Structure and Properties of Crosslinked Polymers

Structure and Properties of Crosslinked Polymers Gasan Magomedov Georgii V Kozlov Gennady E Zaikov iSmithers – A Smithe...

Author: Gasan M Magomedov | Georgii V Kozlov | Gennady E Zaikov

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Structure and Properties of Crosslinked Polymers Gasan Magomedov Georgii V Kozlov Gennady E Zaikov

iSmithers – A Smithers Group Company Shawbury, Shrewsbury, Shropshire, SY4 4NR, United Kingdom Telephone: +44 (0)1939 250383 Fax: +44 (0)1939 251118 http://www.ismithers.net

First Published in 2011 by

iSmithers Shawbury, Shrewsbury, Shropshire, SY4 4NR, UK

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ISBN: 978-1-84735-559-1 (Hardback) 978-1-84735-560-7 (Softback) 978-1-84735-561-4 (ebook)

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P

reface

Crosslinked polymers have recently become widespread in various ﬁelds of industry as adhesives, matrices of polymer composites and so on. Nevertheless, their theoretical investigation and subsequent practical applications have remained on the level of an approximately 40-year standing. It is generally accepted that the main parameter controlling the properties of crosslinked polymers is their crosslinking density. As before, semi-empirical kinetic theories prevail in the receiving ﬁeld of crosslinked polymers. In the present monograph these problems were studied from the principally new point of view, when object structure, either macromolecular coil (microgel) or condensed state structure, was considered in terms of chemical or physical processes. Such an approach is possible with the availability of appropriate structural models, of which fractal (multifractal) analysis and the cluster model of the amorphous state structure of polymers were used. The ﬁrst of the indicated approaches is a general mathematical calculus, whereas the second represents itself as a purely ‘polymeric’ concept. This circumstance deﬁnes the excellent combination and addition to one another of crosslinked polymers. Application of fractal analysis and irreversible aggregation models for the description of crosslinked polymer curing processes allows it to be elucidated that macromolecular coil (microgel) structure, characterised by its fractal dimension, plays a larger role than purely chemical aspects. Such an approach allows a quantitative description of both curing process kinetics and its ﬁnal results to be received. The combined use of fractal analysis and cluster models for the structure of the condensed state of crosslinked polymers allows their quantitative treatment on different structural levels, molecular, topological and suprasegmental, to be obtained for the ﬁrst time and also the interconnection between the indicated levels to be determined. In turn, elaboration of solid-phase crosslinked polymer structure quantitative models allows structure–properties relationships to be obtained for the ﬁrst time, which is one of the main goals of polymer physics. It is necessary to dwell on new and therefore little-studied aspects of crosslinked polymer applications in practical supplements. This is related in the ﬁrst place to a new iii

Structure and Properties of Crosslinked Polymers class of polymer nanocomposites (polymer–polymer nanocomposites) and crosslinked polymer treatment as natural nanocomposites. The polymer–polymer nanocomposites research, in which polyethylene is the matrix and nanoﬁller-epoxies are the polymer nanoparticles, allowed a polymer physics theoretical aspect to be elucidated and the proof of the expediency of their application in practical applications to be given. The consideration of epoxy polymers as natural nanocomposites within the frameworks of cluster models of polymer amorphous state structure allows improvement in the operating characteristics resources of these materials to be revealed, which is realised at present at best at 30–40%. In conclusion let us highlight practical aspects of theoretical postulates, expounded in the present monograph. First of all let us indicate that crosslinking density does not deﬁne crosslinked polymer properties unequivocally, but the parameter controlling these properties is the polymer nanostructure state, which can be very different for the same crosslinking density. The equations were obtained allowing the prediction of crosslinked polymer properties not only within the frameworks of their structure integral characteristics, but also by nanolevel data, i.e., separate nanoclusters. Practical methods for the regulation of the nanostructure were proposed, which suppose the characteristics of crosslinked polymers (e.g., epoxy polymers) exceed the corresponding parameters for polymer/organoclay nanocomposites, which are considered at present as the most perspective ones. The authors believe that the present monograph will be useful both for scientists, concerned with the theoretical research into crosslinked polymers, and engineers using these polymers in practical applications.

Gasan Magomedov Georgii V Kozlov Gennady E Zaikov

iv

C

ontents

1.

2

3

The Main Principles of the Cluster Model ................................... 1 1.1

Fundamentals .................................................................... 1

1.2

Thermodynamics of the Local Order Formation .............. 13

1.3

Polymer Structure Ordering Degree and Cluster Model ... 18

1.4

Thermoﬂuctuational Origin of Clusters ........................... 33

1.5

Functionality of Clusters and Methods of its Estimation . 43

The Main Physical Concepts used in Fractals Theory ................ 61 2.1

The Fractal Analysis of Polymeric Media ......................... 61

2.2

The Fractal Models of Polymer Medium Structure .......... 68

2.3

Polymer Medium with Scaling Theory Positions .............. 70

2.4

The Fractal Analysis in Molecular Mobility Description Questions ..................................................... 74

The Fractal Models of Epoxy Polymers Curing Process ............. 83 3.1

Two Types of Fractal Reactions at Curing of Crosslinked Epoxy Polymers .......................................... 83

3.2

Scaling Relationships for Curing Reactions of Epoxy Polymers ............................................................ 100

3.3

Microgel Formation in the Curing Process of Epoxy Polymers ............................................................. 119

3.4

Synergetics of the Curing Process of Epoxy Polymers ... 138

3.5

The Nanodimensional Effects in the Curing Process of Epoxy Polymers into Fractal Space ............................ 145

v

Structure and Properties of Crosslinked Polymers

4

5

6

vi

The Description of Crosslinked Rubbers within the Frameworks of Fractal Analysis and Local Order Models ....... 159 4.1

Molecular and Structural Characteristics of Crosslinked Polymer Networks ..................................... 159

4.2

The Polychloroprene Crystallisation .............................. 169

4.3

The Cluster Model Application for the Description of the Process and Properties of Polychloroprene Crystallisation ............................................................... 179

4.4

Inﬂuence of Polychloroprene Crystalline Morphology on Its Mechanical Behaviour ..................... 186

Structure of Epoxy Polymers ................................................... 197 5.1

Application of Wide Angle X-ray Diffraction for Study of the Structure of Epoxy Polymers ...................... 197

5.2

The Curing Inﬂuence on Molecular and Structural Characteristics of Epoxy Polymers ................ 209

5.3

The Description of the Structure of Crosslinked Polymers within the Frameworks of Modern Physical Models ........................................... 220

5.4

Synergetics of the Formation of Dissipative Structures in Epoxy Polymers ........................................ 253

5.5

The Structural Analysis of Fluctuation Free Volume of Crosslinked Polymers ................................................ 261

The Properties of Crosslinked Epoxy Polymers........................ 283 6.1

The Glass Transition Temperature ................................. 283

6.2

Elasticity Moduli ........................................................... 288

6.3

Yield Stress .................................................................... 293

6.4

Fracture of Epoxy Polymers ........................................... 304

6.5

The Other Properties ..................................................... 317

6.6

The Physical Ageing of Epoxy Polymers ........................ 327

Contents

7

8

9

10

Nanocomposites on the Basis of Crosslinked Polymers............ 347 7.1

The Formation of the Structure of Polymer/Organoclay Nanocomposites ........................... 347

7.2

The Reinforcement Mechanisms of Polymer/Organoclay Nanocomposites ........................... 352

7.3

The Simulation of Stress-strain Curves for Polymer/Organoclay Nanocomposites within the Frameworks of the Fractal Model ............................ 367

7.4

The Multifractal Model of Sorption Processes for Nanocomposites ...................................................... 371

Polymer-polymeric Nanocomposites ........................................ 381 8.1

The Fractal Analysis of Crystallisation of Nanocomposites ............................................................ 381

8.2

The Melt Viscosity of HDPE/EP Nanocomposites ......... 386

8.3

The Mechanical Properties of HDPE/EP Nanocomposites ............................................................ 389

8.4

The Diffusive Characteristics of HDPE/EP Nanocomposite ............................................................. 398

Crosslinked Epoxy Polymers as Natural Nanocomposites ....... 411 9.1

Formation of the Structure of Natural Nanocomposites 411

9.2

The Properties of Natural Nanocomposites ................... 432

The Solid-phase Extrusion of Rarely Crosslinked Epoxy Polymers ....................................................................... 467

Abbreviations .................................................................................... 479 Index ............................................................................................... 483

vii

1

The Main Principles of the Cluster Model

1.1 Fundamentals The well-known experimental observations have become the prerequisites for the development of the polymer amorphous state structure cluster model. As shown by Haward and co-workers [1, 2], on deformation of glassy polymers beyond the yielding region (on the plateau of forced high elasticity) they obey the laws of rubber-like elasticity theory. In this case the polymer behaviour under high strains is described by either the Langevin equation [3, 4] or the Gaussian interpretation [5], when the polymer chain does not approach completely a stretched state, and correlation between the true stress σtr and the drawing ratio λ for the uniaxial tension is presented as follows [5]:

(1.1)

where Gp is the so-called strain hardening modulus. Formally, the Gp value allows determination of the macromolecular entanglements network density, νe, in accordance with the well-known formulae of the rubber-like elasticity [5]:

(1.2)

and

(1.3)

1

Structure and Properties of Crosslinked Polymers where ρ is the polymer density; R is the universal gas constant; T is the testing temperature; Me is the molecular weight of the chain segment between entanglements; NA is Avogadro’s number. However, the attempts to estimate Me (or νe) via Gp values, determined according to Equation 1.1, have given unlikely low values of Me (or unrealistically high values of νe), which contradicts the requirements of Gaussian statistics. This statistics assumes the availability of, at least, 13 links per chain part between entanglements [6]. Possible reasons for such a discrepancy under consideration of entanglements as traditional macromolecular ‘binary-hooking contacts’ are discussed in detail in paper [5]. The principally different treatment of this problem, proposed in [7], assumes that besides the above-mentioned ‘binary hooking contacts’ network, there is another type of entanglement in the polymer glassy state, the structure of nodes, which is similar to crystallites with extended chains. Such an entanglements node possesses quite high functionality F (the entanglements node functionality means the number of chains coming out of it [8]). Hereafter, this entanglements node will be named ‘cluster’. The cluster consists of different segments of macromolecules, and each segment length is postulated to be equal to the statistical segment length, lst (‘rigidity segment’ of the chain [9]). In this case, the effective (real) molecular weight of the chain part between clusters, , can be calculated as follows [8]:

(1.4)

where Mcl is the molecular weight of the chain part between clusters, calculated according to Equation 1.2. It is obvious that at quite high values of F one can obtain reasonable values of meeting the requirements of Gaussian statistics. Hereafter, for the purpose of distinguishing parameters of the cluster entanglement network and macromolecular hooking network indices, ‘cl’ and ‘e’ will be used respectively. Thus, the proposed model in paper [7] assumes that the structure of the amorphous state of the polymer represents domains consisting of collinear densely packed segments of different macromolecules (clusters), submersed into a loosely packed matrix. Simultaneously, the clusters play the role of multifunctional nodes of physical entanglements nodes. The value F can be estimated (again within the frameworks of the rubber-like elasticity concept) as follows [10]:

2

The Main Principles of the Cluster Model

(1.5)

where G∞ is the equilibrium shear modulus and k is Boltzmann’s constant. Figure 1.1 shows νcl(T) dependences for polycarbonate (PC) produced from bisphenol A and polyarylate (PAr). These dependences indicate a reduction in νcl with an increase in T that supposes thermoﬂuctuational origin of clusters (the local order domains).

vcl × 10-27, m-3 3

2

2

1 1 0 283

333

383

433

483 T, K

Figure 1.1 The dependences of the macromolecular entanglements cluster network density νcl on the testing temperature T for (1) PC and (2) PAr [7]

Moreover, the mentioned dependences display two characteristic temperatures. The ﬁrst is the polymer glass transition temperature, Tg, determining the complete decay of clusters that corresponds to polymer transition into a rubber-like state. The second, Tg' , corresponds to the infection on νcl(T) curves and is located approximately 50 K below Tg. Earlier, also within the frameworks of the local order concepts, it has been ' shown [11, 12] that the temperature Tg is associated with defreezing of the segmental mobility in the loosely packed regions of the polymer. This means that within the ' frameworks of the cluster model Tg can be associated with devitriﬁcation of the 3

Structure and Properties of Crosslinked Polymers loosely packed matrix. For the same polymers the dependences F(T) have a similar shape (Figure 1.2).

F 40

-1 -2

30 20 10 0 283

323

363

403

443 T, K

Figure 1.2 The dependences of clusters functionality F on testing temperature T for (1) PC and (2) PAr [7]

Two basis models of the local order domains in polymers (with folded and extended chains) possess the point of coincidence: they play the role of nodes of a macromolecular entanglement physical network [13–15, 16]. However, their response to mechanical deformation should be signiﬁcantly different: if at high strains the regions with folded chains (‘bundles’) are capable of unfolding and forming stretched conformations, then clusters are incapable of doing so and polymers can be deformed by stretching of ‘tie’ (clusters connecting) chains, i.e., by their orientation in the applied stress direction. Turning back to the analogy with crystalline morphology of polymers, let us note that high strains of semi-crystalline polymers (especially for polyethylenes, which give 1.000–2.000%) are realised due to unfolding of crystallites [17]. That is why using values of limiting strains of polymers one can obtain arguments for the beneﬁt of one or another type of the local order regions in the polymer amorphous state that was performed in paper [18].

4

The Main Principles of the Cluster Model It was assumed [16] that the ‘bundles’ are capable of delivering parts of chains to ‘interstices’ (i.e., to inter-bundle regions) by unfolding, thus acting similarly to crystallites with folded chains (CFC). On this evidence, special attention is paid to a large difference between limiting draw ratios, λlim, of amorphous plastic polymers and such semi-crystalline polymers as high-density polyethylene (HDPE) and polypropylene (PP), for which unfolding of crystallite folds was proved experimentally (λlim ≈ 1.6 for PC, λlim ≈ 6 for PP and λlim ≈ 13 for HDPE at room temperature [19]). The explanation of this difference requires quantitative estimations [18]. Gent and Madan [19] have supposed that drawing proceeds due to straightening of crystalline and amorphous chain sequences. In this case the value of λlim can be expressed via f, the number of times a molecule passes through the same crystallite (or ‘bundle’):

(1.6)

where K is the degree of crystallinity, nst is the number of equivalent statistical links between entanglement nodes in the polymer melt. Usually, nst varies within the range of 100–300; as this value is not sufﬁcient for the estimation results, for all polymers used in paper [18] it was accepted to be equal to 225. It is clear that the value of f determines the number of folds formed by a macromolecule in CFC (or the ‘bundle’). Formation of folds requires meeting the condition f ≥ 2. Estimation results of f for HDPE and PP are shown in Table 1.1. As expected they indicate that macromolecules of the ﬁrst type are folded over 50 times and the second ~5 times, corresponding well to the known data [19]. For amorphous glassy polymers the equivalent of the degree of crystallinity, K, is the relative fraction ϕcl of clusters. The value of ϕcl can be estimated as the total length of macromolecules L per polymer volume unit under their dense packing [20]:

(1.7)

where S is the macromolecule cross-sectional area.

5

Structure and Properties of Crosslinked Polymers The length of the statistical segment lst is estimated as follows [21]:

(1.8)

where l0 is the skeletal bond length in the main chain, C∞ is the characteristic ratio, which is a statistical ﬂexibility indicator of the macromolecule [22]. The total length of segments in the clusters Lcl per polymer volume unit is estimated as follows [23]:

(1.9)

and ϕcl is estimated as the ratio:

(1.10)

Combination of Equations 1.7–1.10 gives the ﬁnal formula for ϕcl calculation [23]:

(1.11)

The values of ϕcl for PC and PAr, calculated according to Equation 1.11, are shown in Table 1.1. Using experimental values of λlim, these results indicate that in all cases f < 1 (Table 1.1). This proves the absence of folding of macromolecules in local order domains (nodes of macromolecular entanglements physical network) of these polymers. Let us also note that unfolding in semi-crystalline polymers crystallites begins at strains of about 50–100% [24]. Substituting λlim ≈ 2 and ϕcl ≈ 0.20 for amorphous regions of HDPE into Equation 1.6, we obtain f < 1. This means that in semi-crystalline polymers macromolecules are folded in the crystalline regions only.

6

The Main Principles of the Cluster Model Thus the calculations executed in paper [18] have shown that the crystallite with stretched chains analogue – the cluster – is the most probable type of suprasegmental structure in the polymer amorphous state.

Table 1.1 Estimation of folding parameter f for amorphous and semicrystalline polymers [18] T, K

λlim

K

ϕcl

f

HDPE

293

13

0.69

–

53

PP

293

6

0.50

–

4.7

PC

333

1.91

–

0.33

0.70

353

2.23

–

0.29

0.73

373

2.15

–

0.24

0.57

393

2.36

–

0.19

0.52

413

2.75

–

0.11

0.35

293

1.66

–

0.50

0.89

313

1.67

–

0.38

0.68

333

1.66

–

0.37

0.66

353

1.76

–

0.31

0.59

373

1.66

–

0.25

0.45

393

1.70

–

0.19

0.35

413

1.75

–

0.16

0.30

433

1.80

–

0.13

0.25

453

1.86

–

0.11

0.22

473

1.97

–

0.01

0.02

Polymer

PAr

The well-known heterogeneity of plastic deformation of amorphous glassy polymers [25, 26] allows their acceptance as heterogeneous systems. This statement is also true for the amorphous phase of semi-crystalline polymers [27, 28]. Nevertheless, the behaviour of both classes of polymers is successfully described by both continual

7

Structure and Properties of Crosslinked Polymers models (let us be reminded that primarily the known Dugdale model was developed for metals [29]) and molecular concepts. In this connection, the question of the scale, which may be considered to be the lower boundary of applicability of continual models, is raised. One further problem consists in the probability of an inhomogeneous molecular system (which unambiguously the amorphous glassy polymer is) being considered as a two-phase system. If there is such a probability, this will allow application of the so-called ‘composite’ models to the description of the amorphous polymer behaviour (see Chapter 9). These models are developed well and used successfully, for example, for the description of artiﬁcial two-phase systems, including ﬁlled systems. These two problems were discussed in paper [30]. Fellers and Huang [31] have applied the ﬂuctuation statistical theory to the description of crazing in amorphous polymers. They have deduced an expression for assessing the polymer volume V0, in which the ﬂuctuation probability equals the unit value:

(1.12)

where σc is the crazing stress, T0 is the equilibrium temperature, the lower boundary of which range is the glass transition temperature Tg, B is the polymer modulus of dilatation bound to Young modulus E according to the following relationship [32]:

(1.13)

where ν is Poisson’s ratio. The distance between clusters Rcl can be estimated according to the following simple formula [33]:

(1.14)

8

The Main Principles of the Cluster Model Table 1.2 shows comparison of Rcl and linear size L0, at which the ﬂuctuation 1/ 3

probability is equal to one (L0 = V0 ) for ﬁve amorphous glassy polymers. As follows from this table data, parameters Rcl and L0 are close in both absolute values and variation tendencies. This means that in scales of characteristic sizes of the cluster structure the amorphous polymer (or the amorphous phase of a semi-crystalline polymer) may be considered as the heterogeneous system [30].

Table 1.2 Comparison of characteristic sizes of the ﬂuctuation theory L0 and cluster model Rcl for amorphous glassy polymers [30] L0, Å

Rcl, Å

Polystyrene

76.4

36.1

Poly(methyl methacrylate)

31.7

31.6

Polyvinylchloride

54.0

27.1

Polycarbonate

39.7

31.1

Polysulfone

36.4

25.0

Polymer

Katsnelson [34] has given the following deﬁnition of the phases of the substances: they are ‘… states of the substance which, being in touch, can exist simultaneously in equilibrium with each other. Obviously different properties correspond to different phases. Hence, it should be taken into account that by different phases … parts of a body are meant related to the solid phase, but possessing different structure and properties’. Clusters and loosely packed matrices, which in accordance with the cluster model [13–15] are the main structural elements of the polymer amorphous state, meet the above deﬁnition, at least partly. It is known that these elements possess different mechanical properties [35] and different glass transition temperatures [36]. All of these facts together give an opportunity to consider the amorphous state of a polymer to be a quasi-two-phase state, disclaiming full precision of the deﬁnition [30]. Now let us consider the applicability of models developed for two-phase ﬁlled polymers for describing the mechanical behaviour of amorphous glassy and semicrystalline polymers. By the analogy with the dispersion theory of strength the composite (or in the considered case, amorphous polymer structure) shear yield stress τa is given as follows [37]:

9

Structure and Properties of Crosslinked Polymers

(1.15)

where τm is the shear yield stress of the loosely packed matrix, b is the Burgers vector. For polymers τm = 0 [35, 38], and in the case of ﬁller particles aggregation (i.e., association of segments to clusters) Equation 1.15 for the polymer amorphous state can be presented as follows [37]:

(1.16)

where k(d) is the aggregation parameter. Equation 1.16 can be used for the description of the temperature dependence of τa displaying one principal difference from the ﬁlled polymers. As the G value for amorphous and semi-crystalline polymers, the macroscopic shear modulus should be used instead of its value for a loosely packed matrix. This is explained so that, contrary to the τa value, the G value for the polymer amorphous state is determined by the structure of both quasi-phases [35]. The truth is that the application of the G value in Equation 1.16 only for a loosely packed matrix would mean determination of the cluster property (τa) from properties of another structure component only, which is physically meaningless. In Figure 1.3 the comparison of experimental temperature dependences of the shear yield stress τa and those calculated according to Equation 1.16 (under the condition k(d) = const. for every polymer) is adduced, which shows good correspondence, thus proving Equation 1.16 showing the accuracy for the description of amorphous and semi-crystalline polymer properties. The aggregation parameter k(d) value was estimated according to Equation 1.16 for eight polymers at T = 293 K. It is obvious that the physical signiﬁcance of this value must be analogous to the functionality F for the cluster structure of the polymers. The relation between k(d) and F values is shown in Figure 1.4, from which the expected correspondence follows. It is assumed from this observation and the fact that k(d) is independent of temperature that the polymer amorphous state k(d) represents some qualitative measure of the polymer’s ability to form clusters at the segmental level [30].

10

The Main Principles of the Cluster Model τa, MPa 25

-3 -4

20 15

2

10 5

1

0 283

323

363

403

443 T, K

Figure 1.3 The experimental temperature dependences of the shear yield stress τa (1, 2) and those calculated according to Equation 1.16 (3, 4) for HDPE (1, 3) and PC (2, 4) [30]

For the calculation of the ﬁlled polymers strength σf a number of empirical equations was deduced, for example [39]:

(1.17)

where a and c are constants, ϕf is the volume content of the ﬁller. In Figure 1.5 the dependences of the fracture stress σf on the ϕcl value for three polymers (PC, PAr and HDPE) are adduced; for the latter, results were obtained by impact tests [40]. As follows from this ﬁgure, the strength of the mentioned polymers is described by the simple equation [30]:

, MPa

(1.18)

which at a = 0 and c =–119 MPa is analogous to Equation 1.17.

11

Structure and Properties of Crosslinked Polymers

k(d) 4

PC P Ar x PSF

3 PHE 2

* PP

PTFE HDPE

1 LDPE 0 0

10

20

30

F

40

Figure 1.4 The relation between aggregation parameter k(d) and cluster functionality F for eight amorphous and semi-crystalline polymers [30]

σf , MPa 60 40

× 20

×× 0

××

-1 -2 ×- 3

0,1 0,2 0,3 0,4 0,5

ϕcl

Figure 1.5 The dependence of the fracture stress σf on the relative fraction ϕcl of clusters for PC (1), PAr (2) and HDPE (3) [30, 40]

12

The Main Principles of the Cluster Model Thus, under some conditions, the ‘composite’ models can be successfully applied to the description of the behaviour of amorphous and semi-crystalline polymers within the frameworks of the quasi-two-phase cluster model. In this case, an obvious possibility for obtaining quantitative correspondence between theory and experiment exists under the condition of heterophase structure of amorphous state and the accuracy of the cluster model [41]. Finishing this section, let us note that for the presence of the local order in the polymer amorphous state (unrelated to a particular model of its domains) there are the most general strict mathematical proofs. In accordance with the Ramsey theorem proved in the theory of numbers, any amount great enough of numbers, points or objects, i > R(i, j) (in the case under consideration, statistical segments) necessarily contains a highly ordered subsystem from Nj ≤ R(i, j) such segments. That is why the absolute disordering of large systems (structures) is impossible [42, 43].

1.2 Thermodynamics of the Local Order Formation In this section, it will be shown that cluster structure formation is the constituent part of much more general concepts, for example, the theory of hierarchic systems evolution [44–49]. Correlation between the speciﬁc Gibbs function of intermolecular

~ im

interactions ΔG (im means intermolecular or, in our case, intersegmental type of interactions; ‘–’ indicates the speciﬁc type of value; ‘~’ outlines the heterogeneous type

~ im

of the system; hereinafter, for the sake of simplicity it will be denoted as ΔG ) and the melting temperature Tm [46, 48, 49] was chosen for experimental examination of the physicochemical theory of evolution of chemical systems [44]. The choice of these parameters is stipulated in papers [46–49]. Firstly, we should be reminded of the fundamentals stated in [48], which are necessary for better understanding of the following material. It is known that the Gibbs– Helmholtz equation is valid for the processes proceeding in simple closed systems:

(1.19)

where ΔG and ΔH are the changes of Gibbs function and enthalpy during the process, respectively, T and p are temperature and pressure, respectively. If we wish to accept that in a deﬁnite temperature range ΔH is independent of T, then

13

Structure and Properties of Crosslinked Polymers for a non-equilibrium phase transition (self-assembly of an individual substance) at temperature T the following relationship is valid:

(1.20)

where ΔGim is the Gibbs function change during crystallisation (self-assembling) of im the studied substance from an overcooled state at T = Tm – ΔT, ΔH m is the enthalpy im change during crystallisation (solidiﬁcation); ΔS m is the crystallisation entropy (change of entropy at phase transition). It has been proposed [44, 46, 48] to use Equation 1.20 for open systems, the composition and Tm values of which vary negligibly. Further, the possibility of application of this equation was displayed for various chemical compounds melting at Tm < 373 K and condensing at constant standard temperature T = T0 = 298 K [46–49]. In a more strict approach, for these cases Equation 1.20 should be represented as follows:

(1.21)

where index i = 1, 2, …, n indicates different substances. In this form, Equation 1.21 represents the analogue of Equation 1.20. Simultaneously these equations are principally different as follows. In Equation 1.20 ΔGim is a variable characterising non-equilibrium transition of an individual substance in the system at any temperature im im T < Tmi . Values of ΔH m , ΔS m and Tm belong to this individual substance and are accepted as constant values. As a whole, Equation 1.20 represents the functional dependence ΔGim = f(T). In Equation 1.21 ΔGim is the variable related to non-equilibrium transitions of different compounds with different melting temperatures Tmi at standard (constant) im temperature T0. In this case Equation 1.21 represents the function ΔGi = f( Tmi ) ~ im [48]. The method of ΔG calculation for polymers is described in paper [50].

~ im

The dependence of ΔG

on ΔT = Tg – 293 K for 15 amorphous glass, semi-crystalline

~ im

and crosslinked polymers is adduced in Figure 1.6, where the ΔG

~ im

kcal/mol. As has been expected, the linear decrease of ΔG 14

value is given in

with an increase in ΔT

The Main Principles of the Cluster Model (or Tg) is observed. A more important fact is that the straight line plotted in Figure 1.6, which approximates well the obtained results, corresponds to the data of the

~ im

papers [46–49] shown in ΔG

– ΔT coordinates for absolutely different chemical

~ im

compounds, but at a 10 : 1 scale by the ΔG axis. The latter circumstance is due to the roughly order of magnitude difference between molar volumes of the segments (which are also kinetically independent fragments) in compounds, used in paper [46],

~ im

which is proved by the plot in Figure 1.7, where ΔG

values are given in cal/g.

Δ Gim 200 -1 -2 ×- 3

100 0

×

- 100

× ×× ×

- 200 - 100

0

100

200 Δ T, K

Figure 1.6 The dependence of the Gibbs speciﬁc function of non-equilibrium phase

~ im

transition ΔG

on ΔT = Tg – 293 K for (1) semi-crystalline; (2) amorphous glassy

~ im

polymers; and (3) crosslinked polymers. The value ΔG

is given in units of kcal/

mol. The straight line is plotted in accordance with data from [46] at a 10 : 1 scale

~ im

by the ΔG

axis [50]

~ im

As follows from this plot, ΔG (ΔT) dependence for the mentioned polymers corresponds, both qualitatively and quantitatively, to the data adduced in papers [46, 48, 49]. Principally, this allows calculation of the segment size, which is different for various polymers. Deviation from the plot of the data for polymers with high Tg indicat es the features of suprasegmental structure for these substances [46]. The data of Figures 1.6 and 1.7 indicate that the cluster model postulated in [7, 13–15], based

15

Structure and Properties of Crosslinked Polymers on the local order existence in the amorphous state of polymers, qualitatively and quantitatively conforms with a much more general macrothermodynamic hierarchical model [44–50], occupying the corresponding energetic niche in structures in the real world hierarchy. The plots in Figures 1.6 and 1.7 demonstrate the direction of the polymer structure evolution during its physical ageing. Striving of the polymer

~ im

~ im

structure to equilibrium means ΔG striving to the minimum (i.e., ΔG shifts towards lower negative values) and the local order degree increasing, respectively, which is accompanied by Tg increasing [51]. Polymer ‘rejuvenation’, which is the opposite process by its thermodynamic directivity to the one considered above, is also possible. In practice, this is realised by ‘injection’ of energy (for example, mechanical) to the polymer [52].

Δ Gim 10

× -1 -2 -3

5

×

0

××

-5 - 100

0

100

200 Δ T, K

Figure 1.7 The dependence of the Gibbs speciﬁc function of non-equilibrium phase

~ im

transition ΔG

on ΔT = Tg – 293 K for (1) 15 semi-crystalline; (2) amorphous

~ im

glassy; and (3) crosslinked polymers. The value of ΔG

is given in cal/g [50]. The

straight line is plotted according to data [46]

The regularities (Gibbs–Helmholtz–Gladyshev equation) shown in Figures 1.6 and 1.7 are true for both different polymers and a single polymer at varying temperature

~ im

T (Gibbs–Helmholtz equation). In Figure 1.8 the dependence ΔG (ΔT) for PC is shown, which also quantitatively corresponds to the one shown in Figure 1.6.

16

The Main Principles of the Cluster Model Therefore, Equations 1.20 and 1.21 are fulﬁlled for polymers simultaneously, i.e., suprasegmental structure formation represents a non-equilibrium transition resulting in formation of non-equilibrium structures. It is signiﬁcant that the beginning of their formation corresponds to the glass transition, i.e., to the transition from an equilibrium devitriﬁed state to a weakly non-equilibrium glassy one.

~ im

Finally, it should be noted that ΔG values ‘regulating’ formation of suprasegmental structures in polymers are deﬁnitely connected with molecular characteristics of the latter. Since a polymer is a solid consisting of long chains of macromolecules, it should be expected that the most important (or at least one of the most important) property is the polymer chain ﬂexibility, which can be expressed with the help of the

~ im

characteristic ratio C∞ [22, 53]. That is why the dependence ΔG

(C∞) shown in

~ im Figure 1.9 clearly displays the tendency of ΔG to increase (and, consequently, Tg

to reduce) with raising of the chain ﬂexibility. The sole noticeable deviation, detected for polystyrene, may be due to the well-known speciﬁcity of its chemical structure

~ im

[54]. The correlation ΔG

(C∞) fully corresponds to the previously postulated [54]

Tg increase with raising of the polymer chain rigidity (that is now thermodynamically substantiated).

Δ Gim 0 -1 - 100

2

- 200 0

50

100

150 Δ T, K

Figure 1.8 The dependence of the Gibbs speciﬁc function of non-equilibrium

~ im

phase transition ΔG

on ΔT = Tg – T for PC (1). The straight line 2 is plotted in accordance with Figure 1.6 [50]

17

Structure and Properties of Crosslinked Polymers

Δ Gim 200

×

×- 1 -2 -3

100 0

× ×××

- 100 - 200 - 300

0

2,5

5,0

7,5 10,0

C∞

Figure 1.9 The dependence of Gibbs speciﬁc function of non-equilibrium phase

~ im

transition ΔG

on characteristic ratio C∞ for (1) semi-crystalline polymers; (2),

amorphous glassy polymers; and (3) crosslinked polymers [50]

~ im

Thus the dependences ΔG (ΔT) for suprasegmental structure of polymers within the frameworks of a macrothermodynamic hierarchical model qualitatively and quantitatively correspond to previously obtained analogous correlations for a wide set of substances [46, 48, 49, 55]. This proves the reality of these structures for the polymer amorphous state. Equations 1.20 and 1.21 are equally applicable to the description of the thermodynamic behaviour of these structures and can be used for their quantitative simulation. If more strict calculations are required, corrections for heat capacity change at phase transitions must be introduced [55].

1.3 Polymer Structure Ordering Degree and Cluster Model It should be expected that the formation of the local order domains will affect the general ordering degree of the polymer amorphous state structure. At present, there are some methods allowing characterisation of the polymer structure order (or disorder) degree, and they will be compared below with the parameters characterising the cluster structure of these materials. The possibility of using the Mooney–Rivlin equation constants for local order characterisation was considered in papers [56, 57]; therefore, the principal difference

18

The Main Principles of the Cluster Model from the methods previously used for rubbers only is the application of this approach to solid-phase polymers [56]. The simplest form of Mooney–Rivlin empirical equation can be written as follows [58]:

(1.22)

where f* is the reduced stress, 2C1 and 2C2 are the equation constants, λ is the drawing ratio. The value of f* is determined as the following ratio [59]:

(1.23)

where σ is the nominal stress, i.e., the one calculated using sample initial crosssectional area. On the grounds of the number of works in the literature, Equation 1.22 is widely applied to the study of mechanical properties of rubbers; Boyer [59, 60] has made the assumption that the ratio 2C2/2C1 can be the measure of short-range order in crosslinked rubbers and has presented a summary table for polymer numbers in the rubber-like state, which proves this assumption. In Figure 1.10 the typical dependences corresponding to Equations 1.22 and 1.23 for PC (testing temperature T = 403 K) and HDPE (T = 293 K) are shown. As one can see, the Mooney–Rivlin equation is applicable to both amorphous glassy PC and semi-crystalline HDPE and gives reasonable 2C1 and 2C2 values. The latter statement is based on the following observation. As it is known [58], the 2C1 constant can be expressed as follows:

(1.24)

where A is the coefﬁcient determined by the functionality of the entanglements network nodes.

19

Structure and Properties of Crosslinked Polymers

f +, MPa 15 1

10

2 5

0

0,1 0,2

0,3

0,4 0,5 λ-1

Figure 1.10 The dependences of the reduced stress f* on the drawing ratio λ for (1) PC and (2) HDPE [56]

Equation 1.24 gives the possibility to estimate Me with respect to the known 2C1 values. These estimation results have shown good correspondence of the obtained values to analogous ones, calculated according to Equations 1.2 and 1.3. It is signiﬁcant that the obtained values of 2C1 correspond to magnitudes Mcl, but not Me for a macromolecular binary-hookings network, which are larger than Mcl by one or two orders of magnitude [56]. In Figure 1.11 the dependence of Mcl on the 2C2/2C1 ratio value, obtained from Mooney–Rivlin plots, is adduced. Data from this ﬁgure display a good linear correlation between the mentioned parameters that conﬁrms the Boyer assumption about the possibility of using the ratio 2C2/2C1 as a measure for a short-range (local) order in polymers. However, Boyer has also assumed [59] that the increase in the absolute value of 2C2/2C1 displayed the growth of the short-range order degree in rubbers. For the studied polymer [56, 57] an increase in Mcl with 2C2/2C1 growth means the entanglements network density increases (Equation 1.3), the number of segments in clusters decreases and, consequently, the local order degree reduces (Equation 1.11). In other words, for amorphous glassy and semi-crystalline polymers the increase of 2C2/2C1 reﬂects the opposite effect to that observed in rubbers. Such a discrepancy is not accidental and reﬂects the difference in structures for these classes of polymers. If the crosslinked network density and possibility of packing of chain segments between crosslinking nodes in rubbers correspond to different structural elements and display opposite tendencies of variation [59], then for the studied

20

The Main Principles of the Cluster Model polymers both the entanglement network density and local order degree increases possess symbate tendencies that follow from the cluster model [13–15]. In other words, analogies between structural and mechanical properties of true crosslinked rubbers and linear polymers, studied in paper [56], are correct only to a certain extent.

Mcl , kg/mol 3 2

-1 -2 -3

1

0

10

20

30

2C2 2C1

Figure 1.11 The dependence of the molecular weight Mcl of the chain part between clusters on the Mooney–Rivlin equation constants ratio 2C2/2C1 for (1) PC; (2) poly(arylate sulfone) (PAS); and (3) HDPE [56]

Within the frameworks of Landau phenomenological theory of second-order transitions [61], the order parameter ψor, unequivocally connected with one of the most important thermodynamic properties which is entropy change ΔS, is determined as follows:

(1.25)

where a and C are parameters, Ttr is the transition temperature.

21

Structure and Properties of Crosslinked Polymers The experimental proof of the accuracy of application of the Landau theory is the correspondence of the temperature dependence of ψor to (Ttr – T)1/2 shape [61]. The plots in Figure 1.12 indicate the same shape of νcl temperature dependence for semicrystalline HDPE [62]. In this case, the glass transition temperature Tg (as is usual for Landau theory [61]), the melting temperature Tm and the temperature of ‘liquid 1–liquid 2’ transition Tll [63] were accepted for Ttr. The temperature of the ‘liquid 1–liquid 2’ transition can be estimated as follows [9]:

(1.26)

vcl × 10-27, m-3 8 6

-1 -2 -3

4 2 0

5

10

-3 15 (Ttr-T)1/2 , K

Figure 1.12 The dependences of cluster entanglement network density νcl on the parameter (Ttr – T)1/2 for HDPE, corresponding to the Landau equation for secondorder phase transitions. The following are selected as transition temperatures Ttr: (1) Tg; (2) Tm; and (3) Tll [62]

The data of Figure 1.12 assume that the cluster entanglement network density νcl is an analogue to the order parameter ψor and, consequently, characterises the local ordering degree in non-crystalline regions of polyethylenes. Two interesting features of the dependences shown in Figure 1.12 should be indicated. First of all, on the dependence νcl(Ttr – T)1/2 at T = 333 K an inﬂection is observed. As it is known [64],

22

The Main Principles of the Cluster Model a relaxation transition of polyethylenes, which Boyer named the ‘glass transition I’, was detected at this temperature. Secondly, the condition ψor = 0 at T = Ttr is assumed in Equation 1.25. The plot in Figure 1.12 indicates that the identity νcl = 0 is reached at Ttr = Tm, but not at Ttr = Tg. In other words, the temperature Tm corresponds not only to crystallites melting, but also to segments ‘melting’, and above Tm only the macromolecular binary-hookings network remains [33]. This situation can be explained by the local order formation speciﬁcity in non-crystalline regions of semicrystalline polymers similar to polyethylenes, for which this process is of ‘forced’ type due to chain tightness in the amorphous phase during crystallisation [65, 66]. Let us note that the similarity of the ψor and νcl parameters eliminates consideration of entanglement nodes as traditional binary-hookings [33] and supposes that local order domains (clusters) play their role [62]. It is obvious that the structure of amorphous polymers (or the amorphous phase of semi-crystalline polymers) gives grounds to assume the availability in them of a deﬁnite chaos degree. It is also quite obvious that the chaos degree of an amorphous phase structure represents an important parameter determining structural characteristics and, consequently, a polymer’s properties. That is why the question about interconnection of these parameters arises. The second important problem is the physical nature of chaos in polymers: is it random (and unpredictable) or deterministic chaos? Among possible measures of the chaos degree of a system (in the given case, in polymer structure), the common one is Lyapunov’s index λL [67]. It represents the measure of exponential rate of divergence or convergence of neighbouring trajectories in phase space along the given coordinate direction. Chaotic processes are characterised by exponential divergence of neighbouring trajectories and by virtue of this, at least, by one positive Lyapunov’s index. The technique for λL estimation is described in paper [68]. So far as the main distinguishing characteristic of a polymer is its composition from long-chain macromolecules, and the basic characteristic of the latter is their ﬂexibility, the interconnection between Lyapunov’s index λL and the characteristic ratio C∞ has been studied [68]. There are two reasons for choosing parameter C∞ as the ﬂexibility measure. The ﬁrst reason is that C∞ is determined more precisely compared with other similar parameters [69]. The second is that it can be estimated based on the polymer macromolecule chemical structure only [70]. First of all, it should be noted that the presence of positive λL [68] indicates the existence of chaos in the polymer structure. Further, a systematic increase of λL (i.e., chaos intensiﬁcation) with chain ﬂexibility raising is observed (Figure 1.13). Obviously an increase in C∞ means growth in chain mobility and, as a consequence, intensiﬁcation of chaotic processes in the system. Two polymers are outside of the general dependence: polystyrene (PS) and poly(methyl methacrylate) (PMMA), which was observed before [69]. This assumes that the C∞ value is not always the sole molecular parameter that determines structural characteristics and properties of polymers, as suggested in paper [70]. It is known

23

Structure and Properties of Crosslinked Polymers [71] that both PS and PMMA have bulk side groups that result in a sharp increase of the cross-sectional area S of their macromolecules. Therefore one may suggest that applying S as the normalising factor will improve correlation λL(C∞). Actually, the dependence λL(C∞/S) does not give fall out results (Figure 1.14) and, despite a deﬁnite data scattering, it can be approximated by a straight line. The most important result following from the above-mentioned data is regular variation of the chaos degree indicator λL for the structure of polymers possessing molecular characteristics C∞ (or C∞/S), which presumes deterministic (predictable) chaos [68].

λL 0,15 0,10 0,05

0

2

4

6

8

10 C∞

Figure 1.13 The dependence of Lyapunov’s index λL on the characteristic ratio C∞ for amorphous glassy and semi-crystalline polymers [68]

λL 0,15

0,10

0,05

0

0,1

0,2

0,3

0,4

0,5

C∞ S

, Å-2

Figure 1.14 The dependence of Lyapunov’s index λL on the molecular parameter (C∞/S) for amorphous glassy and semi-crystalline polymers [68] 24

The Main Principles of the Cluster Model In Figure 1.15 the dependence of the density νcl of the macromolecular entanglements cluster network on λL is shown. As might be expected, chaos intensiﬁcation (λL increase) reduces the νcl value, i.e., the local ordering degree in the polymer amorphous state structure [68]. More precise interpretation of the polymer structure chaotic character within the frameworks of multifractal formalism will be given below.

vcl × 10-27, m-3 4 3 2 1

0

0,05 0,10 0,15

0,20 λ L

Figure 1.15 The dependence of the macromolecular entanglement cluster network density νcl on Lyapunov’s index λL for amorphous glassy and semi-crystalline polymers [68]

Finishing this section, let us consider from the position of thermodynamics the interconnection between polymer structure disorder degree and local ordering degree. One of the possible characteristics of disorder can be the ﬂuctuation free volume fg value [72], which is connected with entropy change ΔS as follows [73]:

(1.27)

One further quantitative interpretation of disorder is proposed in paper [74], where

25

Structure and Properties of Crosslinked Polymers the disorder parameter δ, connected with the thermal mobility in a liquid, at the melting point Tm is determined as follows:

(1.28)

~

where Pi is the internal pressure, Vm is the reduced molar volume at melting point, p* is the characteristic pressure. The value of Pi can be determined as follows [74]:

(1.29)

and ﬁnally, after substitution of Formula 1.29 into Equation 1.28, the equation for δ determination is obtained as follows [75]:

(1.30)

Taking into account successful application of liquid theories to the description of the behaviour of an amorphous polymer [9], the range of application of Equation 1.30 was ~ ~ extended to the glassy state of these polymers (with replacement of Vm on V ) [75]. It is obvious that the problem of quantitative estimation of δ is now reduced to ~ V determination, which for polymers can be implemented by using the following equation [76]:

(1.31)

where α is a linear thermal expansion coefﬁcient.

26

The Main Principles of the Cluster Model The temperature dependence of α can be estimated by different methods. At the ﬁrst approximation, the α value can be accepted as constant, and direct experimental measurements (which is naturally the most precise method) or the known Barker equation can be applied [77]:

(1.32)

where E is the elasticity modulus, C1 is a coefﬁcient, the average value of which is equal to 15 N/m2 K and its variation limits make up 7.5–24 N/m2 K [77]. Equation 1.32 gives an opportunity of simple enough estimation of α according to the mechanical test results. Variation of C1 does not change the δ estimation result qualitatively, but gives the possibility of improving the quantitative correspondence of parameters, calculated in [75], by approximately 15–20%. Preliminary estimations indicate the best correspondence at C1 equal to 7.5, 15 and 24 N/m2 K for PAr, PC and PMMA, respectively [75]. The relative ﬂuctuation free volume value fg is a function of Poisson’s ratio ν, expressed by the following equation [78]:

(1.33)

The constant C2 value may be varied, but this has no effect on the quality of fg estimation; thus it was accepted that [29, 79]:

(1.34)

In Figure 1.16 the temperature dependences of disorder parameter δ, estimated by the method considered above, for three amorphous glassy polymers are adduced. As one can see, the dependences δ(T) are symbate and indicate the degree of disorder increasing with increasing temperature. It should be noted that this result is not

27

Structure and Properties of Crosslinked Polymers trivial. There are several concepts considering the disorder degree as ‘frozen’ at the amorphous polymer transition to the glassy state. According to Equation 1.27, the condition fg = const. also reﬂects this point of view.

δ 0,12 × ×

0,10 ×

0,08 0,06

×

0,04 173

×

× ××

273

× -1 -2 -3 373

473

Figure 1.16 The temperature dependences of the disorder parameter δ for (1) PMMA; (2) PC; and (3) PAr [75]

The similarity of curves δ(T) (Figure 1.16) indicates the possibility of their superimposition by shifting along the temperature axis (i.e., applying the principle of temperature superposition). Actually, the dependence δ(ΔT), where ΔT represents the difference between the glass transition temperature Tg and the testing temperature T indicated the reality of such a superposition (Figure 1.17). Practically, this means that the degree of disorder in amorphous glassy polymers is independent of their chemical structure and is determined by temperature only (more precisely, by the difference between Tg and the testing temperature). This gives grounds to the assumption that disorder in the structure of amorphous polymers, determined by the above-mentioned methods, has thermoﬂuctuational origin only (which actually follows from Sharma’s determination of δ [73]). In Figure 1.18 the comparison of δ and fg values, calculated for PMMA, PC and PAr, is adduced. This comparison displays the approximate equality of the mentioned parameters, i.e., ﬂuctuation free volume is also the disorder indicator for amorphous glassy polymers [75]. Thus as the νcl value determines the number of densely packed

28

The Main Principles of the Cluster Model segments in local order domains and, consequently, is the indicator of this order, an inversely proportional correlation between δ and νcl must be observed. Obviously for the variant of ‘binary-hookings’ network, none of these correlations is expected, because it is more probable that ‘binary-hookings’ will loosen the structure of amorphous polymers [80], i.e., will intensify disorder.

δ 0,12 0,10 0,08

×

× ××

× -1 -2 -3 × ×

0,06 0,04 0

50

100

×

×

150

× 200 ΔT, K

Figure 1.17 The dependence of the disorder parameter δ on the temperature difference ΔT = Tg – T for (1) PMMA; (2) PC; and (3) PAr [75]

In Figure 1.19 the relationship between δ and ϕcl (i.e., between disorder and order parameters, respectively) is shown. As can be expected, the inversely proportional correlation approximated well by a straight line is observed for these parameters. This line is plotted according to the following considerations. Obviously at ϕcl = 1.0 the system possesses an ideal order and δ = 0, whereas at ϕcl = 0 δ is accepted as being equal to fg at Tg according to the Figure 1.18 data (in the given case the value fg = 0.113 is used [81]). The indicated result unequivocally testiﬁes in favour of the cluster treatment of macromolecular entanglement network nodes. For linear polymers δ and fg are approximately equal (Figure 1.18) and for glassy crosslinked polymers δ > fg [72]. This discrepancy will be explained in Chapter 5.

29

Structure and Properties of Crosslinked Polymers

fg 0,15 ×× ×× ×× ××

0,10

× × -1 -2 -3

0,05

0

0,05

0,10

0,15

δ

Figure 1.18 The relationship between the disorder parameter δ and the ﬂuctuation free volume fg for (1) PMMA, (2) PC and (3) PAr [75]

δ 0,15

0,10

××

× × × ×

× ×

0,05

0

1 -2 -3

0,2

0,4

0,6

0,8 0,10 ϕcl

Figure 1.19 The relationship between the disorder parameter δ and the relative fraction ϕcl of clusters for (1) PMMA, (2) PC and (3) PAr [75]

30

The Main Principles of the Cluster Model Simultaneously, the data of Figure 1.19 suppose a tight interconnection between the cluster model [13–15] and the ﬂuctuation free volume theory [78, 79]: separation of a segment from a cluster means formation of a free volume microvoid, whereas the addition of a segment to a cluster means its ‘collapse’. If it is correct, the entropy variation ΔSf due to ﬂuctuation free volume formation (Equation 1.27) must be equal to entropy change ΔScl at partial decay of thermoﬂuctuational clusters. The value of ΔScl can be estimated within the frameworks of the Forsman theory [82] according to following relationship:

(1.35)

where C3 is the polymer concentration in the system (in the case under consideration C3 = 1.0), M0 is the molecular weight of the repeating unit, m is the number of repeating units in the cluster, ρ is the polymer density. The comparison of ΔSf and ΔScl values for PC and PAr is adduced in Figure 1.20, which shows an interesting feature. Let us note ﬁrst of all that the change in the ' slopes of the curves ΔSf(T) corresponds to the glass transition temperature Òg of the loosely packed matrix of the polymers [12]. Thus in the case of a glassy loosely packed matrix the disorder increase due to partial cluster decay is transformed into disorder of ﬂuctuation free volume microvoid formation. For a devitriﬁed loosely packed matrix ΔScl is partly transformed into ΔSf and partly increases disorder by other mechanisms (possibly owing to conformational changes [83]). Hence, strictly speaking, the δ value (and the parameters fg, νcl and ϕcl associated with it) characterises thermoﬂuctional ‘structural’ disorder only, but not its entire spectrum [75]. Let us note that the study of the degree of disorder of the structure of amorphous polymers is not only of theoretical signiﬁcance. In Figure 1.21 the dependence of the yield stress of the three studied polymers on the reciprocal value δ is presented. As one can see this dependence has the simplest shape: it is linear and passes through the coordinates origin. That is why it is quite suitable for prediction of the mechanical properties of polymers. Similar dependences were obtained for the yield strain εY and the elasticity modulus E [75].

31

Structure and Properties of Crosslinked Polymers

ΔS, J/(mol-K) 15 -3 -4

10

2 1 5 0 293 333 373 413 453 493 T, K Figure 1.20 The temperature dependences of entropy change due to ﬂuctuation free volume formation ΔSf (1, 2) and partial decay of clusters ΔScl (3, 4) for PC (1, 3) and PAr (2, 4) [75]

σY, MPa 60

-1 -2

40 20

0

5

10

15

20 δ-1

Figure 1.21 The dependence of the yield stress σY on the disorder parameter δ for (1) PC, (2) Par and (3) PMMA [75]

32

The Main Principles of the Cluster Model The results stated above allow the following main conclusions to be made. Firstly, for amorphous glassy polymers the disorder degree is a function of temperature. The dependence of δ on temperature only suggests the purely thermoﬂuctuational nature of this disorder. Secondly, for amorphous polymers the ﬂuctuation free volume fg is the quantitative characteristic of disorder degree. Thirdly, nodes of macromolecular entanglements cluster network personify the degree of local regulating of amorphous polymer structure as postulated in the cluster model. Fourthly, disorder in the systems under consideration is not necessarily of thermoﬂuctuational origin, but the latter component of it deﬁnes the polymer mechanical properties (Figure 1.21) [75].

1.4 Thermoﬂuctuational Origin of Clusters A set of parameters indicating thermoﬂuctuational origin of local order domains (clusters) was mentioned above, for example, Equation 1.12, Figure 1.8 and Figure 1.19. In the present section this question will be considered in more detail. Firstly, the interconnection between entanglement cluster network parameters and density ﬂuctuations should be studied. Density ﬂuctuations 〈Δρ/ρ〉2 represent the measure of disorder in polymers and are determined as follows [84, 85]:

(1.36)

where N is the number of electrons in an arbitrarily selected volume, 〈N〉 is the average value of N. For a liquid in the equilibrium state, statistical mechanics gives the following expression for 〈Δρ/ρ〉2(V), where V is the standard volume in the limit V®∞ (i.e., in the thermodynamic limit) [85]:

(1.37)

where χT is the isothermal compressibility. Equation 1.37 shows that density ﬂuctuations are due to thermal mobility of atoms with energy kT, but limited by volumetric rigidity (χT)–1.

33

Structure and Properties of Crosslinked Polymers '

Based on the literary data for ρ and χT estimation of 〈Δρ/ρ〉2(∞) at Tll temperatures ( Tll' is the analogue of Tll for semi-crystalline polymers connected with Tm similarly to Equation 1.26) it has been shown that the mentioned value is approximately constant at these temperatures. This observation assumes that as some critical value of 〈Δρ/ρ〉2 is reached, formation of local order domains, i.e., thermoﬂuctuational cluster networks of macromolecular entanglements, is impossible owing to high thermal mobility of macromolecules. On the contrary, below the critical 〈Δρ/ρ〉2 values local order domains (clusters) in the polymer melt are formed, which according to the present treatment are identiﬁed as nodes of the macromolecular entanglements network, i.e., a state is formed, which was deﬁned by Boyer as a ‘liquid with ﬁxed structure’ [59]. Proceeding from the arguments stated above, the critical temperatures Tcr at which critical density ﬂuctuations are reached were calculated. A 〈Δρ/ρ〉2 value for PS at 433 K was accepted. The calculation results are quoted in Table 1.3, from which it ' follows that Tcr and Tll ( Tll ) for a set of amorphous and semi-amorphous polymers are very close: the greatest discrepancy of these temperatures does not exceed 6% [87, 88]. Let us note that in the considered case we are dealing with dynamic (shortliving) local order, which is ‘frozen’ below Tg(Tm) [9].

Table 1.3 The calculated temperatures Tll ( Tll' ), Tcr and their relative discrepancy Δ [88] Polymer

Tcr, K

Tll ( Tll' ), K

Δ, %

Poly(methyl methacrylate)

398

415

4.1

Polystyrene

462

433

6.2

Polycarbonate

488

502

2.8

Low-density polyethylene

488

480

1.7

High-density polyethylene

496

480

3.3

Polypropylene

510

528

3.4

Polyamide-6

635

600

5.8

34

The Main Principles of the Cluster Model Sanditov and Bartenev [78] have shown that the 〈Δρ/ρ〉2 value is connected with relative ﬂuctuation free volume according to the following relationship:

(1.38)

where Vh is the volume of the ﬂuctuation free volume microvoid, Vat is the atomic volume. The 〈Δρ/ρ〉2 value can also be determined by using Poisson’s ratio ν [78]:

(1.39)

The temperature dependences of 〈Δρ/ρ〉2 calculated according to Equations 1.38 and 1.39 for high and low-density polyethylenes (HDPE and LDPE, respectively) are shown in Figures 1.22 and 1.23. Despite different absolute values of density ﬂuctuations, calculated according to the above-mentioned equations, the shapes of their temperature dependences correspond to the data known at present [9, 85]. It is signiﬁcant that the dependences 〈Δρ/ρ〉2(T) do again display inﬂections at T ≈ 333 K, as in Figure 1.12. The higher values of 〈Δρ/ρ〉2 for LDPE are due to higher values of Vh for this polymer [90]. Between order (νcl) and disorder 〈Δρ/ρ〉2 indicators the inversely proportional correlation should be observed, conﬁrmed by the data of Figure 1.24, where the values of 〈Δρ/ρ〉2 were calculated according to Equation 1.38. The correlation 〈Δρ/ρ〉2( ) is one more argument for the beneﬁt of thermoﬂuctuational origin of local domains (clusters) in the polymer amorphous state [89].

35

Structure and Properties of Crosslinked Polymers

Figure 1.22 The temperature dependences of density ﬂuctuations 〈Δρ/ρ〉2, calculated according to Equation 1.38, for (1) HDPE and (2) LDPE [89]

〈Δρ/ρ〉2 × 103 20 15

-1 -2

10 5 0 293 313 333 353 373 T, K Figure 1.23 The temperature dependences of density ﬂuctuations 〈Δρ/ρ〉2, calculated according to Equation 1.39, for (1) HDPE and (2) LDPE [89]

36

The Main Principles of the Cluster Model The thermoﬂuctuational origin of the cluster entanglement network allows theoretical estimation of its density νcl temperature variation. For this purpose the authors [91] have used the model [92] in which the following expression for relaxation time τr was obtained (for the range ofTg < T < Tm):

(1.40) where Ns is the segments number per oscillating macromolecule part and the τ(T) dependence is described by the Frenkel-Eyring-Arrhenius formula [92]:

(1.41)

where τ0 is a constant, ΔU is the potential barrier of segment transition from one quasi-equilibrium state to another; the viscous ﬂow activation energy Uﬂ is suitable for estimation of this value [92]. For the function F(Ns) the following scaling relationship was accepted [92]:

(1.42)

Following the general way of deducing the model [92], the authors [92] have assumed that at all temperatures the size of the macromolecule part between nodes of the macromolecular entanglements network Ne (expressed by the number of statistical segments) is equal to the Ns value, at which τr (T, Ns) becomes equal to some value τcl. The latter parameter can be treated as the time required for packing a chain segment possessing a deﬁnite axial collinearity with the ones already packed to an entanglements mode (by analogy with crystallisation process). It is easy to see that in the given variant the node of macromolecular entanglement thermoﬂuctuational network is considered as a densely packed cluster of macromolecules segments with approximately parallel disposition. Solving the following equation for Ne:

(1.43)

37

Structure and Properties of Crosslinked Polymers we obtain that:

(1.44)

〈Δρ/ρ〉2 4 -1 -2

3 2 1

0

0,2

0,4

0,6

0,8

0,10

(vcl-1) ×1026, m3

Figure 1.24 The relation between density ﬂuctuations 〈Δρ/ρ〉2, calculated according to Equation 1.38, and the reciprocal value of the entanglements cluster network density νcl for (1) HDPE and (2) LDPE [89]

This dependence reﬂects the fact that as the temperature decreases the mobility character becomes of more and more reduced scale; in this case Ne can be considered as an effective parameter characterising the probability of the slowing of chain mobility ﬂuctuations [91]. As at present the ratio τcl/τ0 can hardly be subjected to substantiated theoretical or experimental estimation, the conditions mentioned in paper [92] were followed in further calculations. Taking Ne = 1 at T = Tg and x = 3.3, estimation of the ΔU (or ΔUﬂ) value was accepted according to the data [93]. For the PC segment molecular weight Ms is equal to 726 g/mole, i.e., it is supposed that the PC statistical segment includes three monomeric units. Then Me = NeMs. The comparison of the values of Me calculated by the method indicated above and experimental Mcl values of the

38

The Main Principles of the Cluster Model chain segment molecular weight between entanglement nodes for PC as a function of temperature is adduced in Figure 1.25, where good correspondence between them is observed. Such correspondence can be accepted to be the proof of the validity of the condition Ne = 1 at T = Tg, given in paper [92].

Mcl, g/mol 800 600

-1 -2

400 200 0 273

323

373

423 T, K

Figure 1.25 Theoretical (1) and experimental (2, 3) dependences of the molecular weight Me (Mcl) of chain part between entanglements nodes on the temperature (T) for initial (2) and annealed at 393 K (3) PC [91]

The ﬂuctuation theory of glass transition allows estimation of the volume Va of the so-called cooperatively rearranging regions (CRR) [8, 94]:

(1.45) −1

where Δ( CV ) is the reciprocal of the heat capacity jump at the glass transition, ρg is the polymer density at Tg, δT is the average temperature ﬂuctuation per CRR. Bittrich [94] has estimated the CRR characteristic length ξA, which varies within the range of 0.2–2.5 nm in the temperature interval of 323–348 K and reduces with growth in temperature. Since both CRR [95] and clusters [41] are postulated to be of

39

Structure and Properties of Crosslinked Polymers thermoﬂuctuational origin, consistency of the two models mentioned above should naturally be determined by comparing the temperature dependences of ξA and the size of the clusters for the considered polymers [96, 97]. As has been mentioned above, the cluster model assumes reduction of a number of segments in clusters with temperature increase. As the cluster is considered to be the analogue of a crystallite with extended chains consisting of several collinear densely packed segments, a decrease in the number of segments in the cluster with temperature increase assumes a reduction in its cross-sectional area, which is shown schematically in Figure 1.26 (insertion) and, correspondingly, the cluster diameter Dcl. In papers [96, 97], comparison of the temperature dependences of ξA and Dcl as characteristic sizes of structure microheterogeneity in both models was carried out. In Figure 1.26 the comparison of the indicated sizes is adduced, where ξA values are accepted according to the data [94] and the Dcl calculation technique was given in paper [97]. As one can see, the temperature dependences of these structural characteristics display good correspondence. This allows the identity of such structural elements as CRR and clusters to be assumed and also conﬁrms the thermoﬂuctuational origin of clusters [96, 97].

ξA ,Dcl , nm 3

3 Dcl

2

1

2 1

0 328

×

×

×

-1 5 -6

×× ×

333

338

343

348 T, K

Figure 1.26 The dependences of characteristic length of CRR ξA (1–3) and cluster diameter Dcl (4–6) on the temperature T for PMMA (1, 4), PVC (2, 5) and PS (3, 6). The insertion shows an image of the cluster cross-section [97]

40

The Main Principles of the Cluster Model To conclude this section let us consider the temperature stability of the cluster structure. In paper [98] a simple empirical approximation was obtained, which connects νcl and Poisson’s ratio ν:

, m–3

(1.46)

In accordance with the Le Chatelier–Brown principle (the principle of the least constraint) the ν value determining stability of a solid falls within the following range [42]:

(1.47) Combination of Equations 1.46 and 1.47 gives the condition of cluster structure stability for the amorphous state of the polymers [98]:

, m–3

(1.48)

In Figure 1.27 the temperature dependence of νcl for polyhydroxyester (PHE) is shown, where the lower boundary ( ) of the cluster structure stability is indicated by a dashed horizontal line. Good correspondence of and νcl is observed, at which accelerated thermoﬂuctuational decay of clusters is initiated. Similar dependences were obtained for PC and polysulfone (PSF) [98]. '

As follows from Equation 1.46, ν values become negative at νcl < (i.e., at T ≥ Tg ). The classical elasticity theory allows negative ν values limiting the range of their variation as follows [42]:

(1.49)

41

Structure and Properties of Crosslinked Polymers

vcl × 10-27,m-3 3 vclmin 2

1

0 293

313

333

353

373 T, K

Figure 1.27 The temperature dependence of the entanglement cluster network density νcl for polyhydroxyester. The horizontal dashed line indicates the lower boundary ( ) of the stability of the clusters [98]

It is quite obvious from Equation 1.46 that as ν approaches –1.0, νcl approaches zero. Negative values of ν observed for some substances near phase transitions testify to a change of their volume. Making a deﬁnite analogy, let us note that the glass transition is also accompanied by a discrete change of the polymer volume [78]. The maximum possible value of νcl permitted by the molecular structure of the polymer ( ) can be determined by the following procedure. The macromolecule length L per polymer unit volume is determined according to Equation 1.7, and the segment length in the cluster is taken to be equal to the statistical segment length lst according to Equation 1.8. Thus the value of will be [98]:

(1.50)

values estimated by the method mentioned above, are For PHE, PC and PSF approximately equal to 8.45 × 1027, 10.78 × 1027 and 11.0 × 1027 m–3, respectively. According to Equation 1.48, these values are much higher than the upper border

42

The Main Principles of the Cluster Model of νcl. This means that the condition of the cluster structure stability does not allow complete packing of polymer chain segments in the local order domains (clusters). Other criteria of this effect will be discussed below. The obtained results [98] allow the following picture of temperature variation of the macromolecular entanglement cluster network density to be assumed (Figure 1.27). As the lower border of stability of the clusters ( ) is reached, their accelerated decay is initiated, which leads to a much more abrupt decrease of νcl. It is supposed [41] that a deﬁnite part of clusters with low functionality F (i.e., with a small number of segments in a cluster) is present in a loosely packed matrix. These clusters keep this matrix in the glassy state. This supposition is conﬁrmed by experiments of polyarylate ' annealing, in which decreases in νcl and F were observed simultaneously. When Tg (or ) is reached, clusters with small F (thus thermodynamically unstable ones) decay completely and the loosely packed matrix devitiriﬁes promoting an abrupt decrease in F of stable clusters due to the high molecular mobility of the environment. Abrupt decrease of νcl is ﬁnished by complete thermoﬂuctuational decay of the clusters ‘frozen’ in the glassy state at T = Tg [41].

1.5 Functionality of Clusters and Methods of its Estimation As has been mentioned above, clusters consist of several collinear densely packed segments of different macromolecules and are considered to be multifunctional nodes of a macromolecular entanglement thermoﬂuctuational network [41]. Such a node functionality F is accepted as a number of chains emerging from it [8]. Thus full characterisation of the polymer amorphous state requires knowledge of two parameters: macromolecular entanglement cluster network density νcl and the number of segments ncl in a single cluster (local characteristic). Since the cluster represents an amorphous analogue of a crystallite with extended chains, then ncl = F/2. Several possible techniques for estimation of the value of F (or ncl) were considered in paper [99]. The ﬁrst of the possible methods of ncl estimation is of a semi-quantitative type. It is supposed [9] that the volume of local order domains in polymers is equal to about 10 nm3. Dividing this value by the statistical segment volume we obtain the following values: ncl ≈ 88 for PC and ncl ≈ 40 for HDPE. These values display the correct order (several tens of segments in clusters) and can be the upper estimation border. One more estimation method for F involves Equation 1.5, proposed by Graessley [10]. The temperature dependences of cluster functionality F, calculated for HDPE and PC according to Graessley’s equation, are shown in Figures 1.28 and 1.29, respectively.

43

Structure and Properties of Crosslinked Polymers It follows from the plots of these ﬁgures that the number of segments in one cluster varies from 7 to 12 for HDPE and from 3 to 18 for PC and, as it should be expected [7], the F value decreases as T increases.

F 30 -1

-2 -3

20 10 0 293

313

333

353

373

T, K

Figure 1.28 The dependences of the cluster functionality F on the temperature T for HDPE calculated as follows: (1) according to Equation 1.5; (2) according to Equation 1.51 and (3) according to Equation 1.52 [99]

There are two more approximate methods of F estimation. Several conditions, well approved at present, must be carried out for application of any model of macromolecular entanglement network (cluster or ‘binary-hooking’ [21]). One of these conditions is the fulﬁlment of Gaussian statistics of polymer chains. The fulﬁlment of this condition requires the presence of at least 20 repeating units between the nodes of the entanglement network [82]. In Table 1.4 the values of the molecular weight of a chain part between entanglement nodes in a variant of a ‘binary-hooking’ network for HDPE and PC are adduced, which are accepted from the literary sources [20, 21, 33, 71, 100–102] or calculated in accordance with them. Though different authors have presented signiﬁcantly differing Me values, on the whole, it is obvious that the Gaussian statistics condition for chains is fulﬁlled. For multifunctional nodes Flory [8] has given the relation between Me, determined experimentally, and effective molecular weight of the indicated chain part, expressed by Equation 1.4. Mcl for the entanglement cluster network is estimated according to Equation 1.2, and Me

44

The Main Principles of the Cluster Model

is taken as . Taking into account that the node of a ‘binary-hooking’ network is four-functional, it is ﬁnally obtained that [99]:

(1.51)

Figure 1.29 The dependences of cluster functionality F on temperature T for PC calculated as follows: (1) according to Equation 1.5; (2) according to Equation 1.51; and (3) according to Equation 1.52 [99]

Values of F estimated according to Equation 1.51 are also adduced in Figures 1.28 and 1.29, from which their good correspondence to F values, calculated according to Graessley’s equation, and somewhat poorer correspondence for HDPE follows. Nevertheless, the order of magnitude of F and the temperature dependence tendencies conform in both cases.

45

Structure and Properties of Crosslinked Polymers

Table 1.4 The values of molecular weight Me of chain parts between nodes of the ‘binary-hooking’ network for HDPE and PC according to the literary data [99]

Me, g/mole

Polymer HDPE

1900 [20]

2936 [71]

950–2140 [101]

737 [33]

PC

1780 [21]

2426 [102]

2453 [101]

3214 [100]

As it is known [8], macromolecular coils in θ-solvent and in the condensed state of polymers possess the same shape as that expressed by approximately equal meansquare distance between the ends of a macromolecule. Boyer [59] has schematically shown that this circumstance does not prevent formation of local order domains. More strictly this conclusion is proved in paper [99]. It is obvious that this experimentally proved condition should also be true for a cluster model. For ncl estimation in polymer clusters, Forsman [82] has proposed the following equation:

(1.52)

where θ2 is a dimensionless (normalised by kT) value characterising the interaction energy of segments, the technique of determination of which is given in paper [99], c is the polymer concentration, ρ is the polymer density. The value of A describes the macromolecule size change before and after cluster formation and is determined according to the relationship [82]:

(1.53)

where the parameter α is given as follows [82]: 46

The Main Principles of the Cluster Model

(1.54)

In this equation r0 and r are the mean-square distances between the macromolecule ends before and after cluster formation, respectively. If the distance between the macromolecule ends before and after cluster formation is the same (r = r0), α = 1 and estimation results according to Equation 1.52 are also shown in Figures 1.28 and 1.29. Similar to the previous case, despite a deﬁnite quantitative discrepancy for HDPE, they give the correct order of magnitude for F and the expected course of the temperature dependence. Strictly speaking, the F value is not an indicator of the local ordering degree for polymer structure, since clusters are formed by segments of different macromolecules, but it can be an indicator of mutual penetration of macromolecular coils. As has been shown in papers [22, 100], the same role can be played by the characteristic ratio C∞. If this assumption is correct, a deﬁnite correlation between F and C∞ must be observed. The data of Figure 1.30 show that such a correlation is really observed for nine amorphous and semi-crystalline polymers (the F value is calculated for T = 293 K) [99]. Hence, the number of segments in one cluster depending on polymer characteristics and temperature can vary from several segments up to several tens of segments. It is supposed that the most precise F estimation can be obtained according to Graessley’s equation (Equation 1.5). This conclusion follows from the fact that, for example, it describes the νcl decrease at the glass transition temperature of a loosely packed ' matrix Tg (Figures 1.28 and 1.29). For Equations 1.51 and 1.52 the same precision should not be expected, since the data of the ﬁrst are highly scattered, and the second represents an approximation. These conclusions were proved in paper [103] in the example of two series of crosslinked epoxy polymers (EP) on the basis of epoxidiane resin ED-20, hardened by 3,3′-dichloro-4,4′-diaminodiphenylmethane (EP-1 composition) and isomethyltetrahydrophthalic anhydride (EP-2) with variable curing agent : oligomer ratio. Figure 1.31 shows the comparison of cluster functionality F1 and F2 values, calculated according to Equations 1.4 and 1.52, respectively, for epoxy polymers EP-1 and EP2. As one can see, good correspondence of F1 and F2 values is obtained, which again

47

Structure and Properties of Crosslinked Polymers indicates that the invariability of subchains statistics in epoxy polymers during their curing process does not prevent formation of local order domains (clusters) [103].

F 40 30 20 10

0

0,1

0,2

0,3

0,4

-1 0,5 (C∞)

Figure 1.30 The relation between cluster functionality F and characteristic ratio C∞ for nine amorphous and semi-crystalline polymers [99]

F2 30

20

-1 -2

10

0

10

20

30

F1

Figure 1.31 The correlation between cluster functionalities F1 and F2 calculated according to Equations 1.4 and 1.52, respectively, for epoxy polymers (1) EP-1 and (2) EP-2 [103]

48

The Main Principles of the Cluster Model Special attention is drawn to a much worse correspondence of F values, calculated by different methods for semi-crystalline polymers, in comparison with amorphous ones (Figures 1.28, 1.29 and 1.31). This discrepancy is explained by the different mechanisms of the cluster formation for the mentioned classes of polymers . If clusters in amorphous glassy polymers display thermoﬂuctuational origin and their relative fraction ϕcl is a temperature function [13–15], for non-crystalline regions of such semicrystalline polymers as polyethylenes and polypropylene the situation is somewhat more complex. As it is known [51], clusters represent a ‘frozen’ local order of glassy state, and at temperatures above the glass transition temperature of a polymer their complete decay is observed. Nevertheless, for polyethylenes and polypropylene at room temperature or higher, i.e., at T > Tg of the amorphous phase, the local order presence in this phase is assumed [104–106]. It is postulated that in the indicated case, the cluster formation mechanism is associated with polymer chain tightness in the crystallisation process [32, 107]. The mentioned tightness can be estimated with the help of parameter β, which is determined according to the relationship [32]:

(1.55)

where E is the elasticity modulus, K is the degree of crystallinity. In Figure 1.32 the dependence of functionality F on parameter β for HDPE and PP is adduced. As one can see this dependence is approximated well for the two indicated polymers by a straight line passing through the coordinates origin. Hence, the correlation F(β) proves the supposition made that the tightness of amorphous parts of chains in the crystallisation process is the basic mechanism of formation of local order domains in the devitriﬁed amorphous phase of semi-crystalline polymers [109]. To complete this section, let us consider one more principal question associated with entanglement cluster network formation in semi-crystalline polymers. Equation 1.24 is valid for polymers in which the entire volume is involved in the formation of a macromolecular entanglement network. In this case, if only the part of the polymer with the volume fraction ϕnet is involved in this process, Equation 1.24 should be rewritten as follows [110]:

(1.56)

49

Structure and Properties of Crosslinked Polymers

F 40

20

0

-1 -2

0,15

0,30

β, mol/kg

Figure 1.32 The dependence of the cluster functionality F on parameter β characterising the chain tightness in the crystallisation process for (1) HDPE and (2) PP [109]

At present, there are two points of view on the network structure of macromolecular entanglements in semi-crystalline polymers. One of these [101] supposes that the nodes of macromolecular entanglement networks in polyethylene are crystalline areas, which are lamellar crystallites. Another point of view [111] presumes that the nodes are concentrated in non-crystalline regions of semi-crystalline polymer. It is quite obvious that the condition ϕnet = 1.0 (i.e., the entire polymer represents the network) corresponds to the former opinion, and ϕnet is equal to the volume fraction of polymer non-crystalline regions, i.e., about 0.3 for HDPE and 0.5 for LDPE [86]. Prior to the estimation of the correctness of one or another ϕnet value, one should understand the physical signiﬁcance of the front-factor A in Equations 1.24 and 1.56. One variant [112] provides that the A value can be changed within the range of 0.5– 1.0, where A = 0.5 corresponds to the case of the so-called ‘phantom’ network, which displays full freedom of ﬂuctuations of macromolecular entanglements nodes around their middle positions, and A = 1.0 corresponds to the ‘afﬁne’ network, in which such freedom is completely suppressed. An alternative [113] provides a quantitative relation between value A and functionality F of macromolecular entanglement network nodes:

(1.57)

50

The Main Principles of the Cluster Model It is obvious that the ﬁrst alternative allows estimation of the nodes entanglement network constant based on the A value, calculated using an experimental 2C1 value and Equation 1.56. The second alternative provides for quantitative estimation of functionality F [57]. Table 1.5 shows the results of calculation of the value of A at conditions ϕnet = 0.3 and 1.0 for HDPE and ϕnet = 0.55 and 1.0 for LDPE, from which it follows that the use of the condition ϕnet = 1.0 gives A = 0.33–0.48, i.e., physically meaningless values, whereas application of conditions ϕnet = 0.30 and ϕnet = 0.55 provides for A values in the expected range of 0.5–1.0. In other words, the calculation results indicate the validity of the choice of the condition in which the network is represented by an amorphous phase of semi-crystalline polymer and, consequently, there are no grounds for taking crystallites as nodes of a macromolecular entanglement network [57].

Table 1.5 Front-factor A and functionality F calculated for HDPE and LDPE [57] Polymer HDPE

LDPE

ϕnet

A

F

0.30

0.75

8.0

1.0

0.33

3.0

0.55

0.64

5.6

1.0

0.48

3.8

In Figure 1.33 the dependence of the front-factor A value on molecular weight Mcl is shown. As follows from the adduced plot, the cluster ﬂuctuations constraint is systematically changed from 1.0 at Mcl = 0 to 0.5 at Mcl = 3600 g/mole. In other words, if the amorphous phase of a semi-crystalline polymer represents a single cluster (supercluster), ﬂuctuations are completely suppressed, and its behaviour corresponds to the ‘afﬁne’ model [110]. The value Mcl = Me = 3600 g/mole corresponds to polyethylene melt [20], where constraints imposed by crystallites are absent, and in this case entanglement network behaviour for polyethylenes corresponds to the ‘phantom’ alternative [113].

51

Structure and Properties of Crosslinked Polymers

A 1,0 -1 -2

0,8 0,6 0,4 0

1000 2000 3000 4000

Mcl, g/mol

Figure 1.33 The dependence of the front-factor A value on molecular weight Mcl of a chain part between clusters for (1) LDPE and (2) HDPE [57]

Let us note that according to Equation 1.57 an increase in A is equivalent to raising the cluster functionality F. As Flory has shown [114], the mean square of ﬂuctuations of the chain vectors , induced by joint ﬂuctuations, is given as follows:

(1.58)

where 〈r2〉0 is the mean-square distance between the chain ends. As follows from Equation 1.58 ﬂuctuations in the system must be suppressed with an increase in F, which is supposed from the data of Figure 1.33. Let us also note that the condition F = 2 (or ncl = 1), which determines decay of clusters, occurs at = 〈r2〉0, i.e., in the case of equality of the mean-square ﬂuctuations of chain vectors and the distance between chain ends. As a matter of fact, this condition is the thermoﬂuctuational criterion of the cluster formation, another alternative of which is considered in Section 1.4 (refer to Table 1.3). Thus the results obtained within the frameworks of the rubber high-elasticity theory suggest that the macromolecular entanglement network in semi-crystalline polymers is

52

The Main Principles of the Cluster Model limited by non-crystalline regions, and this network node represents clusters consisting of 2–9 densely packed collinear segments [57].

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The Main Principles of the Cluster Model 31. J.F. Fellers and D.C. Huang, Journal of Applied Polymer Science, 1979, 23, 8, 2315. 32. G.V. Kozlov and D.S. Sanditov in Anharmonic Effects and Physicomechanical Properties of Polymers, Nauka, Novosibirsk, Russia, 1994, p.261. 33. Y-H. Lin, Macromolecules, 1987, 20, 12, 3080. 34. A.A. Katsnelson in Introduction to Physics of Solids, Publishers of MSU, Moscow, Russia, 1984, p.293. 35. V.N. Shogenov, V.N. Belousov, V.V. Potapov, G.V. Kozlov and E.V. Prut, Vysokomolekulyarnye Soedineniya Seriya A, 1991, 33, 1, 155. 36. O.V. Startsev, Y.A. Abeliov, V.N. Kirillov and M.G. Voronkov, Doklady Akademii Nauk SSSR, 1987, 293, 6, 1419. 37. M. Sumita, Y. Tsukumo, K. Miyasaka and K. Ishikawa, Journal of Material Science, 1983, 18, 5, 1758. 38. N.I. Mashukov, V.N. Belousov, G.V. Kozlov, E.N. Ovcharenko and G.P. Gladyshev, Izvestiya Akademii Nauk SSSR: Seriya Khimicheskaya, 1990, 9, 2143. 39. M. Schrager, Journal of Applied Polymer Science, 1978, 22, 8, 2379. 40. A.K. Malamatov and G.V. Kozlov, Doklady Adygskoi (Cherkesskoi) Mezhdunarodnoi Akademii Nauk, 1998, 3, 2, 78. 41. G.V. Kozlov and G.E. Zaikov in Cluster Model Interrelation with Modern Physical Concepts, Nova Science Publishers, Inc., New York, NY, USA, 2009, p.104. 42. A.S. Balankin in Synergetics of Deformable Body, Publishers of Ministry Defence SSSR, Moscow, Russia, 1991, p.404. 43. A.S. Balankin, A.L. Bugrimov, G.V. Kozlov, A.K. Mikitaev and D.S. Sanditov, Doklady AN, 1992, 326, 3, 463. 44. G.P. Gladyshev in Thermodynamics and Macrokinetics of Natural Hierarchic Processes, Nauka, Moscow, Russia, 1988, p.290. 45. G.P. Gladyshev, Journal of Theoretical Biology, 1978, 75, 4, 425.

55

Structure and Properties of Crosslinked Polymers 46. G.P. Gladyshev and D.P. Gladyshev in On Physicochemical Theory of Biological Evolution (Preprint), Olimp, Moscow, Russia, 1993, p.24. 47. G.P. Gladyshev, Journal of Biological Systems, 1993, 1, 2, 115. 48. G.P. Gladyshev and D.P. Gladyshev, Zhurnal Fizichskoi Khimii, 1994, 68, 5, 790. 49. G.P. Gladyshev, Izvestiya Akademii Nauk, Seriya Biologicheskaya, 1995, 1, 5. 50. G.V. Kozlov and G.E. Zaikov in Fractal Analysis of Polymers: From Synthesis to Composites, Eds., G. Kozlov, G. Zaikov and V. Novikov, Nova Science Publishers, Inc., New York, NY, USA, 2003, p.89. 51. V.A. Beloshenko, G.V. Kozlov and Y.S. Lipatov, Fizika Tverdogo Tela, 1994, 36, 10, 2903. 52. S. Matsuoka and H.E. Bair, Journal of Applied Physics, 1977, 48, 10, 4058. 53. S. Wu, Journal of Applied Polymer Science, 1992, 46, 4, 619. 54. V.P. Privalko and Y.S. Lipatov, Vysokomolekulyarnye Soedineniya Seriya A, 1971, 13, 12, 2733. 55. G.P. Gladyshev, Journal of Biological Physics, 1994, 20, 2, 213. 56. V.N. Belousov, G.V. Kozlov and Y.S. Lipatov, Doklady Akademii Nauk SSSR, 1991, 318, 3, 615. 57. G.V. Kozlov, V.Z. Aloev and Y.S. Lipatov, Ukrainskii Khimicheskii Zhurnal, 2001, 67, 10, 115. 58. J.E. Mark and J.L. Sullivan, Journal of Chemical Physics, 1977, 66, 3, 1006. 59. R.F. Boyer, Journal of Macromolecular Science: Physics, 1976, B12, 2, 253. 60. R.F. Boyer, Macromolecules, 1992, 25, 20, 5326. 61. R.M. White and T.H. Geballe in Long Range Order in Solids, Academic Press, New York, NY, USA, 1979, p.447. 62. G.V. Kozlov, L.D. Mil’man and A.K. Mikitaev in Manuscript Deposited to Viniti Ras, Moscow, Russia, 1997, p.622.

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The Main Principles of the Cluster Model 63. A.M. Lobanov and S.Y. Frenkel, Vysokomolekulyarnye Soedineniya Seriya A, 1980, 22, 5, 1045. 64. R.F. Boyer, Polymer Engineering and Science, 1968, 8, 3, 161. 65. N.I. Mashukov, V.D. Serdyuk, G.V. Kozlov, E.N. Ovcharenko, G.P. Gladyshev and A.B. Vodakhov in Stabilization and Modiﬁcation of Polyethylene by Oxygen Acceptors (Preprint), IKhF Akademii Nauk SSSR, Moscow, Russia, 1990, p.64. 66. V.N. Belousov, G.V. Kozlov and N.I. Mashukov, Doklady Adygskoi (Cherkesskoi) Mezhdunarodnoi Akademii Nauk, 1996, 2, 1, 74. 67. M. Zhenyi, S.C. Langford, J.T. Dickinson, M.H. Engelhard and D.R. Boyer, Journal of Materials Response, 1991, 6, 1, 183. 68. V.U. Novikov and G.V. Kozlov, Materialovedenie, 1999, 12, 8. 69. R.F. Boyer and R.L. Miller, Macromolecules, 1977, 10, 5, 1167. 70. R.L. Miller and R.F. Boyer, Journal of Polymer Science, Part B: Polymer Physics Edition, 1984, 22, 12, 2043. 71. S.M. Aharoni, Macromolecules, 1985, 18, 12, 2624. 72. G.V. Kozlov, V.A. Beloshenko and V.A. Lipskaya, Ukrainskii Fizicheskii Zhurnal, 1996, 41, 2, 222. 73. S. Matsuoka, C.J. Aloisio and H.E. Bair, Journal of Applied Physics, 1973, 44, 10, 4265. 74. B.K. Sharma, Acoustic Research Letters, 1980, 4, 2, 19. 75. G.V. Kozlov and G.E. Zaikov, Izvestiya KBSC RAS, 2003, 9, 132. 76. B.K. Sharma, Acustica, 1981, 48, 2, 121. 77. R.E. Barker, Journal of Applied Physics, 1963, 34, 1, 107. 78. D.S. Sanditov and G.M. Bartenev in The Physical Properties of Disordered Structures, Nauka, Novosibirsk, Russia, 1982, p.256. 79. D.S. Sanditov in Nonlinear Effects in Fracture Kinetics, Publishers FTI Akademii Nauk SSSR, Leningrad, Russia, 1988, p.140.

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Structure and Properties of Crosslinked Polymers 80. V.P. Privalko and Y.S. Lipatov, Vysokomolekulyarnye Soedineniya Seriya A 1974, 16, 7, 1562. 81. R.F. Boyer, Journal of Macromolecular Science: Physics, 1973, 7B, 3, 487. 82. W.C. Forsman, Macromolecules, 1982, 15, 6, 1032. 83. E. Hornbogen, International Materials Review, 1989, 34, 6, 277. 84. J.J. Curro and R.-J. Roe, Polymer, 1984, 25, 10, 1424. 85. J. Rathje and W. Ruland, Colloid Polymer Science, 1976, 254, 3, 358. 86. E.L. Kalinchev and M.B. Sakovtseva in Properties and Processing of Thermoplastics, Khimiya, Leningrad, Russia, 1983, p.288. 87. N.I. Mashukov, V.D. Serdyuk, V.N. Belousov, G.V. Kozlov and M.A. Khatsukova, Manuscript Deposited to Viniti Ras, Moscow, Russia, 1994, 1537, 94. 88. V.N. Belousov, G.V. Kozlov and N.I. Mashukov, Doklady Adygskoi (Cherkesskoi) Mezhdunarodnoi Akademii Nauk, 1995, 2, 2, 76. 89. V.D. Serdyuk, D.S. Sanditov, N.I. Mashukov, G.V. Kozlov, V.U. Novikov and G.E. Zaikov, International Journal of Polymer Materials, 1998, 42, 1, 65. 90. V.V. Afaunov, N.I. Mashukov, G.V. Kozlov and D.S. Sanditov, Izvestiya Vuzov, Severo-Kavkazskii Region, Estestvennye Nauki, 1999, 4, 69. 91. N.I. Mashukov, V.D. Serdyuk, G.P. Gladyshev and G.V. Kozlov, Voprosy Oboronnoi Techniki, 1991, 15, 3, 97, 11. 92. G.K. El’yashevich, Vysokomolekulyarnye Soedineniya Seriya A, 1988, 30, 8, 1700. 93. G.V. Kozlov, V.N. Shogenov, A.M. Kharaev and A.K. Mikitaev, Vysokomolekulyarnye Soedineniya Seriya B, 1987, 29, 4, 311. 94. H.-J. Bittrich, Acta Polymerica, 1982, 33, 12, 741. 95. G. Adam and J.H. Gibbs, Journal of Chemical Physics, 1965, 43, 1, 139. 96. G.V. Kozlov and Y.S. Lipatov, Vysokomolekulyarnye Soedineniya Seriya B, 2003, 45, 4, 660.

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The Main Principles of the Cluster Model 97. G.V. Kozlov and Y.S. Lipatov in Perspectives on Chemical and Biochemical Physics, Ed., G. Zaikov, Nova Science Publishers, Inc., New York, NY, USA, 2002, p.205. 98. G.V. Kozlov, V.N. Belousov, V.D. Serdyuk, A.K. Mikitaev and N.I. Mashukov, Doklady Adygskoi (Cherkesskoi) Mezhdunarodnoi Akademii Nauk, 1997, 2, 2, 88. 99. G.V. Kozlov, V.N. Belousov, A.K. Mikitaev and N.I. Mashukov, Doklady Adygskoi (Cherkesskoi) Mezhdunarodnoi Akademii Nauk, 1997, 2, 2, 94. 100. S.M. Aharoni, Macromolecules, 1983, 16, 9, 1722. 101. P.J. Mills, J.N. Hay and R.N. Haward, Journal of Material Science, 1985, 20, 2, 501. 102. D.C. Prevorsek, B.T. De Bona, Journal of Macromolecular Science: Physics, 1981, B19, 4, 605. 103. G.V. Kozlov, V.U. Novikov and G.E. Zaikov, Plasticheskie Massy, 2002, 5, 33. 104. G.V. Kozlov, K.B. Temiraev, A.Kh. Malamatov and G.B. Shustov, Izvestiya KBSC RAS, 1999, 2, 95. 105. N.I. Mashukov, V.D. Serdyuk, V.N. Belousov, G.V. Kozlov, E.N. Ovcharenko and G.P. Gladyshev, Izvestiya AN SSSR, Seriya Khimicheskaya, 1990, 8, 1915. 106. N.I. Mashukov, G.P. Gladyshev and G.V. Kozlov, Vysokomolekulyarnye Soedineniya Seriya A, 1991, 33, 12, 2538. 107. V.D. Serdyuk, G.V. Kozlov, N.I. Mashukov and A.K. Mikitaev, Journal of Materials Science and Technology, 1997, 5, 2, 55. 108. W.R. Krigbaum, R.-J. Roe and K.J. Smith, Polymer, 1964, 5, 3, 533. 109. G.V. Kozlov and G.E. Zaikov, Izvestiya KBSC RAS, 2003, 9, 126. 110. J. Sanjuan and M.A. Llorence, Journal of Polymer Science, Part B: Polymer Physics Edition, 1988, 26, 2, 235. 111. R. Popli and L. Mandelkern, Journal of Polymer Science, Part B: Polymer Physics Edition, 1990, 28, 11, 1917.

59

Structure and Properties of Crosslinked Polymers 112. C.Y. Jiang, L. Carrido and J.E. Mark, Journal of Polymer Science, Part B: Polymer Physics Edition, 1984, 22, 12, 2281. 113. J.R. Falender, G.S.Y. Yeh and J.E. Mark, Journal of Chemical Physics, 1979, 70, 11, 5324. 114. P.J. Flory, Journal of Chemical Physics, 1977, 66, 12, 5720.

60

2

The Main Physical Concepts used in Fractals Theory

The fractals theory and its application to various physical and chemical processes have recently undergone a large amount of development [1–7]. For simpliﬁcation of understanding of the results represented in subsequent chapters some main notions and deﬁnitions are brieﬂy considered and reasons for the application of fractal analysis (and connected with it irreversible aggregation models) for description of the structure and properties of polymer materials and composites on this basis are shown.

2.1 The Fractal Analysis of Polymeric Media Self-similar objects, invariant about local dilatations, i.e., objects which in observation processes at various magniﬁcations repeat the same form, are called fractals. Mandelbrot [1] introduced the notion of fractals as self-similar sets, deﬁning a fractal as a set for which the Hausdorff–Bezikovich dimension always exceeds the topological dimension. The fractal dimension df of the object, adopted in d-dimensional Euclidean space, varies from 1 to d. Fractal objects are natural ﬁllings of sets between known Euclideans with whole number dimensions 0, 1, 2, 3, … The majority of objects existing in nature turn out to be fractal ones, which is the main reason for the vigorous development of fractal analysis methods. According to Family’s classiﬁcation [8], fractal objects can be divided into two main types: deterministic and statistical. The deterministic fractals are self-similar objects, which are precisely constructed on the basis of some basic laws. Typical examples of such fractals are the Cantor set (‘dust’), the Koch curve, the Serpinski carpet, the Vichek snowﬂake and so on. The two most important properties of deterministic fractals are the possibility of precise calculation of their fractal dimension and the unlimited range (– ∞; +∞) of their self-similarity. Since a line, plane or volume can be divided into an inﬁnite number of fragments by various modes then it is possible to construct an inﬁnite number of deterministic fractals with different fractal dimensions. In this connection the deterministic fractals are impossible to classify without introduction of their other parameters in addition to the fractal dimension. Statistical fractals are raised by unordered (random) processes. A disorder element is

61

Structure and Properties of Crosslinked Polymers typical for the majority of physical phenomena and objects. The fact of sufﬁciency of the fulﬁlment of the disorder condition for formation of fractals was ﬁrst noted by Mandelbrot [1]. A statistical walk is a typical example of such a fractal. However, purely statistical models often inadequately describe real physical systems. One cause of this is an excluded volume effect. The essence of this effect consists in a geometrical restriction, forbidding two different system elements to occupy the same space volume. This restriction should be taken into account in corresponding model constructions [9]. The best known examples of such models are walks without self-intersections, ‘lattice animals’ and statistical percolation. In the range of deﬁnite scales fractals have different topological structures depending on the maximum number of elements that are connected with the given system element. If each element can be connected, as a minimum, with two other ones, then the received structure has no branches. By analogy with linear polymers Family [8] calls this type of fractal linear. If branching occurs, then the resulting fractal has a network-like structure; this type of fractal was called a branched fractal. A polymer’s microstructure can possess a high degree of natural or artiﬁcially created self-ordering [10], which is one limiting case. The other limiting case is chaos, as opposed to order. As a rule, such systems were obtained in conditions far from thermodynamical equilibrium; they ﬁll the gap between periodic structures and completely unordered systems [11]. In other words, fractal structures should possess a deﬁnite level of intermediate order, therefore in studying thermodynamically nonequilibrium solid media (which, as a rule, polymers are [12]) possessing local order [13], the question about interrelation of structures of local order level and their fractality degree is very important. At present there can be no doubt about the fact that solid medium approximation cannot serve as an adequate model for real polymers [14]. Just as in the synthesis of polymers a large number of micro-, mezo- and macrodefects appears in them, which can develop in the exploitation process of products from polymers. What is more, it is empirically established that even at initially homogeneous media deformation they acquire hierarchical block structure, characteristic space scales of which Li satisfy the relationship with a sufﬁcient degree of precision:

(2.1)

where ΛiLVWKHDXWRPRGHOLW\FRHI¿FLHQWi = 0, 1, 2, … [15, 16].

62

The Main Physical Concepts used in Fractals Theory One of the main characteristics of solid body structure features is the Euclidean dimension d, which can accept the following values: d = 0 – point defects, d = 1 – linear defects (dislocations), d = 2 – planar defects (grains boundaries, doubles and so on), d = 3 – three-dimensional (spatial) formations. The Euclidean dimensions can serve as characteristics of high-ordered symmetrical microstructures, which are not often formed even in the materials obtained in quasi-equilibrium conditions. Nonequilibrium systems, prepared in highly non-equilibrium conditions and presenting themselves as a peculiar replica of dynamic dissipative structures, cannot be described adequately within the frameworks of metal- and X-ray studies in general [15–19]. This is all the more true for solid-like or glassy polymers – thermodynamically non-equilibrium systems by deﬁnition. So, the atomic structure of non-equilibrium materials can have quasi-crystalline order possessing symmetry elements of the ﬁfth, seventh, thirteenth and higher orders [19] forbidden by the Brave theorem, lying in the basis of classical methods of X-ray analysis. Moreover numerous fractographic [5, 17, 19] and geophysical [16] studies indicate in many materials structure fractality and the essentially non-Euclidean geometry of the fracture and deformation of solid bodies. By virtue of this traditional solid medium, mechanics methods, based on the strains homogeneity assumption, considered as reﬂections of deforming body samples in Euclidean space, cannot describe adequately the rheological behaviour and fracture of real materials, in particular polymers displaying such properties as high elasticity and viscoelasticity. The essential progress in the solution of this problem is connected with synergetics (synergetics is a border ﬁeld of knowledge, devoted to the general laws revealing the formation, stability and decay processes of ordered temporal and spatial structures in complex non-equilibrium systems of various natures [4]) and fractal analysis ideas and methods using [4, 5, 7, 18, 19]. Speciﬁcally, it has been shown that at ﬁrst sight absolutely unordered chaotic systems, forming in non-equilibrium conditions (amorphous phases, fracture surfaces and so on), display actually ordering peculiar elements [2, 18]. If crystals are characterised by deﬁnite symmetry and translational invariance, then non-equilibrium structures can possess enantiomorphism properties [4], even if they do not possess quasi-crystalline structure; these systems are scaleinvariant in the deﬁnite self-similarity range [1, 2]. The so-called Renie dimensions dq serve as a self-similarity quantitative characteristic, which, unlike the topological dimension, can have both even and fractional magnitudes [1], determined by the relationships:

63

Structure and Properties of Crosslinked Polymers

⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪⎩

⎡ ⎢ ⎢ ⎢ ⎢⎣ ⎡ ⎢ ⎢ ⎢ ⎢⎣

⎤ ⎥ ⎥ ⎥ ⎥⎦ ⎤ ⎥ ⎥ ⎥ ⎥⎦

(2.2)

where M is a minimum number of d-dimensional cubes with the side ε, necessary for covering all structure elements; Pi(ε) is the probability of the event that a structure point belongs to the i-th volume εd covering the element; q is a measure transformation parameter (‘magniﬁcation parameter’). For Euclidean objects (smooth curves, regular lattices on planes and in volumes and so on) the identity is carried out [14]:

(2.3)

For regular mathematical fractals of Cantor sets, Koch curves and Serpinski carpets, constructed by recurrent procedures, the Renie dimension dq does not depend on q, but on [14]:

(2.4)

where dH is the Hausdorff–Bezikovich dimension (Hausdorff dimension). According to the mathematical deﬁnition [3], the Hausdorff dimension dH is a local characteristic of the set in the chosen scales range, which can be covered with ‘balls’ not necessarily of the same size under the condition that all ‘balls’ diameters are smaller than the chosen diameter. The structure fractal dimension df is a physically determined value within the scale range in which structure elements are self-similar,

64

The Main Physical Concepts used in Fractals Theory i.e., they are fractals. Since df and dH scale ranges coincide, then df = dH is called the structure fractal (Hausdorff) dimension. Natural fractals, such as clouds, polymers, aerogels, porous media, dendrites, cracks, solid fracture surfaces and so on, possess only statistical self-similarity, which takes place only in the restricted range of the spatial scales [1–3, 14]. For solid-like polymers it has been experimentally shown [20] that such a range spreads from several angstroms up to several tens of angstroms. The interrelation of local order and the fractality degree of disordered solids can be described by common mathematical terms. Speciﬁcally, speaking about the structure of solid polymers, one should note that the majority of researchers assume the availability in them of segmental scale structural formations has been proved, although the packing concrete type in these formations remains debatable [13, 21]. One should underline that the notion about an amorphous state being an absolutely disordered one is incorrect. According to the Ramsey theorem [22], any sufﬁciently large number i > R(i, j) of points or objects (in the considered case – structure elements) contains without fail a high-ordered subsystem from Nj ≤ R(i, j) elements. Therefore absolutely disordered systems (structures) do not exist. It has been similarly shown that any structure consisting of N elements at Nj > BN(j) presents itself totally of a ﬁnite number k ≤ j of self-similar structures inserted into each other, the Hausdorff dimension of which in the general case can be different. This means that any system independent of physical nature, consisting of a sufﬁciently large number of elements, is multifractal (in the special case – fractal) and is characterised by Renie dimensions dq spectrum, q = –∞ − ∞ [23]. The tendency of the condensed systems to self-organisation in scale-invariant multifractal forms is the consequence of the main principles of open systems thermodynamics and dq are deﬁned by competition between short- and longrange interatomic interactions, deﬁning bulk compressibility and shearing stiffness of solids, respectively [24]. Another very important property of fractals, distinguishing them from traditional Euclidean objects, is the necessity of at least three dimension determinations: d – seized Euclidean space dimension, df – fractal (Hausdorff) dimension and ds – spectral (fracton) dimension, characterising object connectivity. For Euclidean spaces d = df = ds and this circumstance allows Euclidean objects to be considered as special (‘degenerated’) cases of fractal objects. We will meet the given rule herein again and again [25]. This means that for fractal objects (for example, polymers) structure description in two fractal dimensions (df and ds) is required even at the ﬁxed d value. Such a situation corresponds to the non-equilibrium thermodynamics rule, according to which for thermodynamically non-equilibrium solids (polymers), for which the Prigogine–Defay criterion is not fulﬁlled, the description of two order parameters, as a minimum, is necessary [12].

65

Structure and Properties of Crosslinked Polymers Fractal objects are characterised by the following relationship between mass M (or density ρ) and measure linear scale L [23]:

(2.5)

where dm is the mass scaling exponent. Unlike mathematical fractals, real fractals (including polymers) have two natural length scales Lmin and Lmax (Figure 2.1); objects below and above are not fractal [23]. The lower limit Lmin is connected with the ﬁnite size of the structural elements and the upper one Lmax with uneven aspiration for the limit df. As was noted above, for polymers it was experimentally stated [20, 26, 27] that Lmin has an order of several angstroms and Lmax of several tens of angstroms.

OQρ

LPLQ

Lmax

OQ L

Figure 2.1 The dependence of the density ρ on the linear scale L of a real fractal. The range Lmin–Lmax is the region of object fractal behaviour [23]

In polymer research, situations often occur when strictly derived relationships describe high-elastic media, for example, elastomers, which are usually analysed at

66

The Main Physical Concepts used in Fractals Theory temperatures higher than their glass transition temperature Tg, but do not describe the behaviour of polymers, which are in a glassy state. At ﬁrst it was explained by the sharp reduction in the lower Tg of the molecular mobility of the chains or by structure ‘freezing’ [28]. Strictly speaking, the principal distinction between higher and lower Tg of the molecular structure does exist –in both cases it consists of long-chain macromolecules [29]. The distinction consists in the transition of a polymer from a thermodynamically non-equilibrium state with lower Tg to a quasi-equilibrium state with higher Tg. Within the frameworks of fractal analysis this means that a polymer structure at T ≥ Tg ceases to be fractal and becomes a Euclidean body (or, at any rate, approaches it sufﬁciently closely). According to [30], an ability to sustain a large deformation with the following complete restoration of characteristics after stress removal is a property that actually all polymers, consisting of long-chain macromolecules, show at corresponding conditions. The given property (stresses relaxation) in one way or another is found beyond temperature range bounds, corresponding to displaying high-elasticity. In other words, a polymer’s macromolecular nature exercises a prevalent inﬂuence on its physical properties at deformation, forcing the medium to display such qualities as yielding, high-elasticity and vitreousness. In virtue of this it can be supposed that the same fractal dimension can be used for object description in different physical states [29, 30]. It is known that polymers possess multi-level different scales structure (molecular, topological, supermolecular, ﬂoccular or block levels), elements of which are interconnected [7, 43]. As a result of a force on a polymer new (secondary) structural elements can be formed – cracks, fracture surfaces, plastic deformation zones and so on. The indicated different structural elements and also the processes forming them are characterised by heterogeneous parameters, therefore up to now only empirical correlations between them were established. If each of the indicated elements (processes) is characterised by a uniform parameter, for example, by fractal dimension, then analytical relationships connecting them, not containing ﬁtting parameters, can be obtained. It is important for properties and behaviour prediction of high-molecular compounds in exploitation. Let us note that fractal analysis was applied and turned out to be useful in high-elasticity [14, 32] and yielding [22, 33–35] phenomena descriptions. It is obvious that for the correct use of methods based on fractal dimension estimation it is necessary to apply physically well-founded parameters describing a polymer’s structure. In this sense Euclidean and fractal objects are principally different: for the ﬁrst only one dimension of space (Euclidean) is required and for fractal objects no fewer than three dimensions [25].

67

Structure and Properties of Crosslinked Polymers

2.2 The Fractal Models of Polymer Medium Structure The cluster model will be used below in structure analysis of amorphous polymers as a local order model [13, 36, 37]. This model establishes that local order domains (clusters), immersed in a loosely packed matrix, consist of several densely packed collinear segments of different macromolecules, i.e., they are amorphous analogues of a crystallite with drawing chains (CDC). The length of the segments which the cluster consists of is accepted to be equal to a polymer statistical segment length lst. At the same time clusters play a role of multifunctional nodes of physical entanglements network with density νcl. It is accepted in the ﬁrst approximation that the number of segments in clusters per polymer volume unit is equal to νcl. This assumption allows determination of the relative fraction of clusters ϕcl [37]. The main factor that deﬁnes interconnection of local order and the fractal nature of the structure of solid polymers is the fact that both these features are a reﬂection of the key property of these polymers– their thermodynamical non-equilibrium nature. The scales of fractal behaviour Lmin and Lmax indicated above correspond very well to cluster structure border sizes: Lmin – to statistical segment length lst, Lmax – to distance between clusters Rcl [38]. However, one should not forget that fractal analysis gives only a common mathematic description of a polymer’s structure, i.e., it does not identify those structural units (elements) that any real polymer consists of. The cluster model of polymer amorphous state structure allows one to obtain a physical description of a thermodynamically nonequilibrium polymer’s structure with local (short-range) order representations drawing and molecular characteristics usage, which identiﬁes its element quantitatively. Since these models consider polymer structure from different positions, they are a very good complement of one another [7, 29]. As it is known [39], structures that are fractal on small scales and homogeneous on large scales are called homogeneous fractals. Such fractals are percolation clusters at the percolation threshold. As it was shown in papers [40, 41], the cluster structure is a percolation system and by virtue of the above is a homogeneous fractal. In other words, the availability of local order in polymers, in a condensed state, is deﬁned by their structure fractality. The structure fractal dimension df can be expressed as follows [39]:

(2.6)

68

The Main Physical Concepts used in Fractals Theory where b and n are critical indices, the deﬁnitions of which were introduced in percolation theory [39]. Hence, the condition β ν 0, which follows from the percolation theory [39] and experimental data for cluster structure of polymers [40, 41], also deﬁnes the fractality of the structure of polymers. It is obvious that the indicated condition and, consequently, the fractality of the structure of polymers deﬁne the dependence of ϕcl on the testing temperature, i.e., local order of thermal ﬂuctuation nature. In other words, the structure fractality and local order ‘frozen’ lower than Tg are interexcluded notions. The rules considered above allow quantitative interconnection of fractal dimension df and local order characteristic ϕcl (or νcl) to be obtained. The analytical relationship between indicated parameters has the following form [7]:

(2.7)

where l0 is the polymer main chain skeletal bond length. In Equation 2.7 l0 is given in m. Equation 2.7 gives two forms of analytical relation between df and ϕcl (or νcl), the physical grounds for which were considered above. Let us note the important conclusion, following from Equation 2.7. The fractal dimension df depends not only on the order parameter value ϕcl, but also on the polymer’s molecular characteristics (C∞ and S), where C∞ is the characteristic ratio, which is an indicator of polymer chain statistical ﬂexibility [42], S is a macromolecule cross-sectional area. This is the reﬂection of the above-mentioned rule, that for description of a polymer’s condensed state structure, as a minimum two order parameters are required [12, 25]. An experimentally proved solid polymers structure fractality serves as another reason in favour of the local order model. As follows from Figure 2.1 data, fractality requires density ρ gradient on deﬁnite length scales L (Lmin–Lmax). The cluster model corresponds to this requirement, since the density of clusters is higher than that of the loosely packed matrix, which ensures ρ as a necessary gradient. Flory’s ‘felt’ model [43], which assumes that polymer structure is presented as chaotically tangled statistical coils, denies local order availability. It is obvious that such a model does not force the ρ gradient and, hence, does not correspond to a polymer’s experimentally stated structure fractality. Therefore, a close interconnection exists between notions of local order and fractality

69

Structure and Properties of Crosslinked Polymers in the case of a polymer’s condensed state having fundamental physical grounds and expressed by a simple analytical relation.

2.3 Polymer Medium with Scaling Theory Positions Researchers of different trends have various scaling notions (scaling invariance). The authors of the present work mainly share de Gennes’ point of view [44], which assumes that it is necessary, as far as it is possible, to distract from the considered system structure details and to select simple universal features, which are typical for a wide class of systems. A scaling law always deﬁnes only some asymptotics, the applicability of which, taking into account system speciﬁc features, it is necessary to analyse concretely for each case. By his works de Gennes offers a new level of study of a polymer’s structural-physical properties, as Flory had done before [45]. Fractal dimension can serve as a scaling characteristic (scaling index) example, which as has been shown for regular fractals, depends on the system formation mechanism only and deﬁnes the global structure of the system, the mass distribution depending on scale. The universality hypothesis is closely connected with the scaling hypothesis, although if it is to be approached more strictly, they should be considered as independent suppositions [46]. The universality hypothesis essentially states that if the same limiting conditions are typical (system parts interactions) for different mechanisms of the formation of systems, then these systems come to be in one universal class of physical phenomena. Such a class is characterised by a deﬁnite set of scaling constants (indices). The fact that universality classes contain not only physical systems (sometimes those that belong to different natural sciences ﬁelds) but also theoretical models gives the possibility of simulation and imitation on a wide range of computer applications. It should be noted that the universality of different classes can also have different forms of scaling laws – different types of relations, describing the system and the equations – connecting scaling constants between them [47]. One of the examples of the scaling representation of macromolecules is the ‘reptation’ model [48], according to which the ‘tube’ diameter Φ, in which the macromolecule is conﬁned (equal to the distance between entanglements nodes), can be estimated according to the relationship [49]:

(2.8) where Men is the molecular weight of the chain part between entanglements nodes, m0 is the molecular weight of the main chain skeletal bond.

70

The Main Physical Concepts used in Fractals Theory The main distinction between physical entanglements (‘sweeps’) nodes and chemical crosslinking nodes is chains sliding in the former. The density of both can be determined according to the high-elasticity theory and for calculation of chains sliding in entanglements nodes the coefﬁcient 0.8 is added [49]. Another variant of macromolecular chain simulation is considered as a fractal below. In paper [50] it has been noted that with reference to the condensed media the fractal concept is used, but not the observed geometrical image description. As for the macromolecule, simulated by a chain with the ﬁxed valent angles or with the braced internal rotation [42] and consisting of a statistical segment, it is difﬁcult to ﬁnd a more visual geometrical image for it, than a fractal broken line [29], the more so, such chains self-similarity is strictly proved [51]. It is also necessary to note that macromolecule presentation by both one-dimensional (in a form of a curve line) and three-dimensional (in a form of a cylinder) objects looks like a crude approximation, particularly if its real structure is taken into account – side branching availability, ﬂexible bonds, existence of stiff segments and so on. For crosslinking epoxy polymers series with varied ratio of hardener reactive groups and epoxy oligomer amounts Kst within the range of 0.5–1.5 chemical crosslinking nodes, the change in density νc results in a characteristic extreme change in ratio C∞ with a minimum at Kst = 1.0 (the stoichiometric ratio) [52]. To simulate the macromolecule part between chemical crosslinking nodes within the frameworks of the models indicated above [42], then using the known values of this part length Le and lst, the typical fractal dependence can be obtained [29]:

, ξ<1

(2.9)

where Nst is the number of statistical segments per macromolecule fragment with molecular weight Mc, ξ is the similarity parameter, D is the fractal dimension of a chain part between chemical crosslinking nodes. At the conditions Lc = const. and Kst = 1.0, ξ = 0.85 was obtained; at Lc = const. and Kst > 1.0, ξ = (0.85)1/2 = 0.926 was found. Estimation of dimension D gave the value 1.17. Hence, at the condition indicated above, the fulﬁlment macromolecule part between chemical crosslinking nodes can be presented as a fractal [29]. With variation in Kst the crosslinking density νc changes and, as a consequence, Lc also changes. This results in the fact that the chain is not quite a self-similar fractal. Nevertheless, it can be simulated by a heterogeneous fractal with the same fragmentation step, but with variable D value, determined according to the known Richardson equation [23]:

71

Structure and Properties of Crosslinked Polymers

(2.10)

where Rc is the distance between crosslinking nodes. The values calculated by the indicated mode D are listed in Table 2.1:

Table 2.1 The calculated molecular characteristics of epoxy polymers [7] Polymer

EP-1

EP-2

Kst

D

C¥ Data of paper [52]

Calculation according to Equation 2.8

Calculation according to Equation 2.14

0.50

2.06

5.23

4.83

5.28

0.75

1.60

4.32

6.97

3.50

1.0

1.25

3.38

9.94

3.68

1.25

1.35

3.75

8.89

3.19

1.50

1.50

4.05

7.68

3.34

0.50

1.85

5.27

6.54

5.35

0.75

1.43

3.98

8.89

4.69

1.0

1.32

3.40

9.24

3.37

1.25

1.35

3.70

9.24

3.90

1.50

1.55

4.10

7.96

4.29

The results of calculation of the characteristic ratio C∞ according to Equation 2.8 do not qualitatively correspond to the paper [52] data: at Kst = 1.0 the value of C∞ is a maximum, not a minimum. Since:

(2.11)

72

The Main Physical Concepts used in Fractals Theory

then, according to Equation 2.8, the C∞ value should be proportional to ν c . In other words, an increase in the density of crosslinking nodes should result in chain ﬂexibility enhancement that does not correspond to the extreme growth of epoxy polymers glass transition temperature [52, 53]. Let us consider the causes of the mentioned disparity. Since: 1/ 2

(2.12)

2

then, multiplying and dividing Equation 2.8 on lst we will obtain the relationship [7]:

(2.13)

which is similar to Equation 2.10. Comparison of Equations 2.10 and 2.13 shows that the scaling relationship in Equation 2.8 is a particular case of the fractal formula in Equation 2.10 with D = 2. For the considered epoxy polymers the condition D = 2 is not fulﬁlled (see Table 2.1) and this can be a cause of the found disparity. Using the relationship between Equations 2.8, 2.10 and 2.13 one can write the relationship in Equation 2.8 in a more common form [7]:

,1
(2.14)

Calculations of C∞ according to Equation 2.14 give values that correspond well to the data in paper [52]. For a linear polymer the macromolecular entanglements (‘sweeps’) network, the density of which νe is determined at T > Tg [49], is an analogue of the chemical bonds network. Therefore, the relationship found in Equation 2.8 is a particular case of a more common fractal relationship in Equation 2.14 and is applicable only to rubbers, for which it was actually derived [29].

73

Structure and Properties of Crosslinked Polymers

2.4 The Fractal Analysis in Molecular Mobility Description Questions Special attention is always paid to the questions of estimation of molecular mobility of polymer chains [54–56]. The reasons are obvious: thermodynamically non-equilibrium solid-like media, particularly relaxation media, and their physical properties are deﬁned by passing relaxation molecular processes in them, which in turn depend on features of the chemical constitution of the molecular chains and the structural organisation of the polymers [56]. As for parameters, there exist different points of view in describing these processes. So, for example, it is assumed that fast relaxations are deﬁned by the mobility of free chains placed between densely packed domains, which are at the same time nodes of macromolecules physical entanglements network. Such treatment corresponds to the main postulates of the cluster model of the structure of polymers in the amorphous state [13], with the aid of which structure elements can be quantitatively described. Let us note one more important aspect. The treatment of the structure of amorphous polymers adduced above belongs to elastomers [56]. Transference of these notions on amorphous glassy polymers assumes the description of densely packed domains ‘freezing’, i.e., a sharp increase in their life time. In addition, fractal forms of macromolecules (statistical macromolecular coils), formed in non-equilibrium physical-chemical processes, are preserved (‘frozen’) in polymers. This assumes that in a glassy state the mobility of chain parts between their ﬁxation points will be the main factor deﬁning molecular mobility [57]. The fractal dimension D in paper [57] is chosen as a structural indicator, characterising molecular mobility level for the following reasons. Firstly, between its ﬁxation points the molecular chain part possesses the self-similarity property and has a different dimension from its topological dimension, i.e., a fractal by deﬁnition [51]. Secondly, it has been shown [58] that the D value with the variation range of 1 < D ≤ 2 characterises precisely the molecular mobility of the chain part in loosely packed regions [56]. The condition D=1 supposes that this chain part loses its fractal nature and mobility loss. Within the frameworks of relaxation spectroscopy this means tg δ = 0, where tg δ is the tangent angle of the mechanical (dielectric) losses. The condition D = 2 supposes the maximum possible mobility of the chain part, corresponding to a high-elastic state of the polymer, i.e., it corresponds to maximum value tg δ at glass transition temperature Tg [29]. In Figure 2.2 the dependence of tg δ on D for the number of copolymers at a measurement frequency of 1 kc [57] is presented. The dependence has clearly deﬁned limits. For D = 1 it extrapolates to tg δ = 0. At D = 2 the tg δ value is approximately equal to the corresponding value at Tg. Thus, linear dependences with indicated

74

The Main Physical Concepts used in Fractals Theory limiting values tg δ can be used for prediction of fractal dimension [29]. Let us demonstrate the examples of dimension D usage for the applied problems settlement. The limiting draw ratio λlim of polymer tension within the framework of high-elasticity theory is determined according to the formula [59]:

(2.15) where nst is the number of statistical segments between entanglements nodes. This equation was applied repeatedly to glassy polymers as well. In paper [60] the following fractal relationship was obtained:

(2.16)

tgδ ⋅ 10

2

10 8 -1 -2 -3 -4 -5 -6

6 4 2

0

1,0

1,2

1,4

1,6 D

Figure 2.2 The dependence of the dielectric losses angle tangent tg δ on the fractal dimension D of the macromolecule part between entanglements for copolyethersulfoneformale with formal blocks contents: (1) 0; (2) 5; (3) 10; (4) 30; (5) 50; and (6) 70 mol%. The measurement frequency is equal to 1 kc [57]

75

Structure and Properties of Crosslinked Polymers It is easy to see that an increase in D at nst = const results in deformability growth of the polymers. Equations 2.15 and 2.16 are identical at D = 2. The important distinction of Equations 2.15 and 2.16 is the fact that λlim depends on one parameter in the ﬁrst equation, which is typical for equilibrium Euclidean objects, and in the second on two parameters, which is typical for thermodynamically non-equilibrium fractal objects, therefore Equation 2.15 using the latter is incorrect. In paper [61] the relationship between chain statistical ﬂexibility, characterised by C∞, and dimension D was deﬁned:

(2.17)

Using the literature data for C∞ [13, 62, 63] the lengths of chain parts between entanglements Le were calculated. The calculation results correspond to the experimental data (Table 2.2) [29]. Therefore, within the frameworks of fractal analysis an increase in network density with reduction in chain statistical ﬂexibility was obtained. The increase in the number of topological ﬁxation points of macromolecules in the glassy state in comparison with the high-elastic state can be predicted by using fractal analysis methods [29, 61]. The fractal dimension D of the chain part between its topological ﬁxation points (entanglements, clusters, crosslinking nodes) is the most important structural parameter, checking molecular mobility and deformability of polymers. Two of the main factors, due to application of dimension D, are its clearly deﬁned variation limits (1 < D ≤ 2) and the dependence on polymer supersegmental (supermolecular) structure. Let us especially note that all fractal relationships contain, at any rate, two variables. The obligatory use of, as a minimum, two parameters (for example, ds and df at ﬁxed d) is the key condition, following from fractality of macromolecular networks of glassy (solid-like) polymers [29].

76

The Main Physical Concepts used in Fractals Theory

Table 2.2 The values of characteristic ratio C¥ and chain part length Le between entanglements [29] Polymer

C¥ [62, 63]

Le, nm Experimental [64]

Calculation

13.4

226

208

Poly n-octyl methacrylate

10

177

149

Polystyrene

10

104

149

Polyvinyl acetate

9.4

88

136

Poly n-butyl methacrylate

9.1

130

131

Poly methyl methacrylate

8.6

97

92

Polyethylene

6.8

56

88

Polyvinyl chloride

6.7

31

87

Poly tetraﬂuoroethylene

6.3

41

80

Polyamide-6

5.3

46

61

Polypropylene oxide

5.1

60

58

Polyethylene oxide

4.2

45

43

Polyethylene terephthalate

4.2

21

36

Polyester terephthalate

3.3

28

30

Polycarbonate

2.4

29

17

Poly n-dodecyl methacrylate

References 1.

B.B. Mandelbrot in The Fractal Geometry of Nature, W.H. Freeman and Co., San-Francisco, CA, USA, 1982, p.459.

2.

B.M. Smirnov in The Physics of Fractal Clusters, Nauka, Moscow, Russia, 1991, p.136.

3.

E. Feder in Fractals, Plenum Press, New York, NY, USA, 1990, p.251.

4.

V.S. Ivanova in Synergetics: Strength and Fracture of Metal Materials, Nauka, Moscow, Russia, 1992, p.160.

77

Structure and Properties of Crosslinked Polymers 5.

V.S. Ivanova, A.S. Balankin, I.Z. Bunin and A.A. Oksogoev in Synergetics and Fractals in Material Science, Nauka, Moscow, Russia, 1994, p.383.

6.

A.N. Bobryshev, V.N. Kozomazov, L.O. Babin and V.I. Solomatov in Synergetics of Composite Materials, NPO ORIUS, Lipetsk, Russia, 1994, p.154.

7.

G.V. Kozlov and V.U. Novikov in Synergetics and Fractal Analysis of Crosslinked Polymers, Klassika, Moscow, Russia, 1998, p.112.

8.

F.J. Family, Journal of Statistical Physics, 1984, 36, 5/6, 881.

9.

Proceedings of the 6th International Symposium on Fractals in Physics, Eds., L. Pietronero and E. Tosatti, North-Holland, 1986, p.643.

10. E. Hornbogen, International Materials Reviews, 1989, 34, 6, 277. 11. M. Sahimi, M. McKarnin, T. Nordhal and M. Tirrel, Physical Review A Rapid Communications, 1985, 32, 1, 590. 12. H.-H. Song and R.-J. Roe, Macromolecules, 1987, 20, 11, 2723. 13. G.V. Kozlov and D.S. Sanditov in Anharmonic Effects and PhysicalMechanical Properties of Polymers, Nauka, Novosibirsk, Russia, 1994, p.261. 14. A.S. Balankin, A.D. Izotov and V.B. Lazarev, Neorganicheskie Materialy, 1993, 29, 4, 451. 15. V.S. Ivanova and A.A. Shanyavskii in The Quantitative Fractography, Metallurgiya, Chelyabinsk, Russia, 1988, p.400. 16. M.A. Sadovskii and V.F. Pisarenko in The Seismic Process in Block Medium, Nauka, Moscow, Russia, 1991, p.96. 17. V.B. Lazarev in Studies on Inorganic Chemistry and Chemical Technology, Nauka, Moscow, Russia, 1988, p.225. 18. A.S. Balankin, The News of Science and Engineering, Viniti, Moscow, Russia, 1991, 5, 66. 19. A.S. Balankin and A.A. Kolesnikov, The News of Science and Engineering, Viniti, Moscow, Russia, 1991, 9, 45.

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The Main Physical Concepts used in Fractals Theory 20. M.G. Zemlyanov, V.K. Malinovskii, V.N. Novikov, P.P. Parshin and A.P. Sokolov, Zhurnal Eksperimental’noi i Teoreticheskoi Fiziki, 1992, 101, 1, 284. 21. V.A. Berstein and V.M. Egorov in Differential Scanning Calorimetry in Physics-Chemistry of Polymers, Khimiya, Leningrad, Russia, 1990, p.256. 22. A.S. Balankin, A.L. Bugrimov, G.V. Kozlov, A.K. Mikitaev and D.S. Sanditov, Doklady RAN, 1992, 326, 3, 463. 23. A.S. Balankin in Synergetics of Deformable Body, Publishers of Ministry of Defence SSSR, Moscow, Russia, 1991, p.404. 24. A.S. Balankin, Pis’ma v ZhETF, 1991, 17, 6, 84. 25. R. Rammal and G.J. Toulouse, Physics Letters, 1983, 44, 1, L13. 26. V.A. Bagryanskii, V.K. Malinovskii, V.N. Novikov, L.M. Pushchaeva and A.P. Sokolov, Fizika Tverdogo Tela, 1988, 30, 8, 2360. 27. B. Chu, D. Wu and J. Phillips, Macromolecules, 1987, 20, 10, 2642. 28. S.F. Edwards and T.A. Vilgis, Polymer, 1987, 28, 3, 375. 29. V.U. Novikov and G.V. Kozlov, Uspekhi Khimii, 2000, 69, 4, 378. 30. P.J. Flory, Polymer Journal, 1985, 17, 1, 1. 31. V.I. Irzhak, B.A. Pozenberg and N.S. Enilokopyan in Crosslinked Polymers, Synthesis, Structure, Properties, Nauka, Moscow, Russia, 1979, p.248. 32. A.S. Balankin, Doklady RAN, 1992, 325, 3, 465. 33. A.S. Balankin and A.L. Bugrimov, Vysokomolekulyarnye Soedineniya Seriya A, 1992, 34, 3, 129. 34. V.A. Beloshenko and G.V. Kozlov, Mekhanika Kompozitnykh Materialov, 1994, 30, 4, 451. 35. G.V. Kozlov, V.D. Serdyuk and V.A. Beloshenko, Mekhanika Kompozitnykh Materialov, 1994, 30, 5, 691. 36. V.N. Belousov, G.V. Kozlov, A.K. Mikitaev and Y.S. Lipatov, Doklady Akademii Nauk SSSR, 1990, 313, 3, 630.

79

Structure and Properties of Crosslinked Polymers 37. D.S. Sanditov, G.V. Kozlov, V.N. Belousov and Y.S. Lipatov, Ukrainian Polymer Journal, 1992, 1, 3/4, 241. 38. G.V. Kozlov, V.N. Belousov and A.K. Mikitaev, Fizika i Technika Vysokikh Davlenii, 1998, 8, 1, 101. 39. I.M. Sokolov, Uspekhi Fizicheskikh Nauk, 1986, 151, 2, 221. 40. G.V. Kozlov, M.A. Gazaev, V.U. Novikov and A.K. Mikitaev, Pis’ma v ZhETF, 1996, 22, 16, 31. 41. G.V. Kozlov, V.U. Novikov, M.A. Gazaev and A.K. Mikitaev, InzhenernoFizicheskii Zhurnal, 1998, 71, 2, 241. 42. V.P. Budtov in Physical Chemistry of Polymer Solutions, Khimiya, SaintPetersburg, Russia, 1992, p.384. 43. P.J. Flory, British Polymer Journal, 1976, 8, 1, 1. 44. P. De Gennes in Scaling Ideas in Polymers Physics, Plenum Press, New York, NY, USA, 1980, p.351. 45. P.J. Flory in Statistical Mechanics of Chain Molecules, Publishers of Ithaka University, Ithaka, Greece, 1970, p.424. 46. R. Bakster in Precisely Decided Models in Statistical Mechanics, Mir, Moscow, Russia, 1985, p.488. 47. A.G. Kokorevich, Y.A. Gravitis and V.G. Osol-Kalnin, Khimiya Drevesiny, 1989, 1, 3. 48. M. Doi and S.F. Edwards in The Theory of Polymer Dynamics, Clarendon Press, Oxford, UK, 1986, p.392. 49. Y.-H. Lin, Macromolecules, 1987, 20, 12, 3080. 50. A.I. Olemskoi and A.Y. Flat, Uspekhi Fizicheskikh Nauk, 1993, 163, 12, 1. 51. S. Havlin and D.J. Ben-Avrahm, Journal of Physics Part A, 1982, 15, 3, L321. 52. G.V. Kozlov, V.A. Beloshenko, E.N. Kuznetsov and Y.S. Lipatov, Doklady NAN Ukraine, 1994, 12, 126. 53. V.A. Beloshenko, G.V. Kozlov and Y.S. Lipatov, Fizika Tverdogo Tela, 1994, 36, 10, 2903.

80

The Main Physical Concepts used in Fractals Theory 54. G.M. Bartenev and S.Y. Frenkel in Physics of Polymers, Khimiya, Leningrad, Russia, 1990, p.432. 55. D.S. Sanditov and G.M. Bartenev in Physical Properties of Disordered Structures, Nauka, Novosibirsk, Russia, 1982, p.259. 56. G.M. Bartenev and Y.V. Zelenev in Physics and Mechanics of Polymers, Vysshaya Shkola, Moscow, Russia, 1983, p.392. 57. G.V. Kozlov, K.B. Temiraev, R.A. Shetov and A.K. Mikitaev, Materialovedenie, 1999, 2, 34. 58. G.V. Kozlov, V.A. Beloshenko, M.A. Gazaev and V.N. Varyukhin, Fizika i Technika Vysokikh Davlenii, 1995, 5, 1, 74. 59. Y. Termonia and P. Smith, Macromolecules, 1988, 21, 7, 2184. 60. G.V. Kozlov, V.A. Beloshenko, V.N. Shogenov and Y.S. Lipatov, Doklady NAN Ukraine, 1995, 5, 100. 61. G.V. Kozlov, V.U. Novikov and A.K. Mikitaev in the Proceedings of the AllRussian Symposium, Russian Federation, Moscow, Russia, 1996, p.226. 62. S.M. Aharoni, Macromolecules, 1983, 16, 9, 1722. 63. S. Wu, Journal of Polymer Science, Part B: Polymer Physics Edition, 1989, 27, 4, 723. 64. S.M. Aharoni, Macromolecules, 1985, 18, 12, 2624.

81

3

The Fractal Models of Epoxy Polymers Curing Process

3.1 Two Types of Fractal Reactions at Curing of Crosslinked Epoxy Polymers The irreversible aggregation models, which were prepared for such practically important process descriptions as ﬂocculation, coagulation, polymerisation and so on, have recently been widespread in physics [1–6]. Several examples of successful application of these models for the description of a number of real processes were obtained [1–11]. Therefore the use of these models for the description of polymerisation processes, in particular curing processes of crosslinked polymers, is of undoubted interest. It must be noted that application of percolation and some other models for this goal did not give expected results [12]. Such general concepts for physical processes as scaling and universality classes were used widely in the indicated irreversible aggregation models [13]. The sense of scaling (scaling invariance) consists in abstraction from structure details and selection of simple universal features, which are typical for a wide class of systems. The fractal dimensions are often-used scaling indices. The essence of universality hypothesis is in the following: if for different systems formation the same limiting conditions (system parts interaction) are typical, then these systems enter the same physical phenomena universality class. This hypothesis is closely connected with scaling hypothesis. In irreversible aggregation processes the particle–cluster [14] and cluster–cluster [15, 16] models are two of the most widespread universality classes (types of aggregates), the difference in aggregation mechanisms of which follows from their names. The application of the models indicated above for description of the curing kinetics of two haloid-containing epoxy polymers was considered in paper [17]. For this purpose let us describe brieﬂy the experimental methods used for the study of the epoxy polymers considered in the present chapter. Firstly, the curing kinetics of a haloid-containing oligomer, having chemical structure [17] was studied. This oligomer (conditional designation EPS-4) was cured by 4,4′-diaminodiphenylmethane (DDM) at stoichiometric ratio DDM : EPS-4.

83

Structure and Properties of Crosslinked Polymers

ɇ2ɋ – ɋɇ – ɋɇ2 – Ɉ ⎯ Ɉ

ɋɇ3 ⏐ ⎯ɋ ⎯ ⏐ ɋɇ3

⎯ Ɉ⎯ ɋl4 2

The curing kinetics of the system of EPS-4/DDM was studied by a method of inverse gas chromatography (IGC) [17]. The basic parameter received from processing of the experimental data was the constant of the reaction rate kr determined for an interval of reaction conversion degree α = 0.1–0.7 of kinetic curve α-t part, where t is the curing reaction duration. Ketones (methyl ethyl ketone, 1,4-dioxane, cyclohexanone) were chosen as the standard substances for the retention time determination and argon as the gas-carrier [17]. The system EPS-4/DDM curing was studied at three curing temperatures Tcur: 383, 393 and 403 K [17, 18]. The curing kinetics of the haloid-containing oligomer on the basis of diphenylpropane and hexachloroethane (2DPP+HCE) was also studied. This oligomer was also cured by DDM at the stoichimetric ratio DDM : 2DPP+HCE. The study of the curing kinetics of the system 2DPP+HCE/DDM was carried out by the method of IR-spectroscopy using a PerkinElmer spectrometer. As the optical density of analytical band 920 cm-1 was not accepted as a measure of the contents of the epoxy groups, then but its ratio to the optical density of a standard, for which IR band of skeletal vibrations for the concentration was constant during the curing process. The optical densities of an analytical band and the band of the standard were determined by method of baseline. The optical densities of an analytical band and the bands of the standard were determined by a baseline method. The curing studies of the system 2DPP+HCE/ DDM were carried out at Tcur values: 333, 353, 373, 393 and 513 K [17, 18]. In Figures 3.1 and 3.2 the kinetic curves α(t) for systems 2DPP+HCE/DDM and EPS-4/DDM are given correspondingly. Two main distinctions of curves α(t) for the indicated systems attract our attention. Firstly, for system 2DPP+HCE/DDM the smooth decrease of a slope of curves α(t) is observed in the process of increased t, whereas for system EPS-4/DDM the linear dependence α(t) up to large (about 0.8) values of α is observed. Secondly, if the limiting conversion degree of the curing reaction for the system 2DPP+HCE/DDM is the function of Tcur (the greater Tcur, the greater is this degree), then for the system EPS-4/DDM such a dependence is not present and at all used Tcur a maximum value of α is observed, close to one.

84

The Fractal Models of Epoxy Polymers Curing Process

α 1.0

5 4 3

0.5

2 1

0.5

0

1.0

1.5 t×10-3, s

Figure 3.1 The dependences of the curing reaction conversion α on the reaction duration t for system 2DPP+HCE/DDM at curing temperatures: (1) 333; (2) 353; (3) 373l; (4) 393; and (5) 513 K [17]

α 1.0

3

2

1

0.5

0

2

4 t×10-3, s

Figure 3.2 The dependences of the curing reaction conversion α on the reaction duration t for system EPS-4/DDM at curing temperatures: (1) 383; (2) 393; and (3) 403 K [17]

85

Structure and Properties of Crosslinked Polymers Let us consider the reasons for the mentioned distinctions, which involve the representations of the irreversible aggregation models and of fractal analysis. Within the frameworks of fractal analysis polymerisation kinetics is described according to the general relationship [5, 6]:

1.0

5

(3.1)

where D is the fractal dimension of the clusters forming at curing, the so-called microgels [19]. The curing formal kinetics within the frameworks of traditional approaches is formulated as follows:

(3.2)

Differentiating the relationship (3.1) on t, we receive:

(3.3)

Combination of Relationships 3.2 and 3.3 allows the equation for the estimation of D according to the curing process kinetic parameters to be received [17]:

(3.4)

where c is the constant found from the boundary conditions. Now it is possible to calculate the value of D, using the reaction rate constants kr, obtained according to the experimental methods described above as a function of t or α. The calculation has shown the principal distinction of the behaviour of D, which

86

The Fractal Models of Epoxy Polymers Curing Process is the structure characteristic of the microgels, for systems 2DPP+HCE/DDM and EPS-4/DDM. For the ﬁrst of the indicated systems the value of D does not depend on t at the initial part of the curve α(t), but is the function of Tcur. So, at Tcur = 353 and 373 K D ≈ 1.5, at Tcur = 393 K D ≈ 1.7 and at Tcur = 513 K D ≈ 2.3. This means that the increase of curing temperature determines the formation of more compact microgels during smaller intervals of t. For the system EPS-4/DDM a similar (but weaker) dependence of D on Tcur is observed, but simultaneously there appears a clearly expressed dependence of D on t. So, in a curing temporal interval 300–3600 s the value of D varies as follows: at Tcur = 383 K D = 1.60–2.38 and at Tcur = 403 K D = 1.51–2.42 [17]. In Figure 3.3 the modelling calculations of a curve α(t) are given at Tcur = 383 K for different situations. So, curve 1 is an experimental curve α(t). The calculation according to Relationship 3.1 under condition D = const. = 1.76 gives curve 2, which does not agree with the experimental curve, but it is qualitatively very similar to curves α(t) for the system 2DPP+HCE/DDM (Figure 3.1), which was expected proceeding from the condition D = const. The calculation according to Relationship 3.1 at the condition D = const. = 2.24 gives curve 3, which does not agree once again with the experimental curve, but it is very similar to curve α(t) for the same system at Tcur = 403 K (Figure 3.2). This fact proves to be true for the comparison of curve 3 with the experimental points 4 for the indicated experimental curve α(t). In other words, this comparison shows that the form of the curve α(t) at the initial parts of the curing reaction is controlled by the structure of the forming microgels, which are controlled in the present chapter by their fractal dimension D. Lastly, the calculation according to Relationship 3.1, but with the variable value D, determined according to Equation 3.4, gives an excellent correspondence with experiment (points 5). The modelling curves α(t) and their comparison with the corresponding experimental curve adduced in Figure 3.3 conﬁrm the assumption made above about the reason for the different form of the kinetic curves for the systems 2DPP+HCE/DDM and EPS-4/DDM. Let us consider the reasons for the different dependences D(t) of the studied systems. As has been pointed out above, the general D variation for both systems as the function of t and Tcur makes up D = 1.51–2.42. This interval D corresponds approximately to the cluster–cluster aggregation mechanism [13]. It was shown in paper [20] that the dimension D of the cluster, formed by joining of two clusters with dimensions D1 and D2 (D1 ≥ D2) in the case of the mentioned aggregation mechanism was determined as follows:

(3.5)

87

Structure and Properties of Crosslinked Polymers where d is the dimension of Euclidean space in which the clustering process is considered. It is obvious that in our case d = 3.

α 1.0

1 3 2

0.5 -4 -5

0

2

4 t×10-3, s

Figure 3.3 The comparison of experimental (1, 4) and modelling (2, 3, 5) curves α(t) for system EPS-4/DDM. 1 – experimental curve for Tcur = 383 K; 2 – calculation according to Relationship 3.1 under the condition of D = const.= 1.76; 3 – calculation according to Relationship 3.1 under the condition of D = const. = 2.24; 4 – experimental data for Tcur = 403 K; 5 – calculation according to Relationship 3.1 with D calculated according to Equation 3.4 [17]

From Equation 3.5 it follows directly that for realisation of the condition D = const. irrespectively from t the fulﬁlment of the criterion [17] is needed:

(3.6)

For realisation of the increase of D with t raising it is required that at a previous curing stage there should be clusters (microgels) corresponding to the condition D1 ≠ D2 or in other words dimension distribution of clusters is required. So, in order to receive an average dimension D = 1.64 at the previous stage an interval D1 ÷ D2 = 1.70 – 1.58, i.e., ΔD = D1 – D2 – = 0.12 is needed. To obtain the average dimension D = 2.24

88

The Fractal Models of Epoxy Polymers Curing Process already needs an interval D1 ÷ D2 = 2.35 – 2.12, i.e., ΔD = 0.23. Hence, for systems which are similar to EPS-4/DDM, in process of t increasing not only does the average value of dimension D of the microgels increase, but a raise in their distribution width also occurs. Proceeding from the obtained results, the authors [17] determined curing kinetics similar to those observed for the system 2DPP+HCE/DDM (Figure 3.1) and corresponding to the condition D = const. as homogeneous and similar to that observed for the system EPS-4/DDM (Figure 3.2) and corresponding to the condition D = variant as heterogeneous. One of the probable reasons determining the distinction of the mentioned types of curing kinetics is the different level of density ﬂuctuations in these systems that will be considered in detail below. Consequently, the results stated above showed the beneﬁt of irreversible aggregation models and fractal analysis notions for description of curing kinetics of haloidcontaining epoxy polymers. There are two different curing regimes (homogeneous and heterogeneous) corresponding to the criteria D = const. and D = variant as the function of reaction duration. The ﬁrst condition corresponds to the same dimension of forming microgels in the curing process and the secondto the distribution of these dimensions. The authors [17] showed the inﬂuence of the structure of microgels, characterised by their fractal dimension D, on the shape and parameters of the curing kinetic curves of epoxy polymers. The authors [21-23] studied the reasons of the mentioned obvious difference of curing kinetic curves for epoxy polymers with using fractal analysis [3, 4, 24] and scaling approach [25] methods. One of the probable methods of the analytical description of the kinetic curves α(t) is the following general fractal equation (more complete analogue of Relationship 3.1) [26]:

(3.7)

where K1 is a constant, c0 is the initial concentration of reagents, η0 is the initial viscosity of the reactive medium. Equation 3.7 is presented in a good light for the description of curves α(t) for the synthesis of linear polymers in the cases of both radical polymerisation [27] and polycondensation [5, 6]. The D value can be determined according to the slope of a linear plot of α as a function of t in double logarithmic coordinates as follows from Equation 3.7 and complex constant K1c0η0 by a ﬁtting method. In Figure 3.4 such a

89

Structure and Properties of Crosslinked Polymers simulation was shown for the system 2DPP+HCE/DDM as points at the following conditions: K1c0η0 = const. = 8.06 × 10–3 and D = const. = 1.78. As can be seen, the good correspondence of experimental and theoretical curves was obtained up to t = 2400 s, where a system universality class change occurs owing to gel formation and a corresponding increase in dimension D from 1.78 to ~ 2.5 [28, 29]. Obtaining an analogous description for the system EPS-4/DDM failed since for this system the value of D is a function of the reaction duration t. In principle, curve 2 in Figure 3.4 can be described with variable D and using η0, but such an approach is a formal one, since the equation with two variables describes practically any smooth monotonous curve [22].

α 2 1

1.0

0.5 -3

0

2

4 t×10-3, s

Figure 3.4 The experimental kinetic curves for systems (1) 2DPP+HCE/DDM and (2) EPS-4/DDM, (3) calculation according to Equation 3.7 [21]

Proceeding from this, the authors [21–23] attempted to describe the curves α(t) shown in Figure 3.4 (see also Figures 3.1 and 3.2) within the frameworks of scaling approaches for reactions of low-molecular substances [30]. Let us consider a reaction in which particles P of a chemical substance diffuse in the medium containing random located static non-saturated traps T. At the contact of particle P with a trap T the particle disappears. Non-saturation of a trap means that the reaction P + T ® T can be repeated an inﬁnite number of times. It is usually considered that if the concentration of particles and traps is large or the reaction occurs with intensive stirring, the process can be considered as the classical reaction of the ﬁrst

90

The Fractal Models of Epoxy Polymers Curing Process order. In this case it can be considered that the concentration of particles c(t) decay law will be the following [30]:

(3.8)

where A is a constant, t is the reaction duration. However, if the concentration of random located traps is small, space areas necessarily exist, practically free from traps. Particles entering these areas can reach the traps only after a quite a long time and, hence, their number decay with time will be slower. The formal analysis of this problem shows that the concentration of particles falls down according to the law [30]:

(3.9) being dependent on the space dimension d (B is constant). If the traps can move, their mobility averages the inﬂuence of spatial heterogeneity, so the assumption resulting from Relationship 3.8 will be carried out better. In this case the concentration of particles falls down according to the combined law [30]:

(3.10)

The authors [21–23] attempted to describe kinetic curves α(t) for systems 2DPP+HCE/ DDM and EPS-4/DDM within the frameworks of Relationships 3.8–3.10. In this case it was assumed that

(3.11)

In Figure 3.5 the dependences of ln (1-α) on t, corresponding to Equation 3.8, are given

91

Structure and Properties of Crosslinked Polymers for both indicated systems. As follows from the adduced plots, the curing kinetics of the system 2DPP+HCE/DDM is well described by linear dependence in coordinates of Figure 3.5, whereas the dependence [ln (1-α)](t) for system EPS-4/DDM deviates from linearity. This means that the curing reaction of the system 2DPP+HCE/DDM, described by Equation 3.8, at the condition indicated above is a classical reaction of the ﬁrst order, occurring in a reactive medium with small density ﬂuctuations.

ln(1-α) 0

-2 1

-4

2 0

2

4 t×10-3, s

Figure 3.5 The dependences of (1–α) on the reaction duration t in logarithmic coordinates corresponding to Equation 3.8 for the systems (1) 2DPP+HCE/DDM and (2) EPS-4/DDM [23]

Attempts by the authors [21–23] to linearise the dependence of (1–α) on t for the system EPS-4/DDM by Equations 3.9 and 3.10 were not successful. Therefore the following assumption was made [21–23]. Equation 3.9 describes reaction kinetics of low-molecular substances at large density ﬂuctuations in Euclidean space with dimension d, which is equal to 3 in the considered case. If we assume that the formation of fractal clusters (microgels) with dimension D deﬁnes the reaction curing course in a fractal space with dimension D, the dimension d in Equation 3.9 should be replaced by D. The dependence of ln (1–α) on tD/(D+2), corresponding to Equation 3.9 with the indicated replacement, is adduced in Figure 3.6. In such treatment the indicated dependence is linear and this circumstance points out that the curing reaction of

92

The Fractal Models of Epoxy Polymers Curing Process system EPS-4/DDM proceeds in the conditions of large density ﬂuctuations in fractal space with dimension D [21–23].

ln(1-α) 0

-2

-4

0

40

80 tD/(D+2)

Figure 3.6 The dependence of (1–α) on parameter tD/(D+2) in logarithmic coordinates corresponding to Equation 3.9 for the system EPS-4/DDM [23]

Hence, the fractal reactions of polymerisation can be divided, as a minimum, into two classes: reactions of fractal objects whose kinetics is described similarly to curve 1 in Figure 3.4 and reactions in fractal space whose kinetics is described by curve 2 in Figure 3.4. The second class reactions correspond to formation of structures on fractal lattices [31, 32]. The principal difference in the indicated reaction classes is the dependence of their proceeding rate on the fractal dimension D of the occurrence of products forming in the reaction (macromolecular coils, microgels). The ﬁrst class of reactions is well described by Equation 3.7 (see Figure 3.4). The indicated equation was received on the basis of theoretical conclusions of paper [33], where it is assumed that the smaller D is the less compact is the fractal and there are more sites on it, which are accessible to the reaction. This means that D decreasing results in reaction rate growth. In Figure 3.7 three modelling curves α(t) corresponding to Equation 3.7 at c0 = const., η0 = const. for D = 1.5, 1.8 and 2.1 are adduced. As follows from the adduced plots, D increasing reduces the reaction rate very sharply and decreases α at t comparable values.

93

Structure and Properties of Crosslinked Polymers As to reactions in fractal spaces, here the situation is the opposite. As it is known [34], if we consider the diffusive movement of a trajectory of an oligomer and curing agent molecules as a random walk trajectory, the number of sites 〈s〉 visited by such a walk is written as follows:

(3.12)

where ds is the spectral dimension of space, characterising its connectivity degree [35]. For Euclidean spaces ds = 2 [36], for crosslinked microgels ds ≈ 1.33 [35]. From Relationship 3.12 it follows that the 〈s〉 value, which can be treated as a contacts number of reacting macromolecules (microgels), is proportional to t in Euclidean and to t0.665 in fractal spaces. At the same t the greater number of indicated contacts in Euclidean space determines the faster curing reaction rate in comparison with a fractal space [26]. In this connection we shall note an interesting detail. As it was shown in paper [37], for an ideal phantom network the relationship is true:

(3.13)

It is easy to see the obvious analogue between Equation 3.9 (at replacement of d on D) and Relationship 3.12 exponents. In Figure 3.8 the kinetic curves α(t), calculated according to Equation 3.9 at B = const. for D = 1.5, 1.8 and 2.1 and also for d = 3, are adduced. It is easy to see that according to the treatment quoted above reaction rate grows as dimension D increases and reaches the largest value in Euclidean space at d = 3. t is necessary to note that in reactions of fractal objects according to the Equation 3.7 at d=D=3 α=const, and accounting for boundary condition α=0 at t=0, α=const=0. This means that such reactions for three-dimensional Euclidean objects do not proceed at all. Consequently, the results stated above have shown that fractal reactions at crosslinked polymers curing can be of two classes: reactions of fractal objects and reactions in fractal space. The main distinction of the two indicated reaction classes is the dependence of their rates on the fractal dimension D of reaction products. Knowledge of such theoretical dependences allows the real curing reaction course to operate.

94

The Fractal Models of Epoxy Polymers Curing Process

α 0.8

1

2

0.4

3

0

40

80 t, min

Figure 3.7 The modelling curves α(t) for reactions of fractal objects calculated according to Equation 3.7 at (1) D = 1.5; (2) D = 1.8; and (3) D = 2.1 [21] The authors of papers [38–40], using the coefﬁcient A and B values determined from slopes of corresponding linear plots of Figures 3.5 and 3.6, described theoretically kinetic curves α(t) with the aid of Equations 3.8 and 3.9 for systems 2DPP+HCE/ DDM and EPS-4/DDM, respectively. The comparison of curves α(t) calculated by the indicated method and those obtained experimentally for both considered systems is adduced in Figure 3.9. As follows from the plots shown in Figure 3.9, good correspondence was obtained between theory and experiment.

α 0.4

4 3 2 1

0.2

0

40

80 t, min

Figure 3.8 The modelling curves α(t) for reactions in fractal space, calculated according to Equation 3.9 at (1) D = 1.5; (2) D = 1.8; (3) D = 2.1 and (4) D = 3 [22]

95

Structure and Properties of Crosslinked Polymers

α 5

1.0

2 1

0.5 -3 -4

0

2

4 t×10-3, s

Figure 3.9 The experimental (1, 2) and theoretical (3–5) kinetic curves α(t) for systems 2DPP+HCE/DDM (1, 3) and EPS-4/DDM (2, 4, 5). Curve 5 corresponds to the curing reaction of system EPS-4/DDM in Euclidean space [38]

As to plotting a theoretical curve α(t) for system EPS-4/DDM one important remark should be made. As follows from Equation 3.9, at t = 0 c(t) = 1 or (1–α) = 1 (Equation 3.11), i.e., at t = 0 α = 0, which is obvious. However, from the plot of Figure 3.6 it follows that the straight line [ln (1–α)](tD/(D+2)) extrapolates to ln (1–α) = 0 (or α = 0) at ﬁnite value tD/(D+2), which is equal to ~ 12. This means that the non-homogeneous reaction of curing for the system EPS-4/DDM has some induction period tin, estimated approximately as equal to 150 s. The physical basis of such an effect is clear. At the beginning of the curing reaction (at t = 0) fractal clusters (microgels) are absent in the reactive mixture and the available curing agent and oligomer can be considered as either points (zero-dimensional objects) or short rods (one-dimensional objects). For the formation of microgels, creating reaction fractal space, a certain time is necessary, which is an induction period tin. Therefore at theoretical curve α(t) calculation one should use not the reaction duration t, but the difference (t – tin) [40]. As was noted above, formal curing kinetics of crosslinked polymers could be described by Equation 3.2. There are two modiﬁed variants of the indicated equation accounting for the curing reaction proceeding with auto-acceleration or auto-deceleration [41]:

(3.14)

96

The Fractal Models of Epoxy Polymers Curing Process

(3.15)

where c and ξ are characteristics of auto-acceleration and auto-deceleration effects, respectively. It was shown above that with fractal analysis and irreversible aggregation models methods using two types of curing reactions exist: homogeneous and nonhomogeneous. The ﬁrst of the indicated reaction types is characterised by fractal dimension D of the microgels, a constant value almost up to the gel formation point, which is treated as the formation of network tightening of the entire system [28] or the transformation of liquid ﬂuid system with crosslinked microgels in an elastic polymer [29]. In non-homogeneous reaction processes on the same part of the kinetic curve monotonous growth in D is observed [17]. The authors of paper [42] compared the two indicated types of curing reaction in the example of two haloid-containing epoxy polymers for elucidation of the availability of the physical basis of the autoacceleration (auto-deceleration) effect in the curing process of crosslinked polymers. As is shown above, polymerisation processes can be described by the general fractal Relationship 3.1. Differentiating the indicated relationship on time t and equating the derivative dα/dt to available ones in Equations 3.14 and 3.15, we receive [42]:

(3.16)

(3.17)

for curing reactions with auto-acceleration and auto-deceleration, respectively, where c1 is a constant determined from boundary conditions. In turn, the dimension D value can be determined according to Equation 3.4. From a comparison of Equations 3.4, 3.16 and 3.17 it is easy to see that in the case D = const. (homogeneous curing reaction) the members in the numerator of Equations 3.16 and 3.17 [(1 + cα) and (1 – ξα), respectively] are equal to 1. In the case D = D(t) (non-

97

Structure and Properties of Crosslinked Polymers homogeneous curing reaction) two methods of description of the kinetic curves α(t) are possible (see Figure 3.10 and also Figures 3.1, 3.2 and 3.6). The ﬁrst of these, used in paper [21], provides for application of the function D(t), determined by any independent method, and then c = 0 and ξ = 0. The second method is used below and supposes D = const.; the D value can be selected arbitrarily from an interval D(t) for the given concrete curing reaction. The comparison of the kinetic curves adduced in Figure 3.10 for systems 2DPP+HCE/DDM and EPS-4/DDM shows that at t < 1200 s the curing reaction for the ﬁrst system proceeds faster than for the second one and vice versa for t > 1200 s. This means that at curing of the system EPS-4/DDM relative to 2DPP+HCE/DDM at t < 1200 s the auto-deceleration effect is observed and at t > 1200 s the auto-acceleration effect is observed. Calculating the value D at t = 1200 s by the known kr and α values according to Equation 3.4 and assuming it to be constant, one can estimate parameters c and ξ in Equations 3.16 and 3.17, respectively. Strictly speaking, in such an approach use of the two indicated equations is not required: if in Equation 3.16 c < 0, then this means availability of the auto-deceleration effect and if c > 0 availability of the auto-acceleration effect. In Figure 3.11 the dependence D(t) is adduced for the system EPS-4/DDM, which was calculated according to Equation 3.4 and the D value for the system 2DPP+HCE/ DDM was shown by a dashed line. From the Figure 3.11 data it follows that at t = 1200 s D values for both systems are equal, at t < 1200 s the D value for system EPS-4/DDM is lower than the corresponding magnitude for system 2DPP+HCE/ DDM and at t > 1200 s it is higher. Comparison of the plots in Figures 3.10 and 3.11 allows the two following conclusions to be made. Firstly, the auto-acceleration (auto-deceleration) effect can be realised only in non-homogeneous curing reactions. In homogeneous curing reactions (D = const.) c = 0 and ξ = 0. The c (or ξ) value is deﬁned by values of the D ratio in homogeneous (Dhom) and non-homogeneous (Dnon) curing reactions. In the case where Dhom > Dnon c < 0, only the auto-deceleration effect is observed; in the case where Dhom > Dnon c > 0, the auto-acceleration effect is also observed. Therefore, in the proposed treatment both the absolute value and the sign of the constant c are deﬁned by the difference Dhom – Dnon – = ΔD. In Figure 3.12 the relationship c(ΔD) for the system EPS-4/DDM is adduced, where the value of c was calculated according to Equation 3.16 at the condition D = const. = 1.75. And as was supposed, negative values of ΔD correspond to negative magnitudes of c (auto-deceleration) and positive values of ΔD correspond to positive magnitudes of c (auto-acceleration). Secondly, hence, absolute values of ΔD and c relationship are well approximated by a linear correlation (Figure 3.12) [42].

98

The Fractal Models of Epoxy Polymers Curing Process

α 1.0

1 2

0.5

2

0

4 t×10-3, s

Figure 3.10 The kinetic curves α(t) of the curing process for the systems (1) EPS-4/ DDM and (2) 2DPP+HCE/DDM [42]

D 2.4

2.0

1.6 0

2

4 t×10-3, s

Figure 3.11 The dependence of the fractal dimension D of microgels on the curing reaction duration t for the system EPS-4/DDM. The condition D = const. for the system 2DPP+HCE/DDM is shown by a dashed line [42]

99

Structure and Properties of Crosslinked Polymers

c 13

8

3

-2 0

0.4

0.8 ΔD

Figure 3.12 The relationship between the auto-acceleration (auto-deceleration) characteristic c and the difference ΔD = Dhom – Dnon for the system EPS-4/DDM [42]

Therefore, the authors of paper [42] have demonstrated that the introduction in the consideration of the structure of microgels forming in the curing reaction process, characterised by its fractal dimension D, and also the introduction of the indicated two types of reaction allowed determination of the physical nature of the auto-acceleration (auto-deceleration) effect. This effect is realised only for non-homogeneous curing reactions and its sign and intensity are deﬁned by the ratio of D values at D = const. and D = variant = D(t). In other words, the indicated factors are deﬁned by the structure change character of microgels in the curing process as a reaction duration function.

3.2 Scaling Relationships for Curing Reactions of Epoxy Polymers It has been indicated in the previous section that the curing kinetics of the system EPS4/DDM can be described according to the scaling Relationship 3.9 with dimension d replacement of D. The authors of paper [43] considered this question in more detail. In Figures 3.13–3.15 the dependences of α on t (α is given as a percentage) are adduced, corresponding to Equations 3.8 and 3.9 and Equation 3.9 with d replacement of D. As follows from the adduced plots, the dependences [ln (100 – α)](t) are curvilinear and, hence, the curing kinetics of the system EPS-4/DDM can not be described according to Equation 3.8, therefore the curing reaction is not a classical reaction of the ﬁrst order.

100

The Fractal Models of Epoxy Polymers Curing Process This also means the availability of large-scale density ﬂuctuations (heterogeneity) in the reactive medium [30]. The specimens for IGC studies were prepared in the following sequence: the oligomer and the curing agent were dissolved in acetone, the mixture from the solutions was piled up on the substrate with subsequent drying. Existing in solution, the heterogeneity of the mixture was ﬁxed in a solvent evaporation process and it was preserved in the solid-phase curing reaction. Nevertheless, one should note that as Tcur increases from 383 to 403 K, the curvilinearity of the plots shown in Figure 3.13 is somewhat smoothed out and the curve [ln (100 – α)](t) for Tcur = 403 K can be approximated by a straight line. Therefore one can assume that temperature increase results in intensiﬁcation of certain diffusive processes [44, 45], both at mixture drying and in the solid-state curing course, which smoothes out the spatial heterogeneity inﬂuence [30].

ln(100-α) 5.0

2.5

3 0

2

2

1 4 t×10-3, s

Figure 3.13 The dependences of the concentration decay of the reacting substances (100 – α) on the reaction duration t in logarithmic coordinates for system EPS-4/ DDM at Tcur: (1) 383; (2) 393 and (3) 403 K [43]

In Figure 3.14 the dependences ln (100 – α) on the parameter td/(d+2) corresponding to Equation 3.9 at d = 3 are adduced for the same epoxy system. It was again not possible to obtain linear correlations and this means that despite the essential density ﬂuctuations (see Figure 3.13) Equation 3.9 does not describe the curing reaction of system EPS-4/DDM.

101

Structure and Properties of Crosslinked Polymers

ln(100-α) 5.0

2.5

3 0

100

2

1 200

td/(d+2)

Figure 3.14 The dependences of the concentration decay of the reacting substances (100 – α) on the parameter td/(d+2) in logarithmic coordinates for system EPS-4/ DDM at Tcur: (1) 383; (2) 393 and (3) 403 K [43]

As has been shown above, crosslinked clusters (microgels), forming in the initial stages of the curing process, have fractal structure, characterised by fractal dimension D [37]. It is possible to determine the D value with application of the kinetic parameters of the curing process according to Equation 3.4. Calculation of the dimension D has shown that its value is actually a function of t (or α) and for the system EPS-4/DDM it varies within the limits 1.51–2.38 (see Figure 3.11). This means that the aggregation process deﬁning the formation of microgels occurs according to the cluster–cluster mechanism [13]. As was noted above, in the polymerisation processes case, dimension d in Equation 3.9 should be replaced with D and then the indicated equation acquires the following form [43]:

(3.18)

Figure 3.15 shows the dependences of ln (100 – α) on the parameter tD/(D+2) for the studied system at three Tcur values, from which it follows that in such treatment the dependences are linear and, hence, the modiﬁed Equation 3.18 describes the system EPS-4/DDM curing reaction as a reaction with large spatial ﬂuctuations and reaction fractal products [43].

102

The Fractal Models of Epoxy Polymers Curing Process

ln(100-α) B

5.0 -0.05 -0.06 373 383 393 403 T cur

2.5

3 0

40

2 80

1 tD/(D+2)

Figure 3.15 The dependences of the concentration decay of the reacting substances (100 – α) on the parameter tD/(D+2) in logarithmic coordinates for the system EPS-4/ DDM at Tcur: (1) 383; (2) 393 and (3) 403 K. Insert: the dependence of coefﬁcient B in Equation 3.18 on Tcur [43]

The adduced linearity of the plots in Figure 3.15 allows determination of the coefﬁcient B value in Equation 3.18 according to their slope. In the insert to Figure 3.15 the dependence of B on the curing temperature Tcur was shown. As is expected, an increase in Tcur results in growth in the absolute value of B that causes α growth at Tcur increase at the same values of t. The results stated above demonstrated once more that in polymerisation reactions in general and speciﬁcally curing reactions it is necessary to take into account the structure of the forming products. The structure of the crosslinked clusters (microgels) can be characterised by its fractal dimension D. Equation 3.18 modiﬁed to account for this factor describes well the curing process of haloid-containing epoxy oligomer EPS-4 at various curing temperatures [43]. As was noted above, for analysis of polymerisation reactions in general and curing reactions of crosslinked polymers in particular a number of physical concepts can be used, of which two will be considered below. Equations 3.1 and 3.8 were used as basic relationships in the indicated concepts. In Figure 3.16 the dependences of ln (100 – α) on t in logarithmic coordinates, corresponding to Equation 3.8, are adduced for the system 2DPP+HCE/DDM at six Tcur values, where α is given as a percentage. As follows from the plots adduced in Figure 3.16, the dependences [ln (100 – α)](t) proved to be linear, i.e., corresponding to Equation 3.8. This means that the curing

103

Structure and Properties of Crosslinked Polymers reaction of system 2DPP+HCE/DDM at all used Tcur can be considered as a classical reaction of the ﬁrst order, proceeding in the medium with small density ﬂuctuations [30]. The linearity of the plots allows determination of the coefﬁcient A value in Equation 3.8 from their slopes. It is easy to see from the data of Figure 3.16 that an increase in Tcur results in growth in A. Therefore in Figure 3.17 the dependence of A1/2 on Tcur for the studied system is adduced, which is sufﬁciently approximated by linear correlation and it can be expressed analytically as follows [46]:

(3.19)

The temperature T0 = 253 K in Equation 3.19 corresponds to the condition A = 0 (see Figure 3.17) and this means that at all T < T0 the curing reaction of system 2DPP+HCE/DDM does not occur. The comparison of coefﬁcient A values, determined from linear plots of Figure 3.16 slope Ae and calculated according to Equation 3.19, was adduced in Table 3.1, from which follows their satisfactory correspondence.

ln(100-α) 5 1 2

4

3 3

4 5

2

6 0

2

4 t×10-3, s

Figure 3.16 The dependences of the concentration decay of the reacting substances (100 – α) on the parameter t in logarithmic coordinates for system 2DPP+HCE/ DDM at Tcur: (1) 295; (2) 333; (3) 353; (4) 373; (5) 393 and (6) 513 K [46]

104

The Fractal Models of Epoxy Polymers Curing Process

A1/2 1.5

1.0

0.5

0 253

353

453

553

Tcur, K

Figure 3.17 The dependence of parameter A1/2 on the curing temperature Tcur for system 2DPP+HCE/DDM [46]

Let us consider further application of Relationship 3.1 for the description of the curing kinetics of system 2DPP+HCE/DDM. In Figure 3.18 the dependences α(t) in double logarithmic coordinates corresponding to Relationship 3.1 were adduced. As one can see, they are linear, which allows determination from their slope of the value of fractal dimension D of microgels forming in the curing process. As the calculations have shown, the D value grows as Tcur increases and varies within limits 1.20–1.95. D increasing means that raising Tcur results in formation of more compact microgels [36]. The interval of the received values of D indicates that formation of microgels occurs according to the irreversible aggregation cluster–cluster mechanism or by joining small microgels to larger ones [2]. For the interval Tcur = 353–513 K the plots of α(t) in double logarithmic coordinates dissociate into two linear parts with different slopes. The part for larger t has a smaller slope and, hence, a higher D magnitude. D calculation for these parts shows an approximately constant D value within the range of 2.35–2.50. Such D magnitudes correspond to the particle–cluster aggregation mechanism [13]. At present theoretically [28] and experimentally [29] it is shown that polymerising system transition to dimension ~ 2.5 means the gel formation point is reached, which is understood as a network formation tightening the system [28]. Hence, the intersection points of two linear parts in Figure 3.18 (at the same Tcur) correspond to the system 2DPP+HCE/ DDM gel formation time in the sense indicated above. Knowing the D values, a proportionality coefﬁcient in Relationship 3.1 can be determined. As was shown

105

Structure and Properties of Crosslinked Polymers earlier [5, 6], the indicated coefﬁcient consists of three factors: K1c0η0, where K1 is a constant, c0 is the initial concentration of the reacting substances, η0 is the initial viscosity of the reacting medium (see Equation 3.7). Since for the studied system the value of c0 is constant, then the proportionality coefﬁcient obtained by the indicated method can be considered as the η0 value in relative units. The increase of Tcur from 295 to 513 K results in the growth of η0 from 0.053 to 2.58, i.e., an increase of about 50 times.

Table 3.1 The comparison of experimental Ae and theoretical AT constants in Equation 3.8 for the system 2DPP+HCE/DDM [46] Tcur, K

Ae

AT

295

0.047

0.049

333

0.147

0.179

353

0.260

0.128

373

0.410

0.400

393

0.740

0.550

513

1.16

1.89

ln α 5

3 -1 -2 -3 1 5.5

6.5

7.5

8.5

ln t

Figure 3.18 The dependences of the degree of reaction conversion α on the reaction duration t in double logarithmic coordinates for system 2DPP+HCE/ DDM at Tcur = (1) 333; (2) 373 and (3) 393 K [46] 106

The Fractal Models of Epoxy Polymers Curing Process Using the values of constant A and η0 determined by the indicated methods, theoretical kinetic curves α(t) for each Tcur can be calculated according to Equations 3.1 and 3.8. The example of comparison of one experimental and two theoretical curves α(t) for the system 2DPP+HCE/DDM at Tcur=353 K is adduced in Figure 3.19. As one can see, both models considered in the present chapter simulate well the experimental kinetic curve α(t) [46].

α 0.8

0.4 -1 -2 0

2

4 t×10-3, s

Figure 3.19 The kinetic curves α(t) for the system 2DPP+HCE/DDM at Tcur = 353 K. The line is experimental data, points – calculation according to Equations 3.8 (1) and 3.1 (2) [46] Since Equations 3.7 and 3.8 describe well the experimental kinetic curve, then one should expect that between these equation parameters (A in Equation 3.8, D and η0 in Equation 3.7) a certain interconnection should exist. For checking of this supposition in Figure 3.20 the correlations between D, η0 and A are adduced for the system 2DPP+HCE/DDM at six used Tcurs. As one can see, a supposed interconnection does actually exist: an increase in A results in growth of both D and η0 (and vice versa: growth in D and η0 results in an increase). The dependences of D and η0 on A can be expressed analytically as follows [46]:

(3.20)

(3.21) 107

Structure and Properties of Crosslinked Polymers It is easy to see that the dependence of η0 on A is much stronger than the similar dependence for D. In Tables 3.2 and 3.3 the comparison of the obtained data values e De and η0 according to the graphic and calculated magnitudes of these parameters according to Equations 3.20 and 3.21, respectively, was adduced. As follows from the indicated comparison, satisfactory correspondence of parameters D and η0, estimated by both considered methods, is again obtained.

D

η0, rel. un.

2

3

2.5 1

2.0

1.5

1.0 0

2

1

0.5

1.0

1.5 Ⱥ

0

Figure 3.20 The dependences of the fractal dimension D (1) and initial viscosity η0 (2) of the reactive medium of microgels on the parameter A value for the system 2DPP+HCE/DDM [46]

Table 3.2 The comparison of experimental De and theoretical DT fractal dimension of microgels for system 2DPP+HCE/DDM [46]

108

Tcur, K

De

DT

295

1.22

1.212

333

1.20

1.376

353

1.53

1.500

373

1.68

1.630

393

1.78

1.840

513

1.95

2.050

The Fractal Models of Epoxy Polymers Curing Process

Table 3.3 The comparison of experimental η0e and calculated theoretically reactive medium initial viscosity (in relative units) for the system 2DPP+HCE/DDM [46] Tcur, K

η0e

ηÒ0

295

0.053

0.007

333

0.019

0.065

353

0.161

0.203

373

0.40

0.50

393

2.40

1.64

513

2.58

4.0

The dependences of parameters D and η0 on the same coefﬁcient A, described by Equations 3.20 and 3.21, suppose D andη0 interconnection of such a kind: D increasing should result in strong growth of η0. Such a dependence does actually exist and for linearisation of correlation η0(D) the use of η0 to the power of 1/8 (Figure 3.21) is required. The relationship between η0 and D can be written analytically as follows [46]:

(3.22)

Therefore, the results stated above have demonstrated that both scaling Equation 3.8 and fractal Relationship 3.1 (or 3.7) describe to an equal extent curing reaction kinetics of haloid-containing epoxy polymer 2DPP+HCE/DDM at different curing temperatures. By virtue of this circumstance there exists an interconnection between the parameters included in the indicated equations. The fractal Relationship 3.7 introduces in consideration of the the kinetics problem the structure of the reaction products (in the given case the structure of microgels and condensed state after the gel formation point), characterised by its fractal dimension D that makes this concept physically more informative [46].

109

Structure and Properties of Crosslinked Polymers

η10/ 8

1.5

1.0

0.5

0

1

2

3 D

Figure 3.21 The dependence of parameter η0 on fractal dimension D of microgels for the system 2DPP+HCE/DDM [46] 1/ 8

The classical problem in chemical kinetics is the inﬂuence of diffusive processes on this kinetics [3, 4, 44, 45]. In reactions controlled by diffusion, their rate is deﬁned by the diffusion time, which is the time necessary for reagents to reach one another. Simulation of similar reactions on Euclidean lattices gives the following results. These reactions were considered [47]:

(3.23)

(3.24)

where A and B are reacting particles, O is an inert product. The following dependences of the density ρA of reacting particles A on the reaction duration t for the indicated reactions were received [47]:

(3.25)

110

The Fractal Models of Epoxy Polymers Curing Process

(3.26)

for Reactions 3.23 and 3.24, respectively, where d is the dimension of Euclidean space in which a reaction proceeds. As it is known [48], a change in space type from Euclidean to fractal strongly changes the chemical reaction course. In this case the dependences ρA(t) have the following form [47]:

(3.27)

(3.28)

for Reactions 3.23 and 3.24, respectively, where ds is the spectral dimension of the reactive medium (in computer simulation – lattice), deﬁning its connectivity [35]. Let us consider the physical model application described above for the curing process treatment of a haloid-containing epoxy polymer (the system 2DPP+HCE/DDM) [49–51]. It is obvious that in this case Relationships 3.26 or 3.28, corresponding to Reaction 3.24 are to be used. Using the Relationships of type 3.25–3.28 in double logarithmic coordinates, the exponent value for them can be estimated according to the slope of the obtained linear plots. The difference (1 – α) was used as the density ρA of particles that ‘survived’ in the curing process (oligomer molecules) [51]. In Figure 3.22 the dependences (1 – α) on t for ﬁve kinetic curves α(t), shown in Figure 3.1, plotted by the indicated method are adduced. As follows from Figure 3.22 data, a change in the curing temperature results in essential growth in the slope of the linear plots and, hence, the exponent in Relationships 3.26 or 3.28. In turn, an exponent increasing in these relationships results in growth of the dimension controlling the studied curing process of epoxy polymers. In Figure 3.22 linear plots were also shown by dashed lines, corresponding to the slope for reactions in fractal (ds = 1.33,

111

Structure and Properties of Crosslinked Polymers the slope Δ = 0.333 [47]) and Euclidean (d = 3, Δ = 0.75 [47]) spaces, obtained by a computer simulation method. As one can see, the straight lines obtained for the system 2DPP+HCE/DDM have a slope that does not correspond to two of the indicated cases, except for Tcur = 513 K, where the curing process is close to the reaction in Euclidean space with dimension d = 3. Such behaviour of dimensions, which changes within the range of 0.296–2.964, allows it to be supposed that it ' is the effective spectral dimension d s accounting for availability in real systems of spatial and energetic disorder [52]. This means that Tcur raising increases the effective connectivity of the solvent-oligomer-curing agent system and diffusion of the reacting particles in this case is unusual (anomalous) [53]. For diffusivity Ddif of particles in a liquid matrix, which is a Euclidean object with dimension d = 3, in supposition of Newtonian rheology the Einstein relationship can be written [54]:

(3.29)

where k is Boltzmann’s constant, η0 is the medium viscosity, Rp is the particle size, α is some numerical coefﬁcient, determined by boundary conditions on the particle surface. From Equation 3.29 it follows that Tcur raising, accompanied by η0 reduction, should result in strong growth in Ddif or intensiﬁcation of diffusive processes, which controls the curing of the studied system. Hence, at small D and ﬁnite t reagent particles can not enter some system regions and, as a consequence, curing is not realised in these regions. Using Relationship 3.1, the D value can be determined, replotting the indicated relationship in double logarithmic coordinates, after which the D value is calculated by the obtained linear plot slope [51]. D values calculated by the indicated method for the system 2DPP+HCE/DDM are adduced in Table 3.4 together with the effective ' spectral dimension d s magnitudes, estimated according to the linear plots slope in ' Figure 3.22. As the authors [55] show, the dimension d s is connected with dimensions of space (reactive medium) d and microgels D according to the following equation:

(3.30)

112

The Fractal Models of Epoxy Polymers Curing Process

ln(1-α) 0.333 0 1 2 3

-1

4 0.750 -2 5 5

6

8

7

ln t

Figure 3.22 The dependences of (1 – α) = ρA on t in double logarithmic coordinates up to the gel formation point for the system 2DPP+HCE/DDM at Tcur: (1) 333; (2) 353; (3) 373; (4) 393; and (5) 513 K. The upper dashed line gives a slope for the reaction in fractal space, the lower one in Euclidean space [51]

Table 3.4 The main dimensions of the system 2DPP+HCE/DDM curing process [51] Tcur, K

D

d s'

d

dw

DT

333

1.20

0.296

3.27

20.27

1.27

353

1.53

0.748

2.73

8.02

1.60

373

1.68

1.012

2.91

5.93

1.76

393

1.78

2.0

3.0

3.0

2.40

513

1.95

2.964

3.0

2.02

2.50

'

Using the d s and D values adduced in Table 3.4, the d magnitude for a reactive medium, which is the analogue of a lattice in computer simulation of chemical reactions, can be calculated. d values are adduced in Table 3.4, from which it follows

113

Structure and Properties of Crosslinked Polymers that within the limits of an error of estimations their average value is equal to 3. Therefore, a reactive medium in the curing case of system 2DPP+HCE/DDM represents a ‘virtual’ fractal with a Hausdorff dimension, which is equal to a Euclidean space dimension, but with connectivity degree typical for fractal objects. This distinguishes ' real chemical reactions from computer simulation, where the condition d s = ds = 4/3 ' for fractal lattices and d s = ds = d = 3 for Euclidean ones is assumed a priori. The authors [56] obtained a result similar to that described above in the case of the scaling treatment of radical polymerisation. The dimension dw of a random walk trajectory of reagents (oligomer and curing agent molecules) can be estimated according to the equation [35]:

(3.31)

Further the dimension theoretical magnitude DT of microgels can be calculated according to the generalised model of diffusion-controlled irreversible aggregation [57]:

(3.32)

where η is the parameter which the authors [57] interpreted as the ratio n/m (n, m are whole positive numbers), characterising the chemical reaction of n statistically walking particles with m aggregate perimeter sites. Since in a curing reaction case one curing agent molecule reacts with one oligomer molecule, then as the ﬁrst approximation it was accepted that n = 1, m = 1 and n/m = 1. In Table 3.4 the dimensions dw and DT are adduced and the comparison of the latter with estimation of this dimension according to Relationship 3.1 shows their good correspondence, accounting for the approximations made. Therefore, up to the gel formation point the curing reaction ' is controlled completely by the dimension d s [51]. In Figure 3.23 the dependences of (1 – α) on t in double logarithmic coordinates are adduced for the curing reaction after the gel formation point. As one can see, in this case the slope of the obtained linear plots is independent of curing temperature and is equal to about 0.333. Such plots are in agreement with Relationship 3.28, describing the reaction on a fractal lattice with dimension ~ 2.5 and superuniversal

114

The Fractal Models of Epoxy Polymers Curing Process exponent ds/4 ≈ 0.333, which is independent of the spreading Euclidean space dimension [47]. This is explained by the formation in the gel formation point of a cluster with dimension ~ 2.5 tightening the reactive space [28, 29]. As one can see, the type of change of space in which the curing reaction occurs, from Euclidean (up to the gel formation point) to fractal (after it) reduces sharply the curing rate of system 2DPP+HCE/DDM. The fact that after the gel formation point the slope of the plots of Figure 3.23 is independent of Tcur indicates unequivocally a spatial (due to diffusion ' conditions of reagents) disorder nature, deﬁning the replacement of ds on d s . The characteristic size r(t) of the space region in which a particle (reagent molecule) was to time moment t can be estimated as follows [53]:

(3.33)

'

In Figure 3.24 the dependence of d s on r(t) is adduced, which proves to be linear and passing through the coordinates origin. This means that at zero level of reagent ' diffusion zero connectivity ( d s =0) of reactant space was reached, which does not allow realisation of the curing process. Figure 3.25 shows the dependence of the experimentally received proportionality coefﬁcient c in Relationship 3.28 on r(t), showing that c is also a function of r(t) and is described analytically according to the empirical equation [51]:

(3.34)

At r(t) = 0 c = 0 again and the curing reaction can not be realised. Hence, the results stated above have shown accuracy in the scaling approach to the description of the curing reaction of the haloid-containing epoxy polymer. Application of the indicated concept allows elucidation of the physical aspects of this process and the main distinctions of real chemical reactions from those obtained by computer simulation. Up to the gel formation point spatial disorder, deﬁned by the different intensities of diffusion of reagents at various curing temperatures, completely controls the curing reaction course. After the gel formation point, tightening cluster formation levels these distinctions [51].

115

Structure and Properties of Crosslinked Polymers

ln(1-α) 4.5

0.750 0.333

4.0 1 2 3.5 3 4 3.0 7.5

8.3

7.9

ln t

Figure 3.23 The dependences of (1 – α) on t in double logarithmic coordinates after the gel formation point for the system 2DPP+HCE/DDM at Tcur: (1) 353; (2) 373; (3) 393 and (4) 513 K. The upper dashed line gives a slope for the reaction in Euclidean space, the lower one in fractal space [51]

ds 3

2

1

0

5

10

15

r(t), rel. un.

Figure 3.24 The dependence of the reactive medium effective spectral dimension d s' on region size r(t), visited by reagents, for the system 2DPP+HCE/DDM [51]

116

The Fractal Models of Epoxy Polymers Curing Process c2/3 10

5

0

5

10

15 r(t), rel. un.

Figure 3.25 The dependence of the proportionality coefﬁcient in Relationship 3.28 on region size r(t), visited by reagents, for the system 2DPP+HCE/DDM [51]

Let us consider the physical signiﬁcance of the unusual (anomalous) diffusion of particles at the curing of epoxy polymers [3, 4, 53]. As it is known [53] unusual transport processes are described according to the equation:

(3.35)

where is the mean-square removing of a particle from its movement start, D′ is the generalised transport coefﬁcient and the power exponent at t is equal to [53]:

(3.36)

where df is the fractal (Hausdorff) dimension of the system, θ is the system connectivity index. At 0 ≤ μ < 1 one can speak about subdiffusive transport processes, at 1 < μ ≤ 2 about superdiffusive processes and μ = 1 corresponds to classical (Gaussian)

117

Structure and Properties of Crosslinked Polymers diffusion [53]. In turn, the exponent μ is connected with the Hurst exponent H by the equation [53]:

μ = 2H

(3.37)

The value of H deﬁnes the trajectory fractal dimension dw of the particles [53]:

, (dw ≥ 1)

(3.38)

The dimension dw can also be expressed through the system connectivity index [53]:

(3.39) '

Assuming ds = d s and df = d, one can calculate μ and θ values according to Equation 3.36. The exponent μ changes within the limits of 0.099–0.988, the index θ within the limits of 18.3–0.04 at Tcur change from 333 to 513 K. This means that the dimension dw varies within the interval of 20.27–2.02 (see Table 3.4) for the system 2DPP+HCE/ DDM under the same conditions [59]. Let us note that high dw values at small Tcur suppose a repeated return of non-reacted particles to the start of their movement. Re-movement of a particle from its movement start can be described with the aid of Relationship 3.33 and also in the following way [53]:

(3.40) For t = 1700 s (see Figure 3.1) calculation according to Relationships 3.33 and 3.40 shows that the r(t) value increases about 28 times with growth in Tcur within the range of 333–513 K. It is obviously the case that the smaller r(t), the fewer reagent molecules can be diffused up to curing process realisation and the more regions there are in the system where oligomer and curing agent molecules (jointly) do not enter, which decreases the curing rate and the conversion degree limiting value αlim. 118

The Fractal Models of Epoxy Polymers Curing Process The values μ = 0.099–0.988 quoted above suppose that at the curing of system 2DPP+HCE/DDM only slow diffusion (subdiffusive transport) is realised and reaching ' Tcur = 513 K of the condition d s = ds = d means that in this case classical Gaussian diffusion (μ = 1) is an upper limit and superdiffusive transport realisation is impossible. In Figure 3.26 the dependence αlim(μ) for the system 2DPP+HCE/DDM is adduced, from which the expected linear relationship between the indicated parameters follows. This relationship conﬁrms that the curing process of system 2DPP+HCE/DDM is controlled by diffusion [58].

αlim 1.0

0.5

0

0.5

1.0

μ

Figure 3.26 The dependence of the conversion limiting degree αlim on the exponent μ for the system 2DPP+HCE/DDM [58]

3.3 Microgel Formation in the Curing Process of Epoxy Polymers The ﬁrst theory of gel formation of crosslinked polymers, elaborated by Carothers and Flory, considered the gel formation point as formation of an inﬁnite network of chemical nodes [19]. Since this theory does not always agree with experimental data then the ‘gel formation period’ concept was proposed. According to the indicated concept two gel formation points exist. The ﬁrst corresponds to an appearance moment in a reactive medium of crosslinked clusters (microgels), characterised by non-fusibility

119

Structure and Properties of Crosslinked Polymers and non-solubility. The second gel formation point corresponds to an essentially later stage of reaction – transformation of a liquid ﬂuid system with crosslinked clusters in an elastic polymer [60]. The authors of papers [61, 62] considered theoretical conditions of realisation of the ﬁrst gel formation point with fractal analysis and percolation theory methods in the example of system 2DPP+HCE/DDM. In Figure 3.27 the kinetic curves α(t) for the system 2DPP+HCE/DDM at ﬁve curing temperatures Tcur are adduced. As it follows from this ﬁgure data, the reaction rate grows with an increase in Tcur. The curves α(t) can be described theoretically within the frameworks of the general fractal Relationship 3.7. As was noted above, if the dependence α(t) is plotted in double logarithmic coordinates, then from these plots in the case of their linearity a fractal dimension D value of microgels could be estimated. In Figure 3.28 such dependences for the system 2DPP+HCE/DDM at three curing temperatures are adduced. As follows from these plots, Tcur growth is accompanied by the slope reduction of linear plots α(t) in double logarithmic coordinates, or increasing D. Within the range Tcur = 295–513 K D increasing from 1.20 to 1.95 is observed (see Table 3.4). For the highest Tcur = 513 K plot slope a discrete change is observed, which corresponds to D growth from 1.95 to ~ 2.68. Such a transition within the frameworks of fractal analysis corresponds to the second gel formation point [28, 29], i.e., to network formation which tightens the whole sample. From the Figure 3.28 data it also follows that the second gel formation point for Tcur = 353 and 373 K in the t scale of Figure 3.27 is not reached.

α, %

5 4

80

3 2 40 1

0

0.5

1.0

1.5 t×10-3, s

Figure 3.27 The kinetic curves α(t) for the system 2DPP+HCE/DDM at Tcur: (1) 333; (2) 353; (3) 373; (4) 393 and (5) 513 K. The vertical arrows indicate the ﬁrst gel formation point [61]

120

The Fractal Models of Epoxy Polymers Curing Process e

In Table 3.5 the experimental values of gel formation time t g for the ﬁrst gel formation e point, determined by the IR-spectroscopy method, are adduced. These values t g are indicated by vertical arrows in Figure 3.27. It is interesting to note that the indicated gel formation point is reached for all Tcurs at approximately the same α value, which is equal to ~ 19%. For the explanation of this observation the authors [61, 62] used the percolation theory [63] and irreversible aggregation model [64], corresponding to simultaneous growth of many clusters that agrees with the real situation at curing of epoxy polymers. According to the model [64], growth of such clusters ceases in the case of their contact. Therefore it is possible to consider the ﬁrst gel formation point, e characterised by time t1 , as the point in which contact of many spherical microgels is realised. According to the percolation theory [63], the volume fraction f of such spheres can be determined according to the relationship:

(3.41)

where xc is the percolation threshold.

ln α 0

-0.5

-1 -2 -3

-1.0

-1.5 6

7

8

9

ln t

Figure 3.28 The dependences of the curing reaction conversion degree α on reaction duration t in double logarithmic coordinates, corresponding to Equation 3.7, for the system 2DPP+HCE/DDM at Tcur: (1) 353; (2) 373 and (3) 513 K [61]

121

Structure and Properties of Crosslinked Polymers

Table 3.5 The dependence of ﬁrst gel formation point reaching time t1e on curing temperature Tcur for the system 2DPP+HCE/DDM [62] Tcur, K

t1e × 10–3, s

295

5.04

333

1.44

353

0.36

373

0.30

393

0.18

513

0.12

e

If we suppose that time t1 corresponds to the percolation threshold of spherical microgels, closely ﬁlling reactive space, then xc = 0.19 and f = 0.79. Such a value of f actually corresponds to close packing of spheres of approximately equal diameter [65]. From this it follows that the ﬁrst gel formation point is characterised by the stopping of growth of the microgels, closely ﬁlling the reactive space at their contact. From Relationship 3.7 at the condition K1 = const. and c0 = const. it can be written [62]:

(3.42)

T

where t1 is the theoretical magnitude of the ﬁrst gel formation point reaching time, e α1 is the α value, corresponding to t1 and equal to ~ 19%. The combination of Relationships 3.22 and 3.42 allows estimation of the value of t1T . The comparison of experimental t1e and theoretical t1T dependences of the gel formation time for its ﬁrst point on the fractal dimension D of microgels for the system 2DPP+HCE/DDM is adduced in Figure 3.29. As follows from this comparison, good correspondence is received between theory and experiment (the logarithmic scale for t1 was used for convenience).

122

The Fractal Models of Epoxy Polymers Curing Process

ln t1 8

-2 1

6

4 1.0

1.5

2.0 D

Figure 3.29 The comparison of (1) experimental and (2) theoretical dependences of the ﬁrst gel formation point time t1 on fractal dimension D of microgels in logarithmic coordinates for the system 2DPP+HCE/DDM [61]

Therefore, application of fractal analysis and percolation methods allows elucidation that the ﬁrst gel formation point of crosslinked polymers in model [60] is a structural transition, which is realised at reactive space ﬁlling by microgels. The gel formation time in the indicated point is controlled by the fractal dimension D of microgels. Between the D value and the reactive medium viscosity η0 the correlation exists: D increasing causes strong growth in η0 [62]. It was shown above that curing of epoxy polymers can proceed in both Euclidean (three-dimensional) and fractal spaces. In the last case on conversion degree-reaction duration, continuous change occurs in the part of the kinetic curve α(t) almost up to the gel formation point in the structure of microgels, which is characterised by its fractal dimension D and, more precisely, monotonous increase in D occurs. At such D variation a curve α(t) has qualitative distinctions from a similar curve for curing of epoxy polymers in Euclidean space, namely practically linear growth of α as a t function is observed in the indicated part up to the gel formation point (α < 0.8). The authors of paper [66] studied the reasons and mechanism of the structure of microgels, which indicated variation on the system EPS-4/DDM example. The estimations of dimension D of microgels according to the methods described above with use of Relationship 3.1 showed that the D value changes within the limits of

123

Structure and Properties of Crosslinked Polymers 1.61–2.38. Such a D variation range assumes that the formation process of microgels proceeds according to the cluster–cluster mechanism, i.e., large microgels are formed from smaller ones, but not directly from oligomeric units [13]. In this case Equation 3.5 holds for large microgel dimension D which is formed from smaller ones with dimensions D1 and D2 (D1 ≥ D2). From the indicated equation it follows that in the case of absence of distribution of microgels by their fractal dimensions, i.e., where D1 = D2= … = Di a large cluster differs from the smaller ones it is formed from by size only and Equation 3.5 gives D = D1 = D2 = … Di. Hence, distribution of the structure of microgels, characterised by distribution of dimensions D, i.e., D1 ≠ D2 ≠ … ≠ Di, is required for D variation at curing reaction course (of t growth). A possible change in D(t) can be estimated theoretically as follows. If we assume that at time moment ti D of the distribution of microgels is equal to Di – Di+1, then according to Equation 3.5 Di = D at the condition Di+1 = D1, Di–1 = D2 can be calculated. The time range Δt = ti – ti–1 = ti+1 – ti is accepted to be equal to 500 s. The corresponding experimental D values are accepted as Di–1, Di+1. In Figure 3.30 the comparison of experimental and calculated dependences D(t) according to the indicated method for the system EPS-4/DDM is adduced. As one can see, the cluster–cluster aggregation mechanism, for which Equation 3.5 was obtained [20], explains quantitatively the variation in experimentally observed D(t) [66].

D 2

3.0 -3 2.5

1

2.0

1.5

0

1

2

3

t×10-3, s

Figure 3.30 The dependences of fractal dimension D(Dsp) of microgels on reaction duration t for the system EPS-4/DDM. (1) Calculation of D according to Equation 3.5; (2) calculation of Dsp according to Equation 3.43; and (3) D experimental values [66]

124

The Fractal Models of Epoxy Polymers Curing Process As was noted above, the reason for the variation in D(t) is the curing reaction proceeding in fractal space. This process by its physical signiﬁcance is similar to the formation of clusters with dimension D on lattices with dimension Dsp [31]. In paper [58] it was supposed that Dsp = D. The relation between D and Dsp is given by the following equation [31]:

(3.43)

where dl and ds are the chemical and spectral dimensions of the cluster (microgel), respectively. The following interconnection exists between dimensions D, dl and ds [67]:

(3.44) and

(3.45) where d is the dimension of Euclidean space in which a fractal is considered (it is obvious that in our case d = 3). From Equations 3.43–3.45 it follows that the condition Dsp = D is realised only at the following values of the indicated dimensions: D ≈ 1.67, dl = 1.0 and ds = 1.0. The last condition means that the microgel is formed by a linear macromolecule [35]. However, as the curing process proceeds polymeric chain branching occurs and this results in ds growth [35]. In turn, as follows from Equation 3.45, this will cause a corresponding increase in dl. In order to accept as ds this dimension value for a branched polymer chain (ds = 1.33 [35]) and as dl this dimension value determined for the cluster–cluster aggregates formed by the mechanism in computer simulation (dl = 1.42 [67]), then Equation 3.43 can be written as follows [66]:

(3.46)

125

Structure and Properties of Crosslinked Polymers Relationship 3.46 deﬁnes the following condition for branched polymers [66]:

(3.47)

Condition 3.47 allows it to be supposed that the largest cluster in the system is the cluster, forming fractal space in the curing process of system EPS-4/DDM (Figure 3.30). As it is known [28], D growth is observed as the macromolecular coil (microgel) molecular mass increases. The plots of Figure 3.30 reveal precisely such a tendency. The tightening cluster, i.e., spreading from the end of one system to another, is such a cluster after the gel formation point [28]. Such a cluster has dimension D ≈ 2.5 [28], which is shown in Figure 3.30 by a horizontal shaded line. In previous sections curves α(t) were described within the frameworks of scaling approaches for reactions of low-molecular substances [30, 47]. Equation 3.9 is used for the description of a reaction in a medium having large density ﬂuctuations. In Section 3.1 it was shown that the dependence (1 – α) on parameter td/(d+2), corresponding to Equation 3.9, is not linear and it can be linearised by the use of microgel dimension D instead of d in the indicated equation. This served as a reason for the assumption that the curing reaction proceeds in fractal space. In Figure 3.31 the dependence of ln (1 – α) on the parameter is adduced for the system EPS-4/DDM, which also proves to be linear. The data of Figure 3.31, together with the assumptions stated in previous sections, suppose that the curing reaction of the system EPS-4/ DDM proceeds in fractal space with dimension Dsp. Let us note that at t = 3 × 103 s the deviation of the dependence from linearity adduced in Figure 3.31 is observed. From the comparison with the data of Figure 3.30 one can see that this deviation corresponds to Dsp ≈ 3, i.e., to transition to non-fractal behaviour at Dsp = d that was expected. Therefore, the results stated above have again conﬁrmed that the distribution of D values is the main reason for the variation in the structure of microgels, characterised by its fractal dimension D. The change in D with increased reaction duration is well described quantitatively within the frameworks of the cluster–cluster aggregation mechanism. The fractal space in which the curing reaction proceeds is formed by the structure of the largest cluster in the system [66].

126

The Fractal Models of Epoxy Polymers Curing Process ln(1-α) 0

-1

-2

-3

20

60

100

in logarithmic Figure 3.31 The dependence of (1 – α) on the parameter coordinates, corresponding to Equation 3.9 for the system EPS-4/DDM [66]

In Section 3.1 the interconnection of the initial viscosity η0 of the reactive medium and the fractal dimension D of the microgels was shown in the system 2DPP+HCE/ DDM case, which is given analytically by Equation 3.22. Let us consider the same question for the system EPS-4/DDM. In Figure 3.22 the dependence D(t), where the D value was calculated according to Equation 3.4, is adduced for the system EPS-4/ DDM. As was noted above, according to the concept [60] the ﬁrst gel formation point (theoretical) corresponds to the moment of appearance in the reactive medium of the ﬁrst crosslinked microgels, characterised by non-fusibility and non-solubility. This point corresponds to a slope change at t = 1200 s in the dependence D(t) (Figure 3.22). The second gel formation point corresponds to a considerably later stage in the reaction – the transformation of the liquid ﬂuid system with crosslinked particles in an elastic polymer. In the dependence D(t) this point corresponds to the condition D ≈ 2.5, which is reached at t ≈ 3800 s. The second gel formation point corresponds to the notion generally accepted in physics of sol-gel transition as a network formation tightening the sample [28]. Within the frameworks of fractal analysis and irreversible aggregation models it has been shown [28] that this transition is characterised by the system universality class change, namely by transition from cluster–cluster aggregation (D ≈ 1.68) at t = 1200 s to particle–cluster aggregation (D ≈ 2.5) at t = 3800 s. The known α and D values allow estimation of the parameter K1c0η0 as t function according to Equation 3.7. Since the values of K1 and c0 in the experimental conditions of paper [68] are constant, then the indicated parameter can be considered as the

127

Structure and Properties of Crosslinked Polymers reactive medium current viscosity η, expressed in relative units. In Figure 3.32 the dependences η(t) for the system EPS-4/DDM at Tcur = 383 and 393 K are shown. As one can see, up to the point of formation of the microgels (t = 1200 s) very weak η growth is observed and then a sharp increase in η (of about two orders) occurs. Such an increase in η can be explained theoretically within the frameworks of the model proposed in paper [37], which uses percolation theory representations. The η value gives as follows [37]:

(3.48) where αc is the reaction conversion critical degree, corresponding to the second gel formation point or percolation network formation, m is the critical index. The αc value was accepted as being equal to the maximum value of α (αc ≈ 0.94 for Tcur = 383 and 393 K).

η, rel. units

D 2

400

2.5

2.0

3 200

1 1.5

0

1

2

3

1.0 t×10-3, s

Figure 3.32 The dependences of viscosity η of the reactive medium (1, 2) and fractal dimension D of microgels (3) on the reaction duration t for the system EPS4/DDM, cured at Tcur = 383 (1) and 393 K (2, 3) [68]

In turn, the complex critical index m is determined as follows [37]:

(3.49)

128

The Fractal Models of Epoxy Polymers Curing Process where ν is the percolation critical index, which is equal to 0.8 [37], dw is the random walk dimension, ds is the spectral dimension. According to the Aharony–Stauffer rule the dw value is estimated as follows [34]:

(3.50) and the ds value for crosslinked (branched) polymers is accepted to be equal to ~ 1.33 [35]. In Figure 3.33 the comparison of experimental (calculated according to Equation 3.7) and theoretical (calculated according to Relationship 3.48) η values, which are plotted for convenience in logarithmic scale, is adduced.

ln η 4

6

3

-1 -2 4

2

0

1

2

3

t×10-3, s

Figure 3.33 The comparison of the dependences of viscosity η of the reactive medium on reaction duration t calculated according to Equation 3.7 (1, 2) and Relationship 3.48 (3, 4) in logarithmic coordinates for the system EPS-4/DDM, cured at Tcur = 383 (1, 3) and 393 K (2, 4) [68]

As one can see, good correspondence of values of η calculated by the indicated methods was obtained. It is particularly important to note that Relationship 3.48

129

Structure and Properties of Crosslinked Polymers explains the sharp growth of η at t > 1200 s. The discrepancy in experimental and theoretical η values is due to the number of approximations used in the calculations. Relationship 3.48 indicates a reason for the unequivocally sharp growth in η at t > 1200 s. From this relationship it follows that at t growth, the exponent m increases owing to increasing D and, hence, dw at ν = const. and ds = const. This means that a decrease in αc – α÷ or an increase in the degree of conversion in the curing reaction is the only reason for the sharp growth in η. This postulate can be conﬁrmed within the frameworks of the Muthucumar concept [69], describing the viscosity of solutions of branched polymeric fractals. According to the concept [69], the viscosity increment Δr depends on the molecular mass M of such a fractal (microgel) as follows:

(3.51)

where d is the dimension of Euclidean space in which a fractal is considered. It is obvious that in our case d = 3. The dependences M(t) for the system EPS-4/DDM, cured at two Tcurs, are adduced in Figure 3.34. As one can see, weak growth in M is observed up to the ﬁrst gel formation point, indicated in Figure 3.34 by stroke-dotted lines, and in the range of t = 1200–3600 s a strong increase (ﬁve-fold) in M occurs. This means that growth in M and, hence, in η is due to the curing process of the space between microgels and microgels themselves in the indicated range t. Let us consider the change in the curing reaction process of the gyration radius of cured microgels Rg. The Rg value is connected with M according to the following fractal relationship [36]:

(3.52)

In Figure 3.34 the dependence Rg(t) for the system EPS-4/DDM, cured at Tcur = 393 K, is adduced. It is interesting to note the absence of similarity between the dependences M(t) and Rg(t). This observation requires special explanation. The D value in solution with consideration of the excluded volume interactions can be determined according to Equation 3.44. Gel formation transition (the range of t = 1200–3800 s, see Figure 3.32) or transition in the condensed state is characterised by a change in

130

The Fractal Models of Epoxy Polymers Curing Process the environment of the microgels and now instead of solvent, the oligomer and lowmolecular molecules of the curing agent are in a similar environment to the microgels. This results in fractal dimension change and for the condensed state its value df is determined as follows [70]:

(3.53)

M, rel. units 10

2

Rg, rel. units 1

2’ 1’

3

3 2 5 1

0

2

0 4 t×10-3, s

Figure 3.34 The dependences of molecular mass M (1, 2) and gyration radius of microgels Rg (3) on reaction duration t for the system EPS-4/DDM, cured at Tcur = 383 (1) and 393 K (2, 3). The lines 1’ and 2’ show the ﬁrst gel formation point at Tcur = 383 (1’) and 393 K (2’) [68]

Combination of Equations 3.44 and 3.53 at the conditions ds = 1.33 and d = 3 indicated above gives for branched (crosslinked) polymers [68]:

(3.54)

131

Structure and Properties of Crosslinked Polymers From Equation 3.54 it follows that at D ≈ 1.81 the df value will be larger than d. Such growth in microgels is restricted since they do not ‘enter’ the three-dimensional space and their density grows with increasing size (the effect is similar to blood coagulation) [71]. This results in the compactness of the microgels increasing, expressed by D growth (Figure 3.32) and formation of denser morphological formations from them in the condensed state (ﬂoccules) [72]. Let us note that D growth begins at the point t ≈ 1200 s, where D ≈ 1.81. The Figure 3.34 data show that within the range of t = 1800–3600 s, at two-fold increase in M very weak Rg growth occurs that is conﬁrmed by the conclusion made above. Hence, the results stated above demonstrated that the reason for the sharp increase in reactive medium viscosity after formation of microgels was curing of the space between them and the corresponding growth of the degree of curing reaction conversion. The growth of molecular mass of microgels after the mentioned point occurs practically at their constant sizes. These data conﬁrm the details of the concept of ‘gel formation period’. As it is known [73], in the case of different chemical reactions proceeding, including curing of crosslinked polymers, an essential role is played by the so-called steric factor p (p ≤ 1), showing that not all collisions of reacting molecules occur with the proper orientation for chemical bond formation of these molecules. The importance of this factor in such treatment is deﬁned by its proportionality to the reaction rate constant kr – the smaller p the smaller kr and the reaction proceeds with a lower rate. As it is elucidated within the frameworks of computer simulation of irreversible aggregation processes, the steric factor p plays an important role in them, deﬁning in essence both the mechanism of the aggregation process and the ﬁnal aggregate structure, characterised by its fractal dimension D [74]. So, at larger values of p, close to one, the diffusion-controlled mechanism of aggregate growth with relatively small values of D (~ 1.65) is realised and at small p of order 0.01 the mechanism of chemically limited aggregation with more compact ﬁnal aggregates (D ≈ 2.11) is realised [13, 75]. Proceeding from the importance of the steric factor p considered above, the authors of paper [76] performed the study of the change of character of the indicated parameter value in the curing process of epoxy polymers on the system EPS-4/DDM example. Let us consider the interconnection of the steric factor p value with the structure of the reaction products, which can be characterised by its fractal dimension D [37]. Within the frameworks of model representations of chemical reactions the reaction rate constant kr can be expressed as follows [73]:

132

The Fractal Models of Epoxy Polymers Curing Process

(3.55)

where R is the universal gas constant, T is the temperature at which the reaction proceeds, η is the reactive medium viscosity, [A] and [B] are two chemical reagents concentrations. Within the frameworks of fractal analysis [51] the concept of ‘accessibility’ of particles or clusters to active (reactive) centres of other particles or clusters for either reaction realisation is introduced. The process of formation of reaction cured products can be described as follows [37]. During an initial period of curing reaction at epoxy oligomer and curing agent interaction clusters are formed, which have dimension D, which can be calculated, for example, according to Equation 3.4. The D value characterises the degree of cluster structure ‘opening’– the smaller D the more intensive the penetration of clusters (or particles) into another cluster (the more ‘accessible’ it is) [77]. This is expressed analytically by Relationships 3.1 or 3.7. Differentiating Relationship 3.1 on t, let us obtain the reaction proceeding rate and then it can be written [76]:

(3.56)

where c1 is a constant. Assuming the multiplier after p in the left part of Relationship 3.56 as constant, the constant c1 can be estimated in this relationship from the known boundary conditions of irreversible aggregation model [74, 75]. This model assumes growth in D with a decrease in p that agrees with the tendency expressed in Relationship 3.56. Supposing that p = 1 is reached at a very high degree of cluster ‘accessibility’ and estimating this degree by the minimal value D = 1.5 [78] and supposing also that the indicated D value is realised at small t of order 100 s [68], let us obtain a simple relationship [76]:

(3.57)

The other limiting case of the model [74] corresponds to D = 2.11. Assuming that the indicated D value is reached at t = 3000 s [68] (see also Figure 3.32) let us obtain

133

Structure and Properties of Crosslinked Polymers from Equation 3.57 that these conditions correspond to p ≈ 0.037, which is close enough to p = 0.01 according to the model [74]. Using Equation 3.4, D variation can be calculated as a function of α or t. In Figures 3.35–3.37 the dependences p(t), calculated according to Equation 3.57 for the system EPS-4/DDM at curing temperatures Tcur = 383, 393 and 403 K, respectively, are adduced. Comparison of these ﬁgures shows qualitative identity of the dependences adduced in them – as the curing reaction proceeds the p value decreases, as the t part is small this decay is rapid and then it decelerates sharply on reaching p ≤ 0.1. Tcur change has clearly expressed quantitative consequences. So, the initial values of p are higher for larger Tcur. In other words, the larger Tcur is the more pronounced reaction diffusive regime is expressed. Tcur increasing results in faster p decay with time t that is due to increasing reaction rate.

h, rel. units

p 0.4

4

1 2

0.2 2

0

A

2

0 4 t×10-3, s

Figure 3.35 The dependences of the steric factor p (1) and chromatographic peak height h (2) on reaction duration t for the system EPS-4/DDM at Tcur = 383 K [76]

In Figures 3.35–3.37 the dependences of chromatographic peak IGC height h (in relative units) as a function of t for corresponding Tcur are also shown. It is easy to see that the minimum on the curve h(t), designated by the letter A and identiﬁed as corresponding to the gel formation point, coincides on a time scale with a bend in curve p(t) for all three used Tcurs. As was noted above, the bend point in curve p(t), corresponding to the condition p ≤ 0.1, answers the formation of compact clusters with large fractal dimension D ≥ 2.11 [75].

134

The Fractal Models of Epoxy Polymers Curing Process

h, rel. units

p 0.8

8

2 A 0.4

4 1

0

0 4 t×10-3, s

2

Figure 3.36 The dependences of the steric factor p (1) and chromatographic peak height h (2) on reaction duration t for the system EPS-4/DDM at Tcur = 393 K [76]

h, rel. units

p 0.8

8 B 2

0.4

4 A 1

0

2

0 4 t×10-3, s

Figure 3.37 The dependences of the steric factor p (1) and the chromatographic peak height h (2) on the reaction duration t for the system EPS-4/DDM at Tcur = 403 K [76]

135

Structure and Properties of Crosslinked Polymers In Figure 3.38 the kinetic curves α(t) for the system EPS at three used Tcurs are adduced. The points in the curves α(t) corresponding to point A in curve h(t) (Figures 3.35–3.37) are indicated by horizontal arrows. The dashed lines in Figure 3.38 show tangents to initial parts of curve α(t). Lastly, the vertical arrows in Figure 3.38 show points in curve α(t) corresponding to point B in curve h(t) – the so-called glass transition (see Figure 3.37). Let us note that the point in curve α(t) corresponding to A deﬁnes the slope reduction of tangents to the indicated curves or a decrease in reaction rate. The point in curve α(t) corresponding to B signiﬁes the change in system fractal dimension (that is easy to deﬁne out of the dependence α(t) in double logarithmic coordinates, corresponding to Relationship 3.1) or system universality class change [13].

α 3 2 1

1.0

0.5

0

4

8 t×10-3, s

Figure 3.38 The kinetic curves α(t) for the system EPS-4/DDM at Tcur: (1) 383; (2) 393 and (3) 403 K. The dashed lines are tangents to the initial parts of curves α(t). Horizontal arrows indicate completion of the formation of microgels, vertical arrows indicate the spatial network over the entire sample formation [76]

The data of Figures 3.35–3.38 allow the system EPS-4/DDM curing process within the frameworks of irreversible aggregation models and fractal analysis to be described as follows. The formation of separate crosslinked clusters (microgels) occurs in the initial stage of the curing reaction [19]. The range of the fractal dimensions of these clusters (D = 1.60–2.22) indicates that this formation is realised by a cluster–cluster mechanism, i.e., small microgels form a larger microgel and so on [13]. This excludes the formation of a chemical crosslinking homogeneous network over the entire sample

136

The Fractal Models of Epoxy Polymers Curing Process (by the ‘tennis-racket’ type) [79, 80]. When the dimension of the microgels reaches the limiting value (D ≈ 2.20 [81]), their growth ceases. This situation in curve p(t) corresponds to a bend, in curve h(t) to point A (to minimum), in curve α(t) to a reduction in reaction rate. During this stage systematic growth occurs in the viscosity of the reactive medium that is reﬂected in h reduction, which is proportional to the square root of diffusivity [76]. The bend point in curve p(t) can be identiﬁed as the transition from a diffusive regime of curing reaction to a kinetic one and this transition is characterised by the following parameters: p ≈ 0.1 and D ≈ 2.1. It follows to note that a similar classiﬁcation of curing regimes is highly conditional. It is signiﬁcant that if the indicated transition corresponds to a different t on a temporal scale for various Tcurs, then on the scale α it is reached at the condition α ≈ const. ≈ 0.57 (Figure 3.38). After formation of the microgels, the curing reaction of the space between them begins, which, as it is noted above, occurs according to a particle–cluster mechanism. The smaller sizes of the oligomer and curing agent molecules in comparison with the size of the microgels result in growth in h and, hence, to an increase in diffusivity. At point B in curve h(t) this process is complete and now the system EPS-4/DDM structure is in the condensed state and is described within the Witten–Sander model with fractal dimension df ≈ 2.5 [82]. Therefore, a universality class replacement from cluster–cluster to particle–cluster occurs [13]. Gel formation by a similar mode is conﬁrmed experimentally in paper [29] in the example of polystyrene physical gels. It is obvious that the proposed curing mechanism of the system EPS-4/DDM corresponds completely to notions about gel formation phenomena as the ‘gel formation period’, but not the ‘gel formation point’ [19, 60]. This period occupies the temporal range A–B in curve h(t) (Figure 3.37). Strictly speaking, gel formation is the critical structural transition and it should be identiﬁed as a spatial network, the formation tightening the entire reactive system [28]. It has been shown both experimentally [29] and theoretically [28] that in the gel point the fractal dimension of the gel-forming system structure is equal to ~ 2.5. Therefore the gel formation point is identiﬁed in such (physically the most strict) treatment as point B in curve h(t). Reduction in p with growth in t (Figures 3.35–3.37) is due to purely steric reasons, caused by complexity of the structure of the reacting objects. The more complex this structure is the more difﬁcult it is to realise mutual orientation of such reacting objects, which allows chemical reaction [83]. The most powerful factor in this sense is screening of aggregate internal regions by external ones, which is typical for fractal objects [64]. In Figure 3.39 the dependences p(α) for the system EPS-4/DDM at three used values of Tcur are adduced. These dependences demonstrate clearly the change in p in the course of the curing reaction: as the reaction proceeds p reduces linearly and at α = 1 p = 0 [76].

137

Structure and Properties of Crosslinked Polymers Hence, the results stated above have shown that the change of structure of the microgels, characterised by its fractal dimension, in the system EPS-4/DDM curing reaction course inﬂuences both the steric factor value and the degree of conversion of the curing reaction. The irreversible aggregation models and application of fractal analysis allows complete description of the curing reaction of the system EPS-4/DDM to be received and physical treatment of the gel formation process in them to be given.

α 0.8 -1 -2 -3 0.4

0

0.5

1.0

α

Figure 3.39 The dependences of the steric factor p on the degree of conversion of the curing reaction α for the system EPS-4/DDM at Tcur: (1) 383; (2) 393 and (3) 403 K [76]

3.4 Synergetics of the Curing Process of Epoxy Polymers As it is known [84], synergetics studies the universal rules of self-organisation of spatial structures in dynamic systems of a different nature. This discipline is based on the physical essence of the process of adaptation of systems to external inﬂuences by way of self-organisation of structures and is universal for animate and inanimate nature systems. Adaptation is the process of reforming lost stability of the structure with new, more stable self-organisation of the structure. Fractal (multifractal) structures are formed in the reformation process, which is impossible to describe correctly within the frameworks of Euclidean geometry [84].

138

The Fractal Models of Epoxy Polymers Curing Process Everything said above relates to synthesis processes of polymers of various types – in these processes by way of self-organisation fractal objects (macromolecular coils, microgels) are formed, for which temperature, synthesis (curing) duration and so on are external inﬂuences. The authors of paper [85] studied the inﬂuence of the indicated factors on the curing process of epoxy polymers within the frameworks of synergetics. The D values calculated with the aid of Relationship 3.1 for the system 2DPP+HCE/ DDM are adduced in Table 3.6. As it follows from these table data, an increase in Tcur results in fractal dimensions of microgels within a sufﬁciently broad range (D = 1.20–1.95). The spectral dimension ds, characterising the degree of connectivity of microgels [35], can be determined with the aid of Equation 3.44. The ds values calculated according to the indicated equation for the studied epoxy polymers microgels are also adduced in Table 3.6. It is signiﬁcant that for two of the least curing temperatures (333 and 353 K) linearly connected microgels are not formed (ds < 1), at Tcur = 373 K a macromolecular coil is formed (ds ≈ 1) and only at Tcur = 393 and 513 K are completely cured microgels formed (ds ® 1.33). This means that in the cases where Tcur = 333 and 353 K curing of epoxy polymers is realised only at the gel formation stage, i.e., at t > 1.5 × 103 s [51].

Table 3.6 The dimensions of cross-linking structures for the system 2DPP+HCE/DDM [85] Tcur, K

Up to gel formation point

After gel formation point

D

ds

df

333

1.20

0.74

2.30

353

1.53

0.86

2.34

373

1.68

1.01

2.45

393

1.78

1.24

2.79

513

1.95

1.32

2.80

As it is known [84], the adaptivity universal algorithm is realised at the selforganisation of structures to transition from the previous point of structure instability to the following one:

139

Structure and Properties of Crosslinked Polymers

(3.58)

where Zn and Zn+1 are critical values of the governing parameter, controlling the structure formation process (in our case, curing); their ratio deﬁnes the system adaptivity measure Am to structural reformation; m is the number of reformations; Δi is a structure stability measure. As a governing parameter in the curing of epoxy polymers two factors can be used: curing temperature Tcur and curing duration t, which reﬂect the external inﬂuence on the self-organisation process of crosslinking spatial structures. Let us consider the t ﬁrst Am estimation using t ( Am ) as a governing parameter. For this purpose let us accept Zn and Zn+1 equal to α values at t = 500 and 1500 s, respectively. In Figure 3.40 the dependence D of microgels and df of the condensed state structure after the t gel formation point on the Am value are adduced. The data of this ﬁgure allow the t following conclusions to be made. Firstly, for crosslinked structures the Am value is min necessary, which is larger than the minimal adaptivity measure Am , equal to 0.213 t according to the gold proportion rule [84]. Secondly, the maximum value of Am ≈ 1.0 gives Euclidean structures with dimension df = d = 3. Thirdly, the dependence of t fractal dimension of formed cured structures on Am has the following form [85]:

(3.59)

From Equation 3.59 it follows that in the considered case the adaptivity measure is equivalent to the system’s curing ability. In turn, the limiting conversion degree αlim is deﬁned by dimension D of the microgels, formed up to the gel formation point, which follows directly from the Figure 3.41 data. As one should expect, for oligomer molecules (D = 1) αlim = 0, i.e., the curing reaction does not occur. The dependence αlim(D) can be described according to the following equation [85]:

(3.60)

140

The Fractal Models of Epoxy Polymers Curing Process

D, df 3

2

1

-1 -2

0 0.2

1.0

0.6

Amt

Figure 3.40 The dependence of the fractal dimension of crosslinked structures up t to D (1) and after df (2) the gel formation point on the value Am for the system 2DPP+HCE/DDM [85]

αlim 1.0

0.5

0 1.0

1.5

2.0

2.5

D

Figure 3.41 The dependence of the limiting conversion degree αlim on fractal dimension D of microgels for the system 2DPP+HCE/DDM [85]

141

Structure and Properties of Crosslinked Polymers Let us note that the maximum value αlim = 1.0 is reached at D = 2.27, i.e., at the greatest possible dimension of the macromolecular coil (microgel) of branched (crosslinked) polymer [81]. t

Combination of Equations 3.59 and 3.60 allows the dependence of αlim on Am to be obtained [85]:

(3.61) t

It is signiﬁcant that for the maximum value αlim = 1.0 Am = 0.70, which according to the gold proportion rule corresponds to Δi = 0.232, m = 4 and for the smallest t value αlim of order 0.05 Am = 0.232, i.e., it corresponds to Δi = 0.232, m = 1. In other words, a sharp increase in the limiting degree of conversion is realised at constant and small microgel stability measure, corresponding to its ability to change structure (curing) owing to possible change in the reformation number m (increasing) only. The proposed synergetic model allows temperature boundaries of realisation of the curing process to be estimated and its optimal temperature regime to be selected. T For this let us estimate the system adaptivity measure by curing temperature Am as follows [85]:

(3.62) ∞

where the maximum value Tcur = 513 K was used as Tcur . t

T

In Figure 3.42 the relation of adaptivity measure Am and Am for the system 2DPP+HCE/DDM up to and after the gel formation point is adduced. As follows T t T from this ﬁgure data, at Am growth an increase in Am is observed and at Am ≈ 0.785 t the Am value reaches its asymptotic value (~ 0.60 up to the gel formation point and T T ~ 0.95 after it). The indicated Am value corresponds to Tcur = 403 K. At Am , which t is equal to 0.60 up to the gel formation point and 0.54 after this point, Am = 0, i.e., T the curing process is not realised. The indicated Am magnitudes allow determination of the lower value of Tcur as equal to 287 and 318 K, respectively. Let us note that curing after the gel formation point can proceed at lower Tcur than before it (even at room temperature).

142

The Fractal Models of Epoxy Polymers Curing Process

Amt 1.0

0.5 -1 -2 0 0.50

0.75

1.0

AmT t

Figure 3.42 The relation of adaptivity measures by curing duration Am and curing T temperature Am up to (1) and after (2) the gel formation point. The vertical T dashed line indicates the Am value corresponding to curing stationary temperature regime transition for the system 2DPP+HCE/DDM [85]

From the data of Figure 3.42 it follows that at Tcur ≥ 403 K for the system 2DPP+HCE/ DDM the curing stationary temperature regime is reached, corresponding to the t condition Am = const. As it is known [84], for synergetic systems the general law is observed: at variation of the external parameter the system behaviour changes from simple to chaotic. However, a certain interval in the external parameter exists, in which system behaviour is ordered and periodic. The ordering consists in the fact that system behaviour is reproduced in each time moment. The number a can be doubled from the following equation [84]:

(3.63)

where Zn is the governing parameter value at which the period doubles a times, Z∞ is the limiting value of this parameter, δ is Feigenbaum’s constant (δ ≈ 4.67 [84]).

143

Structure and Properties of Crosslinked Polymers

∞

∞

∞

Presenting Z∞ as Tcur / Tcur = 1 and Zn = Tcur/ Tcur and assuming also a = 1, we receive [84]:

(3.64)

T

∞

From Equation 3.64 the value of Tcur = Am Tcur , corresponding to the transition from simple to chaotic behaviour , can be estimated. This value of Tcur = 403 K is in excellent agreement with the empirically established temperature boundary of transition to curing stationary regime (Figure 3.42). Hence, the indicated transition is deﬁned by synergetics laws and is a general effect. So, the analogous phenomenon of transition to chaotic behaviour at a = 1 was found for the structure of polymer composites [86].

ds 1.5

1.0

0.5

0 0.6

0.8

1.0

AmɌ

Figure 3.43 The dependence of the spectral dimension ds of microgels on the T

adaptivity measure Am for the system 2DPP+HCE/DDM. The horizontal dashed line indicates the linear connectivity of microgels [85]

144

The Fractal Models of Epoxy Polymers Curing Process T

In Figure 3.43 the dependence of the spectral dimension ds of microgels on Am is adduced for the system 2DPP+HCE/DDM. As one can see, the value ds = 1.0 is reached at Tcur ≈ 363 K, which corresponds to a linear macromolecule [35] and at Tcur < 363 K linearly connected chains do not form in the curing process. At Tcur ≈ 403 K, corresponding to the transition to stationary temperature regime T

( Am ≈ 0.785), the ds value is close to 1.33, i.e., to the maximum spectral dimension for tightly cured macromolecules (microgels) [35]. Hence, the results stated in the present section show the applicability of synergetics principles to the description of curing processes of epoxy polymers. This means that both microgels up to the gel formation point and the tightening of the reactive space cluster after it are self-organising fractal structures, whose dimension is deﬁned by the curing system adaptivity measure. The optimal temperature regime of the curing process was established [85].

3.5 The Nanodimensional Effects in the Curing Process of Epoxy Polymers into Fractal Space In paper [40] it has been shown that the curing of epoxy polymers can occur in both Euclidean three-dimensional space and in fractal space. In the last case the space dimension is equal to the fractal dimension D of microgels, formed in the curing process. The main difference of conversion degree-reaction duration (α – t) kinetic curves in the last case is practically linear dependence α(t) almost up to the gelation point and variation (increase) of the D value on this part of curve α(t). The authors of papers [87–90] carried out a further study of curing of epoxy polymers in fractal space, in particular the reaction rate constant kr and the self-diffusivity Dsd of microgels changing character in the example of the curing of the system EPS-4/DDM [51]. For the system EPS-4/DDM the average value kr = 0.97 × 10–3 mol l/s was determined by the IGC method [51]. From Equation 3.4 the value of c can be determined at the average magnitudes of the parameters included in it: t = 1.5 × 103 s, D = 1.99 and α = 0.35. In this case c = 0.0244 mol l/s. As the calculation showed, kr reduction from 4.16 to 0.76 × 10–3 mol l/s was observed within the range of t = 500–2500 s. The range of the obtained values of D assumes that the formation of microgels occurs according to the cluster–cluster mechanism [91]. In this case the molecular mass M value of microgels is determined according to the following scaling relationship [5]:

(3.65)

145

Structure and Properties of Crosslinked Polymers The gyration radius Rg of microgels is connected with M by the following relationship [5]:

(3.66)

The obtained results allow analysis of the curing kinetics of the system EPS-4/DDM to be carried out within the frameworks of irreversible aggregation models [92]. In the general case the relationship between kr and Rg can be presented as follows [92]:

(3.67)

In turn, the exponent ω is deﬁned by the parameters describing motion of clusters (microgels) in space and their structure. This interconnection has the following form [92]:

(3.68)

where γ characterises the dependences of self-diffusivity Dsd of microgels on their −γ sizes (Dsd ~ R g ), d is the space dimension in which the curing reaction occurs, dw is the dimension of the random walk trajectory of microgels. For the reactions in Euclidean space d = 3, dw = 2 (Brownian motion of microgels), γ = 1 and then ω = 0. This means that in the given case the condition should be fulﬁlled [89]:

..

(3.69)

Condition 3.69 is conﬁrmed experimentally (the kr value does not change with Rg growth [51]) and is general for any macromolecular reactions in Euclidean spaces [93]. For the curing reaction proceeding in fractal space the situation differs completely from

146

The Fractal Models of Epoxy Polymers Curing Process that described above. This aspect attains a special meaning within the frameworks of nanochemistry [94], and therefore deserves consideration in more detail. As it is known [94], in nanochemistry there are two fundamental notions – nanoparticles and nanoreactors: the ﬁrst characterises the dimensional parameter while the second deﬁnes the function of nanoobjects. Thus, an iron cluster loses almost completely its speciﬁc properties (ionisation energy, magnetism) and approaches metallic iron with the number of atoms in a cluster n = 15. At n > 15 it remains a nanoobject in the dimensional sense, but it loses ‘nanoreactor’ qualities, for which properties become a size function. In Figure 3.44 the dependence of the curing rate constant kr on the diameter 2Rg of microgels, which has a very speciﬁc form, is adduced. Within the range of microgels (although the term ‘nanogel’ is more precise) with diameters less than 10 nm, the value of kr is a clearly expressed, rapidly decreasing function of diameter 2Rg and at 2Rg ≥ 100 nm the indicated dependence is practically absent. Let us note that the size 100 nm is assumed as an upper limit (although conditionally) for nanoworld objects [94]. Hence, the data in Figure 3.44 clearly demonstrate that a microgel at 2Rg < 100 nm is a nanoreactor in which the reaction (curing) rate is a strong function of its size, and at 2Rg ≥ 100 nm the microgel loses this function and in essence becomes a chemically inert particle. Let us note that the indicated nanoreactor–nanoparticle transition is possible only in the fractal space. In Euclidean space these notions do not differ (kr = const.).

kr×104, mol⋅l/s 40

20

0

100

200

300

2Rg, nm

Figure 3.44 The dependence of the reaction rate constant kr on diameter 2Rg of microgels for the system EPS-4/DDM [88]

147

Structure and Properties of Crosslinked Polymers In Figure 3.45 the dependence kr(Rg) for the system EPS-4/DDM in double logarithmic coordinates is shown, which is well approximated by a straight line. From the slope of this straight line the value 2ω = –0.58 can be determined. As was noted above, the space dimension in which the curing reaction occurs is equal to D and the value of dw can be determined according to the Aharony–Stauffer rule (Equation 3.50). Hence, Equation 3.68 for the considered case can be rewritten as follows (for any D value) [88]:

(3.70)

Then according to Equation 3.70 γ = –0.42 can be obtained. This means that the self-diffusivity value Dsd decreases with Rg growth much more slowly (Dsd ~ ) −1 compared with the reaction in Euclidean space (Dsd ~ R g ). The indicated difference is demonstrated in Figure 3.46 in a graphic form.

ln kr -5

-6

-7

-8 2

3

4

5 ln Rg

Figure 3.45 The dependence of the reaction rate constant kr on the gyration radius Rg of microgels in double logarithmic coordinates for the system EPS-4/DDM [89]

148

The Fractal Models of Epoxy Polymers Curing Process

Dsd, rel. units 4

2 2 1 0

1

2

3

t×10-3, s

Figure 3.46 The dependences of the self-diffusivity Dsd of microgels on the curing reaction duration t in Euclidean (1) and fractal (2) spaces for the system EPS-4/ DDM [90]

The molecular mass M of microgels depends on the curing duration t as follows [92]:

(3.71)

As was noted above, in Euclidean space 2ω = 0 and the exponent in Relationship 3.71 is equal to 1. This assumes M ~ t. For the reaction in fractal space 2ω < 0 and the exponent in Relationship 3.71 is less than 1. This means that in fractal space the value of M grows more slowly than in Euclidean space. This relation for the system EPS-4/DDM is shown in Figure 3.47. Since the α value in the second case is larger than in the ﬁrst this means that the reaction in fractal space gives a larger number of small clusters (microgels).

149

Structure and Properties of Crosslinked Polymers

M, rel. units 1

3

2

1

0

2

1

2

3

t×10-3, s

Figure 3.47 The dependence of the molecular mass M of microgels on the curing reaction duration t in Euclidean (1) and fractal (2) spaces for the system EPS-4/ DDM [90] The kr change indicated above can be obtained immediately from the Smoluchowski formula, which has the following form [92]:

(3.72) −1

For the reaction in Euclidean space Dsd ~ R g and kr = const., for the reaction in fractal space for the system EPS-4/DDM Dsd ~

and kr ~

, i.e., it is supposed that

kr is reduced as the curing process proceeds (the growth of Rg or M) [87]. Let us note in conclusion the strong dependence of kr on the structure of the microgels, characterised by the fractal dimension D (Figure 3.48). As follows from Figure 3.48, a sharp decay in kr is observed for D growth at D < 2 and the values kr on the asymptotic branch are attained at D > 2. As it is known [92], within the frameworks of irreversible aggregation models the following relationship is true:

(3.73)

150

The Fractal Models of Epoxy Polymers Curing Process

kr×103, mol⋅l/s 4

2

0 1.5

1.9

2.3 D

Figure 3.48 The dependence of the reaction rate constant kr on the fractal dimension D of microgels for the system EPS-4/DDM [89]

If for Euclidean space, for example, with d = 3, the exponent in Relationship 3.73 is constant and equal to 1, then for fractal space with variable value D the situation will be essentially different. For D < 2 the exponent in Relationship 3.73 is less than zero and Rg growth results in kr reduction under other equal conditions. At D = 2 kr does not depend on Rg. And ﬁnally, at D > 2 the kr value should increase with Rg growth. This is expressed by the sharp decay of kr at D < 2, since both Dsd and

RgD −2 reduce this parameter as M of the microgels increases. At D > 2 Dsd reduction D −2 growth and kr decay as the increase is compensated for to a certain extent by R g in M is decelerated. It is easy to see that at D = 2.48 for the system EPS-4/DDM the condition kr = const. is realised. Analytically the correlation kr(D) can be presented as follows [88]:

(3.74)

Therefore, the results presented in this section have shown that for a curing reaction in fractal space the reduction in reaction rate constant is typical as this reaction

151

Structure and Properties of Crosslinked Polymers proceeds. The formation of a large number of microgels with smaller molecular mass in comparison with the reaction in Euclidean space at the same conversion degree is also typical for such a reaction. The dimensional border between a nanoreactor and a nanoparticle for the considered curing reaction has been obtained.

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158

4

The Description of Crosslinked Rubbers within the Frameworks of Fractal Analysis and Local Order Models

The application of fractal analysis for the description of the behaviour of rubbers is difﬁcult because of the fact that these materials are (or are close to) Euclidean objects. Nevertheless, at present the theory of elasticity and entropic high-elasticity of fractals is developed, which differs principally from the classical theory. The change of molecular mobility, characterised by fractal dimension of a chain part between crosslinking nodes, is of interest for rubbers. Lastly, local order models can be used successfully for quantitative description of the nucleation process of crystalline regions and the melting temperature of rubbers. These and some other questions will be considered in detail in the present chapter.

4.1 Molecular and Structural Characteristics of Crosslinked Polymer Networks The main postulate on which exposition of material in the present section is based is the ‘dynamic polymer network’ concept [1]. This concept assumes that molecular and structural characteristics of crosslinked polymer networks are deﬁned not only by the chemical constitution of their chains, but also by boundary conditions (e.g., crosslinking density), external inﬂuence (for example, tension strain) and so on. As a rule, at present crosslinked polymer networks are characterised within the frameworks of entropic rubber high-elasticity concepts [2, 3]. However, in recent years works indicating a more complex structure of crosslinked rubbers have appeared. Flory [4] demonstrated the existence of dynamic local order in rubbers. Balankin [5] showed principal inaccuracy of the entropic high-elasticity theory and proposed a high-elasticity fractal theory of polymers. These observations suppose that more complete characterisation of these materials is necessary for the correct description of the structure of rubbers and their behaviour at deformation. In paper [6] this was carried out by the combined use of a number of theoretical physical concepts, namely the rubber high-elasticity entropic theory, the cluster model of the amorphous state structure of polymers [7, 8] and fractal analysis [9]. Let us consider the theoretical determination of the main parameters of crosslinked polychloroprene (PCP) using the models mentioned above. 159

Structure and Properties of Crosslinked Polymers The density of the chemical crosslinking nodes network νc was determined according to the rubber high-elasticity theory [10]:

(4.1)

where ρ is the polymer density, NA is Avogadro’s number, Mc is the molecular weight of a chain part between chemical crosslinking nodes. For PCP νc = 3.55 × 1025 m–3. The distance Rc between chemical crosslinking neighbouring nodes was calculated according to Equation 1.14. In paper [6] F = 4 was accepted. The estimations showed that using values F = 3 or 5 the discrepancy did not exceed 10%. At F = 4 for PCP Rc = 6.9 nm. The chain part length Lc between chemical crosslinking neighbouring nodes was determined according to the equation [11]:

(4.2)

where S is the macromolecule cross-sectional area. Let us note that Equation 4.2 supposes dense packing of macromolecules [11]. For rubbers it is necessary to introduce a correction, accounting for the large free volume c fg, the limiting value of which is equal to 0.159 [12]. Then the value of Lc corrected by the indicated method is equal to [6]:

(4.3)

c

For PCP it was received: Lc = 81.4 nm [6]. The degree of rolling-up β of macromolecules can be calculated according to the equation [13]:

160

The Description of Crosslinked Rubbers within the Frameworks of Fractal Analysis and Local Order Models

(4.4)

Simulating the PCP macromolecule as a free-rotated chain (this means that its fractal dimension Dch = 2 [14, 15]), the statistical segment length lst according to the relationship can be calculated [16]:

(4.5)

For PCP lst = 0.585 nm and then the characteristic ratio C∞, which is a polymer chain statistical ﬂexibility indicator [16], can be calculated according to Equation 1.8. The C∞ value for PCP is equal to 4.0, which is close enough to the C∞ value for cis-polyisoprene, which varies within the limits of 5.0–5.3 [10, 17]. The statistical segment number Nst per chain part between chemical crosslinking nodes is equal to [6]:

(4.6)

For PCP the estimation according to Equation 4.6 gives Nst = 139. This is also close enough to the estimation in paper [18], where it was accepted that one statistical segment contains ~ 2.30 monomer links. In this case Nst = 108 [18]. We shall receive Nst = 139 using that the statistical segment contains on average ~ 1.80 monomer links. The relative fraction of dynamic local order (clusters) ϕcl can be estimated within the frameworks of the cluster model of the amorphous state of polymers [11]:

(4.7)

We shall obtain ϕcl = 0.0144 for PCP according to Equation 4.7.

161

Structure and Properties of Crosslinked Polymers The structure fractal dimension df of PCP is estimated according to the relationship [15]:

(4.8)

where S is given in Å2. Let us note one important methodological aspect. The macromolecule cross-sectional area S can be either accepted according to the literary data [19] or calculated from X-ray diffractograms as follows (the example of a similar diffractogram for PCP is adduced in Figure 4.1). Bragg’s interval dB can be determined by the angular position 2θ of the main peak according to the formula [20]:

(4.9)

where λr is the X-ray wavelength, which is equal to 1.5418 Å for CuKα [21].

I, rel. units

2θ Figure 4.1 The X-ray diffractogram of PCP [6]

162

The Description of Crosslinked Rubbers within the Frameworks of Fractal Analysis and Local Order Models The macromolecule diameter dM can be calculated further (under the condition that it is simulated as a cylinder) according to the equation [22, 23]:

(4.10)

Proceeding from the value of dM it is easy to calculate the macromolecule crosssectional area S, which turns out to be equal to ~ 29.83 Å2 for PCP. This value is close to experimentally determined values of S for polymers, similar by chemical structure to PCP. So, for cis-polyisoprene S = 28 Å2 [19]. The density νcl of a physical entanglement cluster network, the nodes of which are local order regions, is calculated according to Equation 1.11 and the molecular weight Mcl of the chain part between clusters according to Equation 1.3. The functionality F of the clusters was calculated according to Equation 1.51. All the parameters calculated for non-deformed PCP are adduced in Table 4.1. Let us consider further the calculation of the same characteristics of a PCP network, stretched up to some ﬁxed drawing ratio λ, for example, λ = 3. Assuming network s afﬁne deformation the value of Rc at λ = 3 ( Rc ) becomes [6]:

(4.11)

It is obvious that the parameters determined in points 1–3 are the same for nondeformed and stretched networks. The remaining characteristics were determined as follows. The value of βs is equal to [1]:

(4.12)

163

Structure and Properties of Crosslinked Polymers

Table 4.1 The characteristics of non-deformed and stretched crosslinked PCP networks [6] Characteristics and their measuring units

Non-deformed network

Stretched (λ = 3) network

Mc

22

22

ρ, g/cm3

1.30

1.30

S, Å2

29.83

29.83

νc×1025, m–3

3.5

3.5

Rc, nm

6.9

20.7

81.4

81.4

b

0.086

0.254

lst, nm

0.585

1.12

C∞

4.0

7.59

Nst

139

73

nmon

1.80

3.41

ϕcl

0.0144

0.0274

2.934

2.934

8.5

8.2

Mcl

9.18

9.40

Fcl

2.40

2.35

Lcc , mm

df νcl × 10 , m 25

–3

The fractal dimension of a chain part between chemical crosslinking nodes for the was determined according to the following formula, which is stretched network similar to Equation 2.10 [15]:

(4.13)

164

The Description of Crosslinked Rubbers within the Frameworks of Fractal Analysis and Local Order Models or

(4.14)

In this case Dch decreases because of stretching of the chain part between chemical crosslinking nodes and the value of at λ = 3 is equal to 1.38, which points to the essential reduction of network molecular mobility [11]. This prediction is conﬁrmed experimentally [24]. s

The number of statistical segments N st can be determined with the aid of the relationship [15]:

(4.15)

= 62.8.

Assuming for real rubbers df = 2.95 [25], let us obtain

Then from Equation 4.6 the statistical segment length for the stretched PCP was obtained, which is equal to ~ 1.30 nm. This means lst increases with growth in λ. In other words, chain statistical ﬂexibility is determined not only by the chemical structure of the PCP chain, but also by network deformation conditions. Since

was calculated according to Equations 4.13 or 4.14, the value of lst for a

non-stretched network was used, therefore one should recalculate this parameter using . Such a recalculation results in a somewhat larger value of according to Equation 4.15. Then

= 1.48 and

= 85

= 0.96 nm and C∞ = 6.51 according to Equation

1.8. The precise calculation can be carried out by a successive approximations method, but for our purposes the average value of sufﬁcient. The parameters

,

,

C∞s

,

s nmon ,

,

d sf

, which is equal to 1.43, is ,

,

and

obtained in

such an approximation are adduced in Table 4.1. As one can see, the sole essential consequence of network stretching is an increase in local order degree ϕcl by about 1.9 times by virtue of lst growth in comparison with a non-deformed network [1, 6]. The fractal dimensions Dch and df have strictly deﬁned limits: 1 < Dch ≤ 2 and 2 ≤df < 3 165

Structure and Properties of Crosslinked Polymers [15, 25]. This circumstance allows to carry out the estimation of limiting values Nst and ( ) according to Equation 4.15. For the minimum value (for a crosslinked network with limiting density) we suppose Dch = 1.0, df = 2.0 and then = 2. For the maximum value (for a weakly crosslinked network) we assume Dch = 2.0, df = 2.95 and then ≈ 445. These values of and are in excellent agreement with paper [26] data (1–2 and 500, respectively). Therefore, the complete methods of calculation of the characteristics of crosslinked networks was proposed, which combines the rubber high-elasticity entropic theory, the cluster model of amorphous state structure of polymers and fractal analysis methods. The proposed method has shown that growth in statistical segment length is observed as the drawing ratio increases. This supposes that the chain statistical ﬂexibility depends not only on its chemical constitution, but also on the network deformed state. The considered method can be used for computer simulation and prediction of the structure of crosslinked polymer networks [6]. As it was shown earlier [13, 18, 27], as the degree of molecular orientation increases in PCP the transition from crystallisation with folded chains to crystallisation with their straightening (or to some mixed crystallisation type [27]) occurred. This effect is realised at some critical value (β*) of the degree of rolling-up of the macromolecule β [13]. For PCP the value of β* according to the data of paper [28] turns out to be equal to ~ 0.04. The parameter β is determined according to Equation 4.4. The value of β is also connected with the number of statistical segments Nst between chemical crosslinking nodes [29]:

(4.16)

Lastly the parameter β can be connected with the sample drawing ratio λ as follows [28]:

(4.17)

If we suppose for PCP the complete recovery after tension removal that is typical for rubbers, then the changes in Rc in the orientation process can be deﬁned as λRc (see Equation 4.11). In this case Equations 4.4 and 4.17 allow determination of the β*

166

The Description of Crosslinked Rubbers within the Frameworks of Fractal Analysis and Local Order Models value, proceeding from experimental values of λ (λcr), at which the transition indicated c above is realised, as well as the crosslinked network molecular characteristics Rc, Lc and Nst. It is obvious in the case of accuracy of such estimations that the values of β*, obtained according to Equations 4.4 and 4.14, should be self-congruent. The authors of paper [30] carried out the indicated calculations and received self-congruency checking of the parameters on the example of two PCP samples with different degrees of crosslinking (samples with larger and smaller degrees of crosslinking were c designated as PCP-1 and PCP-2, respectively). The values Mc, νc, Rc and Lc , calculated according to the rubber high-elasticity theory, are adduced in Table 4.2 [1].

Table 4.2 The molecular characteristics of polychloroprene crosslinked networks [30] Polymer

Mc × 10–3

νc × 10–25, m–3

Lc, nm

Rc, nm

PCP-1

22.0

3.54

71.4

6.9

PCP-2

144.5

0.54

469.0

12.9

In paper [28] the experimental magnitude β* ≈ 0.04 for PCP-2 was obtained, which was realised at λcr ≈ 1.5 [18, 28]. If we use the quoted PCP-2 value of Nst = 710 [18], then from Equation 4.17 it follows that β* = 0.056. At the same time Equation 4.4 c gives the correct value of β* at λcr = 1.5, Rc = 12.9 nm and Lc = 469 nm (see Table c 4.3). For PCP-1 at λcr = 2.9 [27], Rc = 6.9 nm, Lc = 71.4 m and Nst = 108 (Table 4.2), both indicated equations give the same value of β* ≈ 0.28. Let us consider the reason for the discrepancy in values of β* (of about 40%), which were obtained according to Equations 4.4 and 4.16 for PCP-2.

Table 4.3 Comparison of the macromolecules rolling-up degree critical values β* for polychloroprene [30] Polymer

lst, nm

Nst

β*, Equation 4.4

β*, Equation 4.17

PCP-1

0.661

108

0.279

0.280

PCP-2

0.661

710

0.041

0.059

0.354

1326

0.041

0.041

167

Structure and Properties of Crosslinked Polymers In papers [18, 28, 31] it is supposed that the statistical segment length lst is deﬁned by the chemical structure of the PCP chain and the indicated segment includes on average ~ 2.3 monomer links with molecular weight ~ 88.5 [31]. However, the value of lst can also be calculated according to Relationship 4.5. The crosslinking density change will result in a corresponding variation of the value of lst, namely in c its increase. This is due to the fact that Lc decreases proportionally to the growth in the crosslinking density νc and Rc proportionally to the cube root of νc [10]. 1/ 3 Therefore, an increase in lst proportionally to ν ñ is supposed. Let us also note that if both parts of Equation 4.5 are divided by , then we will obtain Equation 1.13, which describes the fractal chain with dimension Dch = 2 [15]. The indicated value of Dch characterises the maximum possible molecular mobility of the chain, typical for a rubber-like state [11]. The following values of lst were obtained according to Equation 4.5: 0.661 nm for c PCP-1 and 0.354 nm for PCP-2. In this case at a constant value of Lc for PCP-2, an increase in Nst from 710 according to the data of paper [18] to ~ 1326 should be observed. Using the value of Nst the values of β* calculated according to Equations 4.4 and 4.17 are in excellent agreement both between one another and with the value of this parameter received experimentally (see Table 4.3). Let us note just one more characteristic detail. For the non-deformed state the following values of β were obtained according to Equation 4.4: for PCP-1 β = 0.096 and for PCP-2 β = 0.0275. Comparison of the values of β and β* (Table 4.3) shows that the ratio β*/β is about equal to λcr. This is the correct result, demonstrating afﬁnity (homogeneity) of PCP deformation on the molecular level. At the same time Equation 4.16 gives the following values of β: for PCP-1 β ≈ 0.184 and for PCP2 β ≈ 0.08. If in the ﬁrst case (without accounting for quantitative discrepancy) a physically reasonable value of β was obtained, then for PCP-2 the obtained value of β has no physical signiﬁcance since it assumes chain rolling-up in the polymer orientation process. This is probably due to the fact that in Equation 4.16 the value of β depends to a considerable extent on the distribution of the degree of rolling-up of the chains [29]. At the same time the correct values of β can be obtained according to Equation 4.17 for non-deformed PCP at the condition λ = 1: β = 0.096 for PCP-1 and β = 0.0275 (at Nst = 1326) for PCP-2 [30]. Hence, the results stated above have conﬁrmed that PCP crosslinking density variation results in a change in the statistical segment length lst and, as a consequence, the value of Nst. In other words, statistical ﬂexibility of the PCP chain is deﬁned not only by its chemical structure, but also by the topological level of the PCP structure. A similar effect was observed earlier for glassy epoxy polymers [32]. If the indicated

168

The Description of Crosslinked Rubbers within the Frameworks of Fractal Analysis and Local Order Models effect is not accounted for then Equations 4.4 and 4.17 give incongruent results and calculation according to Equation 4.17 gives an estimation that is differentent from the experimentally determined value of β* [1].

4.2 The Polychloroprene Crystallisation As it is known [3, 33], crystallisation process kinetics can be described with the aid of the Kolmogorov–Avrami equation:

(4.18)

where Kt is the degree of crystallinity at time t, z is the crystallisation rate constant, n is the Kolmogorov–Avrami exponent, characterising nucleation and type of growing crystalline structures for the given polymer. For oriented polymers it is shown that the value of n depends also on the drawing ratio λ [18, 27]. Reduction in n with growth in λ is a general tendency. A considerable number of factors inﬂuencing the value of n makes its description within the frameworks of structural and molecular models difﬁcult. Therefore the authors of papers [34–36] generalised the inﬂuence of the indicated factors on the value of n with the application of fractal analysis methods in the example of uniaxially stretched PCP. Molecular characteristics of PCP crosslinked networks are adduced in Table 4.2. As it was shown earlier [11, 15], the fractal dimension Dch of a chain part between chemical crosslinking nodes is characteristic of its molecular mobility and changes within strictly appointed limits: 1 < Dch ≤ 2. A number of methods for the estimation of the value of Dch exists and here Richardson’s equation analogue, modiﬁed in reference to polymers (Equations 4.13 and 4.14), was used [25]. In Figure 4.2 the dependences of the exponent n on Dch for two PCP series are adduced. The limiting theoretical dependence is shown by a dotted line. This dependence is deﬁned by the following conditions. At Dch = 1 (a chain is stretched completely between crosslinking nodes) n = 1 was supposed, i.e., crystal uniaxial growth at athermic nucleation (at simultaneous beginning of growth of all crystals) [33]. At Dch = 2 (chain mobility is maximum) the maximum value n = 4 was supposed, i.e., threedimensional growth of crystals at thermic nucleation (nucleation of all new crystals in crystallisation process) [33]. Such plotting of theoretical straight line 3 supposes that the experimental points for PCP will be located below this straight line [18, 27].

169

Structure and Properties of Crosslinked Polymers

n 3

4

2

-1 -2 0 1.0

1.5

2.0

Dch

Figure 4.2The dependences of the Kolmogorov–Avrami exponent n on the chain fractal dimension Dch for (1) PCP-1 and (2) PCP-2. The straight line 3 gives the dependence n(Dch) limiting variant. Transition of the dependence n(Dch) at the change in crystalline phase morphology is indicated by an arrow [34]

As the experimental data adduced in Figure 4.2 showed, that was actually true, at any rate for PCP. For PCP, subjected to uniaxial tension, the dependence n(Dch) is linear and virtually parallel to straight line 3. This supposes that at Dch a mixed type of nucleation (n ≈ 3.4) occurs – athermic and thermic – with the ﬁrst prevalence resulting in three-dimensional growth of the crystals. This conclusion is supported by the data of paper [27], namely by Figure 4.3, on which X-ray diffractogram of samples of PCP, crystallised at different drawing ratios λ, are adduced. According to Equation 4.15 λ = 2 corresponds to Dch ≈ 1.55 and, as follows from the Figure 4.2 plot, this corresponds to n = 2, i.e., to transition either to growth of two-dimensional crystals, or to growth of three-dimensional ﬁbrils [33]. The dependence n(Dch) extrapolates to n ≈ 0.3 which corresponds to growth of ﬁbrils at athermic nucleation [33]. As is shown in papers [13, 18, 28], the transition from crystallisation with folding of chains to crystallisation with straightening of chains occurs at some critical value of β*. As was shown above, the parameters β and Nst are connected by Relationship 4.16. Combination of Equations 4.6 and 4.16 gives the following relationship [34]:

(4.19)

170

The Description of Crosslinked Rubbers within the Frameworks of Fractal Analysis and Local Order Models

a

b

c

d

Figure 4.3 The X-ray diffractogram of samples of PCP, crystallised at (a) λ = 1; (b) 2; (c) 3 and (d) 4. The tension direction is vertical [27]

The critical value of β* is within the range of ~ 0.2–0.3 [37]. Assuming β* = 0.2, let us obtain λ ≈ 1.1 for uniaxially stretched PCP at Dch = 1.5, i.e., the transition to axial structure occurs within the range of λ = 1–2, which is conﬁrmed by the data of paper [27]. In Figure 4.2 the results for PCP-2 are also adduced, which demonstrate the dependence n(Dch) displacement at small λ (λ < 2) in respect of both straight line 3 and the similar dependence for PCP-1. Extrapolation of the dependence n(Dch) to Dch = 2 for PCP-2 gives n ≈ 2.5, which corresponds to the two-dimensional with nucleation type of growth of crystals, which is intermediate between athermic and thermic [33]. At λ = 2 the dependence n(Dch) on parallel transference (it is indicated by an arrow in Figure 4.2) is observed and now the dependences n(Dch) for PCP-1 and PCP-2 are described practically by the same straight line. The last transference reason is obvious: extrapolation of the dependence n(Dch) at λ < 2 to Dch = 1 gives a negative value of n that has no physical signiﬁcance. Therefore, growth of λ and the corresponding decrease in Dch should result inevitably in a change in the nucleation mechanism. Proceeding from the data of paper [37], one should suppose that in this case (at critical magnitude β*) the transition from monomolecular nucleation to multimolecular nucleation occurs.

171

Structure and Properties of Crosslinked Polymers The data of Table 4.3 show that at different crosslinking degrees the values of β* or λcr can be different essentially even for the same polymer. The existing modes of estimation of the critical parameters of β* and λcr do not allow their interconnection with structural characteristics and crystallisation conditions of polymers to be elucidated. Therefore in paper [38] theoretical estimation methods of critical parameters as a function of the characteristics of a non-stretched polymer were proposed and the physical essence of the parameters controlling critical values β* and λcr was elucidated. This method is based on the fractal model stated above [34–36]. If we suppose for β* the average value from the interval indicated above (β* = 0.25) and use Equation 4.17, then, replacing in this equation λ and β on their critical magnitudes, λcr can be calculated [38]. In Table 4.4 the comparison of experimental values of λcr and those calculated by the indicated mode is adduced. As one can see, if for PCP-1 and low-density polyethylene (LDPE) good correspondence is obtained, then for PCP-2 a discrepancy between theory and experiment of more than four times is observed.

Table 4.4 The experimental and theoretical values of critical parameters λcr and β* for semi-crystalline polymers [38]

Polymer

Experimental values [27, 28]

λcr, Equation 4.17

λcr, λcr, β*, Equations Equation Equation 4.17 and 4.19 4.21 4.24

λcr

β*

PCP-1

2.90

0.28

2.60

2.19

0.262

2.72

PCP-2

1.55

0.04

6.70

1.77

0.050

1.33

LDPE

1.73

0.20

1.85

1.63

0.185

1.37

Just one more variant of the theoretical estimation of critical parameters is based on the application of Equation 4.19. Supposing that the crystallisation method transition occurs at n = 2 [33], the corresponding values of fractal dimension Dch ( ) can be determined from the Figure 4.2 plot and the value of λcr can be calculated according to Equation 4.19. Comparison of experimental values of λcr and those calculated according to Equation 4.19 adduced in Table 4.4 has shown good correspondence.

172

The Description of Crosslinked Rubbers within the Frameworks of Fractal Analysis and Local Order Models Hence, the critical parameters β* and λcr are deﬁned to a considerable extent by crystallisation conditions [38]. −1

As was noted above, since with a change in crosslinking density νc Lc changes as ν ñ −1 / 3

(Equation 4.2) and Rc as ν ñ

ν 2ñ / 3

and the ratio Rc/lst as

(Equation 1.14), then the value of lst will be changed as

ν ñ−2 / 3 .

Then, using Equation 4.19, it can be written [38]:

(4.20)

Accounting for constant coefﬁcients in Equation 1.14, Relationship 4.20 adopts ﬁnally the following form [38]:

(4.21)

where νc is given in 1027 m–3. according to the data of Figure 4.2 in The substitution of critical values of Equation 4.21 allows calculation of the corresponding values of β*, which are also adduced in Table 4.4. As one can see, Equation 4.21 gives the values of β*, corresponding well to the experimental data. Then from the plots of Figure 4.2 the similar dependence of exponent n and Dch can be received, expressed as follows [38]:

(4.22)

where n0 is the Kolmogorov–Avrami exponent for a non-deformed polymer (at Dch = 2).

173

Structure and Properties of Crosslinked Polymers The combination of Equations 4.21 and 4.22 allows the relationship for calculation of β as a function of νc, n and n0 to be received [38]:

(4.23)

Lastly, assuming that transition from crystallisation with folding of chains to crystallisation with straightening of chains occurs at n = 2 [33], we will obtain according to Relationship 4.23 [38]:

(4.24)

Equation 4.24 allows clear identiﬁcation of two factors, deﬁning the value of β*. The ﬁrst is the crosslinking density νc. Increasing νc always results in growth of β*. This explains to a considerable extent the increase in β* of ~ 7 times for PCP-1 in comparison with growth in PCP-2 with νc of approximately 6.5 times. Conditions of nucleation and crystallisation for a non-deformed polymer are the second factor. Decreasing n0 means rises, as follows from the data of Figure 4.2, and β* reduces, as follows according to Equation 4.21. Therefore, reduction in dimensionality of growth of crystals or transition from thermic nucleation to athermic, resulting in a reduction in n0, cause a decrease in β*. Using the values of β*calculated according to Equation 4.24, the magnitudes of λcr can be calculated according to Equation 4.19. Comparison of the values of λcr calculated by such a method with experimental data is also adduced in Table 4.4, from which their good correspondence follows. Hence, besides the parameters νc and n0 indicated above, Nst or chain statistical ﬂexibility also has an inﬂuence on the value of λcr. Increasing Nst with other conditions equal results in growth in λcr [38]. Therefore, the fractal analysis application stated above allows elucidation of the interconnection of parameters deﬁning the value of the Kolmogorov–Avrami exponent n. The increase in the tension extent λ always results in a reduction in chain molecular mobility, characterised by its fractal dimension Dch. In turn, reduction in Dch results in a linear decrease in n. Change in the nucleation mechanism deﬁnes the parallel displacement of the straight lines n(Dch). The fractal concept stated in the present 174

The Description of Crosslinked Rubbers within the Frameworks of Fractal Analysis and Local Order Models section allows identiﬁcation of parameters deﬁning the critical values of β* and λcr at crystallisation type change in the orientation process of crosslinked polymer networks and also quantitative estimations of the indicated critical magnitudes to be carried out. The value of β* is controlled by polymer network crosslinking density and by nucleation type and non-deformed polymer crystallisation process dimensionality as well. Besides the indicated factors, a chain statistical ﬂexibility, characterised by the value Nst, inﬂuences the value λcr. The theoretical estimations showed good correspondence to experimental results [38]. One other mode of PCP crystallisation kinetics within the frameworks of the thermodynamic approach [39] was considered in paper [40]. It has been shown by the indicated approach that crystallisation kinetics is deﬁned to a considerable extent by the drawing ratio λ of polymers. At a certain value of λ (λcr) a crystalline phase morphology change from lamellar to ﬁbrillar occurs that deﬁnes the transition to linear growth of the one-dimensional volume of crystallites. In paper [39] theoretical dependences of the parameter (1 – x/Kf) on the crystallisation duration t were obtained, corresponding to the equation:

(4.25)

where x and Kf are the current and ﬁnal crystallinity degrees, respectively, kn is the crystallisation rate constant, n is the Kolmogorov–Avrami exponent. The curves (1 – x/Kf)(t) for lamellar and ﬁbrillar morphology crystallisation differ and give two limiting cases of crystallisation kinetics [39]. Therefore the comparison of kinetic curves of crystallisation of real crosslinked polymers with the indicated limiting case allows determination of morphology type, realised at crystallisation. In papers [24, 27] it is supposed that the crystalline morphology of the mixed type is formed at PCP uniaxial tension. The authors of paper [40] used the model [39] for conﬁrmation of this supposition. The Kolmogorov–Avrami equation for the case when the crystallisation parameter is the stress in a uniaxially stretched sample can be presented as follows [39]:

(4.26)

175

Structure and Properties of Crosslinked Polymers where σt is the stress in the sample at time t, σ0 is the stress in the sample at time t0, σ∞ is the stress in a crystallised sample. Comparison of Equation 4.26 with the Kolmogorov–Avrami equation in Relationship 4.18 [41, 42] allows determination of the function x(t). The values of Kf are accepted according to the data of paper [24]. In Figure 4.4 the kinetic curves x(t) for PCP samples at drawing ratios λ = 1.5, 2, 3 and 4 are adduced. As follows from the adduced plots, an increase in λ gives two important effects: raising of the rate and a decrease in the crystallisation induction period tin. However, these curves do not allow analysis of PCP crystalline phase morphology as a function of λ to be carried out. Therefore, the paper [39] concept was used and kinetic curves for PCP were plotted at the indicated magnitudes x in the form of Equation 4.25. With this purpose the parameter (1 – x/Kf) was plotted as a function of relative duration of crystallisation, which is accepted to be equal to a unit for the entire crystallisation process. If crystallisation completion is indicated as tf, then the period tf – tin is accepted to be equal to a unit irrespective of its duration in real time units. The same procedure was used at plotting of limiting theoretical curves for lamellar and ﬁbrillar growth of crystallites according to the data of paper [39].

x 4

0.3

3 2 1

0.2

0.1

0

2

4

t×10-4, s

Figure 4.4 The dependences of current crystallinity degree x on crystallisation duration t at λ = (1) 1.5; (2) 2; (3) 3 and (4) 4 for PCP [40]

176

The Description of Crosslinked Rubbers within the Frameworks of Fractal Analysis and Local Order Models In Figure 4.5 the curves (1 – x/Kf) as a function of trel plotted by the indicated method are adduced. From this plot it follows that the indicated curves for PCP are restricted from above and below by crystallisation kinetics corresponding to limiting theoretical curves. In other words, crystalline morphology of a mixed type is formed at all values of λ indicated above in the course of orientational crystallisation of PCP. However, at small t drawing ratios (λ = 1.5–2.0) the experimental curve disposes closely to theoretical curve 1, describing lamellar growth of crystallites. This means that at small λ in the PCP orientational crystallisation process crystallites with folded chains are mainly formed. At λ increasing within the range of 3–4, i.e., at values λ > λcr (see Table 4.4) the experimental curve is a qualitatively precise reﬂection of the theoretical one [39]. This means that at sufﬁciently large λ, exceeding λcr, ﬁbrillar type crystallites are mainly formed in the PCP orientational crystallisation process. Therefore, the dependences adduced in Figure 4.5 completely conﬁrm the conclusions of paper [27] relative to mixed type crystallite formation in the PCP orientational crystallisation process.

(1-x/Kf) 1.0 1

-3 -4 -5 -6

0.5 2

0

0.5

1.0

trel

Figure 4.5 The dependence of parameter (1 – x/Kf) on the relative duration trel of crystallisation: (1, 2) – theoretical limiting curves for lamellar and ﬁbrillar growth of crystallites, respectively [39]; the experimental curves at λ= (3) 1.5; (4) 2; (5) 3 and (6) 4 for PCP [40]

The plots of Figure 4.5 allow an approximate estimation of one or another fraction of crystallites in mixed type morphology to be carried out. So, at x = 0.5Kf the ﬁbrillar

177

Structure and Properties of Crosslinked Polymers crystallites fraction makes up ~ 20% for λ = 1.5–2.0 and ~ 65% for λ = 3–4, if proceeding from relative coordinates of corresponding kinetic curves. As tin increases the ﬁbrillar crystallites fraction decreases at λ = 1.5–2.0 and increases at λ = 3–4 practically up to 0 and ~ 80%, respectively. Calculation of the Kolmogorov–Avrami exponent n showed its reduction from ~ 3.4 up to ~ 1.3 within the range of λ = 1–4 [27]. Let us consider the change of crystallisation rate constant in Equation 4.25 as a function of λ for PCP in the case of one-dimensional linear growth of crystals. In Figure 4.6 the dependence kn(λ) in logarithmic coordinates is adduced, which turns out to be linear and passing through the coordinates origin. This circumstance allows an analytical form of the relationship between kn and λ to be written as follows [40]:

(4.27)

ln kn 8

4

0

1

2

3

4

λ

Figure 4.6 The dependence of the crystallisation rate constant kn on the drawing ratio λ in logarithmic coordinates for PCP [40]

Hence, increasing λ results in strong (exponential) growth of the crystallisation rate constant kn. For a non-stretched polymer (λ = 1) kn = 1, which is smaller than kn for

178

The Description of Crosslinked Rubbers within the Frameworks of Fractal Analysis and Local Order Models λ = 3–4 by two or three orders. This means that a certain uniaxial tension is required for somewhat noticeable one-dimensional linear growth of PCP crystals and at λ = 1 crystallisation with formation of lamellar crystallites should occur. This conclusion is conﬁrmed by paper [27] X-ray data (see Figure 4.3). Therefore, the crystallisation kinetics study of uniaxially stretched PCP has shown that a crystalline phase with a mixed type morphology is formed in it. A ﬁbrillar crystallites fraction in this morphology increases with both growth of the drawing ratio λ and raising of the crystallisation duration t (for λ ≥ 3). The increase in λ results in a reduction in the crystallisation induction period and the rate of this process increases. The crystallisation rate constant kn is an exponential function of λ in the case of one-dimensional linear growth of crystallites [40].

4.3 The Cluster Model Application for the Description of the Process and Properties of Polychloroprene Crystallisation In the present section notions of the amorphous state structure of a cluster model of polymers [7, 8] were used for the description of a change in nucleation mechanism and melting temperature of an oriented PCP. These notions are developed in the ‘dynamic polymer network’ concept, since supramolecular (more precisely, suprasegmental) structure changes, described by a cluster model, are a logical result of changes on the molecular level, which are due to network tension (see Table 4.1). The cluster model application allows the physical effects predicted within the frameworks of general thermodynamic theory to be concretised and identiﬁed and also to be described quantitatively. A thermal mobility in liquid results in continuous formation and disappearance of heterophase ﬂuctuations, which are potential centres of future crystalline formations [43]. At a temperature decrease to below the melting temperature the chemical potential of molecules in a crystal is smaller than in a liquid and therefore the crystallisation nucleus in heterophase ﬂuctuation can be stable and begins to grow spontaneously [43]. It is supposed that geometry, morphology and crystalline phase nucleation and growth mechanism will be deﬁned to a considerable degree by the nature of heterophase ﬂuctuations in the amorphous state. So, the supposition of heterophase ﬂuctuation from folded chains (Yech model [44]) results in the formation of crystallites with folded chains (CFC). In paper [43], an alternative model of heterophase ﬂuctuation supposes the availability of parallel parts of different chains of macromolecules in it. The use of these models supposes the possibility of formation in the crystallisation process of CFC, crystallites with stretched chains (CSC) or some intermediate morphology

179

Structure and Properties of Crosslinked Polymers [45]. By its physical essence the heterophase ﬂuctuation type, proposed in paper [43], is similar to local order domain (cluster) in the cluster model of the amorphous state structure of polymers [7, 8]. The last model application allows a quantitative description of such heterophase ﬂuctuations and, as a consequence, analysis of the change in crystallisation morphology and nucleation mechanism [46]. In papers [47, 48] such analysis was carried out on the example of PCP and LDPE orientational crystallisation. It is postulated [7, 8] that the local order domain (cluster) consists of several collinear densely packed segments of different macromolecules and the length of these segments is equal to the length of polymer statistical segments lst. It has been shown above within the frameworks of the ‘dynamic polymer network’ concept that an increase in the drawing ratio λ results in growth of lst (see Table 4.1). In papers [45, 46] reduction in the crystallisation critical nucleus length l* with λ growth was demonstrated. Therefore, at criterion lst = l* the cluster becomes a crystallisation nucleus and crystalline phase morphology depends on cluster morphology. Combination of Equations 4.6, 4.12 and 4.17 allows calculation of lst as a function of λ. In Figure 4.7 the dependences l*(λ) for polyethylene, plotted according to the data of paper [46] (curves 1–4), and the dependences lst(λ) calculated by the method mentioned above (curves 5 and 6 for LDPE and PCP, correspondingly) are adduced , corresponding to the nucleation type change (transition from and the values of monomolecular nucleation to multimolecular), which were determined experimentally for LDPE and PCP (the straight lines 7 and 8, respectively), are also indicated by vertical lines. From comparison of these data it is obvious that values of estimated according to the criterion lst = l* correspond well to experimental magnitudes of this parameter. For LDPE, crystallisation with morphology of CSC (experimental value of corresponds to the intersection point of curves 4 and 5) is supposed and for PCP, formation of mixed crystalline morphology, where crystallites with stretched chains (ﬁbrillar) make up approximately 30% (the experimental value of corresponds to the intersection of curves 1 and 6). Let us note that for PCP such an estimation is approximate, since curves 1–4 were plotted according to the data for polyethylene [46]. Nevertheless, the data of Figure 4.7 demonstrate the accuracy of the identiﬁcation of clusters as heterophase ﬂuctuations, which later become the nucleus of crystalline phase growth. From the data of Figure 4.7 it also follows that experimental values of indicate reduction of the fraction of ﬁbrillar crystallites in mixed crystalline morphology at the expense of an increase in the fraction of crystallites with folded chains . The obtained data conﬁrm the conclusion made above, supposing formation of crystallites with mixed morphology at PCP orientational crystallisation [40]. Besides the longitudinal size l* of critical heterophase ﬂuctuation, one more critical size of this ﬂuctuation exists – the transverse size [45, 46]. This size is deﬁned as

180

The Description of Crosslinked Rubbers within the Frameworks of Fractal Analysis and Local Order Models (Sν*)1/2, where S is the macromolecule cross-sectional area, ν* is the critical nucleus functionality. In paper [46] the dependence of (Sν*)1/2 on λ for polyethylene was adduced. A similar parameter for a cluster can be deﬁned as (SFcl)1/2, where Fcl is the cluster functionality. For polyethylenes S = 18.9 Å2 [49] and the method of Fcl calculation was adduced above (see Equation 1.51). In Figure 4.8 the dependences (Sν*)1/2 according to the data of paper [46] and (SFcl)1/2 calculated by the indicated method on the drawing ratio λ for LDPE are adduced. As one can see, according to the condition of equality of transverse sizes the cluster becomes the crystallisation critical nucleus at = 1.75, which is close to the experimental value of this parameter (~ 1.60 [28]).

l*, lst, nm 5

8 7

6

8

4

0

1 2 3 4

1

2

3

4

Figure 4.7 The dependences of the critical nucleus size l* (1–4) and statistical segment length lst (5, 6) on the tension degree λ for LDPE (1–6) and PCP (6). The straight lines 7 and 8 indicate experimental values of for LDPE [46] and PCP [18]. Curves 1–4: 1 – nucleus consisting of ~ 30% chains and ~ 70% loops forming; 2 – nucleus consisting of ~ 50% chains and loops forming; 3 – nucleus consisting of ~ 70% chains and ~ 30% loops forming; 4 – shaft-like nucleus [48]

Analysis of the thermodynamic properties of crosslinked polymer networks shows the crystalline phase melting temperature Tm rising with growth in the tension extent [27, 46, 50]. The results stated in the present chapter suppose that this effect can be

181

Structure and Properties of Crosslinked Polymers due to chain statistical ﬂexibility change in the uniaxial tension process, namely to an increase in statistical segment length lst. This effect can be described theoretically by the modiﬁed model [51], in which the dependence of Tm on network crosslinking density was originally considered. Let us consider the model [51] somewhat in detail in reference to the dependence Tm(λ). Supposing small degrees of supercooling one should select around each crosslinking node a spherical region of radius:

(4.28) where l*(Tc) is the crystallisation nucleus critical size at the given crystallisation temperature Tc.

(Sν*)1/2, (SFcl)1/2, Å 40

2

20

1

0 1.0

1.5

2.0

λ

Figure 4.8 The dependence of critical nucleus sizes (Sν*)1/2 (1) and clusters transverse size (SFcl)1/2 (2) on the drawing ratio λ for LDPE [48]

In order that crystallisation does not proceed at the given Tc in crosslinked polymers, the crosslinking density should be such that the joining of spherical regions of radius r is observed. The joining condition can be written as follows [51]:

(4.29) where c is constant.

182

The Description of Crosslinked Rubbers within the Frameworks of Fractal Analysis and Local Order Models For a statistically crosslinked polymer network, constant c in Equation 4.29 should be smaller than a unit. Because of the statistical character of the arrangement of crosslinking nodes in the system, regions from joining non-crystallising spheres will exist even at c < 1, penetrating into the entire sample, i.e., in the system there exist inﬁnite non-crystallising clusters, hindering crystalline order formation. The critical crosslinking density for inﬁnite non-crystallising cluster origin, hindering the (2r)3 = B ≈ 2.7 crystallisation process, is determined from the condition (4π/3) [51]. Therefore, if Tc is assumed as the long-range crystalline order disappearance temperature (at the given crosslinking density νc), then constant c in Equation 4.29 is to be accepted as approximately equal to 0.86 in accordance with paper [51]. At a crosslinked network tension in supposition of its affine (homogeneous) deformation mentioned above spherical region of radius r assumes the form of ellipsoid of revolution with large axis oriented parallel to tension axis. As was shown above, increasing λ results in lst growth and the dependence lst(λ) is estimated according to combination of Equations 4.6, 4.12 and 4.17 (see Figure 4.7). As the calculations have shown, the growth of lst from 0.585 to 3.32 nm with λ increasing from 1 to 4 is observed. Further, taking into consideration that according to the crystallisation theory of polymers [52] the crystallisation nucleus critical size l*(Tc) is equal to:

(4.30)

and substituting Expression 4.28 in Formula 4.29 with the consideration of the shape change of region or radius r, i.e., replacing r on λr, can be received [48]:

(4.31)

0

where Δh and Tm are the melting enthalpy per volume unit and the melting temperature of an ideal non-crosslinked polymer, respectively, σs is the speciﬁc buttending surface energy.

183

Structure and Properties of Crosslinked Polymers If the melting process is considered as long-range crystalline order disappearance in the system, then the melting temperature Tm can be considered to be equal to Tc at the given network crosslinking density νc [51]. Then Tm for crosslinked PCP networks can be calculated according to Equation 4.31, assuming that the ratio 2σ3/Δh is independent of λ and considering it as a ﬁtting coefﬁcient. For a PCP network the 0 crosslinking density νc = 3.55 × 1025 m–3 (Table 4.1). The value of Tm was accepted to be equal to 333 K [52]. The experimental data for Tm(λ), received by the differential thermal analysis (DTA) method, were used for comparison with theory [51]. The value of the ratio 2σs/Δh for PCP turned out to be equal to ~ 0.506 × 10–10 m [51]. Comparison of the experimental and calculated data is adduced in Figure 4.9, from which follows their good correspondence, particularly at λ ≥ 2 [47].

T m, K 345

4 -5

335 2 325

1 3

315 305 1

2

3

4

λ

Figure 4.9 The dependences of the melting temperature Tm on the drawing ratio λ for PCP. 1 – calculation according to Equation 4.31 at the condition of increasing lst with growth in λ; 2 – calculation according to Equation 4.31 at the condition lst = const.; 3 – calculation according to the Flory equation; 4 – calculation according to the Krigbaum–Roe equation; 5 – experimental data [48]

184

The Description of Crosslinked Rubbers within the Frameworks of Fractal Analysis and Local Order Models Let us consider possible variants of changes of network parameters and their inﬂuence on the value of Tm. As the data of Figure 4.9 show, the assumption of a region with radius shape change from a sphere to an ellipsoid of revolution with larger radius λr and increasing lst with growth of λ gives a good correspondence to experiment. The condition lst = const. and invariance of region of radius r shape, i.e., using coefﬁcient 0.43, but not 0.43λ, in Equation 4.31 gives a constant value of Tm that does not correspond to experimental data [51]. Calculation of the region of radius r shape change at the condition lst = const. gives overstated values of Tm in respect to experimental values as was shown in Figure 4.9 (curve 2). Lastly, the assumption of growth in lst with increasing λ at an invariable shape of region with radius r results 0 very rapidly in the physically unreal condition Tm > Tm [48]. At present two other simple theories exist, describing the dependence Tm(λ). They are the Flory theory [53] and the Krigbaum–Roe theory [54]. In the indicated treatments the value of Δh for PCP was accepted to be equal to 22.7 cal/g (~ 2 × 103 cal/mole). Calculation according to the Flory theory at the condition of increasing lst with growth in λ gives a good correspondence to experiment (Figure 4.9, curve 3) and calculation according to the Krigbaum–Roe theory, as was expected [55], gives too rapid growth in Tm with increasing λ, which does not correspond to the experimental data (Figure 4.9, curve 4). Let us note that the Flory theory was received by supposing crystals ﬁbrillar morphology and the Krigbaum–Roe concept in supposition of lamellar morphology (crystallites with folded chains) [55]. At the same time in the derivation of Equation 4.31 some assumptions in respect to morphology of crystalline phase, forming in the orientational crystallisation process, were not given. As follows from the data of Figure 4.9, the dependence Tm(λ), obtained according to the model [48], disposes at λ > 2 between two limiting curves, corresponding to the Krigbaum–Roe and Flory theories. This means that the dependence Tm(λ), obtained according to Equation 4.31, corresponds to the mixed type of crystalline phase morphology, moreover with λ growth this dependence disposes farther from the dependence calculated according to the Krigbaum–Roe theory and nearer to the dependence calculated according to the Flory theory. This supposes an increase in the fraction of ﬁbrillar crystallites with growth in λ for PCP. Hence, the results of the present section demonstrate that local order domains (clusters) in the cluster model of the amorphous state structure of polymers by both physical signiﬁcance (let us be reminded that clusters have thermoﬂuctuational origin [7, 8]) and their critical sizes correspond to heterophase ﬂuctuations, which become nucleus crystallites. Let us note that dynamic local order domains in the devitriﬁcated state can consist partly of folded chains unlike those in a glassy state. Results obtained for

185

Structure and Properties of Crosslinked Polymers PCP showed that this polymer crystalline morphology at orientational crystallisation was a mixed morphology, including both crystallites with folded chains and ﬁbrillar crystallites. An increase in the last fraction results in reduction of the tension critical degree , at which the transition from monomolecular nucleation to multimolecular nucleation is observed. The results from the present section showed that the correct theoretical prediction of increase of the melting temperature of crosslinked PCP with growth in tension degree λ was possible with the simultaneous use of two assumptions: change in shape of local regions around chemical crosslinking nodes and increase in statistical segment length with function λ. The proposed Equation 4.31 gives the intermediate dependence Tm(λ) between limiting cases, which are described according to the Krigbaum–Roe and Flory equations, characterising the mixed type of PCP crystalline morphology [47, 48].

4.4 Inﬂuence of Polychloroprene Crystalline Morphology on Its Mechanical Behaviour Crystallisation of oriented polymers can occur by the two most probable modes: forming crystals with folded chains and axis of macromolecules perpendicular mainly to the tension direction (CFC-⊥) or forming crystals with folded chains with axis of macromolecules parallel mainly to the tension direction (CFC-II). The individual case of the second mode of crystallisation is crystallisation without folding (CWF). These schemes were considered in detail in paper [27] on the example of PCP crystallised at different drawing ratios λ. In papers [18, 27, 28] the jump-like change of crystalline morphology and properties of crosslinked PCP, crystallised in the stretched state, at critical tension degree λcr was discovered, which is approximately equal to 2.8 [27]. Analysis of these changes showed that the jump-like change in the folding degree of macromolecules in crystallites from folded chains occurred at λcr [27]. These conclusions were conﬁrmed both theoretically [58] and experimentally [40]. In paper [27] it has also been shown that PCP samples strengthening with tension in the plastic deformation process occurs faster the larger degree of tension at which the sample was crystallised. The strengthening rate of samples on this part of the stress–strain (σ–ε) curve starts increasing sharply at λ = λcr [27]. The authors of paper [59] carried out quantitative analysis of the structure–properties relation with tension of PCP samples, crystallised at different λ, and also conﬁrmed the changes in crystalline morphology of these samples, which were described in paper [27].

186

The Description of Crosslinked Rubbers within the Frameworks of Fractal Analysis and Local Order Models In Figure 4.10 the σ–ε curves are shown for PCP samples, crystallised at λ = 2 (λ < λcr) and λ = 4 (λ>λcr) (conventional signs of samples are PCP-2 and PCP-4, respectively). As was noted above, at λ > λcr the strong strain hardening of PCP samples is observed, which can be characterised by the strain hardening modulus Gp [60]. As it is known [60, 61], such growth in Gp is due to the increase in the number of chains connecting crystallites. In turn, the change in the number of tie chains can be the consequence of a change in the crystalline morphology of the PCP samples, since it is obvious that CWF fraction growth gives such an increase in the number of chains (unlike CFC, each segment of a macromolecule in CWF gives two tie chains). Analysis of these changes within the frameworks of the rubber high-elasticity concept [60, 61] can be carried out using Relationship 1.1. The indicated relationship allows calculation of Gp according to the characteristics of the σ–ε curve in the case of linear dependence σtr(λ2 – λ–1), which is adduced in Figure 4.11. σtr is a true stress in tension testing, which in the homogeneous stretching case is determined as follows [60]:

(4.32)

where σ is the nominal stress (without accounting for a change in the sample crosssectional area in the tension process), λt is the drawing ratio in uniaxial tension tests of crystallised samples, ε is the strain corresponding to nominal stress σ. The estimations according to Equation 1.1 showed that for PCP-2 Gp = 2.3 MPa and for PCP-4 considerable growth in Gp up to 11.8 MPa was observed (Figure 4.11). The entanglements network density νt, which creates the tie chains network, can be determined according to Equations 1.2 and 1.3. Values of νt estimated by the indicated mode at ρ = 1.296 g/cm3 [62] and T = 295 K turned out to be equal to 2.67 × 1027 m–3and 8.07 × 1027 m–3 for PCP-2 and PCP-4, respectively. Let us note that the chemical crosslinking network density νc is equal to ~ 0.035 × 1027 m–3 (see Table 4.1), i.e., approximately two orders lower than νt and, hence, this network can not create the observed effects of strain hardening. After subtraction of νc from νt we will obtain the tie chains network density ~ 2.63 × 1027 m–3 and ~ 8.04 × 1027 m–3 for PCP-2 and PCP-4, respectively. In other words, as was expected earlier [27], a sharp increase of the CWF fraction at the expense of reduction of the CFC fraction and corresponding growth of the number of tie chains occurred. This effect is realised at practically invariable crystallinity degree K ≈ 0.28 at λ = 2 and 4 [27]. The indicated change in νt supposes the fraction of CWF increasing by approximately 3 times, which corresponds well to the theoretical [58] and experimental [40] change estimations.

187

Structure and Properties of Crosslinked Polymers

σ, MPa 12

2

6

0

1

0.1

0.2

0.3

ε

Figure 4.10 The stress–strain (σ–ε) curves for samples of (1) PCP-2 and (2) PCP-4 [59]

For further analysis of PCP crystalline morphology changes the Mooney–Rivlin equation (see Formulae 1.22 and 1.23) can be used. In Figure 4.12 the dependences f( λ−t 1 ), corresponding to the Mooney–Rivlin equation, are adduced for PCP-2 and PCP−1 4. As follows from the plots of Figure 4.12, the dependences f*( λ t ) have a linear part at sufﬁciently high values of λt (λt > 1.5) and a nonlinear part at λt < 1.5, which is the main difference of similar dependences of linear polymers from crosslinked rubbers. As Edwards and Vilgis [63] showed, the nonlinear part was due to macromolecules slipping through entanglement nodes in the deformation process. It is obvious that macromolecules slipping through chemical crosslinking nodes is impossible, therefore one has to suppose that it is due to slipping of the macromolecules segment including in CWF and creating tie chains at the last corresponding stretching [59]. An interesting observation follows from the plots of Figure 4.12: the constant 2C1 value in the Mooney–Rivlin equation is equal to zero for PCP-2 and 2C1 ≈ 8 MPa > 0 for PCP-4. The value of 2C1, corresponding to f* at λt = ∞, can be described by Equation 1.24. It is obvious that in this equation the front-factor A is the only parameter that is capable of accepting zero values. Therefore, for PCP-2 2C1 = 0 and, consequently, A = 0. For PCP-4 the value of υ is the volume fraction of CWF forming the tie chains network and then makes up ~ 0.65 of the crystallinity degree K (K ≈ 0.28 [27]) and, hence, υ ≈ 0.182 according to the data of paper [40]. For the value of Mt (tie chain molecular mass), which is equal to ~ 0.097, the calculation of A according to Equation 1.24 gives a value of 0.76.

188

The Description of Crosslinked Rubbers within the Frameworks of Fractal Analysis and Local Order Models σtr, MPa 15

2

10 1 5

0

0.5

1.0

( λ2t - λ−t 1 ) −1

Figure 4.11 The dependences of true stress σtr on the invariant ( λ t – λ t ) value, corresponding to Equation 1.1, for samples of (1) PCP-2 and (2) PCP-4 [59] 2

f*, MPa 30 2 1

20

10

0

0.5

1.0

λ−t 1

Figure 4.12 The dependences of the reduced stress f* on the reciprocal value of −1 the drawing ratio λ t , corresponding to the Mooney–Rivlin Equation 1.22 for samples of (1) PCP-2 and (2) PCP-4 [59]

189

Structure and Properties of Crosslinked Polymers In paper [64] it has been shown that an interconnection exists between the value of A and the physical entanglement nodes functionality F, being described by Equation 1.57. As was noted above, for PCP-2 A = 0 and, hence, F = 2. Since F is considered as a chains number, emerging from one node, then this means that each node itself represents a segment from which two tie chains emerge. Such a crystallite with a mixed morphology scheme is adduced in Figure 4.13a. For PCP-4 A = 0.76 and, hence, we will obtain F = 8.33 according to Equation 1.57. Therefore, in PCP-4 one node itself represents CWF consisting of four segments on average. Such a crystallite scheme is adduced in Figure 4.13b.

a)

b)

Figure 4.13 The schematic representation of mixed type crystallites for (a) PCP-2 and (b) PCP-4 [59]

The schemes shown in Figure 4.13 allow estimation of the CWF fraction in mixed type crystalline morphology. So, under the assumption that CFC has three folds and per such block one segment of CWF is available, we will obtain that the CWF fraction in PCP-2 makes up ~ 25%. For PCP-4, the CFC fraction should be decreased at the expense of the CWF fraction increasing at the invariable crystallinity degree K that in Figure 4.13b is reﬂected by a folds reduction in CFC from 3 to 2. At such mixed morphology crystallite representation the CWF fraction in PCP-4 will exceed 50%. Both adduced estimations correspond well to the results [40, 58] obtained earlier. Hence, the results stated in the present section demonstrated accuracy in the description of mechanical behaviour for PCP samples, crystallised at different tension degrees, within the frameworks of the rubber high-elasticity concept. Such a description is

190

The Description of Crosslinked Rubbers within the Frameworks of Fractal Analysis and Local Order Models possible only within the frameworks of a structural model, supposing PCP mixed crystalline morphology and its jump-like change at λcr ≈ 2.8. The proposed approach allows a quantitative estimation of CFC and CWF ratio change at uniaxial tension extents to be received, which are smaller and larger than λcr.

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Structure and Properties of Crosslinked Polymers 15. G.V. Kozlov and V.U. Novikov in Synergetics and Fractal Analysis of Crosslinked Polymers, Klassika, Moscow, Russia, 1998, p.112. 16. V.P. Budtov in Physical Chemistry of Polymer Solutions, Khimiya, SanktPeterburg, Russia, 1992, p.384. 17. S.M. Aharoni, Macromolecules, 1983, 16, 9, 1722. 18. V.G. Baranov, G.T. Ovanesov, K.A. Gasparyan, Y.K. Kabalyan and S.Y. Frenkel, Doklady Akademii Nauk SSSR, 1974, 217, 1, 119. 19. G.V. Kozlov, G.E. Zaikov and A.K. Mikitaev in Fractal Analysis of Gas Transport Process in Polymers, Nauka, Moscow, Russia, 2009, p.199. 20. N.S. Murthy, S.T. Correale and H. Minor, Macromolecules, 1991, 24, 5, 1185. 21. C. Lin, L.E. Busse, S.R. Nagel and J. Faber, Physical Review: Part B, 1984, 29, 9, 5060. 22. R.L. Miller and R.F. Boyer, Journal of Polymer Science, Part B: Polymer Physics Edition, 1984, 22, 12, 2043. 23. G.V. Kozlov, V.A. Beloshenko, E.N. Kuznetsov and Y.S. Lipatov, Ukrainskii Khimicheskii Zhurnal, 1997, 63, 7, 52. 24. O.F. Belyaev, V.Z. Aloev and Y.V. Zelenev, Vysokomolekulyarnye Soedineniya Seriya B, 1985, 27, 1, 63. 25. A.S. Balankin in Synergetics of Deformable Body, Publishers Ministry of Defence SSSR, Moscow, Russia, 1991, p.404. 26. V.A. Bershtein and V.M. Egorov in Differential Scanning Calorimetry in Physics-Chemistry of Polymers, Khimiya, Leningrad, Russia, 1990, p.256. 27. O.F. Belyaev, V.Z. Aloev and Y.V. Zelenev, Vysokomolekulyarnye Soedineniya Seriya A, 1986, 28, 2, 260. 28. V.G. Baranov, A.A. Martirosyan and S.Y. Frenkel, Vysokomolekulyarnye Soedineniya Seriya B, 1975, 17, 4, 261. 29. S.Y. Frenkel, Khimicheskie Volokna, 1977, 3, 11.

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The Description of Crosslinked Rubbers within the Frameworks of Fractal Analysis and Local Order Models 30. V.Z. Aloev, G.V. Kozlov and Z.I. Afaunova in Interconnection Between Crosslinking Density, Statistical Flexibility and Chain Critical Rolling-up Degree for Polychloroprene, Viniti Ras, Moscow, Russia, 1999, p.3933. 31. V.G. Baranov, R.S. Zurabyan, I.K. Atakhodzhaev and S.Y. Frenkel, Mekhanika Polimerov, 1970, 6, 963. 32. G.V. Kozlov, V.A. Beloshenko, E.N. Kuznetsov and Y.S. Lipatov, Doklady NAN Ukraine, 1994, 12, 126. 33. B. Wunderlich in Macromolecular Physics, Volume 2, Academic Press, New York, NY, USA, San Francisco, CA, USA, London, UK, 1976, p.574. 34. V.Z. Aloev, G.V. Kozlov and G.E. Zaikov, Russian Polymer News, 2001, 6, 4, 63. 35. V.Z. Aloev, G.V. Kozlov and G.E. Zaikov, Kautsuk i Resina, 2004, 3, 38. 36. V.Z. Aloev, G.V. Kozlov, A.K. Mikitaev and B.I. Kunizhev in the Proceedings of International Conference by Rubbers, IRC’04, Moscow, Russia, 2004, p.32. 37. V.G. Baranov, Khimicheskie Volokna, 1977, 3, 14. 38. V.Z. Aloev, G.V. Kozlov and Z.I. Afaunova, Elektronnyi Zhurnal ‘Issledovano v Rossii’, 2000, 75, 1051. 39. R.A. Gasparyan, V.G. Baranov, M.A. Martynov and S.Y. Frenkel, Vysokomolekulyarnye Soedineniya Seriya A, 1992, 34, 9, 68. 40. V.Z. Aloev, A.I. Burya, G.V. Kozlov, V.F. Vargalyuk and G.B. Shustov, Vestnik Dnepropetrovskogo Universiteta, Khimiya, Russia, 2000, 5, 102. 41. A.N. Kolmogorov, Izvestiya Akademii Nauk SSSR, Seriya Matematicheskaya, 1937, 1, 3, 335. 42. M. Avrami, Journal of Chemical Physics, 1939, 7, 6, 1103. 43. V.A. Borokhovskii, K.A. Gasparyan, R.G. Mirsoev and V.G. Baranov, Vysokomolekulyarnye Soedineniya Seriya A, 1976, 18, 11, 2406. 44. G.S.Y. Yeh, Vysokomolekulyarnye Soedineniya Seriya A, 1979, 21, 12, 2433. 45. S.G. Vladovskaya and V.G. Baranov, Vysokomolekulyarnye Soedineniya Seriya A, 1983, 25, 2, 258.

193

Structure and Properties of Crosslinked Polymers 46. S.G. Vladovskaya and V.G. Baranov, Acta Polymerica, 1982, 33, 2, 125. 47. V.Z. Aloev, G.V. Kozlov, E.N. Ovcharenko and G.E. Zaikov, Vestnik KBSU, Khimicheskie Nauki, 2003, 5, 169. 48. V.Z. Aloev, A.I. Burya and G.V. Kozlov, Doklady NAN Ukraine, 2003, 6, 123. 49. S.M. Aharoni, Macromolecules, 1985, 18, 12, 2624. 50. A.N. Gent, Journal of Polymer Science, Part A: Polymer Chemistry Edition, 1965, 3, 11, 3787. 51. R.A. Gasparyan, K.A. Gasparyan, V.G. Baranov, A.M. Ovsipyan and S.Y. Frenkel’, Vysokomolekulyarnye Soedineniya Seriya B, 1988, 30, 12, 896. 52. L. Mandelkern in Crystallization of Polymers, Khimiya, Moscow-Leningrad, Russia, 1968, p.336. 53. P.J. Flory, Journal of Chemical Physics, 1974, 15, 2, 397. 54. W.R. Krigbaum and R-J. Roe, Journal of Polymer Science, Part A: Polymer Chemistry Edition, 1964, 2, 12, 4391. 55. V.G. Kalashnikova, L.P. Lipatova and M.B. Konstantinopol’skaya in The Features of Polymers Crystallization Under Action of Mechanical Stress: Summarizing Information, NIITEKhIM, Moscow, Russia, 1980, p.32. 56. O.F. Belyaev and E.M. Belavtseva, Vysokomolekulyarnye Soedineniya Seriya A, 1974, 16, 1, 141. 57. R.J. Gaylord and D.J. Lohse, Polymer Engineering and Science, 1976, 16, 3, 163. 58. O.F. Belyaev, V.Z. Aloev and Y.V. Zelenev, Plaste und Kautschuk, 1988, 35, 11, 420. 59. V.Z. Aloev, G.V. Kozlov and G.E. Zaikov, Plasticheskie Massy, 2003, 7, 30. 60. R.N. Haward, Macromolecules, 1993, 26, 22, 5860. 61. G.V. Kozlov, V.A. Beloshenko, V.N. Varyukhin and Y.S. Lipatov, Polymer, 1999, 40, 4, 1045.

194

The Description of Crosslinked Rubbers within the Frameworks of Fractal Analysis and Local Order Models 62. M.F. Bukhina in Crystallization of Rubbers, Khimiya, Moscow, Russia, 1973, 233. 63. S.F. Edwards and T. Vilgis, Polymer, 1986, 27, 4, 483. 64. J.R. Falender, G.S.Y. Yeh and J.E. Mark, Journal of Chemical Physics, 1979, 70, 11, 5324.

195

5

Structure of Epoxy Polymers

5.1 Application of Wide Angle X-ray Diffraction for Study of the Structure of Epoxy Polymers Wide angle X-ray diffractometry is a perspective method of the study of the structure of amorphous polymers on a molecular level [1, 2]. However, the obvious discrepancy between its large possibilities and information received according to the experimental data requires further development of this method. As a rule, the wide angle X-ray halos of amorphous polymers have declinations from the ideal shape (asymmetry, indistinctly expressed maximum and so on) that allow [3] the availability of superposition of several simpler by shape scattering curves to be supposed. One expects that using strictly monochromatic K α1 scattering, puriﬁed from K α 2 constituent, will raise the capability of solvable X-ray diffractometry and as a result the possibility to conﬁrm the stated supposition will be given. In paper [3] this was demonstrated on the example of epoxy polymers (EP), which are representatives of glassy crosslinked polymers. There have been studies carried out on EP, obtained by the curing of diglycidyl ether of bisphenol A (DGEBA) by 3,3′-dichloro-4,4′diaminodiphenylmethane with different ratios of the oligomer curing agent in moles (equivalents) of Kst (EP-1), that allows variation of their topology and structure over wide limits [4]. In Figure 5.1 the typical wide angle X-ray diffractogram for EP-1, corresponding to the case where Kst = 1.0, is adduced. Study of the shape of the observed halos allows to establish that their best description is reached under the assumption of the existence of two components (Figure 5.1b). This fact conﬁrms once more the known principle that the amorphous state of polymers is structurally heterogeneous. At the same time the availability of the two components allows to make use of the cluster model of the notions of the amorphous state structure of polymers [5, 6], which supposes the availability in an amorphous glassy polymer of two structural constituents: local order domains (clusters) and a loosely packed matrix. It is natural to connect the halo, the peak of which is located at smaller scattering angles θ (corresponding to a larger Bragg’s interval dB), with a loosely packed matrix, and the halo disposed at larger θ with clusters, which are the more densely packed component (see Equation 4.9).

197

Structure and Properties of Crosslinked Polymers

I 600 a

400 b

200

0 5

10 15 20 25 30 35 2θ

Figure 5.1 The experimental curve of wide angle X-ray scattering (a) and its computer resolution on forming components (b) [3]

Knowledge of Bragg’s interval values for clusters and the loosely packed matrix gives the possibility to calculate characteristic values of the intermolecular distance dm for them according to Equation 4.10. The estimations given by the authors [3] showed that for clusters it was ~ 6 Å, for a loosely packed matrix ~ 7 Å and within experimental error was independent of Kst. The absolute magnitude of obtained values of dm corresponded reasonably to the known literary data [7]. Integral and amplitude intensities of both the general halo (Figure 5.2) and its components depending on Kst change similarly, revealing a minimum at Kst = 1.0. This circumstance supposes that they reﬂect similar structural changes occurring in EP at Kst variation and therefore they can be used to an equal extent in their treatment. In Figure 5.3 the comparison of the scattering integral intensity from a loosely packed matrix Il.m. with its relative fraction ϕl.m., estimated according to the methods of [6], is carried out. A quite deﬁnite tendency for Il.m. growth with increasing ϕl.m. is observed. This dependence is linear and passes through coordinates origin. A similar clearer correlation was obtained for amplitude intensity. This gives additional reasons to connect the scattering curve, corresponding to a larger Bragg’s interval, with the concrete structural component, namely with the loosely packed matrix [3].

198

Structure of Epoxy Polymers

I, rel. units 6

4

1

2

2

0

0.5

1.0

1.5

Kst

Figure 5.2 The dependences of integral (1) and amplitude (2) X-ray scattering intensities on curing agent : oligomer ratio Kst for epoxy polymer EP-1 [3]

Identiﬁcation of the second structural component is more complex. In the lower insert of Figure 5.3 the dependence of the reciprocal value of the scattering integral of this component intensity on the relative fraction of clusters is adduced. It is linear and passes through the origin of the coordinates. A similar dependence was obtained for amplitude intensity. This circumstance allows to connect the indicated scattering component to clusters. If to proceed from analogy with crystalline materials, then the reciprocal dependence Icl(ϕcl) (or I(ϕcl)) presents itself as an unexpected one, since increasing ϕcl, reﬂecting growth in ordering degree, should result in growth in I (or Icl). However, the obtained result is explained completely by the concept of the amorphous state structure defect of polymers [8, 9]. The latter supposes that the well-known model of interpenetrating tangled macromolecular coils (Flory model of ‘felt’ [10]) describes ideal, but not real, amorphous polymer structure. In such treatment local order domains (clusters) should be considered as a departure from completely chaotic (ideal) structure, i.e., as defects. In the indicated treatment increasing ϕcl means intensiﬁcation of the ideal structure ‘distortion’ that should result in I decreasing. There is a number of experimental facts in favour of the approach considered above. For example, in paper [11] reduction in amorphous halo intensity I for nitrocellulose with increase in its ageing temperature was noted. Since the ageing process of amorphous polymers is associated with their ordering degree growth [12], then the latter can be connected with I reduction. The decrease in I at formation of clusters was also noted in paper [13]. It is possible that this process mechanism is similar to

199

Structure and Properties of Crosslinked Polymers the I reduction mechanism at the formation of precrystalline zones in metals (having local ordering, like the considered clusters) by type of Guinier-Preston zones.

Il.m.

ln Icl 1.6

8 0.8

0 2.5

ln lcl

2.7

I cl−1

4

5.0 2.5

0

0

0.4

0.3

0.6 ϕcl

0.8 ϕ l.m.

Figure 5.3 The dependence of the integral intensity of X-ray scattering curves for a loosely packed matrix Il.m. on its relative fraction ϕl.m. for EP-1. In the upper insert: the dependence of the integral intensity reciprocal value Icl of X-ray scattering curves of clusters on amorphous halo half-width lcl in double logarithmic coordinates; in the lower insert: the dependence of of clusters [3]

on the relative fraction ϕcl

Since the halo integral intensity can be considered as a measure of the number of scattering particles and its half-width as a measure of their effective size then elucidation of the interconnection of these parameters is of interest. In the upper part of Figure 5.3 the dependence of ln on ln lcl, reﬂecting the indicated interconnection in the clusters case, is adduced. Here lcl is the half-width of the halo component, corresponding to the cluster constituent. The indicated dependence is transformed in [14]:

(5.1)

200

Structure of Epoxy Polymers where df ≈ 2.7 is the cluster structure fractal dimension. Dependences of type 5.1 are typical for fractal structures and are an experimental conﬁrmation of the fractality of the structure of polymers [16] and the value of df corresponds well to the known earlier values of the fractal dimension of the structure of polymers [17]. The numeric magnitude of df gives the possibility to suppose the cluster structure formation type. As it is known [18], Witten–Sander clusters, formed by limited aggregation of diffusion of particles, have dimension df = 2.5 ± 0.26, which is close to the result obtained above. This allows the cluster structure forming in EP to be attributed to the type mentioned above. Hence, use of monochromatic Cu K α1 irradiation allows a more precise wide angle X-ray halo nature for crosslinked polymers to be made and its constituents within the frameworks of the cluster model of the amorphous state structure of polymers to be identiﬁed [5, 6]. It is known that the introduction of adamantane fragments in epoxy polymers exercises an essential inﬂuence on their characteristics. In papers [19, 20] the effect of such network structures is examined in the example of EP modiﬁed by adamantane acids. The interpretation of the results obtained in [19, 20] within the frameworks of the cluster model of the amorphous state structure of polymers [5, 6] allows to suppose availability of two types of clusters in the studied EP: stable ones, formed by main chain segments, and unstable ones, formed at the expense of the interaction of adamantane fragments. The authors of papers [21–23] studied the problem of how much the indicated notions corresponded to the real structure of the studied EP. This can be carried out with the aid of the methods of [3], based on the study of wide angle X-ray halos. In papers [21–23] the epoxy polymers based on diglycidyl ether of bisphenol A resin (ED-20), modiﬁed by mono- (MCA) and dicarbon (DCA) acids of adamantane were studied. Three EP systems were used, prepared by different modes: EP-2 was prepared by ED-20 curing by a mixture of iso-methyltetrahydrophthalic anhydride (IMTHPhA) and catalyst UP 606/2 (IMTHPhA – 0.85 moles, UP 606/2 – 0.10 mass parts); EP-3 was prepared by ED-20 mixing at 423–453 K with a preliminary prepared curing composition (IMTHPhA and DCA total 0.85 moles and 0.05–0.10 mass parts of UP 606/2); EP-4 by preliminary combination of ED-20 with MCA and curing by the same composition as EP-2. Study of the shape of the observed halos allows to establish [23] that their best description is reached in the assumption of the existence of two components of the amorphous halo for non-modiﬁed EP (EP-2) and two or three for EP containing adamantane fragments (EP-3 and EP-4) (Figure 5.4). The appearance of the third

201

Structure and Properties of Crosslinked Polymers component of the amorphous halo for EP with network fragments testiﬁes that the structural state of EP-3 and EP-4 is more heterogeneous compared to EP-2. To make use of the notions of the cluster model [5, 6], then availability of three halo components can be connected with the existence of three structural constituents: a loosely packed matrix and clusters of two types. The latter are formed by crosslinked network EP collinear segments and adamantane fragments.

I

I 1500

1500

1000

1000

500

500

0

9

29

19

(a)

0 2θ

9

19

29

2θ

(b)

Figure 5.4 The wide angle X-ray scattering curve and its computer resolution on components for EP-2 (a) and EP-4 (b) [23]

In Table 5.1 values of Bragg’s interval dB, calculated according to the position of an experimentally observed halo, and its components, distinguished after computer processing of diffractograms d B1 , d B 2 and d B 3 , and also the intensities of the amplitude Ia and integral Ii corresponding to them are adduced. With growth in modiﬁer concentration cm the tendency of Bragg’s interval to increase is observed. Hence, introduction of network segments results in loosening of the EP structure [21].

202

Table 5.1 The modiﬁer concentration on X-ray structural characteristics of EP [23]

Ia

cm, mass percentage

dB

EP-2

0

4.76

4.84

3.86

–

1.65

1.22

0.67

–

7.7

6.8

–

EP-3

1.0

5.00

5.07

3.86

3.03

1.86

1.53

0.96

0.14

10.2

7.9

0.3

2.5

4.66

4.55

–

3.18

1.91

1.84

–

0.41

15.6

–

2.1

5.0

5.05

5.04

4.21

3.52

2.12

1.75

0.79

0.57

11.5

3.9

3.3

10.0

4.93

4.76

–

3.31

1.98

1.99

–

0.42

8.3

–

0.8

15.0

4.98

5.15

4.27

3.34

1.92

2.12

1.45

0.44

4.3

3.0

0.6

1.4

4.78

4.89

4.05

3.29

1.80

1.63

0.47

0.17

14.6

2.6

0.4

2.9

4.83

4.91

3.96

3.09

1.70

1.69

0.47

0.25

11.6

1.4

0.5

4.3

4.98

4.83

–

3.16

2.37

2.21

–

0.75

18.1

–

4.4

9.2

5.19

5.12

3.95

–

1.97

1.88

0.47

–

13.5

3.0

–

13.2

5.28

5.36

4.54

3.45

2.06

1.43

0.11

0.25

7.6

6.6

1.3

Epoxy polymer

EP-4

d B1

dB2

dB3

I1a

I 2a

I 3a

103 imp/s

Å

I 2i

I1i

I 3i

Relative units

Structure of Epoxy Polymers

203

Structure and Properties of Crosslinked Polymers As earlier, we attribute the halo component, which corresponds to the largest value of Bragg’s interval d B1 , with a loosely packed matrix. In this case d B 2 and d B 3 characterise physical entanglements of the cluster network [23]. Since the values of d B 2 for EP-2, EP-3 and EP-4 are sufﬁciently close (particularly at small modiﬁer concentrations) then this parameter can be attributed to entanglements (clusters) formed by EP network segments. Then d B 3 corresponds to the network of physical entanglements formed at the expense of interaction of adamantane fragments [22]. Firstly let us consider those structural changes that can be elucidated from analysis of the parameters of the EP total amorphous halo. However, let us note ﬁrst of all that introduction of adamantane fragments decreases the crosslinking effective density (Figure 5.5), obviously because of the steric hindrances created by them [19, 20]. The intensity and degree of reduction νc are independent of modiﬁer type and its introduction mode and at concentration cm ≈ 15 mass percentage the value of νc reaches its asymptotic value. Therefore the range 0 ≤ cm ≤ 15 mass percentage represents the greatest interest for study. In paper [6] it has been shown that the glass transition temperature Tg is connected with the macromolecule cross-sectional effective area S and the characteristic ratio C∞ as follows:

(5.2)

In turn, the macromolecule effective diameter dM can be determined from Bragg’s interval value dB according to Equation 4.10. Simulating the crosslinked epoxy polymer macromolecule as a cylinder and using experimental values Tg and dB, the important molecular characteristics S and C∞ can be estimated [23]. The dependences S(νc) for the studied systems and for EP of anhydride and amine curing are also adduced in Figure 5.6. As follows from the data of this ﬁgure, different characters of the dependences S(νc) are observed for the compared EP. The increase in S with growth of νc in the case of non-modiﬁed systems can be explained as follows. In the insert of Figure 5.6 a schematic representation of a crosslinked polymer network is adduced. If we ‘cut out’ a fragment from this network, as it is shown by the dashed lines, then we will obtain a branched macromolecule. The higher the νc the more side branches such a molecule will have. As it is known [24], the availability of side groups in a polymer chain results in a considerable increase in S. Hence, with increasing effective value of νc, S should raise, which reﬂects the dependence S(νc) for non-modiﬁed EP adduced in Figure 5.6. For EP-3 and EP-4

204

Structure of Epoxy Polymers a decrease in νc is accompanied by an increase in the concentration of side groups formed by adamantane fragments (see Figure 5.5), which inﬂuences more strongly the value of S, which results in the dependence S(νc) observed for EP-3 and EP-4.

C∞ 4.5

vc×10-27, m-3 ×-1 -2 -3 1.0

4.0 × ×

0.8

3.5

0.6 3.0 0

5

10

15

0.4 Cm , %

Figure 5.5 The dependence of the crosslinking effective density νc and the characteristic ratio C∞ on modiﬁer contents cm for EP-2 (1), EP-3 (2) and EP-4 (3) [23]

The value of C∞ characterises the polymer chain statistical ﬂexibility and the degree of macromolecular coil compactness [25]. In Figure 5.5 the dependences C∞(cm) for modiﬁed EP are adduced. As follows from the data of Figure 5.5, more intense raising of C∞ is observed for EP-4 than for EP-3, which is due to the differences in the structures of these epoxy polymers. In the EP-4 case the adamantane fragment is built by two bonds into the network or partly forms a card polymer structure. For EP-3 the card polymer structure with one bond at the adamantane fragment is the most probable. Therefore, for macromolecular coil card elements of EP-4 chain structure are more effective in the role of steric hindrances. They do not allow to achieve the same degree of macromolecular coil compactness as do non-modiﬁed EP-1 and EP-2 or EP-3.

205

Structure and Properties of Crosslinked Polymers

S, Å2 45

40 ×× ×

× ×

35

×

× 30

0

0,5

× × ×-1 -2 -3

×

1,0

1,5

vc×10-27, m-3

Figure 5.6 The dependences of macromolecule cross-sectional effective area S on chemical crosslinking effective density νc for non-modiﬁed EP-1 and EP-2 (1), EP-3 (2) and EP-4 (3). In insert: schematic representation of crosslinked polymer network [23] Proceeding to the study of parameters of amorphous halo components, we will also use the comparison of results for non-modiﬁed and modiﬁed EP. In Figure 5.7 the a dependences of the amplitude intensity I1 of the ﬁrst halo component on the loosely packed matrix fraction, calculated according to the methods of [6], are shown. For a all EP growth in I1 with increase in ϕl.m. (structure disordering degree) is observed. a The reasons for this effect were considered above. Unlike the dependence I1 (ϕl.m.) for non-modiﬁed EP the similar dependence for EP-3 and EP-4 extrapolates at ϕl.m. = 0 to a the value I1 different from zero. It is obvious that this difference is connected with the availability of adamantane fragments in EP-3 and EP-4 loosely packed matrix forming a third component of the amorphous halo (see Table 5.1). The extrapolated a value I1 at ϕl.m. = 0 for EP-3 and EP-4 is approximately equal to 0.6 × 103 imp./s a and the amplitude intensity I 3 of the third component of the amorphous halo is varied within the range of (0.14–0.75) × 103 imp/s (Table 5.1), i.e., these values are sufﬁciently close. Let us consider further the interconnection of the second component of the amorphous a halo I 2 , d B 2 and local order domains. As has been already noted, including in clusters interpretation of segments as linear defects supposes that an increase in ϕcl a should result in reduction in I 2 . Such a tendency is actually observed (Figure 5.8), moreover the data for EP-2 are set on a common straight line. Increase in modiﬁer concentration results in growth of d B 2 (Table 5.1). The formation of clusters from segments with increasing distance between the last axis, i.e., d B 2 , means that local

206

Structure of Epoxy Polymers order formation facilitates at cm growth. Therefore one should expect the correlation a between parameters I 2 and d B 2 which is conﬁrmed by the results of the paper [23].

I1a 3

2

1

0

-1 -2 -3

0.2

0.4

ϕl.m.

0.6

a

Figure 5.7 The dependences of amplitude intensity I1 on the loosely packed matrix relative fraction ϕl.m. for EP-1, EP-2 (1), EP-3 (2) and EP-4 (3) [23]

I 2a × 10 −3 1.5

1.0

0.5

0

1

2

3

−1 ϕ cl

a

Figure 5.8 The dependence of amplitude intensity I 2 on the reciprocal value of relative fraction ϕcl of clusters for EP-2 (1), EP-3 (2) and EP-4 (3) [23] 207

Structure and Properties of Crosslinked Polymers Comparison of the dependences of dB and d B 2 on νc for modiﬁed and non-modiﬁed EP shows the essential difference in their behaviour. The values of d B 2 as a function of νc are set in one curve for all studied EP [23]. At the same time dB with increasing νc shows weak growth in the non-modiﬁed EP case and intense reduction for modiﬁed EP (see Figure 5.6). It is probable that adamantane fragments, though they change chain structure, are not included directly in local order domains, which form epoxy polymer network segments. In Figure 5.9 the dependences of the amplitude intensity of the third component of the amorphous halo and Bragg’s interval value, corresponding to the entanglements network of adamantane fragments, on modiﬁer contents cm are adduced.

a

I3 ×10-3

d B3 , Å

-1 -2 -3 -4

0.6

3.6

0.4

3.4

0.2

3.2

0

5

10

15 C ,% m a

Figure 5.9 The dependences of amplitude intensity I 3 (1, 2) and Bragg’s interval value d B 3 (3, 4) on modiﬁer contents cm for EP-3 (1, 3) and EP-4 (2, 4) [23]

The indicated parameters are changed extremely. The dependence d B 3 (cm) shows that in a small cm range modiﬁer concentration increase and corresponding reduction in νc (see Figure 5.5) result in a noticeable simpliﬁcation of the formation conditions for the adamantane fragments network – it can be formed at larger and larger values of d B 3 . However, at cm > 5 the mass percentage growth of d B 3 ceases and some reduction of this parameter, testifying to the opposite effect, is observed. The greatest

208

Structure of Epoxy Polymers stability of clusters, consisting of adamantane fragments, is reached at modiﬁer optimal concentration cm ≈ 5 mass percentage. At cm < 5 mass percentage lack of adamantane fragments makes creation of stable clusters difﬁcult and at cm > 5 mass percentage their surplus creates hindrances for the indicated process [22]. Therefore, the results stated in the present section have shown that the wide angle X-ray diffractometry and mechanical testing using the cluster model of the amorphous state structure of polymers attraction as a theoretical basis allows profound analysis of the structural organisation of complex polymer systems to be carried out, for example, such as epoxy polymers, modiﬁed by network fragments.

5.2 The Curing Inﬂuence on Molecular and Structural Characteristics of Epoxy Polymers At present it is well known [26–28] that a change in the crosslinking density νc for crosslinked polymers results in a number of important variations in characteristics: glass transition temperature Tg, mechanical properties and so on. In turn, for linear polymers correlations between molecular characteristics exist, for example, between macromolecule cross-sectional area S, chain rigidity σ and their macroscopic properties [2, 24, 29, 30]. It can be expected that chemical crosslinking exercises inﬂuence the molecular characteristics of crosslinked polymers thereby inﬂuencing their structure and properties. To test the accuracy of this supposition the authors [31] used the integral molecular parameter (S/C∞)1/2. By its physical essence it is similar to parameter a/σ (where a is chain thickness), used earlier in papers [29, 30]. However, the parameter C∞ is not as vague as σ [24]. This is due to determination of the parameters σ and C∞, the ﬁrst of which depends on the angle of the macromolecule valence bonds and the second on the length of these bonds. The bond lengths are usually determined considerably more precisely than the valence angles. It is probable therefore that in recent years C∞ has been widely applied as a molecular characteristic [32, 33]. In Figure 5.10 the dependence of Tg on (S/C∞)1/2, plotted on the basis of literary data [25, 32, 34–36], is adduced. From the data of Figure 5.10 it follows that even polystyrene and poly(methyl methacrylate), usually showing anomalous behaviour, corresponded to the proposed correlation [24, 29]. The dependence presented in Figure 5.10 is expressed analytically by Equation 5.2. For independent estimation of the value of S the methods stated above [2], based on the application of wide angle X-ray diffractometry, can be used. Within the frameworks of this method an interconnection between Bragg’s interval dB, calculated

209

Structure and Properties of Crosslinked Polymers by amorphous halo maximum according to Equation 4.9, and the macromolecule diameter dM exists, which is described according to Relationship 4.10.

Tg, K 11 10

500

4

250

6

5

9 8 7

3 2 1

0

2

4 (S/C∞)1/2, Å1/2

Figure 5.10 The dependence of the glass transition temperature Tg on the integral molecular parameter (S/C∞)1/2IRUOLQHDUSRO\PHUVSRO\WHWUDÀXRURHWK\OHQH polyethylene (2), polypropylene (3), poly(ethylene terephthalate) (4), poly(vinyl chloride) (5), polystyrene (6), poly(methyl methacrylate) (7), polycarbonate (8), polysulfone (9), polyarylate (10) and polyarylatearylenesulfonoxide (11) [31]

In Figure 5.11 the dependence of dB on the ratio curing agent : oligomer Kst is adduced for epoxy polymers EP-1 and EP-2. The data of Figure 5.11 demonstrate that the dependence dB(Kst) is a maximum at Kst = 1.0, corresponding to the maximum in similar dependences νc and Tg [28]. Hence, both the value of dM and the cross-sectional area of the macromolecule, simulated by a cylinder, are also extremely changed. Tg estimation according to Equation 5.2 at the condition C∞ = const. (C∞ ≈ 5 [35]) shows that the obtained glass transition temperature values are smaller than those observed experimentally. For instance, at Kst = 1.0 for EP-1 the theoretical magnitude Tg = 347 K, whereas the experimental one is 423 K [28]. For EP-2 these values are equal to 349 and 399 K, respectively [28]. In other words, the Tg change is impossible to explain by variation in S only. This observation gives reasons for the assumption that the value of C∞, characterising the degree of compactness of the macromolecular coil [25], changes simultaneously with S. The value of C∞ can be found according

210

Structure of Epoxy Polymers to Equation 5.2, using experimental magnitudes of Tg [28]. The calculations have shown that the extreme decrease in C∞ is observed within the limits of ~ 5.0–3.3 with minimum at Kst = 1.0. Later these results were completely conﬁrmed by the calculations within the frameworks of independent fractal model [37] (see Table 2.1). Hence, chemical curing of epoxy polymers results in increasing cross-sectional effective area of the macromolecule and, as a consequence, its rigidity [24] and also raising of its macromolecular coil compactness [25].

dB, Å 4.9 -1 -2 4.8

4.7

4.6

0.5

1.0

1.5 Kst

Figure 5.11 The dependence of Bragg’s interval dB on the ratio curing agent : oligomer Kst for epoxy polymers EP-1 (1) and EP-2 (2) [31]

Independent checking of the accuracy of quantitative estimations of values of S and C∞ can be carried out as follows. Within the frameworks of the plasticity fractal concept [38, 39] and the cluster model of the amorphous state structure of polymers [5, 6] it has been shown that the yield process is realised in densely packed regions of the structure (clusters). In addition, the relative fraction ϕcl of the clusters is equal to the yield process realisation probability (1 – χ) [39]. The probability (1 – χ) calculation method is given in paper [39] and the value ϕcl can be determined according to Equation 1.11. If the values of S and C∞ were calculated correctly, then the identity should be carried out [39]:

(5.3)

211

Structure and Properties of Crosslinked Polymers As follows from the data of Figure 5.12, this condition is actually carried out. Therefore one can assume that the values of S and C∞ obtained in paper [31] correspond to the real state of the studied epoxy polymers. In Figure 5.13 the dependences of network density of the macromolecular entanglements cluster on the integral molecular parameter (S/C∞)1/2 received for the studied epoxy polymers are adduced. One can see that a clearly expressed interconnection exists between degree of local order (νcl) and molecular characteristics of crosslinked polymer networks.

ϕcl 0.6

0.4

0.2

0

-1 -2

0.2

0.4

0.6 (1-χ)

Figure 5.12 The relation between relative fraction ϕcl of clusters and yield process realisation probability (1 – χ) for epoxy polymers EP-1 (1) and EP-2 (2) [31]

Therefore, the results stated above allow the following scheme of chemical crosslinking inﬂuence on the structural organisation of crosslinked polymers to be supposed. The curing results in a change of the molecular parameters – increasing S and decreasing C∞, – and these changes are stronger the higher νc is. The indicated changes deﬁne physical suprasegmental structure formation of the polymer, namely they control the degree of local order (νcl). In turn, the value of νcl deﬁnes the physical-mechanical properties of crosslinked polymers. The proposed model allows not only qualitative but also quantitative estimation of the considered changes [31].

212

Structure of Epoxy Polymers

νcl×10-27, m-3 2 3 1 2

1

0 2.5

2.9

3.3 (S/C∞)1/2, Å1/2

Figure 5.13 The dependences of the macromolecular entanglements cluster network density νcl on the integral molecular parameter (S/C∞)1/2 for epoxy polymers EP-1 (1) and EP-2 (2) [31]

The computer simulation and prediction of the structure and properties of polymers and polymer composites is one of the most perspective problems of polymer and polymeric materials physics [40, 41]. However, for solution of this problem fulﬁlment of a number of conditions is necessary. One of these conditions is precise knowledge of a number of parameters, used for structure characterisation of polymers, and their identiﬁcation. This requirement is deﬁned, as a minimum, by two fundamental physical postulates. Firstly, as it is well known [42], for glassy polymers the Prigozhin–Defay criterion is not fulﬁlled and by virtue of this the non-equilibrium thermodynamics laws require description of the structure for such polymers using, as a minimum, two order parameters. Secondly, the structure of glassy polymers, including crosslinking polymers, is a homogeneous fractal within the range of linear sizes (scales) of 5–50 Å [43, 44]. This means that, unlike objects with Euclidean geometry, for characterisation of such a structure the application of three dimensions is required: Euclidean dimension d of the environment, fractal (Hausdorff) df and spectral (fraction) ds structure dimensions [45]. In the simplest case the value of d can be accepted to be equal to 3 and then knowledge of df and ds is required (this question was considered in Chapter 2 in more detail). However, the fractal concept is a mathematical model, which does not allow identiﬁcation of structure speciﬁc features of polymers. Therefore, for application of dimensions df and ds within the frameworks of computer simulation it is necessary to elucidate which elements of the structure of polymers (in the considered case crosslinked ones) they personify. In the general case it is known [46] that the value of ds characterises the degree of network connectivity and the value of df (probably

213

Structure and Properties of Crosslinked Polymers dependent on time) of the structure of this network, i.e., the relative arrangement of its elements in space. As to the second dimension, it can be determined according to the following relationship [47]:

(5.4)

where ν is Poisson’s ratio, determined according to the mechanical testing results with the aid of the equation [48]:

(5.5)

where σY is the yield stress, E is the elasticity modulus. In paper [49] it was demonstrated that in the case of crosslinked polymers the discrepancy between experimental and values of ν calculated according to Equation 5.5 at 293 K does not exceed 7%. In the amorphous state of polymers the value of ν is determined by the degree of local order [39, 50] and, hence, the same structural parameter deﬁnes dimension df. As to the spectral dimension ds in the case of crosslinked polymers it is connected with the chemical crosslinking nodes network. In paper [51] the elucidation of such possible interconnection details was undertaken. The results of papers [31, 52] have served as the basis for the supposition accepted above, where the interconnection of the chemical crosslinking nodes density νc and the degree of local order, which also exists between df and ds [45, 46], was shown, and paper [53], where the change in properties of epoxy polymers in due course (at physical ageing) was shown because of a change in the degree of local order (or df), but at invariable value νc,. For the suprasegmental structure of crosslinked polymers the authors [51] selected the cluster model of the amorphous state structure of polymers [5, 6], which is the physical model, allowing to connect suprasegmental structure parameters and molecular characteristics of polymers (for instance, see Equations 1.11 and 4.8). This model presents itself well in the description of the structure and properties of crosslinked polymers [54–57] and, as was shown in the previous section, for the indicated polymers experimental conﬁrmation was obtained by X-ray structural analysis methods [23, 31, 58]. The spectral dimension ds value can be calculated according to Relationship 3.31 214

Structure of Epoxy Polymers '

with replacement of d with df and d s with ds. For estimation of the random walk dimension in Equation 3.31 there is a number of relationships [43–47, 59, 60] that give somewhat differing (within of limits of ~ 20%) values of this parameter. The authors [51] selected the constant value dw variant for the model of ‘lattice animals’ (d = 3) [59], which is usually used for description of branched polymers [60]. The dependence of the values of ds on νc thus determined is adduced in Figure 5.14. As one can see, this dependence has somewhat unexpected character – the increase in crosslinking density results in reduction in ds and, hence, to a decrease in macromolecular network connectivity of chemical crosslinkings. Let us note that at large νc the asymptotic value of ds (see Figure 5.14) approaches the experimentally determined values of spectral dimension for similar EP [43, 44, 61]. Such behaviour can be explained within the frameworks of generally accepted views on structure (or, more precisely, morphology) of crosslinked polymers [62]. In densely crosslinked polymers the globule is a basis morphological unit [62]. The globule represents local condensation of a three-dimensional molecular network with larger than average packing density, which at EP etching is revealed as a discrete particle-globule. The reasons for such heterogeneity of crosslinked polymers are found in three-dimensional network formation process kinetics according to both the polymerisational and the polycondensation mechanism and in the availability of associates in the initial oligomers [62].

ds, δs 1.75

1.65 2

1.55

1 4

3

1.45

0

10

20

νc×10-26, m-3

Figure 5.14 The dependences of the spectral dimension of the entire network ds (1, 2) and interglobular gaps δs (3, 4) on chemical crosslinking density νc for epoxy polymers EP-1 (1, 3) and EP-2 (2, 4) [51]

215

Structure and Properties of Crosslinked Polymers One can proceed from two variants of supramelocular (globular) structure construction: 1) globules grown together, packed hexagonally or cubically, form a strong continuous network, in which the role of interglobular defect gaps is insigniﬁcant; 2) densely crosslinked globules (grains) are localised in a loosely crosslinked matrix, not coming into contact with one another. It can be supposed that in the considered case the morphology second variant is realised. The increase in curing agent concentration results in the simultaneous appearance of a large number of chemical reaction (curing) centres, which is conﬁrmed indirectly by decreasing size of the globules and their number increasing at Kst growth [4]. This results in a decrease in thickness of the defect interglobular zones and an increase in their number that reduces network connectivity (defect zones play the role of some kind of network ruptures) and, hence, decreases the value of ds. In the case of realisation of the second of the variants of crosslinked polymer morphology indicated above, its structure can be presented as a mixture of arbitrary polymeric fractals in the gelation point, having spectral dimensions ds (globules) and δs (interglobular zones). Then the following relationship can be used for δs estimation [63]:

(5.6)

In Equation 5.6 the authors [51] supposed as the ﬁrst approximation that the spectral dimensions of the EP structure and of globules were equal. The dependences δs(νc) for both studied EP are also adduced in Figure 5.14 (curves 3 and 4). As is expected, a similar change in ds and δs is observed at νc variation and with the condition ds > δs. The last condition supposes a smaller degree of network connectivity in interglobular zones in comparison with globules themselves that is explained by the inter globular zone larger imperfection. Let us call attention to the fact that the values of ds and δs for amine curing EP (EP1) are smaller than the corresponding magnitudes for anhydride curing EP (EP-2) at the same νc. This can be due to the following reason. Using amines as the curing agent the epoxy ring rupture occurs without discharging of some substance [64]. In the case of curing by anhydrides, ﬁrstly a monoether is formed as a result of the esteriﬁcation reaction:

216

Structure of Epoxy Polymers

O O ⎥⎢ ⎥⎢ –ɋɇ– Ɉ – ɋ C – Oɇ , ⎪ ⎪ ⎪ ɋɇ2 – ɋH2 and then the diether is formed at carbamide and epoxy groups:

O O ⎥⎢ ⎥⎢ –ɋɇ– Ɉ – ɋ C – O –ɋɇ2 –ɋɇ – . ⎪ ⎪ ⎪ ⎪ ɋɇ2 – ɋH2 OH The interaction of the epoxy and hydroxyl groups, catalysed by anhydrides (residues), occurs simultaneously:

⎪ – ɋɇ – ɈH + – ɋH – ɋɇ – → CH – O – ɋɇ2 – ɋɇ – . ⎪ ⎪ ⎪ O OH The availability of end groups, capable of forming physical bonds (interactions), makes possible the creation of such bonds in the additional network in EP-2 that can result in the crosslinked macromolecular network connectivity increasing and, hence, ds(δs) [65]. In paper [66] the degree of network connectivity was estimated with the aid of the probability of network bonds formation ηx. This parameter was deﬁned as follows:

(5.7)

217

Structure and Properties of Crosslinked Polymers where S is the macromolecule cross-sectional area, am is the magnitude, equivalent to Young’s modulus for a macromolecule elastic stretch, E is the polymer elasticity modulus in tension testing. The values of S for the studied EP were accepted according to the data of paper [31], ac = 7.5 × 10–10 N [66] and values of E were obtained by recalculation of the corresponding values of the elasticity modulus at compression according to the method [48]. In Figure 5.15 the comparison of parameters ηx and ds for the studied EP is adduced. As one can see, the growth of ηx with increasing ds is observed and the correlation between these two parameters is described well by a straight line, drawn according to the following obvious opinions. One can suppose that the chemical crosslinking network ‘mechanical’ connectivity ηx will become equal to zero at such network absence, when EP macromolecules will be simply branched macromolecules, for which ds = 4/3 [67]. The greatest value of ηx is reached at the greatest value of ds, which corresponds to the maximal possible value for real polymers df = 2.95 [47] and according to Equation 3.31 is equal to ~ 1.75.

ηx

0.8

0.4 -1 -2

0

1.4

1.5

1.6

ds

Figure 5.15 The relation between the probability of formation of network bonds ηx and the spectral dimension ds for epoxy polymers EP-1 (1) and EP-2 (2) [51] The formation of loops, not contributing to ηx, but contributing to νc, can be another reason for the degree of decrease in connectivity with increase in concentration of

218

Structure of Epoxy Polymers the curing agent. If the chain part between chemical crosslinking nodes is considered as a random walk, then the probability P0 of formation of loops (probability of the random walk returning back to its starting point) can be described by the following relationship [45]:

(5.8) where N is the number of random walk steps. In the considered case N is accepted to be equal to the number of statistical segments Nst between chemical crosslinking nodes, which is determined according to Equation 4.6. The dependence P0(νc) is presented in Figure 5.16, which was obtained by supposing that in Relationship 5.8 ~ is replaced with the equality sign, i.e., the value of P0 is given in relative units. In this case the given assumption presents itself as being true by virtue of extrapolation of P0 to zero in the absence of crosslinking. As one can see from the data of Figure 5.16, for the studied EP an approximately ﬁve-fold increase in P0 with νc growth is observed that can make its contribution to the degree of reduction in chemical crosslinking network connectivity observed in Figure 5.14 with crosslinking density growth.

P0 0.3

0.2 -1 -2

0.1

0

10

20 νc×10-26, m-3

Figure 5.16 The dependence of the probability of formation of loops P0 on the crosslinking density νc for epoxy polymers EP-1 (1) and EP-2 (2) [51] 219

Structure and Properties of Crosslinked Polymers The scenario of structural changes for crosslinked polymers proposed by the authors of paper [51] is not the only possible one. The reduction in ds (and also ηx, see Figure 5.15) with growth in νc shown in Figure 5.14 is due to decreasing E and increasing νc that follows from the comparison of Equations 3.31, 5.4, 5.5 and 5.7. It is possible that monotonous growth will be observed in the variant of increasing E with growth in νc (for instance, the data of paper [68]) and in this case the function ds(νc). One can suppose that this variant will be realised in case of the ﬁrst of the variants described above of supramolecular (globular) structure formation [62]. As the results stated above have shown, the crosslinking density νc cannot serve as an indicator of the degree of connectivity for a macromolecular network of crosslinked polymers. This notion excludes using νc as one of the characteristics of the structure of crosslinked polymers for computer simulation. As follows from the data adduced above, ds or ηx can be such a parameter, allowing determination of the elasticity modulus (Equation 5.7). However, for estimation of other properties use of just one more parameter is required, which characterises the thermodynamic non-equilibrium level of the structure of glassy polymers. The dimension df, physical entanglements cluster network density νcl [55] or relative fraction of clusters ϕcl [6], which are interconnected (see Equations 1.11 and 4.8), can be this last parameter. For instance, the necessity of accounting for νcl for calculation of yield stress σY at known E was shown in paper [55]. Therefore the correlations often occurring in the literature between one or another property of crosslinked polymers and νc can have only an illustrative character. The obligatory use of two order parameters (for example, df and ds at ﬁxed d) is the main condition, following from the fractality of the structure of glassy polymers, which, in turn, is a consequence of the thermodynamic non-equilibrium of this structure [51].

5.3 The Description of the Structure of Crosslinked Polymers within the Frameworks of Modern Physical Models In the present section a number of modern physical concepts for the description of the structure of crosslinked polymers is used: the thermodynamic concept, the cluster model of amorphous state structure of polymers, fractal analysis, irreversible aggregation models and the thermal cluster model. Within the frameworks of the thermodynamic approach the interconnection of structural and molecular characteristics of crosslinked polymers with disorder parameter δ is considered [69]. According to the concept [69] the indicated parameter, connected with the thermal mobility of molecules near the melting temperature, is expressed by Formula 1.28. Since pi is given by Equation 1.29 then Relationship 1.30 can be received from combination of Equations 1.28 and 1.29.

220

Structure of Epoxy Polymers Accounting for successful application of liquid models for the description of the behaviour of amorphous polymers [70], let us extend the action interval of Equation ~ ~ 1.30 on the glassy state region, replacing Vm with Vl , corresponding to this case. The problem of the quantitative estimation of δ now comes from the determination ~ of V , which can be carried out by using Relationship 1.31. In turn, the relative ﬂuctuation free volume fg, which is also a disorder measure [71], can be calculated according to Equation 1.33. In Figure 5.17 the dependence of fg on the loosely packed matrix relative fraction ϕl.m. for both studied epoxy systems is shown. This dependence is linear, which testiﬁes to the ratio fg/ϕl.m. constancy. In other words, the change in fg for the entire epoxy polymer occurs at the expense of the variation in ϕl.m.. At ϕl.m. = 1.0 the value of fg extrapolates to the magnitude ~ 0.11 that corresponds to the concept known as the Simha–Boyer concept [73], supposing polymer devitriﬁcation at fg ≈ 0.113. Hence, polymer devitriﬁcation is a process of cluster thermoﬂuctuational decay. At ϕl.m. = 0 the straight line passes not through the origin of the coordinates, as occurs in the 0 linear polymers case [74, 75], but cuts on ordinates axis a section of f g ≈ 0.024. 0 It can be supposed that f g is connected with the network availability of chemical bonds. Thus, in crosslinked polymers, unlike in linear ones, the relative ﬂuctuation 0 free volume consists of two components. One ( f g ) is constant by absolute value and connected with chemical crosslinking nodes domains and the second ( ) is the linear function of ϕl.m. and is deﬁned by cluster (suprasegmental) structure state [72].

fg 0.15

-1 -2

0.10

0.05

0

0.4

0.8

ϕl.m.

Figure 5.17 The dependence of the relative ﬂuctuation free volume fg on the loosely packed matrix relative fraction ϕl.m. for epoxy polymers EP-1 (1) and EP-2 (2) [72]

221

Structure and Properties of Crosslinked Polymers Let us consider further the relation between parameters δ and fg (Figure 5.18). The value of fg exceeds δ in all cases, although their change is similar. This fact is probably connected with the fact that the disorder level in crosslinked polymers is deﬁned not by the entire ﬂuctuation free volume, but only its part ( ). The relation δ( ) adduced in Figure 5.19 conﬁrms the stated supposition: the values of δ and not only change similarly, but also have approximately the same absolute magnitudes. Since δ ≈ , then it follows according to Equations 1.30 and 1.31 that the heat expansion coefﬁcient is deﬁned completely by the value of [72].

δ 0.06

0.04

0.02

0

-1 -2

0.03

0.06

0.09 fg

Figure 5.18 The dependence of the disorder parameter δ on the relative ﬂuctuation free volume fg for epoxy polymers EP-1 (1) and EP-2 (2) [72]

The density thermal ﬂuctuations can be considered as a general characteristic of the degree of system disorder [76, 77]. It is supposed [78] that their value Δρ / ρ be deﬁned by fg in the polymer glassy state. The value of Δρ / ρ

2 g

2 g

can

can be calculated

according to Equation 1.34, in which the isothermal compressibility is the reciprocal value of the modulus of dilatation KT. In turn, the value of KT can be determined according to the relationship [48]:

(5.9)

222

Structure of Epoxy Polymers

2

-3 -3 Δρ / ρ g /νc×10 , m

f gcl 0.08

1.5

1.0 0.04

0

0.04

-1 -2 -3 -4

0.5

0.08

0 δ

Figure 5.19 The relation between the disorder parameter δ, the ﬂuctuation free volume connected with clusters (1, 2) and the normalised value of density thermal ﬂuctuations value Δρ / ρ

2 /ν g c

(3, 4) for epoxy polymers EP-1 (1, 3) and

EP-2 (2, 4) [72] It has been shown earlier [78], that the dependences E(Kst) for the studied epoxy polymers have extreme character with minimum at Kst = 1.0, where fg has the smallest value. Since in Equations 1.37 and 5.9 the value of T is constant and the magnitudes of ρ and ν are changed much less than E with variation of Kst [4, 78], then from Equation 1.37 the greatest values of Δρ / ρ

2 g

can be expected for Kst = 1.0, which

does not correspond to the smallest value of fg, appropriate to this case. To remove this contradiction the notions [79] were used, according to which chemical crosslinking network nodes restrict ﬂuctuations of segments in the cluster, thereby decreasing the density ﬂuctuations. The inﬂuence of the crosslinking density νc can be accounted for, normalising Δρ / ρ

2 g

by νc. This method allows reasonable correspondence between

the density ﬂuctuations and the disorder parameter to be established (Figure 5.19). One of the most important properties of solids is their shearing stiffness, characterised by the exponents 1/mn of the Mi equation [47]. This parameter can be determined according to the equation [80]:

(5.10)

223

Structure and Properties of Crosslinked Polymers In Figure 5.20 the dependence of the disorder parameter on the reciprocal value of the shearing stiffness is adduced, which shows that increasing δ reduces the shearing stiffness of the studied epoxy polymers.

δ 0.08

0.04 -1 -2

0

10

20

30 (1/mn)-1

Figure 5.20 The dependence of the disorder parameter δ on the reciprocal value (1/ mn)–1 of the shearing stiffness for epoxy polymers EP-1 (1) and EP-2 (2) [72]

It should be noted that the value of δ is closely connected with different levels of structural organisation of crosslinked polymers. Besides the correlation δ( ) adduced above, applying to the suprasegmental level of the structural hierarchy, it is easy to trace the disorder parameter connection with the molecular and topological levels of the structure. The authors [72] showed that raising the crosslinking density νc reduced the disorder level in epoxy polymers. It was found [72] that increasing the polymer chain statistical ﬂexibility, characterised by growth in C∞, results in raising of δ. These laws keep fully within the frameworks of traditional notions [27]. Hence, the results stated above have shown that the disorder parameter of crosslinked polymers, determined within the frameworks of the thermodynamic concept, is the universal characteristic connected with different levels of structural organisation and deﬁning their properties [72]. Further let us consider the theoretical and experimental aspects of application of the cluster model of the amorphous state structure of polymers [5, 6] for description of

224

Structure of Epoxy Polymers the structural features of the crosslinked epoxy polymer. The photochromic labels method has been widespread in the study of free volume parameters in polymers [81, 82], together with such methods as positron spectroscopy [83, 84] and dilatometry [85]. These methods have allowed qualitatively new information about the free volume in polymers to be received and gives the possibility of comparison with different theoretical models (for example, [86]). The authors of paper [87] demonstrated the correspondence of the experimental data of the photochromic labels method and the cluster model theoretical postulates [5, 6]. The photochromic labels method determines the value and distribution of the free volume for different structural regions of crosslinked polymers: different labels are used for free (handing down) chains, between network neighbouring chains and by curing agent fragments (crosslinking nodes) [81, 82]. So, in paper [82] at the studying of similar to the used above epoxy polymer, namely diepoxide of diglycidyl ether of bisphenol A, cured by p,p'-diaminodiphenylsulfone, as a label for the ﬁrst from the indicated regions the reactive monodiamine p,p'-aminoazobenzene (AA) was applied, for the second the derivative of p,p'-diaminoazobenzene (DAA) was applied, in which four amine hydrogens were substituted by ethyl groups (ttDAA), was also used and for the third-DAA, which is introduced in the chemical network as a curing agent. In paper [82] for epoxy networks after the gelation point it was found that the photoisomerisation kinetics of the indicated regions can be characterised within the frameworks of two processes with substantially differing rates (of approximately two orders). The fraction of ‘fast’ process α was interpreted in paper [82] as the integral area under the distribution curve of the free volume for a separate label. It means that the critical size of the free volume void around the label is required for its isomerisation. Therefore the considered method estimates the state of the surrounding label of different regions of the structure of epoxy polymers (its neighbouring environment) [82]. The free volume microvoid radius in epoxy polymers according to the data of paper [82] makes up on average 6.5 Å (according to the data of positron spectroscopy it is somewhat smaller [82, 83]). The estimations carried out by the authors [87] showed that the distance between cluster centres for epoxy polymers EP-1 and EP-2 changed within the limits of ~ 19–31 Å and the size of a loosely packed matrix gap between two neighbouring clusters was still smaller. Thus, the free volume microvoid sizes and distance up to the neighbouring cluster are comparable by absolute value. This circumstance allows (according to the terminology of paper [82]) the loosely packed matrix gap as a neighbouring environment with a free volume microvoid in it to be considered and, hence, the ‘fast’ process fraction α in epoxy polymers as the loosely packed matrix relative fraction ϕl.m. to be identiﬁed. One more reason for similar estimation is the proportionality of ϕl.m. and the relative ﬂuctuation free volume fg in epoxy polymers (see Figure 5.17) [72, 78]. Let us note one more circumstance

225

Structure and Properties of Crosslinked Polymers conﬁrming the identiﬁcation of α as ϕl.m.. It was noted [81, 82] that such inﬂuences on a polymer as temperature raising, plastiﬁcation and deformation resulted in increasing α and reduction in the physical ageing of this parameter. The value of ϕl.m. behaves in precisely the same way under the inﬂuence of the indicated factors on the polymer [6]. In Figure 5.21 the dependences of the value α for AA (straight line 1) and tt-DAA (straight line 2) crosslinked networks according to the data of paper [82] and ϕl.m. according to the data of paper [72] on the glass transition temperature Tg of epoxy polymers are adduced. One can see that the data for ϕl.m. are located between straight lines 1 and 2, displacing at increasing crosslinking density (or Tg growth) from straight line 1 up to straight line 2. As was noted above, label AA (straight line 1) probes the free volume in sections of pendant chains and label tt-DDA (straight line 2) in sections between fragments included in the network chain. It is obvious that with Tg (and νc) growth the number of pendant chains will be decreased owing to their crosslinking and inclusion in a network, respectively the free volume fraction will be reduced due to these and the free volume between chain fragments, ﬁxed from both sides by chemical crosslinking nodes for a sufﬁciently densely crosslinked network (where molecular mass can reach 104 g/mole [88]), will play the main role. This presumed process is indicated by curve 5 in Figure 5.21.

α, ϕl.m. -3 -4

0.8

1

0.6

5

0.4 2

0.2 273

323

373

423 ɌgɄ

Figure 5.21 The dependences of a ‘fast’ fraction α for labels AA (1) and tt-DAA (2) and the relative fraction ϕl.m. of a loosely packed matrix (3–5) on the glass transition temperature Tg for epoxy polymers EP-1 (3) and EP-2 (4). Curve 5 shows the presumed sites of transition of the prevalent contents of the free volume microvoids [87]

226

Structure of Epoxy Polymers In paper [82] the dependence of α on free volume microvoid radius rh was obtained, which has the sense of a free volume distribution curve. A similar curve ϕl.m.(rh) is adduced in Figure 5.22, where the value of rh was calculated according to the free volume microvoid volume Vh (in assumption of its spherical shape), estimated according to the following relationship [80]:

(5.11)

where ν is Poisson’s ratio, k is Boltzmann’s constant, E is the elasticity modulus. ν and fg are the magnitudes determined according to Relationships 5.5 and 1.33, respectively.

ϕl.m. 1.0

-1 -2 -3

0.8

0.6

0.4

0.2 0

1

2

3

rh, Å

Figure 5.22 The relation between the loosely packed matrix relative fraction ϕl.m. and the free volume microvoids average radius rh for EP-1 (1), EP-2 (2) and aged samples EP-2 (3) [87]

Unlike the plot α(rh) [82], the plot ϕl.m.(rh) adduced in Figure 5.22 has somewhat differing physical signiﬁcance. Since the last correlation includes the data for different epoxy polymers, then it indicates a change in regularity of the free volume microvoid size rh of variation of the degree of thermodynamic non-equilibrium of the structure

227

Structure and Properties of Crosslinked Polymers of these epoxy polymers. So, the local order level increasing (reduction in ϕl.m.) results in growth of rh. In Figure 5.22 the data for the EP-2 aged at 293 K for 1.5 years are also adduced. As is expected, reduction in ϕl.m. with simultaneous rh is observed, which indicates the well-known raising of the degree of thermodynamic equilibrium of the structure of epoxy polymers in the physical ageing process [42]. Therefore, the results stated above suppose the possibility of identiﬁcation of the‘fast’ fraction α by data from the photochromic labels method as the relative fraction ϕl.m. of the loosely packed matrix within the frameworks of the cluster model, which corresponds quite well to the physical signiﬁcance of α. The data of Figure 5.22 show that the relative ﬂuctuation free volume fg, treated as a disorder parameter (see Figure 5.19) [42], can not only serve as a measure of the thermodynamic non-equilibrium of the structure of crosslinked polymers, but also as a free volume microvoid average size (rh or Vh). Let us note that the authors of paper [82] assumed also that the parameter α offered a treatment, believing it to be connected with high segmental mobility, associated with less dense regions in solid polymers. As was shown in Chapter 3, computer simulation of the formation processes of crosslinked polymers resulted in the conclusion that at the gelation point crosslinked networks of these polymers were fractal structures, and that an increase in the crosslinking density νc resulted in raising of the dimension of such a network fractal . However, from the practical point of view the last parameter is not very suitable, although lately it has been preferred to characterise glassy crosslinked polymers precisely with the aid of the value νc [27]. Nevertheless, it was shown that for the same crosslinked polymer at invariable value νc it is possible to receive different properties, for instance, different values of the glass transition temperature [53]. This is explained by a change in the crosslinked polymer suprasegmental (cluster) structure in the physical ageing process, namely by raising of the degree of local order [53]. In turn, the degree of local order deﬁnes the fractal dimension df value for polymer suprasegmental structure (Equation 4.8). Thus, the possibility of prediction of structural characteristics (and, hence, properties) appears for crosslinked polymers in the glassy state, proceeding from the chemical crosslinking level up to the gelation point. One of the possible variants of the solution to the indicated problem was considered in paper [89] on the example of amine and anhydride curing epoxy polymers, cured at atmospheric pressure (EP-1 and EP-2) and in thorough compression conditions at a pressure of 200 MPa (EP-1-200 and EP-2-200), respectively. For derivation of the relationship between crosslinking density νc and structure fractal dimension df for the studied epoxy polymers the model proposed in paper [90] was used, the essence of which consists in the following. Competitive processes of crystal growth and nucleation were described within the frameworks of two kinetic equations [90]:

228

Structure of Epoxy Polymers

(5.12)

(5.13)

where R is the crystal size, t is the current time, k1 and k2 are equilibrium constants, N is the number of crystals, c is the concentration of molecules on the surface, m and p are exponents. Using Equations 5.12 and 5.13 for local order domains (clusters) the description of the growth of epoxy polymers and nucleation is based on the fact that clusters by their physical essence are analogous to crystallites with stretched chains (CSC) [5, 6]. The cluster characteristic size is accepted to be equal to the statistical segment length lst, which is calculated according to Equation 1.8. As in paper [90], the value m = 1, since cluster growth represents joining of the sole segment of length lst [57] and p > 1, since cluster nucleation requires the existence of two or more segments in the nucleation site. However, application of Equation 5.12 to one epoxy polymer taken separately requires serious modiﬁcation. It is obvious if to accept R = lst then the Equation 5.12 no longer makes sense, since for each studied epoxy polymer lst = const. and dR/dt = 0, which indicates the impossibility of the growth of clusters. Therefore, designating as ncl the number of segments in one cluster, Equation 5.12 can be written as follows [89]:

(5.14)

The limiting value n cl is restricted by nucleation of new clusters, having thermoﬂuctuational origin, and therefore [90]:

(5.15)

229

Structure and Properties of Crosslinked Polymers It is obvious that the fraction c of macromolecules segments able to form clusters should reduce during the formation of clusters and, accounting for the condition lst = const. for each epoxy polymer, the value of ncl should also decrease. This explains the availability of unstable clusters, which have smaller values of ncl [5, 6, 91], in the amorphous state structure of polymers. This also explains the self-similarity of the structure of a cluster within the scales range of its existence. Let us note one more important circumstance – the system of Equations 5.12 and 5.13 is applicable to all studied epoxy polymers and the system of Equations 5.14 and 5.15 is applicable to each separately [89]. For all the studied epoxy polymers the following relationships can be written [90]:

(5.16)

and

(5.17)

where tgel is the gelation temporal range. Equation 5.17 predicts that with increasing crosslinking density νc there is a reduction in C∞ and, hence, lst occurs (see also Table 2.1). This results in intensiﬁcation of nucleation of new clusters , i.e., to growth in Ncl. Assuming R = lst, m = 1 and dividing Equation 5.12 by Equation 5.13, we obtain [89]:

(5.18)

since k1 and k2 are constants.

230

Structure of Epoxy Polymers From Equation 5.16 in combination with Equation 5.13 at the condition tgel = const., k2 = const. and, as earlier, N = Ncl we obtain [89]:

(5.19)

or, since k2tgel = const.:

(5.20)

Equations 5.18 and 5.19 allow determination of c [89]:

(5.21)

(5.22)

Equating the right-hand parts of Relationships 5.21 and 5.22, we obtain [89]:

(5.23)

or

(5.24)

231

Structure and Properties of Crosslinked Polymers In paper [90] it was indicated that the nucleation classical theory requires p < 2, the consequence of which is the condition [89]:

(5.25)

One more restriction exists – the volume of clusters should not exceed the total volume of polymer, which gives one further condition [90]:

(5.26)

Therefore, the exponent in Relationship 5.24 has precisely the same limits as the object fractal dimension in three-dimensional Euclidean space [92]. The precise interconnection of p/(p – 1) and df can be obtained, comparing Relationship 5.24 with that obtained in paper [93]:

(5.27)

As it is known [5], the value of Ncl can be written as follows:

(5.28)

where νcl is the macromolecular entanglements cluster network density, which is equal in the ﬁrst approximation to the number of segments in clusters per polymer volume unit [6], Fcl is the cluster functionality and, since the last is CSC analogue, then [5]:

(5.29)

232

Structure of Epoxy Polymers Let us pay attention to one more important circumstance. The comparison of Equations 5.13 and 5.27 demonstrates that the greatest rate of formation of clusters is reached at p = 2 or df = 2. In other words, it is supposed that the smallest fractal dimension of polymer structure corresponds to the greatest level of local order [89]. Within the frameworks of the cluster model [5, 6] a crosslinked polymer can be considered as the superposition of two networks – the chemical crosslinking nodes network and the physical entanglements cluster network with densities νc and νcl, respectively. Assuming each of these networks to be perfect (i.e., not accounting for the effect of end chains), the molecular mass of chain parts between clusters can be determined according to Equation 1.4 and between chemical crosslinking nodes according to the similar equation with replacement of the functionality of clusters Fcl on the corresponding parameter for chemical crosslinking nodes f = 4 [89]. Accounting for Equations 1.4 and 5.28, Relationship 5.27 can be written as follows [89]:

(5.30)

In paper [93] the following relation between parameters C∞ and df was obtained:

(5.31)

where d is the dimension of Euclidean space in which a fractal is considered (it is obvious that in our case d = 3). Combination of Relationships 5.30 and 5.31 allows the following equation to be received [89]:

(5.32)

where l0 is the main chain skeletal bond length (for the studied epoxy polymers l0 = 1.25 Å [35]) and the empirical numerical coefﬁcient in the right-hand part

233

Structure and Properties of Crosslinked Polymers of Equation 5.32 is received by equating the values of df calculated according to Equations 5.4 and 5.32 at νc = 1027 m–3. Comparison of the values of df calculated according to Equations 5.4 and 5.32 as a function of νc for the studied epoxy polymers is adduced in Figure 5.23. As one can see, good correspondence of results calculated according to Equations 5.4 and 5.32 was obtained. This means that increasing the network fractal dimension up to the gelation point [94] results in reduction of the fractal dimension df of the cluster structure of a crosslinked polymer in the glassy state. Let us note that knowledge of the value of df allows properties of epoxy polymers to be predicted [78]. Thus, by varying the value of Kst or interrupting the reaction at a certain stage, the polymers with the desired properties can be received. However, from a practical point of view the use of the stoichiometric value Kst = 1.0 is preferable, since it gives the most stable crosslinked systems [47]. In this case variation of the ﬁnal product properties can be reached at the expense of a change in the curing agent functionality f or alteration of the molecular characteristics of the epoxy oligomer.

df 2.9 -1 -2 -3 -4

2.8

2.7

2.6 0

5

10

15 20 νɫ×10−26, m−3

Figure 5.23 The dependence of structure fractal dimension df on crosslinking density νc. Calculation according to Equation 5.4 for EP-1 (1), EP-2 (2), EP-1200 (3), EP-2-200 (4) and according to Equation 5.32 for all enumerated epoxy polymers [89]

234

Structure of Epoxy Polymers Hence, the cluster model together with the application of fractal analysis notions allows the interconnection of crosslinked polymer characteristics up to the gelation point (or at this point) and in the glassy state to be received. Let us note that this is related only to the polymer directly after the curing process. Further, the value of df will be changed because of the occurrence of the physical ageing process, inevitable by virtue of the thermodynamically non-equilibrium nature of the glassy state of polymers. Nevertheless, the offered methodology allows the prediction of the properties of glassy crosslinked polymers already at the curing reaction stage to be realised. The considered model predicts qualitatively the supposed earlier effects, namely the existence of unstable clusters [91, 95], cluster structure self-similarity [96] and reduction in df with growth in the degree of local ordering [93]. As was elucidated within the frameworks of computer simulation of clusters, formed by diffusion-limited aggregation (DLA), the steric factor p (p ≤ 1), showing that not all particle collisions occur with proper orientation for particle ‘sticking’ to the cluster, plays an important role, deﬁning the actual aggregation mechanism and ﬁnal aggregate structure, characterised by its fractal dimension df [97, 98] (the role of the factor p in curing reactions of epoxy polymers was considered in Chapter 3). So, at large p, close to unity, the diffusion-limited mechanism of aggregate growth is realised with relatively small values of df, and at small p of order 0.01 the chemically limited aggregation mechanism is realised with more compact ﬁnal aggregates. Proceeding from this, in paper [99] the degree of inﬂuence of the crosslinking density νc on the value of p and formation conditions of local order domains (clusters) were elucidated for crosslinked epoxy polymers EP-2 and EP-3. For Witten–Sander (WS) clusters, from the set of which the structure of crosslinked polymers is simulated [100], it can be written [60]:

(5.33) where rd is the radius of the central densely packed region of the WS cluster , associated with the local order domain having radius rcl [101], which can be determined according to the equation:

(5.34) where η is the packing density, for a monodisperse circle equal to 0.76.

235

Structure and Properties of Crosslinked Polymers Estimation of the approximate proportionality coefﬁcient in Relationship 5.33 can be carried out as follows. The distance between chemical crosslinking nodes Rc was calculated according to Equation 1.14 and then the proportionality coefﬁcient value was determined according to Relationship 5.33, assuming rd = Rc and corresponding to this condition a minimum value of p = 0.1 [99]. In Figure 5.24 the dependence p(νc) is adduced, which turns out to be linear. As one can see, an increase in νc results in p growth, which is accompanied by reduction in df [102]. This corresponds to the common notions in model [98]. As the results of papers [103, 104] have shown, an increase in νc results in growth of the cluster network density νcl and reduction in the number of segments ncl in one cluster. This means a large number Ncl (= νcl/ncl) of small clusters are formed. In Figure 5.25 the dependence Ncl(p) is adduced, from which follows linear growth Ncl at increasing p (or νc, see Figure 5.24). However, the dependence Ncl(p) does not pass through the coordinates origin and at Ncl = 0 the value of p is ﬁnite and equal to ~ 0.215. This means that at p ≤ 0.215 the cluster network is not formed or, in other words, the crosslinked polymer does not pass on to a glassy state. From Figure 5.24 it follows that the value p = 0.215 corresponds to νc = 2.2 × 1026 m–3 and at values of νc lower than that indicated, the crosslinking polymer does not proceed in the condensed state. The critical magnitude νc ( ) can be estimated theoretically. The condition Ncl = 0 and, hence, νcl = 0 is a criterion of polymer transition in the rubber-like state [54]. For such a state the fractal dimension Dch of a chain part between crosslinking nodes, characterising molecular mobility, is equal to 2 [6]. For the considered epoxy polymer series the greatest value of C∞ (at the smallest value of νc) is equal to 5.35 [37]. This allows determination of the greatest value of df for epoxy polymers according to Equation 5.31 and then, according to Equation 4.15, the number of statistical segments nst between chemical crosslinking nodes. At Dch = 2 and df = 2.737 according to 5.31 from Relationship 4.15 nst = 23.3 was obtained and the chain part length Lc between crosslinking nodes was calculated according to Equation 4.6. Estimating the common length L of macromolecules per polymer volume unit according to Equation 1.7, can be estimated as L/Lc, from where it follows = 2.1 × 1026 m–3. This theoretical magnitude of , corresponds perfectly to the conformable experimental value of estimated graphically from the data of Figures 5.24 and 5.25 ( = 2.2 × 1026 m–3) [99]. The extrapolation of plot p(νc) (Figure 5.24) to the greatest value p = 1 gives the lim limiting value of νc ( ν c ), equal to ~ 26 × 1026 m–3. At this value of νc the following fractal dimension values should be obtained: Dch = 1 and df = 2 [93]. Under these conditions according to Equation 4.15 nst = 2 will be obtained and, according to lim Equation 4.6, Lc = 12 Å. Then the theoretical magnitude of ν c can be estimated as ~ 24.5 × 1026 m–3, which corresponds well once again to the experimental estimation adduced above. The smallest value of nst=2 corresponds very well to a similar estimation for limitedly crosslinked polymers, obtained by the authors [70].

236

Structure of Epoxy Polymers

p 0.60

0.35 -1 -2

0.10 0

5

10

15 ν ×10-26, m-3 c

Figure 5.24 The dependence of the steric factor p on the crosslinking density νc for epoxy polymers EP-2 (1) and EP-3 (2) [99]

Ncl×10-27, m-3 1.0

0.5 -1 -2

0 0.2

0.3

0.4

0.5

p

Figure 5.25 The dependence of the number of clusters Ncl per polymer volume unit on the steric factor p value for EP-2 (1) and EP-3 (2) [99]

237

Structure and Properties of Crosslinked Polymers Therefore, the results stated above have demonstrated that the chemical crosslinking density νc, controlling the chain part between crosslinking nodes conformation, deﬁnes the steric factor p value: the higher νc, the larger p. This means that increasing νc creates chain conformation, which is favourable for stacking of statistical segments in local order domains. In turn, this circumstance deﬁnes the formation of a large number of small clusters at large νc [99]. The experimental observation of the same Gaussian statistics of polymer chains in θ-solvent and condensed state is the main objection against local order availability in amorphous state polymers [105]. The equality of distances between macromolecules or subchains ends in the indicated states is considered as one of the pieces of evidence of this rule. Boyer [106] demonstrated schematically the possibility of local order existence at fulﬁlment of the indicated condition. However, strict conﬁrmation of such a possibility was not obtained. Therefore the authors of paper [107] conﬁrmed analytically Boyer’s concept on the example of two series of epoxy polymers (EP-1 and EP-2). Let us note one more factor inﬂuencing the sizes of clusters in epoxy polymers, which are expressed by parameters Fcl or ncl (Equation 5.29 gives their interconnection). The calculation of effective molecular mass and for a network of chemical crosslinking nodes and a cluster network, respectively, was adduced above and based on Equation 1.4. As it is known [108], the statistics of subchains will be close to Gaussian if it contains ~ 20 or more real bonds, which is true on the whole for the studied epoxy polymers. It is supposed [107] that subchains between chemical crosslinking nodes and clusters in this respect are in equal conditions, from which it follows:

(5.35)

or

(5.36)

Since Mc and Mcl are inversely proportional to the density νc and νcl, of the corresponding networks then from Relationships 5.35 and 5.36 it follows [107]:

238

Structure of Epoxy Polymers

(5.37) at the condition indicated above f = 4. The accuracy of this supposition will be veriﬁed below. Forsman [108] obtained Equation 1.52, allowing the calculation of the number of segments ncl in one cluster. Parameters included in this equation can be calculated as follows [6]. The value θ2 is determined according to the relationship [108]:

(5.38) where k is Boltzmann’s constant, T is the testing temperature, Useg is the energy of segment association (dissociation) to the cluster, accepted as being equal to [6]:

(5.39) where G is the shear modulus, lst is the statistical segment length, b is the Burgers vector, ν is Poisson’s ratio, r1 and r01 are the external and internal radius defects power ﬁeld, respectively, whose ratio is usually accepted as being equal to ~ 10 [5]. The value of G is determined according to the equation:

(5.40) where E is Young’s modulus and the value of b is deﬁned as follows [6]:

(5.41)

239

Structure and Properties of Crosslinked Polymers The parameter A accounts for the change in the distance between ends of subchains up to and after formation of clusters, h0 and h, correspondingly. In this case the parameter α, determined according to Equation 1.54, is introduced. Following the Boyer concept [106], the authors of paper [107] deﬁned r = r0 and α2 = 1. The parameter A was determined according to Equation 1.53 and at α2 = 1 the value of A = 1. Figure 5.26 shows the comparison of functionality of clusters and values, calculated according to Equations 5.37 and 1.52, respectively (in the last case = ncl/2). As one can see, good correspondence between and is obtained, which conﬁrms once more that invariance of subchains statistics (r = r0) in epoxy polymers does not create obstacles to the formation of local order domains [107].

Fcl2

30

20

-1 -2

10

0

10

20

30

Fcl1

Figure 5.26 The relation between the functionality of clusters and , determined according to Equations 5.37 and 1.52, respectively, for epoxy polymers EP-1 (1) and EP-2 (2) [107]

The crosslinking density νc results in a reduction in the functionality of clusters Fcl. One can suppose that this effect is due to the length of the subchains decreasing with νc raising, reducing their mobility [70] and, as a consequence, probably a reduction

240

Structure of Epoxy Polymers in the contacts of the segments and clusters (‘sticking’). Mobility of subchains in epoxy polymers can be described with the aid of their fractal dimension Dch, which is determined according to Equation 2.17. In Figure 5.27 the dependence of on the value of Dch for two studied epoxy polymers is adduced, which conﬁrms the supposition made above. As follows from the data of Figure 5.27, at values of Dch close to 1 (at molecular mobility almost complete freezing), the functionality of clusters reduces to zero, i.e., local order domains in this case cannot be formed. Hence, invariance of the subchains statistics of epoxy polymers in the glassy state in comparison with the state in solution (for example, up to the gelation point) does not prevent formation of local order domains (clusters) virtually without changing their functionality. Increasing mobility of the subchains, expressed by growth in their fractal dimension, results in the formation of a smaller number of larger clusters, i.e., having relatively high functionality [107].

Fcl2

24

12 -1 -2

0

1.2

1.4

2.0

Dch

Figure 5.27 The dependence of the functionality of clusters , determined according to Equation 1.52, on the fractal dimension of subchains Dch for epoxy polymers EP-1 (1) and EP-2 (2) [107]

Let us consider further the application of irreversible aggregation models for the description of epoxy polymer structure. This was partly carried out above (for

241

Structure and Properties of Crosslinked Polymers instance, in paper [99]), since such concepts as a cluster model, fractal analysis and irreversible aggregation models are interconnected and, as a rule, their combination gives the best result in the description of the structure of polymers [109]. As it has been shown in paper [109], WS DLA particle-cluster (P-Cl) allows description of a large number of real physical processes and the structures formed in them. Computer simulation allows, at any rate, to deﬁne structure modiﬁcation tendencies of ﬁnal aggregates of the inﬂuence of either factor that is important in the case of the description of WS model application at real objects. Nevertheless, when it is a question of the latter then computer simulation schemes require clear physical concretisation in reference to the studied object structure. The beneﬁts of either process (aggregate) within the frameworks of model DLA P-Cl (or other aggregation models, see Chapter 3) are obvious. If the cluster model of amorphous state structure of polymers can be described as a WS cluster then it means that they belong to one universality class of physical phenomena. Let us be reminded that the essence of the universality hypothesis is contained in the following: if the same limiting conditions (system parts interactions) for the formation mechanism of different systems are typical, then these systems are in the same universality class of physical phenomena [60]. In turn, this means that the amorphous state structure of polymers comes into a much wider class of physical phenomena, but does not present itself as being isolated from other physics ﬁelds. In this case general laws of the given universality class systems are applicable to it, which makes its description within the frameworks of modern physical concepts well founded theoretically; the universality class is characterised by a certain set of scaling exponents (indices) [60]. Proceeding from this, in paper [100] it was shown that notions of the WS clusters model allowed the structure of crosslinked polymers to be described with the usage of the scaling index, which is fractal (Hausdorff) dimension. Several reasons (one of them is cluster appearance, Figure 5.28) exist for the structure of amorphous polymers within the frameworks of the cluster model and the WS cluster. The ﬁrst of these reasons is the correspondence of the scaling exponent (fractal dimension df) in both models [110, 111]. The second reason consists in the known fact [112] that the gelation point transition from one universality class to another occurs, namely from DLA cluster-cluster (Cl-Cl) to DLA P-Cl, i.e., the polymer beyond the gelation point is a WS cluster. The third reason is the mechanism of the formation of local order domains in the cluster model. It is shown [74] that separation of one segment from such a domain means formation of one microvoid of ﬂuctuation free volume and segment adding – microvoid ‘collapse’. Such a mechanism within the frameworks of the DLA model can be associated with aggregation P-Cl that together with the value of df is the main characteristics of the given universality class [60].

242

Structure of Epoxy Polymers

2r

Rcl

Figure 5.28 A schematic representation of the cluster structure of the amorphous state of polymers [93]

Let us note that the interpretations of the term ‘cluster’ in models [5, 6] and [111] are different. If in the WS model under the cluster both local order domain (or dense packing region [113]) and chains (branches) emerging from it are understood then in the cluster model only the statistical segments are included in the local order domain. It is obvious that it is impossible to consider the structure of the polymer sample as a WS cluster (on such scales it is a Euclidean object). Therefore by analogy with the modiﬁed WS model [114] the authors [100] associated the structure of crosslinked polymers with a large set of WS clusters, where each such cluster has a size of order 2Rcl. According to the model [114] growth of WS clusters is realised in a large number of ‘seeds’ and the value of Rcl is connected with concentration c of freely diffusible particles by the equation [114]:

(5.42)

Let us consider the physical signiﬁcance of the parameter c. As follows from Figure 5.28, only those statistical segments of macromolecules that are in a loosely packed matrix have the possibility of being added to local order domains. Thus, raising of the local order level results in a reduction in c and at the condition Rcl = const. this means a decrease in (df – d) or a reduction in df. In paper [100] the value of c is accepted as being a product of the number of particles (statistical segments) per volume unit

243

Structure and Properties of Crosslinked Polymers and a relative fraction of the loosely packed matrix. In Figure 5.29 the comparison e of fractal dimensions of the structure of epoxy polymers d f [91] and WS cluster d Tf calculated according to Equation 5.42 is adduced. As follows from the data of e T this ﬁgure, good correspondence is obtained between the values of d f and d f . Hence, the irreversible aggregation model DLA P-Cl in its modiﬁed form [114] gives a good description of the structure of crosslinked polymers at the expected scaling indices using df. This allows the structure of crosslinked polymers to be attributed to the indicated universality class of physical systems and its properties within the frameworks typical for this class of relations to be described [100]. Percolation models are widely used for solving a large number [115] of physical problems, including those for polymers [116, 117]. These models are distinguished by their simplicity and visuality [115] and their application for the description of the structure and properties of polymers allows the use of well-developed mathematical calculus of percolation theory for the indicated purposes [118]. Formation of the cluster structure at the glass transition temperature Tg changes sharply the properties of amorphous polymers, giving them a stiffness that is typical for solid polymers [5, 6]. Therefore it can be assumed that in the given case Tg is the percolation threshold [118], at which inﬁnite (within sample bounds) clusters are formed. The authors of paper [102] checked this assumption on the example of two series (EP-1 and EP-2) of epoxy polymers.

d Tf

d ef -3 -4

2.8

2.8 2.7

1 2.7

2 2.6

2.6 2.5

0.5

1.0

1.5 Kst

Figure 5.29 The dependences of structure fractal dimension, experimentally e T determined d f (1, 2) and d f calculated according to Equation 5.42 (3, 4) on the value of Kst for epoxy polymers EP-1 (1, 3) and EP-2 (3, 4) [100] 244

Structure of Epoxy Polymers As it is known [115], the critical behaviour of the power P∞ of an inﬁnite cluster (of the probability of a node belonging to this cluster) approaching the percolation threshold xc is described by the scaling relationship:

(5.43) Within the frameworks of the cluster model the relative volume fraction ϕcl of clusters is an obvious choice for P∞ and, as it was noted above, the glass transition temperature Tg is accepted as xc. In this case the testing temperature T is accepted as the current probability x and Equation 5.43 can be rewritten as follows [102]:

(5.44) where the change in the places of T and Tg is due to inequality Tg > T. Let us note that since all tests were carried out at T = 293 K, then as a matter of fact Relationship 5.44 gives for the studied epoxy polymers the dependence of ϕcl on Tg. In Figure 5.30 the dependences of ϕcl on ΔT = (Tg – T) in double logarithmic coordinates for EP-1 and EP-2 are adduced. The indicated plots are linear, which allows calculation of the exponent β value according to their slope. The value of β turns out to be equal to 0.36 for EP-1 and 0.58 for EP-2, which is sufﬁciently close to the theoretical ‘geometrical’ magnitude β = 0.40 [115]. Hence, the cluster structure of the studied epoxy polymers is a percolation cluster, having a percolation threshold of Tg [102]. This means that the glass transition of crosslinked polymers represents the phase transition and ϕcl is the order parameter [115]. The same circumstance supposes that the general laws of percolation theory can be used for the description of the structure of epoxy polymers [115]. So, the number of ﬁnite cluster nodes s depends on non-dimensional deviation of concentration τ from the critical one (τ = (x – xc)/xc [115]) at τ®0 as follows [115]:

(5.45) And, as earlier, the number of segments in one cluster ncl, which is equal to Fcl/2, should be accepted as s and as τ the parameter (Tg – T)/Tg. In Figure 5.31 the corresponding dependences in double logarithmic coordinates are adduced, which turn out to be linear, which allows determination of the exponent in Relationship 5.45. Its value is equal to 1.28 for EP-1 and 2.28 for EP-2. These values of γ correspond satisfactorily

245

Structure and Properties of Crosslinked Polymers with this exponent theoretical value (~ 1.84) for a three-dimensional percolation cluster [115].

ln ϕcl

ln ϕcl 2

0

-0.5

-0.5

1 -1.0

-1.0

-1.5

-1.5

3.5

4.0

4.5

-2.0 5.0 ln ΔT

Figure 5.30 The dependences of the relative fraction ϕcl of clusters on the difference of temperatures ΔT = (Tg – T) in double logarithmic coordinates for epoxy polymers EP-1 (1) and EP-2 (2) [102]

ln(Fcl/2) -1 -2

3

2

1

0 -2.5

-2.0

-1.5

-1.0 lnτ

Figure 5.31 The dependences of the number of nodes (segments) in a ﬁnite cluster Fcl/2 on non-dimensional deviation of temperature τ = (Tg – T)/Tg in double logarithmic coordinates for EP-1 (1) and EP-2 (2) [102]

246

Structure of Epoxy Polymers As it is known [115], structures that behave as fractal ones on small length scales and homogeneous on large scales are called homogeneous fractals. For amorphous polymers it was shown experimentally [44], that their fractal behaviour is observed from several angstroms up to ~ 50 Å. Let us note that these linear scales correspond precisely to cluster structure boundaries. For the last statistical segment length lst is a lower linear scale and the distance between clusters Rcl is an upper one. For the studied epoxy polymers the value of lst changes within the limits of 4.23–6.59 Å and Rcl within the limits 16.7–35.3 Å [31, 37], which correspond well to the linear limits of fractal behaviour adduced in paper [44]. Percolation clusters are homogeneous fractals at x < xc (i.e., the structure of epoxy polymers is fractal at T < Tg or, accounting for the above, at T < Tg). In other words, the considered data combination supposes that the well-known fractality of the structure of polymers (crosslinked ones among others [61]) is deﬁned by the availability of cluster structure or, more precisely, by freezing local order presence in them. However, in respect to the fractal dimension df value of the structure of polymers, certain disagreements exist at the general restriction 2 ≤ df < 3 [92]. So, in paper [44] on the basis of theoretical postulates of Alexander and Orbach it was concluded that the value of df for an ideal linear disordered polymer was equal to 2 and for real polymers it was 2.0–2.2. The authors [102] carried out several simple estimations of df for the studied epoxy polymers. For a percolation cluster Relationship 2.6 is true, in which one of the critical indices of percolation cluster ν is determined according to the equation [115]:

(5.46) The estimation according to Equation 5.46 at d = 3 using the values of β and γ obtained earlier gives ν = 0.67 for EP-1 and ν = 1.15 for EP-2, which again corresponds well to the theoretical value ν = 0.88 [115]. The dimensions of df calculated according to Equation 2.6 are equal to 2.46 and 2.50 for EP-1 and EP-2, respectively (compare with the value of df for a percolation cluster, which is equal to ~ 2.545 [115]). The calculation according to Equation 5.4 has shown that values of df for epoxy polymers EP-1 and EP-2 are within the limits of 2.56–2.73, which corresponds well to the estimations carried out above, but is essentially higher than was supposed earlier [44, 67]. In paper [44] the following relationship was obtained for percolation clusters:

(5.47) 247

Structure and Properties of Crosslinked Polymers where dsl is the so-called superlocalisational exponent, ds is the spectral dimension. Using the values of dsl and ds for an epoxy diane polymer [44], the authors [102] calculated the value of df, which turned out to be equal to 2.55. This value is again close to the estimations carried out above and it is necessary to note that the use of Equation 5.47 for epoxy polymers becomes possible owing to the identiﬁcation of their structure as percolation systems, fulﬁlled in the present section. In paper [119] it was shown that the fractal dimension is deﬁned by disorder, which is generated by deterministic chaos. This postulate is expressed analytically as follows [119]:

(5.48)

where Ss is the system entropy, λL is the Lyapunov exponent. The entropy change ΔSs can be estimated according to Equation 2.27. In Figure 5.32 the dependence of df, calculated according to Equation 5.4, on value ΔSs, calculated according to Equation 2.27, is adduced for the studied epoxy polymers. This plot has a number of interesting features. Firstly, extrapolation of the linear plot df(ΔSs) to ΔSs = 0 gives the value df = 2. From Equation 2.27 it follows that the condition ΔSs = 0 corresponds to fg = 0. The last condition indicates dense packing of the structure of epoxy polymers. In other words, for an amorphous polymer the condition df = 2 is reached at the greatest possible dense packing, but not for the ideal disordered polymer, as it is supposed in paper [67]. Secondly, the value df = 3 is reached at magnitude ΔSs, which corresponds to fg ≈ 0.157. As it is known [120], the indicated value of fg is the greatest theoretical value of relative free volume at Tg. In other words, the value df = 3 corresponds to the devitriﬁcated state of the polymer, i.e., to local ‘freezing’ order decay [54]. This circumstance indicates again a connection between the local order and fractality of amorphous polymers. Thirdly, the linear dependence df(ΔSs) means that λL = const. according to Equation 5.48. This indicates that the chaotic component of the dynamics of structure formation is the same for all studied epoxy polymers [121]. Hence, the results stated above have demonstrated that at the glass transition temperature Tg in epoxy polymers the percolation cluster structure is formed, which is described well within the frameworks of the cluster model. This circumstance supposes that the glass transition of epoxy polymers is the phase transition and the value of ϕcl is the order parameter. This fact is one more piece of theoretical evidence of the

248

Structure of Epoxy Polymers fractal nature of the structure of crosslinked polymers (at T < Tg) within the range of linear scales ~ 3–50 Å. The fractal dimension of the structure of epoxy polymers is essentially higher than was proposed earlier, and is deﬁned by the degree of structure disorder. Loss of fractality by the structure of epoxy polymers, i.e., df = d, is reached at the glass transition temperature.

df 3.0

2.5 -1 -2

2.0

0

0.4

0.8

ΔSs

Figure 5.32 The dependence of the structure fractal dimension df on entropy change ΔSs for epoxy polymers EP-1 (1) and EP-2 (2) [102]

Attention is drawn to the fact that the order parameter critical indices β obtained in [102, 110], though they are close enough to classical percolation indices (β = 0.37– 0.40 [115, 122]), are different from them by an absolute value. The reason for such a discrepancy is the fact that the percolation cluster presents itself as a purely geometrical construction, the model of which is too oversimpliﬁed for real amorphous polymers, possessing thermodynamically non-equilibrium structure. Therefore there is some inﬂuence of the thermal interactions on the structure of the indicated polymers (let us be reminded that cluster structure is postulated as a thermoﬂuctuational system [123]). This means that the percolation cluster (frame) of the structure of polymers consists of elements (statistical segments) that possess molecular mobility, i.e., they perform oscillations around some quasi-equilibrium position. Such elements create a virtual volume, exceeding the cluster’s volume from immobile segments that facilitates the percolation process and allows percolation frame formation with a smaller number of elements forming it or with higher values of β. Cluster formation is studied not

249

Structure and Properties of Crosslinked Polymers on the concentration scale, as for percolation clusters [115, 122], but on the relative temperatures scale. Hence, the amorphous state structure of polymers is simulated more correctly by the thermal cluster, i.e., by a cluster whose equilibrium conﬁguration is deﬁned by both geometrical and thermodynamic interactions [124]. In this case the structure order parameter ϕcl of polymers is described by the relationship [124]:

(5.49)

where the thermal cluster order index βT is not necessarily equal to the corresponding critical index β in purely geometrical percolation models. The authors of paper [125] studied the variation of absolute values of βT, elucidated this index connection with the structural characteristics of epoxy polymers and determined the factors inﬂuencing value of βT. In paper [126] it was shown that universality of the critical indices of the percolation system was connected directly to its fractal dimension. The self-similarity of the percolation system supposes the availability of the number of subsets having order n (n = 1, 2, 4, …), which in the case of the structure of amorphous polymers are identiﬁed as follows [125]. The ﬁrst subset (n = 1) is a percolation cluster frame or, as was shown above, a polymer cluster network. The cluster network is immersed into the second loosely packed matrix. The third (n = 4) topological structure is deﬁned for crosslinked polymers as a chemical bonds network. In such a treatment the critical indices β, ν and t are given as follows (in three-dimensional Euclidean space) [126]:

(5.50)

(5.51)

(5.52) where df is the fractal dimension of the structure of the polymers determined according to Equation 5.4.

250

Structure of Epoxy Polymers Thus, the values of β, ν and t are boundary magnitudes for βT, indicating which structural component deﬁnes its behaviour. At βT = β the clusters are such a component or, more precisely, a percolation cluster frame, identiﬁed with the cluster network. At β < βT <ν polymer behaviour is deﬁned by the combined inﬂuence of clusters and the loosely packed matrix. At βT = ν the loosely packed matrix will deﬁne the structural component, at βT = t it is deﬁned by the chemical crosslinking network and at ν < βT < t the combined inﬂuence of the two components indicated above is observed. The estimation of the value of βT for the three series of epoxy polymers considered in paper [125] showed that they could be approximated by the following magnitudes: βT ≈ 0.54 for EP-1, ~ 0.55 for EP-2 and ~ 0.38 for EP-3. In Figure 5.33 the comparison of experimental [93] values of ϕcl and values calculated according to Equation 5.49 is shown as a function of Kst with the magnitudes of βT indicated above. As follows from the data of Figure 5.33 sufﬁcient correspondence between experiment and calculation within the frameworks of the thermal cluster model is obtained. The average value of df for the three mentioned series of epoxy polymers is equal to 2.613. This means according to Equations 5.50 and 5.51 that β = 0.38 and ν = 0.77. From the comparison of these estimations and the values of βT received according to Equation 5.49, one can see that in the case of EP-3 the cluster network is the main structural factor and in EP-1 and EP-2 behaviour is controlled by the combined inﬂuence of the cluster network and the loosely packed matrix. Therefore, only for ‘aged’ EP-3 epoxy polymers do the order parameter indices for thermal and geometrical percolation clusters coincide. This means that in the indicated case thermal interactions are unimportant for the structure of epoxy polymers and its equilibrium conﬁguration is deﬁned only by geometrical interactions. As was shown in paper [127], in the heat ageing process the structure of epoxy polymer EP-3 actually reached its quasi-equilibrium state, characterised by the balance between the tendency of the local order level to increase, of which ϕcl serves as the indicator, and entropic tautness of chains. In this case chain parts between clusters are stretched and their fractal dimension Dch is close to 1.0. This means [93, 127] that the molecular mobility of these chain parts is completely suppressed. For EP-1 and EP-2 the value of Dch > 1.0 [93] and, as follows from the data of Figure 5.33, βT > β. Therefore it can be supposed that the condition βT > β is due to non-aero molecular mobility, i.e., 1 < Dch ≤ 2 [93]. For conﬁrmation of this supposition it is necessary to calculate the value of Dch. This can be carried out by different methods (for example, according to Equations 2.17 or 4.16), but in paper [125] the following simple formula was used [128]:

(5.53)

251

Structure and Properties of Crosslinked Polymers In Figure 5.33 the dependence βT(Dch) for the three epoxy polymers series mentioned above is adduced. As it was supposed, increasing Dch results in βT growth. At Dch = 1.0 the value βT ≈ 0.38, i.e., the condition βT = β is fulﬁlled. The condition βT = ν is reached at Dch = 1.42 and extrapolation of the dependence βT(Dch) to Dch = 2.0 adduced in Figure 5.34 gives βT ≈ 1.35. As it is known [93], the value Dch = 2.0 indicates a polymer rubber-like state (df ≈ 3), which is characterised by complete decay of the cluster structure [54]. According to Equation 5.52 at df = 3 t = 1.33, i.e., for the rubber-like state of epoxy polymers the condition βT = t is carried out and, hence, its behaviour is deﬁned completely by a chemical crosslinking network that is a well-known fact [129]. Accounting for the fact that for glassy epoxy polymers β < βT < t, it is necessary to suppose that in the general case their behaviour is controlled by three structural components in total: a cluster network (local order domains), a loosely packed matrix and a partly (at high Dch) chemical crosslinking network (structure topological level).

ϕcl 0.6 0.5

3

0.4

2

0.3

-4 -5 -6

0.2 0.1 0.50

0.75

1.00

1

1.25

1.50 Kst

Figure 5.33 The dependences of the relative fraction ϕcl, of clusters received experimentally (1–3) and calculated according to Equation 5.49 (4–6) on Kst for epoxy polymers EP-1 (1, 4), EP-2 (2, 5) and EP-3 (3, 6) [125]

Hence, the results stated above have shown that the structure of crosslinked polymers can be simulated as a thermal cluster. In this case correspondence of thermal and percolation clusters, expressed by the equality of the critical indices order parameter βT = β, is reached only in the case of the quasi-equilibrium state of the structure

252

Structure of Epoxy Polymers of epoxy polymers. The value βT is a function of the degree of molecular mobility, characterised by the fractal dimension Dch of the chain part between clusters [125].

βT 0.8

5

0.7

0.5

-1 -2 -3

0.4

4

0.6

0.3 1.0

1.1

1.2

1.3

1.4 Dch

Figure 5.34 The dependence of the index βT on the fractal dimension Dch of a chain part between clusters for EP-1 (1), EP-2 (2) and EP-3 (3). Horizontal dashed lines indicate the values of β (4) and ν (5) for the percolation cluster [125]

5.4 Synergetics of the Formation of Dissipative Structures in Epoxy Polymers At present analysis of relations between molecular characteristics, supramolecular (suprasegmental) structure parameters and properties of crosslinked polymers is carried out, as a rule, on the qualitative level [27]. It is connected with the complexity of the structure of spatial networks and the quantitative structural model for absence of these polymers [93, 130]. Therefore receiving quantitative relations between the mentioned parameters is an important goal of polymer physics, which is necessary for prediction of the properties of crosslinked polymers. The authors [130] solved this problem by the application of a number of physical concepts: synergetics of deformable bodies [47], fractal analysis [92, 93] and the cluster model of the amorphous state structure of polymers [5, 6].

253

Structure and Properties of Crosslinked Polymers As has been shown above, a polymeric chain can be simulated with the aid of Equation 2.10. This equation allows both crosslinked polymer topology (the values Lc and Rc are functions of the chemical crosslinking density νc) and its molecular characteristics (the value lst is deﬁned by skeletal bond length l0 and characteristic ratio C∞, which serves as an indicator of the chain statistical ﬂexibility and Lc also depends on l0) to be accounted for [131]. As it is known [47], even in elastoisotropic solids there are no fewer than three independent length scales that deﬁne the complex dynamics of the self-organisation of dissipative structures. For polymers on such scales as lp, lε and lI it is natural to accept the following parameters: l0, lst and Rc. Then it can be written [93]:

(5.54)

where Λi is the automodelity coefﬁcient, which, according to Equation 5.54, is equal to the ratio Rc/lst. The value of Λi is connected with Poisson’s ratio ν by the relationship [47]:

(5.55)

and the structure fractal dimension df can be determined according to Equation 5.54. The combination of Equations 2.10, 5.4, 5.54 and 5.55 allows to receive the correlation of dimensions Dch and df, characterising molecular and suprasegmental levels of the structure of epoxy polymers, expressed by Equation 4.15. The comparison of the values of Dch, calculated according to Equations 2.10 and 4.15, which is adduced in Figure 5.35, shows good correspondence between them. Let us show how the value of Dch (i.e., at the molecular and topological levels) inﬂuences the suprasegmental level of the structure. Within the frameworks of the cluster model [5, 6] the number of densely packed segments in clusters per volume unit is approximately equal to the cluster network density νcl and is connected with ν by Equation 2.7, in which the numerical constant accounts for the necessary molecular constants of polymers.

254

Structure of Epoxy Polymers

Dch2

2.0

1.5 -1 -2 -3 -4

1.0

1.5

2.0 D ch1

Figure 5.35 The relation between the fractal dimensions of the chain part between chemical crosslinking nodes, determined according to Equations 2.10 and 4.15 for epoxy polymers EP-1 (1), EP-1-200 (2), EP-2 (3) and EP-2-200 (4) [130]

Having estimated ν according to Equations 5.4 and 4.15, the suprasegmental structure characteristic (degree of local order) νcl can be calculated as a function of the topological and molecular parameters Lc, lst and Dch (or, more widely, Mc, l0, m, C∞ and Dch). In Figure 5.36 the comparison of experimental values of νcl and those calculated by the mentioned mode for EP-2 is adduced. The good correspondence of theory and experiment shows that the molecular and topological characteristics enumerated above control the suprasegmental structure formation of crosslinked polymers and vice versa (for example, in physical ageing process) [132]. Hence, the analytical relationship, unequivocally connecting the molecular structure of crosslinked polymers at the topological and suprasegmental levels, was obtained. Let us note the close interconnection between the molecular and topological levels, for example, variation in C∞ with a change in νc (see in more detail in Chapter 2). In turn, the suprasegmental (cluster) structure formed deﬁnes the ﬁnal properties of crosslinked polymers. Let us indicate that it is impossible to avoid the introduction of the intermediate suprasegmental level of the structure, since it just responds to the physical inﬂuence on the polymer (for example, physical ageing) [130].

255

Structure and Properties of Crosslinked Polymers

νcl×10-27, m-3 3

2 -1 2

1

0

0.5

1.0

1.5 K st

Figure 5.36 The comparison of experimental values of νcl (1) and those calculated according to Equation 2.7 (2) of the cluster network density on magnitude Kst for epoxy polymer EP-2 [130]

The availability of morphological formations with large sizes in the structure of crosslinked polymers at present is generally acknowledged [68, 133]. As integrated studies of the morphology of epoxy polymers [134] have shown, such formations, named ﬂoccules, have sizes of the order of several tens or hundreds of microns and their density is approximately 15% higher than the polymer density in interﬂoccular space. The inﬂuence of the size of the ﬂoccules on the mechanical properties of crosslinked polymers was found [135, 136]. Formation of ﬂoccules is interpreted theoretically within the frameworks of the ‘gelation period’ concept [137] (the indicated concept is stated in more detail in Chapter 3). The authors of paper [138] explained the formation of ﬂoccules within the frameworks of synergetics and fractal analysis in the example of sulfur-containing epoxy polymers (SCE) cured by aromatic diamines (4,4′-diaminodiphenylsulfone (SCE-DADPS) and 4,4′-diaminodiphenylmethane (SCE-DADPM)) and isomethyltetrahydrophthalic anhydride (SCE-IMTHPhA) [139]. At the formation of the structure of crosslinked polymers one can observe the formation of dissipative structures (DS) of two levels – micro- and macro-DS. MicroDS are local order domains (clusters) and their formation is due to the high viscosity of the reactive medium in the gelation period. As it is known [47], this results in turbulence of viscous media and subsequent formation of ordered regions.

256

Structure of Epoxy Polymers One further fundamental property of turbulent streams is the fractality of structures formed by them [47]. It is accepted that energy dissipation in three-dimensional turbulent streams is concentrated on the set with non-integral (fractal) dimension. However, the experimental data on rates of ﬂuctuation moments testify that smallscaled properties of a turbulent stream cannot be described with the aid of a self-similar fractal. Therefore for description of turbulent dissipative structures ‘heterogeneous fractals’ are used, the formation rules of which on each step by scales hierarchy are chosen at random in correspondence with some probability distribution. In this case energy transfer is described with the aid of random fragmentation by supposing that the correlation is absent among different stages of process. In this case the fractal volume does not possess the property of global invariance in respect to similarity transformation. Nevertheless, the fractal dimension df can be calculated according to the relationship [47]:

(5.56)

where 〈Nn〉 is an active number on n-th step of fragmentation, βn is the volume fraction, occupied by whirls of scale Ln and in this case:

(5.57)

The sign {…} means averaging by distribution P(βn). One should expect that by virtue of the connection of turbulence and order indicated above a certain fraction of active whirls in the long run is transformed in clusters and then the relative fraction ϕcl of clusters, determined according to Equation 4.8, should be proportional to βn. In Figure 5.37 the relationship ϕcl(βn) is adduced, which conﬁrms the inﬂuence of turbulence on formation of local order domains in epoxy polymers. Let us note that the dependence ϕcl(βn) adduced in Figure 5.37 does not pass through the coordinates origin. This means that not all active whirls are transformed in clusters, but only certain parts. The fraction of transformed whirls depends on βn – the higher βn the larger the fraction of such whirls. Let us indicate that the inverse interconnection also exists: local order availability in polymer structure results in turbulence of deformation on the cold ﬂow plateau [91, 140].

257

Structure and Properties of Crosslinked Polymers Formation of microgels results in the formation of macro-DS [141]. The fractal dimension D of such microgels can be determined according to Equation 3.44. From this equation it follows that in the swollen state D = 5/3 for a linear chain and D = 2 for a microgel. The gelation or transition into the condensed state is characterised by a change in the microgel environment and now instead of molecules of low-molecular solvent it is in similar microgels and non-cured fragments of the environment of epoxy chains (Figure 5.38). This results in a change in the microgel fractal dimension and for the condensed state its value df is determined according to Equation 3.53. The combination of Equations 3.44 and 3.53 at the condition ds = 1.33 and d = 3 gives Formula 3.54 for crosslinked epoxy polymers. According to this formula at D = 2 let us obtain df = 3.33, i.e., df > d. Growth of such microgels is restricted since they are ‘not placed’ in three-dimensional space and their density increases with radius rmg as , α = d – df – < 0 (the effect is similar to blood coagulation) [144]. Let us note that similar analysis for linear polymers gave such a result [142]:

(5.58) where at D ≈ 5/3 one can obtain df = 2.5 that corresponds to the condition df < d and therefore the formation of ﬂoccules in linear polymers is not observed.

ϕcl 0.8

0.4 -1 -2 -3 0 0.4

0.6

0.8

βn

Figure 5.37 The dependence of the relative fraction ϕcl of clusters on the volume fraction βn, occupied by active whirls, for epoxy polymers SCE-IMTHPhA (1), SCE-DADPM (2) and SCE-DADPS (3) [138]

258

Structure of Epoxy Polymers The identiﬁcation of clusters as micro-DS is conﬁrmed in paper [143]. If ﬂoccules are also DS (macro-DS), as postulated above, then the scaling relationship should be carried out [141]:

(5.59)

where dﬂ is the ﬂoccule diameter, t is its formation duration and the scaling exponent γ is determined as follows [141]:

(5.60)

Figure 5.38 The optical microphotograph of ﬂoccules on the fracture surface of epoxy polymer SCE-DADPS. Enlargement is 125× [138]

Using the curing cycle duration as t did not conﬁrm the accuracy of Equation 5.59. However, we must not overlook the fact that formation of microgels occurs long before completion of the curing process [133, 137] (see also Chapter 3) and therefore in Relationship 5.59 the duration τ of the formation of microgels must be used. If we suppose, following paper [141], that formation of ﬂoccules is realised by anomalous

259

Structure and Properties of Crosslinked Polymers mutual diffusion, controlled by dissipation quantum channels, with diffusivity Ddif ≈ 10–6 m/s, then we obtain [138]:

(5.61)

The dependence of dﬂ on τγ is adduced in Figure 5.39, from the plot of which it follows that it corresponds to Relationship 5.59. This conﬁrms the supposition about identiﬁcation of ﬂoccules as macro-DS made above.

dfl, mm 0.3

0.2

-1 -2 -3

0.1

0

0.5

1.0

1.5

τγ

Figure 5.39 The dependence of the ﬂoccules diameter dﬂ on the scaling parameter τγ for epoxy polymers SCE-IMTHPhA (1), SCE-DADPM (2) and SCE-DADPS (3) [138]

Hence, the results stated in the present section demonstrated the applicability of synergetics concepts for description of the structure of crosslinked polymers . The formation of ﬂoccules in crosslinked polymers can be considered as the formation of microgels, which at transition to the condensed state are ‘coagulated’, increasing their density by virtue of the condition df > d, typical only for crosslinked polymers. For linear polymers a swollen macromolecular coil corresponds to the condition

260

Structure of Epoxy Polymers df < d and therefore for these polymers formation of ﬂoccules is not observed. The formation kinetics of ﬂoccules, identiﬁed as macro-DS or ‘disproportionated’ DS [141], obeys the scaling laws of formation of dissipative structures.

5.5 The Structural Analysis of Fluctuation Free Volume of Crosslinked Polymers The model of free volume going back to the classical papers of Frenkel and Eiring [48, 80, 144-147] has been widespread in the physics of liquid and solid states of matter. Some concepts allowing improvement in the nature of ﬂuctuation free volume have been offered in the last 15 years [148–150]. Nevertheless, there is one more aspect of the problem, which has not been mentioned earlier. As a rule, the application of free volume theory for the description of the properties of amorphous bodies is based on a notion that the free volume characterises the structure of the indicated bodies. This postulate is due to a considerable extent to the absence of a quantitative model of the structure of the amorphous condensed state, including the structure of amorphous state polymers. Strictly speaking, one should understand that by structure we mean distribution of body elements in space [151]. It is evident that free volume microvoids cannot be structural elements and at best only mirror the structural state of the studied object. Taking the introduction of some structural elements (relaxators, see for example, [148]) into consideration has practically no inﬂuence on the structural representation of free volume. The appearance of the cluster model of amorphous state structure of polymers [5, 6] allows the free volume to be considered just as a mirror of the indicated structure. Such an approach supposes that the free volume is not simply a set of microvoids in a condensed state, but presents itself as some structural construction, reﬂecting an amorphous state structure and connected with it [152–158]. This will be the main problem in the present section. There are some problems of interest in the description of the free volume of polymers. As has already been mentioned, both ﬂuctuation free volume and clusters have a thermoﬂuctuational nature. It is obvious that the energy of thermal ﬂuctuations can be expressed as kT (where k is Boltzmann’s constant, T is temperature), while microvoid formation energy εh is equal to [80]:

(5.62) where Tg is the glass transition temperature.

261

Structure and Properties of Crosslinked Polymers Since the value of fg ≤ 0.159 [120] and T < Tg, then it is apparent that in the case of glassy polymers energy of the thermal oscillations of order ~ kT is not sufﬁcient for microvoid formation (kT < εh). The second problem requiring explanation and repeatedly discussed [48, 80, 147, 148] is the absolute value of fg, which within the frameworks of the kinetic theory is estimated according to Equation 1.33. The values fg ≈ 0.050–0.100 were obtained for different polymers [157], which is much more than the generally accepted value of fg = 0.025 ± 0.003 for most polymers within the frameworks of the Williams, Landel and Ferry concept (WLF) [8, 145, 146]. At present it is known [92, 93, 159] that the polymer structure is fractal (generally multifractal) with fractal dimension df (2 ≤ df < 3). Therefore it is possible to expect, since the ﬂuctuation free volume is its mirror, that it also possesses fractal properties. If this assumption is correct, the general relation should be fulﬁlled [18]:

(5.63)

where Nh is the number of microvoids, rh is their characteristic size, Df is the fractal dimension of ﬂuctuation free volume. 1/ 3

The value of rh can be estimated as Vh , where Vh is the microvoid volume, determined according to Equation 5.11, and Nh is determined as follows [48]:

(5.64)

In Figure 5.40 the dependences Nh(rh) in double logarithmic coordinates corresponding to Relationship 5.63 are adduced. As one can see, they are linear, which is a characteristic sign of fractal behaviour [18] and one may determine the value of Df from their slopes. For series EP-1 and EP-2 Df is equal to ~ 3.75, and for EP-3 Df is equal to ~ 2.70 [155]. Let us consider the physical signiﬁcance of the model and the determined dimension Df. As is well known, both the length of a macromolecule part between the crosslinking nodes, having molecular mass Mc [160], and the sizes of the free volume microvoids [84] for crosslinked polymers possess a certain special distribution. For the studied

262

Structure of Epoxy Polymers series of epoxy polymers the crosslinking density νc variation makes up for EP-1 and EP-3 ~ 15 × 1026 m–3 and for EP-2 ~ 7 × 1026 m–3 [4].

ln Nh -1 -2 -3

2

1

0

-1

0.7

1.1

1.5

ln rh

Figure 5.40 The dependences of microvoids of ﬂuctuation free volume number Nh on their characteristic size rh in double logarithmic coordinates for epoxy polymers EP-1 (1), EP-2 (2) and EP-3 (3) [155]

These data correspond well to the distribution width Mc for one epoxy polymer [160]. Thus, in essence the treatment offered simulates one epoxy polymer with allowance for constancy of its chemical constitution (DGEBA resin), having some distribution Mc and Vh, by epoxy polymers series with varied νc and, hence, Mc and Vh. Let us note that the free volume microvoids do not make the structure for which the condition df < 3 is obligatory in three-dimensional Euclidean space by virtue of the given space ﬁlling condition, although such a condition is obligatory for polymer structure elements, for example, for statistical segments (see previous section) [92]. Once again it underlines the fact that fg is not a direct structural characteristic [93]. Apparently, the values of Df calculated in a similar way are the average characteristics but, nevertheless, they allow an important conclusion to be made: microvoids of ﬂuctuation free volume form Df-dimensional fractal cluster or such microvoids in totality has a certain (fractal) structure in general. The distribution of the sizes of the power particles is widespread in nature. Whirl cascades in developed turbulence [161], caverns on fracture surfaces of ductile materials [162], aggregates of ﬁller

263

Structure and Properties of Crosslinked Polymers particles in polymer composites [163] etc. obey this distribution. However, the exact fractality deﬁnition of an object (or its totality) alongside carrying out the general Relationship 5.63 needs the proof of its self-similarity [164, 165]. This problem will be discussed in detail below. The absolute values of Df themselves are also not random, and have deep physical ' signiﬁcance. They are close to the corresponding values of the dimension D f of extra energy localisation regions, determined as follows [47]:

(5.65)

'

The value of D f is equal to the structure automodelity coefﬁcient Λi for a polymer [93, 166]:

(5.66)

where

and

are spatial scales of dissipative structures (DS) of adjacent levels.

For amorphous state polymers the values of and are identiﬁed as follows. DS, forming in the mentioned state, are clusters [143]. The cluster model [5, 6] postulates that the length of segments, included in a cluster, is equal to the statistical segment length lst and therefore =lst [93] (see also Equation 5.54). One should accept the distance between clusters Rcl as the next structural scale and therefore = Rcl [93]. In such a treatment the fractal dimension Df is closely connected with the polymer structure characteristics (compare Equations 5.55 and 5.65). Lastly, earlier the relationship was obtained [93]:

(5.67)

where C∞ is a characteristic ratio, which is a measure of a polymer chain statistical ﬂexibility [25].

264

Structure of Epoxy Polymers Thus, Relationship 5.67 assumes correlation of the free volume characteristics and molecular parameters of polymers. A similar correlation was obtained in paper [30]. ' The interconnection between Df and D f will be described below in more detail. Now we shall consider the energetic criterion for formation of the microvoid with volume Vh. For this purpose the concept of solids free fracture was used [167], which postulates that it is possible to accumulate a restricted amount of potential energy with density A* in a restricted volume V, and it is necessary to carry out work equal to the product of surface area S on a speciﬁc surface energy σs to form a new surface. The limiting case of free fracture takes place at the condition [167]:

(5.68)

With reference to the creation of a microvoid of size V = Vh the parameters included in Relationship 5.68 can be presented as follows, rewriting this relationship in the form [47]:

(5.69)

The following relationship is correct in the case of fluctuation free volume representation by a Df-dimensional sphere [47]:

(5.70)

In turn, the value of σs can be expressed by the parameters of the ‘holes’ model [80]:

(5.71)

265

Structure and Properties of Crosslinked Polymers And lastly, deﬁning A* = kT/Vh from combination of Equations 5.62 and 5.69–5.71 we receive the ﬁnal relationship [158]:

(5.72)

It follows from the data of Figure 5.40 that the ageing of epoxy polymers EP-1 results in a decrease in Df and the value of Df, close to 3 is obtained for the aged EP-3 systems [155]. Since the physical ageing process of polymers itself can be considered as a tendency to a thermodynamic equilibrium [12], it is necessary to assume that the condition Df = 3 responds to some quasi-equilibrium state of polymer structure [127] and to microvoid simulation as a three-dimensional sphere (Df = d = 3). Such a treatment allows several conclusions to be made. Firstly, increasing T with other conditions remaining the same means increasing Df and, hence, raising of the degree of thermodynamic nonequilibrium of the polymer structure. It follows according to Equation 5.65 that the growth in Df means increasing ν, which explains a well-known increase in Poisson’s ratio for polymers due to temperature raising [50]. Secondly, increasing ν means a reduction of the local ordering degree of the structure of amorphous polymers [50] that is expressed in the reduction of the relative fraction of clusters. In other words, as has been expected [103, 127], the increase in ϕcl (order parameter of the structure of amorphous polymers [110]) reﬂects the tendency of the polymer to reach thermodynamic equilibrium. Thirdly, from Equation 5.72 at the condition of Df = 3 it is possible to calculate the quasi-equilibrium value of fg ( ), which turns out to be equal to ≈ 0.028–0.42 for the studied epoxy polymers, which is sufﬁciently close to the free volume value in the WLF theory mentioned above. Therefore, the estimation of fg according to Equation 1.33 is correct, and large absolute values of fg for the studied epoxy polymers according to it reﬂect a rather high degree of thermodynamic non-equilibrium. Although the parameters fg and Df are not the direct characteristics of structure, there is a relationship between dimensions df and Df [47]:

(5.73)

266

Structure of Epoxy Polymers

'

It follows from Equation 5.65 that the theoretical value of D f can be inﬁnitely large (at ν = 0.5). However, as has been noted above, Poisson’s ratio value for real solids is terminated from above: ν ≤ 0.475 [47]. Therefore, we receive from Equation 5.65 ' that for real solids, including polymers, the limiting value of D f is equal to 21. Thus, ' the dimension D f characterises not the structure, but its energetic excitation degree, i.e., the degree of divergence from thermodynamic equilibrium, and is connected with the structural characteristic df. Using Equations 5.65 and 5.72, it is possible to demonstrate the closeness of ' absolute values of D f and Df postulated above (Figure 5.41). As one can see, close correspondence between them is obtained, and some discrepancy of the absolute values ' of D f and Df is due to the approximation mentioned above. This correspondence shows that fg can be a mirror of polymer structure.

Df’ 5 -1 -2 -3

4

3

2

3

4

5

Df

Figure 5.41 The relation between the dimensions of ﬂuctuation free volume Df ' (Equation 5.72) and extra energy localisation regions D f (Equation 5.65) for epoxy polymers EP-1 (1), EP-2 (2) and EP-3 (3) [155]

In summary we shall consider the problem of self-similarity for a cluster of free volume microvoids. As it is known [164], linearity of the dependences, corresponding to Relationship 5.63, in double logarithmic coordinates and non-integral value of

267

Structure and Properties of Crosslinked Polymers dimension Df, obtained from the slope of these dependences, is not the proof of the indicated cluster fractality. For the strict proof of this property it is necessary to conﬁrm the self-similarity of the object and to indicate the scales range of this selfsimilarity [164, 165]. As was shown in paper [164], for self-similar fractal objects while using methods similar to application of Relationship 5.63 the condition (at rh i / rh i +1 = const.) should be satisﬁed:

(5.74)

In Figure 5.42 the dependences corresponding to Relationship 5.74 for three series of the studied epoxy polymer were adduced. As one can see, the indicated dependences are linear, pass through the coordinates origin and, according to Relationship 5.74, self-similarity of a ﬂuctuation free volume microvoids cluster is conﬁrmed. As for the scales range of self-similarity, then allowing for Equation 5.74, giving correlation Df and df, it is necessary to assume that this range coincides with a similar range for polymer structure. As was shown above, for the structure of amorphous glassy polymers the indicated range is spread from several angstroms up to several tens of angstroms according to the experimental data [44]. According to the treatment of the connection Df and Λi mentioned above (Equation 5.66), this range is spread from lst up to Rcl [93], which quantitatively corresponds to experimental limits of the indicated range [168]. In papers [169, 170] it was shown that the minimum self-similarity range of values of rh i should contain at least one self-similarity interaction. In this case the condition should be carried out [170]:

(5.75) 1/ D

For epoxy polymers EP-1 and EP-2 the value of 2 f ≈ 1.21, for EP-3 this value is equal to ~ 1.29. At the same time the ratio rh max / rh min is equal to ~ 1.42 for EP-2, ~ 1.93 for EP-2 and ~ 1.46 for EP-3. Thus, from a comparison of the estimations mentioned, Criterion 5.75 is fulﬁlled.

268

Structure of Epoxy Polymers

2

4

1 2

3

0

0.02

0.04

−D f

rh

−D

Figure 5.42 The dependences of ( N h i +1 – N h i ) on the value of rh f , corresponding to Relationship 5.74, for epoxy polymers EP-1 (1), EP-2 (2) and EP-3 (3) [155] Alternatively the number of self-similarity iterations n can be determined according to the equation [170]:

⎛ rh max ⎞ ⎜ ⎟ ⎜r ⎟ ⎝ h min ⎠

Df

= 2n (5.76)

The values of n for the studied epoxy polymers are equal to: n ≈ 1.90 for EP-1, n ≈ 3.57 for EP-2 and n ≈ 1.49 for EP-3, i.e., the number of self-similarity iterations is more than unity at experimental [155] conditions, which conﬁrms again the accuracy of the value of the estimation of Df [169–171]. Hence, the results stated above have shown that ﬂuctuation free volume in epoxy polymers possesses fractal structure. Therefor a microvoid forming it should be simulated by Df-dimensional sphere. The size of the microvoids is controlled by the volume which is necessary for accumulation of the thermal ﬂuctuations energy required for their formation. The absolute values of fg can serve as characteristic of polymer structure thermodynamic non-equilibrium and for quasi-equilibrium structures the value of fg coincides with the data obtained according to the William–Landel–Ferry equation. Microvoids of ﬂuctuation free volume form fractal structure, which is a mirror of polymer structure [152–158].

269

Structure and Properties of Crosslinked Polymers To complete this section we will consider one more aspect of free volume [172, 173] speciﬁc for epoxy polymers. The model of ﬂuctuation free volume allows to carry out the theoretical calculation of free volume compressibility χf according to the following approximate formula [80]:

(5.77)

In Figure 5.43 the comparison of values of χf calculated according to Equation 5.77 and experimental values for an epoxy polymer [172] are adduced. Curve 1 was calculated according to the values of fg and Vh received experimentally (174). As one can see, the shapes of the experimental and theoretical dependences χf(P) are in agreement, but the absolute values of χf received experimentally systematically exceed the calculated ones (by approximately 3 times). This discrepancy can be removed as follows. It was shown earlier [72, 127] that free volume in crosslinked epoxy polymers consisted of two components –ﬂuctuation free volume connected with 0 clusters decay (formation), and the constant component f g (≈ 0.024) connected with chemical crosslinking nodes (see Figure 5.17).

χf, Pa-10 80

40 2 -3 1 0

5

10

15

P

Figure 5.43 The comparison of the theoretical (1, 2) and experimental (3) dependences of free volume compressibility χf on pressure P for epoxy polymers [172]

270

Structure of Epoxy Polymers

Since in regions of elastic strains the component deﬁnes the behaviour of an epoxy polymer, then it can be assumed that it is determined according to the testing methods used in paper [174]. If in Equation 5.77 the value of the entire free volume 0 of epoxy polymers (i.e., + 0.024) is used, then one can obtain much + fg ≈ better quantitative correspondence of experimental and theoretical results [172, 173].

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Structure and Properties of Crosslinked Polymers 113. L.M. Brady and R.C. Ball, Nature, 1984, 309, 5965, 225. 114. T.A. Witten and P. Meakin, Physical Review, Part B, 1983, 28, 10, 5632. 115. I.M. Sokolov, Uspekhi Fizicheskikh Nauk, 1986, 150, 2, 221. 116. L.I. Manevich, V.G. Oshmyan, M.I. Gai, E.L. Akopyan and N.S. Enikolopyan, Doklady Akademii Nauk SSSR, 1986, 289, 1, 128. 117. F. Margolina and S. Wu, Polymer, 1988, 29, 12, 2170. 118. G.V. Kozlov and V.Z. Aloev in Percolation Theory in Physics-Chemistry of Polymers, Polygraphservice and T, Nal’chik, Russia, 2005, p.148. 119. J.L. McCauley, International Journal of Modern Physics, 1989, 3B, 6, 821. 120. I.C. Sanchez, Journal of Applied Physics, 1974, 45, 10, 4204. 121. M. Zhenyi, S.C. Langford, J.T. Dickinson, M.H. Engelhard and D.R. Baer, Journal of Materials Research, 1991, 6, 1, 183. 122. B.I. Shklowskii and A.L. Efros, Uspekhi Fizicheskikh Nauk, 1975, 117, 3, 401. 123. V.N. Belousov, G.V. Kozlov, A.K. Mikitaev and Y.S. Lipatov, Doklady Akademii Nauk SSSR, 1990, 313, 3, 630. 124. F. Family, Journal of Statistical Physics, 1984, 36, 5/6, 881. 125. G.V. Kozlov and I.V. Dolbin in the Proceedings of International Conference ‘Physics of Electronic Materials’, Kaluga, Russia, 2002, p.134. 126. A.N. Bobryshev, V.N. Kozomazov, L.O. Babin and V.I. Solomatov in Synergetics of Composite Materials, NPO Orius, Lipetsk, Russia, 1994, p.153. 127. G.V. Kozlov, V.A. Beloshenko and Y.S. Lipatov, Ukrainskii Khimicheskii Zhurnal, 1998, 64, 3, 56. 128. G.V. Kozlov, K.B. Temiraev, R.A. Shetov and A.K. Mikitaev, Materialovedenie, 1999, 2, 34. 129. G.M. Bartenev and S.Ya. Frenkel in Physics of Polymers, Khimiya, Leningrad, Russia, 1990, p.432.

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Structure of Epoxy Polymers 130. S.M. Balaeva, V.Z. Aloev, Y.A. Malkanduev and G.V. Kozlov in the Proceedings of the 7th Region: Science and Practice’, Polygraphservice and T, Nal’chik, Russia, 2005, p.35. 131. Y.-H. Lin, Macromolecules, 1987, 20, 12, 3080. 132. G.V. Kozlov, I.V. Dolbin and G.E. Zaikov, Zhurnal Prikladnoi Khimii, 2004, 77, 2, 271. 133. M.K. Pakter, Y.M. Paramonov and E.S. Belaya in Structure of Epoxy Polymers, Niitekhim, Moscow, Russia, 1984, p.47. 134. L.D. Mil’man, R.A. Shetov, G.V. Kozlov, A.M. Balkarov, A.A. Beev and A.K. Mikitaev in the Proceedings of All-Union Conference ‘Ensuring of Resistors Developments and Processing by New Materials’, PSU, Penza, Russia, 1985, p.88. 135. J. Mijovic and J.A. Koutscky, Polymer, 1979, 20, 9, 1095. 136. L.D. Mil’man, A.M. Balkarov, G.V. Kozlov and A.K. Mikitaev in Polycondensation Processes and Polymers, Ed., V.V. Korshak, KBSU, Nal’chik, Russia, 1985, p.160. 137. T.E. Lipatova in Oligomers Catalytic Polymerization and Polymer Networks Formation, Naukova Dumka, Kiev, Ukraine, 1974, p.212. 138. G.V. Kozlov, M.V. Burmistr, V.A. Korenyako and G.E. Zaikov, Voprosy Khimii i Khimicheskoi Tekhnologii, 2002, 6, 77. 139. S.A. Samsoniya, G.V. Kozlov, A.M. Balkarov and A.K. Mikitaev, Izvestiya Akademii Nauk Gruzinskoi SSSR, seriya Khimicheskaya, 1988, 14, 4, 259. 140. M.A. Gazaev, G.V. Kozlov, L.D. Mil’man and A.K. Mikitaev, Fizika i Tekhnika Vysokikh Davlenii, 1996, 6, 1, 76. 141. A.S. Balankin, V.S. Ivanova, A.A. Kolesnikov and E.E. Savitskaya, Pis’ma v ZhETF, 1991, 17, 14, 27. 142. G.V. Kozlov, K.B. Temiraev, G.B. Shustov and N.I. Mashukov, Journal of Applied Polymer Science, 2002, 85, 6, 1137. 143. G.V. Kozlov, G.B. Shustov and K.B. Temiraev, Vestnik KBSU, Seriya Khimicheskaya, 1997, 2, 50.

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Structure and Properties of Crosslinked Polymers 144. Y.I. Frenkel in Kinetic Theory of Liquids, Publishers of Akademii Nauk SSSR, Moscow, Leningrad, Russia, 1945, p.425. 145. J. Ferry in Viscoelastic Properties of Polymers, Publishers of Foreign Literature, Moscow, Russia, 1963, p.535. 146. Y.S. Lipatov, Uspekhi Khimii, 1978, 42, 2, 332. 147. D.S. Sanditov and G.V. Kozlov, Fizika i Khimiya Stekla, 1995, 21, 6, 547. 148. V.S. Nechitailo, Zhurnal Fizicheskoi Khimii, 1991, 65, 7, 1979. 149. D.S. Sanditov and S.S. Sangadiev, Fizika i Khimiya Stekla, 1998, 24, 4, 417. 150. D.S. Sanditov, S.S. Sangadiev and G.V. Kozlov, Fizika i Khimiya Stekla 1998, 24, 6, 758. 151. V.S. Ivanova, I.R. Kuzeev and M.M. Zakirnichnaya in Synergetics and Fractals, Universality of Materials Mechanical Behaviour, Publishers of USNTU, Ufa, Russia, 1998, p.366. 152. B.D. Sanditov, G.V. Kozlov, D.S. Sanditov and Y.S. Lipatov in the Proceedings of All-Russian Science Conference ‘Mathematical Simulation in Synergetic Systems’, TSU, Ulan-Ude-Tomsk, Russia, 1999, p.330. 153. G.V. Kozlov, S. Ozden and V.N. Shogenov, Russian Polymer News, 2002, 7, 4, 54. 154. G.V. Kozlov, B.D. Sanditov, D.S. Sanditov and A.V. Bainova in Perspectives on Chemical and Biochemical Physics, Ed., G.E. Zaikov, Nova Science Publishers, Inc., New York, NY, USA, 2002, p.175. 155. G.V. Kozlov, D.S. Sanditov and Y.S. Lipatov in Achievements in Polymers Physics-Chemistry Field, Ed., G.E. Zaikov, Khimiya, Moscow, Russia, 2004, p.412. 156. G.V. Kozlov, B.D. Sanditov, D.S. Sanditov and A.V. Bainova in Polymer Yearbook 18, Eds., R. Pethrick and G.E. Zaikov, Rapra Technology Limited, Shawbury, UK, 2003, p.387. 157. G.V. Kozlov, G.E. Zaikov and Y.S. Lipatov in Chemical and Biological Kinetics, New Horizons, Eds., E.B. Burlakova, A.E. Shilov, S.D. Varfolomeev and G.E. Zaikov, Volume 1, Chemical Kinetics, Brill Academic Publishers, Leiden-Boston, 2005, p.484.

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Structure of Epoxy Polymers 158. G.V. Kozlov, G.E. Zaikov and Y.S. Lipatov in Chemistry as Art, Eds., L.S. Liu and G.E. Zaikov, Nova Science Publishers, Inc., New York, NY, USA, 2006, p.213. 159. V.U. Novikov and G.V. Kozlov, Uspekhi Khimii, 2000, 69, 6, 572. 160. G.I. Sandakov, L.P. Smirnov, A.I. Sosikov, K.T. Summanen and N.N. Volkova, Journal of Polymer Science, Part B: Polymer Physics Edition, 1994, 32, 9, 1585. 161. B.B. Mandelbrot in The Fractal Geometry of Nature, Freeman, San Francisco, CA, USA, 1982, p.459. 162. K. Ishikawa, Materials Science Letters, 1990, 9, 4, 400. 163. G.V. Kozlov and A.K. Mikitaev, Mekhanika Kompozitsionnykh Materialov i Konstructsii, 1996, 2, 3/4, 144. 164. D. Farin, S. Peleg, D. Yavin and D. Avnir, Langmuir, 1985, 1, 4, 399. 165. E. Hornbogen, International Materials Review, 1989, 34, 6, 277. 166. A.S. Balankin, Pis’ma v ZhETF, 1990, 16, 7, 14. 167. E.I. Shemyakin, Doklady Akademii Nauk SSSR, 1988, 300, 5, 1090. 168. G.V. Kozlov, V.N. Belousov and A.K. Mikitaev, Fizika i Tekhnika Vysokikh Davlenii, 1998, 8, 1, 101. 169. P. Pfeifer, Applied Surface Science, 1984, 18, 1, 146. 170. D. Avnir, D. Farin and P. Pfeifer, Journal of Colloid and Interface Science, 1985, 103, 1, 112. 171. G.V. Kozlov, Y.G. Yanovskii and A.K. Mikitaev, Mekhanika Kompozitnykh Materialov, 1998, 34, 4, 539. 172. G.V. Kozlov, D.S. Sanditov, E.N. Ovcharenko and A.K. Mikitaev, Fizika i Khimiya Stekla, 1997, 23, 4, 369. 173. B.D. Sanditov, G.V. Kozlov, V.D. Serdyuk and D.S. Sanditov in the Proceedings of All-Russian Science Conference ‘Mathematical Simulation in Synergetic Systems’, TSU, Ulan-Ude-Tomsk, Russia, 1999, p.317.

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Structure and Properties of Crosslinked Polymers 174. Q. Deng, C.S. Sundar and Y.C. Jean, Journal of Chemical Physics, 1992, 96, 1, 492.

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6

The Properties of Crosslinked Epoxy Polymers

In the present chapter the description of the main properties of crosslinked epoxy polymers within the frameworks of structural notions, stated in Chapters 1, 2 and 5, will be given.

6.1 The Glass Transition Temperature It was stated for linear amorphous polymers [1] that the elasticity modulus E and the glass transition temperature Tg change in a similar way. This is explained by the fact that the indicated parameters are deﬁned by similarly changing characteristics – cohesion energy and chain rigidity, respectively [2, 3]. However, for crosslinked polymers, particularly epoxy polymers, different (including an antibate one) characters of changes in E and Tg can be observed. It was shown in papers [4–6] that at a change in the concentration of chemical network nodes, reached by variation in the curing agent : oligomer ratio Kst, the elasticity modulus at compression and longitudinal ultrasonic waves is changed antibately to Tg. The authors [7] studied the reasons for such behaviour within the frameworks of the cluster model [8, 9] and thermal ﬂuctuations theory [10]. According to the last concept for liquids, occurring in an equilibrium state, which corresponds to a state of polymers at temperature T above Tg, density ﬂuctuations ψ(ρ) at υ®∞ (where υ is the system volume) are deﬁned according to Equation 1.37. The estimation ψ(∞) for the number of polymers using literary data [11] shows that the value of ψ(∞) is approximately constant at Tg [7]. This fact allows a conclusion to be made that at some critical value of ψ(∞)(ψcr) freezing of the formation of local order domains (clusters), i.e., the physical macromolecular entanglements cluster network, is impossible because of the high thermal mobility of macromolecules (the indicated network density νcl at Tg is equal to zero [8]). As a result, Equation 1.37 can be rewritten as follows [7]:

(6.1)

283

Structure and Properties of Crosslinked Polymers where KT is the isothermal modulus of dilatation determined according to Equation 5.9. From Equations 5.9 and 6.1 it follows that the constant ϕcr criterion is inapplicable to the glass transition process of the epoxy polymers studied in papers [4–6] – decreasing E should result in a reduction in Tg that does not conform to the experimental data. For an explanation of this apparent contradiction the authors [7] assumed that network nodes of chemical crosslinkings restricted ﬂuctuations of segments in clusters, thus decreasing density ﬂuctuations, and maintained the model main postulate – polymer devitriﬁcation is deﬁned by the decay in freezing of the local order. Similar ideas were developed by Flory [12], who supposed that in rubbers physical entanglements, considered by many authors as domains of ‘short-lived’ local order, restricted ﬂuctuations in the chemical crosslinking nodes. The direct interconnection between chemical crosslinking network density νc and Tg scarcely occurs, since the glass transition is a critical phenomenon at which all polymer properties are essentially changed, whereas the value of νc at transition through Tg is not changed. The cluster network behaves in quite a different way. In this case the criterion νcl = 0 corresponds to polymer devitriﬁcation. Therefore, the supposition that chemical crosslinking inﬂuences formation in the structure of suprasegmental polymers (cluster) looks more real, characterised by the value of νcl and through it the value Tg [7]. To express analytically the inﬂuence of νc on ψcr is difﬁcult, however it can be supposed that the restrictions for ψcr are intensiﬁed with the growth in νc, i.e., ψcr becomes smaller. The authors [7] carried out the allowance of these restrictions empirically n (as was fulﬁlled by Flory [12]), namely by introduction of the the factor C ν c , where C and n are constants, in the right-hand part of Equation 6.1. In this case Equation 6.1 for crosslinked systems has the form [7]:

(6.2) Proceeding from the fact that approximately the same value of ψcr for all polymers was accepted in paper [7], it was calculated according to the known data for polycarbonate [11]. Since, according to the fulﬁlled estimations, n ≈ 1/2 gives the best correspondence with the experiment, this magnitude of n was used in paper [7]. The value of C can be determined, using the value Tg for one of the epoxy polymers studied in [5]. Finally Equation 6.2 is transformed into the form [7]:

(6.3) 284

The Properties of Crosslinked Epoxy Polymers Figure 6.1 shows the experimental [5] dependences and those calculated according to Equation 6.3 Tg(Kst) for the systems EP-1 and EP-2, cured at two different pressures, which showed good correspondence. The discrepancy obtained in the case of system EP-2 is probably connected with the number of approximations being used. The received result conﬁrms the accuracy of the selected approach. Hence, the availability of a chemical crosslinking network not only inﬂuences the local order level (see, for example, Figure 5.33) that was noted earlier [13], but it also restricts thermal ﬂuctuations of segments in clusters. This effect deﬁnes to a considerable extent the properties of crosslinked systems and, in particular, it allows the antibate change of elasticity modulus and glass transition temperature to be explained [7].

ɌgɄ 550

ɌgɄ

500

400

2

450

-5 -6 -7 -8

1 4

400

350 300

3

350

250 0.50

0.75 1.00

1.25 1.50 Ʉst

Figure 6.1 The dependences of the glass transition temperature Tg on the ratio curing agent : oligomer Kst, obtained experimentally (1–4) [5] and calculated according to Equation 6.3 (5–8) for systems EP-1 (1, 2, 5, 6) and EP-2 (3, 4, 7, 8), cured at pressure 0.1 (1, 3, 5, 7) and 200 MPa (2, 4, 6, 8) [7]

The authors of paper [14] used fractal analysis methods for the description of the glass transition process of crosslinked polymers. For this purpose they used the expression for the estimation of beginning time τ* of the accumulation of avalanche-type defects obtained in paper [15]:

(6.4) 285

Structure and Properties of Crosslinked Polymers where R* is the characteristic size, df is the structure fractal dimension. The transference possibility of the notions [15], which results in Relationship 6.4, on the glass transition process can be proved easily by the following reasons [14]. Each segment in the cluster was considered as a linear defect [16] and since Relationship 6.4 was proposed for the description of the accumulation kinetics of the defects then it can be used in the cluster structure case too. By virtue of the temperature, the temporal superposition τ* can be replaced by the value equal to the reciprocal of the glass transition temperature Tg. In real conditions the glass transition process is a complex multiple-stage hierarchical process [17]. For amorphous systems a number of bifurcations is observed, which are objects of qualitative reformations at controlling their parameters change [18] at critical magnitudes of non-equilibrium factor. So, the ﬁrst stage (or bifurcation) is connected with the formation of nucleating microcrystals, which are incapable of growth, and become the core for the formation of amorphous clusters [17]. Within the frameworks of the cluster model this is treated as a dynamic local order formation on reaching the transition ‘liquid 1–liquid 2’ [19]. The second stage is characterised by self-organisation of clusters [18], which is treated here as a local order avalanchetype formation at the glass transition temperature [20]. Accepting lst as the cluster characteristic size, Equation 6.4 can be written in the form [14]:

(6.5)

In Figure 6.2 the dependences Tg[(df – 1)–1], which correspond to Relationship 6.5, for the systems EP-1 and EP-2 are adduced. They present themselves as straight lines, not coinciding with a similar dependence for linear amorphous polymers, plotted according to the literary data [9] and shown in this ﬁgure by a dashed line. At the same values of df the magnitudes of Tg for crosslinked polymers are higher than for linear ones. This can be connected with the availability in the ﬁrst of the chemical crosslinking nodes, which withstand thermoﬂuctuational decay in clusters on reaching Tg [7]. It was shown above that the degree of chemical crosslinking stabilising 1/ 2 action was proportional to ν ñ . Accounting for this fact the authors [14] plotted 1/ 2 the correlation Tg[ ν ñ (df – 1)–1] shown in Figure 6.3. Now the data for both epoxy systems are set in one straight line. This allows to assume the accuracy of the approach offered above [7], which shows that avalanche-type formation (or dissociation) of local order at Tg is accelerated (or decelerated) by the chemical crosslinking network stabilising action [14].

286

The Properties of Crosslinked Epoxy Polymers

Tg, K 450

2 1 3

350

250 5.6

6.0

6.4 (d -1)-1 f

Figure 6.2 The dependences of the glass transition temperature Tg on parameter (df – 1)–1 for EP-1 (1), EP-2 (2) and linear amorphous polymers (3) [14]

Tg, K 450

350

250

0

0.4

0.8

ν1ɫ/ 2 /(df-1)-1

Figure 6.3 The dependence of the glass transition temperature Tg on parameter 1/ 2 (df – 1)–1, normalised by ν ñ . Conventional signs are the same as in Figure 6.2 [14]

287

Structure and Properties of Crosslinked Polymers Therefore, the results stated above testify that the suprasegmental (cluster) structure of crosslinked polymers obeys the general laws of fractal geometry and deﬁnes the properties of the indicated polymers. Let us note that the value Tg is again a function 1/ 2 of parameter ν ñ (see Equation 6.3).

6.2 Elasticity Moduli The ﬁrst models describing the elastic behaviour of fractal structures used, as a rule, simulation within the frameworks of percolation theory [21–25]. A non-homogeneous statistical mixture of solid and liquid displays solid properties (for instance, shear modulus G not equal to zero) only, when the solid component forms a percolation cluster at gelation in polymer solutions. If the liquid component is replaced by a vacuum then the bulk modulus KT will also be equal to zero below the percolation threshold [21]. This model gives the following relationship for elastic constants [21]:

(6.6) where p is the volume fraction of the solid component, pc is the percolation threshold, η is the exponent. For the exponent η at fractal structure simulation as a Serpinsky carpet the following equation was obtained [21]:

(6.7)

where ν is the correlation length index in the percolation theory, d is the dimension of Euclidean space in which the fractal is considered. The comparison of the conductivity index t and η has shown that the condition η > t was carried out and this contradicts the criterion η = t, proposed earlier by De Gennes [21]. And at last Bergman and Kantor discovered that the ratio of bulk modulus and shear modulus had the universal value [21]:

(6.8) 288

The Properties of Crosslinked Epoxy Polymers A similar approach by was used Webman [23], according to which the deformation of fractals under the action of an external force F occurs only on scales exceeding some certain length LF. In the general case elastic moduli of a fractal section with characteristic size LF depend on the structure of the fractal skeleton, which, in spaces with small d, consists of both non-duplicated and repeatedly duplicated bonds. Assuming that region elasticity is deﬁned by non-duplicated bonds only (i.e., considering the sections consisting of duplicated bonds as absolutely stiff ones) the lowest limit of exponent η can be found [23]:

(6.9)

And in this case the condition η > t is fulﬁlled [23]. The authors [26, 27] used Relationship 6.6 for description of the behaviour of the shear modulus G in the case of linear amorphous polymers. They found out that for the correct description of G the indicated relationship required two modiﬁcations. Firstly, in Equations 6.7–6.9 the dimension d should be replaced with the polymer structure fractal dimension df. Secondly, it is required to introduce a variable percolation threshold pc, accounting for the deviation from the quasi-equilibrium state of the loosely packed matrix [27]:

(6.10)

is the relative ﬂuctuation free volume for the indicated state, accepted to where be equal to 0.060 [27]. In this case for linear amorphous polymers at p = ϕcl the linear dependence of G on (ϕcl – pc), corresponding to Relationship 6.6, was obtained. A more complex situation was observed for crosslinked polymers. It was found out [27] that for EP-1 and EP-2 the dependence of G(ϕcl) does not show even qualitative correspondence to the behaviour predicted by Relationship 6.6. However, this is explained not as the principal inapplicability of percolation models to crosslinked polymers, but as the raised degree of non-equilibrium of the structure of the loosely packed matrix for them. For description of the behaviour of G of the systems EP-1 and EP-2 in paper [26] the method already described was used, but with one closer deﬁnition. As was

289

Structure and Properties of Crosslinked Polymers shown for the considered systems in papers [28, 29], their ﬂuctuation free volume 0 0 consists of two components: the ﬁrst ( f g ) is constant ( f g = 0.024 [29]) and is connected with chemical crosslinking nodes, the second is variable and connected with decay (formation) of clusters (see Figure 5.17). Therefore, for crosslinked 0 polymers in Equation 6.10 instead of = 0.060 the value – f g = 0.036 was used. In Figure 6.4 the dependence G(ϕcl – pc) in double logarithmic coordinates for the systems EP-1 and EP-2 is adduced. As one can see, the linear dependence with slope ~ 1.47 was obtained that, according to Equation 6.7, gives df = 2.67 for the considered epoxy polymers, which corresponds well to the average fractal dimension of their structure [30].

ln G 0.6 -1 -2 0.3

0

-0.3 -2.2

-1.4

-0.6 ln (ϕcl-pc)

Figure 6.4 The dependence of shear modulus G on the percolation parameter (ϕcl – pc) for epoxy polymers EP-1 (1) and EP-2 (2) [26]

The relation adduced in Figure 5.15 between the probability of the formation of frame bonds ηx and the spectral dimension ds assumes a certain correlation ds and the mechanical properties of epoxy polymers, particularly the elasticity modulus. In Figure 6.5 the dependence of the elasticity modulus E at compression on the value ds is adduced, which conﬁrms this supposition. According to the data of Figure 6.5 the greatest value of E for the studied epoxy polymers can be estimated, which is equal to ~ 5 GPa at the maximum value ds = 1.73 [31].

290

The Properties of Crosslinked Epoxy Polymers

E, GPa 4

2 -1 -2 0 1.35

1.45

1.55 d s

Figure 6.5 The dependence of the elasticity modulus E at compression on the spectral dimension ds for epoxy polymers EP-1 (1) and EP-2 (2) [31]

The speciﬁc character of the structure-elasticity modulus relation for crosslinked polymers was considered in paper [28] with allowance for the division of the 0 ﬂuctuation of the free volume of epoxy polymers by two components, f g and , described above. In paper [1] the following interconnection between the microhardness Hv, the elasticity modulus E and Poisson’s ratio ν was supposed:

(6.11)

Although Formulae 5.5 and 6.11 are outwardly identical, they give different values of Hv and σY, since the values of ν included in them are different for various regions of polymer deformation. From Equations 1.33 and 6.11 it follows [28]:

(6.12) The authors [28] used this relationship for receiving the theoretical dependences of E on the value of Kst. Comparison of the calculated and experimental dependences

291

Structure and Properties of Crosslinked Polymers E(Kst) for EP-1, cured at different hydrostatic pressures, showed similarity in these dependences (Figure 6.6). In this case the use of the values of in the calculation of E according to Equation 6.12 gives better correspondence than use of the integral value fg. Similar results were obtained for the system EP-2 [28].

E, GPa

E, GPa

6 8

4 3

6 2 4

2

1

2 0.5

1.0

1.5

-2 Kst

Figure 6.6 The experimental (1, 2) dependences and those calculated according to Equation 6.12 (3–6) of the elasticity modulus E on the ratio curing agent : oligomer Kst for EP-1, cured at pressure 0.1 (1, 3, 4) and 200 MPa (2, 5, 6). (3, 5) and fg (4, 6) [28] The calculation was carried out using

The data of Figure 6.6 allow it to be understood why in paper [28] a better correlation between E and νcl than between E and νc for epoxy polymers was obtained. Apparently, in the elastic deformation region mechanical properties of epoxy polymers are deﬁned not by the entire free volume but only by the part ( ) which is connected with association (dissociation) of segments in clusters. One should note that at different stages of deformation properties of polymers are controlled by different structural regions. So, in elasticity (E) and local plasticity (Hv) ranges the loosely packed matrix ( ) mainly works and at macroscopic yield (σY) regions of chemical crosslinking 0 (fg= + f g ) are worked in addition. The different values of σY and Hv, obtained according to Equations 5.5 and 6.11, are explained by this fact despite the last superﬁcial resemblance [28]. As was noted in the previous chapter, the DS feature, forming in a deformable solid, is the existence of the universal hierarchy of structural levels [32]. As Balankin supposed

292

The Properties of Crosslinked Epoxy Polymers [18, 33], the solid key property – its shear stability, deﬁning the difference in spatial characteristic scales for localisation and dissipation regions of energy ‘pumping up’ in the deformable body at external inﬂuence – lay in the basis of DS hierarchical structure. The characteristic scale of regions (ls), in which extra energy dissipates, is proportional to the shear modulus G. Since according to the discussion adduced above these regions within the framework of a cluster model can be associated with clusters, then it is reasonable to use their characteristic size lst as ls. In Figure 6.7 the relation between G and lst is shown, which is approximated by a straight line, passing through the coordinates origin, that conﬁrms the accuracy of the application of lst as ls and allows the description of the elastic modulus G within the frameworks of synergetics of a deformable body [34].

lst, Å 10 8 6 4

-1 -2 -3 -4

2 0

0.5

1.0

1.5 G, GPɚ

Figure 6.7 The relation between the statistical segment length lst and the shear modulus G for epoxy polymers EP-1 (1, 3) and EP-2 (2, 4), cured at pressure 0.1 (1, 2) and 200 MPa (3, 4) [34]

6.3 Yield Stress The yielding process is an important phenomenon in the mechanics of polymers and its main characteristic (yield stress) restricts a plastic polymers exploitation ﬁeld as engineering materials from the practical point of view [35]. Therefore the authors

293

Structure and Properties of Crosslinked Polymers [36] offered a description of the theoretical yield stress of crosslinked epoxy polymers within the frameworks of the cluster model [8, 9]. The epoxy polymers EP-1 and EP-2 studied in paper [36] are characterised by plastic failure type in compression tests. In this case the clearly expressed yield stress, ‘yield tooth’ and cold ﬂow plateau are observed in deformation curves σ–ε. At present the point of view that supposes a simbate change of the elasticity modulus E and the yield stress σY for polymers prevails, which in the end gives the linear correlation σY(E). In Table 6.1 the values E and σY for the studied epoxy polymers are adduced, which show their principally differing character of dependence on Kst. So, if for the elasticity modulus the data of Table 6.1 demonstrate an extreme change with the minimum at Kst = 1.0 or 1.25, then the yield stress in the considered range of Kst remains practically constant. The comparison of change E and σY as a function of Kst excludes the similarity of the behaviour of E and σY for the studied epoxy polymers.

Table 6.1 The structural and mechanical characteristics of epoxy polymers [36] Epoxy polymer EP-1

EP-2

Kst

νc × 10–26, m–3

ϕc

E, GPa

σY, MPa

σYT , MPa

0.50

2

0.300

4.3

140

148

0.75

6

0.334

3.9

140

148

1.0

17

0.611

2.3

142

151

1.25

12

0.655

2.0

131

140

1.50

8

0.405

3.0

128

138

0.50

4

0.351

3.2

120

128

0.75

10

0.407

2.8

120

129

1.0

11

0.508

2.5

131

140

1.25

10

0.409

3.0

129

138

1.50

8

0.429

2.7

121

128

For an explanation of the indicated non-simbateness and theoretical description of the yield stress the authors [36] used the cluster model of amorphous state structure

294

The Properties of Crosslinked Epoxy Polymers of polymers [8, 9]. As has been indicated above, within the frameworks of this model segments included in clusters are considered as linear defects in the structure (analogue of dislocation in crystalline lattices) that allows mathematical calculus of dislocations theory for the theoretical description of yield stress to be used. Within the frameworks of such an approach the relation between σY and E is expressed by the equation [9]:

(6.13)

where b is the Burgers vector, ρd is the linear defects density, ν is Poisson’s ratio. Let us consider the methods of estimation of the parameters in Equation 6.13. The value of the Burgers vector b for polymeric materials is determined according to Formula 5.41 and the magnitude of Poisson’s ratio ν according to Equation 5.5. The linear defects density ρd was calculated according to the equation [9]:

(6.14) where ϕcl is the cluster relative fraction, S is the macromolecule cross-sectional area, which for the studied epoxy polymers is equal to ~ 32 Å2 [38]. The value of ϕcl was calculated according to the following percolation relationship [9]:

(6.15) where Tg and T are the glass transition and testing temperatures, respectively (see Equation 5.44). In Table 6.1 the comparison of the yield stress values σY received experimentally T and those calculated according to the method considered above σY for the studied epoxy polymers was adduced. As one can see, good correspondence of theory and T experiment was obtained (the average discrepancy of σY and σY is 6.6%, which is comparable with the error in mechanical tests).

295

Structure and Properties of Crosslinked Polymers The results stated above suppose that simbateness of the values of σY and E as a function of any parameter (temperature, crosslinking density and so on) can be an individual case only. Equation 6.13 demonstrates that the indicated simbateness realisation condition is the criterion ρd = const. or, as follows from Equation 6.14, ϕcl = const. In other words, invariance in polymer structure is the condition of the realisation of the change in the simbateness of σY and E. Let us be reminded that amorphous polymers are thermodynamically non-equilibrium solids, for the description of which two parameters of order, as a minimum, are required according to the principle of Prigogine–Defay. It is obvious that Equation 6.13 satisﬁes this principle, whereas the linear correlation σY(E) does not [36]. The study of the yielding process of epoxy polymers within the frameworks of the cluster model was continued in paper [39]. In Figure 6.8 the diagrams σ–ε for EP-2 at sample loading on uniaxial compression up to failure (curve 1) and at successive loading up to strains ε exceeding yield strain σY (curves 2–4) are shown. From comparison of these diagrams one can see that successive suppression of ‘yield tooth’ at invariable stress of cold ﬂow plateau σcf is observed. High values of σcf suppose corresponding values of stable clusters network density (see Chapter 5), which is essentially higher than the density of the chemical crosslinking network [36]. Thus, although crosslinked polymer behaviour on a cold ﬂow plateau is described within the frameworks of the rubber high-elasticity theory, the stable clusters network on this part of the diagram σ-ε is preserved. An unstable decay of clusters only occurs, deﬁning the mechanical devitriﬁcation of the loosely packed matrix. This process begins at a stress equal to the proportionality limit that corresponds to the data [40], where the ' effect of this stress and temperature T2 = Tg is assumed to be similar. The analogue between cold ﬂow and glass transition processes is only partial: in the ﬁrst case only one structural component (the loosely packed matrix) is devitriﬁcated. Complete decay of unstable clusters occurs not at the yielding reaching point at σY, but in the cold ﬂow plateau beginning at σcf. This can be seen from diagrams σ–ε, presented in Figure 6.8. Hence, the yielding process is controlled not by devitriﬁcation of the loosely packed matrix, but by another mechanism. As was mentioned above, one can suppose loss of stability by clusters in mechanical stresses ﬁeld that also follows from the well-known fact that the derivative of dσ/dε tends to zero at the yielding point [41]. According to [18] the critical shear strain γ*, resulting in shear stability loss by the solid, is equal to:

(6.16) where m and n are exponents in the Mi equation [1], establishing an interconnection

296

The Properties of Crosslinked Epoxy Polymers between interaction energy and distance between particles. The value 1/mn can be determined according to Equation 5.10. From combination of Equations 5.10 and 6.13 it follows [39]:

(6.17)

σɆPɚ 150 1

2

100

4

3

50

0

0.1

0.2

0.3

ε

Figure 6.8 The stress–strain (σ–ε) diagrams at loading up to failure (1) and load cyclic application (2–4) for EP-2 (Kst = 1.0). 2 – loading ﬁrst cycle, 3 – the second, 4 – the third [39]

Equation 6.17 gives the strain γ* value without allowance for viscoelastic effects, i.e., the diagram of the σ–ε deviation from linearity beyond the proportionality limit. Accounting for the tension strain as being approximately two times larger than the T corresponding shear strain [41], the theoretical yield strain σY , corresponding to T stability loss by the solid, can be calculated. In Figure 6.9 the comparison of εY and the experimental values of the yield strain εY is adduced. The approximate equality of these parameters is observed supposing that the yielding process is associated with loss of stability of clusters. To be more exact it concerns loss of stability of clusters

297

Structure and Properties of Crosslinked Polymers since the value of ν depends on the cluster network density νcl only [9] and the value of σY is proportional to νcl [42].

εTY 0.15

0.10

0.05

0

-1 -2

0.05

0.10

0.15 εY

Figure 6.9 The relation between experimental εY and theoretical εY yield strain values for epoxy polymers EP-1 (1) and EP-2 (2) [39] T

In paper [43] acceleration of the stress relaxation process was found at loading of epoxy polymers under the conditions similar to those described above (Figure 6.8, curves 2–4). The authors [43] explained the observed effect by the partial rupture of chemical bonds. In order to check this conclusion in paper [39] repeated tests on compression of samples, loaded up to the cold ﬂow plateau and then annealed at T < Tg, were carried out. It has been established that in the diagram σ–ε ‘tooth of yield’ is restored. This can occur at the expense of the restoration of unstable clusters, since the restoration of failed chemical bonds at T < Tg is scarcely probable. In this connection it is also necessary to note that ‘yield tooth’ suppression as a result of preliminary plastic deformation was observed earlier for linear amorphous polymers, for example, polycarbonate [44], for which the chemical bonds network is obviously absent. According to the data of paper [40] the ‘yield tooth’ value Δσ of epoxy polymers decreases with growth of Tg. This rule is obeyed in the case of the systems studied 298

The Properties of Crosslinked Epoxy Polymers in [36, 39]. However, the dependence Δσ(Tg) is not universal. For EP-1 the values of Δσ are essentially lower than for EP-2 at the same Tg. In the case of Kst > 1.0 at comparable Tg Δσ is larger than at Kst ≤ 1.0, i.e., a larger number of unstable clusters is formed at a surplus of curing agent [36]. Figure 6.10 shows the relation between the square of the glass transition temperature range width ΔTg, determined experimentally, and the network density of unstable clusters obtained according to the equation [39]:

(6.18)

ΔTg2 , K2 400 300 200 100

0

0.1

0.2

0.3

0.4

νiscl ×10−27,

m−3

Figure 6.10 The dependence of the square of the glass transition temperature range ΔTg on the network density of unstable clusters for epoxy polymer EP-2 [39]

The comparison of the dependence ΔTc(Kst), obtained in paper [9], and the data of Figure 6.10 shows that the deviation of Kst from its stoichiometric value, which is equal to 1.0, is a factor controlling the value of Δσ for crosslinked polymers [39].

299

Structure and Properties of Crosslinked Polymers The behaviour of a deformable solid undergoing mechanical inﬂuence is deﬁned by the formation and evolution processes of dissipative structures (DS), ensuring an optimal regime of energy dissipation, coming from outside [18, 45]. In the case of metals this approach is generally acknowledged, although a single point of view on the reformation mechanisms of a deformable structural body is absent [45]. For polymers the given problem has not been studied practically, despite the fact that DS availability in them has already been discussed (see Section 5.4). At the same time its solution allows to proceed to deformation of polymers from the point of view of the basic physical principles of non-equilibrium thermodynamics [33]. Application of these principles is possible at quantitative structural model availability. The cluster model, within the frameworks of which DS are identiﬁed as local order domains (see Section 5.4), will be used below as such a model. In paper [46] it was shown that the yielding process in amorphous linear polymers is realised when the effective value νY ≈ 0.41 of Poisson’s ratio is reached. Supposing this conclusion to be correct for crosslinked polymers (EP-1 and EP-2) as well and using the relation between ν and νcl, expressed by Equation 1.46, the authors [47] calculated the values of taking place when the yield stress σY is reached. They are shown in Figure 6.11 together with values of νcl for undeformed epoxy polymers EP-1 and EP-2 as a function of Kst. One can see that the values of do not depend on Kst and practically coincide for EP-1 and EP-2 that is deﬁned by the initial choice of νY. At the same time they are noticeably much lower than νcl. This means that a certain amount of cluster (DS) decay is required for realisation of the yielding process of crosslinked polymers. Such a situation is opposite diametrically to deformation processes of metals in which, on the contrary, DS (dislocation substructures) formation is observed in the yielding process [18, 33]. The indicated difference has a principal character and is due to the difference in notions about comparable classes of ideal (defectless) structures of materials (see Chapter 5). It should be expected that the indicated structural changes would deﬁne the parameters characterising the yielding process of crosslinked polymers. Let us consider this rule on the example of yield strain εY. It is natural to suppose that the larger the amount of DS that is subjected to decay in the yielding process the larger the value of εY should be. The indicated amount of DS can be determined as the difference of cluster network densities up to (νcl) and after ( ) yielding Δνcl = νcl – , which can be found easily from the plots of Figure 6.11. In Figure 6.12 the correlation εY(Δνcl), is adduced, which conﬁrms the assumption made.

300

The Properties of Crosslinked Epoxy Polymers

νcl νYcl ×10−27, m−3 3 -1 -2 -3 -4

2

1

0 0.50 0.75 1.00 1.25

1.50 Kst

Figure 6.11 The dependences of cluster network density in the underformed state (3, 4) on the curing agent : oligomer ratio Kst value νcl (1, 2) and after yielding for EP-1 (1, 3) and EP-2 (2, 4) [47]

εY

0.10

0.08 -1 -2

0.06

0.04 0

1

2 3 Δνcl×10−27, m−3

Figure 6.12 The dependence of yield strain εY on the difference of cluster network densities up to and after yielding Δνcl for epoxy polymers EP-1 (1) and EP-2 (2) [47]

301

Structure and Properties of Crosslinked Polymers Relationships 5.54, 5.65 and 5.67 considered in Chapter 5 together with Equation 1.14 allow determination of the functionality values of F of clusters for EP-1 and EP-2 up to and after yielding. The most characteristic distinguishing feature of the comparable dependences F(Kst) is an essential raising of F on reaching the yield stress (Figure 6.13). A simultaneous reduction in νcl and increasing F at the deformation of epoxy polymers up to the yield stress (Figures 6.11 and 6.13) indicates the decay of unstable clusters, having small values of F, as a result of which only stable clusters with large values of F remain at σY. Decay of unstable clusters induces mechanical devitriﬁcation of the loosely packed matrix, which explains the rubber-like behaviour of the polymer on the forced high-elasticity (cold ﬂow) plateau (see Section 1.1). Let us estimate within the frameworks of the expenditure of energy of the offered model, connected with decay of clusters under external load action, the energy Useg, which is necessary for segment association (dissociation) in a cluster and can be calculated according to Equation 5.39. Then the common energy UY required to reach the yield strain εY will be determined as follows [47]:

(6.19)

F

50 40

4

30 3

20 10

1 2

0 0.50 0.75 1.00 1.25 1.50 Kst Figure 6.13 The dependences of the functionality F of clusters in the underformed state (1, 2) and after yielding (3, 4) on the curing agent : oligomer ratio Kst value for epoxy polymers EP-1 (1, 3) and EP-2 (2, 4) [47]

302

The Properties of Crosslinked Epoxy Polymers On the other hand, the value of UY can be determined according to the diagram σ–ε, assuming it to be approximately triangular up to the yield stress [47]:

(6.20)

Equating Equations 6.19 and 6.20, the yield stress theoretical value can be found and compared with corresponding experimental values of σY (Figure 6.14). The good T correspondence of σY and σY means that the yield stress is actually deﬁned by the process energy of the decay of unstable clusters.

σTY ɆPɚ

300

200

100

0

-1 -2

100

200

300 σYɆPɚ

Figure 6.14 The relation between the experimental σY and σY yield stress values calculated according to Equations 5.39 and 6.20 for epoxy polymers EP-1 (1) and EP-2 (2) [47] T

Hence, the crosslinked polymers yielding process can be described within the frameworks of synergetics of a deformable body, namely by dissipative structure evolution. The quantitative identification of DS is obtained again within the frameworks of the cluster model [47].

303

Structure and Properties of Crosslinked Polymers

6.4 Fracture of Epoxy Polymers At present a large number of new polymers is synthesised in scientiﬁc laboratories around the world, from which only a small part reaches the industrial production stage [48]. Naturally, such work requires large expenditures of time and means. These expenditures can be reduced essentially by the development of methods of the prediction of the properties of new polymers, proceeding from their chemical constitution [49]. Mechanical properties of polymers are among the most important, since a certain level of these properties is always required even for polymers of different special-purpose functions [50]. In papers [38, 51] it has been shown that the curing process of the chemical network of epoxy polymers with the formation of nodes of various density results in a change in the molecular characteristics, particularly the characteristic ratio C∞. If such an effect actually exists, then it should be reﬂected in the deformationstrength characteristics of crosslinked epoxy polymers. Therefore the authors [49] offered methods of prediction of the limiting properties (properties at fracture), based on the notions of fractal analysis and the cluster model of the amorphous state structure of polymers, with reference to a series of sulfur-containing epoxy polymers [52, 53] (see also Section 5.4). The authors [49] carried out predictions of two deformation-strength characteristics: strain up to fracture εf and fracture stress σf. For the value of εf two methods of theoretical estimation can be used. The ﬁrst does not include molecular characteristics in its calculation and, hence, does not account for their change, at least directly [54]. This method is based on the notions of the cluster model of amorphous state structure polymers [8, 9] and the limiting drawing ratio λf value in this case is given by Equation 1.6. The theoretical dependence of εf (where εf = λf – 1) on Kst obtained in the manner indicated is adduced in Figure 6.15 (the dashed line). Its comparison with experimental data shows the inadequacy of Equation 1.6 for the estimation εf of a given series of sulfur-containing epoxy polymers (SCE-DADPS). Since the same equation describes well the data for a number of linear polymers [9, 54], comparison of the data of Figure 6.15 and the results of paper [54] supposes equivalent usage of this method only in the case of invariance of the molecular characteristics of polymers [49]. The second method is based on the notions of fractal analysis and the ﬁnal formula has the form [55]:

(6.21) 304

The Properties of Crosslinked Epoxy Polymers where the relative fraction of clusters ϕcl was calculated according to Equation 6.15 and the value of C∞ according to Relationship 5.2.

εf 0.15

0.10 1 -3 0.05

2

0 1.0

1.2

1.4

Ʉst

Figure 6.15 The dependence of the failure strain εf on the value of Kst for epoxy polymer SCE-DADPS. 1 – calculation according to Equation 1.6, 2 – calculation according to Equation 6.21, 3 – the experimental data [49]

In Figure 6.15 comparison of the values of εf calculated according to Equation 6.21 and those received experimentally is adduced. Now good correspondence of theory and experiment, both qualitative and quantitative, is observed. This means that the change in molecular characteristics in the polymer curing process does actually occur and prediction of εf can give incorrect results if this factor is not taken into consideration. It is interesting to note that the greatest deformability of sulfur-containing epoxy polymers is reached at the most dense crosslinking, being realised at Kst = 1.2 [52] (Figure 6.15). This non-trivial observation can be explained within the frameworks of fractal analysis. As it is known [34, 55], the molecular mobility level can be characterised by the value of fractal dimension Dch of a chain part between its ﬁxation points. The indicated dimension value can be determined both according to Formula 2.17 and with the aid of the following equation [55]:

(6.22)

305

Structure and Properties of Crosslinked Polymers The dependence of the values of Dch calculated according to Equation 6.22 on Kst is adduced in Figure 6.16. As one can see, the value of Dch reaches a maximum at Kst = 1.2, i.e., at this value of Kst the epoxy polymer has the greatest molecular mobility [55]. The common rule for polymers is that molecular mobility intensiﬁcation results in the growth of introducing from outside mechanical energy dissipation and, as a consequence, to an increase in polymer deformability [50]. From Equation 6.22 it follows that a decrease in C∞ is the only cause of the extreme growth in Dch, i.e., we are faced again with the necessity of the allowance for a change in molecular characteristics for crosslinked polymers in their curing process for correct description of the properties of these materials.

Dch 1.10

1.05

1.0 1.0

1.2

1.4

Ʉst

Figure 6.16 The dependence of fractal dimension Dch of the chain part between clusters on the value of Kst for epoxy polymer SCE-DADPS [49]

For calculation of the strength (failure stress) σf the following equation was used [56]:

(6.23)

where ρ is the polymer density (equal for the studied epoxy polymers to ~ 1400 kg/

306

The Properties of Crosslinked Epoxy Polymers m3 [53]), Mcl is the molecular mass of the chain part between clusters, M n is an average-number molecular mass of the polymer. For crosslinked polymers the value of M n is equivalent to the molecular mass of crosslinked clusters (microgels), which is obviously much larger than Mcl and so in the ﬁrst approximation the member (1 – Mcl/ M n ) in Equation 6.23 can be neglected and then the combination of Equations 1.3, 1.11 and 6.23 allows the following formula to be obtained [49]:

(6.24)

In Figure 6.17 comparison of the strength of σf values, obtained experimentally and calculated according to Equation 6.24, is adduced for epoxy polymer SCE-DADPS. As follows from the adduced comparison, good correspondence of theory and experiment is obtained. Hence, the offered methods of calculation of the mechanical limiting characteristics allow the estimation of failure stress and strain (strength and deformability) on the basis of only two independent parameters: the glass transition temperature Tg and the macromolecule cross-sectional area S. The value of Tg can be estimated theoretically as well (for example, by a group contributions method [57]). It is necessary to account for the change in their molecular characteristics in the curing process at deformationstrength properties of crosslinked polymers with different crosslinking density (or variation in Kst) [49]. As it is known [58, 59], the main deﬁciency of epoxy polymers, their tendency to brittle fracture, is considered. It was considered for a long time that brittleness is an inalienable property of crosslinked systems, however, a number of papers has appeared recently that show that epoxy polymers can be deformed in tensile tests, displaying macroscopic yielding [60]. The authors [52] showed the possibility of plastic strains in impact test conditions and explained the factors inﬂuencing the type of fracture of epoxy polymers. The sulfur-containing epoxy polymers, cured by both aromatic diamines (DADPS and DADPM) and Iso-methyltetrahydrophthalic anhydride (IMTHPhA) (see Section 5.4), were studied. The fact that two of the studied epoxy polymers showed plastic fracture in impact loading conditions (SCE-DADPS – 1.2 and SCE-DADPS – 1.3, where the ﬁgure in conditional sign indicates the value of Kst). The diagram load-time (P – t),

307

Structure and Properties of Crosslinked Polymers received in instrumented impact tests for one of the polymers (SCE-DADPS – 1.2), is shown in Figure 6.18 and the diagram of P – t for a typical brittle epoxy polymer (SCE-IMTHPhA – 1.2) is adduced in Figure 6.19.

σfɆPɚ 140 -2 120

100 1 80 1.0

1.2

1.4

Ʉst

50 n

Figure 6.17 The dependences of failure stress σf on the value of Kst for epoxy polymer SCE-DADPS. 1 – calculation according to Equation 6.24; 2 – the experimental data [49]

0.5 ms Figure 6.18 The diagram of load-time (P – t) for plastic fracture of epoxy polymer SCE-DADPS – 1.2, plotted by averaging the results of ﬁve experimental diagrams [52]

308

50 n

The Properties of Crosslinked Epoxy Polymers

0.5 ms Figure 6.19 The diagram of load-time (P – t) for the plastic fracture of epoxy polymer SCE-IMTHPhA – 1.2, plotted by averaging the results of ﬁve experimental diagrams [52] As it is known [61], the impact toughness Ap is deﬁned by the area under the diagram of P – t. From comparison of Figures 6.18 and 6.19 it follows that this parameter (and, hence, Ap) is considerably larger for epoxy polymer SCE-DADPS – 1.2. In Figure 6.20 the dependences of Ap on the Kst value for sulfur-containing epoxy polymers, cured by DADPS and IMTHPhA, are adduced. From the data of Figure 6.20 it is obvious that epoxy polymers SCE-DADPS – 1.2 and SCE-DADPS – 1.3 showed the greatest impact toughness, i.e., precisely those that reveal macroscopic yielding, and in the curve Ap(Kst) a pronounced maximum at Kst= 1.2 is observed. The values of Ap for epoxy polymers, cured by IMTHPhA, showing brittle fracture, are considerably lower than for those cured by DADPS and reveal a weak maximum at Kst = 1.3. Another epoxy polymer showed the same shape of the dependence of Ap on curing agent concentration [62]. The absolute values of Ap for epoxy polymer SCE-DADPS – 1.2 are sufﬁciently high and turn out to be higher than those known from literary sources [63]. Individual samples of SCE-DADPS – 1.2 showed the value of Ap up to 45 kJ/m2 that twice exceeds the value of Ap for the most plastic epoxy polymers. The explanation of causes deﬁning the impact toughness of epoxy polymers is of great interest in order to reveal the criteria at which realisation of plastic deformation and, hence, very high impact toughness is possible for epoxy polymers. The Brown approximation can be used for analysis of the behaviour of the value of Ap [64, 65]:

(6.25)

309

Structure and Properties of Crosslinked Polymers

where G I c is the deformation energy release critical rate, Lsp is the distance between Sharpy pendulum supports (span), acr is the critical defect size that initiates fracture.

Ap, kJ/m2 30

20

10

2 1

0

1.0

1.2

1.4 K st

Figure 6.20 The dependences of impact toughness Ap on the curing agent contents, characterised by parameter Kst, for epoxy polymers SCE- IMTHPhA (1) and SCEDADPS (2) [52]

Since in the experiment the constant value of Lsp was used then according to Equation 6.25 the value of Ap is only the function of the plasticity of epoxy polymers, characterised by the value GI c , and their degree of defectness, characterised by parameter acr. Since in the experiment the samples without a notch were used then parameter acr characterises the critical structural defect (CSD) [66, 67]. Electron microphotographs (enlargement 300×) of the fracture surfaces, an example of which is adduced in Figure 6.21, were used for estimation of the value of acr. Study of these microphotographs allows the following conclusions to be made. Firstly, at a sample external surface, corresponding to the greatest stress zone, the bright smooth area, having an approximately circular shape, is observed – the catastrophic crack nucleation site. Such fracture surface features were already noted in papers on samples of epoxy polymers without notch fracture study [62]. The appearance and sizes of these areas allow them to be identiﬁed as CSD. CSD sizes for samples of the studied sulfur-containing epoxy polymers are changed within the limits of

310

The Properties of Crosslinked Epoxy Polymers 50–200 mcm, which corresponds very well to the known literary data [62]. The dependences of acr on parameter Kst for the systems SCE-IMTHPhA and SCE-DADPS are shown in Figure 6.22. These dependences have extreme character with a clearly expressed minimum (particularly for SCE-DADPS) at Kst = 1.2–1.3. Let us note that the maximum of Ap corresponds to the same values of Kst (Figure 6.20).

Figure 6.21 An electron microphotograph of fracture surface in impact tests of a sample of epoxy polymer SCE-DADPS-1.2. Enlargement 300× [52]

acr, mcm 300 2 1

200

100

0

1.0

1.2

1.4 K st

Figure 6.22 The dependences of the critical structural defect size acr on the amount of curing agent, characterised by parameter Kst for epoxy polymers SCEIMTHPhA (1) and SCE-DADPS (2) [52]

311

Structure and Properties of Crosslinked Polymers Secondly, from the bright, smooth area fan-like roughnesses are diverged, which are identiﬁed as shear local deformation traces in the propagating crack tip [68]. The common appearance of fracture surfaces supposes unstable propagation of the crack and it is signiﬁcant that for epoxy polymers the majority of fracture surfaces at such a mode of crack propagation have a mirror appearance [69]. The increase in fracture surface roughness for the studied sulfur-containing epoxy polymers should reﬂect the shear deformation intensiﬁcation and, hence, raising plasticity, characterised by the value GI c [70]. The last parameter can be estimated by using the Orovan–Irwin equation [65]:

(6.26)

The values of GI c calculated according to Equation 6.26 as a function of the amount of curing agent, characterised by parameter Kst, are shown in Figure 6.23. These plots have two main features. Firstly, the values of GI c are anomalously high in comparison with the known literary data for epoxy polymers, where GI c is usually within the limits of 0.05–0.25 kJ/m2 [69, 71]. Proceeding from the above, the authors [52] assumed that the increase in GI c was due to shear local deformation intensiﬁcation, as fan-like roughnesses showed in fracture surfaces of the considered sulfur-containing epoxy polymers (Figure 6.21). The second and most outstanding feature of Figure 6.23 is the fact that the minimum GI c corresponds to the maximum Ap (Figure 6.20). Thus, the sharp reduction of CSD sizes (Figure 6.22) is a crucial factor of the plastic deformation origin in epoxy polymers SCE-DADPS – 1.2 and SCE-DADPS – 1.3, although it is necessary to account for the fact that the value of GI c even at a minimum point has a relatively large magnitude – up to 0.60 kJ/m2 (Figure 6.23). All of that stated above allows it to be supposed that the brittle–ductile transition in the considered epoxy polymers is controlled by the Ludwig–Davydenkov criterion [72]. This very simple criterion assumes that the brittle–ductile transition controls the relation of fracture and yielding stresses, namely if the fracture stress is smaller than the yield stress then the material breaks in a brittle manner, if the other way around by the ductile mode. As follows from Equation 6.26, the failure stress depends on three parameters: E, GI c and acr. The value of E as a function of the amount of curing agent is shown in Figure 6.24, from which it follows that for epoxy polymer SCE-DADPS with an increase in Kst from 1.0 to 1.4 the elasticity modulus increases by about 2.3, which

312

The Properties of Crosslinked Epoxy Polymers

also promotes growth in σf. As for the values of GI c and acr the reduction in GI c is compensated for by a still greater reduction in acr. Let us also note that relatively high values of GI c and relatively small values of E allow it to be supposed that the interglobular matrix of the considered epoxy polymers has crosslinking low density and contains a sufﬁciently large amount of long linear molecular sequences [60, 71] (see also Section 5.2).

GI c , kJ/m2 3

2

2 1

1

0

1.0

1.2

1.4 K st

Figure 6.23 The dependences of the deformation energy release critical rate GI c on the amount of curing agent, characterised by parameter Kst, for epoxy polymers SCE-IMTHPhA (1) and SCE-DADPS (2) [52]

In Figure 6.25 the dependence of the fracture stress σf on the amount of curing agent for the three considered epoxy systems is shown. The common appearance of the dependence is similar to the dependences described in paper [62] for another epoxy system, but a sharper increase in σf is observed at large Kst. In Figure 6.25 the yield stress σY values are also indicated for two epoxy polymers: SCE-DADPS – 1.2 and SCE-DADPS – 1.3. At Kst increasing from 1.2 up to 1.3 the value σY is increased on 27 % that also reduces plastic deformations appearance probability and, hence, has an inﬂuence on the value Ap that is observed experimentally (Figure 6.20). The fracture strain εf value (Figure 6.26), as was expected, passes through a minimum at

313

Structure and Properties of Crosslinked Polymers Kst = 1.2 for epoxy polymers SCE-DADPS and for SCE-IMTHPhA, having shown only brittle fracture, the dependence of εf on the amount of curing agent is similar to that described in paper [62].

E, GPa

3 2 1 2

1

0

1.0

1.2

1.4 K st

Figure 6.24 The dependences of the elasticity modulus E on the amount of curing agent, characterised by parameter Kst, for epoxy polymers SCE-IMTHPhA (1) and SCE-DADPS (2) [52]

In paper [62] it has been shown that the supramolecular structure of epoxy polymers has two size levels: the ﬁrst ~ 10 nm and the second ~ 10–100 mcm and CSD size should be connected with the size of the supramolecular structure elements of the second level. For veriﬁcation of this concept samples of the considered epoxy polymers underwent etching in concentrated sulfur acid for 36 hours. After washing and drying under a vacuum, the received surfaces were studied under an optical microscope and it was found that their structure consisted of large blocks (ﬂoccules) with diameter from 100 to 500 nm (see Figure 5.38). The adduced sizes correspond to the greatest blocks of the system [52]. The surfaces of the samples were studied by laser microscopy methods, the data of which correspond well to the results obtained by optical microscopy on etched samples. The comparison of the sizes of CSD and ﬂoccules showed unequivocal correspondence between them; however, the size of CSD is about twice as small as the diameter of the corresponding ﬂoccules. The probable scheme of CSD formation in the considered epoxy polymers is shown in Figure 6.27.

314

The Properties of Crosslinked Epoxy Polymers

σ, MPa 120

4

3 2

80

1

40

0

1.0

1.2

1.4 K st

Figure 6.25 The dependences of the fracture stress σf on the amount of curing agent, characterised by parameter Kst, for epoxy polymers SCE-IMTHPhA (1), SCE-DADPS (2) and SCE-DADPM (3) and yield stress σY for epoxy polymer SCEDADPS (4) [52]

εf 0.08

0.06

0.04

2 1

0.02

1.0

1.2

1.4 K st

Figure 6.26 The dependences of the fracture strain εf on the amount of curing agent, characterised by parameter Kst, for epoxy polymers SCE-IMTHPhA (1) and SCE-DADPS (2) [52]

315

Structure and Properties of Crosslinked Polymers

A possible sites of KDS formation

“Tie” molecules

Blocks

Figure 6.27 The hypothetical scheme of critical structural defect formation in epoxy polymers [52]

As it is known [73], fracture of polymers is controlled by the main chain bonds breaking, therefore it is probable that ﬂoccules are connected with one another in the closest contact sites by tie chains. Where ﬂoccules are removed from one another by a somewhat larger distance, areas possessing less crosslinking density and/or chains with less dense packing, are formed, i.e., more imperfect (less strong) (see Section 5.2). The estimations fulﬁlled for poly(methyl methacrylate) have shown that the surface energy of such imperfect gaps is two orders smaller than the surface energy of blocks [74]. Therefore CSD is formed in the initial loading stage in such imperfect layers and up to a certain point (the beginning of catastrophic crack propagation) its growth is restrained by ﬁrm contact with the sites of ﬂoccules. Such a hypothetical mechanism explains the experimentally received relation 1 : 2 between the sizes of CSD and ﬂoccules [52]. Hence, the strength of the considered sulfur-containing epoxy polymers at impact loading is deﬁned, at any rate, by four parameters: elasticity modulus, characterising viscoelastic properties of material; yield stress value, characterising the degree of difﬁculty of realisation of plastic deformation; deformation energy release critical rate, characterising material ductility, and size of supramolecular structure blocks (ﬂoccules), characterising the degree of defectness of the epoxy polymer at loading. In turn, these parameters are controlled by the chemical nature of the epoxy oligomer and curing agent, by their relation and also their curing regime. The optimal combination of these parameters allows epoxy polymers capable of becoming deformed plastically and possessing high impact toughness to be obtained [52].

316

The Properties of Crosslinked Epoxy Polymers

6.5 The Other Properties In the present section some other properties of epoxy polymers will be considered brieﬂy, namely thermal expansion, anharmonicity, heat capacity and microhardness. The thermal expansion study of polymers together with receiving the necessary experimental characteristics gives interesting physical information [75]. This is connected with the chain structure of a macromolecular polymer resulting in a sharp distinction of intra- and intermolecular interaction that causes strong anisotropy of polymeric macromolecule thermal dynamics in respect to its axis. The features of thermal expansion of the lattice of polymeric crystallites (strong transverse expansion at longitudinal shortening) were studied in sufﬁcient detail by X-rays [75]. For amorphous polymers tracing of the behaviour of molecular aggregates in polymer bulk temperature is a more complex problem by virtue of the absence of sharp X-ray reﬂections and at present is absent practically [76]. The authors of paper [76] showed the distinction of micro- and macroexpansion in amorphous polymers and explained it by a certain degree of ordering of chain macromolecules. In other words, the authors [76] found interconnection of thermal expansion and supramolecular structure for a number of amorphous polymers. However, the quantitative structural model for absence of the amorphous state does not allow similar interconnection details to be more precise. Therefore the authors [77] carried out the study of interconnection for amorphous epoxy polymers EP-1 and EP-2 of thermal expansion and structure, for the description of which the cluster model [8, 9] was used. The values of thermal expansion linear coefﬁcients were calculated according to m the X-ray data (see Section 5.1). The value of microexpansion α ⊥ in the direction perpendicular to the macromolecule axis was calculated according to the method in m [76], moreover the value α ⊥ for the system EP-1 was counted off from the state at Kst = 1.0 and for the system EP-2 from the state at Ks = 1.25, where Bragg’s interval dB values have the smallest magnitude. As for amorphous polymers in paper [76], a m much stronger variation in α ⊥ is observed in comparison with α for the dependences m α(Kst) and α ⊥ (Kst) adduced in Figure 6.28, although the tendencies of the dependences shown in Figure 6.28 are similar. This distinction can be explained within the frameworks of the cluster model, which is similar to that used in paper [76], excluding the type of local order domains. Slutsker and Filippov [76] supposed that those domains were the analogue of a crystallite with folded chains, whereas clusters are the amorphous analogue of a crystallite with extended chains [9]. Following the reasons in the paper [76] one can say that

317

Structure and Properties of Crosslinked Polymers thermal expansion of the cluster will be strongly anisotropic and for its description the equation can be used [76]:

(6.27)

where α

m

is the microexpansion parallel to the macromolecule axis.

2 αm ⊥ ×10

4

-1 -2 -3 -4

3 2 1 0 0.50

0.75 1.00 1.25 1.50 Ʉ st

Figure 6.28 The dependences of linear macroexpansion α (1, 2) and m microexpansion α ⊥ (3, 4), determined by dilatational (1, 2) and X-ray (3, 4) methods, on Kst for epoxy polymers EP-1 (1, 3) and EP-2 (2, 4) [77]

Equation 6.27 allows calculation of the value of α and all received magnitudes of the characteristics of macro- and microexpansion of epoxy polymers are adduced m m in Table 6.2. As follows from the data of this table, the values of α, α ⊥ and α are close to the absolute value and the tendency to change to similar magnitudes for amorphous linear polymers is adduced in paper [76]. However, an essential distinction m is observed, which consists in the fact that values of α can have both positive and negative sign. The possibility of conformational trans–gauche transitions, in addition to increasing macromolecule longitudinal shortening, was considered in paper [76] as m one of the factors deﬁning negative values of α . Obviously for the studied epoxy m

318

The Properties of Crosslinked Epoxy Polymers

polymers, positive sign and high absolute values of α suppose high probability of reverse (gauche–trans) transitions with a change in Kst [77]. m

It has been shown earlier [78] that an increase in the extended trans-conformations fraction results in raising of the entanglements cluster network density νcl. Hence m one may assume that increasing α , indicating a rise in gauche–trans transition m probability, should also result in increasing νcl. In Figure 6.29 the dependence νcl( α ) is shown, where the values of νcl for the considered epoxy polymers were accepted according to the data of paper [79]. As follows from the plot of Figure 6.29 despite m a certain data scattering the growth of νcl with increasing α is actually observed, which conﬁrms the assumption made above.

Table 6.2 The structural and thermophysical characteristics of epoxy polymers [77] Epoxy polymer EP-1

EP-2

α m⊥ × 105, K–1

α m × 105, K–1

0.50

4.15

-2.03

2.09

0.75

3.94

-2.09

1.93

1.25

1.04

3.83

1.97

1.50

2.07

1.83

1.99

0.75

1.08

2.81

2.23

1.0

0.43

4.30

1.73

1.50

3.03

0.24

2.10

Kst

α × 105, K–1

For an inextensible thread, which simulates a macromolecule part, under the assumption of sinusoidal oscillations, the wavelength λk can be written as follows [76]:

(6.28) where αm is the total polymer macroexpansion. 319

Structure and Properties of Crosslinked Polymers νcl×10−27, m−3

3 -1 -2

2

1 -3

-1

1

3

5

α m ×10

2

Figure 6.29 The dependence of the entanglements cluster network density νcl on m microexpansion α for epoxy polymers EP-1 (1) and EP-2 (2) [77]

The obtained oscillation wavelength values can be compared approximately with local order domain longitudinal length, which in model [8, 9] is accepted to be equal to the statistical segment length lst. In Figure 6.30 the comparison of the values of λk and lst is shown, which indicates both the identical tendency of their change and the satisfactory correspondence of their absolute values. A discrepancy between λk and lst at large magnitudes is probably due to the approximate character of the fulﬁlled calculations [76]. Nevertheless, the data of Figures 6.29 and 6.30 assume that cluster model quantitative parameters (cluster network density νcl and cluster size lsn) can be used for estimation of such an important characteristic of polymers as the thermal expansion coefﬁcient [77]. Let us consider among the thermodynamic characteristics the lattice Grüneisen parameter γL, which accounts for anharmonicity of intermolecular bonds [1, 80] and is applied widely for the description of states of polymers [81]. The value of γL can be determined according to the equation [1]:

(6.29) where ν is Poisson’s ratio.

320

The Properties of Crosslinked Epoxy Polymers

lst, Å

15

10

5

0

-1 -2

5

10

15 λk, Å

Figure 6.30 The relation between wavelength λk and statistical segment length lst for epoxy polymers EP-1 (1) and EP-2 (2). The straight line shows the relation 1 : 1 [77]

In Figure 6.31 the dependence of γL on the relative fraction ϕl.m. of the loosely packed matrix is shown for epoxy polymers EP-1 and EP-2 and also for comparison purposes for glassy amorphous polycarbonate [9]. As one can see, for linear amorphous polycarbonate the value of γL is deﬁned only by the process of association (dissociation) of segments in clusters, since γL = 0 at ϕl.m. = 0. As has been noted above, for crosslinked polymers at ϕl.m. = 0 fg ≠ 0 (see Figure 5.17) and according to Equations 1.33 and 6.29, γL ≠ 0. Hence, it can be supposed that the value of γL for crosslinked polymers is deﬁned by two structural components: clusters and chemical crosslinking network nodes regions [82]. Accounting for this fact, it is interesting to consider the correlation between γL and νc, which is shown in Figure 6.32. The linear decay of γL with growth in νc is observed, since the lattice Grüneisen parameter accounts for all kinds of anharmonicity in a polymer [80]. In this case for νc = 0 we will obtain γL ≈ 5.2, which corresponds well to the data of paper [83].

321

Structure and Properties of Crosslinked Polymers

γL 6

4 -1 -2 -3

2

0

0.4

0.8

ϕl.m.

Figure 6.31 The dependence of the Grüneisen parameter γL on the relative fraction ϕl.m. of a loosely packed matrix for polycarbonate (1) and epoxy polymers EP-1 (2), EP-2 (3) [82]

γL 5

-1 -2

4

3

2

0

10

20

νc×10-27, m-3

Figure 6.32 The dependence of the Grüneisen parameter γL on the crosslinking density νc for epoxy polymers EP-1 (1) and EP-2 (2) [82]

322

The Properties of Crosslinked Epoxy Polymers Let us estimate further the value of the heat capacity at constant pressure jump ΔCp at the glass transition temperature Tg, using the equation given by Bunderlich [84]:

(6.30) where R is the universal gas constant, ϑ0 is the molar volume occupied by micromolecules, ϑh is the molar free volume, Eh is the free volume microvoid formation energy. Estimations were easily carried out, simplifying Formula 6.30 and assuming in it [82]:

(6.31)

(6.32)

(6.33)

The corresponding values of ΔCp are adduced in Table 6.3. The accuracy of calculations of ΔCp according to Equation 6.30 can be conﬁrmed by using the estimation of the ' heat capacity jump ΔÑ p value according to the Boyer empirical relationship [85]:

(6.34)

and assuming the value of ΔÑ p as an experimental one. '

323

Structure and Properties of Crosslinked Polymers The comparison of values of ΔCp and ΔÑ p adduced in Table 6.3 has shown their good correspondence. This allows a conclusion to be drawn that the cluster model [8, 9] gives a sufﬁciently precise ΔCp calculation. '

Table 6.3 Theoretical and experimental ΔCp values for epoxy polymers [82] Kst

Epoxy polymer EP-1

EP-2

ΔCp, kJ/kg×K

ΔÑ 'p , kJ/kg×K

0.50

32.8

32.0

0.75

28.6

28.5

1.0

21.8

24.6

1.25

23.8

23.7

1.50

24.3

26.7

0.50

29.3

30.6

0.75

26.9

28.0

1.0

26.0

26.1

1.25

26.8

27.5

1.50

31.0

30.2

In papers [86, 87] the deformation nature of polymers in microhardness tests was studied. For this purpose linear, crosslinked and semi-crystalline polymers and different methods of their structure modiﬁcation (crosslinking, extrusion, ﬁlling and physical ageing) were used. It is natural to suppose that in tests on microhardness the loosely packed matrix responds ﬁrst of all on the indentor microforcing and in conformity with the deformable body synergetics laws [33] formation of localized strongly non-equilibrium areas occurs, where extra energy is accumulated. The dimension Df of these areas is determined according to Equation 5.65. As follows from the data of Figure 6.33, the linear correlation between microhardness Hv and extra energy localisation regions dimension Df is observed. In addition,

324

The Properties of Crosslinked Epoxy Polymers the studied polymers are divided into three groups. Since regions of extra energy localisation are found in the loosely packed matrix, then the obtained correlation between Hv and Df means that the microhardness of polymers is deﬁned mainly by the properties of the structure of the loosely packed matrix or, in other words, by its thermodynamic non-equilibrium level [86].

Hv, MPa -1 -2 -3 -4 -5

300

200

100

0

4

8

Df

Figure 6.33 The dependences of the microhardness Hv on the extra energy localisation regions dimension Df for EP-1 (1), EP-3 (2), polyarylate (3), ultra-high molecular polyethylene (UHMPE) (4) and Componor UHMPE-bauxite (5) [87]

The introduction of bauxite as a ﬁller in ultra-high molecular polyethylene (UHMPE) results in two-fold growth of the extra energy localisation regions dimension from Df ≈ 4 (UHMPE) to Df ≈ 8 (UHMPE-bauxite). In addition the microhardness increases approximately twice (Figure 6.33). If the points corresponding to UHMPE and UHMPE-bauxite are disposed on one straight line, then the points for samples of epoxy polymers EP-1 and EP-2 are located on different straight lines (Figure 6.33). With ageing of epoxy polymers (at transition from EP-1 to EP-3) the value of Df decreases from Df ≈ 4.5 to Df ≈ 3, since the system has a tendency of transition to a more equilibrium state (Figure 6.33). Under the assumption that three-dimensional regions of extra energy localisation (at dimension of Euclidean space d = 3) correspond to some quasi-equilibrium state

325

Structure and Properties of Crosslinked Polymers of polymer structure [88] (see also Section 5.5) the deviation from this state can be characterised by the ratio (Df – 3)/3. In turn, assuming that some quasi-equilibrium 0

value of microhardness H v (which is found from the plots of Figure 6.33) corresponds to dimension Df = 3, the deviation from it can be determined as the ratio 0

0

(Hv – H v )/ H v . 0

0

As one can see (Figure 6.34) between the ratios (Hv – H v )/ H v and (Df – 3)/3 the linear correlation common for all considered polymeric materials is observed. Hence, the relative change in microhardness characterises the deviation of the structure of polymers from the quasi-equilibrium state.

(Hv- H v0 )/ H v0

3

2

1

0

0.5

1.0

1.5 (Df-3)/3 0

0

Figure 6.34 The dependence of the microhardness relative change (Hv – H v )/ H v on the dimension Df relative change (Df – 3)/3 for EP-1 (1), EP-3 (2), polyarylate (3), UHMPE (4) and Componor UHMPE-bauxite (5) [87]

The notion was developed earlier that the microhardness of glassy solids is deﬁned by the ﬂuctuation free volume microvoid formation (or collapse) work, ascribed to the microvoid volume unit [1]. Such an approach corresponds to the results in the present section since, as has been shown above, the ﬂuctuation free volume is concentrated in the loosely packed matrix of polymer structure. The relative fraction of the ﬂuctuation free volume fg can be estimated according to Equation 1.33. As was to be expected, the linear correlation between the values of fg and Df, calculated according to Equations

326

The Properties of Crosslinked Epoxy Polymers 1.33 and 5.65, was obtained in the EP-1 case (Figure 6.35). One can be convinced that the dependence of Hv on the ratio Df/fg for all considered polymeric materials is described by a common straight line despite certain data scattering. It is clear that the larger fg the higher the degree of structure loosening and the lower the resistance −1 to indentor forcing: Hv~ f g [86].

fg 0.30

0.15

0

5

10

15

Df

Figure 6.35 The correlation between the relative ﬂuctuation free volume fg and the extra energy localisation regions dimension Df for EP-1 [87]

Hence, the results stated above have demonstrated that the microhardness of polymers is connected with one of the components of their structure, namely with the loosely packed matrix in which the ﬂuctuation free volume is concentrated. Glassy polymers are characterised by some quasi-equilibrium state, corresponding to the threedimensionality of the extra localisation energy. The microhardness is controlled by the degree of deviation of the polymer structure from this state [87].

6.6 The Physical Ageing of Epoxy Polymers It is known that the properties of polymers can be changed essentially because of their physical (heat) ageing. This process is considered as a transition to a thermodynamically more equilibrium state [89] and different parameters were used

327

Structure and Properties of Crosslinked Polymers for its description. In particular, in paper [90] heat ageing of crosslinked (epoxy) polymers is characterised by the change in relative ﬂuctuation free volume fg. In the present section such analysis will be carried out within the frameworks of the cluster model of the amorphous state structure of polymers [8, 9] and fractal analysis [34]. At present it is acceptable to consider [91] that the crosslinking density νc increase results in a rise in the glass transition temperature Tg for crosslinked polymers. The availability of such a dependence is connected with the restriction of molecular mobility of chains by chemical crosslinking nodes [91, 92]. This conclusion is conﬁrmed by a number of experimental results, where a simbate change of Tg and νc was observed [92]. At the same time the following facts attract attention. According to the data of paper [90] heat ageing of epoxy polymers at temperature T < Tg increases Tg, retaining an invariable νc. The indicated circumstances allow to suppose that direct interconnection between Tg and νc is absent but it can be realised through the suprasegmental level of structural organisation of crosslinked polymers (see also Section 6.1). The authors [93] studied the concrete character of the indicated interconnection. From the data of Table 6.4 it follows that in the heat ageing process the value of νc decreases for the majority of EP-2 samples and νcl increases, particularly at a large deviation of epoxy compound composition from the stoichiometric ratio (Kst) = 1.0. Thus, within the frameworks of cluster model epoxy polymer EP-2 the observed change in the structure, occurring in the heat ageing process, can be treated as the increasing of local order level, of which the value νcl serves as a characteristic [9]. Since this means the tendency of transition to more equilibrium structure, then such treatment agrees with the existing concepts [89]. The value of Tg for the aged samples of EP-2 is larger than for native ones (Table 6.4). Analysis of the data adduced in Table 6.4 shows that in the ageing process νcl and Tg are changed similarly, whereas between νc and Tg such an interconnection is absent. Hence, the value of Tg does not depend directly on νc, but is deﬁned by suprasegmental structure characteristics (see, for example, Figures 6.2 and 6.3). In Figure 6.36 the dependences Tg(νcl) are adduced for both series of EP-2. They are linear and located on a straight line that allows to consider the devitriﬁcation process of crosslinked polymers as a ‘freezing’ local order thermal decay process (at νcl = 0 the value of Tg ≈ 295 K, i.e., it is equal to the testing temperature at which the magnitudes of νcl are received). As has been shown in Section 6.1, in this case the role of chemical crosslinking nodes comes to the restriction in ﬂuctuations of clusters (see Equation 6.3). As a result the values of Tg for crosslinked polymers are higher in comparison with linear ones having comparable chain rigidity [93].

328

The Properties of Crosslinked Epoxy Polymers

Table 6.4 The structural and thermophysical characteristics of epoxy polymer EP-2 [93] Kst

Tg, K

νc × 10–26, m–3

νcl × 10–27, m–3

0.50

342 363

4.0 3.0

1.3 4.7

0.75

372 383

1.0

399 408

1.25

378 398

2.4 6.0

1.50

343 408

1.6 5.7

2.3 5.4 1 .0 7.4

3.3 5.8

As was shown in Section 4.1, the fractal dimension Dch of a chain part between its ﬁxation points characterises the molecular mobility level of a polymer. In paper [94] the interconnection of molecular mobility and local order was studied in the example of two series of epoxy polymers (native EP-1 and EP-3aged for 3 years in natural conditions). The dependence Dch( ) adduced in Figure 6.37 shows that increasing νcl or raising of the local order level results in a reduction in Dch and, hence, to chain mobility suppression. Any chain mobility reduction, independent of the reasons for it, results in polymer brittleness [50] that is always observed at physical ageing of these materials. The comparison of the dependences Dch( ) and νcl(Kst), adduced in Figure 6.38, allows to make the conclusion that in the given case this process is due to polymer ageing [94].

329

Structure and Properties of Crosslinked Polymers

ɌgɄ 410 390 370 -1 -2

350 330 1

2

3 4 νcl×10−27, m−3

Figure 6.36 The dependence of the glass transition temperature Tg on the cluster network density νcl for native (1) and aged (2) samples of EP-2 [93]

Dch 1.8 -1 -2

1.6 1.4 1.2 1.0 0.5

0.6

0.7

0.8

0.9 −1 / 3 ×109, m ν cl

Figure 6.37 The dependence of the fractal dimension Dch of a chain part between crosslinking nodes on parameter for EP-1 (1) and EP-3 (2) [94]

330

The Properties of Crosslinked Epoxy Polymers As has been noted in Chapter 1, the glass transition interval initial temperature ' corresponds to the loosely packed matrix glass transition temperature Tg and a ﬁnal one to the polymer glass transition temperature Tg [95]. In addition the glass ' transition interval width ΔTg = Tg – Tg is deﬁned by the character of segments in the amount of distribution of the cluster nseg (see Figure 6.10). Proceeding from this and based on the data adduced in Figure 6.39, it can be afﬁrmed that ageing of epoxy polymers contributes to raising of the homogeneity of the cluster nanostructure [94].

νcl × 10−27, m−3 4 3 2

2

1

1

0 0.50 0.75 1.00 1.25 1.50 Ʉst

Figure 6.38 The dependences of the macromolecular entanglement cluster network density νcl on the curing agent : oligomer ratio Kst for epoxy polymers EP-1 (1) and EP-3 (2) [94]

Thus, the physical ageing process of glassy epoxy polymers presents itself as a change in the structure and properties with time and is a consequence of the thermodynamically non-equilibrium nature of the indicated polymers. At sufﬁciently low temperatures (of the order of room temperature) this process occurs sufﬁciently slowly and changes in the essential properties are realised during a period of the order of several years [93, 94]. Therefore prediction of such phenomena should be fulﬁlled by accounting for both spatial and temporal disorder of the systems undergoing ageing [36]. The spatial heterogeneity of the structure of glassy epoxy polymers results in a sufﬁciently wide range of rates of microscopic relaxation transitions, i.e., spatial disorder results in a temporal one [96].

331

Structure and Properties of Crosslinked Polymers

ΔɌgɄ 60

40

20 -1 -2

0 0.50 0.75 1.00 1.25 1.50 Ʉst '

Figure 6.39 The dependence of the glass transition interval width ΔTg = Tg – Tg on the curing agent : oligomer ratio Kst for epoxy polymers EP-1 (1) and EP-3 (2) [94]

Let us note the following in respect to description of the physical ageing process within the frameworks of fractal analysis. The study of the dynamics of reactions in disordered systems showed [96] that allowance for both types of disorder was necessary. The majority of the subordination takes place over large durations for the reactions, expressed by the common relationship [96]:

(6.35) where Φ(t) is an arbitrary function, dependent on the reaction duration t, α and γ are exponents accounting for spatial and temporal disorder availability, respectively. In respect to the physical ageing process the change of structure with time can be described, proceeding from the common Relationship 6.35, as follows [97]:

(6.36)

where νcl and are the network densities of macromolecular entanglements cluster of aged and native samples, respectively, c is the proportionality coefﬁcient accounting

332

The Properties of Crosslinked Epoxy Polymers for replacement of the proportionality sign in Relationship 6.35 by the equality sign in Equation 6.36. It is necessary to deﬁne the physical signiﬁcance of exponents α and γ in Equation 6.36. The exponent α, characterising the spatial disorder degree, can be deﬁned within the frameworks of fractal analysis as the magnitude (df – 2), where df is the fractal dimension of the structure of the initial epoxy polymer (EP-1), according to the following opinions. The change of df within the limits 2–3 characterises the degree of disorder of the polymer structure and the main part of information about this degree the fractional part df performs – the larger the df the lower the local order level and, hence, the higher the system spatial disordering degree [82]. In essence, the whole-number part of df gives information about the dimension of Euclidean space in which a fractal object is immersed. Let us note that the exponent α = df – 2 coincides with the fractional exponent in the theory of fractional differentiation and integration [98], where this exponent characterises the fraction of the system states, without undergoing the changes in evolution process [99]. In other words, within the frameworks of fractional integro-differentiation it is supposed that the ordered part of the polymer structure will be changed in the physical ageing process that Equation 6.36 reﬂects. As has been shown in Chapter 5, for polymers the quasi-equilibrium state exists which the polymer structure tries to attain with growth of its thermodynamic equilibrium degree, characterised by an approximate condition df = 2.5. This condition corresponds to a fully relaxed polymer with a narrow distribution of sizes of free volume microvoids [100] and, hence, to the least temporal disorder, characterised by the spectrum width relaxation times (see Section 5.5). Therefore the following approximation will be used further [97]:

(6.37)

Thus, Equation 6.36 can ﬁnally be rewritten in the form [97]:

(6.38)

where the empirically determined constant c is equal to ~ 0.14 and the ageing duration t is given in seconds.

333

Structure and Properties of Crosslinked Polymers Figure 6.40 shows the dependences of ϕcl calculated according to Equations 6.38 and 1.11 for native (EP-1) and aged (EP-3) epoxy polymers, on curing the agent : oligomer ratio Kst and also obtained experimentally. As follows from the data of Figure 6.40, good correspondence of theory and experiment was received. Even at Kst = 0.50 and 1.50, where the discrepancy is the greatest, it does not exceed ~ 25%. The authors [97] concluded that the indicated discrepancy was connected with the calculation mode of Poisson’s ratio ν and, hence, df (Equation 5.4), namely using for this purpose Formula 5.5, and the strong dependence of νcl on df in Equation 6.38. It is known [40] that error in the estimation of ν according to Formula 5.5 for epoxy polymers increases with growth in the testing temperature and at high T, corresponding to large values of ν, the indicated mode can be given magnitudes of ν ~ 7% larger than determined by other methods. Since the highest values of ν (0.364 and 0.370, respectively) were observed at Kst = 0.50 and 1.50, then for these values of ν the greatest error should be expected. A decrease in both indicated values of ν up to 0.360 (i.e., on 1–3% only) and recalculation of ϕcl according to Equations 5.4, 1.11 and 6.38 give magnitudes much closer to the experimental values of ϕcl (a dashed line in Figure 6.40).

ϕcl

0.8

-2 3

0.6

0.4

4

1

0.2 0.50 0.75 1.00 1.25 1.50 Ʉst Figure 6.40 The dependences of the relative fraction ϕcl of clusters on the curing agent : oligomer ratio value Kst for EP-1 (1) and EP-3 (2-4). 1, 2 – the experimental data; 3 – calculation according to Equations 6.38 and 1.11; 4 – calculation according to the same equations with correction of values of ν for EP-3 at Kst = 0.50 and 1.50 (see text) [97]

334

The Properties of Crosslinked Epoxy Polymers The second important aspect of the physical ageing process also follows from the data of Figure 6.40 so, for different Kst (and, hence, for different crosslinking densities νc [5, 6]) the ageing process intensity, expressed by the difference of values of ϕcl for EP-3 and EP-1, differs essentially. As has been expected [28], the smallest ageing intensity is observed for the system with stoichiometric value Kst = 1.0 and deviation Kst from this magnitude results in growth in the intensity of the physical ageing process [94]. From Equation 6.38 it follows that the observed effect is due to growth of both α and γ for systems with the greatest deviations from Kst = 1.0 or, in other words, with an increase in the structure fractal dimension df [97]. As has been shown in Chapter 5, for epoxy polymers Relationships 5.54, 5.65 and 5.67 are true, which assume a change not only in the suprasegmental structure characteristics (df and Df) during the physical ageing process, but also the molecular parameter – chain statistical ﬂexibility, characterised by C∞. Finally, the physical ageing process results in levelling of the structure of epoxy polymers, irrespective of the value of νc, which varies for them by about an order of magnitude [5, 6, 28]. It is natural that such structure levelling deﬁnes the essential resemblance of properties for EP-3. So, the elasticity modulus E in compression tests changes for EP-1 within the limits of 2.50–4.20 GPa and only 1.60–1.90 GPa for EP-3 [94]. Thus, methods offered by the authors [97] allow calculations of the changes in the structure of epoxy polymers in their physical ageing process using the cluster model and fractal analysis ideas. It is possible that in Equation 6.38 the coefﬁcient c will be the ageing temperature function and in principle an analytical relationship can be obtained for its determination. It is even more possible that the increase in ν (and, then, df) with temperature raising will account for the increase in the ageing process intensity. These assumptions require experimental veriﬁcation. It is important to note that in the physical ageing process structural reformations affect both molecular and suprasegmental levels [28]. For characterisation of structure in the quasi-equilibrium state the approximate condition df = 2.5 was accepted above (see Equations 6.37 and 6.38). The authors of papers [101–103] carried out a more precise deﬁnition of conditions at which the quasi-equilibrium state was reached and they compared estimations of the different modes of those conditions. The degree of energetic excitation of polymer structure or the degree of its removal from thermodynamic equilibrium can be characterised with the aid of the extra energy localisation regions dimension Df [34]. A number of methods exist which allow determination of the dimension Df. So, in paper [100] Equation 5.72 was proposed for this purpose, where the value of fg was accepted as being constant and equal to 0.055 [1]. Another method of calculation of ϕcl uses a thermal cluster idea,

335

Structure and Properties of Crosslinked Polymers namely Equation 5.49. The value of ϕcl can be calculated according to the following equation, similar to Formula 6.38 [101]:

(6.39)

where c is a constant, equal as earlier to ~ 0.14 if the ageing duration is given in seconds, and ϕcl are the relative fractions of clusters in native and aged states, respectively. Unlike the calculation according to Equations 1.11 and 6.38 fulﬁlled above, in papers [101–103] the value of df determined for epoxy polymers EP-1 according to Equation 5.4, not its approximate value, was used. The limiting (lower) df magnitude can be calculated according to Equation 5.31 using for the calculation the value of C∞ determined according to Equation 5.2. If the value of df is known, then the value of ϕcl can be determined according to Equation 4.8. Since the dimensions of Df and df are connected with one another by Relationship 5.73, then, having determined Df according to Equation 5.72, the value of ϕcl corresponding to this method can be calculated with the aid of Equations 5.73, 5.2 and 4.8. In Figure 6.41 the dependences of the order parameter ϕcl on the value of Kst calculated according to Equations 6.39, 5.72, 5.49 are shown and the experimental values of ϕcl obtained according to Equations 5.4 and 5.31, for native and aged epoxy polymers (EP-1 and EP-3, respectively), are adduced. As one can see, all the indicated methods give agreed results with the greatest discrepancy less than 13%. There are two distinctions of the methods applied in papers [101–103] and the approximate mode used in paper [97]. Firstly, its value Df of quasi-equilibrium state and, respectively, ϕcl corresponds to each crosslinking density νc (or to each value Kst) – the smaller the νc the higher the Df and the smaller the ϕcl. In paper [97] for all epoxy polymers Df = 3 was accepted is smaller for the quasi-equilibrium state. Secondly, the distinction between ϕcl and for more precise methods than for the approximate one [97]. As has been noted above, the quasi-equilibrium state supposes tautness of chains between clusters that hinders further growth in ϕcl and makes it impossible to reach the thermodynamically equilibrium structure with criterion ϕcl = 1.0 [88]. The molecular mobility of chains can be characterised by the value of the fractal dimension Dch of a chain part between clusters (1 < Dch ≤ 2) [28, 34]. At Dch = 1 the chain is stretched fully between its ﬁxation points and its mobility is suppressed. At Dch = 2 the molecular mobility of the chain is the greatest and corresponds to a rubber-like state. The value of Dch can be calculated with the aid of Equation 5.53.

336

The Properties of Crosslinked Epoxy Polymers ϕcl 0.8 0.6 0.4

-1 -2 -3

0.2

-4

-5 0 0.50

0.75

1.00

1.25

1.50 Ʉst

Figure 6.41 The dependences of the relative fraction ϕcl of clusters on the value of Kst for epoxy polymers EP-1 (1) and EP-3 (2-5). 1, 2 – the experimental data; 3, 4 and 5 – calculation according to Equations 6.39, 5.72 and 5.49, respectively [101] The dependences Dch(Kst) for EP-1 and EP-3 are adduced in Figure 6.42 (the values of Dch were calculated according to Equations 6.39 and 5.53). As one can see, a sharp reduction in Dch is observed in the physical ageing process and for epoxy polymer EP-3 the values of this dimension are close to the limiting (and expected) magnitude Dch = 1 (shown in Figure 6.42 by a dashed line). Thus, the data of Figure 6.42 conﬁrm the quasi-equilibrium state concept, due to tautness of the chains, the condition of which is the criterion Dch = 1.0 [88].

Dch 1.4 -1 -2

1.3 1.2 1.1 1.0 0.50 0.75

1.00

1.25 1.50 Ʉst

Figure 6.42 The dependences of the fractal dimension Dch of the chain part between clusters on the value of Kst for epoxy polymers EP-1 (1) and EP-3 (2) [101]

337

Structure and Properties of Crosslinked Polymers Further, the authors [101] considered the error introduced by the approximation df = 2.5 for the quasi-equilibrium state. In Figure 6.43 the comparison of the values of df received experimentally (Equation 5.4) and those calculated according to Equations 5.72, 5.49, 4.8 and 5.73 is adduced. As one can see, between the values of df, received experimentally, and df = 2.5 (horizontal dashed line) the discrepancy exists, which can reach ~ 7%. Although this value is small, power dependence in Equations 6.38 and 6.39 can give an essential error at the estimation of values of νcl or ϕcl (see Figure 6.40). The discrepancy between experimental and theoretical values of df is considerably smaller and does not exceed ~ 3%. Therefore, although the approximation df = 2.5 for the quasi-equilibrium state can be used as the ﬁrst approximation, as was shown in paper [97], application of the value of df calculated according to one of the methods described above will give a more precise result [102].

df 2.8 -2 -3

1

2.7

2.6

4

2.5 0.50

0.75 1.00 1.25 1.50 Ʉst

Figure 6.43 The dependences of the fractal dimension df for structure of the quasiequilibrium state on the value of Kst for epoxy polymer EP-3. 1 – the experimental data (Equation 5.4); 2 – calculation according to Equations 5.72, 5.73; 3 – calculation according to Equations 5.49, 4.8; 4 – approximation df = 2.5 [101]

Let us note in conclusion that Equation 5.53 can also be one of the estimation methods for the value of ϕcl for the quasi-equilibrium state. In this case the fulﬁlment of the condition Dch = 1 is required and then Equation 5.53 will have the following form [101]:

338

The Properties of Crosslinked Epoxy Polymers

(6.40)

Thus, the results stated above have shown that a change in the structure of epoxy polymers in the physical ageing process is restricted by reaching the quasi-equilibrium state, which is characterised by the balance between raising of the local order level (tendency to thermodynamically equilibrium structure) and entropic tautness of the chains. This approach allows to predict the change in the structure of polymers (and, hence, their properties) as a function of their initial structure, duration and temperature of physical ageing [101–103].

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The Properties of Crosslinked Epoxy Polymers 59. G.M. Magomedov, G.M. Bartenev and Y.V. Zelenev in Internal Friction in Metals and Inorganic Materials, Nauka, Moscow, Russia, 1982, p.197. 60. R.J. Morgan, Journal of Applied Polymer Science, 1979, 23, 8, 2711. 61. W.E. Wolstenholme, Journal of Applied Polymer Science, 1962, 5, 2, 332. 62. S.L. Kim, M.D. Skibo, J.A. Manson, K.W. Hertzberg and J. Janiezewski, Polymer Science Engineering, 1978, 18, 14, 1093. 63. M.Y. Katsnel’son and G.A. Balaev in Polymeric Materials, Khimiya, Leningrad, Russia, 1982, p.317. 64. H.R. Brown, Journal of Materials Science, 1973, 8, 7, 941. 65. V.N. Belousov, G.V. Kozlov and A.K. Mikitaev, Vysokomolekulyarnye Soedineniya Seriya B, 1984, 26, 8, 563. 66. V.N. Belousov, G.V. Kozlov and A.K. Mikitaev, Doklady Akademii Nauk SSSR, 1983, 270, 5, 1120. 67. A.K. Mikitaev, L.D. Mil’man and G.V. Kozlov, Vysokomolekulyarnye Soedineniya Seriya A, 1987, 29, 2, 383. 68. T. Kunori and P.H. Geil, Journal of Macromolecular Science: Physics, 1980, B18, 1, 135. 69. R.J. Young in Development Polymer Fracture-1, Applied Science Publishers Ltd, London, UK, 1979, p.183. 70. C.B. Bucknall in Toughened Plastics, Applied Science, London, UK, 1981, p.328. 71. J. Mijovic and J.A. Koutsky, Polymer, 1979, 20, 9, 1095. 72. K. Matsushige, S.V. Radcliffe and E. Baer, Journal of Applied Polymer Science, 1976, 20, 7, 1853. 73. V.R. Regel and A.I. Slutsker in Physics Today and Tomorrow, Nauka, Leningrad, Russia, 1973, p.90. 74. N.P. Novikov, A.A. Kholodilov, N.F. Chernyavskii and V.A. Kargin, Doklady Akademii Nauk SSSR, 1968, 183, 6, 1375.

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Structure and Properties of Crosslinked Polymers 75. Yu.K. Godovskii in Thermophysics of Polymers, Khimiya, Moscow, Russia, 1982, p.280. 76. A.I. Slutsker and V.E. Filippov, Vysokomolekulyarnye Soedineniya Seriya A, 1988, 30, 11, 2386. 77. A.I. Burya, G.V. Kozlov, A.I. Sviridenok and G.E. Zaikov, Doklady NAN Belarusi, 2003, 47, 6, 113. 78. V.D. Serdyuk, G.V. Kozlov, N.I. Mashukov and A.K. Mikitaev, Journal of Materials Science and Technology, 1997, 5, 2, 55. 79. G.V. Kozlov, V.A. Beloshenko, V.N. Varyukhin and Y.S. Lipatov, Polymer, 1999, 40, 4, 1045. 80. G.V. Kozlov and D.S. Sanditov in Anharmonic Effects and PhysicalMechanical Properties of Polymers, Nauka, Novosibirsk, Russia, 1994, p.261. 81. B.K. Sharma, Acoustic Letters, 1984, 8, 1, 11. 82. G.V. Kozlov, V.A. Beloshenko and V.N. Varyukhin, Ukrainskii Fizicheskii Zhurnal, 1996, 41, 2, 218. 83. C.G. Delides and A.C. Stergiou, Polymer Communications, 1985, 26, 6, 168. 84. B. Wunderlich, Journal of Chemical Physics, 1960, 64, 8, 1052. 85. R.F. Boyer, Journal of Macromolecular Science: Physics, 1973, B7, 3, 487. 86. B.D. Sanditov, G.V. Kozlov, V.U. Novikov and V.V. Mantatov in the Proceedings of 2nd Baikal School by Principal Physics ‘Interaction of Radiations and Fields with Substance’, ISU, Irkutsk, Russia, 1999, p.346. 87. G.V. Kozlov, D.S. Sanditov, S.S. Sangadiev and A.B. Bainova in the Proceedings of All-Russian Science Conference ‘Mathematical Simulation in Synergetic Systems’, Ulan-Ude, Russia, 1999, p.248. 88. G.V. Kozlov and G.E. Zaikov in Fractals and Local Order in Polymeric Materials, Eds., G.V. Kozlov and G.E. Zaikov, Nova Science Publishers, Inc., New York, NY, USA, 2001, p.55. 89. S.E.B. Petrie, Journal of Macromolecular Science: Physics, 1976, B12, 2, 225.

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The Properties of Crosslinked Epoxy Polymers 90. T.-D. Chang and J.O. Brittain, Polymer Engineering and Science, 1982, 22, 18, 1221. 91. V.I. Irzhak, B.A. Rosenberg and N.S. Enikolopyan in Crosslinked Polymers: Synthesis, Structure, Properties, Nauka, Moscow, Russia, 1979, p.248. 92. D.C. Timm, A.J. Ayorinde and R.F. Foral, British Polymer Journal, 1985, 17, 2, 227. 93. G.V. Kozlov, V.A. Beloshenko, I.V. Stroganov and Y.S. Lipatov, Doklady NAN Ukraine, 1995, 10, 117. 94. G.V. Kozlov, V.A. Beloshenko, M.A. Gazaev and Y.S. Lipatov, Vysokomolekulyarnye Soedineniya Seriya B, 1996, 38, 8, 1423. 95. V.N. Belousov, B.K. Kotsev and A.K. Mikitaev, Doklady Akademii Nauk SSSR, 1983, 270, 5, 1145. 96. A. Blumen, J. Klafter and G. Zumofen in Fractals in Physics, Eds., L. Pietronero and E. Tosatti, Amsterdam, Oxford, UK, New York, NY, USA, Tokyo, Japan, North-Holland, 1986, p.561. 97. G.V. Kozlov, V.A. Beloshenko and Y.S. Lipatov, Ukrainskii Khimicheskii Zhurnal, 1998, 64, 3, 56. 98. G.V. Kozlov, G.B. Shustov and G.E. Zaikov, Zhurnal Prikladnoi Khimii, 2002, 75, 3, 485. 99. R.R. Nigmatullin, Teoreticheskaya i Mathematicheskaya Fizika, 1992, 90, 3, 354. 100. G.V. Kozlov and G.E. Zaikov in Fractal Analysis of Polymers: From Synthesis to Composites, Eds., G.V. Kozlov, G.E. Zaikov and V.U. Novikov, Nova Science Publishers, Inc., New York, NY, USA, 2003, p.141. 101. G.V. Kozlov and G.E. Zaikov, Materialovedenie, 2002, 12, 13. 102. G.V. Kozlov and I.V. Dolbin in the Proceedings of International Conference, ‘Physics of Electronic Materials’, Kaluga, Russia, 2002, p.134. 103. G.V. Kozlov and G.E. Zaikov in Studies in Chemistry and Biochemistry, Eds., G.V. Zaikov and V.M.M. Lobo, Nova Science Publishers, Inc., New York, NY, USA, 2003, p.206.

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7

Nanocomposites on the Basis of Crosslinked Polymers

In the present chapter the structure and properties of the current most popular polymer/organoclay nanocomposites will be considered. By virtue of the present monograph the speciﬁc character of nanocomposites, in which crosslinked polymers serve as the matrix, will mainly be considered and nanocomposites on the basis of linear polymers will be used for comparison purposes.

7.1 The Formation of the Structure of Polymer/Organoclay Nanocomposites Unlike many mineral ﬁllers used in plastics processing (talc, mica and so on), organoclays, particularly montmorillonite, are capable of stratifying and dispersing as separate platelets, having a thickness of ~ 1 nm [1]. Montmorillonite platelets packets that do not stratify after introduction into the polymer are often tactoids (Figure 7.1). The term ‘intercalation’ describes the case when small amounts of the polymer penetrate into the galleries between silicate platelets, which causes stratiﬁcation of these platelets of the value of 2–3 nm. Exfoliation (stratiﬁcation) occurs at the distance between platelets (in X-rays this distance is called the interval d001) of the order of 8–10 nm. A well-stratiﬁed and dispersed nanocomposite consists of separate organoclay platelets, uniformly distributed in the polymer matrix. The indicated cases are shown schematically in Figure 7.1 and electron micrographs of the intercalated and exfoliated Na+-montmorillonite are adduced in Figure 7.2. Let us consider the main aspects of reinforcing of polymer/organoclay nanocomposites. As for all multiphase systems, the level of interfacial adhesion between the polymer matrix and the nanoﬁller is a crucial factor in the degree of reinforcement [2, 3]. In paper [4] it has been shown that good adhesion results in reinforcement of composites, and poor adhesion in the absence of reinforcing and the absence of interfacial adhesion weakens the polymer composite, i.e., the elasticity modulus for the composite is lower than the corresponding parameter for a matrix polymer. In the general case such behaviour is connected with stress transfer conditions in the interfacial boundary. For allowance of this factor an additional aspect appears for nanocomposites: interfacial layer formation in the polymer-nanoﬁller boundary.

347

Structure and Properties of Crosslinked Polymers

+ Clay

Polymer

Blending in melt

Tactoid

Intercalated

Intercalated disordered

Exfoliated

Figure 7.1 A schematic illustration of the terminology applied in the description of polymer/organoclay nanocomposites [1]

a

c

100 nm

b

d

Figure 7.2 Electron micrographs of nanocomposite poly(butylene terephthalate)/ montmorillonite, illustrating exfoliated (a) and intercalated (b, c, d) structure. Enlargement 30000× [8]

The polymer region adjoining the ﬁller surface and having a structure different from the bulk polymer matrix structure is called an interfacial layer. The polymer structure

348

Nanocomposites on the Basis of Crosslinked Polymers change in the interfacial layer supposes a conformation change in the molecular chains on forming it, which for the ﬂat silicate particles is expressed in polymer chain ‘stretching’ on their surface that results in a more dense packing of interfacial regions in comparison with packing in the bulk polymer matrix [5]. This supposition is conﬁrmed by computer simulation results (Figure 7.3) [6].

( )

(b)

(c)

Figure 7.3 The sequence of interfacial layer formation on the polymermontmorrilonite boundary, obtained according to computer simulation at times: (a) 0 ns; (b) 100 ns and (c) 250 ns. Denser packing is seen at the distance approximately equal to the thickness of a platelet of montmorillonite, i.e., ~ 1 nm [6]

However, it is obvious that similar ‘stretching’ of chains can be realised only at the expense of strong interactions of the surface-polymer nanoﬁller, i.e., at the expense of a sufﬁciently high level of adhesion on the interfacial boundary [7]. The absence of adhesion results in the fact that the polymer chain structure in the bulk polymer matrix and on the interfacial boundary will be the same, i.e., in this case it does not make sense to consider interfacial layer formation. Therefore the complexity of the structure of polymer composites (including nanocomposites) is deﬁned by a sufﬁciently large number of factors inﬂuencing their degree of reinforcement, which can be divided conditionally into three groups: the common parameters, i.e., an adhesion level, and structural parameters of each phase. To the last in the ﬁller case one should attribute the form (anisotropy degree or sides ratio) of its particles, surface structure of the particles and their aggregation degree, and for the polymer matrix the main parameter of any polymer, which is its chain rigidity [9].

349

Structure and Properties of Crosslinked Polymers At present in the literature there are enough data to allow general conclusions to be made about the inﬂuence of the mentioned factors on the degree of reinforcement of polymer/organoclay nanocomposites. It is supposed that one of the most important structural characteristics of nanocomposites is the distance between silicate platelets or the interval d001, determined by X-rays [10]. In addition it is assumed that an increase in d001 is a positive factor and as an ideal variant the fully exfoliated nanocomposite is supposed [11]. On the contrary, other authors (see, for example, [12]) assume that the optimal structure of polymer/organoclay nanocomposites is an intercalated one. Realisation of high values of d001 in itself is difﬁcult, particularly at large contents of organoclay.

d001, Å 5% 1%

200 150

10% 15%

100

20% 50 0 0

40

80

120

160

200

240 t, min

Figure 7.4 The dependences of the interlayer spacing d001 of montmorillonite in an epoxy matrix on the curing reaction duration t. Montmorillonite contents mass percentage: 1–1; 2–5; 4–15; 5–20 [11]

In Figure 7.4 the dependence of d001 on the curing reaction duration for nanocomposites on the basis of an epoxy polymer is adduced. One can see that very fast growth in d001 for small montmorillonite contents is observed and it is much slower for the larger ones. For montmorillonite contents of 15 and 20 mass percentage the tendency of achievement of a plateau with d001 value consisting of about 250% from the initial value by this parameter is clearly seen. The kinetics of intragallery material formation for epoxy polymer/montmorillonite nanocomposites was studied in paper [13]. In Figure 7.5 the dependences of d001 on the curing duration t at three different curing temperatures Tcur are adduced. As one can see, the achievement on the plateau of the

350

Nanocomposites on the Basis of Crosslinked Polymers values of d001 is also observed here, which is realised faster the higher Tcur is. Let us note that in this case montmorillonite plays a catalyst role, essentially accelerating (by several times) the epoxy polymer curing process. Similar observations were made in a case of synthesis of linear polymers [14, 15]. For the conclusion of the realisation of either structure for nanocomposites, one can regard the general conclusion of papers [11, 13] on the basis of crosslinked epoxy polymers: if the curing process within montmorillonite galleries proceeds faster, then the result is an exfoliated nanocomposite and if the reaction proceeds faster outside the ﬁller, then an intercalated nanocomposite is produced.

d001, Å 75 65 55 -1 -2 -3

45 35 0

100

200

300

400

t, min

Figure 7.5 The dependences of the interlayer spacing d001 of montmorillonite in an epoxy matrix on the curing reaction duration t at curing temperatures 343 (1), 353 (2) and 373 K (3) at montmorillonite contents of 10 mass percentage [13]

For computer simulation results of linear polymers it is supposed [16] that the value of d001 increases with molecular mass combined with raising of the agent polarity. It has been shown simultaneously [17] that an excess of this agent can result in a decrease in d001 or collapse of silicate layers owing to intragallery material thermal degradation. Attempts were made to obtain the direct correlation of d001 and nanocomposites reinforcement degree E n/E m (where E n and E m are elasticity moduli of the nanocomposite and the matrix polymer, respectively). The authors of paper [17] received an approximately linear increase in the elasticity modulus with raising d001 for intercalated polycarbonate/montmorillonite nanocomposites. However, for the

351

Structure and Properties of Crosslinked Polymers other nanocomposites opposite in signiﬁcance correlations [13, 17] can be obtained. In connection with this the authors [9] plotted the dependence of the reinforcement degree En/Em for four polymer/montmorillonite nanocomposites on the basis of an epoxy polymer [13], poly(ethylene terephthalate) [20], poly(lactic acid) [18] and polyamide-6 [19] on ϕf, from which the ﬁrst nanocomposite had an exfoliated structure and the three remaining had an intercalated structure, for two nanoﬁller concentrations (2 and 5 mass percentage), which was shown in Figure 7.6. As one can see, all these data are approximated by one straight line that shows the absence of correlation between the interlayer spacing value d001 and the reinforcement degree for nanocomposites. Therefore the question about the optimal type of nanocomposite structure or narrower optimal value of d001 remains open.

En/Em 1,4

1,3

1,2 -1 -2 -3 -4

1,1

1,0

0

0,01

0,02

0,03

0,04

0,05 ϕf

Figure 7.6 The dependence of the reinforcement degree En/Em on montmorillonite contents ϕf for nanocomposites on the basis of epoxy polymer (1), poly(ethylene terephthalate) (2), poly(lactic acid) (3) and polyamide-6 (4) [9]

7.2 The Reinforcement Mechanisms of Polymer/Organoclay Nanocomposites As has been noted in the previous section, the distance (interval) between layered silicate platelets d001, characterising its stratiﬁcation degree, is considered as one of 352

Nanocomposites on the Basis of Crosslinked Polymers the most important characteristics of the structure of intercalated nanocomposites [21]. Since nanocomposites are different from polymer microcomposites, having ﬁller particles with sizes of the order of several tens of micrometres, by polymer-ﬁller interfacial interaction level, then in structural meaning the main attention must be paid to the degree of adhesion of the polymer matrix-nanoﬁller and formation of interfacial layers. Therefore the authors of paper [22] studied the reinforcement mechanism of polymer nanocomposites, ﬁlled by layered silicate, from this point of view. Proceeding from the description of intercalated nanocomposites adduced in Section 7.1, it can be supposed that the polymeric material between Na+-montmorillonite platelets is interfacial regions. In this case the interfacial layer thickness lif is determined as follows [23]:

(7.1)

where dpl is the silicate platelet thickness. In turn, the value of lif and relative fraction ϕif of interfacial regions are connected with one another by the following relationship [24]:

(7.2)

where ϕf is the volume contents of the nanoﬁller, L and B are the length and width of the silicate platelet, respectively. The combination of Equations 7.1 and 7.2 allows the following formula to be obtained [22]:

(7.3)

As a rule, for nanocomposites ﬁlled by Na+-montmorillonite, the last contents is given in mass percentage (Wf), therefore certain difﬁculties appear with estimation of the

353

Structure and Properties of Crosslinked Polymers nanoﬁller volume fraction ϕf. These difﬁculties are connected with the notion of the ‘effective particle’ of layered silicate [19]. In the general case the interconnection of ϕf and the silicate mass fraction Wf within the frameworks of model [19] is given by the ratio (ϕf/Wf), which can be varied within wide limits – from 0.25 to 1.80 (see Figure 13 in paper [19]). Proceeding from the intercalation level of the nanocomposites considered in paper [22], characterised by the value d001 (d001 = 1.1–3.2 nm for polycarbonate/Na+-montmorillonite nanocomposites and 6.3–13.0 nm for epoxy polymer/Na+-montmorillonite) ϕf/Wf = 1.0 has been chosen for the ﬁrst and 0.5 for the second from the indicated nanocomposites. According to the data of paper [19] it was accepted that dpl = 1 nm. The degree of reinforcement of nanocomposites En/Em can be determined theoretically according to the following percolation relationship [25]:

(7.4)

Comparison of the ϕif results calculated according to Equation 7.3 at the conditions indicated above and also the theoretical reinforcement degree (En/Em)T (Equation 7.4) with this parameter the experimental magnitude (En/Em)e (Figure 7.7) has shown that for polycarbonate/Na+-montmorillonite nanocomposites good correspondence of theory and experiment is obtained, whereas for epoxy polymer/Na+-montmorillonite nanocomposites the calculation gives overstated values of the reinforcement degree. This allows it to be supposed that polycarbonate/Na+-montmorillonite nanocomposites belong to intercalated nanocomposites and epoxy polymer/Na+-montmorillonite nanocomposites to exfoliated ones. From the above it follows that the criterion for intercalated nanocomposite will be the following inequality [22]:

(7.5)

At present the necessity of d001 increasing is supposed under any circumstances. However, Inequality 7.5 gives the upper limit of the intercalation of the necessary level of layered silicates. So, d001 for epoxy polymer/Na+-montmorillonite nanocomposites is on average four times larger than for polycarbonate/Na +-montmorillonite. Nevertheless the degree of reinforcement for these nanocomposites is comparable at Na+-montmorillonite contents in the ﬁrst as it is three times larger than in the second.

354

Nanocomposites on the Basis of Crosslinked Polymers This indicates the upper limit of both d001 and ϕf, which are necessary for receiving the optimal degree of reinforcement. The stated treatment also explains the absence of a jump in the reinforcement degree at transition from intercalated nanocomposites to exfoliated ones, shown in paper [19].

(En/Em)T 3,0 × 2,5

× ×

2,0

×

× × ×

×

× 1,5

1,0

× -1 -2 -3

× ×

1,2

1,4

1,6

1,8 (En/Em)e

Figure 7.7 The comparison of experimental (En/Em)e and theoretical (En/Em)T degree of reinforcement of epoxy polymer/Na+-montmorillonite nanocomposites (1, 2) and polycarbonate/Na+-montmorillonite nanocomposites (3) with glass matrix. 1, 3 – calculation according to Equations 7.3 and 7.4; 2 – calculation according to Equations 7.3, 7.4 and 7.6 [22]

Taking into account the discrepancy in theoretical and experimental data for epoxy polymer/Na+-montmorillonite nanocomposites, it can be supposed that the interfacial layer thickness for them is smaller than 0.5 (d001 – dpl), i.e., the interfacial layer does not ﬁll the entire space between silicate platelets. The value of lif can be estimated according to the following fractal relationship [26]:

(7.6)

355

Structure and Properties of Crosslinked Polymers where a is the lower linear scale of fractal behaviour, which for polymers is accepted to be equal to the statistical segment length lst [27], Dp is the nanoﬁller particle size, which for organoclay is accepted to be equal to the arithmetic mean of the length, width and thickness of the silicate platelet [28], d is the dimension of Euclidean space in which a fractal is considered (it is obvious that in our case d = 3), dsurf is the dimension of the organoclay surface, which is equal to 2.78 [29]. As the calculations according to Equation 7.6 have shown, for an epoxy polymer/ Na+-montmorillonite nanocomposite the interfacial layer thickness, corresponding to the condition 2lif < (d001 – dpl), was obtained [22]. Further, using Equations 7.3 and 7.4, the values of (En/Em)T can be calculated again and compared with the corresponding experimental parameters. In this case good correspondence of theory and experiment is obtained. Thus, a relatively small degree of reinforcement of epoxy polymer/Na+-montmorillonite nanocomposites at relatively high values of Wf and d001, essentially exceeding these parameters for the polycarbonate/Na+-montmorillonite nanocomposite, is due to the fact that the interfacial regions (with structure differing from the structure of the bulk polymeric matrix) ﬁlls only part of the intragallery space [22]. It is known [13] that the degree of reinforcement of nanocomposites with an elastomeric matrix at other equal conditions essentially exceeds the indicated parameter for nanocomposites with a glassy matrix. The comparison of the results of the theoretical and experimental determination of the ratio En/Em for epoxy polymer/ Na+-montmorillonite nanocomposites with elastomeric matrix are presented in Figure 7.8. The tests for En and Em determination were carried out at Tg + 40 K [13] and (En/Em)T calculation was fulﬁlled according to Equations 7.3 and 7.4 in supposition of 2lif = d001 – dpl. As follows from the data of Figure 7.8, in this case sufﬁcient correspondence is obtained between theory and experiment, at any rate, by order of magnitude. The fraction of larger interfacial regions in the rubber-like state is deﬁned according to Equation 7.6 by larger chain ﬂexibility in comparison with the glassy state or higher values of C∞ [27]. Taking into account the indicated correspondence, epoxy polymer/Na+-montmorillonite nanocomposites with an elastomeric matrix can be classiﬁed as the intercalated type (which is why they have a higher degree of reinforcement). Thus, attribution of nanocomposites to one of the classes mentioned above is deﬁned not only by the layered nanoﬁller state (in particular by the value d001), but also by the characteristics (including molecular) of the polymeric matrix [22]. For conﬁrmation of the treatment offered above the microscopic structure and properties of the interfacial layer in a Na+-montmorillonite-polymer system and the interaction microscopic mechanisms in this system were studied in a direct computer experiment. The semi-empirical (Hartree-Fock) quantum-mechanical (QM) method with RMZ parameterisation, realised in the original package CLUSTER-1, was applied

356

Nanocomposites on the Basis of Crosslinked Polymers within the frameworks of the cluster approach. This method is the most suitable for simulation of sufﬁciently large inorganic-organic molecular systems [22].

(En/Em)T 4

3

2

1

2

3

4

(En/Em)e

Figure 7.8 The comparison of experimental (En/Em)e and theoretical (En/Em)T degree of reinforcement of epoxy polymer/Na+-montmorillonite nanocomposites with an elastomeric matrix. The calculation of (En/Em)T is carried out according to Equations 7.3 and 7.4 [22]

The results of computer simulation of a system consisting of two Na+-montmorillonite platelets and a crosslinked epoxy polymer between them is presented in Figure 7.9. Attention is paid to the fact that the molecule at the surface has a denser and ﬂatter packing in comparison with the initial free conformation of this macromolecule. This means that such macromolecules near the silicate platelet surface form interfacial regions, which are structurally different from the bulk polymer matrix. Knowing the simulated platelet thickness of Na+-montmorillonite (9.7 Å), the value of lif can be estimated according to the data of Figure 7.9 as equal to ~ 6.7 Å. The second method consists of the use of Equations 7.2 and 7.4. For epoxy polymer/ Na+-montmorillonite nanocomposite with Wf = 5 the mass percentage at ϕif = 0.064, dpl = 9.7 Å and, assuming the ﬁller volume and mass fractions are equal, lif will be equal to 6.21 Å [30].

357

Structure and Properties of Crosslinked Polymers

lif

dpl

Figure 7.9 The structure of a QM optimised cluster, consisting of two interacting Na+-montmorillonite platelets, grafted by (CH3(CH2)17) and a crosslinked epoxy polymer model macromolecule. Atoms and bonds are represented as balls and rods. View from the side [22]

The third method uses Equation 7.6 and the value of C∞ was determined according to Formula 5.2. At Tg = 430 K [13] and S = 40 Å2 [31] we shall obtain C∞ = 3.60 according to Formula 5.2 and then at l0 = 1.54 Å from Equation 1.8 we shall obtain lst = 5.55 Å. Further Equation 7.6 allows calculation of the value of lif, which is equal to 6.02 Å. Hence, all three considered models give similar lif values (the discrepancy is within the limits of 11%), i.e., lif estimation according to any of the described methods is equivalent [22, 30]. Let us return to the difference of the reinforcement degree for nanocomposites with a glassy and elastomeric matrix. More precise treatment of this effect can be obtained by using Equation 7.6. Accepting in Formula 5.2 Tg = 430 K [13] for a glassy matrix and Tg = 273 K for an elastomeric matrix, we shall obtain for epoxy polymer/Na+montmorillonite nanocomposites C∞ = 3.60 and 8.92, respectively. Further using the third of the methods of lif calculation considered above and Equations 7.2 and 7.6, the theoretical reinforcement degree (En/Em)T can be obtained, which is compared with the experimental (En/Em)e one in Figure 7.10. One can see that good correspondence of theory and experiment for both types of polymer matrix is obtained [32].

358

Nanocomposites on the Basis of Crosslinked Polymers In paper [13] it was indicated that in the case of nanocomposites with an elastomeric matrix, ﬁlled by montmorillonite, the degree of reinforcement can exceed 10. The same value (En/Em) can be obtained within the frameworks of the offered model, if in Equation 5.2 a typical value for rubbers Tg ≈ 190–200 K is assumed [33]. Then according to this equation we shall obtain C∞ = 18.4 and by the mode considered above the value (En/Em) > 10 at ϕf = 0.15 can be estimated. The important fundamental aspect of the offered treatment is accounting for polymer matrix molecular characteristics. This aspect was considered in detail in paper [34].

(En/Em)T 4

3

2 -1 -2

1

0

1

2

3

4 (En/Em)e

Figure 7.10 The comparison of experimental (En/Em)e and theoretical (En/Em)T reinforcement degree of epoxy polymer/Na+-montmorillonite nanocomposites with epoxy matrix in glassy (1) and rubber-like (2) states [32]

By virtue of the strong anisotropy of the shape of particles of Na+-montmorillonite mentioned above, for theoretical estimation of the degree of reinforcement of nanocomposites ﬁlled by them the models of Halpin–Tsai and Mori–Tanaka are used [19]. For the case of isotropic (spherical) ﬁller particles En/Em estimation can be carried out according to the equation, obtained in paper [35]:

(7.7)

359

Structure and Properties of Crosslinked Polymers In Figure 7.11 the theoretical dependences (lines), corresponding to the Halpin–Tsai and Mori–Tanaka equations for the cases L/dpl = 50 and 100 and also to Equation 7.7 are adduced. The Na+-montmorillonite contents Wf in mass percentage is chosen as the abscissa as is most often used in practice. In the same ﬁgure the experimental values of En/Em (points) for seven nanocomposites are plotted. Let us note ﬁrst of all the transition from theoretical dependences with larger values of L/dpl to smaller ones with increasing Wf (indicated by arrows in the ﬁgure). Such a transition was expected by virtue of the layered silicate, which is Na+-montmorillonite, increasing the aggregation of platelets at Wf. The aggregation means a rise in the number of platelets in the ‘stack’ (tactoid), dpl growth at L = const. and, as a consequence, the ratio L/dpl decreasing. But the most interesting aspect of the Figure 7.11 dependences is that different polymer matrices correspond to different theoretical curves, showing in addition widely differing En/Em values.

En/Em 1 4

2

-6 -7 -8 -9 - 10 - 11 - 12 - 13

3 4

3

2 5

1 0

5

10

15

Wf, mass%

Figure 7.11 The dependences of the reinforcement degree En/Em on the ﬁlling degree Wf for nanocomposites ﬁlled by Na+-montmorillonite. 1–5 – the theoretical dependences corresponding to Halpin–Tsai (1, 2) and Mori–Tanaka (3, 4) equations at L/dpl = 100 (1, 3) and 50 (2, 4) and to Equation 7.7 (5). 6–13 – the experimental data for nanocomposites on the basis of epoxy polymer at T < Tg (6), polyamide-6 (7), poly(butylenes terephthalate) (8), polycarbonate (9), thermotropic liquid crystalline polyester (10), epoxy polymer at T > Tg (11), polypropylene (12) and polyimide (13) [34]

360

Nanocomposites on the Basis of Crosslinked Polymers It is signiﬁcant that the reinforcement degree corresponds to a class of polymer forming a nanocomposites matrix. The largest values of En/Em are obtained for polymers whose chains are able to stretch on the silicate platelet surface (rigid-chain polyimide, crystallising polypropylene and thermotropic liquid crystalline polyester), intermediate values for polymers whose chains are able to stretch only partly (polycarbonate, poly(butylenes terephthalate) and amorphous polyamide-6) and the smallest values for nanocomposites on the basis of epoxy polymer, the capability of chains stretching of which decreases sharply because of the availability of transverse covalent bonds network [30]. Thus, the data of Figure 7.11 indicate that the ability of reinforcement of the nanocomposite is deﬁned not actually by the anisotropy of form of the ﬁller particles, but by the ability of the polymer matrix to reﬂect (reproduce) this anisotropy. In other words, the ﬁller role comes to polymer matrix structure modiﬁcation in comparison with the initial matrix polymer. A similar concept was used for description of the structure of polymer microcomposites [36, 37]. However, distinction of this general treatment consists in the fact that in the microcomposites case the bulk polymer matrix structure changes (its fractal dimension df is increased) [37] and in the nanocomposites case only the structure of the interfacial regions at the common condition df = const. changes [38]. The indicated ability of a polymer matrix or, more precisely, interfacial regions to reﬂect anisotropy of the shape of ﬁller particles can be expressed with the polymer chain statistical segment length lst: the larger lst, the higher En/Em. Let us be reminded that C∞ and, consequently, lst are measures of polymer chain statistical ﬂexibility. In Figure 7.12 the dependences of (En/Em) on lst for the seven nanocomposites adduced in Figure 7.11 at two contents of Na+-montmorillonite (Wf = 2 and 5 mass percentage) are shown. As one can see, these dependences are linear and at lst = 0 are extrapolated to En/Em=1, which means the absence of reinforcement for low-molecular matrices (at any rate, by the formation mechanism of interfacial regions). These dependences can be described analytically as follows [34]:

(7.8)

where A is a constant, which is equal to 1.08 and 1.36 for nanocomposites with Wf = 2 and 5 mass percentage, if lst is given in nm.

361

Structure and Properties of Crosslinked Polymers En/Em 2,5

2,0

1,5 -1 -2 1,0 0

0,8

,6

2,4

lst, nm

Figure 7.12 The dependences of reinforcement degree En/Em on statistical segment length lst for nanocomposites with Na+-montmorillonite contents 2 (1) and 5 (2) mass percentage [34]

A increasing with Wf growth allows the common dependence of En/Em on Wf and lst for exfoliated (non-aggregated) nanocomposites to be obtained [34]:

(7.9)

where Wf is given in mass percentage, lst in nm. In Figure 7.13 the comparison of (En/Em)T calculated according to Equation 7.9 and (En/Em)e reinforcement degree obtained experimentally for the considered nanocomposites at Wf ≤ 5 mass percentage is adduced. From the data of this ﬁgure it follows that the estimation according to Equation 7.9 gives good correspondence to experiment. For nanocomposites with Wf > 5 mass percentage it is required to account for the layered ﬁller platelets aggregation. The authors of paper [40] proposed one possible

362

Nanocomposites on the Basis of Crosslinked Polymers theoretical treatment of this effect. At ﬁrst, the interfacial layer thickness lif was estimated by two modes. The ﬁrst method uses calculation of lif ( Equation 7.6, which does not suppose the dependence of

) according to

on Wf(ϕf) or L/dpl.

(En/Em)T 3,0 2,5

2,0

1,5

1,0

1,5

2

2,5

3

(En/Em)e

Figure 7.13 The comparison of experimental (En/Em)e and (En/Em)T reinforcement degree calculated according to Equation 7.9 for nanocomposites with Wf ≤ 5 mass percentage. Designations are the same as in Figure 7.11 [34]

In the second method of calculation of lif ( ) Equation 7.2 is used, for which the value of ϕf is determined according to Equation 7.4. These estimations showed decreasing with Wf increasing. So, for nanocomposites on the basis of epoxy polymer at T < Tg the value decreases from 0.58 nm at Wf = 2 mass percentage to 0.043 nm at Wf = 15 mass percentage. The authors [40] believe that this seeming decrease is due to aggregation of Na+-montmorillonite platelets and is an effective value of the interfacial layer thickness per aggregate platelet (ﬁller ‘effective particle’ [19]), consisting of N platelets. Let us note that from the point of view of Equation 7.6 the value lif = 0.43 Å has no physical signiﬁcance, since it requires the value lst = 0.3 Å or C∞ = 0.19, whereas the minimum value of C∞ = 2 [39]. Therefore the value of is deﬁned as an effective one. Further the value of N can be calculated according to the equation [40]:

363

Structure and Properties of Crosslinked Polymers

(7.10)

The estimations according to Equation 7.10 have shown that for nanocomposites on the basis of epoxy polymer an increase in N from 1 to 12.6 within the range Wf = 2–15 mass percentage is observed. The aggregation degree can also be expressed by the relative volume contents of Na+-montmorillonite χ in aggregate (‘effective particle’ [19]), determined as follows [19]:

(7.11)

If the proposed model [40] is correct, then ϕif can be calculated according to Equation 7.2, assuming lif= and ϕf=χ, then the theoretical reinforcement degree (En/Em)T can be determined according to Equation 7.4 and compared to the corresponding experimental value (En/Em)e. The results of such a comparison for nanocomposites on the basis of polyamide-6 and epoxy polymer at T < Tg are adduced in Figure 7.14. One can see that good correspondence of theory and experiment is obtained. Aggregation of the layered ﬁller platelets reduces the degree of reinforcement of nanocomposites. So, at Wf = 15 mass percentage this reduction for nanocomposites on the basis of epoxy polymer makes up ~ 30% [40]. Hence, the results stated in the present section have shown that the interfacial regions fraction in polymer nanocomposites plays the same deﬁning role in their reinforcement process as the degree of volume ﬁlling. In addition the strong effect is given by geometry (shape anisotropy) of ﬁller particles or, more precisely, by the polymer-ﬁller contact area, where interfacial phenomena are realised and interfacial regions are formed. The aggregation of ﬁller particles and polymer matrix molecular structure (polymer chain ﬂexibility) make an essential contribution to the reinforcement mechanism of polymer nanocomposites. The offered fractal model allows the quantitative estimation of the relative fraction of interfacial regions and the prediction of the degree of polymer nanocomposites reinforcement [28, 30]. Polyurethanes (PU), widespread in industry, present themselves as two-phase polymeric materials, consisting of rigid and ﬂexible blocks. At room temperature the former are in a glassy state, the latter in a rubber-like state. Introduction of small amounts of organoclay (within the limits of 1–5 mass percentage) essentially improves PU

364

Nanocomposites on the Basis of Crosslinked Polymers properties [41, 42]. So, the modulus of these rubber-like polymers at strain of 100% and organoclay contents 5 mass percentage increases by about 3.6 times, their strength by about 1.5 times [42]. In connection with PU two-phase structure a number of questions about the structure of the nanocomposites received from them appear. The ﬁrst of these is related to reasons of a high reinforcement degree. The second problem is connected with nanoﬁller (organoclay) concentration in either PU phase. And the third question concerns a sharp increase in reinforcement degree at testing temperatures higher than the glass transition temperature of rigid blocks [42]. The authors of paper [43] gave an answer to the proposed questions within the frameworks of the quantitative model of polymer/organoclay nanocomposites offered in the present section [9, 28, 30].

(En/Em)T 1,6

1,4

-1 -2

1,2

1,0

1,2

1,4

1,6

(En/Em)e

Figure 7.14 The comparison of experimental (En/Em)e and (En/Em)T reinforcement degree calculated according to Equation 7.4 for nanocomposites on the basis of polyamide-6 (1) and epoxy polymer at T < Tg (2) [40]

The glass transition temperature of PU and nanocomposites on its basis was determined by the method of dynamical mechanical spectroscopy. It has been found [42] that PU has two glass transition temperatures, which are equal to 233 and 324 K, i.e., the considered PU is a two-phase material, where Tg = 233 K is related to ﬂexible blocks and Tg = 324 K to rigid ones. It has been reported earlier that for PU on the basis of poly(tetramethylene glycol) (PTMG) the value of Tg is equal to 185 K. The higher 365

Structure and Properties of Crosslinked Polymers values of Tg for the considered PU and damping wide peaks, obtained in dynamical mechanical tests, assume partial blending of ﬂexible and rigid segments phases. The glass transition temperatures of nanocomposites on the basis of PU are about equal to these temperatures for the initial matrix polymer [42]. Using the data stated above, the value of C∞ and, correspondingly, lst can be calculated according to Equations 5.2 and 1.8 for each PU phase separately. For rigid blocks C∞ = 9.64, lst = 1.47 nm and for ﬂexible ones C∞ = 18, lst = 2.83 nm. Further the reinforcement degree (En/Em)T calculated according to Equation 9.7 gives values of this parameter using lst = 1.47 nm, if organoclay is concentrated in the rigid blocks phase, and using lst = 2.83 nm, if organoclay concentration is realised in the ﬂexible blocks phase. In Figure 7.15 the comparison of experimental (En/Em) e and (En/Em)T reinforcement degree values calculated by the mentioned mode for PU/montmorillonite nanocomposites is adduced. As follows from the data of this ﬁgure, good correspondence between theory and experiment is obtained in the case if in Equation 7.9 the value lst = 2.83 nm for ﬂexible blocks is used. In the case of a value of lst = 1.47 nm for rigid blocks using Equation 7.9 understated values of the reinforcement degree are given. This comparison demonstrates that in the case of the considered PU, organoclay is concentrated in the devitriﬁcated ﬂexible blocks phase (Tg = 233 K) at testing temperature 293 K and high degree of reinforcement En/Em at small organoclay contents (Wf = 1–5 mass percentage) is deﬁned by high values of lst or C∞ (polymer chain high ﬂexibility), which are typical for a rubber-like polymer [43].

(En/Em)T 3

-1 -2 -3

2

1

2

3 (En/Em)e

Figure 7.15 The comparison of experimental (En/Em)e and (En/Em)T calculated according to Equation 7.9 at lst = 2.83 (1), 4.51 (2) and 1.47 nm (3) reinforcement degree values for PU/organoclay nanocomposites [43]

366

Nanocomposites on the Basis of Crosslinked Polymers Let us consider the effect of a sharp increase in reinforcement degree at testing temperatures which are higher than the glass transition temperature of rigid blocks (Tg = 324 K). It has been shown above that a similar effect was observed in epoxy polymer/organoclay nanocomposites [13]. The authors [42] explained the Tg increase for the ﬂexible blocks phase in comparison with the value obtained earlier (from 185 to 233 K) by a certain degree of blending of the ﬂexible and rigid blocks and, respectively, by the inﬂuence of the last blocks on the ﬁrst ones. The devitriﬁcation of the rigid blocks phase cancels the indicated restrictions and therefore in this case of the calculation of the value of C∞ (at temperatures higher than 324 K) according to Equation 5.2, Tg = 185 K is accepted. Then estimations using the indicated equation give C∞ = 29.7, lst = 4.51 nm. Further the value of (En/Em)T can be calculated according to Equation 7.9. The comparison of experimental (En/Em)e and (En/Em)T reinforcement degree values calculated by the indicated mode forPU/organoclay nanocomposites at temperatures higher than the rigid block glass transition temperature (Tg = 324 K) is also adduced in Figure 7.15, from which good correspondence between theory and experiment follows. Thus, the results obtained allow an answer to the three questions proposed above to be given. Firstly, organoclay particles are concentrated in the devitriﬁcated ﬂexible blocks phase. Secondly, a high ﬂexibility of polymer chains in the indicated phase deﬁnes a sufﬁciently large degree of reinforcement of PU/organoclay nanocomposites. And thirdly, a still larger increase of polymer chain ﬂexibility, realised after devitriﬁcation of the rigid blocks, results in a corresponding raising of the degree of reinforcement for this state of the considered nanocomposites [43].

7.3 The Simulation of Stress-strain Curves for Polymer/Organoclay Nanocomposites within the Frameworks of the Fractal Model The experimental data on elastoplastics deformation are usually interpreted within the frameworks of entropic high-elasticity classical theory, which corresponds well to experiment only in a relatively small strains region (ε < 0.2, where ε is strain) [44]. The co-ordination of theoretical and experimental results at higher ε is realised either within the frameworks of phenomenological modiﬁcations of entropic theory [45], or on the basis of empirical dependences of elastic potential function on strain invariants [44]. In both cases the required precision of experimental dependences of stress σ on ε (or drawing ratio λ) approximation is reached by way of the choice of the proper parameters of co-ordination, which are in essence the ﬁtting ones. It is obvious that the indicated methods have both theoretical and practical deﬁciencies [46]. Therefore in papers [46-48] the fractal treatment of high-elasticity theory was proposed, which does not possess the indicated deﬁciencies. In paper [49] description

367

Structure and Properties of Crosslinked Polymers of the stress-strain curves within the frameworks of the fractal concept [46–48] on the example of PU and PU/organoclay nanocomposites was given. The discrepancy between high-elasticity classical theory and experimental curves σ-ε or σ-λ for elastoplastics indicated above is due to two factors: ﬁrstly, by essentially non-Gaussian statistics of real polymer networks and, secondly, by the lack of coordination of two main postulates, lying in the basis of entropic high-elasticity classical theory – Gaussian statistics and elastoplastics incompressibility. The last condition is characterised by the criterion ν = 0.5, where ν is Poisson’s ratio [46]. Balankin [46–48] obtained the following equation for the description of elastoplastics curves σ-λ within the frameworks of entropic high-elasticity fractal theory:

(7.12) where E is Young’s modulus, ν is Poisson’s ratio, λ is a drawing ratio (λ = 1 + ε). In turn, the entropic high-elasticity classical theory equation, used for the same purposes, has the form [46]:

(7.13) Further, the calculation of curves σ-λ according to Equation 7.12 at ν = 0 and ν = 0.40 and Equation 7.13 was carried out. In Figures 7.16 and 7.17 the comparison of the curves σ-λ calculated by the indicated mode with corresponding experimental data for PU and nanocomposite PU/organoclay with organoclay contents 3 mass percentage was adduced. As follows from the adduced comparison, good correspondence between theory and experiment was obtained at Equation 7.12 and using condition ν = 0, whereas the same equation and application of the condition ν = 0.40 gives overstated values of σ and application of Equation 7.13 gives understated values of this parameter. Another variant of classical and fractal high-elasticity entropic theory using correctness veriﬁcation is adduced in Figure 7.18, where the comparison of experimental σf and σTf tension strength calculated according to Equations 7.12 and 7.13 for the two PU and nanocomposites considered in paper [49] on their basis is given. As follows from the data of Figure 7.18, Equation 7.12 at the condition ν = 0 again gives a sufﬁcient correspondence to experiment, whereas the entropic high-elasticity classical theory (Formula 7.13) ﬁnds essentially understated strength values for all considered materials. Thus, the data of Figures 7.16–7.18 indicate unequivocally that application

368

Nanocomposites on the Basis of Crosslinked Polymers of the fractal theory of entropic high-elasticity (Equation 7.12) at the condition ν = 0 is the only correct description of curves σ-λ for the considered polyurethanes and nanocomposites on their basis. σ, MPa 1

40 -4

20 2

3 0

1

9

5

λ

Figure 7.16 The dependences of stress σ on the drawing ratio λ for polyurethane PTMG. 1, 2 – calculation according to Equation 7.12 at ν = 0.40 (1) and ν = 0 (2) and Equation 7.13 (3). 4 – the experimental data [49]

σ, MPa 1

40

2

20 3

-4 0

1

3

5

7

λ

Figure 7.17 The dependences of stress σ on the drawing ratio λ for nanocomposite PU/organoclay with Wf = 3 mass percentage. 1, 2 – calculation according to Equation 7.12 at ν = 0.40 (1) and ν = 0 (2) and Equation 7.13 (3). 4 – the experimental data [49]

369

Structure and Properties of Crosslinked Polymers From Equation 5.4 it follows that for the considered elastoplastics df = 2.0 at ν = 0. This means that these materials networks obey Gaussian statistics [48]. Thus, the discrepancy between entropic high-elasticity classical theory (Formula 7.13) and experimental data, obvious from the plots of Figures 7.16–7.18, is due to nonfulﬁlment of the second criterion indicated above, i.e., to postulates about Gaussian statistics and the lack of co-ordination of incompressibility of elastoplastics: at fulﬁlment of the the ﬁrst postulate instead of the incompressibility condition ν = 0.5 the condition ν = 0 was obtained. Fulﬁlment of Gaussian statistics is deﬁned by the considered elastoplastics preparation method, namely curing of a concentrated solution of chains [48]. Substituting the condition ν = 0 in Equation 7.12, its simpliﬁed variant will be obtained:

(7.14)

which coincides with the classical result for long polymer chains [44].

σTf , MPa 40 -1 -2 -3 -4 20

0

20

40 σ , MPa f

Figure 7.18 The comparison of experimental σf and σ f strength values calculated according to Equation 7.12 at ν = 0 (1, 2) and Equation 7.13 (3, 4) for nanocomposites on the basis of PU, based on PTMG (1, 3) and poly(butylenes adipate) diol (PBAD) (2, 4) [49] T

370

Nanocomposites on the Basis of Crosslinked Polymers Hence, the results obtained above have shown that behaviour at deformation for the considered polyurethanes and nanocomposites on its basis is described within the frameworks of entropic high-elasticity fractal theory or, equivalently, within the frameworks of the classical theory approximation for long polymer chains. The considered polymer networks obey Gaussian statistics due to their preparation method. The inaccuracy of the application of the entropic high-elasticity classical theory (Equation 7.13) is deﬁned by non-fulﬁlment in the given case of a postulate about elastoplastics incompressibility [49].

7.4 The Multifractal Model of Sorption Processes for Nanocomposites As it is known [50–52], introduction of organoclay in a polymeric matrix results in essential reduction of permeability to gas of the nanocomposites obtained by such a mode in comparison with a matrix polymer. As a rule, such permeability to gas reduction is explained by an increase in the meandering trajectory of the gas-penetrant molecules through the nanocomposite by virtue of the availability of organoclay anisotropic particles within it [50, 52]. So, the relative permeability to gas Prel, characterising a reduction in this parameter for nanocomposites in comparison with a matrix polymer, is deﬁned as follows [50]:

(7.15)

where p is the sides ratio of the organoclay platelet, ϕf is the nanoﬁller volume content. Approximately the same ratio of permeability to gas coefﬁcient (or selectivity coefﬁcients) values for different gases is considered as an argument in favour of the invariable structure of the polymer matrix in comparison with the matrix polymer [52]. Nevertheless, without denying the inﬂuence of the availability of nanoﬁller on the so-called nonlinearity coefﬁcient, characterising the degree of meandering of the trajectory of the penetrant molecule, one should note that the mentioned coefﬁcient is connected with both polymer and penetrant molecule characteristics [53, 54]. Within the frameworks of such a model good correspondence of theory and experiment was obtained for the case of permeability to gas of polyethylene/Na+-montmorillonite nanocomposites [55]. Therefore the authors of paper [56] applied the multifractal model [57] for water sorption in the description of PU/organoclay nanocomposites [42].

371

Structure and Properties of Crosslinked Polymers As it has been shown in paper [58], in stationary diffusion conditions at duration t = const. the relationship is fulﬁlled:

(7.16) where Q is the absorption (sorption) coefﬁcient (in the considered case, of water), Ddif is the diffusivity. The multifractal treatment [57] allows the value of Q to be determined as follows:

(7.17) where Qm is the water sorption coefﬁcient for a matrix polymer (PU), α m is the nanocomposite polymer matrix volume fraction, which is accessible for water diffusion, Dch is the fractal dimension of a chain part between its ﬁxation points [27]. ac

The authors of paper [59] offered the following equation for estimation of the diffusivity of semi-crystalline polymers:

(7.18) where Dam is the diffusivity of a fully amorphous polymer, τn is the nonlinearity (meandering) coefﬁcient, which is due to the complexity of transport methods of diffusate molecules between crystallites, βn is the so-called polymer chains immobility coefﬁcient. The direct analogy between nanocomposites and semi-crystalline polymers is contained in the fact that both indicated classes of polymeric materials have regions which are impenetrable for transport processes (nanoﬁller and crystallites, respectively). Therefore the indicated processes are realised through the polymer matrix and amorphous phase, respectively. Hence, from the comparison of Relationships 7.16– 7.18 it can be written [56]:

(7.19)

372

Nanocomposites on the Basis of Crosslinked Polymers

(7.20)

(7.21) Thus, the offered treatment assumes not only a change in the nonlinearity coefﬁcient τn because of nanoﬁller introduction, but also permits a change in the polymer matrix structure by virtue of variation of the coefﬁcient βn, which depends on its molecular mobility level [54]. Let us consider the estimation methods of parameters included in Equation 7.17. The value of Qm was accepted to be equal to the value of Q for polyurethanes PTMG and PBAD. As it is known [30], in the polymer nanocomposites case actually the nanoﬁller and interfacial regions of the particles surrounding it with relative fractions ϕf and ϕif, respectively, will be impenetrable for diffusion. Then the polymer matrix relative fraction αm can be estimated as follows [56]:

(7.22)

In turn, the following relationship exists between values ϕif and ϕf [9]:

(7.23)

for exfoliated organoclay. The polymer matrix fraction α m accessible for water diffusion nanocomposites is determined according to the equation [57]: ac

(7.24)

373

Structure and Properties of Crosslinked Polymers where d H 2O is a water cluster diameter, which is equal to 7.8 Å [60]. The use of the last value is due to the fact that water is always clustered in contact with a polymer [61]. As has been shown in the previous section, for the considered elastoplastics the structure fractal dimension value df is equal to 2.0. In turn, this means that Dch = 1.0 [27]. Now the water sorption coefﬁcient for the considered nanocomposites can be calculated according to Equation 7.17. In Figure 7.19 the comparison of experimental Q and QT water sorption coefﬁcient values calculated according to the proposed multifractal treatment for two series of PU/montmorillonite nanocomposites (PTMG/ MMT and PBAD/MMT) at experiment duration 1 and 5 days is assumed. As one can see, good correspondence of theory and experiment was obtained – the average discrepancy between QT and Q makes up about 20%, which is quite sufﬁcient for preliminary estimations.

QT 0.8 -1 -2 -3 -4 0.4

0

0.4

0.8

Q

Figure 7.19 The comparison of experimental Q and QT water sorption coefﬁcient values calculated according to Equation 7.17 during 1 (1, 2) and 5 (3, 4) days for nanocomposites PTMG/MMT (1, 3) and PBAD/MMT (2, 4) [56]

Hence, the results obtained in the present section suppose that the change in water sorption coefﬁcient for PU/organoclay nanocomposites is actually connected with variation in the nonlinearity (meandering) coefﬁcient at a change in the nanoﬁller contents. For the considered nanocomposites a special case is observed, when the fractal dimension of a chain part between its ﬁxation points is equal to 1 and

374

Nanocomposites on the Basis of Crosslinked Polymers therefore the water sorption coefﬁcient for them is determined only by the nonlinearity coefﬁcient. The condition df = 2.0 = const. for the considered nanocomposites is direct proof of this. The multifractal treatment of water diffusion in the nanocomposites process allows the correct quantitative estimation of the water sorption coefﬁcient.

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Nanocomposites on the Basis of Crosslinked Polymers 28. A.K. Mikitaev, G.V. Kozlov and G.E. Zaikov in Polymer Nanocomposites: Variety of Structural Forms and Applications, Nova Science Publishers, Inc., New York, NY, USA, 2008, p.319. 29. T. Pernyeszi and I. Dekany, Colloid Polymer Science, 2003, 281, 1, 73. 30. A.K. Mikitaev, G.V. Kozlov and G.E. Zaikov in Polymer Nanocomposites: Variety of Structural Forms and Applications, Nauka, Moscow, Russia, 2009, p.278. 31. G.V. Kozlov, V.A. Beloshenko, E.N. Kuznetsov and Y.S. Lipatov, Doklady NAN Ukraine, 1994, 12, 126. 32. G.V. Kozlov, A.K. Malamatov and Y.G. Yanovskii in the Proceedings of 9th International Symposium ‘Order, Disorder and Oxides Properties’, Sochi, Russia, 2006, p.232. 33. E.L. Kalinchev and M.B. Sakovtseva in Properties and Processing of Thermoplastics, Khimiya, Leningrad, Russia, 1983, p.288. 34. A.K. Malamatov in the Proceedings of International Science-Technology Conference ‘Composite Building Materials. Theory and Practice’, PSU, Penza, Russia, 2006, p.28. 35. I.I. Tugov and A.Y. Shaulov, Vysokomolekulyarnye Soedineniya Seriya B, 1990, 32, 7, 527. 36. A.I. Burya and G.V. Kozlov in Synergetics and Fractal Analysis of Polymer Composites Filled with Short Fibers, Porogi, Dnepropetrovsk, Ukraine, 2008, p.258. 37. G.V. Kozlov, Y.G. Yanovskii and Y.N. Karnet in Structure and Properties of Particulate-ﬁlled Polymer Composites: Fractal Analysis, Al’yanstransatom, Moscow, Russia, 2008, p.363. 38. A.K. Malamatov, A.I. Burya and G.V. Kozlov, Sovremennye Naukoemkie Tekhnologii, 2005, 11, 16. 39. V.P. Budtov in Physical Chemistry of Polymer Solutions, Khimiya, SanktPeterburg, Russia, 1992, p.384. 40. A.Kh. Malamatov and G.V. Kozlov in the Proceedings of International Science-Technology Conference ‘Composite Building Materials, Theory and Practice’, PSU, Penza, Russia, 2006, p.32.

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Structure and Properties of Crosslinked Polymers 41. T.K. Chen, Y.I. Tien and K.H. Wei, Polymer, 2000, 41, 3, 1345. 42. B.K. Kim, Y.W. Seo and H.M. Jeong, European Polymer Journal, 2003, 39, 1, 85. 43. B.Z. Dzhangurazov, G.V. Kozlov and A.K. Mikitaev in the Proceedings of 29th International Conference ‘Composite Materials in Industry’, Kiev, Russia, 2009, p.325. 44. G.M. Bartenev and S.Y. Frenkel in Physics of Polymers, Khimiya, Leningrad, Russia, 1990, p.432. 45. V.E. Gul and V.N. Kuleznev in Structure and Mechanical Properties of Polymers, Vysshaya Shkola, Moscow, Russia, 1979, p.352. 46. A.S. Balankin, Pis’ma v ZhETF, 1991, 17, 17, 68. 47. A.S. Balankin, Metally, 1992, 2, 41. 48. A.S. Balankin, Fizika Tverdogo Tela, 1992, 34, 3, 1245. 49. B.Z. Dzhangurazov, G.V. Kozlov and A.K. Mikitaev, Plasticheskie Massy, 2010. [In press] 50. S. Hotta and D.R. Paul, Polymer, 2004, 45, 21, 7639. 51. S. Arunvisut, S. Phummanee and A. Somwangthanaroj, Journal of Applied Polymer Science, 2007, 106, 8, 2210. 52. N.Y. Kovaleva, P.N. Brevnov, V.G. Grinev, S.P. Kuznetsov, I.V. Pozdnyakova, S.N. Chvalun, E.A. Sinevich and L.A. Novokshonova, Vysokomolekulyarnye Soedineniya Seriya A, 2004, 46, 6, 1045. 53. G.V. Kozlov, G.E. Zaikov and A.K. Mikitaev in The Fractal Analysis of Gas Transport in Polymers. The Theory and Practical Applications, Nova Science Publishers, Inc., New York, NY, USA, 2009, p.238. 54. G.V. Kozlov, G.E. Zaikov and A.K. Mikitaev in The Fractal Analysis of Gas Transport in Polymers, The Theory and Practical Applications, Nauka, Moscow, Russia, 2009, p.199. 55. G.V. Kozlov and G.M. Magomedov in the Proceedings of International Science-Practice Conference ‘New Technologies in Mechanical Engineering’, PSU, Penza, Russia, 2007, p.161.

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Nanocomposites on the Basis of Crosslinked Polymers 56. B.Z. Dzhangurazov, G.V. Kozlov and A.K. Mikitaev in the Proceedings of 29th International Conference ‘Composite Materials in Industry’, Kiev, Russia, 2009, p.322. 57. R.M. Khalikov, G.V. Kozlov, A.I. Burya and Y.M. Pleskachevskii, Kompozitnye Materialy, 2007, 1, 1, 31. 58. Y.G. Yanovskii, L.K. Nafadzokova, G.V. Kozlov and G.E. Zaikov, Fizicheskaya Mezomekhanika, 2007, 10, 4, 49. 59. R. Ash, R.M. Barrer and D.G. Palmer, Polymer, 1970, 11, 8, 421. 60. G.V. Kozlov, E.N. Ovcharenko and G.E. Zaikov, Teoreticheskie Osnovy Khimicheskoi Tekhnologii, 2008, 42, 4, 453. 61. G.L. Brown in Water in Polymers, Ed., S.P. Rowland, American Chemical Society, Washington, DC, USA, 1984, p.419.

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8

Polymer-polymeric Nanocomposites

Nanocomposites in which both matrix and nanoﬁller are polymers (polymerpolymeric nanocomposites) represent themselves as new, the least studied and the most interesting class of polymer nanocomposites, the opportunities of which will become clear, but at present it is obvious that they are great. So, at small ﬁller contents (~ 2 vol%) the reinforcement degree of these nanocomposites makes up ~ 1.3, which is comparable with the results regarded now as the most perspective polymer/organoclay nanocomposites. But the most interesting property of these nanocomposites is the simultaneous increase in the elasticity modulus and decrease in the melt viscosity – the combination of which can be regarded as unique for all polymer composites in general. In the present chapter the main properties of the indicated materials on the example of polyethylene/epoxy nanocomposite polymer will be brieﬂy considered.

8.1 The Fractal Analysis of Crystallisation of Nanocomposites The introduction into high-density polyethylene (HDPE) of small amounts (up to 5 mass percentage) of crosslinked epoxy polymer DGEBA changes essentially the properties of the native polymer [1]. Such a composition can be considered as a nanocomposite for two reasons: ﬁrstly, the size of the particles of the epoxy polymer is of the order of nanometres and, secondly, the elasticity modulus of a crosslinked glassy epoxy polymer is essentially higher than the similar parameter for the rubber-like amorphous phase of polyethylene in which the epoxy polymer is concentrated [1]. There is a linear reduction of the degree of crystallinity of the considered nanocomposites from 0.68 to 0.46 with an increase in the epoxy polymer contents within the range of 0–5 mass percentage. The authors [2] studied the causes of the indicated degree of reduction in crystallinity and the change in crystallisation mechanism parameters for polyethylene/epoxy nanocomposite polymer (HDPE/EP) with fractal analysis methods. Let us consider structural changes in HDPE/EP nanocomposites with increasing epoxy polymer contents cEP at T = 293 K. As structural characteristics the fractal (Hausdorff) dimension df was used as the most general informant of structure state [3]. The value of df can be determined according to Equation 5.4, where

381

Structure and Properties of Crosslinked Polymers Poisson’s ratio ν was determined by results of mechanical tests with the aid of Equation 5.5 [4]. In Figure 8.1 the dependence df(cEP) is shown, from which reduction in df with an increase in cEP follows. On the whole such dependence df(cEP) agrees with the known data, according to which the df value is reduced with crystallinity degree K according to the simple relationship [5]:

(8.1)

df 2,70 2,68 2,66 2,64 2,62 2,60

0

1

2

3

4

5 cEP, mass %

Figure 8.1 The dependence of fractal dimension df of the structure of HDPE/EP nanocomposites on epoxy polymer contents cEP [2]

In Figure 8.2 the relation of parameters df and (2 + K) for the studied nanocomposites is adduced, from which it follows that the value of df is reduced much more slowly than (2 + K) and Relationship 8.1 is fulﬁlled precisely for the initial HDPE only. As it is known [6], there is intercommunication between df and the relative fraction ϕcl of local order domains (clusters) in the amorphous phase of semi-crystalline polymers, which is described analytically by Equation 4.8.

382

Polymer-polymeric Nanocomposites

df 2,8 2,7 2,6 2,5 2,4

2,5

2,6

2,7

2,8 (2+K)

Figure 8.2 The comparison of fractal dimension df and parameter (2 + K) for HDPE/EP nanocomposites [2]

From Equation 4.8 and the data of Figure 8.2 it follows that the value of ϕcl for nanocomposites rises more slowly than is expected for non-modiﬁed HDPE. This effect is due to rejection of epoxy polymer particles during the crystallisation process from crystallites in the interfacial regions [7]. Since formation of clusters in the amorphous phase of semi-crystalline polymers is controlled by the tightness of the amorphous chains [6], then the indicated rejection of epoxy polymer particles violates the structure of interfacial regions, weakening the tightness of amorphous chains and, as a consequence, decreasing the value of ϕcl [2]. Let us consider crystallisation mechanism changes for HDPE/EP nanocomposites in comparison with the initial HDPE, which deﬁne the degree of reduction in crystallinity with an increase in cEP. As it is known [8], the crystallisation kinetics of polymers is often described with the aid of the Kolmogorov–Avrami equation, obtained for lowmolecular substances (Equation 4.19), in which the exponent n value can be changed within the range of 1–4 [9]. The authors [10] demonstrated that the Kolmogorov–Avrami exponent n increased linearly with increase in the fractal dimension Dch of a chain part between clusters, characterising the extent of molecular mobility for a polymer. The value of Dch can be calculated according to Equation 4.15 and parameters C∞ and df, which are necessary for calculation of Dch, are connected with one another by Relationship

383

Structure and Properties of Crosslinked Polymers 5.31. The calculation according to Equations 5.31 and 4.15 shows that the value of Dch decreases from 1.56 to 1.19 with an increase in cEP within the range of 0–5 mass percentage. According to the calibrating plot, adduced in Figure 4.2, this corresponds to n reduction from 2 to 1. Hence, introduction of an epoxy polymer in HDPE results in a change in the crystallisation mechanism, since n = 2 means growth of either two-dimensional crystals or three-dimensional ﬁbrils and n = 1 assumes growth of one-dimensional crystals at athermic nucleation (at the simultaneous beginning of growth of all crystals) [9]. Further, assuming z = 2.08 × 10–3 s–1 and t = 180 s, the value of K can be calculated theoretically and be compared with the experimental data that was made in Figure 8.3. As one can see, the offered model describes precisely the reduction in K with increase in cEP as being due to Dch decreasing and, consequently, to the Kolmogorov–Avrami exponent n. Another method of calculation of the value of K is the following formula [10]:

(8.2)

The dependence K(cEP), calculated according to Equation 8.2, is shown in Figure 8.3 by a dashed line. As one can see, this dependence demonstrates much slower reduction in K with increase in cEP observed experimentally. This means that for the considered nanocomposites the value of K is deﬁned not so much by the polymer chain ﬂexibility, characterised by parameter C∞, as by the variation of the characteristics of nucleation and crystallisation mechanisms, from which the exponent n is the most important. To specify the agreement between theoretical and experimental data, adduced in Figure 8.3, one can use the variable z in Equation 4.18. An increase in z with growth in melt viscosity, characterised by melt ﬂow index MFI reduction (Figure 8.4), was found. It is signiﬁcant that at zero melt viscosity, z = 0 and crystallisation also ceases. Hence, the crystallisation proceeds at polymer structure border state – at its transition from melt to solid-phase state. Therefore both characteristics of forming the solid phase (dimension Dch) and melt (MFI) inﬂuence the value of K. Hence, the results stated above have shown that reduction in the degree of crystallinity of HDPE/EP nanocomposites is due to variation of the characteristics of nucleation and crystallisation mechanisms. The change of polymer molecular characteristics has less inﬂuence on the crystallinity degree. These effects can be accurately described within the frameworks of the fractal model.

384

Polymer-polymeric Nanocomposites

K 0,7 -2 -3

0,6 0,5 1

0,4 0,3 0

1

3

2

4

5 cEP, mass%

Figure 8.3 The dependences of crystallinity degree K on epoxy polymer contents cEP for HDPE/EP nanocomposites. Calculations according to Equation 4.18 (1) and Equation 8.2 (2) and (3) the experimental data [2]

Z×103, s-1 4 3 2 1 0

1

2

3

MFI1, min/dg

Figure 8.4 The dependence of crystallisation rate constant z on melt ﬂow index MFI for HDPE/EP nanocomposites [2]

385

Structure and Properties of Crosslinked Polymers

8.2 The Melt Viscosity of HDPE/EP Nanocomposites As has been noted above, HDPE modiﬁcation by epoxy polymer DGEBA (EP) allows nanocomposites possessing a number of interesting properties to be obtained. So, at EP contents cEP = 2.0 to 2.5 mass percentage the maximum elasticity modulus E is observed at testing temperature T = 293 K, moreover the increase makes up more than 30% in comparison with the initial HDPE. This stiffness maximum of solid-phase nanocomposite is accompanied by extreme growth in MFI of almost 3 times in comparison with the initial HDPE, which corresponds to the similar melt viscosity reduction, that is, by the essential improvement in processing of HDPE/ EP nanocomposites [1]. In paper [11] the causes of extreme rising of MFI or melt viscosity reduction for HDPE/EP nanocomposites were considered. As it is known [1], the polymer melt viscosity is controlled by two parameters: polymer molecular mass MM and molecular mass of the chain part between clusters Mcl. The common dependence of the polymer melt viscosity η from the indicated factors can be written as follows:

(8.3)

Further the polymer melt structure can be characterised by their macromolecular coil fractal dimension Δf [12]. Then the number of chain ﬁxation points (density) in the melt is accepted as being equal to the number of intersections of macromolecular coils N, which is determined according to the following relationship [12]:

(8.4)

where Rg is the gyration radius of the macromolecular coil, d is the dimension of Euclidean space in which a fractal is considered (it is obvious that in our case d = 3). In paper [11] the value of Δf is accepted as being equal to the structure dimension df of solid-phase HDPE/EP nanocomposites, which was calculated according to Equation

386

Polymer-polymeric Nanocomposites 5.4. In turn, the value of N is scaled with the molecular mass of a chain part between entanglements Me according to the following relationship [6]:

(8.5)

where ρ is the polymer density, NA is Avogadro’s number. The calculation of the polymer melt viscosity value η0 at zero shear according to Relationships 5.4 and 8.3–8.5 (in relative units) has shown that it changes approximately proportionally to the reciprocal MFI and the empirical correlation of these parameters has the form [11]:

(8.6)

where MFI is given in dg/min. In Figure 8.5 the comparison of experimental values and those calculated according to Equation 8.6 of melt ﬂow index as a function of epoxy polymer contents cEP in the studied nanocomposites is adduced. As one can see, the offered fractal model gives good correspondence both qualitatively and quantitatively with the experimental data. However, the mentioned model does not explain the causes of extreme reduction of the melt viscosity of HDPE/EP nanocomposites. Therefore for explanation of this effect the authors [11] used another treatment. As it is known [13], the extreme change of properties of blends in the case of their interaction (both chemical and physical) is realised at equimolar (stoichiometric) component contents. Since for the considered nanocomposites the extremum is reached at 2.0–3.0 mass percentage EP, then this means that the polymer matrix interacts not with the entire epoxy polymer, but only a part of it consisting of 4–6 mass percentage HDPE. In this case for estimation of ' η0 (further designated as η0 ) the relationship applied for description of the chemical reaction kinetics of two components can be used [14]:

(8.7)

387

Structure and Properties of Crosslinked Polymers where Vint is the HDPE and EP interaction rate, [HDPE] and [EP] are the concentrations of HDPE and EP interacting parts, respectively, accepted as being equal to 12 mass percentage, t is the interaction duration. Assuming the values of Vint, t and Δf as being constant (Δf variation makes up 2.606– ' 2.686), the η0 value can be estimated in relative units according to Relationship 8.7:

(8.8)

where the values of [HDPE] and [EP] are given in mass percentage and the constant in the numerator of the right-hand part of the relationship can be found by a matching method.

MFI5463, dg/min 0,8 0,6 0,4

1

0,2 0

-2 -3 1

2

3

4

5 cEP, mass%

Figure 8.5 The dependences of the melt ﬂow index MFI on the epoxy polymer contents cEP for HDPE/EP nanocomposites. (1) The experimental data; (2) calculation according to Equation 8.6; and (3) calculation according to Equation 8.9 [11]

388

Polymer-polymeric Nanocomposites The estimations showed that MFI and η0 intercommunication could be described analytically according to the following empirical equation [11]: '

(8.9)

where MFI is given again in dg/min. Comparison of the experimental MFI values and those calculated according to Equation 8.9 is also adduced in Figure 8.5, from the data of which good correspondence of theory and experiment follows. Hence, the results stated above showed that the extreme change in melt viscosity of HDPE/EP nanocomposites can be accurately described within the frameworks of the fractal model. The main structural parameter controlling this effect is the change of the fractal dimension of macromolecular coil in melt. The main physical cause deﬁning the mentioned effect is a partial interaction of the HDPE matrix and epoxy polymer particles. In this case the extreme change in the melt ﬂow index is accurately described within the frameworks of chemical reactions fractal kinetics.

8.3 The Mechanical Properties of HDPE/EP Nanocomposites As has been noted above, in the case of solid-phase HDPE/EP nanocomposites the extreme growth of the elasticity modulus E was observed within the range of 0–5 mass percentage of the EP contents. The linear reduction of the crystallinity degree K and decrease in cEP from 0.68 for the initial HDPE to 0.46 for HDPE/EP nanocomposites at cEP = 5 mass percentage occurs simultaneously. At present it is assumed that the decrease in K should be accompanied by a reduction in E [15], therefore the indicated extreme increase in E cannot be explained within the frameworks of the existing concepts. Proceeding from this, the authors [16] attempted to obtain a quantitative theoretical model describing the dependence E(cEP) for the considered nanocomposites within the frameworks of fractal analysis. The dependence E(cEP) adduced in Figure 8.6 compared with the plot of Figure 8.5 demonstrates that the maximum value of the elasticity modulus corresponds to the minimum melt viscosity. Therefore, the epoxy polymer is a plasticiser in melt and an anti-plasticiser in the solid-phase state for HDPE/EP nanocomposites [16].

389

Structure and Properties of Crosslinked Polymers

E, GPa -1

1,0

0,9 2

0,8

0,7 0

1

2

3

4

5 cEP, mass%

Figure 8.6 The dependences of the elasticity modulus E, determined experimentally (1) and calculated according to Equation 8.11 (2), on epoxy polymer contents cEP for HDPE/EP nanocomposites [16]

The authors [17] showed that the E value for fractal objects, which are HDPE/EP nanocomposites (see Figure 8.1), was given by the percolation Relationship 6.6 (for more details see Section 6.2). At the same time the cluster structure of the polymer amorphous state presents itself as a percolation system [6, 18, 19], for which the sum (K + ϕcl) should be accepted as p, where ϕcl is the relative fraction of the local orders domains (clusters). In turn, for such a system it can be written [20]:

(8.10)

where Rcl is the distance between clusters, lst is the statistical segment length, accepted as the cluster characteristic size. The estimation of the elasticity modulus theoretical values of ET according to Equations 6.6 and 6.7 shows good correspondence with experimental values of E

390

Polymer-polymeric Nanocomposites using the proportionality coefﬁcient 4.17 GPa. Thus, Relationship 6.6 takes the ﬁnal shape for the considered nanocomposites [16]:

(8.11)

In Figure 8.6 the comparison of the dependences of the elasticity modulus E on cEP obtained experimentally (points) and those calculated according to Equation 8.11 (solid curve) is adduced for the considered polymer-polymeric nanocomposites. As can be seen, this comparison demonstrates good correspondence of theory and experiment (average discrepancy 5.2%), conﬁrming the accuracy of the offered treatment. Hence, the results stated above have shown that the integral structural parameters K and ϕcl inﬂuence not only the elasticity modulus value of semi-crystalline polymers, but also their possible distribution in polymer structure. The decrease in cluster characteristic size lst and reduction in the number of segments in it, which is equal to F/2 (see Equation 5.29), result in an increase in the elasticity modulus for the indicated polymers. Let us note that the offered reinforcement mechanism principally differs from that considered earlier for polymer nanocomposites with inorganic ﬁller, where reinforcement is realised at the expense of formation of interfacial regions [24]. The yield stress is the parameter deﬁning the upper boundary of the region of possible application of engineering plastic polymers and this circumstance has deﬁned the attention paid to the yielding process research in the last 50 years. For semi-crystalline polymers Kargin and Sogolova offered a concept at the beginning of the 1950s [25], in which the yielding process was considered as crystalline phase partial melting (mechanical disordering) – recrystallisation. Although in the years since, a number of other concepts appeared, Kargin and Sogolova’s treatment remains the main one at present. Popli and Mandelkern [26] conﬁrmed the concept mentioned above, having used as an experimental ground the linear dependences of the degree of yield stresscrystallinity (or size of crystallites) for the polyethylenes series with widely varied crystallinity and morphology. Without disputing the general tendency of papers [25, 26], in which it is assumed that yield stress is deﬁned by thermal stability of crystallites, it is necessary to note the narrowness of this approach. A semi-crystalline polymer represents a complex dissipative system, consisting of two interconnected phases (amorphous and crystalline) and interfacial regions between them [27]. Therefore it is clear even intuitively that the structural changes in amorphous and interfacial regions cannot help inﬂuencing the behaviour and state of a crystalline phase. The experimental conﬁrmation of such

391

Structure and Properties of Crosslinked Polymers an inﬂuence is given in papers [28, 29]. Although the possibility of the inﬂuence of the structure of non-crystalline regions on the yielding process was assumed earlier [25, 26], any concretisation of such an inﬂuence is absent. The authors of papers [30, 31] considered one of the variants of the increase in the yield stress of industrial HDPE with the help of the state change of its non-crystalline regions (without the crystallinity degree increasing). In Figure 8.7 the dependences of the crystallinity degree K and the yield stress σY, obtained in quasi-static and impact tests, on epoxy polymer contents cEP for the studied nanocomposites is adduced. Despite the sharp difference of conditions of mechanical tests, both dependences σY(cEP) showed a similar extreme increase in σY with increase in cEP. It is particularly important that such a σY change is accompanied by a monotone linear reduction of crystallinity degree that excludes the possibility of linear correlation σY(K). Since any admixtures in the crystallisation process are rejected from crystalline regions and are grouped on their borders [7], the change in structure of non-crystalline regions of nanocomposites was studied by their permeability to gas by oxygen and nitrogen measurements. It was assumed that the diffusion of gases is realised exclusively through non-crystalline regions [32]. In Figure 8.8 the dependences of normalised permeability to gas P/αam (P is permeability to gas coefﬁcient, αam is non-crystalline regions volume contents) on cEP are adduced. From the comparison of Figures 8.7 and 8.8 data it can be seen that growth in σY corresponds to a decrease in P/αam, i.e., to a decrease in relative free volume fg in non-deformed ﬁlm samples of the studied nanocomposites [33, 34]. Such a relation of σY and P/αam supposes that the thermal stability of crystallites depends not only on their sizes, but also on the state of noncrystalline regions surrounding crystallites. In particular, it can be assumed that the partial melting-recrystallisation process is realised only when a certain level of fg is reached for non-crystalline regions. The Fg value of the deformed polymer consists of two parts: the free volume of non-deformed polymer and the dilatational free volume, which is the consequence of the polymer density change in the deformation process [35]. The ﬁrst component of the decrease in fg due to introduction of the epoxy polymer in HDPE (that is illustrated by a reduction in the permeability to gas) will require compensation at the expense of the second component increase, which should result in growth in the yield strain εY [35]. This assumption is conﬁrmed experimentally (see Figure 8.9). Within the frameworks of the yielding phonon concept [4, 19] this means that not only the parameters characterising the initial non-deformed polymer structure should be taken into account, but also its degree of modiﬁcation in the deformation process, which can be realised with the help of the Grünesen parameter γL. In the ﬁrst approximation the ﬁrst factor can be characterised with

392

Polymer-polymeric Nanocomposites the help of the elasticity modulus E and the value of γL can be estimated according to the equation [4]:

(8.12)

σy, MPa

K

1

40 30

2

20 0,70 10

0

3

1

2

3

0,55 4

0,40 5 cEP, mass%

Figure 8.7 The dependences of yield stress σY in impact (1) and quasi-static (2) tests and crystallinity degree K (3) on epoxy polymer contents cEP for nanocomposites HDPE/EP [31]

In Figure 8.10 the dependence σY(E/γL) is shown, from the data of which it follows that the results of both test types used in papers [30, 31] lie on one common straight line, passing through the coordinates origin. Let us note that at plots using σY(E) two different dependences will be obtained, i.e., if the structure modiﬁcation is taken into account then the yielding mechanisms community implied by the data of Figure 8.10 is not already observed [36].

393

Structure and Properties of Crosslinked Polymers

P×1015 αam 15 1

10

5 2 0

1

2

3

4 5 cEP, mass%

Figure 8.8 The dependences of the permeability to gas coefﬁcient P, normalised over amorphous phase volume contents αam, for oxygen (1) and nitrogen (2) on epoxy polymer contents cEP for nanocomposites HDPE/EP [31]

εY 0,125

0,120

0,115

0,110

0

1

2

3

4 5 cEP, mass%

Figure 8.9 The dependence of yield strain εY on epoxy polymer contents for nanocomposites HDPE/EP [1]

394

Polymer-polymeric Nanocomposites

σY, MPa 40 30 -1 -2

20 10

0

100

200

300 E/ , MPa YL

Figure 8.10 The relation between yield stress σY and ratio of elasticity modulus to the Grünesen parameter E/γL in quasi-static (1) and impact (2) tests for nanocomposites HDPE/EP [31]

In Figure 8.11 the dependences of the yield stress σY on the testing temperature T for initial HDPE and nanocomposite HDPE with epoxy polymer content 2.5 mass percentage are shown. From the adduced plots it follows that within the whole range of temperatures 293–373 K the value of σY for a nanocomposite exceeds the corresponding value for the initial HDPE. Practically this means an increase in the heat stability of polyethylene at the expense of its modiﬁcation by an epoxy polymer that is very important for engineering branches, in particular for the cable industry. Since the dependences σY(T) for the initial HDPE and the considered nanocomposite are paralleled (Figure 8.11), then the heat stability increase ΔT can be deﬁned as a displacement of the dependence σY(T) for the initial HDPE over the abscissa axis up to ﬁtting with a similar dependence to that for a nanocomposite. This displacement is indicated in Figure 8.11. In the considered case the value of ΔT is equal about to 18 K, that is an essential increase of heat stability for polyethylenes, particularly in a range of high temperatures. Hence, the results stated above have shown that for the correct description of the yielding process of semi-crystalline nanocomposites in addition to thermal stability of crystallites at least two factors must be taken into account: a) the complexity of the structure of these nanocomposites and the availability of the mutual inﬂuence on its

395

Structure and Properties of Crosslinked Polymers different regions and b) the modiﬁcation of the mentioned structure in its deformation process. Let us also note the practical importance of an increase in σY by 1.3–1.4 times for the considered nanocomposites in comparison with the initial HDPE [31].

σY, MPa 40 -1 -2

30 T

20 10 0

293

313

333

353

373 T, K

Figure 8.11 The dependences of yield stress σY on testing temperature T for initial HDPE (1) and nanocomposite HDPE/EP with EP content 2 mass percentage [1]

The instrumented impact tests according to the Sharpy method also demonstrated a change in the properties of HDPE/EP nanocomposites in comparison with the initial HDPE. Since the samples of HDPE and HDPE/EP nanocomposites are not broken in impact tests at room temperature then tests of samples with a sharp notch by length a = 0.5 mm of the same materials were carried out. The diagrams of load-time (P-t) for samples of HDPE and HDPE/EP nanocomposites with epoxy polymer content 2.5 mass percentage are adduced in Figure 8.12. Attention is drawn to both qualitative and quantitative changes of the P-t diagram for HDPE/EP nanocomposites. Firstly, if the initial HDPE breaks as ductile solids with reaching macroscopic yielding then the samples of nanocomposite showed brittle fracture. This difference is explained within the frameworks of elastic fracture linear mechanics. As it is known [37], a ductility of samples with a notch is controlled by local plasticity, which is realised at a notch tip. The size of the local plasticity zone strongly depends on the σY value and is reciprocal to its square. Increasing σY for HDPE/EP nanocomposites (see Figure 8.7)

396

Polymer-polymeric Nanocomposites results in the volume of plastically deformed material decreasing and, consequently, in the reduction of the impact energy level dissipated in the local plasticity zone and, ﬁnally, in a decrease in impact toughness Ap. The last effect is obvious from the data of Figure 8.12 since the Ap value is proportional to the area under the P-t diagram. Secondly, a decrease in Ap is deﬁned by a sharp reduction in the time up to fracture tf, i.e., a reduction in the HDPE/EP nanocomposite sample deformability. So, the impact toughness of HDPE/EP nanocomposite, containing 2.5 mass percentage epoxy polymer, reduces in comparison with the initial HDPE by about 3.3 times (7.30 and 24.7 kj/m2, respectively) with a decrease in tf of about 2.5 times (0.96 and 2.38 ms, respectively). Thirdly, an increase in the fracture stress σf for HDPE/EP nanocomposite in comparison with the initial HDPE is observed. This observation is explained by the following: by higher values of the elasticity modulus E and by stronger local restriction of plastic deformation for the studied nanocomposite [38]. Let us note that from the practical point of view a decrease in Ap due to introduction of epoxy polymer has no particular signiﬁcance, since the samples of HDPE/EP nanocomposites, as well as the initial HDPE, are not fractured in impact tests of samples without a notch [1].

P, H 400

2 1

300 200 100

0

1

2

3

4 t, ms

Figure 8.12 The diagrams of load-time (P-t) for samples with a sharp notch by length 0.5 mm of initial HDPE (1) and HDPE/EP nanocomposite with epoxy polymer content 2.5 mass percentage (2). Testing temperature 293 K, strain rate 77 s–1 [1]

397

Structure and Properties of Crosslinked Polymers

8.4 The Diffusive Characteristics of HDPE/EP Nanocomposite It was shown above (Figure 8.8) that for HDPE/EP nanocomposites at epoxy polymer contents cEP = 0–5 mass percentage the extreme reduction of permeability to gas P by oxygen and nitrogen with minimum at cEP ≈ 3 mass percentage was observed. It is supposed [30] that this effect is due to ﬁlling of free volume microvoids, through which gas transport processes in polymers are realised [39], by epoxy polymer particles, that excludes the indicated microvoids from the gas transport process and thereby decreases the P value. The authors of paper [40] gave the quantitative description of the mentioned effect within the frameworks of the fractal concept of gas transport processes in polymers [41, 42]. In Figure 8.13 the dependences of the permeability to gas coefﬁcient by oxygen PO2 and nitrogen PN 2 on epoxy polymer contents cEP for HDPE/EP nanocomposites are adduced.

PO2 ×1015, PN2 ×1015 15

-1 -2

10

5

0

1

2

3

4

5 cEP, mass%

Figure 8.13 The dependences of the permeability to gas coefﬁcients by oxygen PO2 (1) and nitrogen PN 2 (2) on epoxy polymer contents cEP for HDPE/EP nanocomposites [40]

398

Polymer-polymeric Nanocomposites As it was noted above, the minimum of the permeability to gas coefﬁcient for both gases at cEP ≈ 3 mass percentage was observed. Let us consider the quantitative analysis of this effect. Within the frameworks of the fractal model of gas transport processes the permeability to gas coefﬁcient P is determined by the following equation [41, 42]:

(8.13)

where P0 is the universal constant, which is equal to 1.35 × 10–14 (moles m)/(m2 s Pa), fg is a relative ﬂuctuation free volume, dh is a diameter of free volume microvoid, dm is a gas-penetrant molecule diameter, Df is the excess energy localisation regions dimension, ds is the structure spectral dimension. Since the values of fg and Df are related to structural characteristics then they were determined through the value of structure fractal (Hausdorff) dimension df, calculated according to Equation 5.4. In turn, for fg estimation the relationship was used [19]:

(8.14)

and for calculation of dimension Df Equation 5.73 was used. The values dm for O2 and N2 are equal to 2.92 and 3.22 Å, respectively [43]. Since gas transport processes are realised on the molecular level [39] then the polymer macromolecule spectral dimension was accepted as ds, which for linear HDPE is equal to 1.0 [44]. Using the indicated estimations and experimental values of PO2 and PN 2 , one can calculate dh values, the dependence of which on cEP is adduced in Figure 8.14. As was expected, at cEP ≈ 3 mass percentage the minimum of dh is observed. Let us note that a small (about 8%) decrease in dh causes two-fold reduction in PO2 and PN 2 . Such behaviour is explained by the power character of the function in Equation 8.13, where the exponent 2(Df – ds)/ds is changed within the range of 5.08–6.36. Introduction of EP in HDPE results in linear reduction of the crystallinity degree K from 0.68 for the initial HDPE to 0.46 for the HDPE/EP nanocomposite with epoxy

399

Structure and Properties of Crosslinked Polymers polymer content 5 mass percentage. The authors [39] proposed the following form of the dependence of free volume microvoid volume Vh on the K value:

(8.15)

where Tm and T are the polymer melting and testing temperatures, respectively. T

T

In Figure 8.14 the dependence of d h (where d h is calculated according to values of Vh obtained from Equation 8.15) on cEP is also adduced. In Figure 8.14 the experimental values of dh according to the data of positron spectroscopy [45] for high (K = 0.71) and low (K = 0.45) density polyethylenes are indicated by horizontal dashed lines, which correspond sufﬁciently well to the estimations according to Equation 8.15.

T

dh, dh,Å

-1 -2 -3

7

5 6 4 5

4

0

1

2

3

4

5 cEP, mass%

Figure 8.14 The dependences of free volume microvoid diameter dh on epoxy polymer contents cEP for HDPE/EP nanocomposites. The calculation of dh T according to Equation 8.13 for oxygen (1) and nitrogen (2) and d h according to Equation 8.15 (3). The horizontal dashed lines indicate experimental values of dh for polyethylenes of high (4) and low (5) density [40]

400

Polymer-polymeric Nanocomposites As follows from the data of Figure 8.14, a large reduction in dh for HDPE/EP nanocomposites in comparison with hypothetical polyethylene having K = 0.46 is observed. As the estimations according to Equation 8.13 have shown, absence of this effect resulted in an increase in PO2 and PN 2 of about one order of magnitude in comparison with the initial HDPE. In papers [46, 47] the multifractality of free volume microvoids for solid-phase polymers and trivial monofractality (with df = d) for these polymers melt was shown. In Figure 8.15 the corresponding diagrams f-α are adduced, where α is a scaling exponent, characterising singularities concentration, f is the singularities α dimension, equal to [48]:

(8.16)

f 2 1

1,0 0,8

3

0,6 0,4 0,2 0

2

4

6

8

10

α

Figure 8.15 Multifractal diagrams f-α for the solid-phase HDPE (1); its melt (2); and HDPE/EP nanocomposite with epoxy polymer content 3 mass percentage at dh > 4.4 Å (3) [40]

401

Structure and Properties of Crosslinked Polymers In paper [46] the dependence df(dm) was also obtained, from which the growth in df with increase in dm or rising of the measurement scale follows. This is a typical sign of solid-phase HDPE structure multifractality [48]. From the plot of df(dm) the value of df corresponding to the minimum value of dh for HDPE/EP nanocomposites (dh = 4.4 Å) can be determined, which is equal to 2.75. Then according to Equation 8.16 the value f = 0.75, which is shown in Figure 8.15 by a horizontal solid line. This means that in the HDPE case of EP nanocomposites all microvoids with dh > 4.5–4.9 Å depending on epoxy polymer contents or df > 2.75 are ‘cut’ from the multifractal spectrum. The multifractal diagram for HDPE/EP nanocomposite with epoxy polymer content 3 mass percentage obtained from spectra 1 and 3 of Figure 8.15 is adduced in Figure 8.16.

f 1,0 0,8 0,6 0,4 0,2 0

2

4

6

8

10

α

Figure 8.16 The combined multifractal diagram f-α for HDPE/EP nanocomposite with epoxy polymer content 3 mass percentage [40]

As one can see, this diagram is multifractal for dh ≤ 4.4 Å and monofractal for dh > 4.4 Å. According to the data of paper [39] the probability of detection in HDPE microvoids with dh > 4.4 Å can be estimated as being equal to 0.18. This means that the exclusion from gas transport processes of 18% (by volume) of free volume of the largest microvoids results in twice the reduction of permeability to gas coefﬁcient. It was indicated above that the HDPE/EP nanocomposite with cEP ≈ 3 mass percentage has melt viscosity about three times smaller than the initial HDPE melt viscosity, i.e.,

402

Polymer-polymeric Nanocomposites much better processing. This allows the indicated nanocomposite to be recommended as the industrial material for the production of articles with low permeability to gas. Therefore, the quantitative analysis fulﬁlled above has conﬁrmed that reduction in the permeability to gas coefﬁcient for HDPE/EP nanocomposites is due to ﬁlling by epoxy polymer the largest free volume microvoids that excludes them from the gas transport process. The multifractal treatment of this effect is offered and the combined diagram in coordinates f-α is constructed, which is multifractal for dh ≤ 4.4 Å and monofractal for dh > 4.4 Å. As was shown in Section 2.6 of the monograph [24], the intensive degradation in air temperature, by which a sample mass 5% loss temperature T5% is accepted, is closely connected with the oxidiser diffusion on the nanocomposite structure. The authors [49] analysed the change in T5% for nanocomposites on the basis of HDPE, containing two sorts of nanoparticles: EP and an ultra-ﬁne mixture of Fe/FeO(Z); the last content was constant and equal to 0.1 mass percentage. The model of unusual (anomalous) diffusion was used as theoretical ground [50]. The two main types of unusual (anomalous) diffusion on fractal objects can be distinguished [51]: slow and fast diffusion. As a basis for such division the dependence of mobile reagent displacement s on time t is assumed [51]:

(8.17)

where for the classical case (Gaussian diffusion) β = 1/2, for slow diffusion β < 1/2 and for fast diffusion β > 1/2. As was shown in paper [52], polymer melt structure in a range of high temperatures, corresponding to T5%, could be characterised most precisely by a macromolecular coil dimension Δf, which was accepted as being equal to the corresponding dimension df of solid-phase polymer structure [53]. Earlier within the frameworks of fractional derivatives theory the interconnection of Δf and β was shown, which was expressed analytically as follows [51]:

(8.18) for slow diffusion and

403

Structure and Properties of Crosslinked Polymers

(8.19)

for fast diffusion. The value Δf = 2.5 should be considered as the structural boundary between the indicated diffusion types at common variation as 2.0 ≤ Δf < 3. At Δf < 2.5 (less compact macromolecular coils) an oxidiser (oxygen) fast diffusion is realised and at Δf > 2.5 a slow diffusion [54]. For estimation of the value of T5% the following equation was used [55]:

(8.20)

where c is a constant, equal to 0.093 for fast diffusion and 0.305 for slow diffusion [55]. Equation 8.20 deﬁnes the three factors effecting the thermal stability of polymeric materials: the polymer chemical structure, characterised by the Tm value, the polymer melt structure, characterised by the dimension Δf, and the type (intensity) of oxidiser diffusion, connected with the structure and characterised by the exponent β [54, 55]. In Figure 8.17 theoretical dependences T5%(cEP) for the cases of oxidiser slow and fast diffusion and also experimental values of T5% are adduced. As one can see, T5% for the initial HDPE corresponds to the slow diffusion case and for HDPE/ EP nanocomposites to fast diffusion. This transition is due to the availability of Z particles in the HDPE structure, which are traps for oxidiser molecules and interrupt their walk trajectory, owing to the structure connectivity being violated and broken into a number of up ‘substructures’ [24]. An oxidiser walk is allowed only within the limits of such ‘substructure’ [50]. Comparison of paper [24] and Figure 8.17 data shows a number of interesting features of the behaviour of HDPE/EP nanocomposites at the thermal degradation in the air. Firstly, the values of T5% for HDPE/EP nanocomposites is about 20–40 K higher than the corresponding values for HDPE+Z at the same Z content. It is obvious that this distinction is due to the structural changes deﬁned by EP introduction. Secondly, the theoretical curves 1 and 2 for HDPE/EP nanocomposites have shown an increase in

404

Polymer-polymeric Nanocomposites T5% with growth in cEP, whereas for HDPE+Z nanocomposites contents an increase in Z results in reduction of the theoretical (purely structural) value of T5% (see Figure 2.42 in monograph [24]). Thirdly, attention is drawn to the fact that the experimental values of T5% exceed the theoretical ones of the value of ΔT5% ≈ const. This effect is also due to availability of Z particles and the value of ΔT5%, and as in paper [24], can be determined according to the equation:

(8.21)

where ΔT5% is given in K, and Z contents cz in mass percentage.

T5%, K 700

I 3

650 2 1 600 4 550

0

1

2

3

4

5 cEP, mass%

Figure 8.17 The dependences of temperature T5% on epoxy polymer contents cEP for HDPE/EP nanocomposites. The calculation: (1) slow diffusion and (2) fast diffusion; (3) according to Equations 8.20 and 8.21; (4) the experimental data [49]

In this case the theoretical T5% estimation gives good correspondence to experiment (the dashed line in Figure 8.17).

405

Structure and Properties of Crosslinked Polymers

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A.K. Mikitaev, G.V. Kozlov and G.E. Zaikov in Polymer Nanocomposites: Variety of Structural Forms and Applications, Nova Science Publishers, Inc., New York, NY, USA, 2008, p.318.

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A.K. Malamatov, G.V. Kozlov, A.I. Burya and A.K. Mikitaev, Novosti Nauki Pridneprov’ya, 2005, 5, 35.

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I.R. Kuzeev, G.K. Samigullin, D.V. Kulikov and M.M. Zakirnichnaya in Complex Systems in Nature and Engineering, USNTU, Ufa, Russia, 1997, p.225.

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G.V. Kozlov and D.S. Sanditov in Anharmonic Effects and PhysicalMechanical Properties of Polymers, Nauka, Novosibirsk, Russia, 1994, p.261.

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G.V. Kozlov, V.A. Beloshenko, V.N. Varyukhin and V.U. Novikov, Zhurnal Fizicheskikh Issledovaniy, 1997, 1, 2, 204.

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G.V. Kozlov and G.E. Zaikov in Structure of the Polymer Amorphous State, Brill Academic Publishers, Utrecht-Boston, 2004, p.465.

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R. Seguela and F. Rietsch, Polymer, 1986, 27, 5, 703.

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L. Mandelkern in Polymers Crystallization, Khimiya, Leningrad, Russia, 1966, p.336.

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B. Wunderlich, Macromolecular Physics, 1977, 2, 570.

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Polymer-polymeric Nanocomposites 15. I. Narisawa in Strength of Polymeric Materials, Khimiya, Moscow, Russia, 1987, p.400. 16. A.K. Malamatov and G.V. Kozlov in the Proceedings of 4th International Interdisciplinary Symposium ‘Fractals and Applied Synergetics’, FaAS-2005, Interkontakt Nauka, Moscow, Russia, 2005, p.119. 17. D.J. Bergman and Y. Kantor, Physical Review Letters, 1984, 53, 6, 511. 18. G.V. Kozlov, M.A. Gazaev, V.U. Novikov and A.K. Mikitaev, Pis’ma v ZhETF, 1996, 22, 16, 31. 19. G.V. Kozlov and V.U. Novikov, Uspekhi Fizicheskikh Nauk, 2001, 171, 7, 717. 20. A.N. Bobryshev, V.N. Kozomazov, L.O. Babin and V.I. Solomatov in Synergetics of Composite Materials, NPO ORIUS, Lipetsk, Russia, 1994, p.153. 21. S. Wu, Journal of Polymer Science, Part B: Polymer Physics Edition, 1989, 24, 4, 723. 22. W.W. Graessley and S.F. Edwards, Polymer, 1981, 22, 10, 1329. 23. S.M. Aharoni, Macromolecules, 1985, 18, 12, 2624. 24. A.K. Mikitaev, G.V. Kozlov and G.E. Zaikov in Polymer Nanocomposites: Variety of Structural Forms and Applications, Nauka, Moscow, Russia, 2009, p.278. 25. V.D. Kargin and T.I. Sogolova, Zhurnal Fizicheskoy Khimii, 1953, 27, 3, 1039. 26. R. Popli and L. Mandelkern, Journal of Polymer Science, Part B: Polymer Physics Edition, 1987, 25, 3, 441. 27. L. Mandelkern, Polymer Journal, 1985, 17, 2, 337. 28. W. Glenz and A. Peterlin, Journal of Macromolecular Science: Physics, 1970, B4, 3, 473. 29. A.R. Wedgewood and J.C. Seferis, Pure and Applied Chemistry, 1983, 55, 5, 873.

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Structure and Properties of Crosslinked Polymers 30. G.V. Kozlov, G.B. Shustov and N.I. Mashukov in the Proceedings of the International Science-Applications Conference ‘Modern Materials and Technologies-2002’, PSU, Penza, Russia, 2002, p.20. 31. A.K. Malamatov, G.V. Kozlov and A.K. Mikitaev, Izvestiya KBNC RAN, 2006, 1, 142. 32. R.M. Khalikov and G.V. Kozlov, Vysokomolekulyarnye Soedineniya Seriya B, 2006, 48, 4, 699. 33. G.V. Kozlov and G.E. Zaikov, Vysokomolekulyarnye Soedineniya Seriya B, 2003, 45, 7, 1197. 34. G.V. Kozlov and G.E. Zaikov, Journal of Applied Polymer Science, 2004, 92, 6, 3761. 35. S. Matsuoka, H.E. Bair, S.S. Bearder and C.J. Aloisio, Polymer Engineering and Science, 1978, 18, 14, 1073. 36. G.V. Kozlov and A.K. Mikitaev, Plasticheskie Massy, 1987, 12, 49. 37. V.U. Novikov and G.V. Kozlov in Analysis of Polymers Fracture within the Frameworks of Fractals Conception, MSOU, Moscow, Russia, 2001, p.136. 38. G.V. Kozlov, V.D. Serdyuk and V.A. Beloshenko, Mekhanika Kompozitnykh Materialov, 1994, 30, 5, 691. 39. G.V. Kozlov, D.S. Sanditov and Y.S. Lipatov in Achievements in Polymers Physics-Chemistry Field, Ed., G.E. Zaikov, Khimiya, Moscow, Russia, 2004, p.412. 40. A.K. Malamatov, G.V. Kozlov, M.K. Ligidov and S.I. Pakhomov, Izvestiya Vuzov, Khimiya i Khimicheskaya Tekhnologiya, 2006, 49, 4, 59. 41. G.V. Kozlov, G.E. Zaikov and A.K. Mikitaev in The Fractal Analysis of Gas Transport in Polymers: The Theory and Practical Applications, Nova Science Publishers, Inc., New York, NY, USA, 2009, p.238. 42. G.V. Kozlov, G.E. Zaikov and A.K. Mikitaev in The Fractal Analysis of Gas Transport in Polymers: The Theory and Practical Applications, Nauka, Moscow, Russia, 2009, p.199. 43. V.V. Teplyakov and S.G. Durgar’yan, Vysokomolekulyarnye Soedineniya Seriya A, 1984, 26, 7, 1498.

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Polymer-polymeric Nanocomposites 44. S. Alexander and R. Orbach, Journal of Physical Letters (Paris), 1982, 43, 17, L625. 45. D. Lin and S.J. Wang, Journal Physics: Condensed Matter, 1992, 4, 16, 3331. 46. G.V. Kozlov, Journal of the Balkan Tribological Association, 2003, 9, 2, 232. 47. G.V. Kozlov, Z.I. Afaunova and G.E. Zaikov, Oxidation Communications, 2005, 28, 4, 856. 48. F. Feder in Fractals, Plenum Press, New York, NY, USA, 1989, p.250. 49. A.K. Malamatov and G.V. Kozlov in the Proceedings of the International Science-Applications Conference ‘Problems of Machinery Investigation and Designing’, PSU, Penza, Russia, 2005, p.163. 50. L.M. Zelenyi and A.V. Milovanov, Uspekhi Fizicheskikh Nauk, 2004, 174, 8, 809. 51. V.K. Shogenov, A.A. Akhkubekov and R.A. Akhkubekov, Izvestiya Vuzov, 2004, 1, 46. 52. G.V. Kozlov, I.V. Dolbin and G.E. Zaikov, Journal of Applied Polymer Science, 2004, 94, 4, 1353. 53. G.V. Kozlov, K.B. Temiraev, G.B. Shustov and N.I. Mashukov, Journal of Applied Polymer Science, 2002, 85, 6, 1137. 54. R.M. Khalikov and G.V. Kozlov, Inzhenernaya Fizika, 2005, 3, 20. 55. I.V. Dolbin, A.I. Burya and G.V. Kozlov, Teploﬁzika Vysokikh Temperatur, 2007, 45, 3, 355.

409

9

Crosslinked Epoxy Polymers as Natural Nanocomposites

At present it is obvious that polymeric systems by virtue of the special features of their structure are always nanostructural systems [1]. However, treatment of such a structure can be different. The authors [2–4] used for this purpose the cluster model of the amorphous state structure of polymers [5–7], which supposes that the indicated structure consists of local order domains (clusters), immersed into a loosely packed matrix. In this case the latter is considered as a natural nanocomposite matrix and the clusters as a nanoﬁller. A cluster presents itself as a set of several densely packed collinear segments of various macromolecules with sizes up to several nanometres [5–7]. It has been shown that such clusters are true nanoparticles – the nanoworld objects (nanoclusters) [2]. In the present chapter the formation of the structure of these materials and the properties deﬁned by it will be studied within the frameworks of such a representation of the structure of crosslinked epoxy polymers using fractal analysis methods.

9.1 Formation of the Structure of Natural Nanocomposites The treatment of an epoxy polymer as a natural nanocomposite or quasi-two-phase system [8] puts in the foreground the interaction of such system components, which for nanocomposites is expressed ﬁrst of all in an interfacial regions formation [2–4]. Let us note that in the reinforcement process (increase in the elasticity modulus of the nanocomposite in comparison with the matrix polymer) interfacial regions play the same role as the nanoﬁller [2–4]. Such a reinforcement mechanism is due to the formation of nanocomposites with inorganic nanoﬁller [2–4] and the structure of natural nanocomposites (linear amorphous polymers) [9] in three-dimensional Euclidean space. Therefore in paper [10] the study of structure formation conditions for crosslinked epoxy polymers, treated as natural nanocomposites, was carried out within the frameworks of fractal analysis. Formation of the structure of polymers (or composites polymeric matrix) can be realised either in Euclidean space or in fractal space. So, for particulate-ﬁlled polyhydroxyether/graphite composites it was shown that their matrix structure

411

Structure and Properties of Crosslinked Polymers formation was realised in fractal space, where ﬁller particles (aggregates of particles) network is formed [11]. To estimate the space dimension (lattice analogue in computer simulation) Dsp, in which the polymer structure is formed, the following equation allows [11]:

(9.1)

where νF is the Flory exponent, which for densely crosslinked systems is determined as follows [12]:

(9.2) where df is the structure fractal dimension. The nanocluster structure formation occurs at lower glass transition temperature than that of a polymer [13] at oligomer-curing agent mixture curing, i.e., the dimension m of the loosely packed matrix structure d f can be considered as Dsp. This dimension can be calculated according to the mixtures rule:

(9.3)

where is the clusters dimension, which in virtue of their dense packing is accepted as being equal to the greatest dimension for real solids, namely, = 2.95 [14], ϕcl is the relative fraction of clusters, which can be calculated with the aid of Equation 4.8. m

In Figure 9.1 the comparison of dimensions d f and Dsp for the studied EP is adduced. Their good correspondence indicates unequivocally that their loosely packed matrix, which serves simultaneously as a natural nanocomposite matrix, is the fractal space where the nanocluster structure of epoxy polymers is formed. Since for linear amorphous polymers Dsp = 3 [9], i.e., their nanostructure formation is realised in three-dimensional Euclidean space, then the conclusion that chemical crosslinking network availability in the considered EP serves as the indicated distinction cause is obvious enough. In Figure 9.2 the dependence of Dsp on crosslinking density νc is 412

Crosslinked Epoxy Polymers as Natural Nanocomposites adduced, which allows interesting conclusions to be made. As one can see, for the considered EP the greater part of the value of Dsp is close to 2.5 and decreases slowly with νc growth. Let us be reminded that the structure of crosslinked polymers is simulated by a set of a large number of Witten–Sander clusters, the dimension of which is equal to about 2.5 [15]. However, for EP 4 samples with values Dsp = 1.67–2.18 the dependence Dsp(νc) changes its character and then faster decay in Dsp with νc reduction is observed. In Figure 9.2 the border value Dsp ≈ 2.12, corresponding to a change in the formation mechanism of the EP loosely packed matrix structure, from a particle-cluster mechanism (Dsp > 2.12) to a cluster-cluster mechanism (Dsp < 2.12) [16], is indicated by a horizontal dashed line. Hence at sufﬁciently high degrees of chemical crosslinking the formation of the loosely packed matrix structure occurs not by particles (statistical segments) joining, but by consolidation of these groups of segments. Let us draw attention to the fact that for EP-1 samples the values of Dsp = 1.67 and 1.88, whereas for EP-1-200 Dsp = 2.0 and 2.24 at the same values of Kst. The values of Dsp < 2 assume the formation of discontinuities or ruptures (defects) in the loosely packed matrix structure that is observed for EP-1 samples prepared at atmospheric pressure. At the same time EP-1-200 samples, cured at hydrostatic pressure 200 MPa, have no such discontinuities (Dsp ≥ 2.0). Let us note that not only the pressure application effects the formation of the indicated discontinuities (and, hence, the value Dsp), but also the curing agent type: for EP-2 samples Dsp ≥ 2.22 [10].

d mf 3.0

2.5 -1 -2 -3 -4

2.0

1.5

2.0

2.5

3.0

Dsp m

Figure 9.1 Comparison of the fractal dimensions of a loosely packed matrix d f and a space in which a nanocluster structure is formed, Dsp, for epoxy polymers EP-1 (1); EP-1-200 (2); EP-2 (3); and EP-2-200 (4) [10]

413

Structure and Properties of Crosslinked Polymers The formation of discontinuities in epoxy polymers EP-1 are conﬁrmed by the data of Figure 9.3, where the dependences of the density ρ of samples of epoxy polymers on Kst are adduced. The solid curve on this ﬁgure gives the theoretical dependence ρ(Kst), calculated according to the mixtures rule [10]:

(9.4)

where ρcl and ρl.m. are the densities of clusters and the loosely packed matrix, which are accepted as being equal to 1300 and 1200 kg/m3, respectively.

Dsp

2.5

2.0

1.5

1.0 0

10

20 ν ×10-26, m-3 ɫ

Figure 9.2 The dependence of dimension Dsp of space, in which nanocluster structure is formed, on the crosslinking density νc. The designations are the same as in Figure 9.1. The horizontal dashed line indicates the border dimension Dsp for particle-cluster and cluster-cluster mechanisms [10]

The accuracy of such a choice is conﬁrmed by the correspondence of experimental values of ρ and those calculated according to Equation 9.4 at Kst = 0.50, 0.75 and 1.50. However, at Kst = 1.0 and 1.25, i.e., at the least values of Dsp ≤ 2.18 a discrepancy

414

Crosslinked Epoxy Polymers as Natural Nanocomposites between theory and experiment is observed, namely the experimental ρ values are smaller than those calculated according to the mixtures rule, moreover the smaller Dsp is the larger is this discrepancy.

ρ, kg/m3 1280

1

1240 -2 -3 1200 0.5

1.0

1.5 Ʉst

Figure 9.3 The dependences of density ρ on the value of Kst for epoxy polymers EP-1. (1) calculation according to Equation 9.4; 2, 3 – the experimental data for EP-1 (2); and EP-1-200 (3) [10]

The state of the structure of the polymeric material, which is deﬁned by its formation mechanisms, always inﬂuences the properties of the indicated materials. Therefore the analysis fulﬁlled above has a purely applied interest as well. For conﬁrmation of this postulate on Figure 9.4 the dependence of the elasticity modulus E of the considered EP on (such a form of the dependence E(Dsp) was chosen for the purpose of its linearisation) is adduced. As one can see, the linear correlation between E and Dsp was obtained, described analytically by the following empirical equation [10]:

(9.5)

415

Structure and Properties of Crosslinked Polymers

ȿ, GPɚ 5.0

2.5

0

4

8

Dsp

Figure 9.4 The dependence of the elasticity modulus E on the dimension Dsp of a space in which a nanocluster structure is formed. The designations are the same as in Figure 9.1 [10]

Hence, to receive the greatest values of E one should press for formation of the structure of epoxy polymers in space with a higher dimension of Dsp. So, the greatest value of E at nanostructure formation in Euclidean space with Dsp = 3 is equal to 5.04 GPa. Let us note that the generally accepted opinion about the increase in the elasticity modulus of cured systems with raised crosslinking density is not correct. So, for the considered EP at νc = (2–4) × 10–26 m–3 the value E = 3.2–4.3 GPa and at νc = (17–19) × 10–26 m–3 E = 2.3–2.8 GPa, i.e., an increase in the crosslinking density of 6 times results in the reduction of the elasticity modulus by about 1.5 times. Hence, the results stated above have shown that nanocluster structure formation for the considered epoxy systems is realised in fractal space (analogue of fractal lattice in computer simulation), which is created by a loosely packed matrix. The inﬂuence of the crosslinking density on the indicated space dimension is not unequivocal and is deﬁned by the aggregation mechanism, which is realised at nanostructure formation. This space dimension deﬁnes unequivocally the elasticity modulus value of the considered epoxy polymers. Nanoclusters do not possess three-dimensional symmetry and it was shown [17] that statistical segments in them could be displaced by each relatively an other lengthwise, which assumes their rough surface. This roughness degree, which can be estimated with the aid of the surface fractal dimension dsurf, affects the contact on the nanocluster

416

Crosslinked Epoxy Polymers as Natural Nanocomposites loosely packed matrix (analogue of the nanoﬁller-polymer matrix contact), which, in turn, inﬂuences the properties of polymer composites on the whole. Therefore the authors of paper [18] developed determination methods of surface fractal dimension for nanoclusters in epoxy polymers. Earlier for a wide set of sooty and silicate ﬁllers for rubbers an increase in the speciﬁc surface Su of their particles with a reduction in the diameter of the particles Dp was shown [2]. It is signiﬁcant that in this case very strong growth (of about one order) in Su is observed within the narrow range Dp = 10–35 nm. This assumes very large values of Su for nanoclusters, which have sizes (diameter Dp) of less than 1 nm. For dsurf determination the authors [3] used the following relationship:

(9.6)

where d is the dimension of Euclidean space in which a fractal is considered (it is obvious that in our case d = 3), a proportionality coefﬁcient was chosen equal to 4.10 × 105 m2/kg and Dp values are given in nm. However, for nanoclusters such parameters in Relationship 9.6 are impossible to apply, since they give a decrease in dsurf with growth in Su or reduction in Dcl using the following formula [19]:

(9.7)

where ρcl is the nanocluster density, which is accepted, as above, as being equal to 1300 kg/m3 for EP. As it was supposed, calculation according to Equation 9.7 gives for nanocomposites very high values of Su within the range of (1.7–4.5) × 106 m2/kg (Figure 9.5). Let us note that for the sooty and silicate nanoﬁllers mentioned above at Dp = 10–100 nm the value of Su varies within the range of (0.4–4.0) × 105 m2/kg. For the choice of proportionality coefﬁcient in Relationship 9.6, where values Dp = Dcl are given in Å, the following method was used. It is assumed that the greatest possible

417

Structure and Properties of Crosslinked Polymers value dsurf = d = 3 is reached at the lowest nanocluster diameter Dcl and in this case the number of segments in the cluster ncl = 2. Further the value of Dcl = 2Rcl for ncl = 2 can be determined according to the equation [19]:

(9.8)

where S is the macromolecule cross-sectional area, which is equal to ~ 32 Å for the considered EP [20], η is the packing coefﬁcient, which is equal to 0.868 in the case of dense packing.

Su×10-6, m2/kg 4

-1 -2 -3 -4

3

2

1 1.2

2.0

2.8 Dcl, nm

Figure 9.5 The dependence of speciﬁc surface Su of nanoclusters on their diameter Dcl for epoxy polymers EP-1 (1); EP-1-200 (2); EP-2 (3); and EP-2-200 (4) [18]

The value of Su for nanocluster EP can be calculated according to Equation 9.7 and used as the proportionality coefﬁcient in Relationship 9.6. The dependence dsurf(ncl) for the considered EP is adduced in Figure 9.6. As one can see, the strong decay in dsurf with an increase in ncl is observed. This shows that an increase in the number of statistical segments ncl (or an increase in Dcl) results in their denser packing and a reduction in the degree of their surface roughness. One should pay attention to one

418

Crosslinked Epoxy Polymers as Natural Nanocomposites principal distinction in the structure of inorganic (sooty and silicate) nanoparticles and polymer nanoclusters. If the ﬁrst from the indicated nanoparticles have both fractal bulk and fractal surface, i.e., they are bulk fractals, then nanoclusters by virtue of their dense packing possess a structure dimension close to Euclidean d = 3, but a fractal surface, i.e., they are surface fractals.

dsurf 2.9 -1 -2 -3 -4 2.7

2.5 0

5

10

15

ncl

Figure 9.6 The dependence of the surface fractal dimension dsurf of nanoclusters on the number of segments in one nanocluster ncl. The designations are the same as in Figure 9.5 [18]

Equation 9.6 at the proportionality coefﬁcient 5.27 × 106 m2/kg used above is correct in the case of inorganic nanoﬁller particles as well. So, for particles of diameter Dp = 35 nm according to Equation 9.7 we obtain Su ≈ 1.70 × 105 m2/kg and according to Equation 9.6 (at ρ = 1000 kg/m3) dsurf ≈ 2.41, which corresponds well to the experimental data for such particles [2]. For the greatest possible diameter of Dcl in the case of a nanocluster consisting of ncl = 100, which is equal to 67 Å, we obtain a similar mode of Su = 7.50 × 105 m2/kg and dsurf = 2.54, which is close to the indicated parameters for Dcl = 30.4 Å. In other words, for Dcl > 25 Å or ncl > 13 (Figures 9.5 and 9.6) the dependences Su(Dcl) and dsurf(ncl) try to attain their asymptotic magnitudes. Let us also note that calculation of dsurf according to Equations 9.6 and 9.7 gives practically the dependence dsurf only on its dimensional parameter (ρ variation is small), i.e., dsurf change for nanoclusters is a true nanoeffect.

419

Structure and Properties of Crosslinked Polymers Thus, the simple method of estimation of the surface fractal dimension dsurf of nanoclusters for the structure of crosslinked epoxy polymers, which are considered as natural nanocomposites, was offered. The lower boundary of dsurf ≈ 2.55 indicates that packing of nanoclusters is less dense in comparison with an ideal one, for which dsurf = 2.0 was expected. Unlike inorganic nanoﬁller nanoparticles, nanoclusters in polymers are surface fractals. As has been indicated above, the treatment of a polymer as a natural nanocomposite puts in the foreground the interaction of such structure components, which for nanocomposites is expressed ﬁrst of all in the formation of interfacial regions [2–4]. Proceeding from this the authors of paper [21] fulﬁlled the identiﬁcation of interfacial regions in natural nanocomposites (crosslinked polymers) on the example of two epoxy systems. For calculation of the relative fraction ϕif of interfacial regions the following formula was used [22]:

(9.9)

This formula has a universal character. It was received for the description of the structure of particulate-ﬁlled polymer composites [23] and applied successfully in the case of polymer/organoclay nanocomposites [3] and natural nanocomposites (linear amorphous polymers) [9]. For calculation of parameters Dsp and dsurf Equations 9.1 and 9.6 were used, respectively. The relative fraction ϕcl of clusters can be calculated with the aid of Equation 4.8 and the loosely packed matrix relative fraction ϕl.m., which is considered as a natural nanocomposite matrix, was determined according to the following simple equation [7]:

(9.10)

In Figure 9.7 the comparison of relative fractions of the loosely packed matrix ϕl.m. calculated by the indicated methods and the interfacial regions ϕif for epoxy systems EP-1 and EP-2 is adduced. As one can see, good correspondence was obtained between the indicated parameters (the average discrepancy between ϕl.m. and ϕif makes up 8%). This means that in natural nanocomposites, which are crosslinked

420

Crosslinked Epoxy Polymers as Natural Nanocomposites polymers, the regions that differ structurally from the loosely packed matrix, i.e., the natural nanocomposite matrix, are absent. Thus, a crosslinked epoxy polymer can be considered as a speciﬁc nanocomposite consisting of two structural components: nanoﬁller (nanoclusters) and matrix (polymer loosely packed matrix).

ϕl.m., ϕif 0.8 2 1

0.4 -3 -4 0 0.5

1.0

1.5 Ʉst

Figure 9.7 The dependences of the loosely packed matrix ϕl.m. (1, 2) and relative fractions of interfacial regions ϕif (3, 4) on the value of Kst for EP-1 (1, 3) and EP-2 (2, 4) [21]

Hence, the results stated above have demonstrated the absence of interfacial regions, structurally differing from the bulk matrix, in crosslinked epoxy polymers, which are considered as natural nanocomposites. This means that the structure of such nanocomposites is represented as a nanoﬁller (nanoclusters), immersed into a matrix (loosely packed matrix of a crosslinked polymer structure), i.e., unlike polymer nanocomposites with inorganic nanoﬁller (artiﬁcial nanocomposites) they have only two structural components. At present it is established that the structures of both natural and many model objects cannot be described with the aid of only one value of fractal dimension. For more precise description of disordered structures, including polymers, it is necessary to calculate a spectrum of different dimensions, i.e., to use the multifractal formalism [23–25]. At present a number of papers exist that show correspondence of either

421

Structure and Properties of Crosslinked Polymers multifractal characteristics or parameters of real materials [22, 26–29]. However, multifractal characteristics are common mathematical terms, which does not allow them to be directly identiﬁed as structural parameters. It is necessary to make such an identiﬁcation for each class of materials that can make multifractal formalism (which in this case it is more correct to call multifractal analysis) by a very useful complex method of investigation of the structure and properties of materials [30]. As was indicated in paper [31], Renyi generalised dimensions D∞ and D–∞ (at indices q = ∞ and q = –∞, respectively) of a multifractal diagram and characterised the most rareﬁed and the most concentrated sets of the system, respectively, and their difference is the degree of chaos in the system. Proceeding from this, the authors of paper [32] elucidated the interconnection of the degree of chaos and the structural characteristics of crosslinked epoxy polymers and its effect on the properties of the indicated materials. As has been noted above, the parameter Δ80, which is equal to the difference of the Renyi limiting dimensions D–40 and D40 (which are used in practice instead of D–∞ and D∞, respectively), characterises the degree of chaos in a system, i.e., in the structure of the considered epoxy polymers [31]. Renyi limiting dimensions D–40 and D40 for polymeric materials can be determined with the aid of the following equations [22]:

(9.11)

(9.12) where the parameters ϕcl and fg were calculated according to Formulae 6.15 and 8.14, respectively. In Figure 9.8 the dependence Δ80(νc) is adduced, according to which the fast growth of Δ80 with a small increase in νc with small νc (< ~ 10 × 1026 m–3) and reaching the plateau at higher values νc follows. Thus, an increase in the crosslinking level means raising of the degree of chaos in the structure of the considered epoxy polymers. The transition to the condition Δ80 ≈ const. at νc > 10 × 1026 m–3 mentioned above is explained as follows. As was shown in paper [33], for a crosslinked polymer two regimes of behaviour are observed: at Nst > 9 (Nst is the number of statistical segments on a chain part between chemical crosslinking nodes) a crosslinked polymer behaves similarly to a linear one and at Nst ≤ 9 segmental mobility suppression is observed (according to the notions of the cluster model of the amorphous state structure of polymers this means transition to the structure quasi-equilibrium state [6]). For

422

Crosslinked Epoxy Polymers as Natural Nanocomposites conﬁrmation of this assumption the authors [32] estimated the value of Nst for the considered epoxy polymers at νc = 10 × 1026 m–3, i.e., at transition to plateau of the dependence Δ80(νc) (Figure 9.8). Calculations according to Equations 1.7, 1.8, 4.2, 4.6 and 5.31 gave the value Nst = 8.33, which is close to the border magnitude Nst = 9, adduced in [33]. This calculation conﬁrms that the dependence Δ80(νc) transition to plateau or the condition Δ80 ≈ const. is due to formation of the densely crosslinked epoxy polymer structure [33] or the indicated structure transition in the quasiequilibrium state [6]. The degree of chaos is the general characteristic of the structure and therefore it should inﬂuence both structural parameters and properties of epoxy polymers. It has been shown earlier that epoxy polymers can be considered as natural nanocomposites, for which the interfacial (intercomponent) adhesion level is one of the most important characteristics [34]. For polymer composites (nanocomposites) this property can be characterised by the parameter b, estimated according to the equation [35]:

(9.13) are linear thermal expansion coefﬁcients of epoxy polymers, where αEP, and obtained experimentally and calculated according to the mixtures rule and Terner’s equation, respectively.

Δ80

3.0

2.5 -1 -2 -3 -4

2.0

1.5 0

10

20 νɫ×10-26, m-3

Figure 9.8 The dependence of the degree of chaos Δ80 of the structure on the crosslinking density νc for epoxy polymers EP-1 (1); EP-1-200 (2); EP-2 (3); and EP-2-200 (4). The vertical shaded line indicates the value of νc at which transition to Δ80 ≈ const. occurs [32] 423

Structure and Properties of Crosslinked Polymers

and The detailed methods of calculation of are adduced in paper [35] and the experimental αEP values are accepted according to the data of paper [7]. In Figure 9.9 the dependence b(Δ80) is adduced, from which approximately quadric growth in b or raising of the intercomponent adhesion level with increasing Δ80 or the degree of chaos in the structure of the considered epoxy polymers follows.

b 0.4

0.2

0 1.5

2.0

2.5

3.0

Δ80

Figure 9.9 The dependence of parameter b on the degree of chaos of the structure Δ80 for epoxy polymers. The designations are the same as in Figure 9.8 [32]

Figure 9.10 gives a typical example of the dependence of the properties of epoxy polymers on the degree of chaos in the structure, where the correlation between elasticity modulus E and Δ80 is adduced. As one can see, the linear reduction in E with growth in Δ80 is observed. This dependence demonstrates clearly the inﬂuence of the the degree of chaos of the structure of epoxy polymers on their properties. The dependence E(Δ80) adduced in Figure 9.10 is approximated by the following empirical equation [32]:

(9.14)

424

Crosslinked Epoxy Polymers as Natural Nanocomposites

ȿ, GPɚ

6

3

0 1.5

2.0

2.5

3.0

Δ80

Figure 9.10 The dependence of the elasticity modulus E on the the degree of chaos of the structure Δ80 for epoxy polymers. The designations are the same as in Figure 9.8 [32]

As follows from Equation 9.14, the greatest value of E = 8.4 GPa can be received at the condition Δ80 = 0 or D–40 = D40, i.e., at the monofractal structure of the considered epoxy polymers. The data of Figure 9.8 demonstrate that even at νc = 0 Δ80 ≈ 1.20, i.e., the multifractal structure with E ≈ 5.9 GPa is realised. In other words, at the chosen curing regimes the monofractality of the structure of epoxy polymers is unattainable even theoretically. Let us consider in conclusion the physical grounds of the change in the the degree of chaos of the structure for the epoxy polymers studied. Figure 9.11 shows the dependence of Δ80 on the dimension Dsp of the space in which the nanostructure of epoxy polymers is formed and which is equal to their loosely packed matrix dimension (see above). As one can see, at small Dsp (less than ~ 2.35) the structures with a high degree of chaos (Δ80 ≈ const. ≈ 2.78) and, respectively, with low elasticity modulus (less than 3.40 GPa, see Figure 9.10) are formed. The smallest Δ80 value (~ 1.20) was obtained for the case of structure formation in Euclidean space that allows an elasticity modulus of the order of 5.7 GPa to be received. As follows from the data of Figure 9.8, it is impossible to obtain such a value of Δ80 by ‘natural’ methods, since the indicated value of Δ80 is reached at νc = 0 only (see Figure 9.8), i.e., at absence of crosslinking in epoxy polymers. This circumstance requires special curing methods of epoxy polymers, allowing the the degree of chaos of their structure to be minimised.

425

Structure and Properties of Crosslinked Polymers In conclusion one important aspect should be noted. As follows from the data of Figure 9.11, the transition from constant and large values of Δ80 to their decay is realised at Dsp ≈ 2.35. This dimension is a border one for loosely packed matrix formation of cluster-cluster structure mechanisms (Dsp = 1.67–2.12) and particlecluster mechanisms (Dsp = 2.5). Thus, a change in the structure formation mechanism results in the dependence Δ80(Dsp) type change [32].

Δ80 3.0

2.5

2.0

1.5

2.0

2.5

3.0

Dsp

Figure 9.11 The dependence of the degree of chaos of the structure Δ80 on dimension Dsp of a space in which a nanostructure is formed, for epoxy polymers. The designations are the same as in Figure 9.8 [32] In connection with the treatment of crosslinked epoxy polymers as natural nanocomposites the question arises about the applied stress concentration in a loosely packed matrix. The stress concentration factor Ks deﬁnes to a considerable extent the strength of composites [36] and depends on their structure change [37]. The last aspect was the most important in paper [38], the authors of which fulﬁlled the study of the structural factors deﬁning the stress concentration factor value and, hence, characterising a change in the structure of crosslinked epoxy polymers. As was noted above, the strength of composites depends to a considerable extent on the stress concentration factor Ks according to the equation [36]:

(9.15)

426

Crosslinked Epoxy Polymers as Natural Nanocomposites

where σ f and σ f are the failure stress of the composite and polymer matrix, respectively, ϕf is the ﬁller volume content. c

m

For natural nanocomposites (crosslinked epoxy polymers) the failure stress σf was c accepted as being σ f , the relative fraction ϕcl of nanoclusters as ϕf and the loosely l .m. m packed matrix failure stress as σ f , which can be estimated in diagram form as σ f . For this purpose the value of ϕcl was calculated according to Relationship 6.15 and then the dependence σf(ϕcl) for the considered epoxy polymers was plotted (Figure 9.12), l .m. which turns out to be linear and the value of σ f was determined by extrapolation of this linear dependence to ϕcl = 0 or ϕl.m. = 1.0 (where ϕl.m. is the loosely packed matrix relative fraction, connected with ϕcl by Equation 9.10. As follows from the l .m. plot of Figure 9.12, σ f = 50 MPa.

σf, MPɚ 200

120 -1 -2 -3 -4 40 0

0.4

0.8 ϕcl

Figure 9.12 The dependence of the failure stress σf on the relative fraction of nanoclusters ϕcl for epoxy polymers EP-1 (1); EP-1-200 (2); EP-2 (3); and EP-2200 (4) [38]

The estimations according to Equation 9.15 have shown that for the considered epoxy polymers the value of Ks varies within the limits of 4.0–12.2 and grows with

427

Structure and Properties of Crosslinked Polymers an increase in ϕcl. In Figure 9.13 the dependence Ks(ϕcl) is adduced, which turns out to be linear and extrapolates to Ks = 0 at ϕcl = 0. This means that the change in structure of epoxy polymers in their curing process is deﬁned fully by formation of local order domains (nanoclusters) [37]. One should note that within the frameworks of the cluster model of amorphous state structure of polymers a chaotically tangled macromolecular coils set, corresponding to Flory’s ‘felt’ model, is considered as a polymer ideal structure and ordered regions are deﬁned as defect ones, i.e., as deviating from the structure ideal state. In these regions polymer chains segments are considered as linear defects (analogue of dislocations in crystalline lattice). In such a treatment structure change and its defectness, characterised by the parameters Ks and ϕcl, respectively, are synonyms. The correlation Ks(ϕcl) adduced in Figure 9.13 can be described analytically as follows [38]:

(9.16)

Ʉs 15

10

5

0

0.4

0.8 ϕ cl

Figure 9.13 The dependence of the stress concentration factor Ks on the relative fraction ϕcl of nanoclusters for epoxy polymers. The designations are the same as in Figure 9.12 [38]

max

Equation 9.16 allows estimation of the greatest limiting value of Ks ( K s ). For this purpose the greatest possible value of ϕcl ( ) should be calculated according to the thermal cluster model [39], i.e., to Equation 5.49 at β = 0.3. Such a calculation gives max = 0.702 and the estimation according to Equation 9.15 assumes K s = 12.5.

428

Crosslinked Epoxy Polymers as Natural Nanocomposites As has been shown above, a crosslinked structure of polymers is formed in the fractal space, which creates a loosely packed matrix of the indicated polymers. Equation 9.1 allows estimation of this space dimension Dsp. In Figure 9.14 the dependence Ks(Dsp) is adduced, which has shown linear decay Ks with growth in Dsp and at Dsp = 3, i.e., at nanostructure formation in Euclidean space, K = 0 and the structure of epoxy polymers does not undergo changes (formation of nanoclusters) in its creation process. Let us note that such treatment is conﬁrmed by the data for particulate-ﬁlled polymer nanocomposites, for which the structure formation proceeds in Euclidean space and the polymer matrix dimension of nanocomposites is constant and equal to this parameter for a matrix polymer [40]. The similar, but weaker, dependence Ks(Dsp) was found for a linear amorphous polymer (polycarbonate, a dashed line in Figure 9.14), which is due to the absence of such a powerful factor as chemical crosslinking nodes network. The indicated structure change and the variation in Ks caused by it strongly affect the strength of epoxy polymers. From Equation 9.15 it follows that the natural nanocomposite (epoxy polymer) strength grows proportionally to Ks. At Ks = 1.0, m i.e., in the absence of structure changes at curing, let us obtain σf = σ f = 50 MPa, whereas the real strength values at Ks > 1 are varied within the limits 99–189 MPa. Combination of Equations 9.15 and 9.16 allows the following formula for σf determination to be obtained [38]:

(9.17) According to this equation the value of σf changes extremely and passes through a maximum at ϕcl ≈ 0.50. In Figure 9.15 the comparison of the calculation of σf according to Equation 9.17 and experimental data is adduced, which shows good correspondence between theory and experiment. There is one more aspect of the change in the structure of the loosely packed matrix. The value of σf can be determined alternatively according to the following equation [36]:

(9.18)

where the parameter bf characterises the interfacial (intercomponent) adhesion level of the loosely packed matrix of nanoclusters. From Equation 9.18 it follows that the

429

Structure and Properties of Crosslinked Polymers smaller bf is, the higher σf is with other conditions equal and, hence, the higher the intercomponent adhesion level.

Ʉs 15

10

5

0 1.5

2.0

2.5

3.0

Dsp

Figure 9.14 The dependence of the stress concentration factor Ks on the dimension Dsp of the space in which a nanostructure is formed, for epoxy polymers. The designations are the same as in Figure 9.12. The dashed line shows the dependence Ks(Dsp) for polycarbonate [38]

In Figure 9.16 the dependence bf(Ks) is adduced, from which follows an increase in bf or a reduction in the intercomponent adhesion level with growth in the stress concentration factor (degree of change in structure of epoxy polymers). As it is known [3], reduction in the interfacial (intercomponent) adhesion level results in the essential decrease in the elasticity modulus E of nanocomposites. Hence, an increase in Ks can give both positive (σf raising) and negative (E reduction) effects. Therefore there is a range of optimal values of Ks ≈ 4.0–5.0 that, on the one hand, give sufﬁciently high strength values σf ≈ 150–170 MPa and, on the other hand, ensure a sufﬁciently high level of intercomponent adhesion (bf ≈ 400-500) that allows elasticity modulus values E ≈ 2.8–3.0 GPa to be obtained.

430

Crosslinked Epoxy Polymers as Natural Nanocomposites

σfɆPɚ 200

150

100

50 0

0.4

0.8 ϕ cl

Figure 9.15 The dependence of the failure stress σf on the relative fraction ϕcl of nanoclusters for epoxy polymers. The designations are the same as in Figure 9.12. A solid line shows the calculation according to Equation 9.17 [38]

bf 800

400

0

5

10

15

Ʉs

Figure 9.16 The dependence of parameter bf on the stress concentration factor Ks for epoxy polymers. The designations are the same as in Figure 9.12 [38]

431

Structure and Properties of Crosslinked Polymers Hence, the results stated above have demonstrated a high level of stress concentration in the loosely packed matrix, which is due to an essential change in its structure, for crosslinked epoxy polymers, treated as natural nanocomposites. Let us note that high values of the stress concentration factor are due to the indicated fractality of the structure and for a loosely packed matrix the Euclidean structure stress concentration is absent. The increase of the stress concentration factor, characterising the degree of change in structure, raises the strength of the epoxy polymers, but simultaneously reduces the intercomponent adhesion level.

9.2 The Properties of Natural Nanocomposites As was noted above, over the last several years numerous and successful attempts of the application of multifractal formalism for the description of the structure and properties of different materials [26, 27, 29], including polymers [28, 41], were undertaken. This indicates the use of the indicated formalism for the applied purposes, for example, in engineering. At present the correlations of multifractal spectrum speciﬁc characteristics (Renyi dimension ‘latent ordering’ parameter and so on) and traditional parameters, describing the properties of polymeric materials (elasticity modulus, Grüneisen parameter and so on) have already been obtained [22, 42, 43]. At present for polymeric materials there exists a simple method of limiting Renyi dimensions by the structural characteristics of these materials (see Equations 9.11 and 9.12) [22, 43]. The authors of paper [44] used the indicated method for determination of the most general property (adaptability resource) of crosslinked epoxy polymers and for obtaining its connection with the structure of these materials. A polymeric material’s structure can be considered as multifractal, possessing a pseudo-spectrum, when Renyi dimensions Dq decrease with growth in the index q [22, 44]. In this case the structure adaptability resource Ra can be determined according to the equation [44]:

(9.19)

where D40 and D–40 are the smallest and the largest Renyi dimensions at qmax = 40 and qmin = –40, respectively, D1 is the structure information dimension (at q = 1).

432

Crosslinked Epoxy Polymers as Natural Nanocomposites The limiting Renyi dimensions D–40 and D40 were calculated according to Equations 9.11 and 9.12, respectively, and dimension D1 was determined as follows [22]:

(9.20)

where D0 is the Renyi dimension at q = 0, estimated according to Formula 8.16 at the condition D0 = f [22]. Figure 9.17 shows the dependence of the adaptability resource Ra on the crosslinking density νc for the considered epoxy polymers. As one can see, νc increasing results in a linear reduction of the adaptability of epoxy polymers to external inﬂuence and at νc ≈ 22.3 × 1026 m–3 the value of Ra = 0. The greatest value of Ra is reached at νc = 0, i.e., for a non-cured epoxy polymer. Let us note that for the considered epoxy polymers the adaptability resource Ra is varied within the limits of 22–64, which is much higher than this parameter for the extruded polymerisation-ﬁlled compositions, where Ra ≤ 11.45 [41].

Ra 80

-1 -2 -3 -4

40

0

10

20 ν ×1026, m-3 c

Figure 9.17 The dependence of the adaptability resource Ra on the crosslinking density νc for epoxy polymers EP-1 (1); EP-1-200 (2); EP-2 (3); and EP-2-200 (4) [44]

433

Structure and Properties of Crosslinked Polymers The reduction in Ra with growth in νc supposes that the adaptability of the structure of epoxy polymers decreases with growth in the relative fraction ϕcl of nanoclusters or increases with a rise in the relative fraction ϕl.m. of the loosely packed matrix. In Figure 9.18 the dependence Ra(ϕl.m.) is adduced, which has an expected character and can be described analytically by the following empirical equation [44]:

(9.21)

From Equation 9.21 it follows that the zero adaptability resource of the structure of epoxy polymers is reached at ϕl.m. = 0.23 or ϕcl = 0.77. This is the greatest possible ϕcl value according to Equation 5.49 at β = 0.3 (see above). The greatest value of Ra max ( Ra ) is reached at ϕcl = 0 or ϕl.m. = 1.0 and it is determined according to Equation max 9.21 as Ra = 89.

Ra 80

40

0 0.2

0.4

0.6

0.8

ϕl.m.

Figure 9.18 The dependence of the adaptability resource Ra on the relative fraction ϕl.m. of the loosely packed matrix for epoxy polymers. The designations are the same as in Figure 9.17 [44]

One can suppose that as the most general property the structure adaptability of epoxy polymers should inﬂuence their individual properties as well. For conﬁrmation of

434

Crosslinked Epoxy Polymers as Natural Nanocomposites this supposition the elasticity modulus of epoxy polymers at compression Ecomp was recalculated in a similar characteristic at tension Etens according to the formula [47]:

(9.22) where ν is Poisson’s ratio, determined according to Equation 5.5. In Figure 9.19 the dependence Etens(Ra) for the considered epoxy polymers is adduced, from which the rise in the elasticity modulus with growth in structure adaptability follows. At Ra = 0 the value of Etens = 0.56 GPa, that is the elasticity modulus of nanoclusters , and at Ra = 89 the similar parameter can be obtained for the tens loosely packed matrix, namely E l .m. = 2.60 GPa.

Etens, GPɚ 3

2

1

0

40

80

Ra

Figure 9.19 The dependence of the tension elasticity modulus Etens on the adaptability resource Ra for epoxy polymers. The designations are the same as in Figure 9.17 [44]

It is easy to see that the plot of Figure 9.19 assumes consideration of epoxy polymers as natural nanocomposites in which the loosely packed matrix plays a matrix role

435

Structure and Properties of Crosslinked Polymers and nanoclusters play a nanoﬁller role. In such a treatment the modulus efﬁciency coefﬁcient kE can be determined according to the equation [22]:

(9.23)

since for epoxy polymers only the loosely packed matrix is a reinforcing element by virtue of the availability of a chemical crosslinkings network in it (see Figure 9.19). In Figure 9.20 the dependence kE(Ra) is adduced, which shows the raise in the efﬁciency of loosely packed matrix ﬁlling by nanoclusters with growth in Ra. The plot of kE(Ra) min max allows determination of the smallest and the greatest values of kE ( k E and k E , min max respectively): k E = 0.63 and k E = 1.33. Let us note the sufﬁciently high values of kE for natural nanocomposites (epoxy polymers) in comparison with nylon-kevlar composites (kE = 0.15–0.23) and polyhydroxiether-graphite (kE = 0.40–1.69) [22].

kE 1.5

1.0

0.5

0

40

80

Ra

Figure 9.20 The dependence of the modulus efﬁciency coefﬁcient kE on the adaptability resource Ra for epoxy polymers. The designations are the same as in Figure 9.17 [44]

436

Crosslinked Epoxy Polymers as Natural Nanocomposites Hence, the results stated above have shown that analysis of structural properties for epoxy polymers, which are considered as natural nanocomposites, can be carried out within the frameworks of multifractal formalism in its most simple variant. The structure adaptability resource is reduced as the crosslinking density increases and is deﬁned by the relative fraction of the loosely packed matrix. The properties of epoxy polymers are a function of their structure adaptability. As the studies of amorphous linear polymers considered as natural nanocomposites have shown, their elasticity modulus is a linear increasing function of the relative fraction ϕcl of nanoclusters [48]. Such behaviour of the elasticity modulus of the indicated polymers conﬁrms the treatment of nanoclusters as nanoﬁllers (reinforcing elements). The authors of paper [49] carried out a similar analysis of elasticity modulus behaviour in compression tests for crosslinked epoxy polymers. For all four considered epoxy polymers in the series a reduction in the elasticity modulus Ecomp at crosslinking density νc or growth in the relative fraction of nanoclusters ϕcl, which was calculated according to Relationship 6.15, was found. Such dependences Ecomp(νc) and Ecomp(ϕcl) suppose the principal difference of elasticity modulus behaviour for linear and crosslinked polymers: if for the ﬁrst the nanoclusters are a system reinforcing element, then for the second they are a loosely packed matrix, i.e., a natural nanocomposite matrix. This is explained by distinction of the type of intermolecular bonds for the indicated classes of polymers: for linear polymers both nanoclusters and the loosely packed matrix have the same type of intermolecular bonds, namely weak van der Waals bonds, and nanoclusters are the reinforcing element of the structure (nanoﬁller analogue) by virtue of macromolecules of polymers in the loosely packed matrix. For net polymers macromolecules are connected in addition by the stronger covalent bonds by virtue of the concentration of chemical crosslinking nodes in this structural component of epoxy polymers [7]. Therefore for crosslinked polymers the loosely packed matrix is a reinforcing element [49]. Figure 9.21 shows the dependence of the elasticity modulus Ecomp on a sample relative cross-sectional area Sl.m., occupied by a loosely packed matrix, which was determined as follows [49]:

(9.24)

As has been expected, the plot of Figure 9.21 demonstrates growth in Ecomp with increasing relative fraction of the structure reinforcing element in the sample crosssection. Special attention should be drawn to two features of the behaviour of the

437

Structure and Properties of Crosslinked Polymers elasticity modulus of epoxy polymers in compression tests. Firstly, the dependence Ecomp(Sl.m.) at Sl.m. = 0 extrapolates to zero elasticity modulus. Since Sl.m. = 0 according to Equation 9.24 means ϕcl = 1.0, then the indicated extrapolation assumes zero elasticity modulus of nanoclusters at compression. Secondly, the correlation adduced in Figure 9.21 can be approximated by the following empirical equation [49]:

(9.25) where the value of Ecomp at Sl.m. = 1.0 or in the complete absence of nanoclusters is equal to 7.8 GPa. In other words, this means that the loosely packed matrix elasticity modulus for the considered epoxy polymers in compression tests is equal to 7.8 GPa.

Ecomp, GPɚ

4

2 -1 -2 -3 -4

0

0.2

0.4

0.6

Sl.m.

Figure 9.21 The dependence of the compression elasticity modulus Ecomp on a sample cross-sectional area Sl.m., occupied by a loosely packed matrix, for epoxy polymers EP-1 (1); EP-1-200 (2); EP-2 (3); and EP-2-200 (4) [49] Dividing both parts of Equation 9.25 into relationship [49]:

= 7.8 GPa, let us obtain the following

(9.26)

438

Crosslinked Epoxy Polymers as Natural Nanocomposites The ratio in the left-hand part of Equation 9.26 should be considered as a reinforcement degree of crosslinked epoxy polymers treated as a natural nanocomposite. Let us note again that a loosely packed matrix is a structure reinforcing element for a crosslinked polymer, unlike for linear amorphous polymers. Nevertheless, for both indicated classes of polymers the reinforcement degree is deﬁned equally, namely as polymer and loosely packed matrix elasticity moduli ratio and in both cases the nanoclusters are assumed as a nanoﬁller. Conﬁrmation of this postulate can be obtained within the frameworks of the model [50] where three basic cases of the dependence of the reinforcement degree of the composites Ec/Em (where Ec and Em are elasticity moduli of the composite and the matrix polymer, respectively) on ﬁller content ϕf were considered. It has been shown that the following basic types of the dependence Ec/ Em(ϕf) exist: The ideal adhesion between ﬁller and polymer matrix, described by Kerner’s equation, which can be approximated by the following relationship:

(9.27)

Zero adhesional strength at large friction coefﬁcient between ﬁller and polymer matrix, which is described according to the equation:

(9.28)

The complete absence of interaction and ideal slippage between ﬁller and polymer matrix, when the composite elasticity modulus is deﬁned practically by the polymer cross-section (compare with the plot of Figure 9.21) and is connected with the ﬁlling degree by the equation:

(9.29)

comp

It is easy to see that at the condition Ec = Ecomp, Em = E l .m. and ϕf = ϕcl Equations 9.26 and 9.29 are identical. This means that in compression tests a crosslinked polymer 439

Structure and Properties of Crosslinked Polymers behaves like a nanocomposite with nanoﬁller (nanoclusters) in the complete absence of intercomponent adhesion and ideal slippage between nanoclusters and the loosely packed matrix. Let us note that the data adduced in Figure 9.9 conﬁrm this conclusion. As it is known [35], the perfect adhesion by Kerner, i.e., the case 1 (Equation 9.27), corresponds to b = 1, whereas for the considered epoxy polymers the obtained values of b vary within the range of 0.03–0.35. Hence, in the considered case the elasticity modulus of epoxy polymers is deﬁned by the loosely packed matrix cross-section [50] (see also Figure 9.21) and the role of nanoclusters comes to its change (Equation 9.24). For conﬁrmation of that stated above in Figure 9.22 the comparison of the theoretical dependences of the reinforcement degree on the relative fraction ϕcl of nanoclusters, treated as nanoﬁller, and those obtained experimentally (points) are adduced, where the ﬁrst dependence was calculated according to Equation 9.26 and comp in the second case the reinforcement degree was deﬁned as the ratio Ecomp/ E l .m. or comp E /7.8 GPa. As one can see, the comparison adduced in Figure 9.22 has shown good correspondence between theory and experiment.

ȿ comp / ȿ lcomp . m.

1.0

0.5

0

0.4

0.8

ϕcl

comp

Figure 9.22 The dependence of the reinforcement degree Ecomp/ E l .m. on the relative fraction ϕcl of nanoclusters for epoxy polymers. The designations are the same as in Figure 9.21 [49]

440

Crosslinked Epoxy Polymers as Natural Nanocomposites As it is known, the main distinction of linear polymers from crosslinked ones is the availability of the chemical crosslinking nodes network in the latter. Therefore in Figure 9.23 the dependence of Ecomp on the crosslinking density νc is shown for the considered epoxy polymers. As one can see, the dependence Ecomp(νc) breaks down into two parts: at relatively small crosslinking density (νc ≤ 10 × 1026 m–3) the value of Ecomp reduces linearly with growth in νc and at νc> 10 × 1026 m–3 the elasticity modulus of epoxy polymers does not change, i.e., as it is not a function of νc. This transition can be explained as follows. It has been shown above that two behaviour regimes are observed for crosslinked polymers: at Nst > 9 (where Nst is the number of statistical segments per chain part between chemical crosslinking nodes) the crosslinked polymer behaves similarly to a linear one and at Nst ≤ 9 segmental mobility suppression is observed (according to the notions of the cluster model of the amorphous state structure of polymers this means transition to the structure quasi-equilibrium state [6]). The calculation adduced above for the considered epoxy polymers gave the value Nst = 8.33 for the transition point (νc = 10 × 1026 m–3), which is close to the border value Nst = 9, cited in paper [33]. This calculation conﬁrms that the dependence Ecomp(νc) transition to plateau or to the condition Ecomp ≈ const. is due to formation of a densely crosslinked epoxy polymer structure [33] or the indicated structure transition in the quasi-equilibrium state [6].

Ecomp, GPɚ

5

3

1 0

10

20 νɫ×10-26, m-3

Figure 9.23 The dependence of the compression elasticity modulus Ecomp on the crosslinking density νc for epoxy polymers. The designations are the same as in Figure 9.21 [49]

441

Structure and Properties of Crosslinked Polymers However, one further treatment of the dependence of Ecomp on νc is possible, namely the use of the ‘effective crosslinking density’ notion. Such a treatment accounts for the concentration of chemical crosslinking nodes only in the loosely packed matrix and therefore it can be written [49]:

(9.30)

In Figure 9.24 the dependence Ecomp( ) for the considered epoxy polymers is adduced. As one can see, at real crosslinking density , but not the nominal one νc, use of Ecomp linear decay with growth in for all considered epoxy polymers is obtained. In other words, in such a treatment the regions of weak (νc ≤ 10 × 1026 m–3) and dense (νc > 10 × 1026 m–3) crosslinking do not differ structurally.

Ecomp, GPɚ

5

3

1 0

20

40

ν efɫ ×10-26, m-3

Figure 9.24 The dependence of the compression elasticity modulus Ecomp on the effective crosslinking density for epoxy polymers. The designations are the same as in Figure 9.21 [49]

Hence, the results stated above have shown that in this case the compression tests of the loosely packed matrix of epoxy polymers, in which chemical crosslinking

442

Crosslinked Epoxy Polymers as Natural Nanocomposites nodes are concentrated, is the only reinforcing component for them. The treatment of epoxy polymers as natural nanocomposites demonstrated a complete absence of intercomponent adhesion between the nanoclusters and the loosely packed matrix for these materials. Using the effective crosslinking density the absence of structural distinctions between weak and dense crosslinking regions was found. As a rule, the data obtained in tension tests are used in analysis of the mechanical characteristics for polymers. Since for the considered epoxy polymers mechanical characteristics were obtained in compression tests, then the authors [51] carried out a recalculation of these results according to Equation 9.22. For the considered epoxy polymers reduction in the tension elasticity modulus Etens with growth in ϕcl was found, which is an effect opposite to that observed for linear polymers [9, 52]. Therefore the dependence of Etens on the relative fraction ϕl.m. of the loosely packed matrix, which was determined according to Equation 9.10, was plotted. The indicated dependence is shown in Figure 9.25 and it demonstrates linear growth in Etens with an increase in ϕl.m.. This dependence can be approximated by the following empirical formula [51]:

(9.31)

Equation 9.31 allows determination of the elasticity modulus values for nanoclusters tens at the condition ϕl.m. = 0 ( = 0.56 GPa) and the loosely packed matrix E l .m. tens at the condition ϕl.m. = 1.0 ( E l .m. = 2.59 GPa). The data adduced in Figure 9.25 demonstrate again that unlike the linear amorphous polymers the loosely packed matrix, not nanoclusters, is a reinforcing element for crosslinked polymers. As above, the last circumstance is explained by the concentration of chemical crosslinking nodes in the loosely packed matrix. Since the interaction energy of chemical (covalent) bonds is higher by several orders of magnitude than that of the intermolecular (van der Waals) bonds, at the expense of which nanoclusters are formed, then this deﬁnes a higher elasticity modulus for the loosely packed matrix in comparison with nanoclusters. Let us note that opposite tendencies in the behaviour of linear and crosslinked polymers have also been observed. So, if for the ﬁrst the change of glass transition temperature and elasticity modulus occurs similarly [47], then for the second ones it occurs antibately [13]. The authors [13] explained the last effect by the inﬂuence of chemical crosslinkings, which restricted thermal ﬂuctuations in the system. Hence, proceeding from that stated above and considering crosslinked epoxy polymers as natural nanocomposites, their reinforcement degree can be deﬁned as the ratio Etens/ . Three basic cases of the dependence of the reinforcement degree Ec/Em on the

443

Structure and Properties of Crosslinked Polymers ﬁlling degree ϕf were considered above. In Figure 9.26 the theoretical dependences (ϕl.m.), corresponding to three indicated types of these correlations (Equations Etens/ 9.27–9.29, solid lines), and the experimental data for two series of the considered epoxy polymers (points) are adduced. As one can see, the experimental data are close to (although located somewhat above) curve 2, calculated according to Equation 9.28, which reveals a sufﬁciently low level of intercomponent adhesion between the nanoclusters and the loosely packed matrix. Simultaneously the plots of Figure 9.26 demonstrate clearly the reinforcement degree (and, hence, elasticity modulus) increase for natural nanocomposites (crosslinked polymers) – realisation of perfect adhesion between nanoclusters and the loosely packed matrix. In this case the loosely packed matrix will give the same reinforcement degree as an inorganic nanoﬁller at the condition of perfect adhesion between the nanoﬁller and the polymer matrix mentioned above [51].

Etens, GPɚ 3

2

-1 -2

1

0

0.4

0.8 ϕ l.m.

Figure 9.25 The dependence of the tension elasticity modulus Etens on the loosely packed matrix relative fraction ϕl.m. for epoxy polymers EP-1 (1) and EP-1-200 (2) [51]

The comparison of the results obtained above in tension and compression tests demonstrates two principal distinctions of behaviour of epoxy polymers treated as natural nanocomposites in these two cases. Firstly, in tension tests the elasticity

444

Crosslinked Epoxy Polymers as Natural Nanocomposites modulus of the nanoclusters has a ﬁnite non-zero value, whereas in compression tests its value is equal to zero. Secondly, although in both cases a low level of intercomponent adhesion is observed, in tension tests the effect of strong friction between components appears (Equation 9.28, see Figure 9.26), whereas in compression tests ideal slippage between nanoclusters and the loosely packed matrix, corresponding to Equation 9.29, is realised.

ȿtens/ ȿ cltens 1

15 -4 -5 10

5 2 3 0

0.4

0.8 ϕ l.m.

Figure 9.26 The dependences of the reinforcement degree Etens/, on the loosely packed matrix relative fraction ϕl.m.. 1–3 – the theoretical dependences, calculated according to Equations 9.27–9.29, respectively, 4, 5 – the experimental data for EO-1 (4) and EP-1-200 (5) [51]

The intercomponent adhesion level can be estimated quantitatively with the aid of the parameter b (Figure 9.9), which is determined by an independent method using the thermal expansion coefﬁcient of epoxy polymers. In Figure 9.27 the curves 1–3 represent three basic types of the dependences of the linear thermal expansion coefﬁcient αEP of the considered epoxy polymers on the relative contents ϕcl of nanoclusters. The straight line 1 illustrates the case when adhesion is absent between two structural components of natural nanocomposite and at αl.m. > αcl (αl.m., αcl are the thermal expansion coefﬁcients of the loosely packed matrix and nanoclusters, respectively) the loosely packed matrix will expand on heating independently from

445

Structure and Properties of Crosslinked Polymers nanoclusters and in this case αEP = αl.m. [35]. The straight line 2 corresponds to a simple rule for mixtures [35]:

(9.32) where

is a mixture thermal expansion coefﬁcient.

Rule 9.32 is correct for an ideal case when all the components expand independently from one another. And lastly, curve 3 corresponds to Terner’s equation [35]:

(9.33) where Kl.m. and Kcl are a bulk elasticity modulus of the loosely packed matrix and nanoclusters, respectively. At the derivation of Equation 9.33 a strains equality method was used for the calculation of the thermal expansion coefﬁcient of mixtures depending on density, elasticity modulus, thermal expansion coefﬁcient and mass relation of composing components. If the assumptions are correct, then Formula 9.33 will also be applicable for natural nanocomposites.

αEP×105Ʉ-1 2.5

1

2.0

1.5

1.0 0

-4 -5

2 3 0.4

0.8 ϕ cl

Figure 9.27 The dependences of the thermal expansion linear coefﬁcient αEP on the relative fraction ϕcl of nanoclusters. 1 – the adhesion absence on the boundary between components; 2 – the mixtures rule; 3 – Terner equation; 4, 5 – the experimental data for epoxy polymers EP-1 (4) and EP-1-200 (5) [51] 446

Crosslinked Epoxy Polymers as Natural Nanocomposites In Figure 9.27 experimentally determined values of αEP are shown by points for two series of the considered epoxy polymers. Let us note that in Terner’s equation, Young’s modulus E magnitudes were used instead of the bulk elasticity moduli K and Kcl. As it was known [35], such a replacement allows the lower boundary of the αEP value for epoxy polymers to be obtained. From the data of Figure 9.27 it follows that αEP values for both series of the considered epoxy polymers approximately correspond to the mixtures rule (curve 2). The parameter b can be estimated with the aid of Equation 9.13. The calculations according to this equation gave the mean value b ≈ 0.20 (Figure 9.9). The parameter b allows precise qualitative gradation of intercomponent adhesion level. So, the condition b = 0 means absence of the indicated adhesion and b = 1.0 indicates perfect (by Kerner) adhesion. The value b ≈ 0.20 deﬁnes a low, but different from zero, intercomponent adhesion level, that completely corresponds to the data of Figure 9.26, where the experimental points are located somewhat higher than the curve 2, calculated according to the condition of zero intercomponent adhesion. The increase in the elasticity modulus Etens at the condition = const. with growth tens in ϕl.m. also means a rise in the reinforcement degree E / with growth in ϕl.m.. The analytical empirical dependence of Etens/ on ϕl.m. can be obtained by dividing all members of Equation 9.31 by = 0.56 GPa and as a result of this we will obtain [51]:

(9.34)

Let us note in conclusion one more special feature of the structure of crosslinked polymers, namely the duality of its components. In the case of mechanical inﬂuence the loosely packed matrix plays the role of the reinforcing element and in the case of thermal inﬂuence the nanoclusters. Thus, the results stated above have demonstrated the principal distinction between linear and crosslinked polymers under their consideration as natural nanocomposites – if for the former nanoclusters are the reinforcing element (analogue of nanoﬁller) at the deﬁnition of reinforcement degree, then for the latter the loosely packed matrix is such an element. This distinction is due to the different type of bonds in the loosely packed matrix – for linear polymers there are relatively weak intermolecular (van der Waals) bonds and for crosslinked polymers covalent bonds. The level of intercomponent adhesion between nanoclusters and the loosely packed matrix for the

447

Structure and Properties of Crosslinked Polymers considered epoxy polymers is low and essentially smaller than the perfect adhesion by Kerner [51]. At present it is known [53–55] that the microhardness Hv is a property that is sensitive to morphological and structural changes in polymeric materials. The ﬁller availability, whose microhardness exceeds the corresponding characteristic of the polymer matrix, is in addition a powerful factor for composite materials [56]. At introduction in the polymer of sharpened indentors, having a cone or pyramid shape, the stressed state is localised in a sufﬁciently small microvolume and it is assumed that in tests of such kind the real structure of polymeric materials is ‘found’ [57]. In connection with the fact that the structure of polymer nanocomposites is complex enough [3, 4], the question arises of what structure component reacts to indentor squeezing and how much this reaction is modiﬁed with the introduction of a nanoﬁller. Another aspect of the problem is the interconnection of microhardness, determined by the results of tests in a very localised microvolume, and such macroscopic properties of polymeric materials as the elasticity modulus E and the yield stress σY. At present a large number of relationships derived theoretically and obtained empirically between Hv, E and σY exist [58, 59]. The authors of paper [60] gave the description of the microhardness of crosslinked epoxy polymers within the frameworks of fractal (structural) models and elucidated the indicated parameter interconnection with structure and mechanical characteristics. Description of the structure of the epoxy polymers is given within the frameworks of the cluster model of the amorphous state structure of polymers [5–7], which allows polymers to be considered as natural nanocomposites in which nanoclusters play the role of nanoﬁller. Let us consider the interconnection of microhardness Hv and other mechanical characteristics, in particular the yield stress σY, for the studied epoxy polymers. Tabor [61] found for metals, which were treated as perfectly stiff plastic solids, the following relation between Hv and σY:

(9.35)

where c is a constant, which is approximately equal to 3. Relationship 9.35 implies that the pressure applied in microhardness tests under an indentor is higher than the yield stress in quasi-static tests because of the restriction

448

Crosslinked Epoxy Polymers as Natural Nanocomposites imposed by the non-deformed polymer surrounding the indentor. However, a number of authors [55, 58, 59] have shown that the value of c can differ essentially from 3 and vary within wide limits: ~ 1.5–30. For analysis spreading on the solids wider range, it was proposed to consider the role of elasticity in the indentation process. For a solid with elasticity modulus E and Poisson’s ratio ν, Hill obtained the following formula [57]:

(9.36)

and the Marsch empirical equation has the form [57]:

(9.37)

Equations 9.36 and 9.37 allow the estimation of the ratio Hv/σY for epoxy polymers at the condition of the known E and σY and the value ν can be calculated by the results of mechanical tests with the aid of Relationship 5.5. Let us consider now the physical nature of the deviation of ratio Hv/σY from the constant c ≈ 3 in Equation 9.35. The structure fractal dimension df can be calculated according to Equation 5.4. Combination of Equations 5.4, 5.5, 9.36 and 9.37 allows fractal variants of the Hill and Marsch equations to be received, respectively [60]:

(9.38) and

(9.39)

449

Structure and Properties of Crosslinked Polymers From Equations 9.38 and 9.39 it follows that the ratio Hv/σY is deﬁned only by the solid structure state, which is characterised by its basic characteristic – the fractal (Hausdorff) dimension df. In Figure 9.28 the theoretical dependences Hv/σY(df), calculated according to Equations 9.38 and 9.39, are adduced. One can see that they are similar, but Equation 9.38 gives higher (approximately 20% larger) absolute values of the ratio Hv/σY than Formula 9.39. Tabor’s condition (Equation 9.35 with c ≈ 3) in the case of Equation 9.39 is reached at df ≈ 2.95, i.e., at the greatest structure fractal dimension value for real solids [14]. This circumstance assumes two consequences. Firstly, Tabor’s condition indicated above is correct only for Euclidean solids. For fractal objects c < 3 and in the typical range for solids df = 2.0–2.95 the value of Hv/ σY changes within the limits of 1.15–3.0. Secondly, the value of c = 3 on reaching df = 2.5 assumes higher precision of Equation 9.39 in comparison with Formula 9.38. This supposition is conﬁrmed by the data of Figure 9.28, where the dependence of experimental values of Hv/σY (points) on the value of df is shown. They correspond well to the curve calculated according to Equation 9.39, but the correspondence with calculation according to Formula 9.38 is not as good.

Hv/σY 4

1 2

3

-3 -4

2

1 2.0

2.5

3.0 df

Figure 9.28 The dependences of the ratio Hv/σY on the structure fractal dimension df. 1, 2 – the theoretical calculation according to Equations 9.38 (1) and 9.39 (2); 3, 4 – the experimental data for epoxy polymers EP-2 (3); and EP-2-200 (4). The horizontal dashed line indicates Tabor’s criterion c = 3 [60]

450

Crosslinked Epoxy Polymers as Natural Nanocomposites It was assumed that the loosely packed matrix of epoxy polymer structure carries the main loading in the indentor squeezing process as its least hard component. Its m dimension d f can be calculated according to the mixtures rule (Equation 9.3). is accepted as being equal to the greatest dimension for real solids The value of (~ 2.95) by virtue of the dense packing of nanoclusters and the value of ϕcl can be m calculated with the aid of Formula 4.8. In Figure 9.29 the dependence of d f on the crosslinking density νc is adduced, from which it follows that an increase in νc results m in the reduction of d f or loosening of the loosely packed matrix. This effect should affect the Hv value as well, which is conﬁrmed by the data of Figure 9.30, in which m the dependence of Hv on d f for the considered epoxy polymers is adduced. As one m can see, the linear correlation Hv( d f ) was obtained, which shows Hv growth with m a decrease in d f , i.e., at densiﬁcation of the structure of the loosely packed matrix. The mentioned correlation can be described analytically by the following empirical equation [60]:

(9.40)

d mf 2.8 -1 -2

2.4

2.0

0

5

10

15 ν ×10-26, m-3 ɫ m

Figure 9.29 The dependence of the fractal dimension d f of the structure of the loosely packed matrix on the crosslinking density νc for epoxy polymers EP-2 (1) and EP-2-200 (2) [60]

451

Structure and Properties of Crosslinked Polymers Equation 9.40 shows that the condition Hv = 0 can be realised for a linear object only with dimension d = 1 (actually, for a separate stretched macromolecule), i.e., a zero value of Hv is unattainable for real solids having dimension within the range of m 2.0–2.95. At the smallest magnitude of d f = 2.0 the value of Hv = 150 MPa, for the greatest densely packed structure of the loosely packed matrix, i.e., in the case when the structures of nanoclusters and the loosely packed matrix are indistinguishable, the value Hv = 293 MPa.

HvɆPɚ 300

200

100 -1 -2 0 1.0

1.5

2.5

2.0

d mf m

Figure 9.30 The dependence of the microhardness Hv on the fractal dimension d f of the structure of the loosely packed matrix for epoxy polymers EP-2 (1) and EP2-200 (2) [60]

m

The microhardness of the loosely packed matrix H v can also be calculated according to the mixtures rule [53, 59]:

(9.41) where is the microhardness of the nanoclusters, which is accepted as being equal to 293 MPa for the reasons indicated above.

452

Crosslinked Epoxy Polymers as Natural Nanocomposites m

In Figure 9.31 the dependences of Hv and H v on Kst are adduced. As one can see, at Kst = 0.50 and 1.50 (the smallest values of νc = (4–9) × 1026 m–3) the magnitudes m of Hv and H v are close and at higher values of νc = (10–14) × 1026 m–3 in the case m of Kst = 0.75–1.25 the difference between Hv and H v can reach 80 MPa, which indicates a high degree of loosening of the loosely packed matrix.

Hv, H νm ɆPɚ -1 -2 -3 -4

300

200

100

0

0.5

1.0

1.5 Ʉst

Figure 9.31 The dependences of the microhardness of epoxy polymers Hv (1, 2) m and the loosely packed matrix H v (3, 4) on the curing agent : oligomer ratio value Kst for EP-2 (1, 3) and EP-2-200 (2, 4) [60]

Thus, the fractal analogues of the Hill and Marsch equations obtained above have shown that the microhardness of crosslinked epoxy polymers is deﬁned only by their structure, characterised by its fractal dimension. Tabor’s criterion is correct for Euclidean (or close to Euclidean) solids only. The degree of increase in crosslinking results in loosening of the loosely packed matrix and to a corresponding reduction of the microhardness of crosslinked epoxy polymers [60]. The thermal expansion is one of the criteria of choice of engineering polymers for their application in either quality [35]. As a rule, polymers have high thermal expansion coefﬁcient, which makes their application in contact with other materials difﬁcult. In paper [62] the structural analysis of the thermal expansion of crosslinked polymers considered as natural nanocomposites was carried out.

453

Structure and Properties of Crosslinked Polymers In Figure 9.32 the dependence of the thermal expansion linear coefﬁcient αEP on the relative fraction ϕcl of nanoclusters, which are considered as nanoﬁller, for epoxy polymers is adduced. As has been expected [35], an increase in ϕcl results in a reduction in αEP, comparable with that observed for polymer composites with the introduction of particulate ﬁllers. So, an increase in ϕcl from 0 to 0.60 reduces αEP by about 1.50 times (Figure 9.32) and with the introduction of calcium carbonate or aluminium powder with volume contents ϕf = 0.60 in the epoxy polymer the thermal expansion linear coefﬁcient value decreases by 1.70–2.0 times [35]. The dependence αEP(ϕcl) adduced in Figure 9.32 can be expressed analytically by the following empirical equation [62]:

(9.42)

αEP×105Ʉ-1 2.5 -1 -2

2.1

1.7 0

0.4

0.8 ϕ cl

Figure 9.32 The dependence of the thermal expansion linear coefﬁcient αEP on the relative fraction ϕcl of nanoclusters for epoxy polymers EP-1 (1) and EP-2 (2) [62]

As it is known [6], within the frameworks of fractal analysis the molecular mobility level can be described with the aid of the fractal dimension of the chain part between nanoclusters Dch, which changes within the limits of 1 < Dch ≤ 2. At Dch = 1 the

454

Crosslinked Epoxy Polymers as Natural Nanocomposites indicated part is stretched completely between clusters and its molecular mobility is suppressed. At Dch = 2 a chain part has a molecular mobility level that is typical for rubbers. The value of Dch can be estimated with the aid of Equation 6.22. In Figure 9.33 the dependence αEP(Dch) for the considered epoxy polymers is adduced, which has an expected character. The growth in the thermal expansion linear coefﬁcient with an increase in molecular mobility level is observed. In Figure 9.33 a solid straight line shows the similar dependence α(Dch) for amorphous aromatic polyamide (phenylone S-2). As one can see, this straight line corresponds well to the data for the considered epoxy polymers. This means that irrespective of the class of polymers, their thermal expansion coefﬁcient is deﬁned by the molecular mobility level, which in paper [62] is characterised by the dimension Dch.

αEP×105Ʉ-1 3 2.3

1.9 -1 -2 1.5 1.0

1.1

1.2

1.3

Dch

Figure 9.33 The dependence of the thermal expansion linear coefﬁcient αEP on the fractal dimension of the chain part between nanoclusters Dch for epoxy polymers. The designations are the same as in Figure 9.32. The straight line 3 shows the dependence α(Dch) for phenylone [62]

Despite the approximate character of Equation 9.42, it can be used for the description of the theoretical temperature dependence of αEP, to calculate the value of ϕcl according to Relationship 6.15. Figure 9.34 shows the dependence αEP(T) for epoxy polymer

455

Structure and Properties of Crosslinked Polymers EP-1, having a glass transition temperature of 423 K, which, as was expected, shows an increase in αEP with growth in testing temperature [35].

αEP×105Ʉ-1 2.6

2.2

1.8 273

353

423 ɌɄ

Figure 9.34 The dependence of the thermal expansion linear coefﬁcient αEP on testing temperature T, calculated according to Equations 6.15 and 9.42 [62]

Hence, the results stated above have shown that formation of nanostructures in natural nanocomposites (epoxy polymers) gives the same effect of the reduction of the thermal expansion coefﬁcient as well as the introduction of a particulate inorganic ﬁller in the polymer matrix. One remark should be made with regard to the estimations adduced above. For polymers (natural nanocomposites) the dependence α(ϕcl) is described by the mixtures rule (see Figure 9.27) and for polymer nanocomposites, ﬁlled by inorganic particulate nanoﬁller, the analogous correlation is given by Terner’s equation, which is due to a much higher level of interfacial adhesion in the latter case [40]. Lastly, the molecular mobility level, characterised by dimension Dch, can serve as the common basis for estimation of the thermal expansion of different classes of polymeric materials. The last parameter choice is obvious: the value of Dch always changes irrespective of polymeric material class within the same limits (1 < Dch ≤ 2 [6]). It has been shown earlier [63, 64] that an amorphous polymers structure, simulated by the cluster model of the amorphous state structure of polymers [5, 6], can be represented as a percolation system with the critical temperature Tcr, which is the glass

456

Crosslinked Epoxy Polymers as Natural Nanocomposites transition temperature Tg. The relative fraction ϕcl of the nanoclusters is connected with Tg by Relationship 4.8. Attention is attracted by the fact that the critical indices of order parameter β obtained in papers [63, 64], though they are close enough to classical percolation indices (which are equal to 0.37–0.40 [65]), still differ from them by an absolute value. The reason for such a discrepancy is the fact that the percolation cluster represents a purely geometrical construction, which is too simpliﬁed a model for real amorphous polymers possessing thermodynamically non-equilibrium structure. Therefore thermal interactions have an inﬂuence on the structure of the mentioned polymers (let us be reminded that nanocluster structure is postulated as a thermoﬂuctuational system [5, 6]). The formation of the nanoclusters is studied not on a concentration scale, as for the percolation cluster [65], but on a relative temperature scale. Hence, the structure of amorphous polymers should be simulated more correctly by the thermal cluster, i.e., by a cluster whose equilibrium conﬁguration is deﬁned by both geometrical and thermal interactions [39]. In polymeric materials under thermal interactions the molecular mobility should be understood, i.e., thermal oscillations of fragments of macromolecules around their quasi-equilibrium positions. In this case the order parameter ϕcl of the polymer structure is described according to Relationship 5.49, in which an order parameter index βT of thermal clusters is not necessarily equal to the critical index β in purely geometrical percolation models. Proceeding from that mentioned above, the authors [66] have estimated absolute values of βT variation, elucidated possible relations of this index with structural characteristics of epoxy polymers and their glass transition temperatures and also deﬁned factors inﬂuencing the βT value, for four series of epoxy polymers. In paper [19] it has been shown that the universality of the critical indices of the percolation system is connected directly with the fractal dimension of this system. The self-similarity of the percolation system assumes the availability of a number of subsets having the order n (n = 1, 2, 4), which in the case of the structure of polymeric materials are identiﬁed as follows. The percolation cluster network or matrix physical entanglement cluster network is the ﬁrst subset (n = 1) in the polymer matrix. The loosely packed matrix, into which the cluster network is immersed, is the second one (n = 2). For polymer composites, the ﬁller particles network, which is naturally absent in epoxy polymers, is the third subset (n = 4). In such a treatment the percolation cluster critical indices β and ν are given as follows (in three-dimensional Euclidean space) [19]:

(9.43)

457

Structure and Properties of Crosslinked Polymers

(9.44) where df is the fractal dimension of the structure of epoxy polymers, which is determined according to Equations 5.4 and 5.5. Thus, the percolation critical indices β and ν are border values for βT, indicating which structural component of an epoxy polymer deﬁnes its behaviour. At βT = β nanoclusters or, more precisely, the percolation cluster network, identiﬁed with the nanocluster network, are such a component. At β < βT < ν epoxy polymer behaviour is deﬁned by the combined inﬂuence of the nanoclusters and the loosely packed matrix. At βT = ν the loosely packed matrix will be a structural component, deﬁning epoxy polymer behaviour. The estimations according to Equations 5.4 and 5.5 have shown that in the considered case the average df value is equal to 2.644 and then according to Equations 9.43 and 9.44 let us obtain β = 0.38, ν = 0.76. Then, using Equation 5.49, βT values for the considered epoxy polymers can be calculated [66]. The calculation of the relative fraction ϕcl of nanoclusters was carried out with the help of Equation 6.15. In Figure 9.35 the dependence of the thermal cluster order parameter index βT on the relative fraction ϕcl of nanoclusters is adduced (let us be reminded that ϕcl is an order parameter in strict physical signiﬁcance of this term for the structure of polymers [63]). As one can see, reduction in βT within the range of ~ 0.76–0.38 with growth in ϕcl from 0.32 to 0.66 is observed. As was noted above, this means that at small ϕcl of order 0.30 the behaviour of epoxy polymers is deﬁned by a loosely packed matrix and at large ϕcl (order of 0.60) by a macromolecular entanglement cluster network, the nodes of which are nanoclusters [5, 6]. It is obvious that the role of the latter in the deﬁnition of the behaviour of epoxy polymers increases with raising of their contents. For particulate-ﬁlled polymer composites the authors [22, 67] have obtained βT values within the range of ~ 0.6–1.1, which is explained by ﬁller (graphite) availability. The dependence βT(ϕcl), shown in Figure 9.35, can be expressed analytically by the following empirical equation [66]:

(9.45) The combination of Equations 5.49 and 9.45 allows the theoretical calculation of the glass transition temperature of epoxy polymers as a function of ϕcl. In Figure 9.36 the T comparison of the Tg values obtained experimentally and Tg values calculated by the indicated mode of glass transition temperature for the considered epoxy polymers

458

Crosslinked Epoxy Polymers as Natural Nanocomposites is adduced. As one can see, good correspondence between theory and experiment is T obtained (the mean discrepancy between Tg and Tg is 4.2%) [66].

βɌ -1 -2 -3 -4

1.0

6 0.6

5 0.2 0.3

0.5

0.7 ϕcl

Figure 9.35 The dependence of the thermal cluster order parameter index βT on the relative fraction ϕcl of nanoclusters for epoxy polymers EP-1 (1); EP-1-200 (2); EP-2 (3); and EP-2-200 (4). Horizontal dashed lines give the values of critical percolation indices β (5) and ν (6) [66]

It has been shown earlier [67] that index βT is a function of the molecular mobility level, characterised by the fractal dimension Dch of a chain part between nanoclusters. At Dch = 1 the indicated part is fully stretched between nanoclusters, its molecular mobility is suppressed and βT = β. At Dch = 2 the molecular mobility is the greatest, which is typical for the rubber-like state of polymers [6]. Calculation of the dimension Dch can be carried out with the help of Equation 6.22. In Figure 9.37 the dependence βT(Dch) for the considered epoxy polymers is adduced. As earlier [6, 22, 67], the linear dependence of βT on Dch is obtained, showing the order parameter index of a thermal cluster, by which the structure of epoxy polymers is simulated, at the intensiﬁcation of the molecular mobility, characterised by dimension Dch (or the intensiﬁcation of the thermal interactions). In the same ﬁgure the dependence βT(Dch) for polyhydroxyester-graphite (PHE-Gr) particulateﬁlled composites is shown by a dotted line [22, 67]. The dependences βT(Dch) for the

459

Structure and Properties of Crosslinked Polymers considered epoxy polymers and PHE-Gr composites have identical character and the most essential distinction between them is the fact that unlike epoxy polymers the dependence βT(Dch) for PHE-Gr is extrapolated to the lower limiting value βT = 0.38, not at Dch = 1.0, but at Dch = 1.10, i.e., at non-zero molecular mobility. The indicated value Dch = 1.10 was explained by loosening of the polymer matrix by a ﬁller (probably in the interfacial layers) even at the greatest possible density of local packing [22]. For polymers simulated as natural nanocomposites, interfacial (intercomponent) regions of the nanoclusters and loosely packed matrix are absent [68], which deﬁnes for them Dch = 1.0 at βT = 0.38.

Ɍ gɌ , Ʉ 450

400

350

300 300

350

400

450 ɌgɄ T

Figure 9.36 The relation between experimental Tg and Tg values calculated according to Equations 5.49 and 9.45 of the glass transition temperature for epoxy polymers. The designations are the same as in Figure 9.35 [66]

Hence, the results stated above have shown that the order parameter index of a thermal cluster, by which the structure of the considered epoxy polymers is simulated, decreases with growth in the relative fraction of nanoclusters and its variation makes up 0.38–0.76 and this means that the loosely packed matrix and nanoclusters are structural components deﬁning the behaviour of epoxy polymers, and the role of nanoclusters grows as their contents increase. The thermal cluster model allows the glass transition temperature of epoxy polymers as a function of the relative fraction of nanoclusters to be predicted. The order parameter index of the thermal cluster

460

Crosslinked Epoxy Polymers as Natural Nanocomposites increases with the raising of the molecular mobility level of the epoxy polymers (thermal interactions intensiﬁcation) [66].

βɌ 1.0

0.6

0.2 1.0

1.1

1.2

1.3

Dch

Figure 9.37 The dependence of the thermal cluster order parameter index βT on dimension Dch of a chain part between nanoclusters for epoxy polymers. The dependence βT(Dch) for PHE-Gr composites [67] is shown by a dotted line. The designations are the same as in Figure 9.35 [66]

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Crosslinked Epoxy Polymers as Natural Nanocomposites 18. Z.M. Amirshikhova, G.V. Kozlov and G.M. Magomedov in the Proceedings of 30th Anniversary International Conference ‘Composite Materials in Industry’ (SLAVPOLICOM), Yalta, Ukraine, 2001, p.268. 19. A.N. Bobryshev, V.N. Kozomazov, L.O. Babin and V.I. Solomatov in Synergetics of Composite Materials, NPO ORIUS, Lipetsk, Russia, 1994, p.154. 20. G.V. Kozlov, V.A. Beloshenko, E.N. Kuznetsov and Y.S. Lipatov, Doklady NAN Ukraine, 1994, 12, 126. 21. G.M. Magomedov, G.V. Kozlov and Z.M. Amirshikhova, Izvestiya DSPU, 2009, 4, 19. 22. G.V. Kozlov, Y.G. Yanovskii and Y.N. Karnet in Structure and Properties of Particulate-Filled Polymer Composites: Fractal Analysis, Al’yanstransatom, Moscow, Russia, 2008, p.363. 23. H.G.E. Hentschel and I. Procaccia, Physica D, 1983, 8, 3, 435. 24. T.C. Halsey, M.H. Jensen, L.P. Kadanoff, I. Procaccia and B.I. Shraiman, Physical Review: Part A, 1986, 33, 2, 1141. 25. J.L. McCayley, Journal of Modern Physics: Part B, 1989, 3, 6, 821. 26. G.V. Vstovskii, I.Z. Bunin, A.G. Kolmakov and I.Y. Tanitovskii, Doklady Akademii Nauk SSSR, 1995, 343, 5, 613. 27. A.G. Kolmakov, Metally, 1996, 6, 37. 28. V.U. Novikov, G.V. Kozlov and A.V. Bilibin, Materialovedenie, 1998, 10, 14. 29. B.I. Semenov, S.N. Agibalov and A.G. Kolmakov, Materialovedenie, 1999, 5, 25. 30. I.Z. Bunin, Metally, 1996, 6, 29. 31. V.U. Novikov, D.V. Kozitskii and V.S. Ivanova, Materialovedenie, 1999, 8, 12. 32. G.M. Magomedov, G.V. Kozlov, Z.M. Amirshikhova and I.M. Alieva, Izvestiya DSPU, 2010, 1, 11. 33. V.A. Bershtein and V.M. Egorov in Differential Scanning Calorimetry in Physics-Chemistry of Polymers, Khimiya, Leningrad, Russia, 1990, p.256.

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Structure and Properties of Crosslinked Polymers 34. Y.S. Lipatov in Interfacial Phenomena in Polymers, Naukova Dumka, Kiev, Russia, 1980, p.260. 35. L. Holliday and J. Robinson in Polymer Engineering Composites, Ed., M.O.W. Richardson, Applied Science Publishers Ltd, London, UK, 1980, p.241. 36. S. Ahmed and F.R. Jones, Journal of Materials Science, 1990, 25, 12, 4933. 37. J. Leidner and R.T. Woodhams, Journal of Applied Polymer Science, 1974, 18, 8, 1639. 38. Z.M. Amirshikhova, G.V. Kozlov and G.M. Magomedov, Deformatsiya i Razrushenie Materialov, 2010, 8, 22. 39. F. Family, Journal of Statistical Physics, 1984, 36, 5/6, 881. 40. G.V. Kozlov, Y.G. Yanovskii, A.I. Burya and Z.K. Afashagova, Mekhanika Kompozitsionnykh Materialov i Konstruktsii, 2007, 13, 4, 479. 41. G.V. Kozlov, V.Z. Aloev and Y.G. Yanovskii, Mekhanika Kompozitsionnykh Materialov i Konstruktsii, 2004, 10, 2, 267. 42. V.U. Novikov, D.V. Kozitskii, I.S. Deev and L.P. Kobets, Materialovedenie, 2001, 11, 2. 43. G.V. Kozlov and E.N. Ovcharenko, Izvestiya, 2001, 2, 81. 44. Z.M. Amirshikhova, G.V. Kozlov and G.M. Magomedov in the Proceedings of the 3rd International Science-Practice Conference ‘Nanostructures in Polymers and Polymer Nanocomposites’, KBSU, Nal’chik, Russia, 2010, p.98. 45. V.S. Ivanova, I.R. Kuzeev and M.M. Zakirnichnaya in Synergetics and Fractals, Universality of Materials Mechanical Behaviour, Publishers USNTU, Ufa, Russia, 1998, p.366. 46. G.V. Kozlov, V.A. Beloshenko, V.N. Varyukhin and Y.S. Lipatov, Polymer, 1999, 40, 4, 1045. 47. G.V. Kozlov and D.S. Sanditov in Anharmonic Effects and PhysicalMechanical Properties of Polymers, Nauka, Novosibirsk, Russia, 1994, p.261.

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Crosslinked Epoxy Polymers as Natural Nanocomposites 48. M.T. Bashorov, G.V. Kozlov, G.E. Zaikov and A.K. Mikitaev, Khimicheskaya Fizika i Mezoskopiya, 2009, 11, 2, 196. 49. Z.M. Amirshikhova, G.V. Kozlov and G.M. Magomedov, Khimicheskaya Fizika i Mezoskopiya, 2010, 12, 4, 543. 50. I.I. Tugov and A.Y. Shaulov, Vysokomolekulyarnie Soedineniya Seriya B, 1990, 32, 7, 527. 51. Z.M. Amirshikhova, G.V. Kozlov and G.M. Magomedov, Nanotekhnika, 2010, 1, 3. 52. G.V. Kozlov, V.A. Beloshenko and V.N. Shogenov, Fiziko-Khimicheskaya Mekhanika Materialov, 1999, 35, 5, 105. 53. F.R. Balta-Calleja and H.R. Kilian, Colloid Polymer Science, 1988, 266, 1, 29. 54. F.R. Balta-Calleja, C. Santa Cruz, R.K. Bayer and H.R. Kilian, Colloid Polymer Science, 1990, 268, 5, 440. 55. V.Z. Aloev and G.V. Kozlov in Physics of Orientational Phenomena in Polymeric Materials, Polygraphservice and T, Nal’chik, Russia, 2002, p.288. 56. A.J. Perry and D.J. Rowcliffe, Journal Materials Science Letters, 1973, 8, 6, 904. 57. D.S. Sanditov and G.M. Bartenev in Physical Properties of Disordered Structures, Nauka, Novosibirsk, Russia, 1982, p.256. 58. D.L. Kohlstedt, Journal Materials Science, 1973, 8, 6, 777. 59. G.V. Kozlov, V.A. Beloshenko, V.Z. Aloev and V.N. Varyukhin, FizikoKhimicheskaya Mekhanika Materialov, 2000, 36, 3, 98. 60. Z.M. Amirshikhova, G.V. Kozlov and G.M. Magomedov, Inzhenernaya Fizika, 2010, 2, 43. 61. D. Tabor in The Hardness of Metals, Oxford University Press, New York, NY, USA, 1951, p.329. 62. Z.M. Amirshikhova, G.V. Kozlov and G.M. Magomedov, Izvestiya VUZov, 2010, 6, 47.

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Structure and Properties of Crosslinked Polymers 63. G.V. Kozlov, M.A. Gazaev, V.U. Novikov and A.K. Mikitaev, Pis’ma v ZhETF, 1996, 22, 16, 31. 64. G.V. Kozlov, V.U. Novikov, M.A. Gazaev and A.K. Mikitaev, InzhenernoFizicheskii Zhurnal, 1998, 71, 2, 241. 65. B.I. Shklovskii and A.L. Efros, Uspekhi Fizicheskikh Nauk, 1975, 117, 3, 401. 66. Z.M. Amirshikhova, G.V. Kozlov, G.M. Magomedov and G.E. Zaikov, Khimicheskaya Fizika i Mezoskopiya, 2010, 12, 2, 235. 67. G.V. Kozlov and Y.S. Lipatov, Mekhanika Kompozitnykh Materialov, 2003, 39, 1, 89. 68. M.T. Bashorov, G.V. Kozlov, G.B. Shustov and A.K. Mikitaev, Izvestiya Vuzov, 2009, 6, 44.

466

10

The Solid-phase Extrusion of Rarely Crosslinked Epoxy Polymers

As was noted in the previous chapter, at present it is generally acknowledged [1] that macromolecular formations and polymer systems by virtue of the special features of their structure are always natural nanostructural systems. In connection with this fact the question of the use of this feature for improvement in the properties and operating characteristics of polymeric materials arises. It is obvious that for a structure, it is necessary to receive the relationships between the properties of the quantitative nanostructural model of the indicated materials. It is also obvious that if the dependence of a speciﬁc property on the structural state of a material will be unequivocal, then there will be quite sufﬁcient modes to achieve this state. The cluster model of the amorphous state structure of polymers [2, 3] is the most suitable for description of this state structure. It has been shown that basis of the structural element (cluster) of this model is a nanoparticle (nanocluster) [4]. The cluster model was used successfully for the description of the structure and properties of crosslinked epoxy polymers [5]. Therefore the authors [6] carried out the study of the regulation modes of nanostructures and the inﬂuence of the latter on the properties of rarely crosslinked epoxy polymers. In paper [6] the studied object is an epoxy polymer on the basis of resin UPS-181, cured by iso-methyltetrahydrophthalic anhydride in the ratio by mass 1 : 0.56. Testing specimens were obtained by the hydrostatic extrusion method. The choice of the indicated method is due to the fact that exertion of high hydrostatic pressure in the deformation process prevents the formation and growth of defects, resulting in the failure of the material [7]. The extrusion strain εe is equal to 0.14, 0.25, 0.36, 0.43 and 0.52. The value εe was calculated according to the formula [7]:

εe =

d12 − d 22 d12

(10.1)

where d1, d2 are the diameters of the intermediate product and extrudate, respectively (the latter is equal to 12 mm).

467

Structure and Properties of Crosslinked Polymers The specimens obtained by hydrostatic extrusion were annealed at a maximum temperature of 353 K for 15 minutes. The hydrostatic extrusion and subsequent annealing of rarely crosslinked epoxy polymer (REP) result in very essential and unexpected changes in its mechanical behaviour and properties. The qualitative changes of REP mechanical behaviour can be monitored according to the corresponding changes of the stress-strain (σ-ε) diagrams, shown in Figure 10.1.

σɆPɚ 150

100 3 50

1 2

0

0.1

0.2

0.3

ε

Figure 10.1 The stress-strain (σ-ε) diagrams for native (1), extruded up to εe = 0.52 (2) and annealed (3) REP specimens [6]

The native REP shows expected behaviour and its elasticity modulus E and yield stress σY are typical for such polymers at testing temperature T, which is about 40 K away from the glass transition temperature Tg [8]. The small (~ 3 MPa) stress decay ΔσY behind yield stress is observed, which is also typical for amorphous polymers [5]. However, REP extrusion up to εe = 0.52 results in disappearance of the stress decay ΔσY (‘yield tooth’) and the essential reduction in E and σY. The σ-ε diagram itself resembles more closely the analogous diagram for rubber than that for a glassy polymer. This specimen annealing at the greatest temperature Tan = 353 K gives no less strong, but diametrically opposite, effect – yield stress and elasticity modulus increase sharply (the latter increases by about 2 times in comparison with the native REP and

468

The Solid-phase Extrusion of Rarely Crosslinked Epoxy Polymers more than one order in comparison with the extruded specimen). The pronounced ‘yield tooth’ appears. It is necessary to note that specimen shrinkage with annealing is small (~ 10%), and makes up about 20% of εe [6]. The general picture of the change of parameters E and σY as a function of εe is presented in Figures 10.2 and 10.3, respectively. As one can see, both indicated parameters showed common tendencies with a change in εe: up to εe≈ 0.36 included a weak increase of E and σY with growth in εe is observed, moreover their absolute values for extruded and annealed specimens are close but at εe>0.36 the pronounced divergence of these parameters for the indicated specimens types is displayed. The cluster model of the amorphous state structure of polymers and the yielding treatment of polymers elaborated within its frameworks [2, 3, 9] allows such behaviour of the studied samples to be explained. The cluster model supposes that amorphous state structure of polymers represents the local order domains (nanoclusters), surrounded by a loosely packed matrix. Nanoclusters consist of several collinear densely packed statistical segments of different macromolecules and by virtue of this they offer the analogue of crystallite with extended chains. There are two types of nanoclusters: stable, consisting of a number of relatively large segments, and non-stable, consisting of a smaller number of such segments [9]. With a temperature increase or application of a mechanical stress non-stable clusters are disintegrated in the ﬁrst place, which results in two wellknown effects. The ﬁrst is known as a two-stage glass transition process [10] and it ' supposes that at Tg = Tg – 50 K disintegration of non-stable clusters, restraining the loosely packed matrix in a glassy state, occurs that deﬁnes devitriﬁcation of the latter [2, 3]. The decrease in the well-known rapid mechanical properties of polymers on approaching Tg is a consequence of this [8]. The second effect consists of decay of non-stable clusters at σY under the action of mechanical stress, loosely packed matrix mechanical devitriﬁcation and, as a consequence, the rubber-like behaviour of glassy polymers on the cold ﬂow plateau [9]. The stress decay ΔσY behind the yield stress is due to decay in non-stable nanoclusters and therefore the ΔσY value is characteristic of the fraction of these nanoclusters [3]. Proceeding from this brief description, the experimental results, adduced in Figures 10.1–10.3, can be interpreted. The rarely crosslinked epoxy polymer on the basis of resin UP5-181 has a low glass transition temperature Tg, which can be estimated according to shrinkage measurements data as being equal to ~ 333 K. This means that the testing temperatures T = 293 K ' and Tg for it are close, which is conﬁrmed by the small ΔσY value for the native REP. This supposes (nanostructures) a small relative fraction ϕcl of the nanoclusters [2, 3] and, since these nanoclusters have arbitrary orientation, an increase in εe results rapidly in their decay, which causes mechanical devitriﬁcation in the loosely packed matrix at εe > 0.36. The devitriﬁcated loosely packed matrix gives an insigniﬁcant

469

Structure and Properties of Crosslinked Polymers contribution to E [11, 12], practically equal to zero, that results in a sharp (discrete) ' decrease in the elasticity modulus. At T > Tg rapid decay in ϕcl is observed, i.e., the number of segments decreases in both stable and non-stable nanoclusters [3]. Since just these parameters (E and ϕcl) deﬁne the σY value (see Section 6.3), their decrease deﬁnes a sharp lessening in the yield stress. Now extruded at εe > 0.36 REP presents the rubber with a high degree of crosslinking that is reﬂected by its σ-ε diagram (Figure 10.1, curve 2).

ȿ, GPɚ 3

2 -1 -2

1

0 0

0.2

0.4

0.6

εe

Figure 10.2 The dependences of the elasticity modulus E on the extrusion strain εe for extruded (1) and annealed (2) REP [6]

The shrinkage in the polymer-oriented chains occurs with the extruded REP annealing at temperatures higher than Tg. Since this process is realised in a narrow range of temperatures and in a small time interval, then a large number of non-stable nanoclusters is formed. This effect is intensiﬁed by the available molecular orientation, i.e., by building of preliminary favourable segments, and it is reﬂected by a strong increase in ΔσY (Figure 10.1, curve 3). The increase in ϕcl results in growth in E (Figure 10.2) and the combined enhancement of ϕcl and E to considerable growth in σY (Figure 10.3).

470

The Solid-phase Extrusion of Rarely Crosslinked Epoxy Polymers

σYɆPɚ 150

100

50

0 0

0.2

0.4

0.6

εe

Figure 10.3 The dependences of the yield stress σY on the extrusion strain εe for extruded (1) and annealed (2) REP [6]

The considered structural changes can be described quantitatively within the frameworks of a cluster model. The relative fraction ϕcl of nanoclusters can be calculated according to the method stated in paper [13]. The dependences ϕcl(εe) shown in Figure 10.4 have the character expected from the description adduced above and are its quantitative conﬁrmation. The dependence of the density ρ of REP extruded specimens on εe adduced in Figure 10.5 is similar to the dependence ϕcl(εe), which should also be expected, since a decrease in the fraction of densely packed segments should be reﬂected in the lessening of ρ. In paper [14] the supposition has been made that a change in ρ can be due to formation of a microcracks network in the specimen that results in lessening of ρ at large εe (0.43 and 0.52), which are close to the limiting ones. The estimation of the relative change of ρ (Δρ) according to the equation is:

Δρ =

ρ max − ρ min ρ max

(10.2)

where ρmax and ρmin are the greatest and the least density values, respectively.

471

Structure and Properties of Crosslinked Polymers

ϕcl 0.6

0.4 -1 -2

0.2

0 0

0.2

0.4

0.6

εe

Figure 10.4 The dependences of the relative fraction ϕcl of nanoclusters on the extrusion strain εe for extruded (1) and annealed (2) REP [6]

ρ×10-3, kg/m3 2.44 2.42 2.40

-1 -2

2.38 2.36 0

0.2

0.4

0.6

εe

Figure 10.5 The dependences of the density ρ of specimens on the extrusion strain εe for extruded (1) and annealed (2) REP [6]

472

The Solid-phase Extrusion of Rarely Crosslinked Epoxy Polymers Equation 10.2 shows that Δρ ≈ 0.01. This value can be reasonable for a free volume increase, which is necessary for devitriﬁcation of the loosely packed matrix [3] ' (accounting for the closeness of T and Tg ), but it is evidently small, if the formation of microcracks is assumed as real. As the experiments have shown, REP extrusion at εe > 0.52 is impossible owing to cracking of the specimens in the extrusion process. Therefore the critical dilatation Δδcr value, which is necessary for formation of microcracks in the cluster, can be estimated as follows [15]:

(10.3)

where ν is Poisson’s ratio. Accepting the average value ν ≈ 0.35, we obtain Δδcr ≈ 0.60, which is essentially higher than the estimation of Δρ made earlier. These calculations assume that a decrease in ρ at εe = 0.43 and 0.52 is due to decay in non-stable nanoclusters and to a corresponding loosening of the REP structure. The data stated above give a clear example of the great possibilities of the operation of the properties of a polymer through its nanostructure change. From the plots of Figure 10.2 it follows that annealing of REP, extruded up to εe = 0.52, results in an increase in the elasticity modulus by more than 8 times and from the data of Figure 10.3 an increase of 6 times in the yield stress follows. From the practical point of view extrusion and subsequent annealing of rarely crosslinked epoxy polymers allow materials to be obtained that are just as good in stiffness and strength as densely crosslinked EP, but exceeding the latter in the degree of plasticity. Let us note that besides extrusion and annealing other modes of polymer nanostructure operation exist: plasticisation [16], ﬁlling [17, 18], obtaining ﬁlms from different solvents [19] and so on. Hence, the results stated above have demonstrated that neither the degree of crosslinking nor the molecular orientation level deﬁne the ﬁnal properties of crosslinked polymers. The factor controlling properties is a state of the suprasegmental (nanocluster) structure, which, in turn, can be purposefully regulated by application of molecular orientation and thermal treatment. In the treatment stated above it is not only the integral characteristics of the nanostructure (macromolecular entanglements cluster network density νcl or relative fraction ϕcl of nanoclusters) that are important, but also the separate nanocluster parameters. In the case of particulate-ﬁlled polymer nanocomposites (artiﬁcial

473

Structure and Properties of Crosslinked Polymers nanocomposites) it is well known that their elasticity modulus increases sharply with a decrease in the size of the nanoﬁller particles [4]. A similar effect has been noted above for rarely crosslinked epoxy polymer (REP), having undergone treatment of different kinds (see Figure 10.1). Therefore the authors [20] carried out a study of the dependence of the elasticity modulus E on the size of the nanoclusters for REP. It has been shown earlier in the example of PC that the value of E is deﬁned completely by the structure of the natural nanocomposite (polymer) according to the formula [7]:

(10.4)

where ncl is the number of statistical segments in one nanocluster. The interconnection described by Equation 1.11 between parameters νcl and ϕcl and nanocluster size, at which its diameter Dcl was accepted, is determined according to Equation 9.8. In Figure 10.6 the dependence E(Dcl) is adduced for REP, having undergone the indicated kinds of treatment at εe values within the range of 0.16–0.52. As one can see, as in the case of artiﬁcial nanocomposites, for REP the strong (about one order) growth in E with reduction in the nanoclusters size Dcl from 3 to 0.8 nm is observed. This fact conﬁrms again that the REP elasticity modulus is deﬁned by neither the crosslinking degree nor the molecular orientation level, but depends only on the nanocluster structure state of the epoxy polymer simulated as a natural nanocomposite [20]. The other method of calculation of the theoretical dependence E(Dcl) for natural nanocomposites (polymers) was given in paper [21]. The authors [22] have shown that the elasticity modulus E value for fractal objects, which polymers are, is given by the percolation Relationship 6.6. As it is known [4], a cluster nanostructure of polymers represents the percolation system, for which p = ϕcl, pc = 0.34 [23] and further it can be written:

(10.5)

474

The Solid-phase Extrusion of Rarely Crosslinked Epoxy Polymers where Rcl is the distance between clusters, which is determined according to Equation 1.14, lst is the statistical segment length, ν is the correlation length index, accepted as being equal to 0.8 [24].

ȿ, GPɚ 3

-1 -2 -3

2

1

0

5

10

15 Dcl, nm

Figure 10.6 The dependence of the elasticity modulus E on the diameter Dcl of nanoclusters for native (1); extruded (2); and annealed (3) REP. 4 – calculation according to Equation 4.10 [20]

Since we are ﬁrst interested in a change in E with variation in ncl, then the authors [21] accepted νcl = const. = 2.5 × 1027 m–3, lst = const. =4.34 Å. The calculation of the value of E according to Equations 6.6 and 10.5 allows determination of the value of E from the formula [21]:

In Figure 10.7 the theoretical dependence (the solid line) of E on the size (diameter) Dcl of the nanoclusters calculated according to Equation 10.6 is adduced. As one can see, the strong growth in E with a decrease in Dcl, identical to that shown in Figure 10.6, is observed. The experimental data for REP adduced in Figure 10.7, which

475

Structure and Properties of Crosslinked Polymers underwent hydrostatic extrusion and subsequent annealing, correspond well to the calculation according to Equation 10.6. A decrease in Dcl from 3.2 to 0.7 nm results again in growth in E of about one order [21].

ȿ, GPɚ 3 -1 -2 -3

2

1

0

4

1

2

3 Dcl, nm

Figure 10.7 The dependence of the elasticity modulus E on the diameter Dcl of nanoclusters for native (1); extruded (2); and annealed (3) REP. 4 – calculation according to Equation 10.6 [21]

A similar effect can be obtained for linear amorphous polycarbonate (PC). The calculation according to Equation 10.6 shows that reduction in ncl from 16 (the experimental value ncl at T = 293 K for PC [3]) to 2 results in growth in E from 1.5 to 5.8 GPa and the structureless (ncl = 1) PC production allows to receive E ≈ 9.2 GPa, i.e., comparable with that obtained for composites on the basis of PC. Hence, the results stated in the present chapter give a purely practical aspect of the application of such theoretical concepts as the cluster model of the amorphous state structure of polymers and fractal analysis for the description of the structure and properties of polymers treated as natural nanocomposites. The goal-directed creation of the necessary nanostructure allows polymers to be obtained whose properties are just as good as (and even exceeding) those of composites . Structureless (defectless) polymers are the most perspective in this respect. Such polymers can be a natural replacement for a large number of polymer nanocomposites elaborated at present.

476

The Solid-phase Extrusion of Rarely Crosslinked Epoxy Polymers

References 1.

S.S. Ivanchev and A.N. Ozerin, Vysokomolekularnye Soedineniya Seriya B, 2006, 48, 8, 1531.

2.

G.V. Kozlov and V.U. Novikov, Uspekhi Fizicheskikh Nauk, 2001, 171, 7, 717.

3.

G.V. Kozlov, E.N. Ovcharenko and A.K. Mikitaev in Structure of Polymers Amorphous State, Mendeleev RKhTU, Moscow, Russia, 2009, p.392.

4.

A.K. Mikitaev, G.V. Kozlov and G.E. Zaikov in Polymer Nanocomposites: Variety of Structural Forms and Applications, Nauka, Moscow, Russia, 2009, p.278.

5.

G.V. Kozlov, V.A. Beloshenko, V.N. Varyukhin and Y.S. Lipatov, Polymer, 1999, 40, 4, 1045.

6.

M.T. Bashorov, G.V. Kozlov and A.K. Mikitaev, Fizika i Khimiya Obrabotki Materialov, 2009, 2, 76.

7.

V.Z. Aloev and G.V. Kozlov in Physics of Orientational Phenomena in Polymeric Materials, Polygraphservice and T, Nal’chik, Russia, 2002, p.288.

8.

A.T. DiBenedetto and K.L. Trachte, Journal of Applied Polymer Science, 1970, 14, 11, 2249.

9.

G.V. Kozlov, V.A. Beloshenko, M.A. Gazaev and V.U. Novikov, Mekhanika Kompozitnykh Materialov, 1996, 32, 2, 270.

10. V.N. Belousov, B.K. Kotsev and A.K. Mikitaev, Doklady Akademii Nauk SSSR, 1985, 280, 5, 1140. 11. V.N. Shogenov, V.N. Belousov, V.V. Potapov, G.V. Kozlov and E.V. Prut, Vysokomolekularnye Soedineniya Seriya A, 1991, 33, 1, 155. 12. G.V. Kozlov, V.A. Beloshenko and V.N. Shogenov, Fiziko-Khimicheskaya Mekhanika Materialov, 1999, 35, 5, 105. 13. G.V. Kozlov, A.I. Burya and G.B. Shustov, Fizika i Khimiya Obrabotki Materialov, 2005, 5, 81. 14. M.K. Pakter, V.A. Beloshenko, B.I. Beresnev, T.P. Zaika, L.A. Abdrakhmanova and N.I. Bezai, Vysokomolekularnye Soedineniya Seriya A, 1990, 32, 10, 2039. 477

Structure and Properties of Crosslinked Polymers 15. A.S. Balankin in Synergetics of Deformable Body, Publishers of Ministry Defence SSSR, Moscow, Russia, 1991, p.404. 16. G.V. Kozlov, D.S. Sanditov and Y.S. Lipatov in Fractals and Local Order in Polymeric Materials, Eds., G.V. Kozlov and G.E. Zaikov, Nova Science Publishers, Inc., New York, NY, USA, 2001, p.65. 17. G.V. Kozlov, Y.G. Yanovskii and Y.N. Karnet in Structure and Properties of Particulate-Filled Polymer Composites: Fractal Analysis, Al’yanstransatom, Moscow, Russia, 2008, p.363. 18. A.I. Burya and G.V. Kozlov in Synergetics and Fractal Analysis of Polymer Composites, Filled with Short Fibers, Porogi, Dnepropetrovsk, Ukraine, 2008, p.258. 19. V.N. Shogenov and G.V. Kozlov in Fractal Clusters in Physics-Chemistry of Polymers, Polygraphservice and T, Nal’chik, Russia, 2002, p.270. 20. M.T. Bashorov, G.V. Kozlov, V.Z. Aloev and A.K. Mikitaev in the Proceedings of the 7th International Science-Technology Conference ‘Materials and Technologies of XXI Century’, PSU, Penza, Russia, 2009, p.20. 21. Z.M. Amirshikhova, M.T. Bashorov, G.V. Kozlov and G.M. Magomedov in the Proceedings of 8th International Science-Technology Conference ‘Materials and Technologies of XXI Century’, PSU, Penza, Russia, 2010, p.7. 22. D.J. Bergman and Y. Kantor, Physical Review Letters, 1984, 53, 6, 511. 23. A.Kh. Malamatov and G.V. Kozlov in the Proceedings of 4th International Interdisciplinary Symposium ‘Fractals and Applied Synergetics’, FaAS-2005, Interkontakt Nauka, Moscow, Russia, 2005, p.119. 24. I.M. Sokolov, Uspekhi Fizicheskikh Nauk, 1986, 150, 2, 221.

478

A

bbreviations

2 DPP+HCE

Haloid-containing oligomer based on diphenylpropane and hexachlorethane

AA

p,p′-aminoazobenzene

CDC

Crystallite with drawing chains

CFC

Crystallite with folded chains

Cl-Cl

Cluster-cluster

CRR

Cooperatively rearranging regions

CSC

Crystallite with stretched chains

CSD

Critical structural defect

CuKα

Monochromatic irradiation with copper ﬁlter

CWF

Crystallisation without folding

DAA

p,p′-diaminoazobenzene

DADPM

4,4′-Diaminodiphenylmethane

DADPS

4,4′-Diaminodiphenylsulfone

DDM

4,4′-Diaminodiphenylmethane

DGEBA

Diglycidyl ester of bisphenol A

DLA

Diffusion limited aggregation

DS

Dissipative structure

479

Structure and Properties of Crosslinked Polymers DTA

Differential thermal analysis

ED-20

Trade mark of an epoxy resin based on diglycidyl ester of bisphenol A

EP

Epoxy Polymer

EP-1

Amine crosslinked epoxy polymer based on bisphenol A diglycidyl ester

EP-1-200 and EP-2-200

EP-1 and EP-2 epoxy polymers cured under hydrostatic pressure 200 MPa

EP-2

Anhydride crosslinked epoxy polymer based on bisphenol A diglycidyl ester

EP-3

EP-1 epoxy polymer aged under natural conditions for two years

EPS-4

Haloid-containing oligomer

Fe/FeO (Z)

Ultraﬁne mixture of iron and iron oxide

HDPE

High-density polyethylene

IGC

Inverse gas chromatography

IMTPhA

Iso-methyltetrahydrophthalic anhydride

LDPE

Low-density polyethylene

MCA

Dicarbon acid of adamanthane

MFI

Melt ﬂow index

MMT

Montmorillonite

PBAD

Poly(butylene adipate) diol

PC

Polycarbonate

P-Cl

Particulate-cluster

PCP

Polychloroprene

480

Abbreviations PHE

Polyhydroxyester

PHE-GR

Polyhydroxyester-graphite

P-t

Load-time

PS

Polystyrene

PTMG

Poly(tetramethylene glycol)

PU

Polyurethane

QM

Quantum-mechanical

REP

Rarely crosslinked epoxy polymer

SCE

Sulfur-containing epoxy polymer

tt-DAA

Derivative of p,p′-diaminoazobenzene

UHMPE

Ultra-high molecular weight polyethylene

UP 606/2

Crosslinking catalyst

UPS-181

Epoxy resin

WLF

Williams-Landel-Ferry

WS cluster

Witten–Sander cluster

481

INDEX

Index Terms

Links

A Affine model

51

Affine network

50

Aggregate growth, diffusion-controlled

132

Aggregation, diffusion-limited

235

Aggregation models, irreversible

242

242

Aggregation parameter

10

12

Aharony–Stauffer rule

129

148

Amorphous polymers, halo

197

199

206

Annealing

468

473

476

Auto-acceleration, effect of

96

100

Auto-deceleration, effect of

96

100

Automodelity coefficient

62

Avogadro’s number

2

208

210

239

261

208

211

160

B Barker equation

27

Binary-hooking

29

contacts

2

network

45

44

Boltzmann’s constant

3

112

227

Bonds, Van der Waals

437

443

447

Boyer concept

323

240

Bragg’s interval

197

202

Brave theorem

63

Brown approximation

309

Brownian motion

146

204

This page has been reformatted by Knovel to provide easier navigation.

Index Terms

Links

Bulk modulus

288

Burgers vector

10

239

61

64

422

426

2

22

37

22

25

42

1

3

13

15

18

21

31

166

179

185

201

211

235

242

245

248

253

261

264

283

286

293

296

303

317

324

328

411

422

456

87

105

126

Cluster-cluster mechanism

102

124

136

145

413

Cluster-cluster structure mechanisms

426

Cold flow plateau

257

294

296

298

302

469

Conductivity index

288

Connectivity, degree of

220

Correlation length index

288

Critical structural defect

310

Crystalline morphology

4

177

191

169

172

179

295

C Cantor set Chaos, degree of Cluster entanglement network network density Cluster model

476 Cluster morphology Cluster-cluster aggregation mechanism

180

Crystalline phase nucleation

179

Crystalline phase, melting temperature

181

Crystalline phase, morphology

170

180

185

5

49

169

14

23

166

183

383

Crystallinity, degree of Crystallisation entropy morphology

14 180

This page has been reformatted by Knovel to provide easier navigation.

Index Terms

Links

Crystallisation (Cont.) rate

385

temperature

182

theory of polymers

183

without folding

186

Crystallite with drawing chains Crystallites with folded chain Crystallites with stretched chains Curing

agent

solvent-oligomer induction period

190

68 5

179

190

179

229

232

312

86

96

99

105

111

114

119

125

130

132

134

136

138

140

142

146

148

197

201

216

259

306

429

131

133

137

199

210

219

225

283

285

299

301

310

315

331

334

413

105

111

142

112 96

rate

118

temperature

103

D Decay process, freezing local order thermal De Gennes principle

328 70

Devitrification, mechanical

469

Diaminodiphenylmethane

307

Diaminodiphenylsulfone

307

Differential thermal analysis

184

Diffusion

403

Diffusion limited aggregation particle-cluster

242

Diglycidyl ether of bisphenol A

197

Dilatometry

225

Dispersion theory

309

263

9

This page has been reformatted by Knovel to provide easier navigation.

Index Terms

Links

Dissipative structures

264

300

macro

256

258

micro

256

259

Drawing ratio

368

Dugdale model

8

E Einstein relationship

112

Elasticity modulus

218

220

227

285

288

312

314

335

347

351

390

393

395

397

416

425

432

435

437

441

443

448

468

470

473

474

Elasticity modulus Entropic high-elasticity theory

159

Epoxy matrix

359

Epoxy polymer, haloid-containing

111

115

Epoxy polymer, rarely crosslinked

468

470

474

extrusion

473

Epoxy polymers

47

72

83

89

97

100

111

117

119

132

138

145

201

205

209

211

215

218

221

226

233

240

244

260

263

266

268

283

290

298

301

306

amorphous curing process

317 83

glassy

168

loop formation

219

sulfur-containing

304

Euclidean dimension

63

307

314

This page has been reformatted by Knovel to provide easier navigation.

Index Terms

Links

Euclidean space

92

94

111

113

123

125

130

146

149

150

232

250

263

288

356

411

416

425

429

457

265

267

269

Exfoliation

347

Extrusion, hydrostatic

468

476

Extrusion strain

467

470

F Feigenbaum’s constant Flocculation

143 83

Floccule diameter

259

Floccules

259

261

316

30

33

261

289

327

399

63

74

76

86

89

120

123

127

133

138

159

169

220

242

253

256

285

304

332

381

389

411

476

249

257

264

Fractal model

68

83

211

387

399

Fractal space

149

411

412

429

Fractal theory

61

369

Flory

70

equation

184

exponent

412

felt model

428

model

199

theory

185

Fluctuation free volume formation microvoid formation Fluctuation theory Fractal analysis

Fractal dimension

32 326 8

This page has been reformatted by Knovel to provide easier navigation.

Index Terms

Links

Fractals, high-elasticity of

159

Fractals, homogeneous

247

Fracture stress

315

Free volume distribution curve

227

Frenkel-Eyring-Arrhenius formula

368

371

402

37

G Gas coefficient

394

398

Gaussian diffusion

119

403

Gaussian interpretation

1

Gaussian statistics

2

Gel formation

137

period

119

point

115

time

121

transition

130

Gelation period

256

Gelation point

228

Gelation, temporal range

230

Gibbs-Helmholtz equation

13

Gibbs-Helmholtz-Gladyshev equation

16

Glass transition temperature

Glass transition, two-stage

44

238

368

370

120

126

134

137

142

234 16

3

8

22

28

67

73

204

209

226

228

244

248

261

283

285

295

299

307

323

328

330

365

367

412

456

460

469

Graessley’s equation

43

45

47

Grüneisen parameter

320

392

395

Guinier-Preston zones

200

432

This page has been reformatted by Knovel to provide easier navigation.

Index Terms

Links

H Halpin-Tsai

359

Hartree-Fock quantum-mechanical method

356

Hausdorff dimension

114

Hausdorff–Bezikovich dimension High-density polyethylene semi-crystalline

64 5

11

19

22

43

49

52

381

19

22

386

396

75

367

35

High-density polyethylene/epoxy polymer nanocomposites High-elasticity, theory of

401

I Indentation

449

Intercalation

347

Inverse gas chromatography Iso-methyltetrahydrophthalic anhydride

354

84 307

Infrared spectroscopy

84

Isothermal compressibility

33

121

K Kerner’s equation Koch curve

439 61

64

Kolmogorov–Avrami equation

169

173

Krigbaum–Roe equation

184

178

383

L Landau theory Langevin equation Laser microscopy Le Chatelier–Brown principle Local order, degree of

21 1 314 41 212

214

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38

Index Terms Low-density polyethylene

Links 35

Ludwig-Davydenkov criterion

312

Lyapunov exponent

248

Lyapunov’s index

38

52

172

180

3

6

25

33

37

23

M Macromolecular entanglements

1 50

cluster network density

232

Macromolecules, degree of rolling-up

160

Marsch equation

449

Melt flow index

384

Melt viscosity

384

387

Melting temperature

184

186

Melting-recrystallisation process

392

Mi equation

296

Microgel

92

96

99

102

105

108

119

122

130

136

139

144

258

259

307

146

148

386

333

398

400

119

gyration radius

130 324

tests

448

Microvoid

269

formation

262

formation energy

261

Modulus of dilatation

8

Montmorillonite nanocomposites Mooney–Rivlin Equation

389

86

formation of Microhardness

166

351

359

350

354

374

18

21

188

Mori-Tanaka equations

359

Muthucumar concept

130

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Index Terms

Links

N Nanocluster

411

416

419

421

434

438

440

443

446

454

457

471

180

228

384

364

371

374

381

127

137 128

288

474 Nanocomposite high-density polyethylene

395

Nanocomposite high-density polyethylene/ epoxy polymer

394

Nanogel

147

Network connectivity, degree of

217

Nucleation

174

classical theory

232

thermic

169

396

O Optical microscopy

314

Organoclay nanocomposites

352

Orovan-Irwin equation

312

P Particle-cluster mechanism

414

Particle-cluster aggregation mechanism

105

Polycarbonate

28

Percolation

69

83

121

cluster

247

250

253

models

244

system

250

390

457

474

Phantom network

50

Polyhydroxyester-graphite

460

Photochromic labels method

225

Photoisomerisation kinetics

225

Physical ageing

228

16

226

228

255

266

329

331

335

337

339

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327

Index Terms

Links

Poisson’s ratio

8

27

35

41

214

227

239

254

267

291

295

300

320

334

368

382

435

449

27

32

40

165

170

176

184

187

221

242

371

429

20

34

473 Poly(methyl methacrylate)

23

Poly(tetramethylene glycol)

365

Polyamide -6

365

Polycarbonate, amorphous

476

Polychloroprene

159 190

crystalline morphology

186

crystallisation

169

179

Polycondensation

89

215

Polyhydroxyester

41

Polyhydroxyester-graphite

459

Polymer, ageing of

329

Polymer, amorphous

8

13

23

26

247

457

468

372

160

306

361

364

Polymer composites

264

Polymer, density of

2

31

Polymer, devitrification of

221

284

Polymer, macroexpansion of

319

Polymer matrix

347

358

456 Polymer network, dynamic

159

179

Polymer network crosslinked

159

166

181

183

Polymer, orientation process

168

Polymer, rejuvenation of

16

Polymer, semi-crystalline

4

9

12

18

47

372

382

391

Polymers, crosslinked Polymers, glassy

18 213

220

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Index Terms Polymers, glassy amorphous

Links 5

7

Polymers, linear

258

260

Polymers, linear amorphous

288

300

9

18

318

439

Polymers, solid-phase

19

Polymerisation

83

102

radical

89

114

Polystyrene

23

40

Polyurethane

364

369

Positron spectroscopy

225

400

65

213

296

Quasi-equilibrium state

335

422

441

Quasi-static tests

448

Prigogine–Defay criterion

27

197

Q

Quasi-two-phase cluster model Quasi-two-phase system

13 411

R Ramsey theorem

13

Recrystallisation

391

Reinforcement, degree of

349

65 354

357

365

439

443

166

187

190

296

447 Reinforcement mechanism

411

Renyi limiting dimensions

422

Reptation model

70

Richardson equation

71

Rubber high-elasticity theory

52

Rubber-like elasticity theory

1

432

160

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Index Terms

Links

S Scaling hypothesis index

83

89

70 242

invariance

70

relationship

145

245

Self-diffusivity

145

148

Serpinski carpet

61

64

Sharpy method

396

Sharpy pendulum

310

Shear modulus

288

macroscopic Shear yield stress

9 221

Smoluchowski formula

150

Sodium-montmorillonite nanocomposites

356

Sodium-montmorillonite platelets

358

Statistics, non-Gaussian

368

Stress-strain curves

293

10

Simha-Boyer concept

Strain hardening modulus

288

359

1 367

Sulfur-containing epoxy polymerdiaminodiphenylmethane

315

Sulfur-containing epoxy polymerdiaminodiphenylsulfone

307

Sulfur-containing epoxy polymer-isomethyltetrahydrophthalic anhydride

308

311

Tabor’s criterion

450

453

Terner’s equation

423

446

Testing, mechanical

209

214

Thermal cluster model

220

250

314

T 450

456

428

460

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Index Terms

Links

Thermal expansion coefficient

445

Thermodynamic, non-equilibrium

300

degree of

453

227

U Ultra-high molecular polyethylene

325

Ultra-high molecular weight polyethylene bauxite

325

V Vichek snowflake Viscosity

61 106

108

112

123

198

202

209

128

130

392

133

W Water sorption coefficient

375

Wide angle x-ray diffraction

197

halos

201

Williams, Landel and Ferry concept

262

269

Witten-Sander clusters

201

235

Witten-Sander model

137

243

63

162

X-ray diffractogram

162

170

X-ray structural analysis methods

214

413

X X-ray analysis

199

Y Yech Model

179

Yield strain

300

Yield stress

214

293

300

302

315

395

448

468

470

473

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Index Terms

Links

Yield tooth

294

296

Yielding process

391

395

phonon concept Young’s modulus

298

468

239

368

392 8

218

447

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