the form 1/ 2

 6   q≤  n1/3   p 

⋅β

The value np is the function of the polymerization degree of the polymer only. Therefrom, it follows that the higher the molecular mass of the polymer, the worse is its solubility. 1/ 3 Let np = ξN, where ξ = 0.24⋅10–3 at N = N0 = 104 [94]. Then n1/3 p = 0.13N

and condition of solubility (XII.35) may be presented in the form

523

q≤

2.1

⋅β .

N1/ 6

(XII.36)

Because calculations of all constants in correlation (XII.36) were performed for N = 104, and for real polymers N may display different values, in more general form, expression (XII.36) must be present as follows: 1/ 6

 10 4   q ≤ 1.374  N   

β,

(XII.37)

where 2.1/(104)1/6 = 1.374 is the value of the coefficient at β in expression (XII.37) at N = 104. The value N0 corresponded to 2ρ = 1.374 is obtained from the graphic shown in Figure 94, which is depicted with no regard to molecular masses of the polymers analyzed. Within the framework of the present approach, this value must correspond to some average polymerization degree N0, at which the polymer solubility in various solvents must be estimated experimentally. Because the polymerization degree N of real polymers may differ from N0, condition (XII.37) will finally change to: 1/ 6

N  q ≤ 1.374 0   N 

β.

(XII.38)

Hence, even if the polymer has the globular supermolecular structure, the condition of its solubility depends on the polymerization degree. For the solubility condition to be fulfilled in the form of expression (XII.20), the whole polymer– solvent system must be reduced by molecular weights, i.e. the solubility criterion must be presented in the form:

µ* ≤ 1.374β,

(XII.39)

where µ* = µ(N/N0)1/6 (µ is of the same meaning as before, i.e. µ = δ p2 δ s2 ). To illustrate the influence of the polymerization degree on solubility of polymers, Figure 95 indicates the curve of dependence of µ*/µ on (N/N0)1/6. If the real polymerization degree N < N0, the solubility increases, and vice versa. In some cases, taking into account the polymer molecular mass improves the predictive power of the criterion.

524

Figure 95. Dependence of µ*/µ on (N/N0)1/6.

Up to now, analysis of the solubility criterion was performed for amorphous polymers with the globular supermolecular structure. However, it is common knowledge that the factor as orientation affects solubility of polymers. For example, polyvinyl alcohol is well soluble in water, but fibers formed from this polymer are quite stable in water. That is why the solubility criterion in the form of expression (XII.21) may not be applied to prediction of the solubility of polymeric substances with stretched macromolecules. In this connection, let us discuss the problem of changing of criterion (XII.21) at the transition to the fibril structure. Let us confine ourselves to the case when fibrils touch by the generatrix. In contrast with ‘connecting globules’, connection of the fibril with other fibrils is performed via ‘connecting cylinders’ (oriented polymer), which are considered above in detail. Figure 96 indicates the cross-section of fibrils and the scheme of application of forces in it.

Figure 96. Cross-section of fibrils and schematic representation of surface forces applied.

According to the Laplace law, the force ff clamping down the given fibril in relation to other ones equals f f = 2 Rp Lf

γ s + γ p − γ s-p R

,

(XII.40)

where 2Rp is the distance between connecting cylinders of fibrils (see Figure 96); Lf is the length of the connecting cylinder generatrix, equal to the fibril length. Taking into account that Rp = R/2, the expression for ff will be presented in the following form: ff = Lf(γs + γp – γs-p).

(XII.41)

525

Here γs-p is the coefficient of the interfacial tension of the polymer–solvent type. The force fc applied to the connecting cylinder from the side of the solvent and striving to detach the fibril equals

θ γ s + γ p − γ s-p = πLf γ s + γ p − γ s - p , f c = 2πrf Lf sin ⋅ R 2

(

)

(XII.42)

where rf is the radius of the connecting cylinder; θ = 60° (see Figure 96). The first condition of detachment, according to which fc – ff > 0, has the same form as in the case of the globular structure (see above)

γ s + γ p > γ s-p . This condition is always fulfilled. The second condition of detachment is associated with the work of detachment: p A = ( f c − f f )ε max r,

(XII.43)

p performed by forces of the surface tension of the solvent; ε max is the deformation at break of the connecting cylinder. This work must exceed the energy of disruption of intermolecular bonds. The work of the solvent consumed for disruption of connective cylinders and related to the volume of two connecting cylinders should be presented in the following form:

(δ s* )2 = A (2Vc ) .

(XII.44)

Here Vc is the volume of the connecting cylinder, equal to Vc = πr2Lf. Using expressions (XII.41) and (XII.42), we obtain

(δ s* )2 = δ s2 ρf γ s + γγp − γ s-p ,

(XII.45)

s

where

s (δ s* )2 = 3γ sε max

4πrs ;

r 2(π − 1) εp ρ f = max ⋅ s ⋅ . s r 3 ε max f Consequently, the second condition may be presented in the following form:

( )2

δ s2 ≤ δ s* or

µ ≤ ρ fβ ,

(XII.46)

526

where µ and β are of the same meaning as in the case of the globular supermolecular structure. This condition differs from condition (XII.24) by the value of coefficient ρ only. The relation ρf/ρgl, where ρgl corresponds to the polymer with the globular supermolecular structure, may be presented in the form:

ρ f 2(π − 1) rgl = ⋅ , rf ρ gl 3 where rgl is the radius of the connecting globule; rf is the radius of the connecting cylinder. Because according to estimations made in ref. [94] rgl/rf ~ (1/3)0.5, consequently,

ρf ≈ 0.82. ρ gl As a consequence, the condition of polymer solubility with the fibril structure is stricter than for the globular structure. The same criterion is valuable, but with lower coefficient 2ρf = 1.125:

µ ≤ 1.125β = 1.125Φ Φ − Φ 2 − 1 + a  .  

(XII.47)

Hence, the polymer with the fibrillar supermolecular structure may not dissolve even under the condition of polymer solubility with the globular structure.

Chapter XIII. Surface properties of organic liquids and polymers

It is important to know the surface tension value and be able to predict it on the basis of the chemical structure of the substance is important, because many individual properties of liquids and solids, as well as miscibility and solubility of them in each other, depend on the surface tension of liquids and surface energy of solids. Among existing methods of calculation of surface tension, the most wide-spread is the additive scheme based on summing up of parachors which characterize the contribution of separate atoms to surface tension. The calculation is performed by the formula  ∑ Pi  P γ =  = i V   V  4

4

   ,  

(XIII.1)

where P is the total parachor of the molecule (or the repeat unit of the polymer); Pi are parachors typical of every atom. Values of atomic parachors Pi, as well as constants which characterize contribution of various types of bonds (double, triple) and cycles (three-component, four-component, etc.) to the general parachor of the substance, are indicated in Table 45. Table 45 Values of atomic parachors Pi and parachors for a series of constants Pj Constants Pj Atom Pi C 4.8 Double bond 23.2 H

17.1

O

20.0

O2*

60.0

N S F

12.5 48.2 27.5

Cl Br

54.3 68.0

I 91.0 * For two oxygen atoms participating in the ester

Triple bond

46.4

Three-component cycle

16.7

Four-component cycle

11.6

Five-component cycle

8.5

Six-component cycle

6.1

C O

group.

O

Formula (XIII.1) enables the value γ to be calculated with high accuracy for organic liquids and polymers. However, the physical meaning of values Pi participating in correlation (XIII.1) is not clear yet. This makes it difficult to analyze the influence of the weak Van-der-Waals interaction and strong polar groups on formation of some surface properties of organic liquids and polymers. This problem is quite important, because analysis of such an influence on the quantitative level allows

528

regulation of the chemical structure of substances to ensure that they have the necessary surface properties.

XIII.1. Surface tension of organic liquids Let us consider the approach to analysis of influence of the chemical structure on surface tension, suggested in ref. [34]. This analysis was based on the following ideas. It is common knowledge that molecules of a liquid located on the surface participate in an intermolecular interaction, different from that in the volume. If the volumetric problem is considered (Figure 97), then at hexagonal packing of spherical molecules present in the volume the coordination number equals 12, and on the surface – 9, i.e. their ratio is 4/3. As the free energy decreases with the coordination number increase, molecules strive to transit from the surface layer into the volume. That is why the formation of a unit of new surface require consumption of energy.

Figure 97. Schematic representation of arrangement and interaction of molecules in the vertical (a) and horizontal (b) planes.

Let us calculate first the number of molecules n located on the surface unit. It may be easily displayed [34] that at hexagonal packing (see Figure 97) the value n per 1 cm2 of surface will be n=

1016 ⋅ 0.2887 r2

,

(XIII.2)

where r is the molecule radius, expressed in Å. Therewith,  ∑ ∆Vi    r = i ⋅ 3  4   

1/ 3 1/ 3

  = 0.6204 ∑ ∆Vi     i 

,

(XIII.3)

where ∆Vi are Van-der-Waals volumes of atoms participating in the molecule composition. With regard to correlations (XIII.2) and (XIII.3), we obtain that

529

n=

1016    ∑ ∆Vi     i 

2/3

⋅ 0.75 .

(XIII.3)

Let us calculate the cohesive energy e* accounted for one molecule. According to the approach for estimating the solubility parameter (see above), the

∑ ∆Ei*

cohesive energy accounted for one mole of the substance equals

k , where

i

∆Ei* are molar constants displaying the meaning of the energy of intermolecular interaction for each type of atoms and separate polar groups participating in the molecule or the repeat unit of the polymer (values of ∆Ei* are shown in Table 43); k is the coefficient of molecular packing in the volume of the substance considered. Then

∑ ∆Ei* ∑ ∆Ei*

e* = i kN A

= i ⋅ 0.166 . k ⋅ 1016

In formula (XIII.5), the value e* is expressed in ergs, and

(XIII.5)

∑ ∆Ei*

– in Joules.

i

The cohesive energy γ*, which would be possessed by all molecules in the volume located in the surface layer of 1 cm2 square equals ne* and, according to correlations (XIII.4) and (XIII.5), equals

γ * = 0.125

∑ ∆Ei* i

  k  ∑ ∆Vi*     i 

2/3

.

(XIII.6)

The value of γ* significantly exceeds the value of surface tension, because the formation of a specific new surface requires overcoming of not the whole cohesive energy, but just a part of it, because in the surface layer the molecule possesses the coordination number different from zero. Figure 97b indicates that approximately 1/4 part of the cohesive energy is consumed for the molecule yield from the volume to the surface, because in this case the coordination number changes from 12 to 9. Moreover, it should be taken into account that packing of molecules in the surface layer is not perfect, as it is observed from Figure 97b. Let us designate the true coefficient of molecules packing in the surface layer as ksrf. For future calculations, let us determine the coefficient of packing ksrf,p in the surface layer at the ideal hexagonal packing of spheres. To determine the maximum possible occupied volume Vocc, let us multiply the number of molecules n in the specific surface layer by the volume of a single molecule:

530

Vocc =

1016 r

2

4 ⋅ 0.2887 πr 3 = 1.2086⋅1016 Å3 3

(r is expressed in ångströms). The total volume of the surface layer Vtot = 2⋅1016 Å3, and then ksrp,p = Vocc/Vtot = 0.604. Hence, the coefficient of packing ksrf,p at the ideal packing of spheres in the surface layer equals 0.604, whereas the appropriate coefficient kvol,p in the volume equals 0.740. Consequently, the part of molecules present on the surface equal ksrf/0.604, where ksrf is the true coefficient of molecular packing in the surface layer. Then for the value γ*, we obtain

γ * = 0.125

ksrf ∑ ∆Ei* i

  0.604 ∑ ∆Vi     i 

2/3

.

(XIII.7)

Let us now take into account the part of the cohesive energy which must be overcome for the molecule to transit to the surface from the volume. This part α stipulated in this transition by the coordination number change, will be determined from the correlation:

α=

12

k k − 9 srf 0.740 0.604 = 1 − 3 ksrf ⋅1.225 . k 4 k 12 0.740

(XIII.8)

Then, for the surface tension of liquids, we obtain:

γ s = αγ * =

0.207ksrf ∑ ∆Ei* i

  k 0.74  ∑ ∆Vi*     i 

 0.919ksrf  1 − . k 

2/3 

(XIII.9)

Basing on correlation (XIII.9), ref. [34] indicates the calculation of values ksrf/k for a great number of organic liquids of various nature. Initial data and results of this calculation for representatives of various classes of liquids are indicated in Table 46. Clearly, for all organic liquids the value ksrf/k < 1. Therewith, it was found that for many organic liquids, especially for unitypical ones, for example, for hydrocarbons, alcohols, etc., the value ksrf/k is approximately constant. The calculations performed indicate [34] that for hydrocarbons, perfluorocompounds, halogen-containing compounds, aldehydes, ketones and nitrocompounds the average value (ksrf/k)avg = 0.851. For alcohols, amides, and acids, (ksrf/k)avg = 0.954, and for nitriles (ksrf/k)avg = 0.910. The values of the coefficient of molecular packing in the surface layer are also displayed in Table 46.

531

Because for some series of organic liquids the value ksrf/k plays the role of a constant, then correlation (XIII.9) may be reduced to the form:

γs = A

∑ ∆Ei* i

   ∑ ∆Vi*     i 

2/3

,

(XIII.10)

where A = 0.207⋅(ksrf/k)⋅(1 – 0.919⋅ksrf/k). Taking into account the above-displayed values of ksrf/k, the following expression for calculation of the surface tension of organic liquids may be presented:

γ s = Aj

∑ ∆Ei* i

   ∑ ∆Vi*     i 

2/3

.

(XIII.11)

For hydrocarbons, perfluorocompounds, halogen-containing compounds, esters, aldehydes, ketones and nitrocompounds (group I), A1 = 0.0287; for alcohols, amides, and acids (group II), A2 = 0.0181; for nitriles (group III), A3 = 0.0229. Recall that ∆Ei* is expressed in Joules, ∆Vi – Å3, and then, with indicated coefficients Aj, the value γs is expressed in dyn/cm. Table 46 Initial data and results of calculation of the surface tension of a series of organic liquids 2/3 γ, dyn/cm ∑ ∆Ei*,  ∑ ∆Vi*  , k ksrf Compound i   Exper. Calc.  i  J/mol 3 Å 1 2 3 4 5 6 7 Hydrocarbons n-Pentane 21.30 0.510 0.453 16.03 17.960 13,903 Isopentane 12,180 21.40 0.510 0.447 15.00 17.330 n-Hexane 23.70 0.527 0.462 18.41 19.895 16,603 Cyclohexane 16,206 21.90 0.567 0.443 25.12 23.760 n-Heptane 19,303 25.90 0.541 0.469 20.21 21.604 n-Octane 22,003 28.20 0.549 0.472 21.75 22.700 Isooctane 20,281 28.20 0.545 0.479 18.85 21.710 n-Nonane 30.30 0.558 0.479 22.91 23.940 24,704 n-Decane 27,404 32.30 0.571 0.489 23.92 25.900 n-Tetradecane 38,209 39.90 0.589 0.506 26.96 28.770 Benzene 19,186 19.80 0.597 0.501 28.78 29.760 Toluene 20,687 22.30 0.592 0.489 28.53 28.680 n-Xylene 23,391 24.70 0.610 0.960 30.03 30.650 m-Xylene 23,391 24.70 0.600 0.515 26.63 29.740 o-Xylene 23,391 24.70 0.595 0.499 28.31 29.250 Ethylbenzene 23,391 24.60 0.598 0.514 29.04 29.599 Isopropylbenzene 24,369 26.80 0.596 0.490 28.20 28.790

532

1 n-Perfluoropentane n-Perfluoroheptane n-Perfluorooctane Dichloromethane Chloroform Tetrachlorocarbon Ethyl chloride 1,1-Dichloroethane 1,2-Dichloroethane 1,1,1-Trichloroethane Tetrachloroethane Isobutyl chloride cis-Dichloroethylene trans-Dichloroethylene Trichloroethylene Tetrachloroethylene Chlorobenzene Bromobenzene Iodobenzene Bromoform Ethyl bromide 1,2-Dibromoethane Ethyliodide Glycol Glycerol Cyclohexanol Methanol Ethanol n-Propanol n-Butanol Isobutyl alcohol n-Amyl alcohol n-Hexanol n-Octanol Acetic Butyric Isovaleric n-Valeric Methyl formiate Ethyl formiate Methyl acetate Ethyl acetate n-Propyl acetate Isopropyl acetate Ethyl propyonate n-Butyl acetate Amyl acetate Ethyl isovaleate Diethyl carbonate Benzaldehyde Acetaldehyde

2

3 4 Perfluorocompounds 12,724 29.50 0.574 15,228 32.80 0.551 17,732 35.90 0.571 Halogen-containing compounds 14,408 15.14 0.554 13,275 17.60 0.555 12,147 19.87 0.553 11,453 15.52 0.527 17,109 17.94 0.544 17,109 17.94 0.579 15,976 20.24 0.555 28,416 22.40 0.607 15,132 20.96 0.550 15,357 17.20 0.571 15,357 17.20 0.563 18,313 19.57 0.582 21,268 21.70 0.605 23,642 21.80 0.605 27,011 23.00 0.634 31,688 24.60 0.661 23,379 20.73 0.650 14,822 16.70 0.542 23,843 20.05 0.628 19,491 18.41 0.590 Alcohols 39,840 15.50 0.659 59,557 19.41 0.705 33,443 22.96 0.658 20,118 11.02 0.544 22,819 14.10 0.565 25,511 17.20 0.563 28,219 19.77 0.578 26,497 19.77 0.573 30,919 22.30 0.587 33,624 24.70 0.590 39,024 21.90 0.597 Acids 23,015 14.60 0.589 28,420 20.00 0.587 29,398 22.80 0.597 31,120 22.50 0.519 Esters 13,376 14.90 0.567 16,076 17.80 0.565 16,076 17.80 0.565 18,781 20.40 0.565 21,481 22.87 0.573 19,758 22.90 0.567 21,481 22.87 0.573 24,181 25.20 0.577 26,882 27.40 0.584 25,159 27.00 0.570 22,075 24.20 0.593 Aldehydes and ketones 27,672 22.50 0.636 12,782 13.20 0.509

5

6

7

0.523 0.478 0.493

9.87 12.60 13.60

16.904 17.330 18.680

0.458 0.424 0.350 0.423 0.475 0.455 0.440 0.518 0.458 0.468 0.484 0.472 0.481 0.479 0.522 0.558 0.609 0.470 0.497 0.535

28.12 27.16 26.75 22.18 24.75 32.23 25.77 36.04 21.99 28.00 25.00 29.50 32.26 33.19 36.34 37.65 31.68 24.15 38.91 28.10

24.800 23.600 20.900 20.500 23.800 28.010 23.700 34.230 22.160 26.090 25.195 28.600 30.720 32.600 36.790 41.753 38.000 22.520 36.800 30.400

0.625 0.660 0.576 0.539 0.558 0.547 0.562 0.564 0.559 0.564 0.557

46.49 62.29 34.37 22.55 22.32 23.70 24.57 22.98 25.68 24.48 27.53

51.900 71.100 44.600 33.100 23.600 23.930 29.200 28.300 30.820 31.460 32.840

0.566 0.556 0.563 0.555

27.42 26.96 25.31 27.35

30.450 30.700 32.640 31.680

0.490 0.495 0.490 0.499 0.505 0.498 0.505 0.506 0.511 0.504 0.502

24.62 23.84 24.49 23.75 24.28 22.35 24.27 25.20 25.88 23.77 26.44

25.696 25.510 25.510 25.650 26.770 25.350 26.770 27.430 28.470 26.350 28.550

0.513 0.468

39.70 23.32

37.890 24.900

533

1 Acetone Methylethylketone Diethylketone Cyclohexanon

2 15,483 18,183 20,883 23,186

Diethyl ether 1,4-Dioxane

11,796 25,564

Aniline Pyridine Formamide

30,008 17,514 31,739

Acetonitrile Propyonitrile n-Valeonitrile Capronitrile Methacrylonitrile

17,021 19,725 25,126 27,826 20,674

Nitromethane Nitroethane 2-Nitropropane Nitrobenzene

15,913 18,613 19,592 31,199

3 16.30 18.98 21.50 22.10 Ethers 20.20 19.70 Amines, amides 21.50 18.50 12.30 Nitriles 13.50 16.40 21.70 24.00 18.50 Nitrocompounds 12.86 15.88 18.70 22.30

4 0.534 0.556 0.570 0.575

5 0.478 0.491 0.503 0.450

6 24.60 25.18 25.18 35.62

7 21.299 24.690 26.660 27.850

0.521 0.614

0.411 0.540

16.49 33.70

18.270 35.590

0.658 0.595 0.640

0.541 0.403 0.585

43.31 37.25 58.35

43.230 29.700 43.700

0.565 0.565 0.589 0.592 0.574

0.514 0.546 0.532 0.533 0.529

29.10 27.25 27.44 27.87 24.40

26.210 26.390 30.600 31.080 27.990

0.518 0.534 0.547 0.615

0.428 0.433 0.471 0.506

36.98 32.06 29.12 43.35

35.500 33.600 32.900 39.700

Calculations performed by formula (XIII.11) indicate quite good agreement of the calculated and experimental values of γ, deviations being approximately the same as in calculations with the help of parachors. Correlation (XIII.11) enables the contribution of separate groups and specific intermolecular interaction to the surface tension to be estimated. For example, usually, of interest is the contribution of hydrogen bonds made in formation of the surface properties of organic liquids. Let us estimate such contribution on the example of alcohols and acids. For this purpose, let us transform correlation (XIII.11) to the following form:

γs =

where

∑ ∆Eih*

  * *   ∑ ∆Eih + ∑ ∆Eid , 2/3     i i    ∑ ∆Vi*     i  0.0181

is the contribution of hydrogen bonds to cohesive energy;

i

∑ ∆Eid* = ∑ ∆Ei* − ∑ ∆Eih* is the contribution of weak dispersion interaction to i

i

i

cohesive energy. For ethanol,

∑ ∆Ei*

* * * = 2 ∆EC + 6 ∆EH + + ∆EO

i

5,460 cal/mol = 22,820 J/mol, and

∑ ∆Eih*

= 2⋅550.7 + 6⋅47.7 + 142.6 + 3929 =

i

∑ ∆Eih*

= 16,423 J/mol. Then a part of the surface

i

tension stipulated by hydrogen bonding will be determined as

534

γ sh =

0.0181

∑ ∆Eih*

= 20.9 dyn/cm.   i  ∑ ∆Vi*     i  The contribution associated with weak dispersion interaction gives 2/3

γ sq =

0.0181    ∑ ∆Vi*     i 

2/3

∑ ∆Eid*

= 8.1 dyn/cm.

i

Total surface tension γs = γsh + γsd = 29 dyn/cm, and the contribution of each component equal 72.0 and 28.0%, respectively. Analogously, for acetic acid γsh = 20.5, and γsd = 8.2 dyn/cm; the contribution of each component equals 71.0 and 29.0%, respectively. This analysis may be performed not only for hydrogen bonds, but also for specific interactions of other types. It should be noted that according to the Fowkes method [143] significantly different contributions of hydrogen bonding to the total value of surface tension are obtained. For example, for formamide γsh = 19 and γsd = 58.2 dyn/cm [143]. The part of the surface tension stipulated by hydrogen bonding is 33%. According to calculations performed with the help of data from Table 46, for

∑ ∆Eih*

formamide

= 16,423 and

i

∑ ∆Eih* ∑ ∆Ei* ≈ i

∑ ∆Ei*

= 31,739 J/mol. Then this part is

i

0.52 or 52%, which is significantly higher than that obtained by

i

Fowkes. Besides estimation of the surface tension γ of organic liquids, of interest is the relationship of the value γ with the specific cohesive energy of the liquid δ2 or solubility parameter δ. This problem has been discussed many times in the literature; there are empirical correlations, which allow setting of this connection. For example, the correlation is known [122]

δ = 4.1(γ/V1/3)0.43,

(XIII.12)

where V is the molar volume of the liquid. The physical meaning of this correlation is not clear and, moreover, it does not describe the properties of all organic liquids. For example, calculation of δ for alcohols and organic acids by this correlation leads to significantly reduced values. With the help of correlation (XIII.11), the dependence connecting the solubility parameter value with the surface tension may be deduced. For this purpose, let us use correlation (XII.3), according to which solubility parameter δ is calculated. First, correlation (XIII.11) is transformed by multiplying the numerator and 1/ 3

  denominator by  ∑ ∆Vi     i 

. Then, the following expression is obtained

535

∑

γs = A i

1/ 3



 ∑ ∆Vi   i  ∆ V ∑ i

∆Ei*  

.

(XIII.13)

i

Substituting correlation (XII.3) into formula (XIII.13), we obtain 1/ 3

  γs = A⋅0.6023  ∑ ∆Vi     i 

δ2 .

(XIII.14)

In correlation (XIII.14), values of ∆Vi are expressed in ångströms, δ2 – in J/cm . Taking into account that A displays several different values for various different groups of organic liquids, we obtain the correlation: 3

1/ 3

  γs = Bj  ∑ ∆Vi     i 

δ2 ,

(XIII.15)

where Bj = 0.6023⋅Aj. For liquids from the group I, B1 = 0.0172, from the group II – B2 = 0.0109, from the group III – B3 = 0.0138. To determine the value γ by correlation (XIII.15), both the calculated and experimental values of δ may be used. The results of calculation of γ performed with application of experimental values of δ are shown in Table 46. Clearly, the experimental values γexper and calculated values γcalc coincide quite well. The opposite task (which is more important) may also be solved, i.e. values of the solubility parameters δ may be calculated from the experimental value of surface tension. For this purpose, let us transform correlation (XIII.15) to the following form (with regard to the fact that N A ∑ ∆Vi = Vk ): i

1/ 2

δ=

(0.6023)1 / 6  γ s    B1j / 2 ⋅ k 1 / 6  V 1 / 3 

.

(XIII.16)

Clearly, correlation (XIII.16) by its form is similar to empirical formula (XIII.12). However, the multiplicand before the value (γs/V1/3) is not constant but depends upon the chemical structure of organic liquids, because the coefficient of packing k depends on it. In the first approximation, it may be assumed that for groups of organic compounds considered above the values of k vary in a narrow range, and their average value may be applied. The calculations performed indicate that for the group I of organic liquids kavg,1 = 0.580, for the group II – kavg,2 = 0.601, and for the group III – kavg,3 = 0.586. Then basing on formula (XIII.16), we obtain:

δ

1/ 2 * γ s   , = C j  1/ 3 

V



(XIII.17)

536

where C1* = 7.67; C2* = 9.58; C3* = 8.56.

XIII.2. Surface tension of polymers Let us now turn to calculations of the surface tension of solid polymers. At first glance, it seems that the surface tension of polymers may be calculated directly from formula (XIII.11), where

∑ ∆Ei*

and

i

∑ ∆Vi

are determined per repeat unit of

i

the polymer. However, some difficulties appear in this case associated with both chain structure of the polymer and somewhat ambiguity of interpretation of the repeat unit. Because the value ∑ ∆Vi participates in this correlation in the degree 2/3, and the value

∑

i * ∆Ei – in the first degree, then the more the number of chains n is considered

i

as a kinetic unit (the length of the whole chain, the segment length), the higher value of γ will be obtained by formula (XIII.11). The second reason is clear from the example of polyethylene and other polymers. If for polyethylene the repeat unit is assumed in the form (CH2CH2), then from correlation (XIII.11) we obtain γ = 14.7 dyn/cm. But if the repeat unit is assumed in the form (CH2), then γ = 11.65 dyn/cm. The same result is also obtained in the case of polyamides. For example, for polyamide 6 (CH2)5

C NH O

n

and polyamide 6,6 (CH2)4

C NH

(CH2)6

NH C

O O n when formula (XIII.11) is applied, we obtain γ = 45.9 dyn/cm (polyamide 6) and γ = 58.0 dyn/cm (polyamide 6,6). From the point of view of the gross formula, the chemical structure of these polyamides is identical (polyamide 6,6 contains twice the number of the same elements, which are contained in polyamide 6). Experimental values of γ for these polyamides are approximately equal and vary in the range from 40 to 47 dyn/cm. The above-discussed problems are analyzed in detail in ref. [33]. Consequently, the following correlation is obtained:

γp =Cj

∑ ∆Ei* i

   ∑ ∆Vi     i 

,

2/3

(XIII.18)

m1 / 3

where m is the number of atoms in the repeat unit of polymer. For non-polar polymers of the group I (hydrocarbons, perfluoropolymers, polyethers), C1n-p = 0.1277; for

537

polar polymers of the group II (polyesters, polymers containing a nitrogroup, etc.), C1p = 0.0751; for polymers containing alcohol, acidic and amide groups (all of them are polar and indicate hydrogen bonding), C2 = 0.0476; for polymers containing nitrile groups (all of them are polar), C3 = 0.0600. If a polymer contains aromatic cycles (for example, polystyrene, polyphenylene, etc.), the value C4 = 0.1014. Calculations performed due to formula (XIII.18) display (Table 47) that in the most of cases, good coincidence of experimental and calculated values of the surface tension γp is observed. The greatest deviations are observed for polymers which crystallize easily; to them polyethylene oxide and polyformaldehyde, for which the coefficient of molecular packing exceeds significantly the average value. Table 47 Initial data and results of calculation of the surface tension of polymers γp, by eqs. γp, by eq. γp, exper. ∆Ei* ∑ (XIII.28) and ∆ V ∑ i [133, 214] (XIII.18) Polymer i (XIII.29) i dyn/cm Polyethylene 5,401 34.2 31.0-35.7 36.0 35.6 Polypropylene 14,885 51.3 29.0-29.6 38.8 39.6 Polyisobutylene 10,801 68.6 27.0-33.6 28.8 34.8 Rubber SCB 9,054 64.3 32.0 33.5 32.9 Rubber SCI 10,030 81.5 31.0 29.0 28.6 Polyvinyl fluoride 12,087 39.3 28.0-36.7 43.2 44.3 Polyvinyl chloride 11,055 49.0 39.0-41.5 34.2 36.3 Polyvinyliden fluoride 5,205 44.6 23.0-32.7 29.1 29.0 Polytrifluoroethylene 11,891 49.7 22.0-23.9 36.3 38.9 Polychlorotrifluoroethylene 10,761 65.7 31.0 27.4 30.7 Polytetrafluoroethylene 5,008 55.0 18.5-19.0 24.5 18.8 Polystyrene 22,990 109.7 33.0-43.0 40.5 38.0 Polyvinyl alcohol 23,675 41.6 37.0 46.5 61.0 Polyacrylonitrile 19,324 54.0 44.0 42.6 50.0 Polymethylacrylate 18,379 79.5 41.0 32.6 33.8 Polyamide II 51,562 201.2 31.0-33.0 34.9 31.4 Polyamide 6 38,059 116.0 40.0-47.0 45.0 39.5 Polyamide 6,6 76.118 231.5 39.3-46.0 45.2 35.1 Polyamide 10,10 97,723 368.3 28.5-32.0 36.1 29.1 Polyethyleneterephthalate 43,547 169.9 41.0-47.0 38.1 35.2 Polyethylacrylate 21,080 96.7 35.0 30.4 31.6 Poly(methyl methacrylate) 21,080 96.7 39.0-40.2 30.4 31.6 Poly(ethyl methacrylate) 23,780 113.8 33.0 29.1 30.0 Poly(vinyl acetate) 18,379 79.6 36.0 32.6 33.8 Polyformaldehyde 3,297 27.2 36.0 29.1 36.0 Poly(ethylene oxide) 5,997 43.8 43.0 32.1 43.0 Poly(propylene oxide) 8,698 60.9 32.0 33.2 31.9

For polymers, it is desirable to calculate the surface energy γp with the help of solubility parameter δ. Clearly, from equation (XII.3) we obtain:

∑ ∆Ei* = δ 2 N A ∑ ∆Vi . i

i

Substituting formula (XIII.19) into equation (XIII.18), we obtain

(XIII.19)

538

1/ 2

   ∑ ∆Vi  δ 2 N A   i  γp =Cj  m1 / 2

(XIII.20)

or

γ p = D jδ

∑ ∆Vi

2

i

m

,

(XIII.21)

where Dj = CjNA (Dj is the parameter which, similar to Cj, depends on affiliation of the polymer to the given group). With regard to values of Cj shown above, it may be stated that for non-polar polymers (hydrocarbons, polyethers, etc.), D1n-p = 0.0769. For polar polymers containing ester and other polar groups, D1p = 0.0452. For polymers containing hydroxyl, acidic and amide groups (strong hydrogen bonding), D2 = 0.0287; for polymers containing nitrile groups, D3 = 0.0361. For polymers containing aromatic cycles only (for example, polystyrene, polyphenylene, etc.), D4 = 0.061. Therewith, if the Van-der-Waals volume is expressed in Å3, and the specific cohesive energy δ2 in J/cm3, then the surface tension calculated by formula (XIII.21) is determined in dyn/cm. The drawback of the calculation scheme described above concludes in the fact that transition from one sequence of liquids or from one group of polymers to another requires application of the own coefficient Aj or Cj in equations (XIII.11) and (XIII.18). Difficulties in this case are associated with the fact that molecules of the liquid or repeat units of polymers will possess larger volumes, and the relative fraction of specific groups, by which they are separated according to the current feature, will be small. For example, nitrile groups may be contained not only in polyacrylonitrile, where their relative contribution to the Van-der-Waals volume and the cohesive energy will be high, but also in other polymeric systems, where their contribution may be rather low. Moreover, difficulties appear in the calculation of the surface energy of copolymers, units of which belong to different classes of homopolymers. In this connection, ref. [37] indicates development of the calculation scheme for estimating the surface tension of organic liquids and polymers, parameters of which depend on the chemical structure of the organic liquid or polymer only, but not on their belonging to any class. In the work cited, the value of kp is expressed via the correlation:

δ* =

∑ ∆Ei* i

   ∑ ∆Vi     i 

2/3

.

(XIII.22)

Here δ* is the specific surface cohesive energy, i.e. the cohesive energy per specific surface. Note that this value differs from the usual density of the cohesive energy determined by correlation (XII.3). The following dependence connects the coefficient of molecular packing of the liquid on its surface kp and the specific surface cohesive energy δ* [37]:

539

Ksrf = –0.4112 + 0.3012 lgδ*.

(XIII.23)

After substituting expression (XIII.9) into formula (XIII.23), we obtain

γs =δ*

0.0461⋅ lg δ * − 0.063  0.2768 ⋅ lg δ * − 0.378  ⋅ 1 − . k k  

(XIII.24)

Substituting expression (II.5) into formula (XIII.24), finally we get:  (0.2768 ⋅ lg δ * − 0.378) M  ⋅ 1 − . ∆ N V ρ ∑ i A   i   

γs =δ

* * (0.0461⋅ lg δ − 0.063) M 

ρN A ∑ ∆Vi i

(XIII.25)

Correlation (XIII.25) enables the surface tension of organic liquids to be estimated with accuracy valuable for practical calculations independently of the class, to which the current liquid belongs. Results of calculation indicate [37] that the calculated and experimental data are in satisfactory agreement in most cases (see Table 46). Let us now turn to estimation of the surface energy of polymers. The initial correlation for deducing equation (XIII.18) has the form [37]: k  k δ*  γ p = α ⋅ 0.1532 srf ⋅ 1 − 0.919 srf  , k  k m1 / 3 

(XIII.26)

where α is the constant dependent on the polymer belonging to the class of polar or non-polar polymers only. Let us assume that the dependence of the coefficient of molar packing in the surface layer ksrf on δ* is the same as for liquids. As often mentioned above, the coefficient of molecular packing of the polymer in the volume is almost independent of the chemical structure of polymers, and its average value kavg = 0.681 for solid polymeric substances. Using the dependence ksrf on δ* in the form of correlation (XIII.23) and kavg = 0.681, basing on correlation (XIII.23) we obtain:

γp =α

δ* 1/ 3

m

(0.067761⋅ lg δ * − 0.0925)(1.5549 − 0.40671⋅ lg δ * ).

(XIII.27)

The analysis performed indicates [37] that values of α are indeed approximately equal; for polar polymers, αavg = 2.097, and for non-polar polymers – αavg = 3.055. Then, the formula for calculation of the surface energy of polar polymers changes to:

γ pp =

δ* 1/ 3

m

(0.1421⋅ lg δ * − 0.194)(1.5549 − 0.4067 ⋅ lg δ * ),

(XIII.28)

540

and for non-polar polymers,

γ pn =

δ* 1/ 3

m

(0.2070 ⋅ lg δ * − 0.2826)(1.5549 − 0.4067 ⋅ lg δ * ).

(XIII.29)

The surface energy values calculated from formulae (XIII.28) and (XIII.29) are displayed in Table 47. Clearly, in most cases, good correspondence between the calculated and experimental surface energy values is observed. It should be noted that experimental determination of the surface energy for solids, including polymers, is rather difficult, although there are various modifications of the method for estimating γp. That is why greatly different data on the surface energy for the same polymer may be observed in the literature. Correlations (XIII.28) and (XIII.29) deduced in ref. [37] enable the surface energy of polymers of any chemical structure to be estimated regardless of their belonging to any class of chemical compounds. This removes the difficulties typical of calculation of the surface tension of organic liquids, mentioned above. In conclusion, let us discuss one more connection between the surface energy of organic liquids and polymers and the solubility parameter δ, the value of which is determined from relation (XII.3). Taking into account that δ * =

∑ ∆Ei* i

   ∑ ∆Vi     i 

2/3

and substituting this expression

into equation (XII.3), we obtain that 1/ 3

  δ = δ N A  ∑ ∆Vi     i  *

2

.

(XIII.30)

Furthermore, substituting expression (XIII.30) into formula (XIII.29) and performing necessary transformations, we get: 

   ∆  − V 0 . 073 ∑ i   i  ×    

 1 / 3  0.09221lg δ − 0.01537 lg

  γ s = N Aδ 2  ∑ ∆Vi     i 

    

k

. (XIII.31)      0.554 lg δ − 0.0923 lg ∑ ∆Vi  − 0.439       i  × 1 −  k       Correlation (XIII.31) links value of the surface tension of liquids γs with the solubility parameter δ. According to the correlation mentioned, the value δ may be obtained if experimental values of the surface tension of liquids are known

541

(estimation is performed by the method of inspection). This is of importance because the surface tension of liquids can be measured quite easily, and the solubility parameter associated with the latent heat of evaporation of the liquid is more difficult to calculate. It is especially difficult for polymers, because their solubility parameter may be determined by indirect methods only – by measurements of swelling in different solvents, viscosity of solutions, etc. It should be noted that the possibility of estimating the surface energy of polymers experimentally is also important, because their surface energy is related by simple correlations with cohesive energy and sublimation energy. Various situations may appear when calculating the surface energy of copolymers or homogeneous mixtures. If components of a copolymer relate to the same group of substances (according to the above-mentioned classification), then, as the value Cj is identical for them, correlation (XIII.18) applied to copolymers obtains the form:

γp =Cj

    α1 ∑ ∆Ei*  + α 2  ∑ ∆Ei*  + ... +      i 1  i 2         α1 ∑ ∆Vi  + α 2  ∑ ∆Vi  + ... + α n  ∑ ∆Vi           i 1  i 2  i n     + α n  ∑ ∆Ei*     i n → , × (α1m1 + α 2 m2 + ... + α n mn )1 / 3

→

2/3

×

,

(XIII.32)

  where α1, α2, …, αn are molar parts of the components 1, 2, …, n;  ∑ ∆Ei*  ,    i 1      ∑ ∆Ei*  , …,  ∑ ∆Ei*  are values of the cohesive energy for the components 1,      i n  i 2       2, …, n;  ∑ ∆Vi  ,  ∑ ∆Vi  , …,  ∑ ∆Vi  are their Van-der-Waals volumes;        i n  i 1  i 2 m1, m2, …, mn are numbers of atoms in repeat units of the components 1, 2, …, n. In the reduced form, correlation (XIII.32) gives

γp = Cj

k =n



k =1

 i



∑ α k  ∑ ∆Ei* 

k

. (XIII.33) 1/ 3 k = n     k =n   ∑ α k  ∑ ∆Vi    ∑ α k mk      k =1  i  k   k =1   If it is desirable to express the surface energy of a copolymer via the surface energies of homopolymers γp,1, γp,2, …, γp,n obtained from components 1, 2, …, n, then equation (XIII.32) is reduced to the following form: 2/3

542

2/3

γp = Cj

  α1γ p,1 ∑ ∆Vi     i 1

2/3



 ∑ ∆Vi  m12/ 3 + ... +  i 2

m11 / 3 + α 2γ p,2  

        α1 ∑ ∆Vi  + α 2  ∑ ∆Vi  + ... + α n  ∑ ∆Vi           i 1  i 2  i n  

2/3

×

→ , (XIII.34)

2/3

→

  + α nγ p, n  ∑ ∆Vi  m1n/ 3    i n

× (α1m1 + α 2 m2 + ... + α n mn )1 / 3

.

In the reduced form, correlation (XIII.34) looks as follows: 2/3

k =n

γp = Cj

  ∑ α k γ p, k  ∑ ∆Vi  k =1  i k

k = n     ∑ α k  ∑ ∆Vi      k =1  i k  

2/3

m1k / 3 1/ 3

.

(XIII.35)

 k =n   ∑ α k mk     k =1 

If all components of the copolymer relate to different groups of substances (they possess different values of Cj), then correlation (XIII.32) for this copolymer (on the example of a double copolymer) is presented in the form:

(

γ p = α1C j , I + α 2C j , II

)

    α1 ∑ ∆Ei*  + α 2  ∑ ∆Ei*       i 1  i 2       α1  ∑ ∆Vi  + α 2  ∑ ∆Vi        i 1  i  2  

(XIII.36)

2/3

(α1m1 + α 2 m2 )1 / 3

or 2/3

(

γ p = α1C j , I + α 2C j , II

)

  α1γ p,1 ∑ ∆Vi     i 1

m11 / 3 C j , I +

      α1 ∑ ∆Vi  + α 2  ∑ ∆Vi         i 1  i 2  

→

2/3

×

(XIII.37)

2/3

→

  + α 2γ p,2  ∑ ∆Vi     i 2

m12/ 3 C j , II

. × (α1m1 + α 2 m2 )1 / 3 In the case of a multicomponent copolymer, various situations may appear: a part of components may belong to the same group of substances, and the rest belongs to another group. In the general case, we obtain:

543

Cj = α1Cj,1 + α2Cj,2 + … + αnCj,n,

(XIII.38)

where Cj,1 is the value of Cj for the group of substances, to which component 1 belongs (it may be Cj,I, Cj,II, Cj,III, or Cj,IV); Cj,2, …, Cj,n is the same for components 2, …, n. Then, with regard to equation (XIII.38), correlation (XIII.33) obtains the form:

γp =

k =1



k =1

 i



∑ α k  ∑ ∆Ei* 

k =n

∑α k C j,k

k =n

k = n     ∑ α k  ∑ ∆Vi     k =1  i  k  

k

2/3

1/ 3

 k =n

,

(XIII.39)

  ∑ α k mk     k =1 

and correlation (XIII.35) is reduced to 2/3

k =n

γp =

k =n

∑α k C j,k

k =1

  ∑ α k γ p, k  ∑ ∆Vi  k =1  i k

k = n     ∑ α k  ∑ ∆Vi      k =1  i k  

2/3

m1k / 3 C j , k 1/ 3

.

(XIII.40)

 k =n   ∑ α k mk     k =1 

The above-described correlations for calculation of the surface energy of copolymers are not quite suitable, because affiliation of the polymer to some group must be taken into account every time. That is why, for computer realization of the method, it is desirable to express values Cj via the surface energies of copolymer components. From equation (XIII.18) we obtain that 2/3

  γ p  ∑ ∆Vi  m1 / 3    Cj =  i . ∑ ∆Ei* i

Then for the copolymer, we obtain: 2/3 2/3       γ  ∆V  m1 / 3   m12/ 3 γ p2 ∑ ∆Vi i p1  ∑ 1     i 1  i 2 + α2 + ... + γ p = α1      * *  ∑ ∆Ei   ∑ ∆Ei         i 1  i 2 

544

2/3          m1n/ 3  α1 ∑ ∆Ei*  + α 2  ∑ ∆Ei*  + γ p, n ∑ ∆Vi         i n 1  i 2 ⋅  i → αn          ∑ ∆Ei*   α1 ∑ ∆Vi  + α 2  ∑ ∆Vi  +         i  i n 1  i 2  

→

  + ... + α n  ∑ ∆Ei*     i n    + ... + α n  ∑ ∆Vi      i  n 

(XIII.41)

,

2/3

(α1m1 + α 2 m2 + ... + α n mn )1 / 3

where all designations are identical to the above-considered formulae. In the reduced form, correlation (XIII.41) will have the form: 2/3

k =n     γ pk  ∑ ∆Vi  m1k / 3 α k  ∑ ∆Ei*  ∑     k =n k =1  i k  i k ⋅ γ p = ∑ αk . 2/3 1/ 3   k = n  k =1    k =n   ∑ ∆Ei*   ∑ α k  ∑ ∆Vi    ∑α k mk         i k  k =1  i  k   k =1  

(XIII.42)

Let us show an example of calculation of the surface energy for polyvinyl alcohol copolymer with poly(vinyl acetate): CH2

CH

CH2

OH

CH O C CH3 O

Polyvinyl alcohol (PVA)

Poly(vinyl acetate) (PVAc)

For PVA we obtain

∑ ∆Ei*

* * * = 2 ∆EC + 4 ∆EH + ∆EO + ∆Eh* = 2⋅550.7 + 4⋅47.7 + 142.6 + 3929 =

i

5,363.8 cal/mol = 22,420.6 J/mol. ∑ ∆Vi = ∆VC,10 + ∆VC,39 + ∆VH,125 + ∆VH,124 + ∆VO,132 = 13.1 + 12.2 + 3⋅2.0 + 4.7 + i

5.6 = 41.6 Å3. The number of atoms m in the repeat unit of PVA equals 7. Substituting values

∑ ∆Ei* , ∑ ∆Vi , and m into equation (XIII.18) and taking into i

i

account that the present polymer belongs to group II of polar polymers (C2 = 0.0476), we obtain that

545

γ p = 0.0476

22,420.6 (41.6) 2 / 3 ⋅ 71 / 3

= 46.5 dyn/cm.

For PVAc we obtain:

∑ ∆Ei*

* * * = 4 ∆EC + 6 ∆EH + 2 ∆EO + ∆Ed* = 4⋅550.7 + 6⋅47.7 + 2⋅142.6 + 1623 =

i

4,397 cal/mol = 18,380 J/mol. ∑ ∆Vi = ∆VC,10 + ∆VC,39 + ∆VC,48 + ∆VC,13 + 6∆VH,124 + ∆VO,129 + ∆VO,139 = 13.1 + i

12.2 + 15.9 + 17.2 + 6⋅2.0 + 3.4 + 5.8 = 79.6 Å3. The number of atoms m in the repeat unit of PVAc equals 12. Using equation (XIII.18) and taking into account that PVAc belongs to polar polymers from group I (Cp1 = 0.0751), we obtain that

γ p = 0.0751

18,380 (79.6) 2 / 3 ⋅121 / 3

= 32.6 dyn/cm.

To calculate γp of the copolymer, let us first use correlation (XIII.36); assume that the molar part of PVA α1 = 0.4, and for PVAc – α2 = 0.6. Substituting all parameters of the copolymer into correlation (XIII.36), we obtain: γ p = (0.4 ⋅ 0.0476 + 0.6 ⋅ 0.0751)

0.4 ⋅ 22,420.6 + 0.6 ⋅ 18,380 (0.4 ⋅ 41.6 + 0.6 ⋅ 79.6) 2 / 3 ⋅ (0.4 ⋅ 7 + 0.6 ⋅ 12)1 / 3

= 37.0 dyn/cm.

Let us now use correlation (XIII.37):

γ p = (0.4 ⋅ 0.0476 + 0.6 ⋅ 0.0751) →

0.4 ⋅ 46.5 ⋅ 41.6 2 / 3 ⋅ 71 / 3 / 0.0476 + (0.4 ⋅ 41.6 + 0.6 ⋅ 79.6) 2 / 3 ×

0.6 ⋅ 32.6 ⋅ 79.6 2 / 3 ⋅121 / 3 / 0.0751 × (0.4 ⋅ 7 + 0.6 ⋅12)1 / 3

→

= 37.0 dyn/cm.

Let us now consider the surface energy of a triple copolymer based on acrylonitrile, butadiene and styrene (ABS–plastic): CH2

CH

–CH2–CH=CH–CH2–

CH2

CH

C N

Polyacrylonitrile (PAN)

Polybutadiene (PB)

Polystyrene (PS)

For PAN,

∑

* * ∆Ei* = 3 ∆EC + 3 ∆EH +

i

4623.2 cal/mol = 19,325 J/mol.

∆E *N + ∆Ed* = 3⋅550.7 + 3⋅47.7 + 1205 + 1623 =

546

∑ ∆Vi

= ∆VC,10 + ∆VC,6 + ∆VC,71 + 3∆VH,124 + ∆VN,154 = 13.1 + 9.0 + 15.9 + 3⋅2.0 +

i

10 = 54 Å3. PAN belongs to polymers of group III, because it contains a nitrile group (C3 = 0.060). The number of atoms, m, in the repeat unit of PAN equals 7. For it, according to formula (XIII.18), we obtain that

γ p = 0.060

19,325 54 2 / 3 ⋅ 71 / 3

= 42.4 dyn/cm.

For PB,

∑

* * ∆Ei* = 4 ∆EC + 6 ∆EH +

∆E≠* = 4⋅550.7 + 6⋅47.7 – 323 = 2,166 cal/mol = 9,054

i

J/mol. ∑ ∆Vi = 2∆VC,10 + 2∆VC,16 + 6∆VH,124 = 2⋅13.1 + 2⋅13.1 + 6⋅2.0 = 64.4 Å3. i

PB belongs to non-polar polymers of the group I (C1n = 0.1277); for it, m = 10. Application of formula (XIII.18) leads to the following result:

γ p = 0.1277

9,054 (64.4) 2 / 3 ⋅101 / 3

= 33.4 dyn/cm.

For PS,

∑

* * ∆Ei* = 8 ∆EC + 8 ∆EH +

∆E * = 8⋅550.7 + 8⋅47.7 + 713 = 5,500 cal/mol = 22,991

i

J/mol. ∑ ∆Vi = ∆VC,10 + ∆VC,7 + ∆VC,19 + 5∆VC,18 + 8∆VH,124 = 13.1 + 8.7 + 8.4 + 5⋅12.7 + i

8⋅2.0 = 109.7 Å3. PS belongs to group IV of polymers (C4 = 0.1014); for it, m = 16. Substitution of all values of PS parameters into formula (XIII.18) gives

γ p = 0.1014

22,991 (109.7) 2 / 3 ⋅ 161 / 3

= 40.4 dyn/cm.

Calculating the surface energy of the copolymer, we assume that the part of PAN units α1 = 0.3, the part of PB α2 = 0.2, and the part of PS α3 = 0.5. Then it is obtained from equation (XIII.37) that γ p = (0.3 ⋅ 0.06 + 0.2 ⋅ 0.1277 + 0.5 ⋅ 0.1014) →

0.3 ⋅ 42.4 ⋅ 54 2 / 3 ⋅ 71 / 3 / 0.06 + (0.3 ⋅ 54 + 0.2 ⋅ 64.4 + 0.5 ⋅109.7) 2 / 3 ×

0.2 ⋅ 33.4 ⋅ 64.4 2 / 3 ⋅101 / 3 / 0.1277 + 0.5 ⋅ 40.4 ⋅109.7 2 / 3 ⋅161 / 3 / 0.1014 × (0.3 ⋅ 7 + 0.2 ⋅10 + 0.5 ⋅16)1 / 3

→

= 40.9 dyn/cm.

Chapter XIV. Miscibility of polymers

The problem of miscibility of polymers is one of the most important problems at present. The point is is that new polymeric materials are now created, as a rule, not by synthesizing new polymers, but by composing mixtures of the known polymers. Therewith, it is essential to consider not only mixed composites, into which components of the mixture are injected in comparable amounts, but also the introduction of microadditions of polymers, their surface modification, etc. There are also problems appearing in this case, such as microphase separation, facilities to control the composition and size of microphases, etc. Having no way of discussing numerous publications which have appeared recently in this field, let us analyze one of possible ways of predicting the miscibility of polymers and estimating the composition of microphases. One of possible ways of looking at the problem of predicting the miscibility of polymers is the application of criterion (XII.21), which is intended for analyzing the solubility of polymers. Therewith, if one polymer is introduced in small amounts into another polymer, the former of them is considered as the ‘polymer’ and the latter – as the ‘solvent’ when criterion (XII.21) is applied. In principle, the following variants are probable. 1. When criterion (XII.21) is used, it turns out that, in all cases, the left part of the criterion is greater than the right part, i.e. total incompatibility takes place. Under all cases we mean those when the first polymer is introduced in small quantities into the second one and, vice versa, the second polymer is introduced in small quantities into the first one. Then the criterion of the miscibility of polymers obtains the form:  at introduction of the first polymer into the second one:

µ1 =

δ p2,1 δ p2,2

> 1.374Φ Φ − Φ 2 − 1 + a1  = 2 ρβ1 ;  

(XIV.1)

 at introduction of the second polymer into the first one:

µ2 =

δ p2,2 δ p2,1

> 1.374Φ Φ − Φ 2 − 1 + a2  = 2 ρβ 2 ;  

(XIV.2)

where δp,1 and δp,2 are the solubility parameters of polymers 1 and 2, respectively; Φ=

(

)

4 Vp,1 ⋅ Vp,2 1 / 3

(Vp,11/ 3 + Vp,21/ 3 )2

,

(XIV.3)

where Vp,1 and Vp,2 are the molar volumes of polymers 1 and 2, respectively; a1 = γp,1;p,2/γp,2; a2 = γp,1;p,2/γp,1, where

(XIV.4)

548

γp,1;p,2 = γp,1 + γp,2 – 2Φ(γp,1⋅γp,2)1/2,

(XIV.5)

where γp,1 and γp,2 are values of the surface energy for polymers 1 and 2, respectively. Criteria (XIV.1) and (XIV.2) mean that the polymers display total incompatibility. 2. When small quantities of the first polymer are introduced into the second one, criterion (XII.21) indicates that they are miscible, i.e.

µ1 =

δ p2,1 δ p2,2

< 1.374Φ Φ − Φ 2 − 1 + a1  = 2 ρβ1 .  

(XIV.6)

However, when the second polymer is introduced into the first one, it may appear that miscibility is not observed, i.e.

δ p2,2

> 1.374Φ Φ − Φ 2 − 1 + a2  = 2 ρβ 2 . (XIV.7)   δ p2,1 This, at first glance, a paradoxic conclusion is indicated, indeed, which will be shown below. Here, it should also be noted that, usually, the miscibility of two polymers is judged by the glass transition temperature of their mixture. A single glass transition temperature is typical of totally miscible polymers and lies between the glass transition temperatures of original components. A mixture of totally incompatible polymers display two glass transition temperatures, each of which is identical to the glass transition temperature of the original component. For partial miscibility, when both components are present in different quantities in the microphase, two glass transition temperatures are also indicated but, compared with the glass transition temperatures of original components, they are shifted towards each other. For the case under consideration, when the first polymer is combined with the second one, the dependence of the glass transition temperature on the composition is of the shape schematically represented in Figure 98, curve 1. This form of the curve will be explained below on the example of a ‘polymer 1–polymer 2’ specific system.

µ2 =

Figure 98. Schematic representation of dependence of the glass transition temperature Tg on the composition of the mixture α: 1 – partially miscible polymers; 2 – totally miscible polymers.

549

3. This case is appropriate to the total miscibility of polymers, i.e. the first polymer is ‘dissolved’ in the second one, and the second polymer is ‘dissolved’ in the first one. The miscibility criterion obtains the form: – at introduction of the first polymer into the second one:

µ1 =

δ p2,1 δ p2,2

< 1.374Φ Φ − Φ 2 − 1 + a1  = 2 ρβ1 ;  

(XIV.8)

– at introduction of the second polymer into the first one:

µ2 =

δ p2,2 δ p2,1

< 1.374Φ Φ − Φ 2 − 1 + a2  = 2 ρβ 2 .  

(XIV.9)

In the case as the one above-mentioned, a single glass transition temperature Tg is displayed, and the dependence of Tg on the composition of the mixture has the shape schematically represented in Figure 98, curve 2. Let us discuss now the behavior of a series of particular well-studied polymeric mixtures. The first of them is the mixture of polystyrene with poly(vinylmethyl ether): CH2

CH

Polymer 1

CH2

CH

Polymer 2.

O CH3 The behavior of this mixture of various compositions is studied in detail in a series of works [131, 153, 168, 198–200]. The dependence of the glass transition temperature Tg of this mixture on the composition is displayed in Figure 99.

Figure 99. Dependence of the glass transition temperature Tg on mole fraction α of polystyrene for the mixture of poly(vinylmethyl ether) with polystyrene (2). Dotted curve (1) represents dependence of Tg on α on the condition, as components are completely miscible.

Clearly, as polystyrene concentration increases from 0 to 40%, the change of Tg is extremely weak and is almost independent of the composition of the mixture.

550

This dependence is not described by equation (IV.55) valid for statistic copolymers and homogenous mixtures of polymers (curve 1). Let us analyze in detail the miscibility of polystyrene (PS) and poly(vinylmethyl ether) (PVME) with the help of criterion (XII.21). Suppose first that PS is the ‘solvent’ for PVME. The initial characteristics necessary for the application of criterion (XII.21) are shown in Table 48. Substituting values of these characteristics into equation (XIV.20), we obtain that

µ2= 0.684 < 1.218 = 2ρβ2. Table 48 Initial data for estimating compatibility of a series of polmers: polystyrene (PS), poly(vinylmethyl ether) (PVME) and poly(2,6-dimethyl-1,4-phenylene oxide) (PPO) Physical characteristics PS PVME PPO 9.12 7.54 8.93 Solubility parameter δ, (cal/cm3)1/2 41.6 33.4 44.4 Surface energy γ, dyn/cm Molar volume V, cm3/mol 97.08 60.7 103.75 22988 8696 23587 ∆Ei* , J/mol Cohesive energy i 109.8 60.7 117.4 ∆Vi , Å3 Van-der-Waals volume i Glass transition temperature Tg , K 373 245 

∑

∑

Hence, because the left part of the miscibility criterion is smaller than the right part, then PVME is ‘dissolved’ well in PS. Suggest now that PVME is the ‘solvent’ for PS. Then, with the help of miscibility criterion (XIV.1) we obtain that

µ1= 1.462 > 1.175 = 2ρβ1. Because the left part of the criterion is greater than the right part, this means that poly(vinylmethyl ether) is not ‘dissolved’ in polystyrene. That is why when PS is injected into PVME, a microphase separation should be expected. However, because PS is the ‘solvent’ for PVME, a part of PVME will be miscible with PS. Consequently, two microphases will be formed, one of which contains PVME, and the second – a mixture of PVME with PS. When the PVME concentration in the second microphase increases, miscibility of this microphase with PVME is improved and at a definite concentration of PVME, the second microphase will be miscible with PVME. Let us determine this critical concentration of PVME. For this purpose, let us write down the correlation for calculating the solubility parameter of the current mixture of polymers. Basing on equation (XII.3), we obtain that     α  ∑ ∆Ei*  + (1 − α ) ∑ ∆Ei*       i 2  i 1 2 = , δ mixture       N A α  ∑ ∆Vi  + (1 − α ) ∑ ∆Vi        i 2  i 1  

(XIV.10)

551

    where α is the molar part of PVME in the mixture;  ∑ ∆Ei*  and  ∑ ∆Ei*  are      i 1  i 2     molar cohesive energies for PS and PVME, respectively;  ∑ ∆Vi  and  ∑ ∆Vi       i 1  i 2 are Van-der-Waals volumes of repeat units of PS and PVME, respectively. Substituting characteristics of polymers shown in Table 48 into correlation (XIV.10), we obtain that 2 = δ mixture

22,988 − 14,292α cal/cm3. 4.18(66.13 − 29.57α )

(XIV.11)

Let us now estimate the surface energy of the mixture according to equation (XIII.39):

γ mixture = [αC1n + (1 − α )C4 ]

  α  ∑ ∆Ei*  +    i 2       N A α  ∑ ∆Vi  + (1 − α ) ∑ ∆Vi         i 2  i 1  

→

  + (1 − α ) ∑ ∆Ei*     i 1

[αm2 + (1 − α )m1 ]1 / 3

2/3

→ (XIV.12)

,

where C1n and C4 are coefficients from equation (XIII.39) for PVME (non-polar polymer) and PS, respectively; m1 and m2 are the numbers of atoms in repeat units of PS and PVME, respectively. Substituting all the values from Table 48 into correlation (XIII.39), we obtain that

γ mixture = (0.0231α − 0.1046)

22,988 − 14,292α (109.8 − 49.1α ) 2 / 3 (16 − 6α )1 / 3

.

(XIV.13)

Future analysis requires estimation of the molar volume of the mixture: Vmixture = α53.68 + (1 – α)97.088.

(XIV.14)

To calculate left and right parts of criterion (XII.21), let us use correlations (XIV.11), (XIV.13) and (XIV.14). Therewith, let us consider a two-component mixture, one of the components of which represents PVME, and the second one is a mixture (microphase) of PVME/PS type with different molar part α of PVME. Calculation results are displayed in Figure 100 in the shape of two dependences of both parts of the criterion (XII.21) on the molar part of PVME. The cross-point of these two dependences corresponds to the PVME concentration in the microphase at

552

which the miscibility of PVME with this microphase appears. This threshold concentration αcr = 0.62. The Van-der-Waals volume of the mixture with the threshold concentration of PVME will equal    ∑ ∆V  = 60.7⋅0.62 + 109.8⋅0.38 = 79.36 Å3. i   i  th

Figure 100. Dependences of µ (2) and 2ρβ (1) on the molar fraction α of poly(vinylmethyl ether) within microphase (for explanation see text).

Now the glass transition temperature of the mixture with the threshold concentration of PVME may be calculated. For this purpose, let us use equation (IV.55):        α th  ∑ ∆Vi  −  ∑ ∆Vi   +  ∑ ∆Vi        i 2  i 1   i 1  = Tg, th . (XIV.15)              ∆V  ∆V  ∑ ∆Vi  ∑ i   ∑ i    i 2  i 1  i 1 − + 2α th (1 − α th ) ⋅ 0.03 α th  + Tg,1  Tg,1  Tg,2     Substituting all parameters of the system from Table 48 into correlation (XIV.15) and taking into account that αth = 0.62, we obtain that Tg,th = 284 K. Now the dependence of glass transition temperature Tg on the composition of the mixture consisting of PVME and ‘critical’ mixture, i.e. a mixture consisting of PS and PVME with the critical concentration αth of PVME equal to 0.62, may be determined. Using equation (IV.55) for this purpose, we obtain that

553

       α  ∑ ∆Vi  −  ∑ ∆Vi   +  ∑ ∆Vi         i 2  i  th   i  th  Tg, th = ,         ∑ ∆Vi    ∑ ∆Vi    ∑ ∆Vi          i  th  i  th 2  i + 2α (1 − α ) ⋅ 0.03 − + α  T T T g,2 g, th g, th      

(XIV.16)

where α is the molar part of PVME. Correlation (XIV.16) is true only in the range of α from 1 down to 0.62. The curve composed in accordance with correlation (XIV.16) is shown in Figure 99 (curve 2). Clearly, experimental points fit well this dependence. To describe the second part of the dependence of Tg on the composition of PVME/PS mixture (inside the range of α from 0.62 down to 0), it is necessary first to determine the Van-der-Waals volume of the mixture, for which Tg = 284 K: 

      + 0.380.62 ∑ ∆Vi  + 0.38 ∑ ∆Vi   ;       2  i 2  i 1   

∑ ∆Vi = 0.62 ∑ ∆Vi   i

i

∑ ∆Vi

= 67.8 Å3.

i

Now, the dependence of Tg on the composition of the mixture, one of the components of which represents PS, and the second – the microphase with Tg = 284 K (see above), may be described. For this system, we obtain that Tg =

α ′(109.8 − 67.8) + 67.8 ,  109.8 67.8  67.8 ′ ′ − + + − ⋅ 2 ( 1 ) 0 . 03 α ′ α α   373 248  248

(XIV.17)

where α′ is the molar part of PS on the scale from 1 down to 0.38. To determine the true molar part of PS in the mixture, it should be presented: (1 – α) = 0.38 + α′⋅0.62, where α is the molar part of PVME, and (1 – α) is the molar part of PS in the total mixture. (1 − α ) − 0.38 into correlation (XIV.17), we find that the Substituting α′ = 0.62 correlation obtained describes well the second part of dependence of Tg in the range of α from 0 to 0.62 on the composition of the mixture at high concentration of PS (see Figure 99). Hence, solubility criterion (XII.21) may be successfully applied to description of the dependence of glass transition temperature Tg of the polymer system, in which one of polymers ‘dissolves’ well the second one, but the first polymer is badly ‘dissolved’ in the second one.

554

Let us analyze now a simpler case, when two polymers are totally miscible with each other. As an example of such mixture, let us consider the polystyrene (PS)–poly(2,6dimethyl-1,4-phenylenoxide) (PPO) system, studied well in a series of works [139, 166, 169, 197, 203, 204, 209]. Suggest that PS is the ‘solvent’ for PPO. Applying solubility criterion (XII.21) with regard to all physical characteristics of the mixture components (see Table 48), it is found that µ2 = δ 22 δ12 = 0.959; Φ = 1.0; γ1,2 = 0.0456; a = 0.001096; 2ρβ2 = 1.328 (δ1 and δ2 are solubility parameters of PS and PPO, respectively). Because µ2 < 2ρβ2, then according to criterion (XII.17) PS is a good ‘solvent’ for PPO, i.e. miscibility must be observed. It should be assumed now that PPO is the “solvent” for PS. Then µ1 = δ12 δ 22 = 1.043; Φ = 1.0; γ1,2 = 0.0456; a = 0.00103; 2ρβ1 = 1.33. Clearly, in this case also, the left part of criterion (XII.21) is smaller than the right part of it, i.e. miscibility is also possible. Experiments indicate that, indeed, polymers in the present pair are totally miscible. This leads to a description of the glass transition temperature of the PS/PVME mixture by the same dependence on the composition as for statistic copolymers, i.e. by correlation (IV.55). This dependence is shown in Figure 101, which indicates that experimental points fit well the calculated curve. It should be noted that the approach considered possesses the advantage before some other approaches that it requires no introduction of ‘fitted’ parameters. To put it differently, this analysis requires understanding of the chemical structure of components, based on which all physical characteristics of polymers are calculated.

Figure 101. Dependence of the glass transition temperature Tg on the molar fraction α of poly(2,6dimethyl-1,4-phenyleneoxide) for the polymer blend with polystyrene.

Chapter XV. Influence of the end groups on the properties of polymers

The problem of influence of the molecular mass of the polymer and the problem of the end groups role and their influence on the properties associated with it has been discussed many times in the literature. For example, there are data on the dependence of glass transition temperature Tg and melting point Tm on the molecular mass in the shape of curves with a plateau, as well as information on other properties dependent on the molecular mass. As an example, Figure 21 displays the dependence of Tg on the mean molecular mass of polystyrene. There is a series of correlations describing these dependences, for example [144], Tg = Tg(∞) – k/M, where Tg(∞) is the glass transition temperature of the polymer at the molecular mass M → ∞; k is the polymer constant. Dependences of this kind do not describe Tg(M) curves along their length. More complicated dependence Tg(M) is of the form [145]: Tg = Tg(∞) – K*/(M + M*), where K* and M* are fitted parameters. Obviously, not only glass transition temperature Tg, but also all other properties depend on the molecular mass. Searching for these dependences is of special interest. This problem is of prime importance for analysis of the molecular mass influence on the properties of propagating chains at a very low number of units n. The circumstance that the chemical structure of end groups may significantly (and even principally sometimes) differ from the structure of the repeat unit of the polymer is not always taken into account in this analysis. This causes that the composition of the dependence of a parameter of any physical property on n indicates every new value of n nonequivalent to the former one, because the chemical structure of the averaged unit changes with transition from n to (n + 1), etc. Only when n ≈ 10 – 20, the influence of end groups becomes low and dependences of the properties on n approach the values of parameters typical of the repeat unit of the polymer. Ref. [8] indicates quantitative description of the dependences of three physical parameters – Van-der-Waals volume, molar refraction and molar cohesive energy – on the number of units in a polymeric chain starting from n = 1. Therewith, the influence of the type of initiator applied to polymerization on the chemical structure of end groups and properties of dimers, trimers, etc. followed from it, is taken into account. Calculations were performed on the example of four polymers – poly(methyl methacrylate) (PMMA), polystyrene (PS), poly(ethylene terephthalate) (PET) and polycarbonate (PC) based on bisphenol A. The following types of systems were considered:

556

CH3 H3C

C

(CH2

C OCH3

CH3

CH3

C)n- 2 CH2 C OCH3

CH C OCH3

O CH3

O CH3 C

C OCH3

C)n- 2 CH C OCH3

O

O

O

O CH3 C

CH

(CH2

CH3 H3C

PMMA-2

C OCH3

CH3

CH3 C

C OCH3

C)n- 2 CH C OCH3

C OCH3

O

O

O

C

(CH2

O C

(CH2

O C

PMMA-1

PMMA-3

CH3

CH3

C)n- 1 CH2 C OCH3

CH C OCH3

O

O

PMMA-4

CH3 O (CH2

PMMA-5

C)n C C OCH3 O

H3C

CH

(CH2

CH)n - 2

CH2

H3C

CH

(CH2

CH)n - 2

CH

CH2

CH

PS-1

PS-2

O H3C

CH

(CH2

CH)n- 1

C

PS-3

557

O

O

C

HO

HO

(CH2

CH)n C

[C

C O

O

O

[C

C O

O

O

(CH2)2

PS-4

(CH2)2

O]n H

PET-1

C OH PET-2

O]n C O

O

CH3 H

C

[O

PC-1

O C] n OH O

CH3 CH3 H

C

[O

O C] n O O

CH3

CH3

CH3 H

[O

C

PC-2

C

O C] n O

CH3

PC-3

CH3

O

For all structures, Van-der-Waals volumes

OH

∑ ∆Vi , molar refractions ∑ Ri i

i

and molar cohesive energies ∑ ∆Ei* have been calculated. As an example, let us i

consider in detail the calculation of these characteristics for PMMA-1 structure.    ∑ ∆Vi  = 3∆VC,13 + ∆VC,1 + 2∆VC,48 + 2∆VC,41 + ∆VC,10 + ∆VC,6 + 18∆VH,124    i  PMMA -1 + 2∆VO,129 + 2∆VO,139 + (∆VC,10 + ∆VC,1 + ∆VC,13 + ∆VC,48 + ∆VC,41 + 8∆VH,124 + ∆VO,129 + ∆VO,139)(n – 2);    ∑ ∆Vi  = 3⋅17.2 + 5.0 + 2⋅15.9 + 2⋅20.3 + 13.1 + 9.0 + 18⋅2.0 + 2⋅3.4 +   i   PMMA -1 2⋅5.8 + (13.1 + 5.0 + 17.2 + 15.9 + 20.3 + 8⋅2.0 + 3.4 + 5.8)(n – 2) = 205.5 + (n – 2)⋅96.7 Å3.

558

  * * * * * *  ∑ ∆Ei*  = 10∆ EC + 18∆ EH + 2∆ Ed* (5∆ EC + 8∆ EH + 4∆ EO + 2∆ EO    i  PMMA -1 + ∆ Ed* )(n – 2);    ∑ ∆Ei*  = 10⋅550.7 + 18⋅47.7 + 4⋅142.6 + 2⋅1623 + (5⋅550.7 + 8⋅47.7 +    i  PMMA -1 2⋅142.6 + 1623)(n – 2) = 10,182 + (n – 2)⋅5,043 cal/mol = 42,561 + (n – 2)⋅21,080 J/mol.    ∑ Ri  = 10RC + 18RH + 2RO< + 2RO= + (5RC + 8RH + RO< + RO=)(n – 2);    i  PMMA -1    ∑ Ri  = 10⋅2.418 + 18⋅1.100 + 2⋅1.643 + 2⋅2.211 + (5⋅2.418 + 8⋅1.100 +    i  PMMA -1 1.643 + 2.211)(n – 2) = 51.688 + (n – 2)⋅24.744 cm3/mol. Besides these characteristics, let us also calculate heat capacity in the liquid state Cpl . For PMMA-1, we obtain:

(Cpl )PMMA-1 =

l Cp, C ⋅(3∆VC,13 + ∆VC,1 + 2∆VC,48 + 2∆VC,41 + ∆VC,10 + ∆VC,6) +

l l l Cp, H ⋅18VH,124 + C p, O ⋅(2∆VO,129 + 2∆VO,139) + [ C p, C (∆VC,10 + ∆VC,1 + ∆VC,13 + l l ∆VC,48 + ∆VC,41) + Cp, H ⋅8∆VH,124 + C p, O ⋅(∆VO,129 + ∆VO,139)](n – 2);

(Cpl )PMMA-1 = 0.34565⋅(3⋅17.2 + 5.0 + 2⋅15.9 + 2⋅20.3 + 13.1 + 9.0) + 0.62289⋅18⋅2.0

+ 0.92998⋅(2⋅3.4 + 2⋅5.8) + [0.34565⋅(13.1 + 5.0 + 17.2 + 15.9 + 20.3) + 0.62289⋅8⋅2.0 + 0.92998⋅(3.4 + 5.8)](n – 2) = 91.8 + 43.2(n – 2) cal/(mol⋅deg).

Values

∑ ∆Vi , ∑ i

i

∆Ei* ,

∑

Table 49 l Ri and Cp for PMMA-5 with different values of the

i

polymerization degree n n 1 2 3 4 5 7 10 15 20

   ∑ ∆Vi  , Å3    i 0

   ∑ ∆Ei*  , cal/mol    i 0

   ∑ Ri  , cm3/mol    i 0

297.0 197.0 163.0 146.5 136.6 115.6 107.0 100.3 97.0

17541 11292 9209 8168 7543 6363 5828 5411 5203

84.42 54.50 44.70 39.80 36.60 31.60 29.07 27.08 26.08

l , Cp,0 cal/(mol⋅deg) 125.0 84.5 70.7 63.8 59.6 50.3 46.8 44.0 42.7

559

Analogous calculations were also performed for all other structures. Then values obtained for

∑ ∆Vi , ∑ ∆Ei* , ∑ Ri i

i

and Cpl were recalculated for a single

i

    unit of polymer by simple division by n, and dependences  ∑ ∆Vi  n =  ∑ ∆Vi  ;      i   i 0         l  ∑ ∆Ei*  n =  ∑ ∆Ei*  ,  ∑ Ri  n =  ∑ Ri  and Cpl n = Cp,0 on the number          i   i 0  i   i 0 of units n have been plotted. These dependences for PMMA with various end groups     are presented in Figure 102. Table 49 showing values of  ∑ ∆Vi  ,  ∑ ∆Ei*  ,      i 0  i 0   l  ∑ Ri  and Cp,0 depended on n for PMMA-5, is also displayed as an example.    i 0



Figure 102. Dependences of 





∑ ∆'  L

L





(a), 





∑ ∆  L

L





(b), 





∑ #  L

L



l (c) and C p,0 (d) as a

function of degree of polymerization n for poly(methyl methacrylate). Numbers of curves correspond to the numbers of PMMA specimens (see text).

Curves in Figure 102 and Table 49 clearly display that dependences of the Van-der-Waals volume, the cohesive energy, the molar refraction and heat capacity   represent by shape the curves asymptotically approaching values  ∑ ∆Vi  ,    i 0     l  ∑ ∆Ei*  ,  ∑ Ri  and Cp,0 typical of the repeat unit of the polymer. Therewith,      i 0  i 0 the shape of curves is different for different end groups of the polymer. In some cases, when end groups make no significant contribution into the characteristics considered,       l even at low n values of  ∑ ∆Vi  ,  ∑ ∆Ei*  ,  ∑ Ri  and Cp,0 differ        i 0  i 0  i 0 insignificantly from asymptotic one, typical of the current chain. In the case, when the

560

end group possesses a significant Van-der-Waals volume, it contributes substantially       to the physical characteristics, and values  ∑ ∆Vi  ,  ∑ ∆Ei*  ,  ∑ Ri  and        i 0  i 0  i 0 l Cp,0 at low n appear sharply different from asymptotic ones. The asymptote is

approached (depending on type of the end group) at n = 10–20, which is assumed sometimes for the value of the chain segment. This situation is typical, for example, of PMMA-4, when benzoyl peroxide is used as an initiator in polymerization of methyl-methacrylate. Analyzing the data obtained in ref. [8], some other regularities of the influence of the chemical structure of end groups on the physical characteristics of the polymer may be observed. For example, the Van-der-Waals volume of the averaged repeat unit of the polymer is generally affected by the volume of end groups. The same may be indicated for molar refraction. Consideration of Figure 102 confirms this conclusion. For systems PMMA-1, PMMA-2 and PMMA-3, the Van-der-Waals volume and the molar refraction weakly depend on n. This is clearly indicated by consideration of curves 1–3 in Figure 102. The same may be also said about the cohesive energy of these systems. This is associated with the fact that in the case of PMMA-1, PMMA-2 and PMMA-3, end groups do not enter any specific intermolecular interaction. That is why they cause no significant change of the cohesive energy. In turn, the presence of bulky end groups significantly different from the repeat unit of the polymer by the chemical structure leads to a sharp change of all the characteristics discussed (see Figure 102, curves 4, 5). All the above-said also relates fully to systems based on polystyrene. Systems containing end groups, although of small volume but possessing a strong specific intermolecular interaction, behave themselves differently. For example, they may be end OH-groups in the case of poly(ethylene terephthalate) and polycarbonate. This specificity of the structure of end groups leads to different regularities: the characteristics associated with volume and molecular refraction change insignificantly with n; in turn, cohesive energy, in which specific intermolecular interaction plays a highly significant role, changes greatly as n increases. All this must be taken into account when the influence of the polymerization degree (especially, on initial stages of the process) on the measured physical characteristics is analyzed. Above, we analyzed the physical characteristics of a series of polymers dependent on the chain length. These characteristics are initial for the estimation of the physical parameters of polymeric substances. For example, due to the Van-der    Waals volume  ∑ ∆Vi  and cohesive energy  ∑ ∆Ei*  important parameters of      i 0  i 0 the substance such as the density of cohesive energy or Hildebrand solubility parameter may be calculated. Of interest is also the dependence of this parameter δ on n at transition from monomer to dimer, trimer, etc. These calculations were performed in ref. [8] for systems PMMA-4, PS-4, PETP-1 and PC-1. Calculation results in the form of dependence of δ on n are displayed in Figure 103. One can recognize that for some systems (PMMA-4, PS-4) regardless of whether they contain bulky end groups or not, the solubility parameter depends significantly on n. This is associated with the fact that the presence of end OH-groups in these polymers causes the appearance of hydrogen bonds that requires

561

introduction of constants ∆Eh* which take into account the contribution of the energy of hydrogen bonds to total cohesive energy. As n increases, the contribution of these bonds becomes weaker and at n = 10 – 20 is almost negligible.

Figure 103. Dependence of solubility parameter δ on the degree of polymerization n for PS-4 (1), PMMA-4 (2), PC-1 (3) and PET-1 (4).

Besides the solubility parameter, of special interest is also consideration of the influence of the chemical structure of end groups on surface tension. Figure 104 indicates the dependence of the surface tension on polymerization degree n for PMMA-5. Obviously, the surface tension decreases as n increases, striving to the border value typical for PMMA.

Figure 104. Dependence of the surface tension γp as a function of the degree of polymerization n for PMMA-5.

Hence, consideration of the influence of the chemical structure of end groups of polymers on their molecular characteristics and macroscopic properties indicates that this influence may be quite different. The presence of the effect of bulky end groups affects mainly the Van-der-Waals volume of the ‘averaged’ unit, cohesive energy and molar refraction. The presence of groups (of the hydroxyl type) with small volume but possessing specific intermolecular interaction has a strong effect on the cohesive energy and solubility parameter, as well as on surface tension, with a small influence on the Van-der-Waals volume and molar refraction. Considering the dependence of any physical properties on the chain length n, these factors must be taken into account.

Chapter XVI. Thermophysical properties of polymers

The thermophysical properties are heat capacity, thermal diffusivity and heat conductivity.

XVI.1 Heat capacity Heat capacity is the amount of heat which must be consumed for heating a body up by 1°C. Molar heat capacity, if a mole of substance is considered, and specific heat capacity in the case of 1 g of substance, differ. Heat capacity at constant pressure Cp equals the rate of enthalpy change with temperature increase, and heat capacity at constant volume Cv – the rate of internal energy change with temperature increase. In a quite wide temperature range, heat capacity increases linearly with temperature, the temperature coefficient of heat capacity increase for solid polymers possessing the average value of 3⋅10–3. At phase or physical transition of the polymer, heat capacity changes in a jump-like manner. For example, transition from the glassy to rubbery state indicates quite a sharp jump-like increase of heat capacity. When the physical transition is passed, heat capacity returns to weak growth with temperature. The heat capacity of polymers depends on their chemical structure (Table 50). Among polymeric hydrocarbons, the lowest heat capacity is displayed by polyethylene and polyoxymethylene. When hydrogen atoms are substituted by polar groups, heat capacity increases. Also, heat capacity rapidly increases in transition from aliphatic to aromatic polymers. Many attempts have been made to calculate the heat capacity of polymers on the basis of the chemical structure of the repeat unit. Let us consider the method of calculation discussed in ref. [22]. In the work cited, it is supposed that the molar heat capacity of a polymeric substance is proportional to the Van-der-Waals volume of atoms of the repeat unit of the polymer. To put it differently, C sp = ∑ C sp, i ∆Vi + As

(XVI.1)

C lp = ∑ C lp,i ∆Vi + Al ,

(XVI.2)

i

and i

where C sp and C lp are the molar heat capacities of the polymer in the glassy state and the rubbery state, respectively; C sp,i and C lp,i are constants for each atom having the meaning of heat capacities reduced to the unit Van-der-Waals volume and applicable for the glassy state and the rubbery state, respectively; As and Al are parameters equal to: As = 0.77 cal/(mol⋅deg), Al = 0.69 cal/(mol⋅deg).

563

Table 50 Calculated and experimental values of molar heat capacities for a series of polymers C sp , cal/(mol⋅deg) C lp , cal/(mol⋅deg) Polymer Calc. Experimental Calc. Experimental Polyethylene 11.02 10.4; 11.1; 11.8; 10.15 14.7 15.1 Polypropylene 16.9 15.7; 16.5 21.7 21.6 Polyisobutylene 22.8 22.4 28.8 26.4 Poly-4-methylpentene-1 34.6 33.6   Polybutadiene 20.0 21.0 26.3 24.3 Polyisoprene 25.8 25.7 33.3 31.3 Polystyrene 32.4 30.5; 29.3 43.0 42.6 Polyoxymethylene 8.8 10.2; 8.9 13.2 15.0 Polyoxytetramethylene 26.4 28.2 34.1 35.7 Polyoxypropylene 19.8 19.7 26.0 26.5 Poly(vinyl chloride) 16.2 14.4; 13.8; 16.2; 18.1 18.1 Poly(chlorotrifluoroethylene) 25.4 25.0; 23.3   Poly(tetrafluoroethylene) 23.4 23.0; 21.1 23.0 23.0 Poly(vinyl alcohol) 16.3 13.6; 15.4   Poly(vinyl acetate) 27.2 30.2; 23.6 37.0 39.5 Poly(methyl methacrylate) 33.1 33.0; 32.0 44.0 43.5 Poly(acrylonitrile) 15.5 15.9   Polyamide-6 40.2 39.1; 39.2 59.4 57.8 Polyamide-6,6 81.1 79.0; 76.6   Polyamide-6,10 104.7 107 146.2 147.0 Poly(ethylene terephthalate) 52.9 52.0; 52.2 73.3 71.0 Polycarbonate based on bisphenol A 72.7 72.4; 71.0 97.6 97.8 Poly(propylene sulfone) 29.4 29.4 38.8 38.8 Poly-2,6-dimethyl-1,4-phenyleneoxide 35.1 36.4; 34.9 47.0 50.5 Table 51

Atom Carbon Hydrogen Oxygen Nitrogen Fluorine Chlorine Sulfur

s l Values C p and C p for different atoms Numerical value, Designation Designation cal/(mol⋅deg) Å3 0.232030 C Cs C Cl s l 0.714129 CH CH l 0.634726 C Os CO s l 0.314997 CN CN 0.543367 C Fs C Fl l s 0.368819 C Cl C Cl 0.273109 CSs CSl

Numerical value, cal/(mol⋅deg) Å3 0.345646 0.622889 0.929977 2.099874 0.444909 0.284693 0.303031

Values of C sp,i and C lp,i were obtained by regression analysis with the help of solving the excessive system of equations deduced on the basis of correlation (XVI.1) or (XVI.2) using the experimental data on heat capacity for well-studied polymers (so-called polymeric standards–polyethylene, polystyrene, poly(methyl methacrylate), etc.).

564

Values of C sp,i and C lp,i obtained for each atom are shown in Table 51. With the help of these values and Van-der-Waals volumes displayed in Table 3, molar heat capacities C sp and C lp may be calculated for many polymers. Satisfactory accuracy of calculation is clear from Table 50. For copolymers, equations (XVI.1) and (XVI.2) are reduced to the following form:       C sp = α1 ∑ C sp,i ∆Vi  + α 2  ∑ C sp,i ∆Vi  + ... + α n  ∑ C sp,i ∆Vi  + As ,        i 1  i 2  i n

(XVI.3)

      C lp = α1 ∑ C lp,i ∆Vi  + α 2  ∑ C lp,i ∆Vi  + ... + α n  ∑ C lp,i ∆Vi  + Al ,        i 1  i 2  i n

(XVI.4)

  where α1, α2, …, αn are the molar parts of the components 1, 2, …, n;  ∑ C sp, i ∆Vi  ,    i 1            ∑ C lp, i ∆Vi  ,  ∑ C sp, i ∆Vi  ,  ∑ C lp, i ∆Vi  , …,  ∑ C sp, i ∆Vi  ,  ∑ C lp, i ∆Vi             i n  i n  i 1  i 2  i 2 are the sets of constants for the components 1, 2, …, n, respectively. If it is required to express the heat capacity of a copolymer via the heat capacity of homopolymers based on the components 1, 2, …, n, it should be written down that C sp = α1C sp,1 + α 2C sp,2 + ... + α n C sp, n ,

(XVI.5)

C lp = α1C lp,1 + α 2C lp,2 + ... + α nC lp, n ,

(XVI.6)

where C sp,1 , C sp,2 , …, C sp, n are the molar heat capacities of polymers based on the components 1, 2, …, n in the glassy state; C lp,1 , C lp,2 , …, C lp, n are the molar heat capacities of polymers based on the components 1, 2, …, n in the rubbery state.

XVI.2 Thermal diffusivity and heat conductivity Thermal diffusivity is a characteristic that describes the rate of temperature propagation under the effect of heat flow under non-stationary temperature conditions. This characteristic is determined from the correlation:

α=

λ Cpρ

,

(XVI.7)

where λ is heat conductivity; Cp is the specific heat capacity under constant pressure; ρ is polymer density.

565

For solid (glassy and crystalline) polymers, thermal diffusivity decreases slowly with temperature increase. However, when a polymer transits from the glassy to the rubbery state, thermal diffusivity decreases abruptly. Thermal diffusivity depends on the chemical structure of the polymer, as well as on the crystallinity degree, molecular mass and pressure. Table 52 shows the values of thermal diffusivity for a series of polymers with different chemical structures. Table 52 Polymer

Thermal diffusivity α for a series of polymers at 293 K Structural formula

High pressure polyethylene Polyisoprene

CH2CH2 CH2

CH C CH2

α⋅107, m2/sec 1.40 0.90

CH3 Poly(ethylene oxide) Poly(dimethyl siloxane)

CH2CH2O

1.08

CH3 Si

0.90

O

CH3 Polystyrene

CH2

Poly(vinyl chloride)

CH2

CH

CH

0.99

1.21

Cl Poly(methyl methacrylate)

CH3 CH2

1.19

C C O CH3 O

Heat conductivity represents the capability of polymeric substances for heat transfer from more heated substances to less heated ones. The coefficient of heat conductivity λ represents the proportionality coefficient between the heat flow and the temperature gradient. Heat conductivity is associated with the propagation and dissipation of elastic waves caused by thermal oscillations of particles in the substance. At temperatures approaching the absolute zero, heat conductivity also strives to zero. The heat conductivity of polymers depends on their chemical structure in the frame of one and the same physical state. For amorphous glassy polymers, heat conductivity is somewhat lower than for crystalline polymers. The temperature dependence of heat conductivity is affected by the chemical structure of the polymer. For one series of polymers, the slope of this dependence is positive, and for another one – negative. In the area of physical transition from the glassy to rubbery state, a low maximum of heat conductivity is observed. Table 53 shows heat conductivity values for a series of polymers with different chemical structures, from which the type of the effect of the chemical structure on heat conductivity is clear. Of special meaning is the anisotropy of the heat conductivity of polymers. This property is typical of oriented polymeric systems, in which heat conductivity is different along

566

and transverse to the orientation axis. The thermal properties of polymers are discussed in more detail in reviews [59, 61, 62]. Polymer Polyethylene of: High pressure Low pressure Polypropylene: Atactic Isotactic Polystyrene

Heat conductivity λ for a series of polymers at 293 K Structural formula

Poly(vinyl chloride)

CH2CH2 CH3

Table 53

λ, W/(m⋅K) 0.380 0.470

CH2

CH

CH2

CH

0.175 0.230 0.130

CH2

CH

0.170

Cl CH3

0.190

Poly(methyl methacrylate)

CH2

C C O CH3

Polytetrafluoroethylene Poly(dimethyl siloxane)

O CF2CF2 CH3

Si CH3

O

0.250 0.167

Chapter XVII. Molecular design and computer synthesis of polymers with predetermined properties

In previous sections of the book, the possibility of calculating the physical properties of polymers on the basis on the chemical structure of the repeat unit of linear polymers or the repeating fragment of polymer networks was demonstrated. At the present time, the above-mentioned calculations are performed in a computer. Therewith, main direct and reverse tasks and secondary tasks may be solved. The direct task concludes in calculating the polymer properties from the data on the chemical structure of the repeat unit of the polymer or a fragment of network. The reverse task (which is more complicated) consists of predicting the structural formula of the repeat unit of the polymer which would have a single or several required characteristics of the physical properties. A solution of both tasks is described in refs. [9, 17, 26, 126]. When setting and solving problems, two border approaches and a series of intermediate approaches are possible. The first border approach concludes in recording in computer memory a large number of polymers with different chemical structure for which the physical characteristics mentioned above were previously calculated. Then, computer’s role will conclude only in the search for necessary structural formulas in the database and layout of the required information, when both direct and reverse tasks are solved. In the case of such approach, the accuracy of layout characteristics is close to 100%, but the predictive power for structures absent in the database equals zero. In fact, such approach enables just to create a database. The second border variant concludes in the following. A table of Van-derWaals volumes of atoms is recorded in the computer memory. A fragment of this Table is shown as an example below. H O H C 1.54 1.54

C

C

1.54

C

1.08 1.54

C

C

1.28

5

2

C

1.54

C

O

C C

1.54

C

C

3

H

C

1.08 1.54

C C

C

O

1.54

C

1.08

4

C

1.54

1.54

H C

C

1.28

1.08

6

C

1.54

1.54

1.54

1.54

1

C

1.50 1.54

1.54

7

C

8

O

In principle, each of the atoms squared by dotted lines may be valently bonded to other atoms; for the above-depicted structures, this possibility is demonstrated by the matrix (1 – joining is possible; 0 – prohibition): Atom, No. 1 2 3 4 5 6 7 8

1 1 1 1 1 1 0 0 0

2 1 1 1 1 1 1 0 0

3 1 1 1 1 1 0 1 0

4 1 1 1 1 1 1 0 0

5 1 1 1 1 1 0 0 1

6 0 1 0 1 0 0 0 0

7 0 0 1 0 0 0 0 0

8 0 0 0 0 1 0 0 0

568

The table representing Van-der-Waals volumes of atoms, shown in the present monograph, contains over 200 values of volumes for atoms valently bonded with other various atoms. The task of building a repeat unit of a polymer from all this multiplicity of atoms, even at their limited amount in the repeat unit, is rather difficult to be computerized, and its solution by a computer requires too much computer time due to the appearance of a tremendous amount of variants. In the present case, we are talking about solving the reverse problem, when polymers with given properties are to be ‘synthesized’ using a computer. The above-discussed situation led to the need to develop an intermediate approach (between the border ones described) to computerized prognosis of the physical characteristics of polymers and their computerized synthesis. This approach concludes in preliminary summation of all constants participating in the initial correlations for calculation of properties for separate fragments (‘basic fragment’), from which then a repeat unit of a polymer is constructed on a computer. The variant of the software described in ref. [17] provides constructing of polymers, repeat units of which contain two fragments only possessing a possibility of chemical bonding with each other. According to the method of mutual joining, all the fragments discussed are subdivided into 8 classes: Class I

Structure C

II

C

III

C

IV

C

1.50

1.37

where R1: CH2;

O O

1.54

1.48

R1 R2

R1,2 R1,2

Class V

1.54

C C

O

O

VI

1.48

O

Structure

1.50

VII

1.37

VIII

C C

O N C

; (CH2)4, etc.; R2:

CH

1.50

1.37

1.37

1.37

1.50

O

R1,2

1.37

R2

O

1.37

N

R1,2 R3

1.37

C

;

OH , etc.;

O O C R3:,

O

O

C

N

C N

;

O O

C N

N

C

C

C

C

O

O

O

O

, etc.

The matrix shown below reflects possibilities of mutual joinings of the fragments related to different classes (1 –joining is possible; 0 – prohibition):

569

Fragment, No. I II III IV V VI VII VIII

I 1 0 0 0 0 0 0 0

II 0 1 0 0 0 0 0 0

III 0 0 0 0 1 0 0 0

IV 0 0 0 0 0 1 0 0

V 0 0 1 0 0 0 0 0

VI 0 0 0 1 0 0 0 0

VII 0 0 0 0 0 0 0 1

VIII 0 0 0 0 0 0 1 0

Generally, ref. [17] presents series of constants calculated for 194 fragments (‘basic fragments’) of various chemical structures. As an example, some of them are shown in Table 54. Dotted squares mark the chemical structure of the fragment itself; also displayed are atoms, joined to the fragment, and lengths of chemical bonds expressed in Å. The possibility of joining of one of 18 fragments to another, taking into account their belonging to different classes, is demonstrated in the following matrix (1 – joining is possible; 0 – prohibition): Fragment, No. 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18

1 1 1 1 0 1 1 1 1 1 0 0 0 0 0 0 0 0 0

2 1 1 1 0 1 1 1 1 1 0 0 0 0 0 0 0 0 0

3 1 1 1 0 1 1 1 1 1 0 0 0 0 0 0 0 0 0

4 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0

5 1 1 1 0 1 1 1 1 1 0 0 0 0 0 0 0 0 0

6 1 1 1 0 1 1 1 1 1 0 0 0 0 0 0 0 0 0

7 1 1 1 0 1 1 1 1 1 0 0 0 0 0 0 0 0 0

8 1 1 1 0 1 1 1 1 1 0 0 0 0 0 0 0 0 0

9 10 1 0 1 0 1 1 0 0 1 0 1 0 1 0 1 0 1 0 0 1 0 1 0 1 0 1 0 0 0 0 0 0 0 0 0 0

11 0 0 0 0 0 0 0 0 0 1 1 1 1 0 0 0 0 0

12 0 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0

13 0 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0

14 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1

15 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1

16 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1

17 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0

18 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 0 0

The program contemplates both the solutions of the direct task concluded in the determination of polymer properties basing on the chemical structure of the repeat unit and the reverse task consisted in searching for the units such that the polymer composed from them would possess the given physico-chemical properties. Solving the reverse task, the computer calculates a parameter of one of the required properties using all probable variants of combinations from ‘basic fragments’ recorded by its memory. For polymers, the property of which appears in the required interval, the parameter of another property is calculated, after which the required polymers are also selected, etc. The drawback of this software is that composition of a repeat unit of a polymer from two large ‘basic fragment’ does not embrace the tremendous amount of probable structures. Obviously, the smaller the size of ‘basic fragments’ used for composing the repeat unit, the more variants of polymers may be obtained due to the great amount and multiplicity of ‘basic fragments’ in the unit and for calculating their properties.

570

That is why A.A. Askadskii, E.G. Galpern, A.L. Chistyakov, and I.V. Stankevich [126] have created a software for computer ‘synthesis’ of polymers from the scarcest ‘basic fragments’, i.e. those which, in principle, cannot be sectioned. These basic fragments are shown in Table 55. To denote the possibility of joining these small fragments, recorded in the computer memory, to each other, definite labels are attributed to every end of ‘basic fragments’. Note that the fragments used for composing the repeat unit possess only 20 different end groups and, consequently, 20 different labels (see Table 55), which may interact with each other in accordance with the matrix recorded by the computer memory: LABELS 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20

1 1 1 1 0 0 0 0 0 0 0 0 0 0 1 0 0 1 0 0 1

2 1 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

3 1 1 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 1 0

4 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0

5 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0

6 0 0 0 0 0 0 0 0 0 0 1 0 0 0 1 0 0 1 0 0

7 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0

8 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

9 10 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0

11 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0

12 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0

13 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0

14 1 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

15 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0

16 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0

17 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

18 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0

19 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

20 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

In this matrix, no difference is made for the given C–C bond of 1.54 Å (in aliphatic fragments) and 1.48 Å (in aromatic fragments), because transition from one bond to another causes only insignificant changes in the Van-der-Waals volume, which display no substantial influence on the accuracy of calculation of physical properties.

Figure 105. Schematic representation of construction of the combinatorial tree when constructing a repeat unit of polymer (see text).

571

Table 54 Fragment No.

Class

1 1

2 I

2

Chemical structure of fragment

Chemical structure and sets of constants for various fragments 3 ∑Vi , M ∑ (ai ∆Vi + bi ) ⋅10 , ∑ K i ∆Vi ⋅103, ∑ (δ i ∆Vi + γ i ) , i

3

C

I

C

1.54

1.54

C

CH2

1.54

i

1.54

Å3 4 17.1

5 14

Å3K−1 6 80.2

43.9

44

79.6

i

i

∑ Ri ,

∑ Ei* ,

∑ Ci ⋅103,

cal/mol 10 646

(cm3/mol)⋅MPa−1 11 –3.090

i

i

i

Å3K−1 7 24.3

8 1.434

cm3/mol 9 4.618

129.0

62.3

3.122

10.879

3058

–9.458

86

195.2

129.5

6.778

15.290

6020

–17.871

82.1

76

135.8

100.7

7.028

24.107

4208

–12.686

24.5

30

33.9

54.2

2.076

6.143

4718

–10.061

124.9

144

449

194.4

10.440

31.016

7502

–16.618

Å

C

CH O CH3

3

I

CH3 C

1.54

1.54

C

C C O CH3 O

4

VI

O 5

1.37

I

C

1.37

O

1.54

CH

1.54

C

OH 6

I

C

1.54

C O O

(CH2)4

O C

1.54

C

O 571

572

2 I

3

C

1.54

C O

O C

O 8

I

C

1.54

C

6 231.8

7 191.6

8 11.090

9 36.651

10 9126

11 –16.926

128.9

142

216.2

226.9

9.911

34.934

14334

–17.106

136.5

162

208.0

226.4

10.655

40.570

15958

–19.113

75.6

76

135.7

96.2

6.463

24.107

4208

–12.686

159.8

168

319.6

200.0

14.521

49.864

8559

–22.174

130.6

164

273.4

190.6

11.047

36.651

9126

–18.626

1.54

C

C HN (CH2)4 NH C

1.54

C

O

I 1.54

5 164

O

O 9

4 131.0

C HN O NH C

1.54

C

O 10

II

C 11

1.48

C

II

C 12

1.48

1.48

1.48

C

O

II 1.54

C

C O O O

C O

1.54

C

572

1 7

573

1 13

2 II

3

C

1.48

C HN (CH2)6 NH C O

14

16

1.50

O

III

C

1.50

O

1.50

C

O

1.37

O

7 274.4

8 12.736

9 44.170

10 15626

11 –23.295

163.1

184

395.4

199.0

14.786

51.500

8701

–18.976

3.4

16

78

0.20

0.211

1.643

143

+3.198

87.5

108

259.7

103

7.363

27.393

4494

–6.290

86.3

108

273

103

7.288

27.393

4494

–7.990

74.6

56

321

104.9

6.318

18.472

2584

–12.378

C

C

O O

18

6 376.6

1.50

IV

C

5 170

1.50

III 1.50

4 162.7

O

O

C 17

C

III

C 15

1.48

V

1.54

(CH2)4

C

1.54

C

573

C

1.37

574

Fragment No. 1 1 2

C

CH

C

Molar refraction Ri, cm3/mol 8 4.618

Cohesive energy ∆Ε*, cal/mol 9 646

1.48; 1.54

1.48; 1.54

2

2

34.2

9.236

2915

1.54

1.54

2

2

51.3

13.854

1938

1.48; 1.54

1.48; 1.54

2

2

51.3

13.854

3561

1.48; 1.54

1.48; 1.54

2

2

43.9

10.879

3058

1.48; 1.54

1.48; 1.54

2

2

31.9

9.485

1999

1.48; 1.54

1.48; 1.54

2

2

46.8

12.752

2705

1.48; 1.54

1.48; 1.54

2

2

92.6

28.725

4854

CH3 3

CH3 C

C

C

CH3 4

C

CH

C

C2H5 5

C

CH

C

O CH3 6

7 8

C

CH

C

Cl C……CCl2……C C CH C

574

Table 55 Smallest basic fragments for the computer synthesis of polymers Length of bonds with Labels Chemical structure of fragment Van-derneighboring atoms, Å Waals volume, Å3 Left Right Left Right 2 3 4 5 6 7 1.48; 1.54 1.48; 1.54 I I 17.1 C……CH2……C

575

1 9

2

CH3

10

11

C

C

C

C

C

C

C

O C

C

3 1.48; 1.54

4 1.48; 1.54

5 2

6 2

7 109.8

8 33.343

9 7123

1.48; 1.54

1.48; 1.54

2

2

18.6

4.629

693

1.48; 1.54

1.48; 1.54

2

2

108.9

32.797

7218

1.48; 1.54

1.48; 1.54

2

2

155.2

40.234

8966

1.48; 1.54

1.48; 1.54

2

2

81.1

14.694

1797

1.48

1.48

3

3

75.6

24.107

4208

O C 12

C

13

O

C

C

CF3 C

C

C

CF3 14

C

C 575

576

2

C

3 1.48

4 1.48

5 3

6 3

7 75.6

8 24.107

9 4208

1.54

1.54

2

2

54.6

13.399

6084

1.48

1.48

3

3

100.3

33.127

8791

1.54

1.50

1

4

20.2

4.618

646

1.54

1.50

2

4

37.8

9.236

2915

1.54

1.50

2

4

54.9

13.854

3561

1.54

1.50

2

4

96.2

24.725

4854

1.48

1.37

3

5

78.8

24.107

4208

C

16

N C

N

C

C

C

O 17

NH

C

C

C

N 18 19

C……CH2……O C CH O CH3 CH3

20

C

C

O

CH3 21

C

CH

O

22

C

O

576

1 15

577

1 23

2

C

25

4 1.37

5 3

6 5

7 78.8

8 24.107

9 4208

1.50

1.50

4

4

23.3

4.618

646

1.37

1.37

5

5

82.1

24.107

4208

1.37

1.37

5

5

82.1

24.107

4208

1.37

1.37

5

5

115.5

33.343

8746

1.50

1.50

8

8

3.4

1.643

143

1.37

1.50

9

8

2.7

1.643

143

1.37

1.37

9

9

2.1

1.643

143

1.54

1.50

2

8

25.1

6.272

2460

1.54

1.37

2

9

24.4

6.272

2460

1.54

1.37

1

6

18.6

4.618

646

1.54

1.37

2

6

20.1

4.629

2316

O

O……CH2……O

24

3 1.48

O

O

26

O

O

27

CH3 O

O CH3

28

31

C……O……C C……O……C C……O……C C C O C

32

C

29 30

33 34

O C O

C

O C……CH2……N C C N

577

O

578

2

C

4 1.37

5 3

6 6

7 77.4

8 24.107

9 4208

1.48

1.37

3

6

77.4

24.107

4208

1.37

1.48; 1.54

11

14

28.4

8.231

5875

1.37

1.37

5

6

80.6

24.107

4208

1.37

1.37

5

6

80.6

24.107

4208

1.37

1.37

6

6

79.2

24.107

4208

1.37

1.37

6

6

79.2

24.107

4208

1.37

1.37

11

11

8.5

3.602

1253

1.76

1.48

7

3

76.8

24.107

4208

N

36

C 37

3 1.48

N

C

NHC

C

O 38

O

N

39

O 40

N

N N

41

N 42

N

C……NH……C

43

S

C

578

1 35

579

1 44

2

S 45

S

S

S S

N

S S

O

S

52

6 3

7 76.8

8 24.107

9 4208

1.76

1.76

7

7

78.0

24.107

4208

1.76

1.76

7

7

78.0

24.107

4208

1.76

1.37

7

6

78.6

24.107

4208

1.76

1.37

7

6

78.6

24.107

4208

1.76

1.37

7

5

80.0

24.107

4208

1.76

1.37

7

5

80.0

24.107

4208

1.76

1.76

12

12

16.5

8.00

1750

1.76

1.76

12

12

26.1

8.870

3656

N

50

51

5 7

S

48

49

4 1.48

C

46

47

3 1.76

O C……S……C O C

S

579

O

C

580

2

C

O

O

C

C

C

C

O

O

N

54

O

C

C

N

C

N C

6 11

7 144.6

8 46.103

9 12542

1.37

1.37

11

11

190.2

61.441

15553

1.37

1.37

5

11

113.4

35.105

8375

1.48

1.37

3

11

110.1

35.105

8375

1.37

1.48

3

5

103.4

33.126

8791

C

O O

O

5 11

C

O 55

4 1.37

C

N

O

3 1.37

C N

C

C O O

56

C

C N

C

C 57

O

O NH

C N

C

580

1 53

581

1 58

2

O

3 1.76

4 1.37

5 7

6 11

7 111.3

8 35.105

9 8375

1.76

1.48

7

3

101.3

33.127

8791

1.64

1.64

13

13

71.6





1.64

1.64

10

10

0.5

1.50

1.50

4

4

24.8

 4.629

 2316

1.37

1.37

11

11

25.6

8.558

1899

1.37

1.37

6

6

124.8

39.445

7219

1.48

1.48

3

3

121.2

39.445

7219

1.48; 1.54

1.48; 1.54

1

1

27.1

4.898

599

C

S

C

N C O NH

59

S

C

C

N CH3

60

O

Si

O

CH3 Si……O……Si O C O

61 62

C

63

O N

C

CH3 64

N

N

65

C

C……CF2……C

581

66

C

582

As an example illustrating building of the repeat unit, Figure 105 shows a combinatorial tree, which is composed from three fragments (‘basic fragments’) with various end labels. The rule, according to which the formation of a bond between end groups is permitted, or prohibited, is reflected in the matrix of their interactions. Clearly, the quantity of probable variants of the repeat unit of the polymer increases with every step of joining a consequent fragment. It should be taken into account that if all ‘basic fragments’ recorded by the computer memory participate in the composition and therewith each of them may join to the propagating unit by both one or another ends, then 2n of variants of the repeat unit (where n is the number of ‘basic fragments’) appear on every stage of joining of the next fragment. Hence, this procedure represents a geometric progression with denominator 2. Besides direct and reverse tasks, software for prognosis of the physical properties of polymers enables other tasks to be solved. For example, the upper and lower ranges of the physical characteristics may be determined which, in principle, organic polymers are able to display; for example, the lower value of the onset temperature of intense thermal degradation of organic polymers, Td, is 150°C, and the upper limit is 575°C. Hence, an organic polymer possessing the onset temperature of intense thermal degradation above 575°C is difficult to produce. ‘Combination’ of various properties in the same polymer may also be estimated, etc. (see below). Software based on using the finest basic fragments enables calculations and ‘synthesis’ of polymers of the following classes: 1. Polyolefins, vinyl polymers, etc. 2. Polyethers and polyetherketones 3. Polyesters 4. Polyamides 5. Polysulfones, polyethersulfones 6. Polyimides 7. Polyoxadiazoles 8. Polybenzimidazoles 9. Polysulfides 10. Organosilicon polymers 11. Polycarbonates The number of polymer classes may be increased, if necessary. This requires introduction of a group determining affiliation of polymers to the chosen class into the database; for example, introduction of urethane group into the database enables to calculate properties of polyurethanes and to synthesize polyurethanes with the given properties in a computer. The drawback of this software is that the above-mentioned tasks may be solved only for polymers containing the basic fragments present in the database. Although the number of polymers is great in this case, some limits do exist. In this connection, A.A. Askadskii and A.F. Klinskikh [22] created software, according to which the chemical structure of a polymer is ‘gathered’ not from ‘basic fragments’, but from separate atoms. In this case, the chemical structure of a repeat unit is displayed on the screen in the form of a structural formula of an organic compound, as a chemist writes it down on paper. After that, the physical properties of the polymer of the given structure are calculated and laid out immediately to the user. Therewith, the molecular design of a polymer may be performed by changing its chemical structure, introducing various groups, etc., and its physical characteristics are obtained immediately. This program enables us to solve the first (direct) task only, because when polymers are synthesized from separate atoms (but not ‘basic

583

fragments’), the number of variants is so high that they cannot be realized on a computer of any power. This program enables to calculate the properties of polymers and copolymers, and polymer networks, listed below: 1. Molecular mass of the repeat unit; 2. Molar volume at room temperature; 3. Temperature dependence of the molar volume; 4. Dependence of molar volume on crystallinity degree; 5. The Van-der-Waals volume; 6. Density at room temperature; 7. Dependence of density on crystallinity degree; 8. Temperature dependence of density; 9. Thermal expansion value in the given temperature range; 10. The Hildebrand parameter of solubility; 11. Surface energy; 12. Refractive index at room temperature; 13. Temperature dependence of refractive index; 14. Dielectric constant at room temperature; 15. Birefringence; 16. Dipole moment; 17. Stress-optical coefficient in the glassy state; 18. Glass transition temperature; 19. Dependence of glass transition temperature on molecular mass; 20. Temperature of the onset of intense thermal degradation; 21. Dependence of temperature of polymer transition to the viscous-flow state on molecular mass; 22. Coefficient of volumetric expansion in the glassy state; 23. Coefficient of volumetric expansion in the rubbery state; 24. Molar heat capacity under constant pressure in the glassy state at room temperature; 25. Molar heat capacity under constant pressure in the rubbery state at room temperature; 26. Temperature dependence of molar heat capacity in the glassy state and the rubbery state; 27. Specific heat capacity at room temperature under constant pressure in the glassy state; 28. Specific heat capacity at room temperature under constant pressure in the rubbery state; 29. Jump of molar heat capacity at transition from the glassy state to the rubbery state; 30. Jump of specific heat capacity at transition from the glassy state to the rubbery state; 31. Total energy of intermolecular interaction; 32. Energy of dispersion interaction; 33. Energy of strong dipole–dipole interaction and hydrogen bonds; 34. The part of energy of dispersion interaction in total energy of intermolecular interaction; 35. The part of energy of dipole–dipole interaction and hydrogen bonds in total energy of intermolecular interaction; 36. Molar refraction; 37. Molar polarizability;

584

38. Total cohesive energy; 39. The part of cohesive energy stipulated by hydrogen bonds; 40. The part of cohesive energy stipulated by dipole–dipole interaction; 41. The part of cohesive energy stipulated by dispersion interaction; 42. Permeability by oxygen; 43. Permeability by carbon dioxide; 44. Permeability by nitrogen; 45. Interface tension on the polymer–solvent interface; 46. Interface tension on the polymer–polymer interface; 47. Elasticity modulus in the area of rubber-like plateau; 48. Value of mechanical segment (molecular mass or polymerization degree, at which the rubbery state is displayed); 49. Activation energy of low-temperature γ-transition; 50. Newtonian viscosity of polymeric melts; 51. Prediction of solubility of polymers in organic solvents; 52. Influence of molecular mass on solubility; 53. Influence of orientation degree on solubility; 54. Prediction of miscibility of polymers. For low-molecular liquids, calculation of the properties listed below is involved in the program: 1. Molecular mass; 2. Molar volume; 3. The Van-der-Waals volume; 4. Density; 5. The Hildebrand parameter of solubility; 6. Surface tension; 7. Refractive index; 8. Dielectric constant; 9. Molar heat capacity in the solid state; 10. Molar heat capacity in the liquid state; 11. Molar refraction; 12. Molar polarizability; 13. Total cohesive energy; 14. The part of cohesive energy stipulated by hydrogen bonds; 15. The part of cohesive energy stipulated by dipole–dipole interaction; 16. The part of cohesive energy stipulated by dispersion interaction. Let us now turn to the principles of computer realization of the method of atomic constants. Preliminarily, it should be noted that practical validity of this method for solving the problem of searching for structures of the repeat unit providing required physicochemical properties of polymer is obvious. For example, in the frames of the method of constants, the values of typical temperatures (glass transition, degradation and melting) of polymers may be calculated and structures fulfilling the requirements on the total complex of the above-mentioned properties may be found. The structures found may form the base for applying software of computer planning of organic synthesis (CPOS). Let us consider in greater detail the calculation scheme of the method of constants on the example of calculation of the glass transition temperature of polymer Tg by the chemical formula of its repeat unit. Tg is calculated from equation (IV.39). The algorithm involves: 1) analysis of the structure by atoms and bonds; 2) separation of groups of atoms responsible for dipole–dipole interaction and hydrogen

585

bonds (including various cases of hydrogen bonding typical of polyamides); 3) determination of ortho-, metha- and para-substitutions for aromatic cycles in the main chain; 4) determination of double ‘carbon–carbon’ bonds in the main chain. As these rules are not in contradiction with each other, they assume a formalized description providing computer realization of the algorithm. Analogous analysis is performed in calculation of all other properties of polymers, networks and organic liquids. Computerization of the method of atomic constants is based on modularity, developed, compatible Windows-interface and conformity with CPOS programs. First two principles are well known and common. That is why let us dwell on the third principle. Being the original object for CPOS application, the structure of the target compound such be such as to ensure the required physico-chemical properties. Because searching for such structure is one of the tasks in the method of constants, coordinated description of structures in the method of constants and CPOS must be provided. Main blocks for CPOS programs are tables of atoms and the table of connectivity, which sets a selection of atoms and types of bonds in structure of the compound. For realization of the method of constants, it has been assumed desirable to form such tables for each calculated formula of the repeat unit. Of special attention was solution of the problem of user communication with the program. To depict the structure of the repeat unit of polymer, the planar (two-dimensional) image of disposition of atoms linked by definite types of bonds was used. If necessary, correspondence to linear Wiswesser formulae or to the writing form by Morgan is set with the help of tables of atoms and connectivity. The problem of the unambiguous interpretation of the structure of the repeat unit displayed on the screen is also important. For this purpose, control of the existence of data on atoms, groups of atoms and bonds in the database is used in the program. The program represents an assembly of modules providing: 1) the regime of editing the structure analyzed (entering, deleting and transposition of symbols of atoms, groups of atoms, data files recording on disk and reading from disk with the structure, etc.); 2) Analysis of the structure, the calculation of parameters and treatment of structural errors. Separation of modules of the second group enables to make the program easily transmitted between various systems. In practice, the program is used since 1994, first applied to DOS, and at present – to Windows’98.

Figure 106. Schematic representation of the region of possible existence for two properties of polymers (see text).

586

The program was tested on all compounds for which, on the one hand, experimental data were known and, on the other hand, calculations by the method of constants had been performed before. For example, the properties of polyolefins, vinyl, acrylic and dienic polymers, aliphatic and aromatic polymers (including polyamides), polyesters, polyimides, polysulfones, polyetherketones, etc. were calculated. It has been shown that the program successfully interprets the structure of the repeat unit in accordance with the calculation scheme of the method of constants, and provides for required accuracy of the calculation method of 3–5%. Examples of solving the direct, reverse and combined tasks of computerized synthesis of polymers based on the above-described programs are indicated in Appendices 1 – 3. Let us consider one more task, which may be solved by the above-described software. At present, in connection with the application of polymers to various spheres, requirements on their physical properties have become much stricter. Problems appear, solutions of which require polymers with a combination of a series of physical properties, simultaneously falling within the given ranges of the values of their characteristics. A question is raised about the combination of properties of polymers. Described in ref. [23] is the method of its solving with the help of ‘miscibility diagrams’, which consists of the following: assume that for a large group of polymers values of parameters of any two physical properties (from experiment or calculation) are known. Let us compose a diagram on which a specific point (Figure 106) corresponds to every polymer, where the abscissa axis (X) represents the values of the parameters of the first property, and on the ordinate axis (Y) – of the second one. All these points are located inside some region, which may be contoured. This region is called ‘the miscibility region’ of the mentioned physical properties for the given group of polymers. Attention should be paid to the fact that the density of points distribution in the region may be rather irregular. Such diagrams enable to estimate possibility of existence of polymers with given properties without additional consumption of labor. Actually, let the ranges (X2 – X1) and (Y2 – Y1) of values of properties, which must be possessed by polymers, be given. If the area determined by these ranges, falls within the miscibility region in the diagram, and some points exist there, i.e. the polymer may exist (case I). In the opposite case, obtaining the required polymer among polymers of these classes is of low probability (case II). To compose miscibility diagrams of this kind, either experimental data on various physical properties of polymers, or calculated data may be used. Combinations of these data may also be used. To calculate various physical properties of polymers, it is desirable to apply the computer program suggested in ref. [17], because with the help of it the properties of polymers either already synthesized, or the ones which may be, in fact, synthesized are considered. This very program was used in ref. [23] for composing combination regions of various properties of polymers. Among these properties the following were considered: density, ρ; refractive index, n; glass transition temperature, Tg; the relation of glass transition temperature to the melting point, Tg/Tm; the onset temperature of intense thermal degradation, Td; the Hildebrand solubility parameter, δ (density of cohesive energy); stress-optical coefficient Cσ. Ref. [23] indicates 27 diagrams built on a computer, which reflect combination of any two of the above-mentioned physical properties. Formulae, with the help of which the connection between two given parameters of properties may be searched, are shown in monograph [6].

587

Figure 107. Combination diagrams of Td and Tg (a), δ and Tg (b), δ and n (c) (see text).

As an example, Figure 107 shows miscibility diagrams of the following properties of polymers: glass transition temperature, Tg, and the onset temperature of intense thermal degradation, Td; Tg and solubility parameter, δ; refractive index, n, and δ. Let us consider several features of these and other similar diagrams. All regions of miscibility display different density. This means that there are sub-regions of miscibility exist containing the main bulk of polymers, and from these very sub-regions the chemical structure of polymers may be most easily chosen, which would fit the two given properties. All the diagrams display the presence of clearly expressed dependences of one property on another. Each of these dependencies characterizes definite series of polymers, for example, polyesters, polyamides, polyimides, etc.

588

Every diagram contains a point from which these dependences start. This is associated with the program feature [17], in which several homologues (CH2)n up to n = 20 are set as original fragments from the total 194 ones, and the point, from which these dependences start, characterizes polyethylene. Other features inherent to each diagram are clearly seen in corresponding figures. Having these diagrams, the possibility of obtaining polymers, which would possess the necessary combination of one of several properties may be predicted. For example, if polymers with the solubility parameter δ = 10 (cal/cm3)1/2 and Tg ≈ 300°C should be obtained, it may be made easily, because the point corresponded to these coordinates falls within the densest part of the diagram in Figure 107b. It would be quite difficult to produce a polymer with the glass transition temperature Tg ~ 500°C at the same solubility parameter, and at Tg = 600°C this becomes practically impossible, because the point corresponding to these coordinates is out of the range of the ‘miscibility’ region. This analysis may be easily performed for any diagram presented in ref. [23], as well as for a set of these diagrams, which enables us to predict the possibility of obtaining polymers with a a set of the required properties. Clearly, that if such diagrams are composed with the help of a computer program, according to which the polymer is composed from the finest ‘basic fragments’, the regions of miscibility of polymers will be significantly expanded by points reflecting properties of the very large number of polymers.

Appendix 1. Examples of solution of direct problems of polymer synthesis

Let us consider solution of the direct problem of estimation of the properties of polymers based on their chemical structure on an example of natural polymers. The specific features of these polymers are the following: 1) These polymers contain cyclic non-aromatic structures with a large number of OH-groups able to form hydrogen bonds. 2) Hydrogen bonds in these polymers may be formed both between neighboring chains and in the range of a single chain, and even in a single repeat unit. The scheme representing these interactions is shown in Figure 41. The presence of interchain and intrachain (including intra-unit) hydrogen bonds depends on the steric factor – dislocation of OH-groups in relation to the plane of circles. 3) Many high-molecular compounds, including cellulose, degrade thermally at heating before the process of devitrification is onset. Consequently, the glass transition temperature (the most important characteristic for polymers) may not be determined experimentally. 4) Many natural high-molecular compounds possess different units, i.e. contain fragments of different chemical nature in their chains. In this Section, we will discuss application of the approach developed for estimating the physical properties of polymers based on their chemical structure to natural high-molecular compounds, which possess quite complicated but well studied chemical structure. Table A-1-1 indicates chemical structures of 16 fragments of hemicelluloses structures, most often met in the structure of natural polymers. Of special attention is the fact that in some cases, despite the identical gross-formula, these fragments included in polymeric chains involve different physical properties of natural polymers. For example, for structures Fs1 and Fs2, gross-formulae are identical, but in the structure of Fs2 fragment OH-groups disposed at one side of the circle plane may form hydrogen bonds between each other. Hence, they participate in the formation of bonds between neighboring chains. This circumstance influences the glass transition temperature and some other properties of polymers containing these fragments. Let consider in greater detail the influence of the chemical structure of the above-mentioned natural polymers on their glass transition temperature. The glass transition temperature is calculated from equation (IV.39). Calculating the glass transition temperature from this equation, all fine features of the structure of the polymeric unit should be taken into account. If hydrogen bonds are formed between chains of the polymer (as a result, a physical network is formed at the sacrifice of intermolecular bonds), then parameter bh = –140⋅10–3 Å3K–1 should be introduced into equation (IV.39). In the case if OH-groups in the repeat unit are disposed in the manner that hydrogen bonds are formed between them in the range of the same unit, then the network of physical intermolecular bonds between chains is not formed (see Figure A-3-1,b), and for other groups, constants bd instead of bh must be introduced into equation (IV.39), which are introduced for every branching from the main chain of the polymer. The use of this approximation leads to values of the glass transition temperature and all other properties, which are displayed in Table A-1-2. It is clearly

590

observed that at the identical gross-formula the glass transition temperature of the structure Fs1 (541 K) is much higher than for the structure Fs2 (319 K). Therewith, such important characteristic of polymers as Mc (molecular mass of the mechanical segment of macromolecule, at which the rubbery state appears) is also changed. Saturation of the structure of hemicelluloses by OH- and COOH-groups leads to a sharp increase of the glass transition temperature, if these groups are capable of forming hydrogen bonds between chains of polymers. If hydrogen bonds are formed inside the repeat unit, the glass transition temperature is abruptly decreased. Let us consider in brief some other properties. The densities of solid samples are approximately equal, the lowest density being possessed by the structure Fs5, in which only a single OH-group is present. Consequently, this structure possesses the highest onset temperature of intense thermal degradation. It should be noted that, in some cases, the onset temperature of intense thermal degradation lies below the glass transition temperature or is close to it, which is the reason that experimental determination of Tg for natural polymers is rather difficult. The surface energy for 16 calculated structures changes in quite wide ranges, the lowest surface energy being typical of the structure Fs5 containing the smallest number of OH-groups, and the highest value is typical of the structure Fs8 containing the greatest number of these groups. Dielectric constants are approximately equal, the solubility parameter is quite high compared with synthetic polymers, refractive indices are approximately equal, and the temperature coefficient of volumetric expansion in the glassy state significantly depends upon the chemical structure of the fragment and changes antibatically to the glass transition temperature. Further on, let us consider the properties of hemicelluloses composed from the above-mentioned fragments. The chemical composition of 8 hemicelluloses and disposition of fragments in them are shown in Table A-1-3. The main properties of hemicelluloses of the given composition are shown in Table A-1-4. All properties were calculated on the basis of the molar content of fragments. Actually, the glass transition temperature of hemicelluloses of various chemical structures varies from 332 to 517 K, and the onset temperature of intense thermal degradation – from 517 to 580 K. The main characteristics indicated in Table A-1-4 show that the values of these temperatures for hemicelluloses C4 are very similar. Of special attention is the extremely low permeability of films from hemicelluloses to various gases. Experimental testing of the calculation results was performed on the basis of 1,4-β-D-glucomannan, the structure of which (F8) is shown on Table A-1-3. The thermomechanical curve for the initial preparation containing an uncontrolled amount of water is shown in Figure A-1-1. After drying in vacuum at a temperature of 80°C during two hours, the form of the thermomechanical curve changed a little (Figure A1-2). The form of Debye patterns (Figure A-1-3) also changes during such thermal processing in vacuum. They indicate that the glucomannan specimens studied were practically amorphous. The calculated value of the glass transition temperature (356 K) with the accuracy, typical of such calculations, coincides with the experimental value which, according to the thermomechanical curve (see Figure A-1-2), is 333 K. It is not inconcievable that after further drying the glass transition temperature might be somewhat increased. The second peak of deformation on the thermomechanical curve appears at a temperature close to the temperature of intense thermal degradation and is probably associated with it.

591

Needless to say that the results of all the calculations performed may be considered only as preliminary ones, and in future they must be related to not only with the primary, but also with the secondary structure of chemicelluoses. Table A-1-1 Chemical formulae of the fragments of hemicellulose structures Designation of structures Chemical formulae of structures fragments fragments 1 2 CH2OH Fs1 O O

OH OH

Fs2

CH2OH O O

OH

HO

Fs3 O O

OH OH

Fs4

COOCH3 O O OH OH

Fs5 O O

COOCH3 OH

Fs6 O O

OH O OH HO OH

O

592

1 Fs7

2 COOH O O

OH O

O

OH HO OH

Fs8 O O

OH

O

OH

O

OH OH

Fs9 O O

OH O

O

COOCH3 HO COOCH3

Fs10 O O

OH

COOH O

O HO OCH3 OH

593

1 Fs11

2 OH HO OH O

O

CH2OH CH2 O O

OH

HO

CH2OH

Fs12

O O

OH

O

OH OH

OH O

CH2OH

Fs13 O O

OH

OH

O

H3CO OH O COOH

Fs14

CH2OH O

HO OH

OH O CH2 HO

O

O

HO

O OH

CH2OH O

OH

594

1 Fs15

2 O O

O OH O

O

OCH3 OH

OH OH

O OH

Fs16

CH2OH

CH2OH O

HO

COOH

O

HO

OH

OH OH

OH OH

O CH2

HO

O

O

OH

O

OH

HO

O

O CH2

CH2 HO

O

O OH

O

OH

O

OH

Table A-1-2 Properties of hemicellulose structure fragments Values of properties of structures fragments Designation of structures ∑ ∆Vi , Å3 ρ, 3 Tg, K Td, K properties MM Vm, cm3/mol g/cm i Fs1 162 116 132 1.39 541 556 Fs2 162 116 132 1.39 319 556 Fs3 132 95 107 1.40 511 571 Fs4 190 135 152 1.41 459 585 Fs5 174 128 145 1.36 403 607 Fs6 264 189 214 1.40 574 527 Fs7 308 212 240 1.45 641 520 Fs8 250 174 197 1.44 427 520 Fs9 348 256 290 1.36 436 570 Fs10 322 229 259 1.41 518 530 Fs11 324 233 263 1.39 431 511 Fs12 324 233 263 1.39 432 511 Fs13 322 229 259 1.41 518 530 Fs14 486 349 395 1.39 388 525 Fs15 454 324 367 1.40 552 517 Fs16 929 661 748 1.41 437 513 Values of properties of structures fragments Designation of E*, E, γ, αh, αdd, αd, ε, structures properties kcal/mol rel. un. rel. un. rel. un. rel. un. kcal/mol dyn/cm Fs1 45.5 23.9 0.72 0.00 0.28 3.36 49.6 Fs2 45.5 23.9 0.72 0.00 0.28 3.36 49.6 Fs3 54.8 18.2 0.63 0.07 0.30 3.34 37.96

595

Fs4 Fs5 Fs6 Fs7 Fs8 Fs9 Fs10 Fs11 Fs12 Fs13 Fs14 Fs15 Fs16 Designation of structures properties Fs1 Fs2 Fs3 Fs4 Fs5 Fs6 Fs7 Fs8 Fs9 Fs10 Fs11 Fs12 Fs13 Fs14 Fs15 Fs16

46.4 37.9 53.0 58.9 62.1 36.5 49.9 45.5 45.5 49.9 45.5 50.3 46.6 Edd+h, kcal/mol 37.60 22.20 27.70 30.18 21.53 64.07 80.77 39.00 48.40 63.65 84.83 84.83 63.65 70.17 99.57 157.40

Designation of structures properties

αL⋅104, K−1

Fs1 Fs2 Fs3 Fs4 Fs5 Fs6 Fs7 Fs8 Fs9 Fs10 Fs11 Fs12 Fs13 Fs14 Fs15 Fs16

4.46 4.46 4.71 5.25 5.97 4.20 3.76 3.59 5.52 4.65 4.04 4.04 4.65 4.18 4.37 3.90

Designation of structures properties

P, cm3/mol

Fs1 Fs2

51.2 51.2

21.5 0.54 0.11 0.35 3.49 16.8 0.34 0.21 0.44 3.33 35.2 0.66 0.03 0.31 3.34 43.3 0.67 0.05 0.28 3.56 37.8 0.61 0.13 0.26 3.45 32.3 0.36 0.18 0.46 3.33 39.7 0.58 0.09 0.33 3.43 47.8 0.72 0.00 0.28 3.36 47.8 0.72 0.00 0.28 3.36 39.7 0.58 0.09 0.33 3.43 71.7 0.72 0.00 0.28 3.36 56.7 0.61 0.06 0.33 3.40 139.1 0.70 0.03 0.27 3.38 Values of properties of structures fragments Ed, n, ∆U, δ, kcal/mol rel. un. kcal/mol (cal/cm3)1/2 12.0 14.30 1.49 27.9 12.0 14.30 1.49 27.9 10.26 13.85 1.49 26.4 15.97 12.65 1.48 23.7 17.23 11.43 1.48 20.8 21.15 13.63 1.49 29.7 22.83 14.29 1.48 33.1 18.74 14.72 1.48 34.7 35.45 11.23 1.48 22.6 26.87 13.16 1.48 26.8 24.60 14.34 1.49 30.8 24.60 14.34 1.49 30.8 26.87 13.16 1.48 26.8 36.61 14.34 1.49 29.8 37.76 13.23 1.48 28.5 69.68 14.51 1.49 31.9 Values of properties of structures fragments n l, R, C ps , C pl , rel. un. cm3/mol cal/(mol⋅deg) cal/(mol⋅deg) 467 52.7 67.6 33.37 487 52.7 67.6 33.37 431 41.5 54.0 27.23 435 57.6 76.1 38.12 376 52.3 69.5 36.59 609 83.8 107.3 54.45 707 95.7 123.7 60.61 693 77.9 100.3 49.83 512 105.4 138.3 73.18 586 99.9 129.4 65.30 678 106.1 134.5 66.74 678 106.1 134.5 66.74 586 99.9 129.4 65.30 750 159.5 201.3 100.11 700 142.1 182.7 92.57 995 302.7 381.4 189.45 Values of properties of structures fragments Cσ, µ, PO 2 , PCO 2 , B D DU DU 104 104

1.55 1.55

0.03 0.03

0.07 0.07

46.15 38.76 85.22 103.60 90.93 83.85 90.52 109.43 109.43 90.52 158.65 137.33 321.32

αG⋅104, K–4 1.78 3.01 1.88 2.09 2.38 1.67 1.50 1.43 2.20 1.85 2.22 2.22 1.85 1.66 1.74 1.55 Ml 79034 46602 56980 82740 65434 160900 217850 110198 178375 188853 219817 219817 188853 245324 318105 653567 PN 2 , DU 0.00 0.00

596

Fs3 Fs4 Fs5 Fs6 Fs7 Fs8 Fs9 Fs10 Fs11 Fs12 Fs13 Fs14 Fs15 Fs16

41.4 61.1 56.0 82.9 97.8 78.3 112.0 102.5 102.3 102.3 102.5 153.5 144.0 292.6

132 91 104 108 99 113 95 108 95 95 108 98 101 96

1.39 1.70 1.63 1.99 2.16 1.93 2.32 2.21 2.21 2.21 2.21 2.72 2.63 3.76

Note: MM is the molecular mass; Vm is the molar volume;

0.04 0.22 1.38 0.11 0.02 0.00 3.17 0.11 0.03 0.03 0.11 0.03 0.15 0.01

∑ ∆V i

0.09 0.56 4.12 0.27 0.03 0.00 10.23 0.26 0.07 0.07 0.26 0.07 0.37 0.03

0.01 0.03 0.24 0.02 0.00 0.00 0.60 0.02 0.00 0.00 0.02 0.00 0.02 0.00

is the Van-der-Waals volume; ρ is

i

the density; Tg is the glass transition temperature; Td is the onset temperature of intense thermal degradation; γ is the surface energy; E* is the cohesive energy; αh is the relation of hydrogen bonds energy to total cohesive energy; αdd is the relation of dipole–dipole interaction energy to total cohesive energy; αd is the relation of dispersion interaction energy to total cohesive energy; ε is the dielectric constant; E is the total energy of intermolecular interaction; Edd+h is the energy of dipole–dipole interaction and hydrogen bonds; Ed is the energy of dispersion interaction; δ is the solubility parameter; n is the refractive index; ∆U is the activation energy of low-temperature γ-transition; αG is the coefficient of thermal expansion in the glassy state; αL is the coefficient of thermal expansion in the rubbery state; nl is the polymerization degree of the polymer when the rubbery state appears; C ps is the molar heat capacity in the glassy state; C pl is the molar heat capacity in the rubbery state; R is the molar refraction; Ml is the molecular mass of polymer when the rubbery state appears; P is the polarizability; Cσ is the stress-optical coefficient; µ is the dipole moment; PO 2 , PCO 2 , PN 2 are the permeabilities by oxygen, carbon dioxide and nitrogen, respectively (the unit of measurement of permeability 1 DU = 0.45⋅10–10 cm2⋅s–1⋅atm–1).

ε

 





ε

 





Figure A-1-1. Thermomechanical curve of initial 1,4-β-D-glucomannan.

597

Table A-1-3 Designations of hemicelluloses 1 C1

Fragmentary composition and structural formulae of hemicelluloses Fragmentary composition and name Structural formulae of hemicelluloses of natural biopolymer 2 3 (Fs12)3–(Fs2)42–(Fs1)23–(Fs11)5 aspen glucomannan [208]

OH OH OH O

O

CH 2O H CH 2O H

CH2 O H O

O

O

OH

CH2

CH2 O H O OH

OH

O

O O

OH

42

OH

O OH

OH

23

5

O

OH OH

OH O

CH2 O H

C2

(Fs12)2–(Fs2)13–(Fs1)5–(Fs12)3 spruce glucomannan [116]

3

CH2 O H

CH 2O H O

O

O OH

OH

CH 2OH

CH2 O H O

O

OH

O 13

O

OH

OH

O OH

5

O

O

OH OH

OH

OH

OH

O

O 2

CH2 O H

3

597

CH2 O H

OH

598

2 (Fs10)19–(Fs3)125–(Fs6)5 aspen 4-0-methylglucuronoxylan [117]

3

598

1 C3

O

O

O

OH

O

OH

O

O OH

OH

125

COOH O

OH

OH

OH

OCH3 OH C4

O

O

O

OH

19

5

Fs3–Fs10–Fs3–Fs6–Fs8–Fs15 spruce arabinoglucuronoxylan [118] O O

O O

OH

O

O O

OH

OH

OH

OH OH

O O

O

O

O

OH

O

O

OCH3

O OH

OH

OH OH

O

OH

OH

OCH3

O

O

COOH

O

O

O

OH

OH

OH

OH OH

OH

O OH

COOH

599

1 C5

2 Fs7–(Fs4)50–Fs7 Galactouronan from plants [130]

3

COOH

COOCH3

COOH

O

O

O OH

O

OH

O

O OH

O

OH

50

O OH

OH

OH

OH

OH

OH C6

O

O

(Fs9)8–(Fs5)21–(Fs3)52–(Fs13)15–(Fs5)3 birch xylan [118] O

O

O

O

O

O O OH

O

CO OCH 3 OH

O 21

O

OH OH

OH COOCH3 3

52 OH

O

COO CH3

O

OH

O

H3CO OH

OH O COO CH3 8

COOH

15

599

600

2 Fs16–Fs14 larch arabinogalactan [60]

3 CH2OH

CH2OH O

OH

CH2OH O

OH

OH

OH

O

CH2OH

CH2OH

OH

OH

O

O OH

O OH

OH

OH

O

CH2OH O

OH

CH2OH O

OH

OH

CH2OH O

O

O

O OH

OH

OH

CH2OH O

O

OH

O

CH2OH O

CH2OH

O

OH

OH

CH2OH O

O

O CH2

O

O

OH

OH

O CH2

OH

O

[Fs2–Fs1–(Fs2)4–Fs1–Fs2]48 1,4-β-D-glucomannan [2]

OH

O

O

OH

OH

C8

OH

CH2

O

O

OH

OH

CH2 OH

O

OH

OH

OH

600

1 C7

OH

OH

O O

OH

O OH

OH

OH

48

Note: chemical formulae of hemicelluloses and the sample of 1,4-β-D-glucomannan shown in the Table were presented by Doctors of Biology V.D. Shcherbukhin and A.V. Bolobova.

601

Table A-1-4 Properties of hemicelluloses Properties of hemicelluloses δ, Tg, Td, γ, αG⋅10–4, (cal/cm3) K K dyn/cm K-1 1/2

Designations of PO 2 , PCO 2 , ∑ ∆Vi , ρ, hemiceli 3 DU DU g/cm luloses Å3 C1 146 1.39 332 546 2.90 14.3 45.5 0.03 0.07 C2 160 1.39 367 538 2.62 14.3 45.5 0.03 0.07 C3 130 1.40 499 557 1.92 13.7 53.4 0.05 0.12 C4 211 1.41 517 530 1.83 13.6 53.3 0.05 0.11 C5 156 1.42 464 580 2.07 12.8 47.2 0.19 0.47 C6 154 1.38 435 566 2.21 12.8 47.0 0.21 0.54 C7 571 1.40 419 517 1.62 14.5 46.2 0.02 0.04 C8 1,052 1.39 356 556 2.70 14.3 45.5 0.03 0.07 Note: Designations of properties of hemicelluloses correspond to those used in Table A-1-2.

PN 2 , DU 0.00 0.00 0.01 0.01 0.03 0.03 0.00 0.00

Figure A-1-2. Thermomechanical curve of dry 1,4-β-D-glucomannan.

Figure A-1-3. Debye patterns of powder-like initial (a) and dry (b) specimens of 1,4-β-D glucomannan

Appendix 2. Examples of solving the reverse problem of polymer synthesis

The reverse task concludes in synthesis, i.e. in finding a chemical formula or formulae of polymers whose properties must fall within the given range. Let poly(ether ketones) be required to be synthesized possessing the glass transition temperature above 493 K and simultaneously with the melting point below 653 K (the latter is important for successful processing). There are no limits on the values of the rest of properties (solubility parameter, density, dielectric constant, refractive index). The given task will be solved with the help of the program operating with the finest basic fragments. Let us select the number of basic fragments in repeat units, equal to 7. The solution of the given task yields the chemical formulae of poly(ether ketones), shown in Table A-2-1 (450 000 structures were calculated, from which only 24 structures fell within the required range of properties). Let us widen a little the range of properties by introduction of the lower border for the glass transition temperature, equal to 483 K. In this case, from the same number of calculated structures 84 structures of poly(ether ketones) were selected, the chemical formulae of which are shown in Table A-2-2.

603

Table A-2-1 Chemical formulae of poly(ether ketones) with the glass transition temperature Tg above 493 K and the melting point Tm below 653 K Properties of poly(ether ketones) Numbers of poly(ether ketones) Chemical formulae of poly(ether ketones) n, Tg, Tm, δ, ρ, ε, structures rel. un. K K (cal/cm3)1/2 g/cm3 rel. un. 1 2 3 4 5 6 7 8 1 495 632 9.78 1.24 2.95 1.62 −COpC6H4OpC6H4phtC10H6mC6H4− 2 495 632 9.78 1.24 2.95 1.62 −COpC6H4OpC6H4phtmC6H4ClOH6− 3 498 652 9.63 1.20 2.83 1.59 −COpC6H4OpC6H4flClOH6mC6H4− 4 498 652 9.63 1.20 2.83 1.59 −COpC6H4OpC6H4flmC6H4C10H6− 5 495 632 9.78 1.24 2.95 1.62 −COpC6H4OpC6H4C10H6phtmC6H4− 6 498 652 9.63 1.20 2.83 1.59 −COpC6H4OpC6H4C10H6flmC6H4− 7 495 632 9.78 1.24 2.95 1.62 −COpC6H4OpC6H4mC6H4phtC10H6− 8 498 652 9.63 1.20 2.83 1.59 −COpC6H4OpC6H4mC6H4flC10H6− 9 495 632 9.78 1.24 2.95 1.62 −COC10H6phtpC6H4ΟpC6H4mC6H4− 10 495 632 9.78 1.24 2.95 1.62 −COC10H6phtmC6H4pC6H4ΟpC6H4− 11 498 652 9.63 1.20 2.83 1.59 −COC10H6flpC6H4ΟpC6H4mC6H4− 12 498 652 9.63 1.20 2.83 1.59 −COC10H6flmC6H4pC6H4ΟpC6H4− 13 495 632 9.78 1.24 2.95 1.62 −COC10H6pC6H4ΟpC6H4phtmC6H4− 14 498 652 9.63 1.20 2.83 1.59 −COC10H6pC6H4ΟpC6H4flmC6H4− 15 495 632 9.78 1.24 2.95 1.62 −COC10H6mC6H4phtpC6H4ΟpC6H4− 16 498 652 9.63 1.20 2.83 1.59 −COC10H6mC6H4flpC6H4ΟpC6H4− 17 495 632 9.78 1.24 2.95 1.62 −COmC6H4phtpC6H4ΟpC6H4C10H6− 18 495 632 9.78 1.24 2.95 1.62 −COmC6H4phtC10H6pC6H4ΟpC6H4− 19 489 652 9.63 1.20 2.83 1.59 −COmC6H4flpC6H4ΟpC6H4C10H6− 20 498 652 9.63 1.20 2.83 1.59 −COmC6H4flC10H6pC6H4ΟpC6H4− 21 495 632 9.78 1.24 2.95 1.62 −COmC6H4pC6H4ΟpC6H4phtC10H6− 22 498 652 9.63 1.20 2.83 1.59 −COmC6H4pC6H4ΟpC6H4flC10H6− 23 495 632 9.78 1.24 2.95 1.62 −COmC6H4C10H6phtpC6H4ΟpC6H4− 24 498 652 9.63 1.20 2.83 1.59 −COmC6H4C10H6flpC6H4ΟpC6H4− Note: Designations are the same as in the Note to Table A-2-2. 603

604

26

489

550

9.71

1.24

2.96

1.62

−COpC6H4ΟpC6H4mC6H4phtmC6H4−

604

Table A-2-2 Chemical formulae of poly(ether ketones) with the glass transition temperature Tg above 483 K and the melting point Tm below 653 K Properties of poly(ether ketones) Numbers of poly(ether ketones) Chemical formulae of poly(ether ketones) n, Tg, Tm, δ, ρ, ε, structures rel. un. K K (cal/cm 3)1/2 g/cm3 rel. un. 1 2 3 4 5 6 7 8 1 489 626 9.71 1.24 2.96 1.62 −COmC6H4ΟpC6H4phtmC6H4pC6H4− 2 489 626 9.71 1.24 2.96 1.62 −COmC6H4ΟpC6H4phtpC6H4mC6H4− 3 493 648 9.55 1.19 2.83 1.58 −COmC6H4ΟpC6H4flmC6H4pC6H4− 4 493 648 9.55 1.19 2.83 1.58 −COmC6H4ΟpC6H4flmC6H4pC6H4− 5 489 626 9.71 1.24 2.96 1.62 −COmC6H4ΟpC6H4mC6H4phtpC6H4− 6 493 648 9.55 1.19 2.83 1.58 −COmC6H4ΟpC6H4mC6H4flpC6H4− 7 489 626 9.71 1.24 2.96 1.62 −COmC6H4ΟpC6H4pC6H4phtmC6H4− 8 493 648 9.55 1.19 2.83 1.58 −COmC6H4ΟpC6H4pC6H4flmC6H4− 9 489 626 9.71 1.24 2.96 1.62 −COpC6H4ΟmC6H4phtmC6H4pC6H4− 10 489 626 9.71 1.24 2.96 1.62 −COpC6H4ΟmC6H4phtpC6H4mC6H4− 11 493 648 9.55 1.19 2.83 1.58 −COpC6H4ΟmC6H4flmC6H4pC6H4− 12 493 648 9.55 1.19 2.83 1.58 −COpC6H4ΟmC6H4flpC6H4mC6H4− 13 498 626 9.71 1.24 2.96 1.62 −COpC6H4ΟmC6H4mC6H4phtpC6H4− 14 493 648 9.55 1.19 2.83 1.58 −COpC6H4ΟmC6H4mC6H4flpC6H4− 15 489 626 9.71 1.24 2.96 1.62 −COpC6H4ΟmC6H4pC6H4phtmC6H4− 16 493 648 9.55 1.19 2.83 1.58 −COpC6H4ΟmC6H4pC6H4flmC6H4− 17 495 632 9.78 1.24 2.95 1.62 −COpC6H4ΟpC6H4phtC10H6mC6H4− 18 495 632 9.78 1.24 2.95 1.62 −COpC6H4ΟpC6H4phtmC6H4C10H6− 19 498 550 9.71 1.24 2.96 1.62 −COpC6H4ΟpC6H4phtmC6H4mC6H4− 20 498 652 9.63 1.20 2.83 1.59 −COpC6H4ΟpC6H4flC10H6mC6H4− 21 498 652 9.63 1.20 2.83 1.59 −COpC6H4ΟpC6H4flmC6H4C10H6− 22 493 578 9.55 1.19 2.83 1.58 −COpC6H4ΟpC6H4flmC6H4mC6H4− 23 495 632 9.78 1.24 2.95 1.62 −COpC6H4ΟpC6H4C10H6phbmC6H4− 24 498 652 9.63 1.20 2.83 1.59 −COpC6H4ΟpC6H4C10H6flmC6H4− 25 495 632 9.78 1.24 2.95 1.62 −COpC6H4ΟpC6H4mC6H4phtC10H6−

605

2 498 493 495 495 498 498 495 498 495 498 489 489 495 480 495 489 489 489 493 493 498 493 498 493 493 493 489 493 489 493 495

3 652 578 632 632 652 652 632 652 632 652 626 626 632 550 632 550 626 626 648 648 652 578 652 578 648 648 626 648 626 648 632

4 9.63 9.55 9.78 9.78 9.63 9.63 9.78 9.63 9.78 6.63 9.71 9.71 9.78 9.71 9.78 9.71 9.71 9.71 9.55 9.55 9.63 9.55 9.63 9.55 9.55 9.55 9.71 9.55 9.71 9.55 9.78

5 1.20 1.19 1.24 1.24 1.20 1.20 1.24 1.20 1.24 1.20 1.24 1.24 1.24 1.24 1.24 1.24 1.24 1.24 1.19 1.19 1.20 1.19 1.20 1.19 1.19 1.19 1.24 1.19 1.24 1.19 1.24

6 2.83 2.83 2.95 2.95 2.83 2.83 2.95 2.83 2.95 2.83 2.96 2.96 2.95 2.96 2.95 2.96 2.96 2.96 2.83 2.83 2.83 2.83 2.83 2.83 2.83 2.83 2.96 2.83 2.96 2.83 2.95

7 1.59 1.58 1.62 1.62 1.59 1.59 1.62 1.59 1.62 1.59 1.62 1.62 1.62 1.62 1.62 1.62 1.62 1.62 1.58 1.58 1.59 1.58 1.59 1.58 1.58 1.58 1.62 1.58 1.62 1.58 1.62

8 −COpC6H4ΟpC6H4mC6H4flC10H6− −COpC6H4ΟpC6H4mC6H4flmC6H4− −COC10H6phtpC6H4OpC6H4mC6H4− −COC10H6phtmC6H4pC6H4OpC6H4− −COC10H6flpC6H4ΟpC6H4mC6H4− −COC10H6flmC6H4pC6H4OpC6H4− −COC10H6pC6H4ΟpC6H4phtmC6H4− −COC10H6pC6H4ΟpC6H4flmC6H4− −COC10H6mC6H4phtpC6H4OpC6H4− −COC10H6mC6H4flpC6H4OpC6H4− −COmC6H4phtmC6H4ΟpC6H4pC6H4− −COmC6H4phtpC6H4ΟmC6H4pC6H4− −COmC6H4phtpC6H4ΟpC6H4C10H6− −COmC6H4phtpC6H4ΟpC6H4mC6H4− −COmC6H4phtC10H6pC6H4ΟpC6H4− −COmC6H4phtmC6H4pC6H4ΟpC6H4− −COmC6H4phtpC6H4mC6H4ΟpC6H4− −COmC6H4phtpC6H4pC6H4ΟmC6H4− −COmC6H4flmC6H4ΟpC6H4pC6H4− −COmC6H4flpC6H4ΟmC6H4pC6H4− −COmC6H4flpC6H4ΟpC6H4C10H6− −COmC6H4flpC6H4ΟpC6H4mC6H4− −COmC6H4flC10H6pC6H4ΟpC6H4− −COmC6H4flmC6H4pC6H4ΟpC6H4− −COmC6H4flpC6H4mC6H4ΟpC6H4− −COmC6H4flpC6H4pC6H4ΟmC6H4− −COmC6H4mC6H4ΟpC6H4phtpC6H4− −COmC6H4mC6H4ΟpC6H4flpC6H4− −COmC6H4pC6H4ΟmC6H4phtpC6H4− −COmC6H4pC6H4ΟmC6H4flpC6H4− −COmC6H4pC6H4ΟpC6H4phtC10H6−

605

1 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57

606

C

–pht–

C

O, C

8 −COmC6H4pC6H4ΟpC6H4phtmC6H4− −COmC6H4pC6H4ΟpC6H4flC10H6− −COmC6H4pC6H4ΟpC6H4flmC6H4− −COmC6H4C10H6phtpC6H4OpC6H4− −COmC6H4C10H6flpC6H4OpC6H4− −COmC6H4mC6H4phtpC6H4OpC6H4− −COmC6H4mC6H4flpC6H4OpC6H4− −COmC6H4pC6H4phtmC6H4OpC6H4− −COmC6H4pC6H4phtpC6H4OmC6H4− −COmC6H4mC6H4flmC6H4OpC6H4− −COmC6H4pC6H4flpC6H4OmC6H4− −COpC6H4phtmC6H4ΟpC6H4mC6H4− −COpC6H4phtpC6H4ΟmC6H4mC6H4− −COpC6H4phtmC6H4mC6H4ΟpC6H4− −COpC6H4phtmC6H4pC6H4ΟmC6H4− −COpC6H4flmC6H4ΟpC6H4mC6H4− −COpC6H4flpC6H4ΟmC6H4mC6H4− −COpC6H4flmC6H4mC6H4ΟpC6H4− −COpC6H4flmC6H4pC6H4ΟmC6H4− −COpC6H4mC6H4ΟpC6H4phtmC6H4− −COpC6H4mC6H4ΟpC6H4flmC6H4− −COpC6H4pC6H4ΟmC6H4phtmC6H4− −COpC6H4pC6H4ΟmC6H4flmC6H4− −COpC6H4mC6H4phtmC6H4OpC6H4− −COpC6H4mC6H4phtpC6H4OmC6H4− −COpC6H4mC6H4flmC6H4ΟpC6H4− −COpC6H4mC6H4flpC6H4OmC6H4−

O

–fl–

, –mC6H4–

, –pC6H4–

.

2) Tg is the glass transition temperature; Tm is the melting point; δ is the solubility parameter; ρ is the density; ε is the dielectric constant; n is the refractive index.

606

1 2 3 4 5 6 7 58 489 550 9.71 1.24 2.96 1.62 59 498 652 9.63 1.20 2.83 1.59 60 493 578 9.55 1.19 2.83 1.58 61 495 632 9.78 1.24 2.95 1.62 62 498 652 9.63 1.20 2.83 1.59 63 489 550 9.71 1.24 2.96 1.62 64 493 578 9.55 1.19 2.83 1.58 65 489 626 9.71 1.24 2.96 1.62 66 489 626 9.71 1.24 2.96 1.62 67 493 648 9.55 1.19 2.83 1.58 68 493 648 9.55 1.19 2.83 1.58 69 489 626 9.71 1.24 2.96 1.62 70 489 626 9.71 1.24 2.96 1.62 71 489 626 9.71 1.24 2.96 1.62 72 489 626 9.71 1.24 2.96 1.62 73 493 648 9.55 1.19 2.83 1.58 74 493 648 9.55 1.19 2.83 1.58 75 493 648 9.55 1.19 2.83 1.58 76 493 648 9.55 1.19 2.83 1.58 77 489 626 9.71 1.24 2.96 1.62 78 493 648 9.55 1.19 2.83 1.58 79 489 626 9.71 1.24 2.96 1.62 80 493 648 9.55 1.19 2.83 1.58 81 489 626 9.71 1.24 2.96 1.62 82 489 626 9.71 1.24 2.96 1.62 83 493 648 9.55 1.19 2.83 1.58 84 493 648 9.55 1.19 2.83 1.58 Note: 1) The following designations of the basic fragments are used in chemical formulae of poly(ether ketones):

Appendix 3. The example of solving the complex problem – analysis of the chemical structure of phenol formaldehyde resin

The example of solving the complex problem of polymer synthesis belongs to the analysis of the chemical structure of one of the representatives of polymer networks – phenol-formaldehyde resin widely applied, in particular, to production of pressed woody articles. Before analyzing the structure of a polymer network based on phenol formaldehyde resin, it should be noted that although phenol formaldehyde resins (PFR) are one of the oldest representatives of high cross-linked polymer networks, their structure is still of great interest [201]. There is a broad list of works devoted to the analysis of the structure and properties of PFR. However, there is still no general point of view on the chemical structure of the network formed by these resins. The reason is, apparently, in the fact that due to proceeding of the PFR curing the network structure obtained may be different. Another reason is associated with the fact that OH-groups capable of forming hydrogen bonds are present in the network structure. These hydrogen bonds may be displayed in both intermolecular interaction of these groups disposed in neighboring PFR chains and at the sacrifice of the same interaction OH-groups disposed in the same repeat unit.

Figure A-3-1. Schematic representation of a network of hydrogen bonds (a) appearing between neighboring chains and (b) within the same repeat unit.

The meaning of the notions of ‘intermolecular interaction’ and ‘intramolecular interaction’ in the context of the approach under discussion should be explained with the help of a simplified scheme represented in Figure A-3-1. Figure A-3-1a demonstrates the case corresponding to the intermolecular interaction, which leads to the formation of a network of physical (hydrogen) bonds linking chains. Clearly, the formation of this network promotes an increase of the glass transition temperature. The case displayed in Figure A-3-1b reflects the situation when two OH-groups are in the same unit of the linear fragment of PFR, and hydrogen bonding then occurs in the frames of the same unit, and the network of physical bonds between neighboring ones is not formed. This problem was analyzed above in detail on the example of poly(imidophenyl quinoxalines) and poly(amidophenyl quinoxalines). It stands to reason that various types of intermolecular interaction (hydrogen bonding) causing a significant influence on PFR properties, as it will be shown below, may coexist in reality. In cured PFR, according to the existing data, the following structures (repeating fragments of the network) may exist:

608

Structure 1:

CH2

CH2

CH2 OH

This structure is included only in the network the repeat unit of which is marked by dotted lines. The glass transition temperature of the network will be calculated using equation (IV.85). According to the definition of the chemical structure of the cross-linked point of the network, the structure of the cross-linked point is formed by a selection of atoms, including the one from which the chain is branched, and the neighboring atoms with their substituents, as well. For structure 1 of the network, the structure of the cross-linked point is marked by dotted lines: CH2

OH Clearly, the whole cross-linked point contains three such selections of atoms. Then for the structure 1, we obtain*:    ∑ ∆Vi  = 3(∆VC,19 + ½∆VC,12 + ½⋅2∆VH,124) + 2∆VC,18 + 2∆VH,124 + ∆VC,20 + ∆VO,133  i 1 + ∆VH,125 = 3(8.4 + ½⋅12.6 + ½⋅2⋅2.0) + 2⋅12.7 + 2⋅2.0 + 11.65 + 5.2 + 4.7 = 101.1 Å3;    ∑ ai ∆Vi + ∑ b j  = aC(2∆VC,18 + ∆VC,20) + aH(2∆VH,124 + ∆VC,125) + aO,s∆VO,133 + bh  i  j  1 = [0.021(2⋅12.7 + 11.65) + 19.93(2⋅2.0 + 4.7) 7.03⋅5.2 – 139.6]⋅10–3 = 71.6⋅10–3 Å3 K–1;    ∑ K i ∆Vi  = KC(∆VC,19 + ½∆VC,12) + KH2⋅1/2∆VH,124 = [1.15(8.4 + ½⋅12.6) +  i 1 2.307⋅2⋅1/2⋅2.0]⋅10–3 = 21.5⋅10–3 Å3K–1.       Substituting values of  ∑ ∆Vi  ,  ∑ ai ∆Vi + ∑ b j  , and  ∑ K i ∆Vi  into j  i 1  i  i 1 1 equation (IV.85), we obtain the following value of the glass transition temperature of the network in the case of the ideal structure 1:

*

Here and below, numbers of atoms correspond to their numbers in Table 3.

609

Tg =

101.1 ⋅103 = 743 K. 21.5 ⋅ 3 + 71.6

(A-3-1)

The calculated value of Tg is significantly above the range of experimental values, Tg,exper, shown in Table A-3-1 and, consequently, the real PFR network will differ by structure from the ideal structure 1. Table A-3-1 Experimental values of the characteristics of cured PFR Property name Unit Limits of measurements Glass transition temperature, Tg,exper K 343–433 1.24–1.33 g/cm3 Density, ρ (exper.) Refractive index, n (exper.) rel. un. 1.57–1.63 K–1 Coefficient of volumetric expansion, αG (exper.) (1.80–2.25)⋅10–4 s 29–48 cal/(mol⋅deg) Molar heat capacity, (exper.)

Cp

Structure 2: CH2

CH2 HO CH2 OH

This possible structure of PFR relates to a linear polymer (the repeat unit is marked by dotted lines). For this structure, V ∆ ∑ i = 3∆VC,19 + 2∆VC,18 + 1/2⋅2∆VC,12 + ∆VC,20 + ∆VC,40 + 4⋅1/2∆VH,124 + 2∆VH,124 i

+ 2∆VH,124 + 2∆VH,125 + ∆VO,133 + ∆VO,132 = 3⋅8.4 + 2⋅12.7 + 1/2⋅2⋅12.6 + 11.65 + 16.0 + 4⋅1/2⋅2.0 + 2⋅2.0 + 2⋅2.0 + 2⋅4.7 + 5.2 + 5.6 = 123.05 Å3; ∑ ai ∆Vi = aC(3∆VC,19 + 2∆VC,18 + 1/2⋅2∆VC,12 + ∆VC,20 + ∆VC,40) + aH(4⋅1/2∆VH,124 + i

2∆VH,124 + 2∆VH,124 + 2∆VH,125) + aO,s(∆VO,133 + ∆VO,132) = [0.021(3⋅8.4 + 2⋅12.7 + 1/2⋅2⋅12.6 + 11.65 + 16.0) + 19.98(4⋅1/2⋅2.0 + 2⋅2.0 + 2⋅2.0 + 2⋅4.7) + 7.03(5.2 + 5.6)]⋅10–3 = 505.4⋅10–3 Å3K–1. If both OH-groups participate in the formation of hydrogen bonds between chains (as shown in Figure A-3-1), then

∑bj j

= 2bh + bM;

∑bj

= (–139.6⋅2 + 16.0)⋅10–3 = –263.2⋅10–3 Å3K–1.

j

Substitution of all values of parameters into equation (IV.41) leads to the following value of Tg of the ideal structure 2: Tg =

123.05 ⋅103 = 508 K. 505.4 − 263.2

(A-3-2)

610

Similar to the structure 1, this value is higher than the experimental temperature interval in which the α-transition is observed for PFR. Structure 3:

CH2

CH2 OH

This structure forms only linear chains, the repeat unit of which is marked by dotted lines. For structure 3, we obtain:    ∑ ∆Vi  = 2∆VC,19 + ∆VC,20 + 3∆VC,18 + 1/2⋅2∆VC,12 + 1/2⋅4∆VH,124 + ∆VH,125 +  i 3 3∆VH,124 + ∆VO,133 = 2⋅8.4 + 11.65 + 3⋅12.7 + 1/2⋅2⋅12.6 + 1/2⋅4⋅2.0 + 4.7 + 3⋅2.0 + 5.2 = 99.05 Å3;    ∑ ai ∆Vi  = aC(2∆VC,19 + ∆VC,20 + 3∆VC,18 + 1/2⋅2∆VC,12) + aH(1/2⋅4∆VH,124 +  i 3 ∆VH,125 + 3∆VH,124) + aO,s⋅∆VO,133 = [0.021(2⋅8.4 + 11.65 + 3⋅12.7 + 1/2⋅2⋅12.6) + 19.98(1/2⋅4⋅2.0 + 4.7+ 3⋅2.0) + 7.03 5.2]⋅10–3 = 331.9⋅10–3 Å3K–1. ∑ b j = 2bh + bM; ∑ b j = (–139.6 + 16.0)⋅10–3 = –123.6⋅10–3 Å3K–1. j

j

Substitution of all these values of parameters into equation (IV.41) leads to the following value of Tg of the ideal structure 3: Tg =

99.05 ⋅ 103 = 476 K. 331.9 − 123.6

(A-3-3)

This value of Tg is somewhat above the temperature range of the α-transition for PFR. Structure 4:

CH2

CH2 O

This structure forms a network, the repeating fragment of which is marked by dotted lines. The structure of the cross-linked point of this network is the following (marked by dotted lines):

H2C

CH2 O

611

For structure 4, we obtain:    ∑ ∆Vi  = 2∆VC,19 + ∆VC,20 + 1/2⋅2∆VC,12 + 3∆VC,18 + 3∆VH,124 + 4⋅1/2∆VH,124 +  i 4 ∆VO,131 = 3⋅8.4 + 11.65 + 1/2⋅2⋅12.6 + 3⋅12.7 + 3⋅2.0 + 4⋅1/2⋅2.0 + 2.1 = 91.25 Å3;    ∑ ai ∆Vi  = aC(3∆VC,18) + aH(3∆VH,124) = (0.021⋅3⋅12.7 + 19.98⋅3⋅2.0)⋅10–3 =  i 4 120.7⋅10–3 Å3K–1; ∑ b j = 0; j

   ∑ K i ∆Vi  = KC(2∆VC,19 + ∆VC,20 + 1/2⋅2∆VC,12) + KH⋅4⋅1/2∆VH,124 + KO∆VO,131 =  i 4 [1.15(2⋅8.4 + 11.65 + 1/2⋅2⋅12.6) + 2.307⋅4⋅1/2⋅2.0 + 0.058⋅2.1]⋅10–3 = 56.5⋅10–3 Å3K– 1 . Substituting these values into equation (IV.85), we obtain the following value of Tg for the ideal network, composed from the structure 4: Tg =

91.25 ⋅103 = 515 K. 56.5 + 120.7

(A-3-4)

Hence, the glass transition temperature of this cross-linked polymer network would be quite high compared with Tg,exper. Structure 5: CH2

CH2 HO

CH2 OH From the viewpoint of the gross-formula, this structure is identical to the structure 2, but in the present case, hydrogen bonding appears within the single repeat unit. Such a structure is described in ref. [99]. If hydrogen bonds are formed between neighboring OH-groups disposed within the same repeat unit, then, as mentioned above, the physical network between chains stipulated by hydrogen bonds is not formed. Then calculating the glass transition temperature, the value of ∑ b j is calculated in the following way: because j

this structure is linear, the constant bM must be taken into calculation when considering the m-substitution in an aromatic cycle; moreover, there are two branches from the aromatic cycle, which requires introduction of two constants bd. Then ∑ b j j

= bM + 2bd = [16.0 – 2⋅55.4]⋅10–3 = –94.8⋅10–3 Å3K–1. Substituting values

∑ ai ∆Vi i

and

∑bj

∑ ∆Vi , i

into equation (IV.85), the following value of the glass transition

j

temperature is obtained for the structure 5:

612

Tg =

123.05 ⋅ 103 = 300 K. 505.4 − 94.8

(A-3-5)

This value of Tg falls below the range of the α-transition for cured PFR. As indicated, no calculated values of the glass transition temperature of the ideal structures of PFR discussed falls within the range of experimental values of the glass transition temperature, Tg,exper. It may therefore be assumed that there is a more complicated structure of cured PFR than the considered ideal ones; this has been indicated by many investigators. Most probably, the real structure of PFR consists of the selection of the considered ideal structures. To state the most probablr real PFR structure, let us calculate, besides Tg, the following characteristics of cured PFR: density ρ, refractive index n, coefficient of volumetric expansion αG, and molar heat capacity C ps . Selection of these characteristics is stipulated by both their importance and sensitivity to changes of the structural formula of the substance and the fact that for various ideal PFR structures, as the glass transition temperature increases, values of other characteristics may increase or decrease, which is important for search of the most probable structure of PFR by the method of experiment design (see below). The calculated values of PFR density for structures 1–5, obtained due to equation (II.6), are indicated in Table A-3-2. Table A-3-2 Calculated values of Tg, ρ, n, αG and C sp for the ideal structures of cured PFR Ideal structure of PFR Structure 1 Structure 2 Structure 3 Structure 4 Structure 5

Tg, K

ρ, g/cm3

n, rel. un.

αG⋅104, K–1

743 507 476 515 300

1.253 1.250 1.205 1.301 1.250

1.621 1.583 1.604 1.645 1.583

1.30 3.20 2.02 1.86 3.20

C sp , cal/(mol⋅deg) 31.9 42.4 31.4 26.1 42.4

Let us calculate the refractive index n by equation (VIII.2), which contains the molar refraction summed up from the refraction indices of separate atoms and constants for different types of bonds (double, triple). For structure 1, we obtain (values of Ri are taken from [28]): R1 = 3⋅(RC + 1/2⋅RC + RH) + 2RC + 2RH + RC + RO′ + RH + 3R≠ = 3⋅(2.418 + 1/2⋅2.418 + 1.10) + 2⋅2.418 + 2⋅1.10 + 2.418 + 1.525 + 1.10 + 3⋅1.733 = 31.459 cm3/mol. For structure 2: R2 = 8RC + 8RH + 2RO′ + 3R≠ = 8⋅2.418 + 8⋅1.10 + 2⋅1.525 + 3⋅1.733 = 36.393 cm3/mol. For structure 3: R3 = 7RC + 6RH + RO′ + 3R≠ = 7⋅2.418 + 6⋅1.10 + 1.525 + 3⋅1.733 = 30.25 cm3/mol. For structure 4: R4 = 7RC + 5RH + RO< + 3R≠ = 7⋅2.418 + 5⋅1.10 + 1.643 + 3⋅1.733 = 29.268 cm3/mol. For structure 5, the value R5 coincides with the value R2. Values of n for various structures calculated by equation (VIII.2) are shown in Table A-3-2. Heat capacity C ps in the glassy state of the polymer was calculated by correlation (XVI.1).

613

The value αG is directly connected with the glass transition temperature Tg by the correlation:

αG =

0.096 . Tg ( K )

(A-3-6)

The calculated values of C ps and αG for all five ideal structures are shown in Table A-3-2. Comparison of the calculated values of PFR characteristics for possible ideal structures, shown in Table A-3-2, with their experimental values (see Table A-3-1) enables us to conclude that cured PFR does not represent an ideal network, but contains a set of structures; this set can be determined using the method of experiment design for multi-component systems with composing simplex networks and polynomial ‘composition – property’ models. For independent variables X of these models, let us assume the above-considered ideal structures the combination of which will determine the real structural composition of cured PFR: x1 for structure 1; x2 for structure 2; x3 for structure 3; and x4 for structure 4. For the response function Y, the following properties of cured PFR will be taken: glass transition temperature, Tg (K); density, ρ (g/cm3); refractive index, n (rel. un.); coefficient of volumetric expansion, αG (K–1); and molar heat capacity, C ps (cal/(mol⋅deg)). As indicated, instead of the ideal structure 2, the structure 5 is assumed as the independent variable; this structure differs by the existence of hydrogen bonding of OH-groups within the same repeat unit which, with all other characteristics being the same, provides for lower glass transition temperatures (see Table A-3-2). The dependence of the response functions Y on variables X will be searched for in the form of the second order model [70]: Y = ∑βij + ∑βijxixj; where

β i = y i;

1 ≤ i ≤ 4;

1 ≤ j ≤ 4,

βij = 4yij – 2yi – 2yj.

(A-3-7) (A-3-8)

Design of the experiment and appropriate simplex lattice are shown in Table A-3-3 and Figure A-3-2a, respectively. Table A-3-3 Matrix of the experimental design and calculated values of PFR at the experimental design Coordinates of Calculated values of PFR Point of the experiment design points experiment C sp , Tg, n, ρ, αG⋅104, design x1 x2 x3 x4 3 –1 rel. un. K g/cm K cal/(mol⋅deg) y1 1 0 0 0 743 1.253 1.621 1.30 31.90 y2 0 1 0 0 300 1.250 1.583 3.20 42.40 y3 0 0 1 0 476 1.205 1.604 2.02 31.40 y4 0 0 0 1 515 1.301 1.645 1.86 26.10 y12 ½ 1/2 0 0 410 1.251 1.600 2.34 37.20 y23 0 1/2 1/2 0 359 1.232 1.594 2.67 36.90 y13 ½ 0 1/2 0 582 1.232 1.613 1.65 31.65 y14 ½ 0 0 1/2 615 1.276 1.633 1.56 29.00 y24 0 1/2 0 1/2 364 1.272 1.614 2.64 34.25 y34 0 0 1/2 1/2 494 1.254 1.624 1.94 28.75

614

Figure A-3-2. Designation of responses at the points of simplex lattice: a) experimental design points; b) control design points.

As indicated by the diagram, the experimental design points with the single index, yi, are located in the tetrahedron apexes, with two indices, yij – in the middle of the tetrahedron edges possessing apexes, i and j. Therewith, values of the response functions Tg, ρ, n, αG, and C ps in the points of the experimental design yi, located in the apexes of the tetrahedron, are calculated by equations (IV.85), (II.6), (VIII.2), (XVI.1) and (A-3-6), and in intermediate points of the experiment, i.e. with two or more indices at y, by the equations: – for the glass transition temperature,       α 1  ∑ ∆Vi  + α 3  ∑ ∆Vi  + α 4  ∑ ∆Vi  +  i 1  i 3  i 4 → Tg =        α 1  ∑ α i ∆Vi + ∑ b j  +  ∑ K i ∆Vi   + α 3  ∑ α i ∆Vi + ∑ b j  + j j 1   i 1  i  i 3   + α 5  ∑ ∆Vi   i 5 → ;        + α 4  ∑ α i ∆Vi + ∑ b j  +  ∑ K i ∆Vi   + α 5  ∑ α i ∆Vi + ∑ b j  j j  i  4  4  i  i 5 –

for the density,

ρ=

–

0.681(α1 M 1 + α 3 M 3 + α 4 M 4 + α 5 M 5 )

;

(A-3-10)

          N A α 1  ∑ ∆Vi  + α 3  ∑ ∆Vi  + α 4  ∑ ∆Vi  + α 5  ∑ ∆Vi   1  i 3  i 4  i  5    i

; (A-3-11)

          N A α1  ∑ ∆Vi  + α 3  ∑ ∆Vi  + α 4  ∑ ∆Vi  + α 5  ∑ ∆Vi     i 1  i 3  i 4  i  5 

for the refractive index,

n2 −1 = n2 + 2

–

(A-3-9)

0.681(α 1 R1 + α 3 R3 + α 4 R4 + α 5 R5 )

for the molar heat capacity,

615

      C ps = α1  ∑ C ps ,i ∆Vi  + α 3  ∑ C ps ,i ∆Vi  + α 4  ∑ C ps ,i ∆Vi  +  i 1  i 3  i 4   + α 5  ∑ C ps ,i ∆Vi  + C 0 ;  i 5

(A-3-12)

C ps = α 1C ps ,1 + α 3C ps ,3 + α 4 C ps , 4 + α 5C ps ,5 + C 0 . The coefficient of volumetric expansion is calculated by correlation (A-3-6), the value Tg being determined from equation (A-3-9). In correlations (A-3-9) – (A-312), α1, α3, α4, and α5 are the molar parts of the structures 1, 3, 4, and 5, respectively; therewith, α1 = x1, α3 = x3, α4 = x4, and α5 = x5. Using the response values in the design points shown in Table A-3-3 and equations (A-3-8) for calculation of the coefficient β for the second order model (A-37), polynomial models for cured PFR are found, in which insignificant coefficients are neglected: – for the glass transition temperature, K: Tg = 743x1 + 300x2 + 476x3 + 515x4 – 446x1x2 – 116x1x3 – – 56x1x4 – 116x2x3 – 174x2x4; –

for the density, g/cm3:

ρ = 1.253x1 + 1.250x2 + 1.205x3 + 1.301x4 + 0.012x1x3 + + 0.018x2x3 – 0.014x2x4; –

(A-3-15)

for the coefficient of volumetric expansion, K–1:

αG = (1.3x1 + 3.2x2 + 2.02x3 + 1.86x4 + 0.36x1x2 + + 0.24x2x3 + 0.44x2x4)⋅10–4; –

(A-3-14)

for the refractive index, rel. un.: n = 1.621x1 + 1.583x2 + 1.604x3 + 1.646x4 – 0.009x1x2;

–

(A-3-13)

(A-3-16)

for the molar heat capacity, cal/(mol⋅deg): C ps = 31.9x1 + 42.4x2 + 31.4x3 + 26.1x4 + 0.2x1x2.

(A-3-17)

Using polynomial models (A-3-13) – (A-3-17) obtained and equations (IV.85), (II.6), (VIII.2), (XVI.1), and (A-3-6) in the control design points shown in Figure A-3-2b (the design point with three indices, yijk, locates in the middle of the tetrahedron designe with apexes i, j and k), the properties of cured PFR are calculated. As indicated in Table A-3-2, the relative error of the properties prognosis for cured PFR does not exceed 1.7%, which is quite acceptable for application to polynomial models both in order to determine the real structure of cured PFR and the degree of influence of the ideal structure type on the PFR properties. In particular, indicated as an example in Figure A-3-3 is graphic interpretation of the polynomial model of the glass transition temperature for PFR (A-3-13) in the area of change of

616

Control design points

Coordinates of control design points

Values of PFR properties in control design points

x1

x2

x3

x4

Tg/T ′g

ρ/ρ′

n/n′

(αG/αG′)⋅104

y123

1/3

1/3

1/3

0

y124

1/3

1/3

0

1/3

y134

1/3

0

1/3

1/3

y234

0

1/3

1/3

1/3

428 431 436 442 559 558 394 397

1.238 1.259 1.265 1.266 1.253 1.254 1.252 1.253

1.603 1.602 1.617 1.615 1.623 1.624 1.611 1.611

2.240 2.235 2.200 2.200 1.720 1.714 2.440 2.436

C sp C sp 35.23 35.23 33.47 33.47 29.80 29.80 33.3 33.3

′

Relative error of the predicted properties of PFR, % δn δαG δTg δρ δCs

p

– 0.7

– 1.7

0.1

0.2

0

– 1.4

– 0.1

0.1

0

0

0.2

– 0.1

0.1

0.4

0

– 0.8

– 0.1

0

0.2

0

Notes: 1. Shown in the numerator are properties of PFR calculated by equations (109), (6), (VII.2), (XVI.1), and (A-3-6). The denominator indicates the properties calculated by polynomial models (A-3-13) – (A-3-17). 2. Relative error of the predicted properties of PFR is calculated by the formula: δy ijjk =

′ y ijk − y ijk ⋅ 100% . ′ y ijk

616

Table A-3-4 Errors of the predicted properties of cured PFRs

617

the ideal structures 1, 3 and 5 ratio (in the absence of the ideal structure 4). As indicated, isolines of the glass transition temperature of PFR represent straight lines, suggesting that contributions of separate ideal structures into formation of properties of the real desired PFR structure are additive and, the more so, the glass transition of PFR is preferably determined by the relative concentration of the ideal structures 1 and 5.

Figure A-3-3. Isolines of the glass transition temperature Tg of PFR in the region of ideal structures x1, x2 and x3 (x4 = 0).

Searching for the real structure of cured PFR includes the determination of the ratio between contents of the ideal structures 1, 3, 4, and 5, at which border conditions on the experimental values of the resin properties, indicated in Table A-3-1, are fulfilled. The acceptable selection of these ideal structures Ω x (at ∑ xi = 1, 0 ≤ xi ≤ 1, i

i = 1 − 4 ) is set on in the following sequence. When the concentration of the structure 4 is fixed (x4 = const) for every tested property of the resin (Tg, ρ, n, αG, and C ps ), regions of acceptable selections of the remaining three ideal structures are determined in the plane of change of variables x1, x2 and x3. The desired value x1x4 =const is obtained by the consequent superimposition of these regions. The example of this procedure at x4 = const is indicated in Figure A-3-4. As indicated, such indices of cured PFR as the refractive index (Figure A-3-4c) and the molar heat capacity (Figure A-3-4e) cause no influence on the formation of the region of the acceptable selection of the ideal structures x1x4 =0 (Figure A-3-4d). Apparently, the influence of these characteristics of PFR is evident only for ideal structure 4 and increases with the concentration of the latter (Figure A-3-5).

618

Figure A-3-4. Schematic representation of the search procedure for an acceptable set Ωx of ideal PFR 7 structures for x4 = 0. Showing domains of the acceptable sets of structures: (a) [[ =

º

º

ρ

[[ = for the density; (c) Q G [[ = for the refractive index; (d) x1x4 =0 for the coefficient of volumetric & expansion; (e) [ [ = for the molar heat capacity; (f) x x =0 for the ideal PFR

for the glass transition temperature of cured PFR: (b)

º

º

 

1 4

structures with the coordinates of vertices T1 (0.35; 0.34; 0.31; 0.00); T2 (0.34; 0.35; 0.31; 0.00); T3 (0.55; 0.45; 0.00; 0.00).

The region of acceptable selection of the ideal structures of cured PFR is determined by consecutive sorting out of the regions x1x4 =const at variation of the structure x4 content with step 0.01 from 0 to 1. Consequently, it is obtained that that the desired region Ω x represents an improper figure and the coordinates of apexes of this figure are shown in Figure A-3-6. Disposition of Ω x regions in the factor space of ideal structures of PFR enables us to conclude that the real structure of cured PFR may be represented only by the combination of linear (structures 3 and 5) and network (structures 1 and 4) fragments of the polymeric chain; this has also been confirmed by other investigators [99].

Figure A-3-5. Effect of the ideal structure 4 on the change of the domain Ωx of acceptable set of ideal structures of cured PFR for the refractive index (I) and for the molar heat capacity (II) at x4 = 0 (a); 0.2 (b); 0.4 (c).

619

For the quantitative estimation of the properties of PFR containing various types of structures, let us substitute all calculated parameters into equations (A-3-13) – (A-3-17). Consequently, we obtain the following expressions for calculation of the glass transition temperature Tg, density ρ, refractive index n, coefficient of volumetric expansion αG, and molar heat capacity C ps : Tg =

ρ=

α1 ⋅101.1 + α 5 ⋅123.05 + α 3 ⋅ 99.05 + α 4 ⋅ 91.25 3 ⋅10 ; α1 ⋅136.1 + α 5 ⋅ 410.6 + α 3 ⋅ 208.3 + α 4 ⋅177.2

0.681(α1 ⋅ 112 + α 5 ⋅ 136 + α 3 ⋅ 106 + α 4 ⋅ 105) ⋅ 103 ; 0.6023(α1 ⋅ 101.1 + α 5 ⋅ 123.05 + α 3 ⋅ 99.05 + α 4 ⋅ 91.25)

(A-3-18)

(A-3-19)

n 2 − 1 0.681(α1 ⋅ 31.459 + α 5 ⋅ 36.393 + α 3 ⋅ 30.25 + α 4 ⋅ 29.268) 3 ⋅10 ; (A-3-20) = n 2 + 2 0.6023(α1 ⋅101.1 + α 5 ⋅123.05 + α 3 ⋅ 99.05 + α 4 ⋅ 91.25)

αG = 0.0096/Tg; C ps = α1⋅31.9 + α5⋅42.4 + α3⋅31.4 + α4⋅26.1,

(A-3-21) (A-3-22)

where α1, α3, α4, and α5 are the molar parts of the structures 1, 3, 4 and 5, respectively. One of the possible relations between relative content of ideal structures, responded to the region of admissible selection (Figure A-3-6) has the following form: α1 = 0.345, α5 = 0.345, α3 = 0.2, and α4 = 0.11.

Figure A-3-6. Coordinates of vetrices of the domain of admissible set of ideal structures Ωx for cured PFR: T1 (0.35; 0.34; 0.31; 0.00); T2 (0.34; 0.35; 0.31; 0.00); T3 (0.55; 0.45; 0.00; 0.00); T4 (0.00; 0.22; 0.13; 0.65).

Substituting these values into correlations (A-3-18) – (A-3-22), for the real network we obtain: Tg = 429 K; ρ = 1.248 g/cm3; n = 1.6057; αG = 2.238⋅10–4 K–1; C ps = 34.8 cal/(mol⋅deg); as indicated (see Figure A-3-3), all these values fall within the range of experimental values.

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Taking into account the presence of a narrow region of the acceptable set of ideal structures for cured PFR (see Figure A-3-6), it may be concluded that the real structure of the network obtained is close to the most probable one. Hence, the approach described in the Section IV.4, combined with the method of experiment design, enables, firstly, to made definite conclusions about the structure of cross-linked systems, basing on the comparison of experimental and calculated characteristics of the network. Secondly, the examined problem of the analysis of phenol formaldehyde resin structure relates to the mixed problem, because the direct problem is solved in the initial stage – the properties of ideal structures of PFR are determined by their chemical structure. In the second stage of investigations, solution of the reverse problem is considered – the chemical structure of PFR is searched, which provides for the desirable set of properties of this resin. In this case, such set of properties is given by their experimental values shown in Table A-3-1. In the present example, note also the application of the approach to solving the reverse problem of polymers synthesis, different from the one described in Appendix 2.

Appendix 4. Application of the approach to multicomponent copolymers

Usually, analysis of the structure and properties of copolymers was performed for the systems containing two components. It has been was performed much less frequently for triple copolymers. In this appendix, applicability of the approach, described in the present monograph, to analysis of the structure and properties of copolymers containing from three to five copolymers will be discussed. Simultaneously, let us compare experimental and calculated values of the physical characteristics for both homopolymers and homopolymer-based multicomponent copolymers. To analyze the properties of multicomponent copolymers, six monomers are chosen in ref. [39], used in reactions of polymerization and copolymerization: methyl methacrylate (MMA), butyl methacrylate (BMA), butyl acrylate (BA), n-nonyl acrylate (NA), 2-ethylhexyl acrylate (EHA), and n-heptyl acrylate (HA). Homopolymers and copolymers based on these compounds are fully amorphous highmolecular substances, which are easily formed in a block and suitable for investigation of thermal and optical characteristics. Indicated in Table A-4-1 is the composition of five copolymers synthesized in ref. [39]. Calculation of the glass transition temperature Tg for multicomponent copolymers is performed by equations (IV.55) and (IV.57). The first of these equations for calculation of the glass transition temperature of copolymers requires no understanding of the experimental glass transition temperatures of homopolymers. Equation (IV.57) contains values of glass transition temperature of homopolymers, for calculation of Tg of copolymers experimental values of Tg for appropriate homopolymers being used. Table A-4-1 Copolymer 1 2 3 4 5

MMA 0.333 0.250 0.250 0.200 0.250

Composition of copolymers Molar parts of units in copolymer BMA BA NA 0.333 0.333  0.250 0.250 0.250 0.250 0.250  0.200 0.200 0.200 0.250 0.250 

EHA   0.250 0.200 

HA     0.250

Density is calculated by equations (II.7), (A-4-1) and (A-4-2). The first of them is approximate, obtained on the base of suggestion that the coefficient of molecular packing is approximately the same for all polymers, and equals kavg = 0.681 for all block polymers at room temperature. However, for polymer and copolymers existing in the rubbery state at room temperature, it is desirable to take into account temperature dependences of the coefficient of molecular packing, k, in the density calculation, because the difference of the real coefficient of packing for them from the average value kavg, increases with a decrease of the glass transition temperature. Equations (A-4-1) and (A-4-2) are obtained with regard to this circumstance and enable, as indicated below, to calculate more accurately the density of polymers and copolymers. For the latter, we obtain:

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Tg, K Td, K Cσ, B ρ, g/cm3 αG⋅104, K-1 −3.30 1.17 − 1.23 378 2.70 623 − 2.85 1.17; 1.17 377 2.55 629 PBMA − − − − 300 1.48 2.82 17.7 − 18.4 1.06 1.09; 1.06 287 1.50 2.71 18.2 27.2 3.05 627 16.6 PBA − − − 221 1.47 18.2 − 18.4 30.7 2.80 1.04 1.11; 1.05 242 1.50 2.76 18.4 28.0 3.05 627 32.2 PNA − − − − − − − − 215 1.04; 0.97 219 1.51 2.58 17.5 25.3 3.05 626 40.5 PEHA  − − − − − − − −  1.05; 0.99 242 1.51 2.60 17.5 3.05 626 39.4 PHA − − − − − − − − 220 1.06; 0.99 225 1.51 2.63 17.7 26.1 3.05 626 38.1 Notes: 1. * - shown in the numerator are experimental values of physical characteristics; indicated in the denominator are calculated values. In the case calculated ρ values, the first of them was obtained by equation (II.7), and the second – by equations (A-4-1) and (A-4-2); 2. ρ is the density; Tg is the glass transition temperature; n is the refractive index; ε is the dielectric constant; δ is the solubility parameter; γ is the surface energy; αG is the thermal coefficient of volumetric expansion in the glassy state; Cσ is the stress-optical coefficient. Homopolymer PMMA

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Table A-4-2 Calculated and experimental characteristics of homopolymers* N δ, (J/cm3)1/2 γ, dyn/cm ε 1.49; 1.55 3.10 18.6 39.0 − 40.2 1.49 2.94 19.0 31.0

623

k =n

ρ G20 =

0.667 ∑α k M k k =1

k =n   [1 + α G ( 293 − Tg )]N A ∑α k  ∑ ∆Vi  k =1  i k

, 293 < Tg;

(A-4-1)

, 293 > Tg.

(A-4-2)

k =n

ρ G20 =

0.667 ∑ α k M k k =1

k =n   [1 + α L (293 − Tg )]N A ∑ α k  ∑ ∆Vi  k =1  i k

Besides the glass transition temperature and density, the following characteristics are calculated for copolymers: the refractive index n, the stress-optical coefficient Cσ, temperature of the onset of intense thermal degradation Td, the dielectric constant ε, the solubility parameter δ, the surface energy γ. Calculations were performed using equations (VIII.3), (VII.7), (VIII.16), (IX.4), (XII.4), and (XIII.39), respectively. First of all, let us compare the calculated and experimental values of the properties of homopolymers. These data are indicated in Table A-4-2. In most cases, good coincidence between experimental and calculated data is observed. Separately, we must dwell on the calculation of such characteristic of homo- and copolymers as density ρ, for which two values are shown in the Table. The first of them is determined with the help of equation (II.7), and the second – by correlations (A-4-1) and (A-4-2), which take into account the temperature dependence of the coefficient of molecular packing. In the case of a glassy polymer (poly(methyl methacrylate)), the calculated value of density obtained by equation (II.7) coincides with the experimental value. For homopolymers with low glass transition temperatures, which are in the rubbery state at room temperature, consideration of the temperature dependence to the coefficient of molecular packing leads to lower values of density compared with those determined on the base of the value kavg. Therewith, the data on density values coincide with experimental ones. The glass transition temperature, determined from equation (IV.41) with the accuracy usual for such calculations, coincides with the experimental values. The refractive index for homopolymers is calculated from equation (VIII.1), as well as from equation (VIII.6), which takes into account the temperature dependence of the coefficient of molecular packing. The latter circumstance enables us to calculate the refractive index with high accuracy. With regard to such characteristics as the solubility parameter, surface energy, temperature of the onset of intense thermal degradation, the thermal coefficient of volumetric expansion in the glassy state, and the stress-optical coefficient, they coincide with the experimental ones with the accuracy usual for such calculations. The properties of multi-component copolymers are indicated in Table A-4-3. Here, the coincidence of the calculated and experimental characteristics is approximately the same as for homopolymers. The density calculated from equation (II.7) using the average value of the coefficient of molecular packing is somewhat higher than the values, determined experimentally. This is stipulated by the fact that all copolymers possess the glass transition temperature below room temperature. Values of density ρ calculated by correlation (A-4-2) with regard to the temperature dependence of the coefficient of molecular packing, is somewhat lower than these calculated by equation (II.7), and coincide well with the experimental values of ρ.

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Table A-4-3 Physical characteristics of homopolymers* Van-der-Waals volume, n Tg, K Td, K Copolymer Cσ, B ρ, g/cm3 Å 1 125 − − 563 298 1.09 1.50 15.5 273;268 627 1.12;1.08 2 148 − − 573 237 1.04 1.50 21.6 247;243 627 1.09;1.03 3 143 − − 570 233 1.04 1.50 21.3 258;255 627 1.09;1.04 4 158 − − 610 240 1.02 1.50 25.2 244;240 627 1.08;1.02 5 139 − − 570 243 1.06 1.50 21.0 252;247 627 1.10;1.04 Notes: 1. * Indicated in the numerator are experimental values, and in the denominator – calculated ones; for density ρ, the first value is obtained from equation (II.7), and the second – from equation (A4-2); for Tg, the first value is calculated by equation (IV.55), and the second – by correlation (IV.57), using experimental values of Tg for homopolymers. 2. ρ is the density; n is the refractive index; Cσ is the stress-optical coefficient; Tg is the glass transition temperature; Td is the onset temperature of intense thermal degradation.

For all copolymers, the refractive index n is approximately the same, and is associated with the close values of the refractive index for all homopolymers, mentioned above. The stress-optical coefficient is also almost identical for all copolymers, excluding the copolymer 1 with increased concentration of MMA units possessing a negative value of Cσ. The glass transition temperature of copolymers indicated in Table A-4-3 is calculated by equations (IV.55) and (IV.57). The former requires no understanding of experimental values of the glass transition temperature of homopolymers, and the latter uses these values. By and large good coincidence of calculated and experimental values of Td is observed. In most cases, when experimental values of Tg for homopolymers are used and substituted to equation (IV.57), correspondence of the calculation to the experiment becomes somewhat better. As for temperature of the onset of rapid thermal degradation Td, good coincidence is observed for the copolymer 4, for which the error is 2.7%. For other copolymers, the calculated values deviated from the experimental ones for this characteristic by 50° to higher values which gives 8%. This phenomenon has not as yet been explained. It should be taken into account only that the characteristic such as the dependence of the temperature of the onset of intense thermal degradation on the presence of admixtures and other reasons is stronger. The calculated values of this characteristic are determined for the ideal polymeric system. Hence, the possibility of predicting various characteristics of multi-component copolymers is obvious. Therewith, the approach used does not require any experimentally determined values. The prognosis is performed on the basis of only the chemical structure of the multi-component copolymer.

Appendix 5. Influence of strong intermolecular interaction occurring between two dissimilar polymers on their miscibility

When two or more polymers are mixed, the situation may appear when an additional strong intermolecular interaction occurs between chains of these polymers, which is not displayed between macromolecules of each of polymers, taken separately. These may be hydrogen bonds or strong dipole–dipole interactions. Analysis of the influence of these interactions on the glass transition temperature will be shown below. In a series of works devoted to the analysis of miscibility of polymers and the properties of mixtures obtained, the main attention is paid to these special interactions occurring between macromolecular chains of the mixed polymers. The scheme of this interaction is demonstrated by the following example:

F3C

CF3 O H O

Let us consider in detail examples of such interactions and experimental methods of their determination. Mixtures of styrene copolymer with vinylphenylhexafluoromethylcarbynol with the polymers as polycarbonate based on bisphenol A, poly(butyl methacrylate), poly-2,6-dimethyl-1,4-phenylene oxide may represent an example of this significant event [209]. Introduction of hydroxyl groups to the mentioned polymer leads to the formation of hydrogen bonds between the copolymer and polycarbonate, which causes a sharp increase of miscibility. The thermograms obtained by the DSC method for polystyrene mixtures with polycarbonate show clearly two glass transition temperatures indicating immiscibility of these polymers. The situation is changed abruptly when polystyrene introduced to the mixture is substituted by a copolymer of the above given structure: thermograms indicate the single glass transition temperature (which indicates miscibility). Therewith, the glass transition temperature regularly increases with the concentration of polycarbonate. Mixtures of copolymers of the above-mentioned structure with poly(ethylene oxide) (PEO), which represents the solid polymer, also display this effect [210]. Crystallinity of PEO in the mixture varies in the manner by which the melting point decreases. The formation of hydrogen bonds between PEO and copolymer has been studied in dependence on temperature. As temperature increases, hydrogen bonds between copolymer and PEO dissociate, but are restored when cooled down. Even in the case of solid polymer such as PEO, the presence of hydrogen bonding between chains of

626

mixed polymers improves their miscibility, suppresses crystallization, and forms onephase systems. Behavior of styrene copolymer mixtures with vinylphenylhexafluoromethyl carbynol was studied for such polymers as poly(vinyl acetate), poly(methyl methacrylate), poly(ethyl methacrylate), poly-n-(butyl methacrylate), poly(methyl vinyl ether), poly-2,6-dimethyl-1,4-phenylene oxide, polycarbonate based on bisphenol A, styrene copolymer with acrylonitrile, as well as with amorphous and solid polyesters and polyamides [193]. For these systems, hydrogen bonding influences the miscibility of components, which has been confirmed by measurements of the glass transition temperature, as well as by FTIR-spectroscopy. Ref. [165] represents a detailed investigation of the miscibility of this polymer with a series of aliphatic polyamides, such as nylon-6,12 and N,N’-dimethyl-substituted nylon-6,12. The miscibility criterion was again assumed to be general glass transition temperature for the mixture, therewith, the polymers were used, for which the glass transition temperature is significantly different from that for polystyrene and copolymer. For example, the glass transition temperature of nylon-6,12 is 46°C, and the melting point is estimated in the range of 206–215°C. To estimate the influence of the crystallinity degree on miscibility, besides nylon-6,12, N,N’-dimethyl-substituted nylon-6,12 was used, as well as copolymers with different concentration of it. The main conclusions made according to the results of ref. [165] are that hydrogen bonds between components of the mixture are formed and these bonds dissociate at heating and occur again at cooling down. Therewith, the introduction of a small amount of carbonyl groups to polystyrene improved the miscibility of such immiscible polymers as polystyrene and polyamide. Consequently, the copolymer under consideration may be a compatibilizer. Ref. [192] represents the study of the miscibility of substituted phenol condensed resins with poly(methyl methacrylate) (PMMA). The following resins were studied: OH

R

n

where R = H; t-butyl; NO2; Cl. This work indicates again that formation of the hydrogen bonds between components of the mixture plays a significant role in miscibility. Thermodynamic parameters of dissociation of hydrogen bonds, such as enthalpy and entropy, were determined. The most significant influence is caused by such substituents as NO2 and Cl. All mixtures of these phenol resins with poly(methyl methacrylate) displayed the single glass transition temperature that meant that they are miscible well. Therewith, the dependences of the glass transition temperature on the composition were corresponded to three different cases: 1. The glass transition temperature of mixtures is higher than the mean glass transition temperature. 2. The glass transition temperature of mixtures is always lower than the mean glass transition temperature.

627

3. Dependence of the glass transition temperature on the composition is of the Sshape in relation to the mean dependence. The authors of ref. [192] suggest description of the mixture behavior appropriate to the cases 1 and 2 by the following correlation: Tg = W1Tg1 + W2Tg2 + qW1W2,

(A-5-1)

where Tg1 and Tg2 are the glass transition temperatures of polymers 1 and 2, respectively; W1 and W2 are their weight parts; the parameter q may be interpreted as the contribution of hydrogen bonds, which may be considered as pseudo-crosslinks. Therewith, the value of q characterizes intensity of the hydrogen bonding. The experiments showed that when NO2 and t-butyl are used as substituents, the values of q are negative and approximately equal by the absolute value. If substituents are H and Cl, then the values of q are positive (all the above-said is true at substitution of an aromatic cycle in the para-position). Consequently, mixtures of phenol resins possessing t-butyl and NO2-groups with poly(methyl methacrylate) display the behavior corresponding to the second case. When there is no substitution, the behavior of the system corresponds to the third case. Finally, when Cl is the substituent, behavior of the mixture corresponds to the first case. For the third case, the S-shape of dependence of the glass transition temperature Tg on the composition may be described by the following correlation: Tg =

W1Tg1 + kW2Tg2 W1 + kW2

+ qW1W2 ,

(A-5-2)

which is the most general one compared with expression (A-5-1). Thereafter [190], correlation (A-5-2) was applied to description of the dependence of the glass transition temperature on the weight part of the components for mixtures of substituted phenol resins (see above) with such polymers as poly(ethyl methacrylate) and poly(methyl methacrylate) F and t-butyl being substituents on the phenol resin, and behavior of the mixtures based on non-substituted phenol resin was also studied. All three cases described above were obtained, and all parameters of equation (A-5-2) were found. Studied in refs. [170] and [171] was the phenomenon of hydrogen bonding between macromolecules in mixtures of polymers by the FTIR-spectroscopy, and using the Patterson and Robald theory [173] and by obtaining diagrams with the lowest threshold temperature of dissolution. The solubility and miscibility of poly(ethyl oxazoline) (

N CH2

CH2

)n

C2H5C O were studied; poly(ethyl oxazoline) is the isomer of poly-N,N’-dimethylacrylamide (

CH2

CH

)n

O CN(CH3)2

628

To estimate the miscibility of poly(ethyl oxazoline) with other polymers, two series of experiments were performed. In the first series, the films based on mixtures of pairs of polymers were obtained, one of which was poly(ethyl oxazoline). In the second series of experiments, complexes based on the same pairs of polymers were obtained. These complexes were obtained by mixing polymeric solutions with further separation of residue, its drying in vacuum during a long time to a constant mass. It was found that the composition of the complex differs from that of the initial mixture, the composition of the complex being correspondent to the ratio of polar groups leading to a strong intermolecular interaction. The glass transition temperature of poly(ethyl oxazoline) mixed with polyacrylic acid is below the additive values, whereas for the complexes, the glass transition temperature is above them. The authors associate this with the formation of a network of physical bonds in the case of complexes. The miscibility of poly(vinyl methyl ether) with styrene copolymers with methyl methacrylate [136] was also studied. The threshold composition of the copolymer, at which it is miscible with poly(vinyl methyl ether), was found. Such copolymer must contain about 60% (mol) of polystyrene. Phase diagrams and socalled ‘windows of miscibility’ for these mixtures were obtained. Besides the glass transition temperature, the thermal stability of mixtures was also studied [221] (on the example of poly-p-hydroxystyrene mixtures with poly(vinyl pyrrolidone) and poly(ethylene oxazoline)). It was found that during heating a small loss of polyhydroxystyrene mass (6%) in the temperature range of 200–250°C is caused by the cross-linking reaction leading to formation of ether bonds. Importance of hydrogen bonds formation between chains of polymers is also noted. Intermolecular complexes may be formed directly during polymerization [135]. For example, this is manifested in photopolymerization of acrylic acid, in which poly(ethylene oxide) is dissolved. The glass transition temperature of the complexes exceeds not only the glass transition temperature of mixtures, but also glass transition temperatures of the original components. These complexes are dissolved in such solvents as dimethylformamide and dimethylsulfoxide, and definitely swell in water and methanol; however, they do not swell in dioxane. If the number of groups able to form hydrogen bonds is decreased by copolymerization of acrylic acid with methyl methacrylate, the ability to complex formation is preserved, although it is somewhat reduced. Thermal oxidation also depends [191] on the formation of hydrogen bonds between chains of polymers (for example, for mixtures of poly(vinyl methyl ether) and modified polystyrene). Modified polystyrene represented a styrene copolymer containing 2.5% (mol) of hexofluoro-2-propylstyrene, i.e. it contained hydroxyl groups inclined to formation of hydrogen bonds. As mentioned many times above, this circumstance provided for improved miscibility of the components in the mixture. However, in addition, phenol groups in the copolymer caused an increase of the system resistance to thermal oxidation, i.e. these groups acted as antioxidants. This was expressed in an increase of the induction period of poly(vinyl methyl ether) thermal oxidation, and the rate of the process decreased. The complex formation described above was also observed between poly-N,N’-dimethylacrylamide and phenoloformaldehyde resin [218]. The complexes were precipitated from solutions in acetone, ethyl acetate and dioxane. Therewith, as before, the composition of these complexes corresponded to the molar ratio of the components. The glass transition temperature of complexes was higher than the glass transition temperatures of the original components. Poly(dimethyl acrylamide) also formed complexes with

629

p-methoxyphenoloformaldehyde resin, therewith the glass transition temperature increases significantly compared with that of each of the components [216, 217]. In ref. [162] were synthesized interpenetrating networks (IPN) based on miscible polymers – poly-1-hydroxy-2,6-methylphenylene and poly(methyl methacrylate). Two types of cross-linked agents were used at various temperatures: hexamethylene tetramine and 1,3-dioxolane. As indicated in the cited work, the intensity of hydrogen bonding in mixtures and IPN is determined by variation of the cross-linking temperature and decrease of the concentration of groups able to form hydrogen bonds. The concentration of these groups was varied by using methyl methacrylate copolymer with styrene. Using FTIR-spectroscopy, it was shown that to preserve miscibility in these mixtures, it is necessary to ensure that the value of intermolecular interaction, stipulated by hydrogen bonds, is not lower than a definite threshold value. Hydrogen bonding influences the phase behavior of mixtures of polymers [163]. Methyl methacrylate copolymers with styrene and phenoloformaldehyde resin, in which hydroxyl groups are partially methylated, were chosen as objects for such investigation. Miscibility diagrams were composed and so-called ‘windows of miscibility’ were found where full miscibility and microphase separation were observed in dependence on temperature and concentration of one of the components. Ref. [222] studied the miscibility of mixtures of homopolymers with copolymers. The first pair represented a mixture of poly-4-hydroxystyrene with n-butyl acrylate copolymer with t-butyl methacrylate. The mixtures mentioned are miscible, when butyl acrylate concentration in the copolymer is 64% or higher. The second pair represented poly-t-butyl acrylate and styrene copolymer with 4hydroxystyrene. For this pair, the ‘window of miscibility’ existed at 4-hydroxystyrene concentration in copolymer between 28 and 66% (mol). The glass transition temperature of miscible mixtures of poly-t-butyl acrylate and styrene copolymer with 4-hydroxystyrene is significantly lower than the additive values and, which is of special interest, it is extremely weakly dependent on the mixture composition, being approximately equal to the glass transition temperature of copolymers themselves. Such behavior may be explained only by the formation of hydrogen bonds between components of the mixture, which is confirmed experimentally by IR-spectroscopy methods. Future development of these ideas may be observed in ref. [116], in which hydrogen bonding of IPN, obtained from a miscible polymeric mixture of poly-1hydroxyl-2,6-methylenephenylene and poly(methyl methacrylate), was studied. Copolymers of methyl methacrylate and styrene were also used in order to decrease the amount of carbonyl groups. To obtain IPN, various cross-linking agents were used, such as hexamethylene tetramine and 1,3-dioxolane. Cross-linking was performed at various temperatures, by which the intensity of hydrogen bonding in IPN was varied. The miscible mixture of the two polymers displayed thermal reversibility in relation to hydrogen bonding. Semi-IPN and IPN prepared at temperatures above the glass transition temperature of the mixture contained no initial amount of hydrogen bonds after cooling down to room temperature; but when poly(methyl methacrylate) copolymer with styrene is used instead of pure poly(methyl methacrylate) (i.e. when the amount of carbonyl groups is reduced), semi-IPN did not form a single phase at a significant decrease of the concentration of these groups. However, semi-IPN and IPN synthesized at comparatively low temperatures (below the glass transition temperature of the mixture), preserved high

630

level of hydrogen bonds amount compared with the ones, synthesized at high temperatures, and the single phase [196]. In ref. [186], the authors turned to new objects, such as polybenzimidazoles and poly-4-vinylpyridine. The main object of investigation was poly-2,2’-(mphenylene)-5,5’-bibenzimidazole

NH

NH

N

N

C

C n

All mixtures of different compositions displayed the single glass transition temperature that testified about their miscibility of components and, therewith, the glass transition temperature was above the additive values. As in the previous works, the authors explain this by the formation of hydrogen bonds taking place according to the scheme: N

N

C

C N

N H

H

N

)n The formation of hydrogen bonds is confirmed by experimental FTIRspectroscopy. The main practical conclusion which can be drawn on the basis of the works cited is that miscibility can be improved by introduction of a small amount of functional groups, hydroxyl ones, for example, into one of the components, which lead to the formation of hydrogen bonds between polymers – the mixture components. Therewith, domains of various size may be formed, and to decrease their size it is enough, for example, to introduce 4.4% (mol) of hydroxyl groups into polystyrene so that it would become miscible with poly-n-butyl acrylate. Preliminary investigations of these regions of relaxation and the size of domains by the NMR method were performed in ref. [158]. In studies [159, 178, 195, 207, 223], these investigations were continued. Complexes based on poly-4-hydroxystyrene and poly-N,N’-dimethyl acrylamide were studied by the NMR 13C method with cross-polarization and sample rotation at the magic angle. The size of inhomogeneities determined by this method was found to be equal to ≈ 2.5 nm [207]. Further on, this investigation was performed for IPN. It is found that, in this case, the size of aggregates is smaller than 2.2 nm [195]. The influence of PMMA microtacticity on its miscibility with a styrene copolymer with vinyl phenol was studied by the NMR method [159]. It was found that syndiotactic PMMA is miscible more efficiently with the copolymer mentioned and formed a single phase in a wide range of mixture compositions based on both (

CH CH2

631

methyl methacrylates. As indicated by the NMR method, for most composites, the size of microinhomogeneities is about 2 nm. Investigations of this kind have also been carried out with poly-4-vinylphenyl dimethylsilanol and its copolymers with styrene [178, 185]. In the present case, a significant influence of hydrogen bonding between phenolic and phosphate groups was also observed, as indicated by IR-spectroscopy, NMR 31P and NMR 13C methods [224]. Studied in ref. [137] were mixtures of poly(ethylene oxide) and poly(butyl methacrylate) with modified poly(butyl methacrylate) containing groups, able to form hydrogen bonds [179 – 184]. Introduction of 4-hydroxy-4,4-bis-3-fluoromethylbutyl group CH3 Si

O

(CH2)3 F3C C

CF3

OH n even into a siloxane polymer possessing a low surface energy makes it possible obtaining mixtures miscible with poly(ethylene oxide) and poly(butyl methacrylate). Estimated in ref. [151] is the parameter of the polymer–polymer interaction based on the study of diffusion of water in the mixtures. In all cases, this parameter became more negative, which indicated better miscibility. Correlations between the diffusion coefficient and the specific volume of the polymer were found. It should be noted [194] that functionalization of polystyrene at the sacrifice of introduction of fluoralkylcarbonyl or hydroxyl groups in it improves the miscibility of polystyrene with other polymers and increases its thermal stability, oxidation resistance, and decreases combustibility; this is evident in its mixtures with other polymers. The miscibility of a series of polyamides with other polymers was studied [128, 138, 157]. In ref. [138], poly-p-phenylene terephthalamide sulfonyl was chosen as the test sample: HN

NH CO SO3H

CO n

Based on its mixtures with poly(vinyl pyrrolidone), poly-4-vinylpyridine and polyvinyl alcohol, the so-called molecular composites were obtained, in which the rigid-chain polyamide sulfonyl was the reinforcing element. A significant influence of hydrogen bonding between elements of the molecular composite was detected. Therewith, for mixtures with poly(vinyl pyridine) and IPN, the glass transition temperature was below the additive values that, according to the authors, testified about a small number of contacts between macromolecules. An attempt was also made to modify nylon-6 by preparing its mixtures with a small amount of phenoloformaldehyde resin [157]. Addition of phenoloformaldehyde resin in amounts of 1–2% increases the elasticity coefficient and decreases water absorption. Therewith, larger spherulites were produced. Increase of the phenoloformaldehyde resin content in the composite induces no improvement of properties. Consequently, it is concluded that the miscibility of nylon-6 with

632

phenoloformaldehyde resin is rather restricted and falls within the range up to 3% of phenoloformaldehyde resin. The aim of the work [128] was study of the mechanism of polymer-analogous reaction between poly(trichlorobutadiene) (PTCB) and aliphatic diamines: t-butylamine, diethylamine and triethylamine, which model chemical processes proceeding in separate fragments of polymeric chains during PTCB interaction with branched poly(ethylene imine). With the help of FTIR methods and electronic spectroscopy, it was shown that when poly-1,1,2-trichlorobutadiene interacts with amines, hydrogen-bonded ionic complexes with the charge transfer are formed accompanying reactions of allyl chlorine substitution by amino group and dehydration. The contribution of each of these processes to the total conversion of functional groups of the polymer depends on amine nature (the N-substitution degree) and the type of solvent. The primary and secondary amines are more inclined to form stable hydrogen-bonded complexes with polymers, whereas the tertiary amines cause mostly dehydrochlorination of the polymer and formation of polyene sequences of different length. Dehydrochlorination prevails in dioxane, and substitution and complex formation – in chloroform. Hence, for more objective prediction of the miscibility of polymers with each other, as well as for analyzing the reasons for miscibility, special attention should be paid not only to the properties of individual components, but also to the specific intermolecular interaction between them. As the results of numerous works indicate, there are several ways of improving the miscibility of polymers: 1. Selection of polymeric pairs or modification of polymers, which are performed for strong intermolecular interaction occurring between heterogeneous polymeric chains (for example, hydrogen bonding). This problem will be analyzed in detail below. 2. The second way concludes in performing chemical reactions between components of the mixture that finally leads to obtaining of so-called interpolymers, described in detail in ref. [215]. 3. The third way concludes in introducing compatibilizers into the system, i.e. lowmolecular and high-molecular compounds possessing functional groups, which strengthen the specific intermolecular interaction between chains. Copolymers may also be used as compatibilizers,. Let us discuss in more detail the problems associated with strengthening of the specific intermolecular interaction between mixed polymers. If such an interaction does not occur, the glass transition temperature of a homogeneous mixture of miscible polymers is calculated from equation (IV.57). It contains the glass transition temperatures of homopolymers based on the components 1 and 2, the values of which may be taken as both calculated and experimental for estimating the glass transition temperature of the mixture. In another form, equation (IV.57) is reduced to expression (IV.55), which contains sets of constants for the components 1 and 2, associated with the energy of intermolecular interaction. Suppose now that hydrogen bonding occurs in the case, polymer 2 is added to polymer 1. In this case, constant bh = –140⋅10–3 Å3K–1, which characterizes the   contribution of hydrogen bonds, must be introduced to the value  ∑ ai ∆Vi + ∑ b j  j  i 1 for the component 1. For this case from equation (IV.55), we obtain:

633

    α1  ∑ ∆Vi  + α 2  ∑ ∆Vi   i 1  i 2 Tg = , (A-5-3)     α1  ∑ ai ∆Vi + ∑ b j + 2α 2bh  + α 2  ∑ ai ∆Vi + ∑ b j  + 2α1α 2 ⋅ 0.03 j j  i 1  i 2 because a part of repeat units of the polymer 1 appeared bonded to the component 2 by hydrogen bonds. This equation was deduced on the basis of the assumption that two repeat units are linked by hydrogen bonds, one of them being belonged to the polymer 1, and the second – to the polymer 2. Taking into account equation (IV.57) and the expression α1 = (1 – α2), it should be written down that

Tg =

(1 − α 2 ) ∑ ∆Vi     ∑ ∆Vi  1 (1 − α 2 )  i Tg,1

  + α 2  ∑ ∆Vi   i 1  i 2 ,    ∑ ∆Vi   i 2 + 2(1 − α 2 )α 2bh + α 2 + 2(1 − α 2 )α 2 ⋅ 0.03 Tg,2

(A-5-4)

where Tg,1 and Tg,2 are the glass transition temperatures (calculated or experimental) for the components 1 and 2, respectively. Let us now analyze the dependence of the glass transition temperature on the mixture composition, reflected by equations (A-5-3) and (A-5-4). Three different cases will be considered: 1) The Van-der-Waals volumes of repeat unit of polymers 1 and 2 are approximately     equal:  ∑ ∆Vi  ≈  ∑ ∆Vi  .  i 1  i 2 2) The Van-der-Waals volume of repeat unit of the polymer 1 is significantly lower     than for the polymer 2:  ∑ ∆Vi  <<  ∑ ∆Vi  .  i 1  i 2 3) The Van-der-Waals volume of repeat unit of the polymer 1 is significantly higher     than for the polymer 2:  ∑ ∆Vi  >>  ∑ ∆Vi  .  i 1  i 2       Initial values of  ∑ ∆Vi  ,  ∑ ∆Vi  , Tg,1, Tg,2,  ∑ ai ∆Vi + ∑ b j  , and j  i 1  i 2  i 1    ∑ ai ∆Vi + ∑ b j  for the mixtures corresponded to all three cases mentioned are  i  j  2 shown in Table A-5-1. Indicated in Figure A-5-1,a are dependences of the glass transition temperature of mixtures on the content of component 2 for the case, when      ∑ ∆Vi  ≈  ∑ ∆Vi  . Therewith, these values vary in a wide range. When the Van i 1  i 2   der-Waals volume of the repeat units is low, the value of  ∑ ai ∆Vi + ∑ b j  is also j  i 2

634

low, therewith, the cases are considered when the glass transition temperature is practically the same for both components of the mixture. Since the value bh, which characterizes the influence of hydrogen bonds on Tg, is quite high, it may be observed from equations (A-5-3) and (A-5-4) that the smaller is the Van-der-Waals volume, the more the glass transition temperature exceeds the value appropriate to the usual case, at which additional specific interaction between components is absent. This affects the run of Tg dependences on α2, indicated in Figure A-5-1,a. When the Van-der-Waals volume of repeat units of both components increases, the effect of additional interaction decreases, and the glass transition temperature of the mixture is just a little higher than the value of Tg calculated by equations (IV.55) and (IV.57). Table A-5-1 Initial values of parameters for calculation of the glass transition temperature of miscible mixtures of homopolymers             Number of  ∆Vi  ,  ∆Vi  , , L ∆'L + -M  , , L ∆'L + -M  , ∑ ∑       Figure curve in      i 1  i 2 M M  L   L  Figure A-5-1 3 3 3 –1 3 –1 Å Å ÅK ÅK 260 293 115 110 A-5-1a 1 588 665 260 250 2 814 931 360 350 3 1041 1197 460 450 4 543 279 240 105 A-5-1b 1 769 279 340 105 2 995 279 440 105 3 1222 279 540 105 4 260 598 115 225 A-5-1c 1 260 904 115 340 2 260 1170 115 440 3 260 1436 115 540 4 Note: Tg,1 = 376 K; Tg,2 = 442 K.

∑

∑

∑

∑

    Figure A-5-1,b is appropriate to the case, when  ∑ ∆Vi  <<  ∑ ∆Vi  . Here  i 1  i 2 the glass transition temperature is significantly higher than the value of Tg calculated by expression (A-5-3). It should be noted that even if additional hydrogen bonding is absent, the glass transition temperature of the mixture exceeds the value of Tg calculated on the basis of the molar parts of components.     Figure A-5-1,c describes the situation when  ∑ ∆Vi  >>  ∑ ∆Vi  . In this  i 1  i 2 case, influence of the additional hydrogen bonding decreases, because the total energy of dispersion interaction significantly exceeds the energy of hydrogen bonds. Such situation also appears in the cases, when the excessive intermolecular interaction between components at their mixing is the dipole–dipole interaction, occurred owing to the presence of polar groups in the repeat unit. In this case, the value bh in equations (A-5-3) and (A-5-4) is substituted by the value bd:

635

Figure A-5-1. Dependence of the glass transition temperature Tg on concentration of the second component α2 for miscible mixtures of two polymers when formation of hydrogen













bonding between them takes place: a)  ∑ ∆Vi  ≈  ∑ ∆Vi  ; b)  ∑ ∆Vi  <<

 i 1  i 2  i 1        ∑ ∆Vi  ; c)  ∑ ∆Vi  >>  ∑ ∆Vi  ; numbers on curves correspond to the  i 2  i 1  i 2

numbers in the curves shown in Table A-5-1. The curves 1’– 4’ correspond to the situation when no additional hydrogen bonding between the components takes place.

636

    α1  ∑ ∆Vi  + α 2  ∑ ∆Vi       i 1  i 2 Tg = ;         α1  ∑ ai ∆Vi + ∑ b j + 2α 2 bd  + α 2  ∑ ai ∆Vi + ∑ b j  + 2α1α 2 ⋅ 0.03 j j  i 1  i 2 

Tg =

  + α 2  ∑ ∆Vi     i 1  i 2 .    ∆V  ∑ i   i 2 + 2(1 − α 2 )α 2 ⋅ 0.03 + 2(1 − α 2 )α 2 bd + α 2 Tg,2

(A-5-5)



(1 − α 2 ) ∑ ∆Vi     ∆V  i ∑   1 (1 − α 2 )  i Tg,1

(A-5-6)

    When  ∑ ∆Vi  ≈  ∑ ∆Vi  , the influence of additional intermolecular  i 1  i 2 interaction on the dependence of Tg on the composition is also displayed, but it appears significantly lower than at occurrence of hydrogen bonds between components (Figure A-5-2,a). The reason is that the value bd, characterizing influence of the dipole–dipole interaction on the glass transition temperature, is lower by the absolute value than the value bh. If the Van-der-Waals volume of the component 2 is significantly higher than that of the component 1, positive deviation of Tg values from the average values also takes place, however, the effect is not so high as in the case of occurrence of hydrogen bonds (Figure A-5-2,b). If the Van-der-Waals volume of the repeat unit of the component 1 is much higher than of the component 2, the influence of additional intermolecular interaction is not so high, and dependences of the glass transition temperatures on the composition are quite close to those calculated by equation (IV.55), i.e. with no regard to additional intermolecular interaction between components. This is clearly indicated in Figure A-5-2,c. It is common knowledge that besides the dependences of glass transition temperatures on the composition of the mixtures with maxima (curve 1) and minima (curve 2), the S-shaped dependences are encountered quite often (curve 3) (Figure A5-3). This is associated with the fact that one of the components of the mixture represents a copolymer containing a limited number of polar groups, able to form hydrogen bonds. Let us consider this situation in more detail. Suppose that a copolymer contains a limited amount of repeat units capable of hydrogen bonding (copolymer 2). For example, copolymer 2 may be presented in the form: (CH2

CH)m

(*)

(CH2

CH)n

OH (**)

637

Figure A-5-2. Similar to the situation in Figure A-5-1 but with occurrence of additional dipole–dipole interaction between components.

638

Let us designate the molar part of the repeat unit (**) as β. Suppose that the counter-polymer (polymer 1) is able to form hydrogen bonds with the repeat unit (**), contained in the copolymer 2. Applying these designations, equations (A-5-3) and (A5-4) are reduced to the following forms: 

Tg =



(1 − α 2 ) ∑ ∆Vi  

1

i

+

  (1 − α 2 ) ∑ ai ∆Vi + ∑ b j  + 2(1 − α 2 )α 2 2α 2 βbh + j  i 1

→

  + α 2  ∑ ∆Vi     i 2 → ;   + α 2  ∑ ai ∆Vi + ∑ b j  + 2(1 − α 2 )α 2 ⋅ 0.03   j  i 2





( − α  ) ∑ ∆'  

%J = ( − α 

   ∑ ∆'   ) L

L

%J

L

L

;

(A-5-7)

+



+ ( − α  )α  α  β- + α 

   ∑ ∆'   

K

→

L

L

%J

+

.

(A-5-8)

  + α   ∑ ∆'    → + ( − α  )α  ⋅   L

L

Figure A-5-3. Schematic representation of typical deviation of the glass transition temperature from its average values for the mixtures of miscible polymers (see text).

As above, let us discuss a series of cases: 1) The Van-der-Waals volume of the repeat unit is approximately the same. Initial values of all parameters necessary for the calculation are shown in Table A-5-2. During calculation, let us vary content of the component (**) for the copolymer 2. Calculation results are shown in Figure A-5-4,a. Clearly, the part of the

639

components (**) able to form hydrogen bonds is quite moderate and, therewith, S-shaped dependences of the glass transition temperature of the mixtures on the composition are formed. These dependences were observed in many works (refs. [180, 192, 193, 209]). Table A-5-2 Initial values of parameters for calculating the glass transition temperature of miscible (Tg,1 = 376 K, Tg,2 = 442 K) mixtures of homopolymer (1) and copolymer (2)           , Å3K–1   , Å3K–1  ∆V  , Å3  ∆V  , Å3 , ' , ' ∆ + ∆ + i i ∑ ∑ L L L L M M             i 1  i 2 M M  L   L 

∑

110

115

∑

293

∑

∑

260

Figure A-5-4. Dependence of the glass transition temperature Tg on concentration of the second component α2 for miscible mixtures of homopolymer and copolymer (see text) when: a) hydrogen bonding between components exists; b) dipole–dipole interaction between components prevail.

2) This case corresponds to formation of strong dipole–dipole interactions between copolymer 2 and polymer 1.

640

For calculation of the glass transition temperature, appropriate equations are deduced from equations (A-5-7) and (A-5-8):

  − α  ) ∑ ∆'  +   %J = →   ( − α  ) ∑ , ∆' + ∑ -  + ( − α  )α  α  β- +  

(



L

L

L

L

M

L

G

M

;   + α   ∑ ∆'     →   + α   ∑ , ∆' + ∑ -  + ( − α  )α  ⋅        ( − α  ) ∑ ∆'  +   → =      ∑ ∆'   ∑ ∆'     + + ( − α  )α  α  β- + α  ( − α  ) 

(A-5-9)

L

L

L

L

%J

L

M

M

L

L

L

L

L

L

%J

G

%J

(A-5-10)

  + α   ∑ ∆'    →  + ( − α  )α  ⋅   L

L

The calculation results obtained by equations (A-5-9) and (A-5-10) are indicated in Figure A-5-4,b. Calculated dependences of Tg on the part of copolymer 2 were obtained under various parts of the component (**) in the copolymer. It should be noted that occurrence of the S-shape of Tg dependence on the composition is displayed only in the case, when the copolymer 2 contains a significant part of the component (**). However, in all the cases, these dependences differ from these obtained with no regard to occurrence of an additional dipole–dipole interaction between components of the mixture. In relation to variations of the glass transition temperatures occurred as a result of excessive intermolecular interaction between components, the calculated data indicate that these interactions are of a significant importance. The values of Tg differ from those, when no additional intermolecular interaction occurs. Note also the important circumstance that all equations (A-5-3) – (A-5-10) applied to the analysis of the glass transition temperature dependence on the mixture composition displays no one ‘fitting’ parameter and, hence, all calculations are performed on the basis of the chemical structure of the mixture components. Let us consider behavior of a series of particular mixtures of two miscible polymers. The first of them represents a mixture of poly-n-butyl methacrylate (PBMA) with styrene copolymer with 4-vinylphenyl dimethylsilanol (St-VPDMS) [180]:

641

(CH2

CH)m

H3C

(CH2

CH)n

Si CH3 OH

Firstly, it is necessary to calculate glass transition temperatures for St-VPDMS copolymer at different concentration of VPDMS. Such calculations were performed by equation (IV.55), and their results are presented in Table A-5-3. Good coincidence of calculated and experimental data is observed. For further calculations of glass transition temperatures of PBMA mixtures with the given copolymer, it is necessary     to know values of  ∑ , ∆' + ∑ -  and  ∑ , ∆' + ∑ -  determined       !  $9 '!$ from equation (IV.55). It is also necessary to determine the same values for copolymers of all compositions. These values are also shown in Table A-5-3. They were used for future calculation of the glass transition temperature of a series of PBMA and St-VPDMS mixtures of different compositions. Indicated in Figure A-5-5 are the calculation results for two mixtures in the form of dependences of the glass transition temperature on the composition. These dependences for miscible mixtures containing St-VPDMS-34 and St-VPDMS-9 were obtained by equation (A-5-7). For every mixture, quite good coincidence with the experimental data is observed (see Figure A-5-5,a and b). L

L

L

M

M

L

L

L

M

M

Table A-5-3 Parameters of copolymers of styrene with vinylphenyldimethylsilanol Tg (exp.), ºC Designations The part of Tg (calc.), ºC   ∆Vi ,   VPDMS, , L ∆'L + -M × 3  Å  L  % (mol) M   3 3 -1 ×10 , Å K PS 100 105 103 110 293 PBMA 100 29 19 148 516 PVPDMS-2 1.9 97 103 111 295 PVPDMS-4 4.1 98 103 113 301 PVPDMS-9 8.7 98 103 116 309 PVPDMS-P 11.4 99 103 117 311 PVPDMS-18 18.2 101 104 122 324 PVPDMS-34 33.9 115 109 133 348 PVPDMS-60 60.0 121 122 151 382 PVPDMS-100 100 153 178 418 

∑

∑

∑

Moreover, let us analyze dependence of Tg on composition for PBMA mixture with St-VPDMS-9 with the help o equation (A-5-8). Therewith, let us also introduce experimental data into this equation. Consequently, we obtain a good coincidence of calculated and experimental data (see Figure A-5-5,c). Note that glass transition temperatures of mixtures are below the average values, when concentration of modified styrene (i.e. St-VPDMS) in the mixture is low. In the case, when concentration of this component in the mixture increases, glass transition

642

temperatures exceed the average values. Such exceeding is explained by formation of a great amount of hydrogen bonds between components at their mixing.

Figure A-5-5. Dependence of the glass transition temperature Tg on the mole fraction of copolymer α2 for miscible mixtures of PS with VPDMS/PBMA: a) PVPDMS-34; b) PVPDMS-9. 1 – calculated curve; 2 – experimental dependence; 3 – linear dependence.

Let us now consider behavior of another mixture obtained at the mixing of poly-(2,2’-m-phenylene-5,5’-bibenzimidazol) (PBI): N

N

NH

NH

and poly-4-vinylpyridine:

643

[CH2

CH]

N These mixtures were studied in ref. [186]. It has been indicated that hydrogen bonds are formed between residues of 4-vinylpyridine and NH-groups in PBI. This is the reason that homogeneous mixtures based on these two polymers are formed. Experimental dependences of the glass transition temperatures on the composition of these mixtures, taken from ref. [186], are indicated in Figure A-5-6. Glass transition temperatures of the mixtures exceed values of Tg for the original components. Calculated dependences of Tg on composition, determined by equation (A-5-8), are also indicated in Figure A-5-6. It is obvious from this figure that experimental points fit well the calculated curve, when the molar fraction of PBI in the mixture does not exceed 0.3. Deviation of the experimental points from the calculated curve at high concentration of PBI is associated with the onset of poly-4-vinylpyridine degradation at heating over 375°. Note one more time that all calculations are performed by the equations containing no ‘fitting parameters. Hence, all the results are obtained only on the base of chemical structure of the components of the mixture.

Figure A-5-6. Dependence of the glass transition temperature Tg on the composition α2 for miscible mixtures of poly(vinyl pyridine) with poly(benzimidazole). 1 – calculated curve; 2 – experimental dependence; 3 – linear dependence.

Let us make some conclusions from the above analysis. In the case of miscible mixtures of two homopolymers, the dependence of the glass transition temperature on the composition of the mixture may obtain different forms /shapes/. If the Van-derWaals volumes of the repeat unit of homopolymers are approximately equal, and no additional strong intermolecular interaction occurs between the components, the glass transition temperature falls below the average values. If the Van-der-Waals volume of

644

the unit of homopolymer 1 is significantly smaller than that of homopolymer 2, positive deviations of the glass transition temperature from the average value take place even if the additional intermolecular interactions are absent. But if such interactions take place, a positive deviation of Tg values from the average values always takes place. In the case of homopolymer mixtures with a copolymer containing a limited amount of polar groups, capable for strong intermolecular interaction with other components (dipole–dipole interaction, hydrogen bonds), dependences of Tg on the composition are S-shaped. All the mentioned features of behavior of such miscible mixtures may be described with the help of the equations mentioned above, which allow estimation of the properties of the mixtures based on the chemical structure of the components. It is self-evident that the present approach cannot pretend to be general, and more general investigations are required to take into account the influence of various specific interactions and fine features of the structure of components of mixtures on their thermomechanical behavior.

Appendix 6. On formation of super-molecular structures in amorphous polymers

1 Scheme of formation of the super-molecular structure The experimental data obtained up to the present time, as well as theoretical investigations of polymers enable us to appoint a scheme of the formation of supermolecular structures and, yielding from this scheme, to make an attempt in future to set a connection between the chemical structure of a polymer, its super-molecular structure and physical properties [236]. The question about super-molecular structures and their forms was originally formulated in ref. [237]. Further on, various models of super-molecular structure appeared [238, 239], which allow an explanation of a series of properties of amorphous polymers in solution, melt and solid state. Simultaneously with model ideas, numerous experiments with obtaining super-molecular structure by various methods were performed, with the method of electron microscopy being the chief method. The most typical situation in amorphous polymers observed is the picture of globules the sizes of which correspond to aggregates of tens and hundreds of macromolecules. Let us dwell on problems of super-molecular structure formation and description of this process with the help of an analytical apparatus, developed in refs. [240 – 242]. Before passing to the essence of the problem, let us note in brief the main results of these works, which will be necessary for us for future discussion. Suggested by Lifshitz in ref. [240] was the equation: gˆψ = Λexp{µ*(n)/RT}Ψ, n = ψ2exp{µ*(n)/RT}, N = ∫ nd 3 r ,

(A-6-1) (A-6-2)

where gˆ is the integral operator; gˆψ = ∫ g (x − x′)ψ (x′)d 3 x′ ; g(x – x′) is the

conditional probability of existence of two units in points x and x′, respectively; ψ and Λ are the characteristic function and the characteristic value of equation (A-6-1), respectively; n(x) is the function of the smoothed density which defines the macroscopic state of the polymeric globule; µ* is the chemical potential of the system of broken chains; N is the total number of units of the macromolecule. Besides description of the ‘coil – globule’ transition, this equation allows analysis of the structure of various physical factors (temperature, solvent quality, interaction with the solvent). Basing on the analysis of solutions of equation (A-6-1), a phase diagram of the system of interacting particles of the polymeric macromolecule in solution was composed (Figure A-6-1), which, further on, will be widely used for explaining the formation of the super-molecular structure. As the ‘starting state’, consider a diluted solution of a polymer, in which overlapping of coils is absent, and a monomacromolecular transition is possible. It may be either a diluted solution, specially prepared for this purpose, or a solution of propagating chains of the polymer in the medium in which synthesis proceeds. The synthesis may proceed both in the monomer medium and in the medium of a specially

646

selected solvent (then, in the first case, the amount of macromolecules formed at the onset of the reaction will be low, and the solution may be considered diluted). The solution temperature T will be lower than some critical temperature Tc (which will be determined below), but is greater than the triple point T (Figure A-6-1). Then, macromolecules with molecular mass above the BA line (Figure A-6-1) will transit into the globular state with a dense center and loose ‘periphery’, and macromolecules with the molecular weight below the BA line will remain in the coil state.

Figure A-6-1. Phase diagram of a system of interacting particles (beads) of the chain in solution. EDC lines separate the area of globule without periphery, EDBA – globules with periphery, CBA – the coil. T is the temperature of the triple point; Tcr is the critical temperature of broken chains. Figure A-6-2. Scheme of formation of the initial element of super-molecular structure: 1, 2 – centers of globules of macromolecules; 3, 4 – peripheries of globules of macromolecules; 5 – bond globule (or coil).

To solve the task set, let us limit by the volumetric approximation when the density of a macromolecule at the nucleus–periphery interface changes jump-like to zero (periphery density). Let us discuss an interaction of two such globules with the periphery when they approach each other due to diffusion motion or sedimentation (interaction of coils is out of interest for us, because in this case either globules with a periphery may appear, if the total number of units in coils will appear above the coil– globule transition on the phase diagram, or a coil with higher molecular weight is formed, i.e. the problem is reduced to the same cases, which will be discussed below). When peripheries of globules of two macromolecules interpenetrate, units of peripheries will interact with each other. Therewith, the following variants are possible. If the total number of units in peripheries under given temperature is above the coil–globule phase transition line (Figure A-6-1), a bond globule with periphery is formed (Figure A-6-2). If the total number of units in peripheries is low and appears below the phase transition line, a bond coil is formed at the capture of periphery units of one globule by periphery units of another one, which at a future decrease of temperature transits either to the bond globule with the periphery or to the bond globule without the periphery (Figure A-6-1). Hence, in any case, the initial element forming the super-molecular structure will be two globules connected with each other by a globule or a bond coil. Depending on the size of the globule center and bond globule (or coil), these elements of super-molecular structure in the limit form two most typical types of elements of the super-molecular structure.

647

If a bond globule (coil) is small compared with nucleus of globules of two initial macromolecules, i.e. r << R (Figure A-6-2), an element of the super-molecular structure of the ‘dumbbell’ type is formed. If a bond globule (coil) is comparable with globules of original macromolecules, i.e. r ~ R (Figure A-6-2), an element of the super-molecular structure of the ‘ellipsoid’ type is formed. The rest of elements of the super-molecular structure will be of an intermediate type between the ellipsoid and the dumbbell. When the element of the super-molecular structure formed interacts with each other or with globules of macromolecules, new bond globules or coils appear, which lead to formation of super-macromolecules composed of globules– macromolecules. Depending upon the size of bond globules (coils), these super-macromolecules will possess different mobility. In the case of small bond globules (coils), their mobility will be maximal. Consequently, the super-macromolecule forms a giant globule representing the final element of the super-molecular structure. In the case when bond globules are large (the original element of the super-molecular structure, on average, approaches the ellipsoid), the mobility of separate elements of the supermacromolecule will be low, and a drawn fibrillar super-molecular structure is obtained. The size of the super-molecular structure will be defined by the length of the super-macromolecule. Let us estimate the length of a segment1 of the super-macromolecule in the case of a fibrillar super-molecular structure (in the case of the globular supermolecular structure, the size of the segment equals the size of the globulemacromolecule). Making estimations, we will proceed from the assumption that the sizes of super-macromolecular chains of globular and fibrillar super-molecular structures are equal. The number of globules–macromolecules by radius nR and length nL of cylinder of the fibrillar structure may be determined from the condition: 2 Vfib = πRfib Lfib , Rfib = nRRmol, Lfib = 2nLRmol, Vglob =

4

3 3 πRmol n ,

where n is the total number of globules–macromolecules in the final element of the super-molecular structure. From the condition Vfib = Vglob, it follows that 2nL nR2 = 4 3 n .

(A-6-3)

Let us deduce the second condition for nL considering a super-macromolecule, which forms the fibrillar super-molecular structure, as the Porod–Kratki chain. Suggest that a/L << 1, where a and L are sizes of the segment and the whole supermacromolecule, respectively. Then, according to [243], the length of the Porod–

( )

Kratki chain will be h 2 = 2La, where L = 2nRmol, a = βRmol, and h 2

1/ 2

=2nLRmol.

2

Then the condition for h may change to: 2nL =

1

2 nβ .

(A-6-4)

For the segment of super-macromolecule, we assume its part, the ends of which may travel independently, i.e. this is the same definition as for chains of macromolecules, but in relation to superchains.

648

From conditions (A-6-3) and (A-6-4), it follows that β = 8n 9nR4 . The number of globules–macromolecules n will be calculated on the base of the size of the supermolecular structure globule. The calculations will be performed on the example of polyarylate F-2, for which the electron-microscopic image of the film, taken from refs. [4, 244], is displayed in Figure A-6-3. Let us determine the number of macromolecules in super-globules. As indicated in Figure (A-6-3), super-globules are principally of two types – comparatively small and homogeneous, and larger. The former possess the radius Rcr = 125 Å, and the latter ~500–1200 Å. Polyarylate studied has M = 40 000, and polymerization degree is equal to 90. The Van-der-Waals volume of the repeat unit ∑ ∆Vi is 390.2 Å2; taking into account that the coefficient i

of molecular packing in the film is ~0.695, the volume of a single unit is 390.2/ 0.695 = 561 Å3. The volume of the macromolecule consisting of 90 units equals 50 860 Å3. At Rcr = 125 Å the number of macromolecules forming the super-globule will be ~150, and at Rcr = 500 Å it is ~10 000.

Figure A-6-3. Electron–microscopic image of polyarylate F-2 film according to data from refs. [4, 244].

The mean radius of the globule, 〈Dglob〉, determined by the microphotography method [4, 244], is ~800 Å. The number n appropriate to it at Rmol ~ 20 Å is n ~ 8 000. The value nR will be determined on the basis of the mean diameter of fibril 〈Dfib〉 ~ 192 Å; nR ~ 5. Then β ≅ 11. From this it is clear that the transversal size of the fibril will be similar by order of magnitude of the segment length of the supermacromolecule. The latter result is quite natural, because the condition a/L << 1 is true at β ≅ 11, and in this case, Dfib ~ a. In accordance with the above-discussed ideas, the fibrillar or globular structure may be formed by selecting the molecular mass distribution, solvent quality, temperature solution and rigidity of molecules. To analyze the influence of the abovecounted factors on the type of super-molecular structure separately, let us consider changing of the periphery thickness depending on the properties of macromolecules and their solutions. According to refs. [240, 241], when passing the coil–globule transition, the dependence of the globule density n on the coordinate x at T > Tc will obtain the form indicated in Figure A-6-4. As volumetric approximation will be considered further, the dependence for the periphery thickness obtained below may be also used in the temperature range T < T ≤ Tc. Considering the globule consisting of

649

a dense center with density nn and a periphery, let us determine the periphery thickness as the area where density changes linearly with distance x, counted from the dn x , where Rn is the radius of the central zone. As center border. Then, n(x) = nn − dx Rn nn is known, and the coil density equals zero, then to determine the periphery,

dn dx Rn

must be found. From expression (A-6-2), it may be found that dn d lnψ 2n = ⋅ . n dµ * dx dx ⋅ 1− RT dn

Figure A-6-4. Dependence of the globule density n on coordinate x (volumetric approximation is cross-hatched).

For the derivative dn/dx to be independent on x, it must be taken Ψ ~ eαx. Then, dn 2n = α. n dµ * dx Rn ⋅ 1− RT dn Let us consider the initial equation (A-6-1). For the condition ψ ~ eαx to be met in the area of the periphery, it is necessary that exp{µ*(n)/RT} = const. For the d2 area of periphery, let us present the operator gˆ in the form gˆ → 1 + a 2 2 , as it was dx made in ref. [240]. Then equation (A-6-1) will obtain the form: a2

d 2ψ Λ const − 1 + (1 − Λ const )ψ = 0 , α = ± 2 dx a2

and dn Λ const − 1 = 2n( Rn )α = 2n( Rn ) . dx a2

650

The dependence of the periphery density on coordinate x will be expressed as follows: 2n( Rn ) n ( x ) = nn − Λ const − 1x . a The periphery thickness 2 l will be found from the conditions n(Rn) = nn/2 and n(2l) = 0, and will be equal 2l =

a . Λ const − 1

(A-6-5)

In the volumetric approximation, when profile of the globule is substituted by a jog, the periphery thickness will be equal to l. Let us estimate const in expression (A-6-5) in the volumetric approximation. B2 As in the volumetric approximation, according to [242], Λ = 1 + , where B and C 4C are the second and the third virial coefficients in the equation of the globule state, at θ-temperature the periphery size is l = a / 2 const − 1 . The size of the globule Rglob is composed of the center size Rn and the periphery size. That is why, Rglob = Rn +

a . 2 const − 1

When T = θ, the globule transforms into a coil, Rn = 0, and 2Rglob = Na 2 , i.e. the square size of the globule equals to the mean square of the coil size (amplitude of fluctuations). That is why, const = 1 + 1/N. Hence, when T = θ, free energy and density of the globule, according to [242], will be equal to zero, and the periphery thickness 2l = N1/2a. When T < θ, the value 1/N may be neglected, and const ≅ 1. N 1/ 2 a aC 1/ 2θ − When T < θ, the size of the center will be Rn = , and its density is b(T − θ ) 2 3  1/ 2   N θ  4 3 1/ 3   C  nn = = 1  πa N 1 +    , 4 πR 3 n 3  3   N  b(θ − T )  

i.e. the density of the center increases as temperature decreases. To analyze the influence of temperature, solvent quality, and chain rigidity on the periphery thickness and, respectively, on the formation of the super-molecular structure, let us consider expression (A-6-5) at T < θ. The problem of the sequence of effects (heating–cooling, rate of solvent elimination, etc.) on the system and hysteresis phenomena associated with it will not be discussed. As in this temperature range const ≅ 1, nglob = –B/2C (according to ref. [242]), the expression for the periphery thickness may obtain the form: l=

a . − 2nglob B

(A-6-6)

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Rigid-chain polymers possessing large value of the segment, a, will be characterized by extended periphery l, and all other factors being the same, the fibrillar structure will be typical of them. In the limit, when the segment becomes equal to the macromolecule by length, and the latter obtains a stick-like form, the idea of the center becomes senseless, and the super-molecular structure may be of the fibrillar type only. The border of rigidity, out of which globules will not be already formed, may be determined from the phase diagram (Figure A-6-1) for the equivalent chain, macroscopic characteristics of which coincide with the characteristics of the original macromolecule [245]. Let us now estimate the influence of the solvent quality, all other factors being the same (constant concentration and temperature). As virial coefficients A2 are used in studying solutions of polymers, determined from the concentration dependence of osmotic pressure and light scattering, to analyze the influence of the solvent quality, it will be assumed that A2 ~ B. For the periphery to form, according to expression (A-6-6), the value of B must be negative, i.e. a bad solvent is favorable for forming the periphery. The worse is the thermodynamic quality of the solvent (the absolute value of B is higher), the smaller is the periphery size. If the solvent quality is significantly worse, the size of the periphery will be minimal and polymer will be precipitated out as compact globules without a periphery. The problem of formation of a super-molecular structure in a polymeric substance obtained from the solution without precipitation of the polymer will be largely of interest for us. It follows from expression (A-6-6) that when the film is cast from a bad solvent (large absolute value of B), globules with periphery l of small size will be formed. This, in its turn, will lead to formation of a globular super-molecular structure, and the sample will possess increased friability. Experimentally, this fact is confirmed well. When the quality of the solvent is increased (|B| → 0), the size of periphery will increase, and the probability of formation of a fibrillar structure increases. Therewith, samples possess better ability to deformation. When the thermodynamic quality of the solvent is increased significantly, the value of B becomes positive, and then the globule with center and periphery is not formed at all, and at formation of a solid macromolecules remain in the form of coils. In this connection, the experimental data become clear; according to these data, the addition of a small amount of a polymer precipitator to the solution improves the mechanical properties of films. Let us now discuss the influence of temperature on the character of formation of the super-molecular structure. If the absolute value of the second negative virial coefficient B increases as temperature decreases, then the quality of the solvent deteriorates; such solution has the upper critical temperature of dissolution. Therewith, according to correlation (A-6-6), the size of the periphery l decreases, and a compact globular structure will be formed during the formation of a film from the solution. This will lead to deterioration of the mechanical properties of the film. In the case when the absolute value of the second negative virial coefficient B increases with temperature, the quality of the solvent deteriorates; such solvent displays the lower critical temperature of dissolution. According to expression (A-6-6), in this case, the size of the periphery l will also be low, which will lead to the formation of a compact globular super-molecular structure. Consequently, in the first case, to eliminate formation of the globular super-molecular structure and to obtain globules with extended periphery, able to form bond globules and formation of a fibrillar structure, the film must be cast under increased temperature. In the second case, the film must

652

be cast at reduced temperature. Then, according to expression (A-6-6), the absolute value of B will be lower, and the size of periphery l will be greater. It is common knowledge that when the transitions occurs (glass transition temperature, melting point and temperature of degradation), the super-molecular structures weakly affect the appropriate critical temperatures. However, if the problems of plasticization and transfer (diffusion, conductivity, etc.) are considered, the contribution of the super-molecular structure may appear decisive. For example, let us consider plasticization of polymer. According to existing notions, plasticization is subdivided into two types: intrastructural one (by Zhurkov and Kargin–Malinski) and interstructural one (by Kozlov). Reasoning from the above-developed notions on the super-molecular structure, plasticization via the globule center corresponds to the intrastructural plasticization, when the plasticizer penetrating into the globule center, changes the temperature of the coil–globule transition (more accurately, the glass transition temperature). Therewith, the glass transition temperature will change smoothly with the concentration of the plasticizer, because in most cases (especially, in globular super-molecular structures), the volume of the globule center significantly exceeds the periphery volume. In the case of interstructural plasticization, the process of plasticization proceeds at the sacrifice of peripheries, because the plasticizer does not penetrate inside the center. Therewith, peripheries of those globules will be saturated by the plasticizers, which are located on the surface of superglobules. As the volume of these peripheries represents a small part of the volume of the superglobule center, saturation of the periphery by a plasticizer will proceed faster than in the case of intrastructural plasticization, which is observed experimentally and is indicated by an abrupt decrease of the glass transition temperature. The mechanism of interstructural plasticization suggested enables plasticization and anti-plasticization phenomena to be explained. As the elasticity modulus decreases during plasticization at the given temperature, within the framework of the model of the super-molecular structure considered this event may be explained by the fact that the periphery extends under the effect of the plasticizer, and the contribution of the more dense center to the elastic properties decreases. When anti-plasticization proceeds, the periphery thickness decreases first, contribution of the center to the elastic properties increases, and the elasticity modulus increases. Further increase of the plasticizer concentration indicates the same effect of it as in the case of plasticization. Doping of polymer (for example, polyacetylene by doping components I2 and AsF5) is, apparently, analogous to interstructural plasticization. That is why in spite of the low concentration of the dopant components in proportion to the total volume of the polymer, 100% doping of the periphery appears, and the total conductivity of the polymer will be determined by the conductivity of I2 and AsF5 vapors multiplied by the concentration of these components in the polymer (because the doped periphery represents a mixture of ‘gases’ of units and doping components). Therewith, transition to the dielectric state will proceed due to collapse of the periphery. Consequently, atoms of doping components appear disseminated in the solid matrix. Due to the same reasons, the diffusion coefficient of the polymer will be equal to the diffusion coefficient of gas of the penetrating substance multiplied by the concentration of peripheries. When temperature decreases, the coefficient of diffusion similar to conductivity must display a minimum in the temperature range of the metal–dielectric transition.

653

A diluted solution at temperatures below the θ-temperature was considered below as the onset state for forming a super-molecular structure. Therewith, a significant role affecting the stability of the super-molecular structure as the polymer concentration increases is played by surface tension, appearing in both the globule itself and in the solvent affecting the globule after the coil–globule transition. If a ‘good’ solvent at temperatures above the θ-point is taken for the onset state, formation of the super-molecular structure is rather problematic and requires further consideration with regard to the data obtained in ref. [246]. It should be noted that the majority of super-molecular structures were obtained from solutions of polymers at room temperature, whereas their θ-temperatures exceed the room temperature by 10–20° and more [247]. That is why the mechanism of formation of a super-molecular structure discussed above may be assumed as quite a typical one of polymers. As shown below, the mechanism of formation of a super-molecular structure suggested may be used for explaining and predicting the solubility of polymers.

2 Calculation method of evaluation of dimensions of elements of super-molecular structure of polymers The physical characteristics of polymers calculated by the approach discussed in the monograph may be subdivided conditionally into two types. The characteristics mainly determined by the chemical structure of the repeat unit relate to the first type. These characteristics are the glass transition temperature, temperature of the onset of intense thermal degradation, the refractive index, density, etc. The second type embraces the characteristics not only by the chemical structure of the repeat unit, but also by the geometric size of elements of the supermolecular structure. These are the elasticity modulus, solubility, conductivity, etc. To calculate these characteristics, one must know how to determine the geometric size of elements of the super-molecular structure in relation to conditions of molding of the polymeric sample. In the case of casting film samples, the super-molecular structure will depend on the chemical structure of the solvent and the polymer, the temperature of solvent evaporation, the rate of evaporation, adding of precipitators, etc. Before we turn to discussion of the analysis results, several general notes must be made. Due to its stoichiometric configuration, not every system of macromolecules is capable of spontaneous formation of either a globular or a fibrillar structure. If we start preparation of the sample from a solution, then the flexible chains in static conditions counteract in size and in the corresponding region of scaling diagrams of state the interaction of the blocks may lead to the formation of globular density agglomerations. Semi-rigid chains, vice versa, strive to straighten as the concentration increases, which will lead to the formation of a fibrillar super-molecular structure. Moreover, flexible molecules in a straightening field (whatsoever in the presence of strong gradients of rate or force) may behave themselves as rigid ones and form a fibrillar super-molecular structure. Selection of a bad or good solvent determines both thermodynamic and kinetic factors. Therewith, the rate of solvent elimination (or precipitator injection) may play the decisive role, because the system of macromolecules must manage to rearrange itself to one or another side. Moreover, it is common knowledge that amphiphilic or diphilic macromolecules may interact ‘in parts’ separately with the precipitator. For example [254], when poly(methyl methacrylate) is precipitated by methanol, a fine

654

suspension is formed, drying of which leads to a typically granulated (globular) structure, because the precipitator solvates ester groups, but causes a collapse of nonpolar backbone of the chains. In contrast, saturated hydrocarbons as precipitators lead to the formation of a continuous gel, because they precipitate, in the strict sense, only polar groups, but not the backbone. External fields and the regularity of chains (as well as the absolute values of the size of the repeat unit) contribute to the kinetics of structure formation. Following the results from ref. [94], the special attention in this Appendix will be paid to the thermodynamic aspect of the problem of formation of super-molecular structure proceeding at casting of a polymeric substance from solution. Performed in ref. [255] are calculations on the determination of sizes of superstructures appearing in block-copolymers. Analogous types of elements of supermolecular structure may also be observed in homopolymers (according to technology in ref. [255], lamellar; fibrillar, appropriate to the cylindrical one; globular, appropriate to the spherical one). Let indicate that the original physical assumptions used in calculating the size of super-structures in block-copolymers are maintained at formation of the super-molecular structure according to the scheme discussed in ref. [92]. That is why correlations from ref. [255], appropriately corrected, may be applied to the determination of the size of elements of the super-molecular structure. In spite of ref. [255], discussed below are super-molecular structures in the amorphous state of the homopolymer. The physical base of structure formation in block-copolymers is immiscibility of blocks, their tendency to segregation. Therewith, morphology is determined by the ratio of the molecular mass of blocks. In the case of forming a super-molecular structure in homopolymers, it is based on super-macromolecules, formed from globules – macromolecules existing below the coil – globule transition temperature2, Tcr [92]. Below Tcr, the polymer and the solvent become immiscible, and segregation appears (the globule presses out the solvent). We are interested in the result of segregation at decrease of the solvent concentration, as well as in the influence of the molecular mass on the results obtained. When determining sizes of elements of the super-molecular structure, let consider the final stage of appearance of the super-molecular structure, when the concentration (volumetric part) of the solvent, cs ~ 0, and the concentration of supermolecules, csm ~ 1. Let us consider a change of the conformational free energy of the supermolecule of the α-type, ∆Fα, where α = 1 in the case of the lamellar structure, α = 2 in the case of the fibrillar structure, and α = 3 in the case of the globular structure. According to ref. [255], let us confine ourselves to two summands and write down that ∆Fα = ∆Fα,s + ∆Fα,e,

(A-6-7)

Rα2 2 kT ; H sm is 2 H sm the mean square distance between ends of the super-macromolecule placed into the 2 medium of the same super-macromolecules with concentration csm; H sm = l2M/m at csm ~ 1 according to ref. [256], where M is the molecular mass of the where ∆Fα,e is the elastic component of the free energy; ∆Fα,e =

2

3

2

In ref. [244], such super-macromolecules were named ‘nails’ consisting of many twisted globules – macromolecules.

655

supermacromolecule; m and l are the mass and the size of the globule – macromolecules; Rα is the size of the element of the super-molecular structure of the α-type; ∆Fα,s is the surface component of the free energy appearing due to the effect of the surface tension forces of the solvent on the super-macromolecule: ∆Fα,s = γαMv Rα ,

(A-6-8)

where γ is the coefficient of the surface tension of the solvent; v is the specific volume of the super-macromolecule. Influence of the solvent on the supermacromolecule is taken into account in the value of ∆Fα,s. Reducing ∆Fα to minimum by Rα, let deduce the value of  αγ 2  l  Rα =   3kT 

1

3

M2   m 

1

3 v  . 

Taking into consideration that M/m = Nc, where Nc is the number of globules– macromolecules forming the super-macromolecule, M v = Ncl (with an accuracy of a constant coefficient), we obtain the final expression for Rα: 1

 αγ 2  3 2 3 Rα =  l  Nc l .  3kT 

(A-6-9)

Let us discuss now in more detail the mechanism of twisting of the supermacromolecule into a globule or a fibril in order to refine the value of the surface tension participating in expression (A-6-9). For an example, let us consider a globular super-molecular structure, although the result obtained may also be applied to other types of the super-molecular structure.

Figure A-6-5. Initial stage of interaction of globules–macromolecules and formation of supermolecular structure (the scheme is presented according to data of ref. [92]).

According to the notions developed in ref. [92], globules–macromolecules possess a denser center and friable periphery (Figure A-6-5). When globules– macromolecules interact, bond globules are formed in the periphery, which is clearly observed from Figure A-6-5. If the bond globules appear on two opposite ends of diameters of the globules–macromolecules, no motion of elements of the supermacromolecule would appear, because forces stipulated by the surface tension from the side of opposite hemispheres of the globule–macromolecule would be equilibrated. But as the radii connecting centers of globules locate at an angle β, the

656

value of which is random, a non-equilibrated force appears, by which the solvent acts on elements of the supermacromolecule. The value of this force is proportional to a non-equilibrated square equal to the doubled square cut-off from the surface of the globule–macromolecule by the bond globule (Figure A-6-5). Hence, the surface component of the free energy for the globule–macromolecule may be presented in the 2 form: (∆Fs)glob = γSbond, where Sbond = π Rb.glob. ; Rb.glob. is the radius of the bond globule. The density of the surface component of the free energy of the globule– macromolecule (sub-index gm), calculated per specific surface, will be equal to  ∆F  s   = ξγ , 2   4πRgm   2

 Rb.glob.   ; Rgm is the radius of the globule–macromolecule. From the where ξ = 2   Rgm    condition of equality of the density of the surface component of the free energy on the globule of super-molecular structure and on the globule–macromolecule (because effect of the surface tension on the globule of super-molecular structure is composed of effects on separate globules–macromolecules), it is obtained that 1

∆Fαs ∆Fs = = ξγ . 2 2 4πRα 4πRgm Hence, influence of the solvent on globules of the super-macromolecule may be taken into account via the efficient coefficient of surface tension γ′, which is γ′ = ξγ. The value of Nc will be found from the condition of equilibrium of volumes of a freely linked super-macromolecular chain and a globular super-molecular structure  αγ ′ 2  3 2 N c3 2l 3 =  l l N c  3kT  or −1

 αγ ′ 2  N c1 2 =  l  .  3kT  In the case of a globular super-molecular structure, R3 kT = . l γ ′ b2 Let us estimate the value of ξ for the case of the globular super-molecular structure. If, for example, R3/l = 20 [92], l ~ 20 Å, T = 300 K, γ = 0.04 N⋅m,

657

2

1 ξ= , 2ξ = 800

 Rb.glob.    = n ,  R  N  glob 

where n is the number of units in the bond globule; N is the number of units in the 2 2 = Nlc2 , where lc is the value of the macromolecule, because Rb.glob. = nl2, Rgm macromolecule segment. Let N ≅ 10,000. Then n ≅ 24 units, and the periphery of every macromolecule gives 12 units. In the case of fibrils, R2/l ≅ 5, and n ≅ 72 units. These results allow refining of the model of super-molecular structure formation. If it was suggested in ref. [92] that the fibrillar structure was formed by elements of the ellipsoid type, it followed from the above estimations that the dumbbell was the universal element of the super-molecular structure. As the number of units forming the bond globule at formation of fibrillar and globular structures is much less than the amount of units in the globule–macromolecule, a dumbbell-shape figure is formed, when two globules–macromolecules unite. In the case of the fibrillar structure, the bond globule is 3-fold greater. As the bond globule contains a low number of units, it displays rather low (entropic) elasticity almost in the whole temperature range that enables the super-macromolecule to adhere in the required shape. Let us estimate R3. Because γ = kT/a2 [257], where a is the typical size in the a2 2 2 liquid, then R3/lb.glob. = 2 , a2 = N s ls2 , lb.glob. = nlun. , and R3/l = (Ns/n)(ls/lun.), lb.glob. where Ns/n is the number of solvent molecules per one unit of polymer. Let us determine the value Ns/n from the condition that the polymer unit forms a sphere, and solvent molecules locate in the spherical layer near the sphere–unit: 2

2 R  R  N s 3Run. 2 Rs + 3Run. 4 Rs2 + 8Rs3 = = 6 un.  + 12 un.  + 8 . n R Rs3  s   Rs 

Initial data and calculation results of sizes of elements of the super-molecular structure for a series of polymers at polymerization degree N = 100 ∑ ∆Vi , Å3 R,Å æ R /l Solvent i

Chloroform DCE Benzene Toluene Dioxane

Chloroform Benzene Dioxane

3

  = 96.8 Å3, l = 21 Å Poly(methyl methacrylate),  ∑ ∆Vi     i  un. 69.6 0.896 23.17 76.2 0.923 23.89 88.2 0.969 25.14 105.1 1.028 27.13 87.7 0,968 25.11   = 109.7 Å3, l = 22 Å Polystyrene,  ∑ ∆Vi    i   un. 69.6 0.859 22.21 88.2 0.930 24.08 87.7 0.928 24.03

3

487 502 527 569 527

488 530 528

658

Chloroform Dichloroethane Tetrachloroethane Nitrobenzene Tetrahydrofuran

  Aromatic polyester,  ∑ ∆Vi  = 390.2 Å3, l = 34 Å    i  un. 69.6 0.178 8.39 76.2 0.195 8.64 105.8 0.271 9.84 105.1 0.269 9.81 84.2 0.215 8.96 2

2

286 292 333 333 306

2

 l   R   R  R R Since  s  =  s  , then 3 = 6 + 12 s + 8 s  . The value Rs/Run. l Run.  Run.   lun.   Run.  1

may be presented in the form: æ = Rs/Run

 3 =  ∑ ∆Vi   i s

1

 3  ∑ ∆Vi  , where  i  un.

     ∑ ∆Vi  and  ∑ ∆Vi  are the Van-der-Waals volumes of the solvent molecule  i s  i  un. and the repeat unit of the polymer, respectively. If we assume from the experimental      ∑ ∆Vi  ≈ 2.2. data that Ns/n = 20(Run/Rs)2, then it is obtained that  ∑ ∆Vi   i  un.  i s On a series of particular examples, let us calculate the dependence of R3/l on æ. Initial data and calculation results for three polymers – poly(methyl methacrylate), polystyrene, and phenolphthalein and isophthalic acid polyester – are shown in the Table above. It is indicated that if the Van-der-Waals volumes of the repeat unit of polymer and the solvent molecule are similar, then æ ~ 1, and the ratio of the size of the element of the super-molecular structure R3 to the globule–macromolecule size l (R3/l) changes in a narrow range from 22 to 27. If the Van-der-Waals volume of the solvent molecule is significantly smaller than the volume of the repeat unit of polymer, then æ and R3/l are significantly lower (0.18–0.27 and 8.4–9.8, respectively). However, if the absolute size of the element of the super-molecular structure R3 is calculated, these values are close for all polymers (see the Table). It should be noted on this subject that electron micrographs for various amorphous polymers, observed in numerous works, display approximate equality of these sizes, which was one of the reasons to ascribe this structure to artifacts and to doubt in reality its existence. As indicated by estimations for polymers of different classes made in ref. [94], this very picture must be observed. In the case of fibrils, from the condition of equality of the volumes of the cylinder formed by a freely linked chain of the supermacromolecule and a fibril, we obtain that 2

 αγ ′ 2  3 4 3 2 l  N c πl L , πl LN c =   3kT  2

where L is the length of the element of the cylinder (fibril). 1  2γ ′ 2  Nc 3 =  l   3kT 

−2 3

, R2 =

3kTl , 2γ ′l 2

659

and the ratio R2/R3 =

2 3Rb.glob. g 2 2 Rb.glob. f

=

3ng 2nf

, where ng is the number of units in the bond

globule of the globular super-molecular structure; nf is the number of units in the bond globule of the fibrillar super-molecular structure. Basing on the experimental data [767], it is obtained that R2/R3 ≈ 1/4 and ng/nf ≈ 1/6 (in the same solvent). Let now determine the value of the free energy of the domain (supermacromolecule) ∆Fα. With the value of Rα calculated by equation (A-6-9), expression (A-6-7) gives ∆Fα = 3 2

αγ Mv . Rα

Let us now estimate free energy of globular and fibrillar super-molecular structures: ∆Fg = ∆Fg ∆Ff

=

9γ g 2 R3

V , ∆Ff =

9γ f V; 2 R2

3γ g R2 3ξ g R2 3ng R2  3ng = = = 2γ f R3 2ξ f R3 2nf R3  2nf

2

  . 

∆Fg 1 3 1 = , and = << 1, i.e. the fibrillar structure is ∆Ff 16 2nf 2 ⋅ 6 4 energetically less profitable than the globular one in the current solvent. To obtain a fibrillar structure, the following condition must be fulfilled: Estimations give

3ng

=

∆Fg + ∆F ≈ ∆Ff, where ∆F may be associated with heat delivery in the case of solution or with application of mechanical energy (in the solid state). To obtain a bond globule appropriate to the fibrillar structure, a polymer must be exposed in the area of the coil–globule transition temperature Tcr. The value of the bond globule must depend on the difference of temperatures Tcr – T and the rate of evaporation of the solvent. To obtain the desired super-molecular structure, Tcr for the given polymer– solvent system must be known. Usually, Tcr is significantly higher than the experiment temperature, or it rapidly shifts to the side of high temperatures at solvent evaporation. That is why peripheries of globules–macromolecules do not manage to penetrate into each other. If this process is decelerated artificially, formation of a fibrillar structure may be expected. Obtaining ratios for R3/l and R2/R3, the ratio for R2/l may be obtained: 2  Rs   Rs R2 ng    . = 9 + 18 + 12 l nf  Run.  Run.   

Let assume that ng/nf = δg/δf, where δg is the thickness of the periphery of the globule–macromolecule for the globular super-molecular structure; δf is the thickness

660

of the periphery of the globule of macromolecule of the fibrillar super-molecular structure. The main factor affecting the periphery thickness is the difference of temperatures, Tcr – T. If this value is extremely high, then δ → δg. In the general case, if the results from refs. [92, 257, 258] are used, it may be indicated that

δ = δg.+ (Rα − δg) e

β

T −Tcr T

.

When T = Tcr, δ = Rα, and when (Tcr – T)/T → ∞, δ → δg:

δg

=

ng

=

1

,  Rα   T − Tcr    − 1 exp β 1+    δg T      β∆T  Rα 7T β∆T  = 5 , ≅ −7 , ∆Tlim ≅ − cr . exp T T β δg  cr  cr

δf

nf

If the initial temperature of solvent evaporation at casting of the film corresponds to the condition ∆T > ∆Tlim, then the polymer will transit into the globule state. When ∆T < ∆Tlim, formation of a fibrillar super-molecular structure may be expected. Therewith, it should be taken into account that when the solvent concentration changes, Tcr will also change, and to obtain a fibrillar structure, Tcr must always be controlled. Determined above were the sizes of various types of super-molecular structure, their relation to the chemical structure of the polymer and the solvent was stated. However, the question remains, how a globule or fibril is formed from a supermacromolecule existing in a relatively diluted solution that activates formation of closed forms, the process of super-molecular structure formation. Data accumulated up to now enable to answer this question. Usually, the film is formed in a definite volume at gradual evaporation of the solvent. Therewith, temperature gradients occur between the solvent surface and the vessel bottom. Moreover, gradients of surface tension forces occur on surfaces of elements of supermacromolecules. Consequently, a convection motion occurs in the solution, and at some concentration of the solvent, the Benar cells are formed (either spheres or cylinders). The moment of appearance of the Benar cells is determined by the Marangoni number, because in thickness of the solution considered a significant role is played by gravitation–capillary effects [259]. The type of excited motion is determined by rigidity of the chain of super-macromolecule, i.e. by the size of the bond globule. Note that in contrast to simple viscous liquids and their mixtures, in the case of macromolecules, only the first modes of convection motion will be excited (rotation of the whole sphere or cylinder), because other modes of the motion cannot be excited due to bulk of the super-macromolecular chain. This convection motion may ascribe the shape to super-macromolecules, which they possess in the supermolecular structure. The fact that formation of the super-molecular structure is associated with convection motions may be observed experimentally. Introducing spin labels into a polymer and observing the solution behavior in an alternating magnetic field, one may observe a resonance absorption in the range of low frequencies associated with particular motions of the super-macromolecule. This resonance

661

absorption is absent in diluted solutions, appears at the moment of super-molecular structure formation and will last up to complete evaporation of the solvent.

3 Phase state of polymers as a result of formation of the supermolecular structure by one-cavity bond hyperboloids To describe the amorphous state of a substance, the space of constant negative curvature (the Lobachevski space) has been used in some works [260–262]. When structures given in the Lobachevski space are mapped on the Euclidean plane, an irregularity appears, usually registered by X-ray analysis. Moreover, as indicated in ref. [263], in a space with a negative curvature crystalline lattice considered in the harmonic approximation becomes unstable at quite high temperature – melting is observed. We have described the model of the super-molecular structure of polymers, formed on the account of interaction (bonding) of globules–macromolecules via periphery (Figure A-6-5). Application of this model to the description of solubility [264] demanded revision of the structure formed by peripheries when globules– macromolecules are bonded. It was found that this structure must possess the surface of a negative curvature. Owing to this very negative curvature of the structure surface (‘bond belt’) formed between globules of the super-molecular structure, the solvent affects the globules by forces, which (under definite conditions) cause their detachment. In this Section, let us determine the curvature of the surface of the bond structure formed between globules–macromolecules and super-molecular structures via refinement of the structure and associate the former with parameters of the supermolecular structure (thickness of peripheries, size of elements of super-molecular structure, globules–macromolecules, etc.). Therewith, let us base on the results obtained in ref. [93]. Furthermore, proceeding from particular expression for curvature, let us analyze the influence of the parameters of the super-molecular structure on the phase state of the polymer. Let us also determine the size of the elementary structure (amorphone) producing the amorphous state [265]. In turn, influence of the curvature of bond hyperboloid on the phase state of polymer appears after potential mapping of atoms interaction (for example, the Lennard–Jones potential) on a surface with negative curvature (the Lobachevski space). Consequently, additional terms appear in the interaction potential, which depend on the space curvature. In ref. [264], the bond region of globules–macromolecules and globules of super-molecular structure is considered as an element of a structure with negative curvature. To perform further calculations, let us refine the shape of the surface of this element. Let us assume that at quite good approximation (the latter fact is confirmed by calculations of various characteristics of polymers stipulated by the supermolecular structure) this surface may be considered as a one-cavity hyperboloid. Then within the framework of the globule (or fibril) of the super-molecular structure, onecavity hyperboloids form the full surface of negative curvature when crossed [266]. Let us determine curvature K of the one-cavity hyperboloid via the sizes of the periphery and the globule–macromolecule. Therewith, let us assume that bonding regions between globules–macromolecules and globules of the super-molecular structure possess the same size, because in both cases bonding is performed via the globule–macromolecule. Figure A-6-6 depicts the one-cavity bond hyperboloid formed by mutual penetration of peripheries of two globules–macromolecules. The main curvature radii R1 and R2 will be respectively equal to: R1 = CB, R2 = AB. If R1

662

and R2 are expressed via radii of globules – macromolecules Rgl-m and periphery thickness l, we obtain that

Figure A-6-6. Scheme for calculating curvature of the one-cavity bond hyperboloid: 1, 2 – globules– macromolecules 1 and 2; 3 – one-cavity bond hyperboloid; dotted lines mark borders of peripheries of globules 1 and 2.

R1 = βl,

R2 = Rgl-m

 1 − l Rgl−m  4 Rgl−m l

 .  

(A-6-10)

Since l 4 Rgl−m

<< 1, then R2 ≈

lRgl−m .

The value of coefficient β will be refined in future calculations. According to the definition, the curvature value K = 1/R1R2, where R1 and R2 are determined by formulae (A-6-10). In real polymeric systems, the curvature of one-cavity bond hyperboloids will fluctuate owing to molecular-mass distribution (because the thickness of the periphery depends on the molecular mass). That is why it is necessary to average the curvature by quite a great number of macromolecules within the framework of the element of the super-molecular structure and to operate with the mean curvature in future. Let us consider now the potential of interaction of atoms forming the bond hyperboloid in the negative curvature space. As indicated in ref. [263], critical temperatures do not depend on the metrics selection. The most important is that the metrics must correspond to the space with a constant negative curvature. That is why, furthermore, conformal Euclidean metrics will be used, because in this case, expressions for potential and appropriate calculations are simpler.

663

According to ref. [263], in the present case, the potential of interaction of atoms in the space with negative curvature is of the following form: R

Φ = Φ 0 + K ∫ Φ ′0 ( R) R 2 dR ,

(A-6-11)

R0

where Φ0 is the desired potential in the absence of curvature; R0 is the coordinate at which Φ0(R0) = 0. Let Φ0 be described by the Lennard–Jones potential:  1 2  Φ0(ξ) = D  12 − 6  . ξ  ξ Here ξ is the non-dimensional coordinate; ξ = R/a, a is the distance between neighboring lattice points. Then the potential (A-6-11) in the non-dimensional system of coordinates ξ may be presented in the following form: Φ(ξ) = Φ0(ξ) + Ka2

ξ

∫ Φ′0 (ξ )ξ

2

dξ ,

(A-6-12)

ξ0

where ξ0 is the coordinate, at which Φ0(ξ0) = 0, ξ0 = R0/a; K is the curvature of the surface possessing negative curvature in the Lobachevski space. Integrating equation (A-6-12), we obtain that  1  1 Φ(ξ ) 1 2 1  1  = 12 − 6 + 12 Ka 2  − 4  − 12 Ka 2  − 4  . 10 10 D 4ξ 0  4ξ 0  ξ ξ  10ξ 0  10ξ 0

(A-6-13)

From the condition Φ0(ξ0) = 0, we obtain that ξ 06 = 1/2. Substituting ξ0 into the expression for potential (A-6-13), we finally obtain that  1 Φ(ξ ) 1 2 1  = 12 − 6 + 12 Ka 2  − 4  + Ka 2 . 10 D 4ξ 0  ξ ξ  10ξ 0

(A-6-14)

From the condition Φ′(ξ) = 0, we find extreme values of the potential (A-614):  1  1   − 6 + 1 2 + Ka 2  = 0.  ξ  ξ     In the case, if K > 0, Φ(ξ) possesses a single maximum at ξ = 1, which coincides with the minimum of the potential Φ0(ξ). This result is not of interest. When K < 0, Φ(ξ) possesses the minimum at ξ1 = 1, and the maximum at ξ2 = 1/ a − K , if ξ2 > ξ1. When ξ2 < ξ1, potential (A-6-14) possesses the maximum at ξ1 = 1 and the minimum at ξ2:

664 6 3 Φ(ξ1 ) Φ(ξ 2 ) Φ(∞ ) 4 1 = −1 + Ka 2 , = − Ka 2 − Ka 2 + Ka 2 , → Ka 2 . D D D 5 5 It is clear that at ξ2 > ξ1, i.e. at |Ka2| < 1, the atom existing in the fields of two neighboring atoms will possess only one state of equilibrium. When ξ2 < ξ1, i.e. |Ka2| > 1, the atom existing in the field of two neighboring atoms possesses two states of equilibrium. Therewith, if all three atoms are of the same type, or the atom located in the point ξ = 1 interacts with similar (but of a different type, than in the point ξ = 1) atoms located in the points ξ = 0 and ξ = 2, the difference of energies of potential pits in points ξ 2′ and ξ 2′′ (the difference in energies of rotary isomers) ∆E = 0. But if the atom located in the point ξ = 1 interacts with different atoms located in points ξ = 0 and ξ = 2, then ξ1 − ξ 2′ ≠ ξ 2′′ − ξ1 and ∆E ≠ 0 (Figure A-6-7). Summing up potentials, we will proceed from the condition that positions of the maximum of the potential Φ(ξ) of the left and the right atoms coincide (although this condition is not obligatory).

( ) ( )

Figure A-6-7. Potential of interaction of the given atom (ξ1 = 1) with two neighboring atoms (ξ = 0 and ξ = 2) under two different conditions: ξ2 > ξ1 (a) or ξ2 < ξ1 (b).

Hence, when |Ka2| > 1, summation of potentials (A-6-14) will give two states of equilibrium for every atoms, which is typical of systems of the order–disorder type. Systems of similar type display a phase transition of the second kind [267]. To provide transitions of atom from one state of equilibrium into another, the presence of defects (of the disclination type) in the polymer is necessary, the size of which (angular values) may be determined via coefficients of molecular packing of amorphous and crystalline states. When |Ka2| < 1, summation of potentials (A-6-14) will give only one state of equilibrium for each atom, which is typical of regular systems of the crystal type. In such systems, the phase transition of the first order takes place [267]. Let estimate the value β basing on the following ideas. According to formula (A-6-10), K = 1 βl lRgl−m . If assume that a ~ 2 Å (this corresponds to two Van-der-

(

)

(

)

Waals radii of hydrogen atom), and Rgl-m ≈ 20 Å, then at Ka2 = 4 βl 20l ~ 1, β2/3 ≅ l ,β –1

2/3

(

)

≅ 1 lun. nper . For the globule, nper ~ 24 [94], β

2/3

≅ 1/(5lun.), lun. ~ 3 Å, β ≅

0.45, i.e. is approximately equal to the half of the periphery thickness. That is why, R1 in expression (A-6-10) may be presented in the following form: R1 = 0.5lun. nper . If nper > nf or nper > ngl (where nf and ngl are the numbers of units in the bond

665

hyperboloid in the case of fibrillar or globular super-molecular structure [94]), the polymer exists in the crystalline state. On the base of data from ref. [94], let us determine

(Ka ) (Ka ) 2

a

2

c

l =  c  la

  

3

2

n  =  c   na 

3

4

= 3 4 ≅ 2.3 . 3

1

Since (Ka2)c = 1, (Ka2)a = 2.3 and ξ2 ≅

≅ 0.7. The value ξ2 is appropriate to − Ka 2 the size of the molecular defect and may be estimated via the coefficient of molecular packing in amorphous, ka, and crystalline, kc, state by the relation [94]:ξ2 = (kc – ka)1/3. Usually in the crystalline state, the mean packing coefficient kc = 0.74, and in the amorphous one ka = 0.68; then ξ2 ≅ 0.45. Let us now determine the size of amophone3. For scattering of X-rays or electrons on the globule–macromolecule to be fully isotropic, it is necessary that the whole surface of the globule is overlapped by cross-sections of the rotation hyperboloid with radius R2, i.e. 4πRgl2 −m = NπR22 . N=

Rgl2 −m R22

=

4 Rgl−m l

=4

M mol ~ 40 , ndet

L2a = 4 NRgl2 −m ~ 160Rgl2 −m , Na ~ 12Rgl-m ~ 240 Å. The cross size of the element of the super-molecular structure in a polymer fibril approaches by the order of value the size of amorphone estimated above. That is why the fibril may be approximated as a tube possessing the radius of amorphone and filled with amorphones along the generatrix. In the globule of the super-molecular 4 4 structure, the amount of globules–macromolecules equals πRgl3 = N m πRgl3 -m , 3 3 Nm =

Rgl3 Rgl3 -m

3

 5 ⋅ 10 2   ~ 16 ⋅ 103 , ~   ⋅ 2 10  

Nm ~ 400 , Na m

where Nm is the number of globules–macromolecules forming the globule of the super-molecular structure; Nam is the number of globules–macromolecules forming the amorphone. Hence, the globule of the super-molecular structure contains ~400 amorphones. Given here were statistical estimates of amorphone. Determining the number of amorphones in the globule of the super-molecular structure, it was suggested that it has the spherical shape. In reality, to determine its shape, it is necessary to find, from the condition of dense packing of globules in the element of the super-molecular structure, the shape of the bordering surface providing isotropic scattering of X-rays 3

The assumption used is excessive and guarantees fully isotropic scattering; more precise solution of the problem requires determination of the amount of cross-sections and their projections appearing at overlapping of bond hyperboloids under condition of formation of element of the super-molecular structure.

666

or electrons on globules–macromolecules. This sort of problems may be solved by a computer. The model of the super-molecular structure considered enables to refine the physical meaning of elements of the nonlinear mechanical model of polymeric substances, suggested in ref. [87]. This model is depicted in Figure A-6-8. According to this model, relaxation processes are described by the spectrum of relaxation times, which contains two main times (short and long) and a number of other relaxation times dependent on these two main times. In the right part of Figure A-6-8 shown the globule–macromolecule and radial lines connecting the center O of the globule– macromolecule with canters of neighboring globules–macromolecules. Let us separate the typical element, deformation of which determines the deformation behavior of the whole polymeric substance. In Figure A-6-8, this element is marked by the dotted line. Two thresholds may be indicated in the mentioned element: one threshold (OA) includes a bond hyperboloid, and another one (OB) does not include it. Therewith, OA ≠ OB, but taking into account the small thickness of the periphery, it may be assumed that OA ≈ OB, i.e. the supposition about equal lengths of elements in the nonlinear mechanical model made in ref. [87], may be assumed to be true.

Figure A-6-8. Element of super-molecular structure (right) and nonlinear mechanical model of polymeric body appropriate to it (See text).

Let us estimate the angle β between the axis lines of elements of the supermolecular structure (pairs of bonded globules). This may be made proceeding from the following ideas. In the case of spatial contacts, β = 4π/n, where n is the number of globules present in the first spherical layer near the present globule–macromolecule of the radius Rgl-m. Let us determine the value of n from the following expression: 2 Rgl−m + 2 Ri )3 − Rgl3 −m R (  Rgl−m   + 12 gl−m , n= = 8 + 6

Ri3

 R   i 

Ri

where Ri is the radius of the i-th globule present in the spherical layer near the globule of radius Rgl-m. In the general case, Rgl-m/Ri is a random value, because Ri may obtain values depended on molecular mass and molecular-mass distribution. Let us estimate values of n and β from different border cases. When Rgl-m/Ri = 1, n = 26, β = π/6; Rgl-m/Ri → 0, n = 8, β = π/2;

667

Rgl-m/Ri → ∞, n → ∞, β = 0. Suggesting these cases to be equally probable, let us calculate the value of β as 〈 β〉 =

1π π  2π .  + + 0 = 3 6 2  9

Hence, 〈β〉 corresponds to Rgl-m/Ri ~ 1 (more accurate, to Rgl-m/Ri = 0.63). The model of the super-molecular structure suggested allows complete definition of the physical meaning of relaxation mechanisms of the Alexandrov– Lazurkin elements, participating in the nonlinear model: one mechanism associated with long relaxation times takes place in bond hyperboloids and may be described with the help of the order–disorder system in the isomeric approximation by Volkenstein. The second mechanism with shorter relaxation times is analogous to the mechanism associated with stagnation of internal rotation and is developed in the globule–macromolecule itself (the Bresler–Frenkel mechanism). So, one of the main relaxation times (the short one) is typical of the globule of super-molecular structure of a positive curvature (corresponds to the time in ferroelectric of the mixing type), and the second one is typical of the bond hyperboloid with a negative curvature (long time in the system of the order–disorder type). The time and frequency dependences of polymer behavior in the transitional region can be described satisfactorily with the help of the Isakovich–Chaban [268, 269] and Irzhak–Rozenberg–Rostiashvili [270] models. In the one-dimensional case, it follows from these models that the main relaxation process is stipulated by the mutual diffusion of five to seven atoms. In the case of the model of the suggested super-molecular structure, these regions of rearrangement of atoms may be concretely defined – they must proceed in the bond hyperboloids.

669

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1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12.

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689

Subject Index

A Acetone 454 Acid, -, acetic 454 -, butyric 454 -, isophthalic 70 -, isovaleric 454 -, polyacrylic 422 -, sebacic 70 -, n-valeric 454 Alcohol polyvinyl 9, 39, 70 Aldehyde(s) 532 Amide(s) 533 Amine(s) 533 Amino group 632 Angle tangent -, of dielectric losses 86 -, of mechanical losses 86 Anharmonic oscillator 405 Anhydride, -, polysebacic acid 473 Aniline phthalein 70 Aniline phthalein polypyromellitimide 70 Anisotropy, -, polarizability 426 Annihilation -, positrons 52 Arabinogalactane 599 Arabinoglucuronoxylane 597 Arsenic 17 Artifact(s) 509 B Barrier, -, energetic 88 -, potential 88 Benzene 454, 516 – 518 Benzoyl peroxide 560 Birefrigrance 426 Bispheno-A-polycarbonate 70, 87, 449, 625 Bisphenol 358, 555 Bond(s) hydrogen 63, 101, 399, 588 intrachain 588 intermolecular 101, 399 interchain 588 intraunit 588 physical 102 Van-der-Waals 101 Boron 17, 63 Bromine 17 Bromobenzene 454 Butanale 454 Butylethyl ether 454

n-Butanol 454 t-Butyl 626, 627 n-Butylacetate 454 t-Butylamine 644 Butyl acrylate 621 Butyl methacrylate 621 C Carbon 17, 63, 399 Chlorine 17, 64 Chlorobenzene 454 Chloroform 516 Coefficient bond elasticity 459 deformation-optical 426, 427 depolarization 446 diffusion 480 of heat capacity increase 562 of elasticity of rotary-isomeric subsystem 459 of linear expansion 59, 60 of molecular packing 29 – 34 of molecules packing in the surface layer 530 of oscillator anharmonism 405 of packing, partial 398 of the reaction equation, stoichiometric 103 of the repeating unit friction 62 oscillator elasticity 405 partial 62 Poisson 427 segment friction 385 stress-optical 426, 427 volumetric expansion 59, 60, 63 weight 460 Conformation of macromolecules 88, 497 Connection cylinder(s) 524 Connective globules 524 Constant, Boltzmann 88 equilibrium 103 Morse potential 457 reaction rate 477 universal gas 89 Constant(s) (increments), -, fragments 567 – 568 -, linear chains 342 -, network cross-linked point 342 -, self (Van-der-Waals) volume 17 -, strong intermolecular interaction 63 -, weak intermolecular interaction 63 Contraction 47 Coordination bonds 10 Copolymer alternating 12

690

block-copolymer 12 grafted 13 of n-butyl acrylate with tert-butyl methacrylate 629 statistical of styrene with acrylonitrile 626 of styrene with divinylbenzene 14 of styrene with hexafluorine-2propylstyrene 628 of styrene with 4-hydroxystyrene 628 of styrene with methyl methacrylate 629 of styrene with 4-vinylphenyl dimethylsilanol 630 of styrene with vinylphenylhexafluoromethylcarbiol 625 Compatibilizer 632 Creep -, mechanical 441 -, optical 441 of polymer 475 Criterion miscibility 547 solubility 513, 514 Cross-linked point -, cross-linked chains 14, 343 -, four-functional 345 -, isocyanurate 469 -, network 346 -, trifunctional 337, 345 Curve dilatometric 58 recovery 56 stress relaxation 56, 496 thermogravimetric 78, 79 with saturation 555 Cyclohexane 454 Cyclohexanol 454 Cyclohexanon 454

D Debye pattern 589 n-Decane 454 Defects in the form of branchings 375 hanged chains 375 isolated loop 375 Degradation, -, thermal 78 -, thermooxidative 78 Degree -, cross-linking 363 -, polymerization 520 Dendrimer(s) 333 Density -, real 45 -, seeming 45

of cohesive energy 504, 512 of polymer 18 Diamond 353 1,2-Dibromoethane 454 Diamyl ether 454 Dibutyl ether 454 Dielectric constant 418, 447 Dielectric(s) 447 Diethyl ether 454 Diethylamine 632 Diethylketone 454 Diisoamyl ether 454 Diisocyanate 467 Diisopropyl ether 454 Difference of main normal stresses 427 of run, optical 427 Diffusion, -, Fick’s 479 -, kinetic units 475, 479 -, relaxants 475, 479 Dimer 560 1,3-Dioxolane 629 Dipropyl ether 454 Domain(s) 471 E Elastomer networks 456 Energy -, anharmonic oscillator, free 405 -, chemical bonds dissociation 409 -, cohesive 504, 538, 555 -, dipole-dipole interaction 62 -, dispersion interaction 595 -, free 61 -, heat 89 -, hydrogen bonds 61 -, interaction activation 494 -, intermolecular interaction 61 -, internal 88 -, low-temperature gamma-transition, activation 595 -, particle 89 -, relaxants interaction 495 -, rotary isomers, difference 459 -, rotation 88 -, surface 469 -, transition 88 Entropy 89, 475 Epoxy resin 14 Equation -, Arrhenius 103 -, Boltzmann 88 -, Boltzmann-Volterra 475 -, chemical kinetics 103 -, Clausius-Mossotti 418, 446 -, Fick 503 -, Flory 72 -, Fulcher-Tamman 75

691

-, Kargin-Slonymskii 74 -, Lorenz-Lorentz 418 Ethanale 454 Ethanol 454 Ether -, butyethyl 454 -, diamyl 454 -, dibutyl 454 -, diethyl 454 -, diglycidyl 358 -, diisoamyl 454 -, diisopropyl 454 -, polyvinylethyl 69 -, polyvinylisobutyl 405 -, polyvinylmethyl 69, 628 Ethyl acetate 454 Ethyl cellulose 449 Ethylbenzene 454 Ethylbromide 454 Ethylformate 454 2-Ethylhexyl acrylate 621 Ethyliodide 454 Ethylpropionate 454 F Fibrils 524 Fluorine 17, 61 Formamide 534 Force constant 409 Fragments -, basic 602 -, intercrosslinked, mean 347 -, linear 343 -, network, repeating 345 Frequency -, D-line of sodium 446 -, oscillation, natural 89 Front-factor 456 Function -, memory 476 -, response 613 G Galactouronane 598 Gibbs triangle 383 Glass transition mechanical 85 structural 85 temperature interval 91 Globule 509 Glucomannan 599 H Halogen-containing compound(s) 530 Heat capacity 562 Height of the sphere segment 17 Hemicellulose 589

n-Heptane 454 n-Heptylacrylate 621 Hexagonal packing 528 Hexamethylene tetramine 629 n-Hexane 454 n-Hexanol 454 Hydrocarbons -, monosubstituted, unsaturated 11 -, substituted, saturated 11 Hydrogen 17, 63 I Interaction cylinder 457 Interaction, -, chemical 16 -, dipole – dipole 61, 101 -, dispersion self 61 -, dispersion weak 61 -, intermolecular (physical) 16, 607 -, intermolecular strong 61 -, intramolecular 607 Iodine 17 Isobutanol 454 cis-Isomer 11 Isomer(s) 11, 320 Isopentane 453 Isopropylbenzene 454 J Joining of units -, anomalous (“head-to-head”, “tailto-tail”) 10 normal (“head-to-tail”) 10 K Ketone(s) 533 Kinetic unit 87, 475 L Law (rule), -, additivity 405 -, Il’ushin 494 -, Laplace 511, 524 -, mass action 103 -, Maxwell 447 -, molar parts, Zhurkov’s 105 -, Wertheim 426 Length, -, chemical bond 17 -, Cune segment 385 -, intermolecular radius 17 -, wave 418 Lifetime, -, intermediate component 52 of atom, settled 89 of long living component 51, 52

692

-, positron 51 Lifetime spectrum of positron(s) 51 Long-living component intensity 52 Lorentz correction 418 M Macrodiisocyanates 468 Macromolecule branched 13 flexible 73 linear 13 rigid 73 Main principal stress(es) 427 Mechanism glass transition 87 of relaxation process 87 of sorption, relaxation 497 relaxation 54, 55 swelling, relaxation 497 Melting enthalpy 398 entropy 398 point 398 Memory function 477 Metha-, para-phenylene diamine 109 Methanol 454 Method absorption, equilibrium 43 calorimetric 83, 85 capillary condensation 43 dilatometric 45, 83, 85 DSC 626 electron microscopy 469 electron spectroscopy 632 extrapolation 398 Foukes 535 Fourier-transformation 630 IR-spectroscopy 626, 630 neutron scattering 509 NMR, 13C NMR, 31P NMR 630, 631 of atomic approaches 76 of atomic constants 584 of deformations freezing 439 of dynamic mechanical analysis 86, 469 of experiment design 612 of gradient tubes 45 of dynamic tests 83 of least squares 108 of local Lorentz field 417, 418 of mercury porometry 43 of miscibility diagrams 587 of polymerization in solution 45 of registration of positron lifetime spectra 51 of small-angle X-ray scattering 509 of X-ray photoelectronic spectroscopy 469

optical 83 photocreep 443 photoelasticity 439 polarization-optical 439 polycondensation 45 positrons annihilation 43 pycnometric 45 reverse variable step 496 Runge-Cutt, numerical 482 Simpson 496 thermogravimetric 79 thermomechanical 66, 78 Methyl acetate 454 Methyl formate 454 Methyl methacrylate 628 Methylethylketone 454 4-O-Methylglucuronoxylane Methyliodide 454 Microdomain(s) 471 Microphase 471, 547 Miscibility of polymers 547, 625 Mobility translation 102 vibration 102 Modulus elasticity 79, 86, 440, 460, 474 equilibrium 456 of cross-linked points 460 of linear fragments 460 relaxation 494 rubbery 79 storage 473 Molecular mass of chain 385 of cross-linking bridges 350 of cross-linking point 347, 350 of intercross-linking point fragment 456 of linear chain between cross-linking points 348 of polymer 555 of segment 71 of the repeating unit 35 of unit neighboring to the crosslinking point 347 Moment dipole 595 magnet 445 Monomers bifunctional 14 disubstituted 11 polyfunctional 14 N N,N’-dimethylacrylamide 630 N,N’-dimethyl-substituted Nylon-6,12 626 Naphthalene 454 Network(s) high cross-linked 343

693

less cross-linked 343 Nitrile acetate 455 butyrate 455 isoheptanate 455 pentanate 455 propionate 455 Nitrobenzene 518 – 520 Nitrogen 17, 63, 505, 563 Nitroethane 454 Nitromethane 454 1-Nitropropane 454 2-Nitropropane 454 n-Nonane 454 n-Nonyl acrylate 454 Number -, coordination 64 -, repeating units in a segment 385 Nylon-6 626 O n-Octane 454 n-Octanol 454 Oscillator 446 Osmotic pressure 497 Oxygen 17, 63, 399

P Parachor of atom 527 of frequency 527 n-Pentane 453 n-Pentanol 454 Perfluorocompound 532 Permeability 595 Phase transition -, the first kind 87 -, the secondary kind 87 Phenolic resins 626 Phenoloformaldehyde resin 10, 607, 631 Phenolphthalein 70 Physical state -, glassy 29, 41, 67 -, rubbery 41, 67 -, viscous flow 67 Polarizability 446, 595 Polarization, atomic 418 deformational 447 electronic 418, 446 microscopic 446 orientational 418, 447 1,2-Polybutadiene 37 1,4-Polybutadiene 68, 449 1,4-Polyisocyanurate(s) 439 1,4-Polyisoprene 68, 449

1,4-cis-Polybutadiene 37 1,4-cis-Polyisoprene 37 1,4-trans-Polybutadiene 37 Poly(α,α,α,α-tetrafluorine-p-xylylene) 449 Poly(α-methyl styrene) 449 Poly(α-vinyl naphthalene) 449 Poly(β-vinyl naphthalene) 449 Poly(1,1,2-trichlorobutadiene) 632 Poly[1,1-cyclohexane bis(4-phenyl)carbonate] 449 Poly[1,1-ethane bis(4-phenyl)carbonate] 449 Poly(1,4-cyclohexylidene dimethylene terephthalate) 449 Poly(1-butene) 449 Poly(1-hydroxy-2,6-methylphenylene) 629 Poly(1-hydroxyl-2,6-methylenephenylene) 629 Poly[2,2’-(m-phenylene)-5,5’bibenzimidazole] 450 Poly(2,6-dimethyl-1,4-phenylene oxide) 449, 625 Poly(3,4-dichlorostyrene) 449 Poly[4,4’-diphenoxy di(4-phenylene)sulfone] 450 Poly[4,4’-isopropylidene diphenoxy di(4phenylene)sulfone] 450 Poly[4,4’-sulfone diphenoxy di(4-phenylene) sulfone] 450 Poly(4-hydroxy styrene) 629 Poly(4-methyl-1-pentene) 449 Poly-4-methylpentene-1 38, 404, 424 Poly(4-vinyl pyridine) 631, 642 Polyacetylene 9 Polyacrylamide 72 Polyacrylonitrile 72, 450 Polyamide 9, 38, 75, 109, 399, 405, 537 Polyamide 6,6-α-isomer 38, 537 Polyamide 6,6-β-isomer 38 Polyamide based on sebacic acid and anilinephthalein 70 Polyamidophenylquinoxaline(s) 320 Polyarylate F-1 74 Polybenzimidazole(s) 582 Polybenzoxazole 484 Polybutadiene 449 Poly(butyl acrylate) 69 Poly(butyl methacrylate) 625, 449 Polycarbonate 582 Polycarbonate based on bisphenol A 70, 87, 449, 625 Polychloroprene 37, 41 Poly(chloro-p-xylylene) 449 Poly(chlorotrifluoroethylene) 449 Poly(cyclohexyl methacrylate) 449 Poly(dimethyl siloxane) 68, 449 Polyester(s) based on secacic acid and phenolphthalein 70 isophthalic acid and phenolphthalein 70 Polyether(s) 9, 468

694

Poly(ether ether ketone) 449 Poly(ether ketone(s)) 449 Poly(ether sulfone(s)) 449 Poly(ethyl α-chloroacrylate) 449 Poly(ethyl methacrylate) 69, 449 Poly(ethyl oxazoline) 627 Polyethylene 9, 449 Poly(ethylene adipate) 38 Poly(ethylene isophthalate) 38 Poly(ethylene oxazoline) 628 Poly(ethylene oxide) 9, 537, 625 Poly(ethylene terephthalate) 70, 450 Polyformaldehyde 537 Poly(hexamethylene adipamide) 450 Poly(hexamethylene sebacamide) 449 Poly(hexamethylene terephthalate) 37 Poly(hexyl methacrylate) 69 Polyimide 69 Poly(isobutyl methacrylate) 449 Polyisobutylene 69, 449 Polyisoprene 449 Polymer chains -, branched 12, 13 -, cross-linked 14 -, linear 13 -, main 13 Polymer structure -, fibrillar 524 -, globular 523 -, macroporous 44 -, microporous 43 -, nodular 509 -, porous 43 -, supermolecular 509 Polymeric unit, -, elementary 10 -, repeating 10 Polymers amorphous 34, 46, 84 atactic, irregular 11 branched 13 carbochain 9, 10 cardo-type structure 514 chelate 10 crystalline 46, 84 dendrite 13, 335 elementorganic 9, 10 erythro-diisotactic 12 glassy-like 68 globular, amorphous 510, 520 gradient 465, 472 heat-resistant 68, 514 heterochain 10 heteromodular 465 homo- 35 inter- 15 isotactic 11 isotropic, amorphous 520 ladder 15 network 10, 608

non-globular, amorphous 510 non-polar 538 optically sensitive 426 oriented 524 partially crystalline 34, 84 polar 538 rigid-chain 73 rubber-like 68 semi-crystalline 45, 46, 84 star-like 335 stereoregular 11 syndiotactic 11 treo-diisotactic 12 Poly(m-chlorostyrene) 449 Poly(methyl α-chloroacrylate) 450 Poly(methyl acrylate) 69 Poly(methyl methacrylate) 69, 449, 626, 629, 630 Polymethylidene phthalide 46 Poly(N,N’-dimethyl acrylamide) 627 Poly[N,N’-(n,n’oxydiphenylene)pyromellitimide] 450 Poly(naphtoylene benzimidazole) 68 Poly(n-butyl acrylate) 68, 629 Poly(n-butyl methacrylate) 36, 626, 640 Poly(n-butyraldehyde) 41 Poly(n-propyl methacrylate) 36 Poly(N-vinyl carbazole) 449 Poly(o-methyl styrene) 449 Poly(oxy-2,2-dichloromethyl trimethylene)449 Poly[oxy(2,6-diphenyl-1,4-phenylene)] 449 Polyolefins 582 Polyoxadiazole 9, 484 Polyoxymethylene 39 Poly(p-chlorostyrene) 449 Poly(phenyl quinoxaline) 514 Poly(phenyl sylsesquioxane) 15 Polyphenylene 9 Poly(phosphonitrile chloride) 9 Poly(p-hydroxybenzoate) 450 Poly(p-hydroxystyrene) 628 Polypropylene 9, 37, 449 Poly(propylene oxide) 40 Poly(p-xylylene) 449 Polystyrene 69, 449 Polysulfide(s) 582 Polysulfone(s) 582 Poly(t-butyl methacrylate) 629 Poly(tert-butyl acrylate) 629 Poly(tetramethylene adipate) 405 Poly(tetramethylene oxide) 405 Poly(tetramethylene terephthalate) 449 Polytetrafluoroethylene 39, 50 Poly[thio(p-phenylene)] 449 Poly(trichlorobutadiene) 15 Poly(trimethylene adipate) 405 Polyurethane(s) 9, 582 Poly(vinyl acetate) 69, 450 Poly(vinyl alcohol) 70 Poly(vinyl butyral) 449

695

Poly(vinyl chloride) (PVC) 39, 69, 449, 537 Poly(vinyl cyclohexane) 449 Poly(vinyl fluoride) 39, 537 Polyvinylmethyl ether 69 Polyvinylethyl ether 69 Poly(vinyl pyrrolidone) 426, 628 Poly(vinyliden chloride) 449 Pore(s) distribution by sizes 43 specific surface 43 total volume 43 Porosity factor 45 Potential chemical interaction 409 Lennard-Jones 406 Morse 409 Principal strains 427 Process of adhesive wetting 511 of copolymerization 12 of creep 475 of desorption 500 of microphase separation 470 of polymerization 46 of stress relaxation 475 of swelling 497 of syneresis 403 Propanal 454 n-Propanol 454 n-Propyl acetate 454 Pyromellitic dianhydride 70 R Rate of diffusing particle 497 of system cooling 91 of system heating 91

-, of the method of constants 585 Segment -, molecular mass 72, 73 Cune 385 mechanical 73 of macromolecule 51 Silicon 17, 63 Simplex lattice 613 Size of bond, typical 459 of crystallites 398 of solvent, typical 513 typical 385 Solubility criterion 513 -, Hildebrand parameter 504 Sorbate 44, 45 Sorbent 44, 45 Spherical segment height 17 Square (surface) of interaction cylinder 457 Van-der-Waals, of atom 459 Steric factor 103, 588 Structural element 89 Styrene 454 Sub-micropore(s) 50 Substituent, side 73 Substitution, -, metha- 63 -, ortho- 63 -, para- 63 Sulfonated polyamide 631 poly(p-phenylene terephthalamide) 631 Sulfur 17, 63 T

Reaction of polycyclotrimerization 15, 468 of trimerization 15 of urethane formation 468 Refraction molar 448 molecular 418 of atoms, by Eizenlor 419 of atoms, specific 418 Relaxant(s) 475 Relaxation of microdefects 55 stress 475 Rubber, -, butadiene 514 -, isoprene 514 -, synthetic 10 S Scheme -, additive 527

Teflon 51 Temperature (point) -, critical 405 -, dissolution, lowest critical 627 -, flow 67 -, glass transition 36, 66, 589 -, intensive thermal degradation 408 -, melting 398 -,onset of intense thermal degradation 408 -, phase transition 406 Tension interfacial 510 surface 510 Theory (concept, rule) -, Flory- Haggins 520 -, Iliushin 494 -, Patterson and Robard 627 -, rotary-isomeric 521 -, Simha-Boyer 60 -, Volkenstein-Ptytsin 90

696

-, Zhurkov 101

Vector,

Unsaturated hydrocarbons disubstituted 11 monosubstituted 11

-, electric field intensity 445 -, electric induction 445 -, polarization 445 Viscosity, -, shear, Newtonian 385 -, system 75 Volume empty 17 fluctuation 42 free 42 molar 17 of atom 17 of defect 457 of end groups 555 of expansion 42 of ideal crystal 44 of interaction cylinder 457 of the repeating unit 17 self (real) 17 specific 17 total molar 17 Van-der-Waals, free 17 Van-der-Waals, of polymer 557 Van-der-Waals, of the repeating fragment of the network 344

V

W

Value,

Wisevesser formula 585 Work of adhesion 510

Tests, -, thermogravimetric 78 -, thermomechanical 78 -, X-ray structural 79 with periodical loading 84 Thermodynamic function 87 probability 88 Time, -, observation 89 -, relaxation 51, 89 of transition from the glassy state into the rubbery state 88, 89 Tin 17 Toluene 454 2,4-Toluylenediisocyanate 468, 472, 473 Trans-isomer 11 Triethylamine 632 Trimer 568 U

-, defect 457 -, mechanical segment 73 Van-der-Waals, -, radius of atom 17, 457 – 459 -, surface of atom 459 -, volume of atom 17 – 29, 453 – 455, 457 – 459

X m-Xylene 454 o-Xylene 454 p-Xylene

454