Multiscale Modelling of Polymer Properties

Preface

This book is a direct result of project PMILS (Polymer Molecular Modelling at Integrated Length/time Scales) - a European project funded by the European Commission's 5^^ Framework Programme. The PMILS project addressed the fundamental question of predicting macroscopic properties of polymeric materials from their molecular constitution and processing history. It combines a wide range of modelling tools (quantum mechanical, molecular modelling, mesoscopic, macroscopic) and experimental methods. Project PMILS was a pioneering project in the area of Industrial Technologies Materials. In PMILS a multidisciplinary team of highly qualified partners from Belgium, Denmark, France, Greece, Norway, United Kingdom and Spain contributed expert know-how on areas which at first sight might seem unrelated (quantum chemistry, molecular modelling, polymer synthesis, viscoelasticity, process optimization, etc.) but which could be brought to bear on multiscale, hierarchical modelling of polymers in a complementary fashion. The project was co-ordinated by the Universidad Politecnica de Madrid (UPM). It is the UPM's principal investigator (Prof. M. Laso) together with Dr. Eric Perpete, of the Laboratoire de Chimie Theorique Appliquee at F.U.N.D.P. Namur, who have undertaken the editorial work of the present volume in the "Computers Aided Chemical Engineering" series. It had been a challenge to bring the team together on a European level, because people coming from different modelling backgrounds had to understand each others 'modelling language' and had to find out which parameters were needed as input for the other partners. The team used a systematic hierarchical approach for the organisation of these diverse methods and applied the tools for the understanding of the macroscopic properties. It became clear that these modelling tools are very versatile and could be adapted to a multitude of problems.

vi

Preface

The support given to PMILS clearly demonstrates the relevance the EC gives to advanced combined approaches to complex problems in Materials Science. Furthermore, in the spirit of continued support of European Research, the 6^^ Framework Programme the NMP priority (Nanotechnologies, Materials and Production) subsequently opened a call in the area of "Modelling and design of Multifunctional Materials". The objective was to use the powerful tools of materials modelling for the understanding of the complex behaviour of new knowledge-based materials and their industrial use. As a result of this call, projects are now funded in a variety of areas (e. g. design of polymers of controlled permeability, thin films for energy storage, corrosion of coatings, modelling of noise and vibration) which showcase the Commission's committment to the development of acvanced multi-scale simulation methods. Furthermore, a coordinated call with the National Science Foundation of the United States on Computational Materials Science was also organised to fund joint project proposals which address properties and phenomena that span multiple time and length scales and require multiscale modelling involving a balanced participation of partners from EU and US. It is thus very satisfactory to see that promising lines of research initiated in PMILS have resulted in successful proposals to this EC-NSF coordinated call. Materials research is very much alive in Europe. The work programme for Materials Sciences in the 7^^ Framework Programme is being prepared as this book goes to press. The contents of this book may inspire and motivate readers to create new ways of applying modelling for the improvement of Materials Science and for meeting the needs of European Industry.

Dr. Astrid-Christina Koch European Commission Directorate General for Research Industrial Technologies - Materials Brussels, May 2006

IX

Introduction Polymeric materials constitute a very widespread and economically important family of materials. It is no exaggeration to say that they are ubiquitous in our daily lives. Yet, in spite of their very widespread use, a good deal of their physical and chemical behaviour and not a little of the phenomena underlying their production technology is still far from being understood. While the basic concept of a polymer goes back to the 1920s and very great advances have continuously been made since, some of the most fundamental questions regarding their properties and performance remain unanswered. It is not difficult to select a few prominent factors responsible for this unsatisfactory but also challenging state of affairs: • Unlike low-molecular-weight substances, bulk amounts of polymers invariably contain molecules of widely different molecular size: they are poly disperse and this polydispersity is has a major impact on most macroscopic static and, above all, dynamic properties. • Even in the ideal monodisperse case, the dynamic processes that take place in polymers at the molecular level involve not just one or a few characteristic time and length scales, but a whole hierarchy, ranging from the very fast, very small scales (e.g. single-bond vibration) to the slow and large ones (e.g., whole-polymer-chain relaxation). Characteristic spatial and temporal scales can easily span 10 orders of magnitude. • Only exceptionally do polymers form well-developed crystalline solids. In

addition, the existence or even prevalence of amorphous domains is almost universal in the solid state. Polymers lend themselves ideally to the formation of "hybrid" molecules or copolymers, the properties of which frequently depend in a very complex fashion on the nature, proportions and sequence of the intervening monomers.

X

Introduction

Depending on the specific case, i.e., on which property of which polymer and for which application, efforts to understand and predict the effect of these factors, (plus several others we have omitted) on final material properties and general dynamic behaviour face barriers which are more often than not unsurmountable. On the other hand, it is precisely this complexity that makes polymers "tailorable" materials par excellence; hence the great interest and relevance of methods of prediction of polymer properties and behaviour in the most general sense. Although widely differing in their subject matter and in their methodological approaches, the contributions collected in this multi-author book share a common unifying theme: the combined use of two or more techniques and the communication between description levels which reside at very different spatial or temporal characteristic scales. The authors are of course not alone in this effort: "multiscale", "hierarchical", "multilevel" are words that, over the last decade, have attained considerable visibility. They turn up in the most cursory search as buzzwords in almost any conceivable field of materials science but also in physics. A little historical reflection shows that "multiscale" views of materials have existed for a long time. What are well-established fields like statistical mechanics, continuum mechanics, electronics, plasticity theory, etc if not the ultimate two-level approach: one in which the atomistic and electronic reality, with its unimaginably large number of microscopic degrees of freedom, gets condensed down to a few equations and numerical values of parameters. In this admittedly narrow sense, multilevel is not that new. There is however more to these terms than meets the eye: while Materials Science long ago adapted and developed tools to perform the atomisticcontinuum jump, the driving force was the blatant impossibility of handling the myriad of atomistic/electronic details lurking behind the single number that quantifies a macroscopic property. In many cases, there was also no necessity to do so, since phenomenology complemented by good experiments was sufficient to cover most design needs. What then is new about present-day "multiscale" methods? At a very obvious level, the drive to be able to condense and carry the information made available by powerful computers at the smallest scales all the way to the macroscopic level, where its usefulness is presumably greatest. At a deeper level, and this is

Introduction

xi

especially meant for dynamic properties and behaviour away from equilibrium, we dare to say that the reason for crossing scale barriers by coarse-graining the more-detailed description level is the undesirability to simulate reality in its fiiU detail. Carrying out a large-scale molecular-dynamics simulation of say, 10^^ atoms up to the macroscopic observation time of one second may be feasible in the not-too-distant ftiture. Such brute-force simulations can be considered as computer experiments that can closely mimic the problem at hand. Computer experiments require few assumptions and not much in the way of insight, which may be considered as an asset or as a liability. It is our personal opinion, and we hope this book proves it to a certain extent, that the most promising path for far reaching advances in modelling polymeric materials is that of understanding through simplicity, that is, looking for the coarsest possible description of phenomena of interest while avoiding oversimplification. It is in this sense that our statement about the undesirability to simulate reality in its full detail should be understood. In some chapters of this book the reader will find examples of maximal scale jumping: for example, going directly from the electronic structure of a given compound to something as macroscopic as the value of a parameter in an equation of state. Although the two areas involved in this work, quantum chemical methods and equations of state (EOS) in classical equilibrium thermodynamics, apparently are totally unrelated, a judicious and intuitively appealing coarse-graining makes it possible to predict numerical values for constants appearing in the EOS which have been hitherto almost invariably obtained by regression. Other sections will introduce mesoscopic descriptions of polymers and link them either to the atomistic level or to the macroscopic level or even to both. In these cases, the goal is to be able to predict not only static thermodynamic properties but also to do so for transport properties (diffusivity, viscosity, etc) and even to use these directly in complex flow calculations. Again, straightforward coarse-graining rules are applied, mostly based on intuition or on the well established basis of linear response theory. It is however essential to emphasize in this preface that, complex though the calculation/simulation work at a single description level may be, the major and frequently unrecognized challenge facing multiscale or coarse-graining efforts lies elsewhere: in the need to guarantee the thermodynamic consistency of the simplification process. While intuition is a vahd and often reliable tool, the danger of leaning too heavily on what seem natural ways of extracting and passing information to higher hierarchical levels of description should not be

xii

Introduction

underestimated. Fortunately, recent developments in the area of non-equilibrium thermodynamics, that go beyond linear response theory, furnish a reliable guide to consistency. If a prediction can be risked at this stage, it is the well-nigh certainty that we will witness a healthy growth in thermodynamically guided simulations and coarse-graining. Einstein's, perhaps apocryphal, words are especially fitting: "Everything should be made as simple as possible, but not simpler". Although the subject of this book is integration, as a matter of presentation the material it contains must be divided. We have chosen to divide the book into two sections: the first contains chapters focusing on methodological aspects while chapters in the second section are concerned with the study of practical cases. The chapters in each section are arranged in ascending order according to the scale at which the individual studies are primarily addressed, e.g., from quantum or atomistic scales to the macroscopic. We would like to express our sincere gratitude to all the authors for their excellent contributions, to Dr. V. Wathelet for her efficient help in assembling this volume, as well as to Dr. J.-L. Valles and Dr. A.-C. Koch who continuously supported our project, and consequently share its success.

Madrid and Namur, May 2006

M. Laso, E. A. Perpete

Multiscale Modelling of Polymer Properties M. Laso, E.A. Perpete (Editors) © 2006 Elsevier B.V. All rights reserved.

Chapter 1

Calculation of Hartree-Fock Energy Derivatives in Polymers Denis Jacquemin/ Eric A. Perpete,^ and Bernard Kirtman^ ^Laboratoire de Chimie Theorique Appliquee, Facultes Universitaires Notre-Dame de la Paix, Rue de Bruxelles, 61, 5000 Namur, Belgium. ^Department of Chemistry and Biochemistry, University of California, Santa Barbara, CA 93106-9510 USA 1. Introduction 1.1. General framework Hermann Staudinger, the precursor of polymer's chemistry, received the 1953 Nobel Prize for his discoveries in macromolecular chemistry. Today, these compounds have invaded numerous fields like medicine, informatics, aeronautics, ... such that world production (in volume) of polymers is now larger than that of steel. This success originates from the ability of designing macromolecules with very diverse properties. Indeed, the mechanical, optical and electrical properties of polymers can easily be tuned by chemical transformation. New polymers with high performance in electro-optic, microelectronic and nonlinear optics are intensively looked for. In this framework of multidisciplinary research, theoretical chemistry can be viewed as an initial and often essential step. Indeed, it allows the evaluation of material properties, as well as the design of structure-property relationships, so that the synthesis can be driven to the most promising structures. To satisfy such criteria, it is necessary to be able to accurately determine the structures and properties of polymers. For instance, the knowledge of the ground-state equilibrium geometry is often a necessary prerequisite to the calculation of other properties. Many textbooks and recent reviews describe several aspects of quantum-mechanical calculations on polymers [1-4]. In the present contribution, we only summarize the results of our recent work aiming at the

4

D. Jacquemin et al.

accurate determination of Hartree-Fock (HF) energy derivatives in stereoregular polymers [5-17]. 1.2. Oligomer versus polymer approach Two methods can be considered for evaluating the properties of infinite periodic chains. In the oligomeric approach, one uses standard quantum chemistry packages to compute the desired properties on increasingly large chains and extrapolates to the infinite chain limit. For instance, one calculates the difference of polarizability between consecutive chains in the alkane series (methane, ethane, propane, butane, ...) and obtains, once sufficiently large compounds are used, a valuable approximation for the response of the infinitely long polyethylene. This is illustrated in Figure 1 for hydrogen fluoride chains. As can be seen, the saturation with respect to oligomer size can be very slow. In general, the more the investigated property is related to higher-order derivatives of the energy, the slower the saturation. Indeed, in Figure 1, the pentamer allows to determine the energy with a very small error (0.001 %) whereas the inaccuracy on the force is still larger than 5% [5]. Similar effects are obtained for hyperpolarizabilities: the higher the derivative order, the slower the saturation. 10^

£

W

^ c ^ »-H

UJ D

- ^- 2

10

> _ j

:;s

^^
10

0

20 30 N u m b e r of u n i t c e l l s

Figure 1: Evolution with oligomer size of the relative errors on the total energy and force for model fluorhydric acid chains. The polymer value is used as reference to determine the errors.

Calculation of Hartree-Fock energy derivatives in polymers

The alternative strategy, i.e. the polymeric approach, enables the evaluation of this limit by a one-shot calculation. This methodology relies on the explicit use of translational symmetry, which permits to fasten the calculations (as does point group symmetry in molecular calculation), and to circumvent extrapolation problems. Indeed, in the oligomeric scheme, one needs to be close enough to the saturation regime to correctly estimate the polymeric value. The drawback is that polymeric techniques are up to now available only for a small number of properties: the development of original methods and algorithms are required. While, the computation of the total (restricted) Hartree-Fock (HF) energy per unit cell, or the band structure and related density of states and static of stereoregular macromolecules is now wellestablished [1,2], routine calculations did appeared only recently for properties related to energy derivatives, i.e. geometries, vibrational spectra, nonlinear optics properties,... 2. Hartree-Fock Energy In Polymers 2.1. Polymer Hamiltonian To obtain a stationary solution of a system by means of quantum mechanics, one needs to determine its wavefunction, W, by solving the timeindependent Schrodinger equation, //W(r,R)=£W(r,R);

(1)

where r and R are the electronic and nuclear coordinates, respectively. E is the total energy of the system described by the Hamiltonian H. In most of the cases, Eq. (1) is simplified by putting foward the Bom and Oppenheimer (BO) approximation which decouples the electronic and nuclear displacements. In the BO approach, one solves an electronic Schrodinger equation, ^ e l e c ^ e l e c ^ j . . j ^ ^ = E'''''^'''\r\K);

(2)

to obtain the electronic energy (that only parametrically depends of the nuclear position). By adding E^^^ to the (classical) nuclear repulsion energy, £""", one gets the total energy of the system. As Eq. (2) is to complex to be solved exactly for non-trivial systems, further approximations are required. However, it is necessary that the approached Hamiltonian contains as much as possible the "essential" information of the exact system. In stereoregular polymers, the chains are highly symmetric and it appears useful to optimize the form of the electronic Hamiltonian and nuclear repulsion energy to take this into account:

D. Jacquemin et al. A^e

jm-X

-I

/•!

;_

00

00

^ g Ng

-^^Q,Qs ^ i._

^

y.-00/l--OC /_1 ;'_1 '/.,-^

^ ^

^

(X

<xi N^ Nf. ^~

y.-W/,--00 |»l A-1 /^A^

(4)

/v- >»_i D _ i

^^. /«_oo/i=_«/i.lB-l ^MyB^

with A^^ (A^J the number of electrons (nuclei) per unit cell and Q^the atomic number of atom A. In these equations the summations over the cell index, j and h appear explicitly. They can be simplified by noting that the interaction of one cell with all others is the same whatever the cell chosen, i.e. one can consider only the interactions between a reference unit cell (7 = 0) and all the other unit cells and obtain the complete information. In this framework Eq. (4) becomes. 00

N,

N,

(5) and similarly for Eq. (3). 2.2. Crystalline orbitals After having defined the Hamiltonian in agreement with the macromolecule symmetry, it is time to turn to ^^•^. Within the HF scheme, W^'^ is represented by a Slater determinant built with one-electron wavefunctions (orbitals). It is Bloch's theorem which, by using the periodicity of the electronic density, allows to define the orbitals for infinitely long systems, ,{k,x,y,z + ja) = e'^>^(/:,;c,3;,z);

(6)

with a the unit cell length and k the wave vector associated to the electron described by 0^ denominated crystalline orbital (CO). The energies associated to these orbitals are fc-dependent and present the periodicity of the reciprocal space, i.e. (7) ^.W = ^p(fc+/^; a with I an integer. In the reciprocal space, one can therefore restrict the investigation to an interval of wave vector [-jt/a.+Jt/a], the first Brillouin

Calculation of Hartree-Fock energy derivatives in polymers

7

zone, that corresponds to the Wigner-Seitz unit cell of the direct space. As the energy bands are symmetric,

^{k)'^{-k);

(8)

half the first Brillouin zone is sufficient. The relation between the energy and the wave vector is called, the dispersion and its representation, the energy band. If one plots all the energy bands in half the first Brillouin zone, one obtains the well-known band structure of the polymer [1,2]. An alternative form of Bloch's theorem allows to express the CO as a product between plane-wave and a function presenting the periodicity of the direct space, 0^(/:,jc,>;,z) = e'\(fc,x,>;,z);

(9)

The Bom - von Karman cyclic conditions which impose the Bloch's functions to be equal in cell 0 and 2iV + l (N-^^) are often used for periodic calculations: (l>p{k,x,y,z + {2N + l)a) = (l>p{k,x,y,z);

(10)

By combining the two latter equations, we obtain (l)^{Kx,y,z-^{2N+l)a) = ^''t^'^'^'^'K^ik^x^y^z) = (l>,{Kx,y,z);

(U)

which is only satisfied for a discrete set of k values given by k^—^^K;

with

(12)

KG['N,N].

2.3. Roothaan-Hall equations The integro-differential HF equations for stereoregular polymers are simply obtained by using the above mentioned Hamiltonian and crystalline orbitals and applying the standard HF approximations, i.e. considering that each electron is moving in the mean-field created by the other electrons. This leads to a Fock operator depending on its own solutions, that is determined by an iterative selfconsistent field (SCF) procedure. Such HF equations are almost always solved by representing CO as a linear combination of atomic orbitals (LCAO): it is the well-known Roothaan-Hall approach. In LCAO, the crystalline orbitals (CO, 0^(ifc,r)) used, to build the Slater determinant are expanded as linear combinations of AO's centred in each unit cell ^If^R^-

jae\:

D. Jacquemin et al.

where r is the position of the electron, R^^ the position of the nuclei in the reference unit cell, e^ a unit vector pointing in the longitudinal direction, n is the band index and co the number of atomic orbitals per unit cell. l/^j2N +1 is the normalization factor. Note that the LCAO coefficients, C^^{k), depend on k. The fc-dependent overlap and Fock matrices read: (14) N

(15)

F.M' 2^''"^;."' j—N

and the direct-space Fock matrix in Eq. (15) can be expanded in terms of the different integrals: N

poj ^ffo.j

(O m

i \1 ^ l~-N p

jM.V

/^\X/i|^

N

<»

v« V Vpw ^ o

l-^v/

o) (u

-J

YG";.*-**' I. «,^ /J--00

/^

_i ^

/ v \ /C/i

»

N

N

I A7 U^ XT ^ - l^-Nh'-N p

r - /?^ - hde^ 1

N

Q)

^

o)

Y Y Y\*P».'*>-''G"''•*••'*';

(16)

a^

ycv

r/z;

w oi

.>j+' 4- ^ ^ ^ P^' \c^'J'^'^-'^ _ i . V V V V pO./+7-/'/-^o,/i.7j+ ^ ZJ Zj^i^ l^-N p o

^ At.v.p,a h~-<x>

^ ^ ^ ZJ ZJ^'P ^ I'-N h—N p a

^ti,p,v,o

H^^l being the one-electron interaction matrix obtained by summing the kinetic energy and electron-nuclear attraction. The Coulomb integral indices have been chosen to highlight the convergence of the lattice sums (see next section). The density matrix elements are obtained by integration: ''dob

K:--M

2c„„(fc)c:„(fc)k

^ikja '

(18)

with N^^^ the number of doubly-occupied bands. The two-electron integrals are defined as:

Calculation of Hartree-Fock energy derivatives in polymers

9

r -r An SCF procedure is adopted to determine the ground state wavefunction and energy. It consists in solving the ^-dependent generalized eigenvalue problem, F(A:)C(A:) = Cik)S{k)eik);

(20)

by imposing the orthonormality condition: (C\k))S{k)C{k)^l;

(21)

The SCF-LCAO-CO process mimics its molecular counterpart but with two additional steps: a direct-to-reciprocal space transformation, Eqs (14) and (15), and a reciprocal-to-direct space integration, Eq. (18). The total energy per unit cell is the sum of the electronic and nuclear repulsion energies:

^

j~-N

fi

V

In the section 3 we differentiate E w.r.t. the coordinates of the nuclei and the unit cell length in order to obtain general analytic formulas for the forces and cell pressure. In section 4, derivatives w.r.t. external electric fields are determined. 2.4. Lattice summations and long-range corrections If the polymeric HF procedure is very similar to the molecular scheme, several aspects of HF calculations are, nevertheless, specific to the use of the polymer symmetry: 1. Pseudo-linear dependencies 2. Energy bands numbering 3. Integration of the density matrix 4. Use of the polymer symmetry for the integrals 5. Calculation of lattice summations. The first four considerations do not play a significant role in the following, and we redirect the interest of the reader to the appropriate literature [1-6]. However, the latter is essential in the following, and we describe this issue with more details here. The lattice summations included in section 2.3. can be divided into two categories. On the one hand, we find summations presenting an exponential convergence: J c, exp(-V^? + (c, - jay- J;

(24)

10

D. Jacquemin et al.

They are easy to evaluate and a small number of cells is sufficient to reach an accurate estimation. One the other hand, we have summations corresponding to Coulombic interactions: 00

(25) which diverge when considered individually. In the first category, one finds the summations over the j and / indices with the exception of the j summation of the exchange which belongs to the second group as do the h sums of the electron-nuclei attraction, nuclei-nuclei and electron-electron repulsions. The convergence speed of j exchange sum depends on the gap of the polymer. The larger the gap, the faster the saturation. As most polymers are insulators or semi-conductors, a dozen of unit cells are generally sufficient to obtain accurate estimations of the exchange contribution. Nevertheless, for macromolecules with very small band gap, the convergence of the exchange could be very slow. On the contrary, the Coulomb type sums are always problematic. This is due to the long-range (LR) character of charge interactions : it is so huge that when considering repulsive and attractive Coulombic sums separately, they diverge, i.e. individual terms of Eqs. (17) and (22) are not finite. However, the diverging terms cancel each other in the expression of the total energy provided neutral structures are considered. For estimating these effects, the Namur group has developed (among others) a multipolar technique [1]. The basic idea of this approach is that unit cells far apart interact via their multipoles. In this procedure polygamma functions, that are directly related to Rienmand zeta's, provide the values of LR summation towards infinity. During practical calculations, the overlap-type sums which converge exponentially are evaluated in a short-range region (containing 2 ^ +1 unit cells), the Coulombic summations are computed exactly in the middle-range region (containing 2M + 1 unit cells) and approximated by multipoles in the LR region which goes towards infinity (see Fig. 2). For insulating materials, A^ is selected in a way that the overlap between cells 0 and A^ + 1 is very close to zero and a limit of M ^ 2N has been shown to be suitable. Teramae demonstrated that this cut-off procedure (so-called Namur cut-off) is the most successful amongst the different procedures proposed by theoreticians [18]. It is also striking to note that the multipole approach has been transposed to molecular computation to speed up the calculation of the Fock matrix and to reach, for large molecules, a cpu-time requirement which is linearly proportional to the number of basis functions.

Calculation of Hartree-Fock energy derivatives in polymers

11

Separated Charge Distributions

Zero overlap with cell 0 Y -M

-N

->^^

->-^

0

T M

N

-^-^

Approximate Exact Complete Exact Coulomb Coulomb Hamiltonian Coulomb Figure 2: Schematic representation of the Namur cut-off procedure.

Approximate Coulomb

3. Geometrical derivatives First, let us describe the (analytic) geometry optimization and vibrational analysis schemes developed for macromolecules. For the optmizations, the approaches proposed in the literature mostly differ by the scheme proposed for the gradient with respect to the unit cell length (so-called cell-stress). Dewar and coworkers first performed an analytic CO geometry optimization of polyethylene but their technique lacked of cell-stress [19]. Teramae and coworkers recognize this limitation and solved the problem by providing a formula for calculating the cell-stress as a combination of the forces on the nuclei centered in different unit cells [20]. This formula, also used by Hirata and Iwata [21] presents the advantage of avoiding the direct evaluation of the cell-stress and subsequently no extra integral calculation is required. However, as seen below, the long-range (LR) Coulombic effects related to the cell-stress are large, and more particularly are larger than these associated with geometrical gradients with respect to nuclei positions [7]. Subsequently, the formulas needed to compute directly (and possibly solely) the cell-stress have been worked out [7]. Kudin and Scuseria designed an efficient geometry optimization algorithm that avoids the need of explicit calculation of the cellstress: the UC length is optimized indirectly through the evaluation of the internal coordinates that extend between adjacent UC [22]. The Crystal group extended the optimization techniques to periodic slabs and crystals [23]. The number of developments dealing with the analytical determination of HF second derivatives are sparse. Indeed, in general, the vibrational frequencies are evaluated by numerical differentiation of the gradients. To the best of our knowledge, Hirata and Iwata have been the first to propose an analytical approach for the Hessian [24]. We have recently developed a similar analytical approach for obtaining the Young modulus [15] and the infrared intensities [16]. An analytic scheme for obtaining the full phonon structure by a single-shot calculation has also been elaborated by Sun and Bartlett [25].

D. Jacquemin et al.

12

3.1. First derivatives 3.1.1. The HF gradients To optimize the geometry of polymers, one needs to determine the forces. To reach this goal, we differentiate the total energy per unit cell, Eq. (22), with respect to the Cartesian coordinates {I^Jy or IS) of one atom (/). During the structural optimization, all equivalent atoms (i.e. atoms in different unit cells related by a translation) are moving in phase, i.e. the modification of the position of one atom is in fact the in-phase displacement of all equivalent atoms. Subsequently, the translational symmetry of the polymer is conserved throughout the optimization procedure, and only derivatives with respect to the nuclei inside to the reference unit cell need to be computed. By differentiating Eq. (22) with respect to a given nuclear position, one obtains after rearrangements of the terms [7], dH 0.7 ' dl pOJ

§•111 h-N ^i

^ ^

(o CD N ZJZJ p

a

l~-N

-^111 ^

p

o

°°

ZJ^^P

ZJ

;3 -ff^f^^y^p^o

/I—00^'

dE^ dl

(26)

"S? pO,Uj-h ZJ^^P l~-Nh-

where the so-called energy weighted density matrix has been defined similarly to the density matrix.

5;c..(^)c::.(^)^.(^)V^^>

(27)

Eq. (26) is completely equivalent to the corresponding molecular equation. It is also the same equation as in Ref. [20]. With respect to the computation of the energy, the only additional components are the integral derivatives and W^^;^ The convergence speed of the former are detailed in section 3.1.2, whereas, for the latter, the integration over the Brillouin zone is performed as for the density, Eq. (18), using various procedures [6]. For the cell-stress one can obtain a completely similar equation,

Calculation of Hartree-Fock energy derivatives in polymers

13

/^w«.A ^a

da

da

N

(o

m

j^-N

fi

V

111

/

2

a

/—AT

dE^ da

da

(28)

N

4

p

a

l~-Nh~-N

da

3.1.2. Derivatives of integrals Integral derivatives can be computed as in molecules, the main concern being the saturation speed of the lattice summations in Eqs. (26) and (28). For the nuclear repulsion energy, which is computed classically, a LR procedure is described below. This approach can be easily generalized to the other electrostatic interactions involved in integrals derivatives [7]. If we differentiate Eq (23) with respect to an atomic position varying along the perpendicular or transverse axis of the polymer (X or Y), we obtain: dE^ dL

=a2e.K-/j|;

1

(29)

(A/K-O^K-^.)^K-/,-/.ar)

that, formally, can be simplified as: dE^

^

1

"df.-Ir, -«lc" + ((i-x)"j

1/2 ~ 2^-^'

(30)

with c and d determined by the distance between the atoms A and / in the reference unit cell, i.e. c and d depend only on coordinates in cell /i = 0. Although, it is easy to compute / , the summation towards infinity converges very slowly with x. For example, one needs to use 2000 unit cells to obtain an accuracy of 10"^ a.u. and one million unit cell are necessary to get a 10"^^ convergence [8]. To circumvent this drawback, we proposed to compute Eq. (30) exactly for a few cells and used multiple Taylor expansions to evaluate the LR contributions, i.e. we proceed consistently with Namur's approach illustrated in Fig. 2. Indeed, when x is quite important, c and d become relatively negligible, and Taylor expansion can be used. This technique is completely general and can be shown to be mathematically equivalent to a multipolar expansion procedure. Moreover, accuracy can be systematically improved: one just needs to increase the order of the expansion.

D. Jacquemin et ah

14

Let's consider / and perform a multiple Taylor expansion around c = 0 and d = 0. By restricting the expansion to the first orders, one successively obtains:

r= [x] 1

3d 1/5 "^

(31)

[x] X 3C^

r\ _ rO _

6d~

r52;

3/2

2(x'] "jc" (x^j jc

/• = /

.

.,»

/^ = /^ + S(xyx'

(33)

+

15^"

2(xYx'

(34)

\3/2

(xYx'

As depicted in Fig. 2 and explained in section 2.4, Eqs (31)-(34) are used in the LR region by performing summations towards infinity. Therefore, the LR contributions to Eq. (30) correspond to polygamma functions which possess tabulated values: 30

—00

(35)

\f = -W{i,U); L.x-(/ 00

x~-U J -00

1*1 \f'-\l*l

lx~V x~-U 00

-00

1*1

Ix'U

X'-U

\f = - f (2,f/) + i

'-W{4,U):

(36)

8

(c'-4d') {c'-l2c'd'^Sd') f = -W{2,U) + ^^ ^ ^-W(A,U) - ^ —^ ^W(6,U): (37)

\_x~U x~-U J

where, for clarity, we have used (/=M + 1. These LR corrections are extremely "cheap" to obtain: in most practical calculations, their computation represents a completely negligible cost. Indeed, on the one hand, polygamma functions are evaluated at the beginning of the calculations because the only required parameter is [/, and, on the other hand, Eqs (35)-(37) depend solely on the coordinates inside the reference unit cell. One can proceed similarly for gradients along the longitudinal axis [7]. For the derivative of the nuclear repulsion energy with respect to unit cell length, one gets:

Calculation of Hartree-Fock energy derivatives in polymers

15

{A^-B^-ha)h

ris;

( ^ ( 4 - B,,f + (A, - fi,)% (A, - B, - Aa)') ^£' o'a

( c - jf)j 1/2 ~

(39)

^ ^ '

JT—»lc' + (rf-jc) j

After using the symmetry and polygamma functions, we obtain the following corrections in increasing order of polygamma functions: 00

-00

(40)

k" = o.X~U

JC—f/

E-Ek=2-i

\g^ = -2W{0,U);

2^2 k= i^i

\g* = -2W(0,U) — i

00

(41)

Mc'-2d')

'-W{2,U);

(42)

-00

1*1 V-\l*l|g« = -2W{Q,U) - ^—^ .v-f/

'- W{2,U)

x~-U

(43)

5(24cV-3c''-8rf'')

n4,u) 96 The diverging behaviour of Eq. (38) is concentrated in Eq. (41) but is annihilated by other contribution from the total forces [7]. Nevertheless, the LR corrections to Eq. (38) are larger than in Eq. (29), illustrating that the lattice summations present in the cell-stress, Eq. (28), are converging more slowly than in other gradients. The LR corrections to the derivatives of the one- and twoelectron integrals can be obtained by following exactly the same procedure [7]. Note that even for forces along the transverse axis, some lattice summations may diverge when considered individually. Note that this LR procedure can be extended to derivatives with respect to the helical angle once "generalized" polygamma functions are used [13]. 3.1.3. Example Examples illustrating this procedure can be found in Refs. [5,7,8,10]. It turns out that the use of LR corrections considerably speed up the procedure. For

D. Jacquemin et al.

16

instance, in zigzag hydrogen chains, one needs M=10^ to obtain perfectly converged gradient without LR corrections but only M=10 when two orders of LR terms are added to the exact Coulomb summation. In fact, the LR corrections are mandatory in order to obtain accurate gradients (10"^ a.u.) with a computationally tractable number of unit cells in the medium-range region (Fig. 2), This long-range procedure helped optimizing accurately hydrogen fluoride chains, the variations with and without corrections being of the order of 0.001 A on the optimized bond length [8]. This procedure also allowed to optimized the ground-state geometry of polyyne [10]. 3,2, Second derivatives 3J,L The Hessian and the Polymeric CPHF As for the first derivatives, there are different types of second derivatives due to the unit cell length parameter [15]. A particularly interesting term is the second derivative with respect to the unit cell length. Indeed, this term is (formally) directly related to the Young modulus. Nevertheless to obtain the Young modulus by a single-shot calculation, one needs to compute the full Hessian [15]. Derivatives with respect to the position of the nuclei have already been given by Hirata and Iwata [24] and we therefore focus on derivatives with respect to a. By combining Hirata and Iwata formula with the expression of the cell-stress, Eq. (28), the "cell Hessian" can be obtained: w -i2 17

da'

AT

fO

n

^

2d2^ 2J^P 2d

ftt

a l^^N

•t

-N

tu

0}

fa

N

N

\-h ^

n

2^0, jM,h-^l \

d'G

da' 0,hJJ+i ^^i,py,Q

da"(a

0.J

m

(a

N

N .«

V

n/-^O^j,h,h+l

da

da

j^-N

<x>

\

(44)

N

*^ ^

p

a

l—N,

da dec

In addition to the first and second derivatives of the one and two-electron integrals with respect to the unit cell length, analytical evaluation of Eq, (44) requires the evaluation of the a-derivatives of the density and energy-weighted

Calculation of Hartree-Fock energy derivatives in polymers

17

density matrices. To determine these latter quantities, a coupled-perturbed Hartree-Fock (CPHF) procedure has been set up [15,16], By differentiating with respect to a the /:-dependent SCF eigenvalue equation and the orthonormalisation conditions (Eqs (20) and (21)), one obtains: ¥\k)C{k) + ¥{k)C\k)«

C\k)S{k)s{k) + C(k)&\k)e{k) + C(it)S(it)e"(it); (45)

CHk)S(k)C(k) + CHk)SHk)C(k) + CHk)S(k)a{k) * 0;

(46)

where the superscript a stands for differentiation with respect to the unit cell length, M^{k)^^^: (47) ^ ^ da Following the molecular CPHF procedure, the unknown is rewritten under the form of a product including the unperturbed LCAQ matrix: e(ifc)«C(ifc)U^(ik);

(4^)

The elements of the new unknown {\J^{k)) requested to evaluate the derivative of the density are given by UUk)-

^^v(fc)-/:v(^K(^)

^^^ ^^^^^^^^.^^^(virt,Qcc);

(49)

^v\k} •^sAk) hi^

^^.vW^

^r (k) "*'"

with (^,v)-.(o€€,0€c)or(virt,virt);

(SO)

At

where we have defined: QtVi) m aim^k)C(k):

(51)

r(k)^(TikWik)C(k);

(52)

similarly to the standard molecular orbital CPHF procedure. As can be seen, U'*(^) depends upon F''(^) which is ftmction of V{k): an iterative procedure is mandatory. As for the energy, the main difference in the CO approach is the ^-space character of the equations, so that a real to it-space transformation is necessary. Noting that ikja is actually independent of a, one has: F^UA^EMM^

f e^^^;

(S3)

18

D. Jacquemin et al.

and similarly for the overlap matrices. Once the fc-space first derivative of the density matrix is obtained from the undifferentiated and differentiated LCAO coefficients, (55) one performs a numerical integration to get back to real space:

da

(56)

Jt \

Therefore the CO-CPHF procedure parallels its molecular counterpart but with two additional steps, Eqs (54) and (56). This procedure is illustrated in Ref. [15]. 3.2.2. Long-Range As Eq. (44) is converging at a slow speed with respect to the Coulombic summation indices, it is also necessary to develop a LR procedure for the second derivatives w.r.t. the atomic positions, i.e. very accurate evaluation of the vibrational spectra or elastic properties of polymeric chains requires the use of a multipole-like approach. To determine LR corrections, one can proceed as shown in section 3.1.2. For the second derivative of the nuclear repulsion energy w.r.t. unit cell length, one obtains: {A^-B.-haf3h^

(^(A,-Sj%(A,-5,f + ( A - B , - / i a f )

11^1 QAQB

da'

dE"' da

3{c-x) X

(57)

00

(58)

After using the symmetry and polygamma functions, we obtain the following corrections for the fu-st two orders of polygamma ftmctions: G{C^ -2d^)W{2,U);

(59)

Calculation of Hartree-Fock energy derivatives in polymers

19

6{c^'l(f)W{l,U)-^-^

^-W{4,U): (60) 16 whereas the diverging behaviour of Eq. (57) is concentrated in a W{QJJ) term that cancels out with contributions of same form but opposite sign originating from other Coulombic terms [15]. 4. Polymers in Electric Fields In this section we deal with the response of stereoregular polymers to uniform static and/or dynamic electric fields at the Hartree-Fock level of approximation. The first three sub-sections present the general formalism for calculating the response whereas the last two sub-sections (4.4 and 4.5) discuss two different types of derivative: (a) electric field derivatives at a fixed geometry and (b) geometrical derivatives in the presence of a finite static field. Our presentation is based on using the vector potential to represent the interaction between the external electric fields and the polymer [9,11]. Although the vector potential is the most straightforward way to develop the theory there is another approach, known as the 'modem theory of polarization' (MTP) [26] that yields similar formulas where comparisons can be made. Indeed, it is readily established that the MTP perturbation treatment of static fields [27] leads to essentially the same fundamental equation (see Sec.4.4) as our earlier vector potential treatment [9], when the latter is applied to the special case of static fields (In order to make this connection one must compare the two formulations before the interaction term is replaced in Ref. [27] by a discrete Berry phase [28] expression.). Similarly, the MTP finite field treatment that has recently been presented [29] is akin to our Sec.4.5, although much different methods of solution are utilized in the two cases. 4.1. Use of vector potential to maintain translational invariance When a uniform static or dynamic electric field is applied to a stereoregular polymer the appropriate interaction terms must be added to the HF Hamiltonian. In the case of an ordinary molecule the interaction in the longitudinal direction, e^, is normally represented by the scalar interaction potential between the instantaneous electronic dipole moment operator (the origin is assumed to be at the center of nuclear charge) and the applied field, EXi)

V^eE^{t)2z,;

(61)

20

D. Jacquemin et al

Here e is the magnitude of the electronic charge and z • is the position of the f^ electron in the longitudinal direction. For stereoregular polymers the scalar potential is inadequate because it destroys translational symmetry and is unbounded from below in the chain direction. The simplest way out of this dilemma is to employ the vector potential \{t) instead of E ( 0 . In that event the momentum operator p = -iV is replaced by p-\-(e /c)\{t) where

mO-c-^:

(62)

If A ( 0 is spatially uniform, then so is E ( 0 and vice versa. Hence, use of the vector potential maintains translational invariance but it also makes the Hamiltonian time-dependent. This is not a problem since we are often interested in time-dependent (or, equivalently, frequency-dependent) electric fields. Then static, or dc, fields correspond to the special case for which the frequency is zero. At the same time that we introduce the vector potential into the kinetic energy expression we must also carry out the corresponding transformation of quasimomentum, k, that appears in the CO of Eq. (13):

p - > p + - A ( 0 ; k^k' c

= k-\'-A(t); c

(63)

Then the periodic function 0^(^',x,y,z) of Eq.(9) becomes y

10

v„{k\x,y.z) = ^"^0.(/:U,y,z) = -^^2^^^„{k')

N

2e'''''-'''xi;

(64)

Note that the LCAO coefficients are now explicitly, as well as implicitly (through A:'), time-dependent. Finally, multiplication of the unit cell periodic function v„ by the plane-wave exp(/fe) yields the time-dependent Bloch orbital 0„(^',x,y,z). 4.2. Crystalline orbital time-dependent Hartree-Fock (CO-TDHF) equation The next step is to substitute the Bloch orbitals into the time-dependent HartreeFock equation [9]:

in order to obtain a relation that can be solved for the coefficient matrix C. As in the conventional TDHF treatment of ordinary molecules [30] it is convenient

Calculation of Hartree-Fock energy derivatives in polymers

21

to use the general form of this equation with off-diagonal Lagrange multipliers, ^m/i(^')' rather than the canonical form used for the field-free problem (see Eq. (7) above). Except for the non-canonical £^,„(^'), and terms due to idldt, it is easy to show [9] that the result is identical to Eq. (20) with the proviso that k is replaced by k'. The idldt factor gives rise to three terms. Two of them arise because the coefficients depend both explicitly and implicitly (through ^')upon f. Thus, dt

/^

dt \

dk' J \

dt

1^.

,

\

dk' j

\

dt

^.

The last term on the rhs of Eq. (66) is the same as in ordinary molecular calculations whereas the first term, which is proportional to the field, arises specifically because of the Bom-von Karman boundary conditions. The latter has a simple physical interpretation. In a static electric field there will be electronic charge transfer between the unit cells of a finite oligomer so that the chain ends become oppositely charged. If the ends are subsequently attached, and periodic boundary conditions are imposed, then charge must flow through the system in order to maintain neutrality for each unit cell. The derivative with respect to A:' in Eq. (66) is associated with this intercellular charge flow. It is just this term that is missing in the traditional sawtooth formulation of the electronic polarization due to an electric field [31]. The third term due to idldt derives from the exponential factor exp[-/A;'(^-ya)] in Eq.(64). This leads to a contribution proportional to (z-ja), which also has a simple physical interpretation. In fact, it is associated with the intracellular charge transfer that accompanies the intercellular charge transfer when a finite oligomer is subjected to an applied field [31,32]. Although neither of the charge transfer terms is Hermitian by itself, the sum of the two does satisfy this requirement. Taking into account the additional terms that arise because of the time-dependent field the Fock matrix, F , defined by Eqs. (15)-(19) gets replaced by: ¥{E,) - / s ( ^ j + MeE^t) + is[^-^^eE^(t);

(67)

Here the field-dependence of F ( £ j , as in the treatment of an ordinary molecule, is due to the field-dependence of the density matrix defined in Eq. (18). The -iS[d/dt) term is also the same as one finds in a molecular treatment. On the other hand, the factor M + iS[d/dk') is associated with the application of the boundary conditions. The matrix M is defined in the same

22

D. Jacquemin et al

way as S except that, in this case, we need matrix elements of z - ja rather than the identity, i.e,

M,Ak')= 2^'"^<^ =

't'"'"'h(-lV-Hxi);

(68)

At this point one can replace A:' by k in Eqs.(67) and (68)as long as there is a finite band gap. For closed shell systems, it can be demonstrated [9] that all properties are correctly obtained by subsequently integrating over A: from -It/a to -\-JT/a. Eq. (67) (with k' = k) can also be obtained by using the periodic scalar potential [33],

V^ieEXt)e'''V,e'''':

(69)

rather than the form given in Eq. (61). Finally, we note that Eq. (67) is also applicable in Kohn-Sham density functional theory by replacing the exchange terms in Eq. (16) with a single particle exchange-correlation potential. At the present time, however, conventional potentials have been found [34,35] to result in a catastrophic overpolarization for finite oligomers due to an inadequate treatment of exchange [36-39]. One possible way around this difficulty is to use exact exchange but, then, an appropriate correlation potential remains to be developed [38]. Another possibility is to make an Ewald split into short- and long-range potentials using an exact treatment of exchange in the latter and standard DFT exchange in the former [40,41]. Although this procedure shows some initial promise, it has yet to be thoroughly tested. If one uses the TDHF Fock matrix of Eq. (67) in Eq. (20) and, at the same time, replaces the diagonal matrix of Lagrange multipliers by a non-diagonal (Hermitian) matrix, then one obtains the CO-TDHF equation. 4.3. Polarization We define P.{t) to be the longitudinal dipole moment per unit cell, i.e. the polarization (dipole moment per unit length) times the cell length. The dipole moment, in turn, is the negative of the energy derivative with respect to the field, which is one of the quantities of interest in this chapter. For an ordinary molecule this property can be determined by calculating the average value of the instantaneous dipole moment operator. In the presence of applied field(s) there will be field-induced contributions due to molecular polarizabilities and hyperpolarizabilities as well as a permanent dipole moment term. For the case of an infinite stereoregular polymer the polarization cannot be calculated in the same manner because the dipole moment operator is unbounded from below, as noted in the above discussion of the scalar potential. A way around that

Calculation of Hartree-Fock energy derivatives in polymers

23

difficulty was described by Blount over 40 years ago [42]. Basically, Blount's method amounts to generating the electronic coordinate z by taking the derivative of exp(//:z) with respect to k.lt is easy to verify that his procedure leads to the expression [9]

-nla

j=-N

n ju,v

i^xi

funik);

(70)

Note that P^[t) is just the average value (per unit cell) of the intracellular plus intercellular charge transfer terms in Eq. (67) as one might have anticipated purely on physical grounds. 4.4. Perturbation theory solution of the CO-TDHF equation For fields that are not too strong the CO-TDHF equation may be solved by perturbation theory, using the field-free solution as the zeroth-order approximation, in a manner similar to that employed for the finite oligomer problem [30]. By expanding Eq.(70) as a power series in the field(s) - which may be static and/or dynamic - this will yield the permanent dipole moment, linear polarizability, and various nonlinear optical properties (hyperpolarizabilities) as coefficients of the different terms in the expansion. The key new term in the CO-TDHF equation that requires further consideration is ieS[dC/dk), which introduces the CO phase problem. 4.4.1. CO phase problem In general the CO coefficients will have both a real and an imaginary component, which may be expressed in terms of a magnitude and a phase angle. On the other hand, each orbital is determined by the CO-TDHF equation only up to an arbitrary phase factor expr/0(n,/:)j. In order to obtain numerically well-behaved dC/dk the phase angle must be a continuous function of k. This will not happen automatically because the orbitals are determined individually for each value of k. However, one is free to multiply by a phase factor in order to satisfy the continuity condition (see further below). In addition, the desired set of phase angles is not unique. One can add any continuous function of k as long as the periodicity requirement


(71)

remains fulfilled. Thus, in general we may use 6[n,k)+mka. Ordinarily, physical properties would be expected to be independent of m. However, that

24

D. Jacquemin et al.

turns out not to be the case in the CO-TDHF treatment. The addition of mka to the phase angle changes the derivative with respect io k hy ma which leads to a change in the permanentdipole moment per unit cell by the same amount. This is a fundamental indeterminacy that cannot be removed a priori. It can be removed after the fact as long as one has some crude estimate (error « all) for the dipole moment. On the other hand, it turns out that all higher-order properties are independent of AW as discussed below. 4.4.2. Evaluation of dCIdk The first step in the perturbation treatment is to obtain the field-free solution including thefield-freephase factors. Except for the contribution due to d9/dk, the derivatives of thefield-freecoefficient matrix, C^, with respect to k may be evaluated analytically by differentiating the field-free Fock equation and normalization condition (Eq. (21)). Writing dC^/dk in the form dCydk = C'Q;

(72)

for each k leads to [33]

with B = ( c « n * ^ C ^ and R = ( c « n * ^ C ^ • (74) ^ f dk ^ ' dk where (C^^) is the hermitian conjugate of C^^. The derivatives in Eq. (74) may be obtained by analytical differentiation of the expressions on the rhs of Eqs. (14) and (15). It is important to note that Eq. (73) leaves the imaginary component of Q^, i.e. 32^, undetermined. This is as it should be since SQ^ depends upon the phase factor chosen for the n^ CO. Indeed, it is straightforward to show that introducing a phase angle 0{n,k) causes 3g„^ to change by dd(n,k)/dk [11]. One can derive a satisfactory analytical formula ***for the phase angle as well as for ^Q„„, with the latter containing the contribution from d9{n,k)/dk. The procedure that has been utilized [11] involves choosing 9(n,k) so that, for each band, the imaginary component of one particular AO coefficient vanishes at all k (this coefficient must originally have a non-zero real component throughout the Brillouin zone but is otherwise arbitrary). It

Calculation of Hartree-Fock energy derivatives in polymers follows that the derivative of the imaginary component, with respect to /:, is zero. In combination with Eq. (72), this condition yields a simple expression for 3Q^^ in terms of the quantities that appear in Eq.(73). Analytical geometric derivatives can, then, readily be obtained and used in the calculation of infrared intensities [43], as determined by geometric derivatives of the permanent dipole moment. On the other hand, Raman and hyperRaman intensities depend only on the induced dipole moment, which is independent of 2Q„„ [11]. 4.4.3. Perturbation equations To obtain equations for each order of perturbation theory one follows the conventional procedure of expanding the LCAO coefficients and the Lagrangian multipliers as power series in the field(s). Equating terms of like order in the field(s) yields the desired perturbation equations. The zeroth-order result is the usual field-free CO equation. In first-order, there are the additional intracellular and intercellular charge transfer terms discussed in Sec. 4.2 as well as the dCjdt contribution, which is identical in form to the dCjdt term that occurs in a finite oligomer treatment. First-order equations for static and dynamic fields are given in Ref [9] as are the second- and third-order equations. The first-order intercellular charge transfer term is determined by dC^/dk, which is obtained as described in Sec. 4.3.2. It turns out that the phase angle term d9{n,k)/dk makes no contribution to thefirst-orderwavefunction if one adopts the non-canonical solution of Kama and Dupuis [30]. More generally, it has been demonstrated [11] that the linear polarizability and all hyperpolarizabilities are independent of the phase angle derivative. Hence, this derivative may be set equal to zero except when calculating the permanent dipole moment. The latter is obtained simply by substituting C = C" in Eq.(70). In second-order the intercellular charge transfer term is determined by the derivative of thefirst-ordercoefficient matrix with respect to k. The latter may be found by differentiating the first-order perturbation equation. This introduces d^C^/dk , which may be evaluated [11,32] by differentiating the corresponding relation for dC^/dk, i.e. Eq. (73). One can proceed in the same manner to whatever order is desired although to our knowledge no-one has gone beyond second-order since that is sufficient to obtain thefirst,second, and even the third hyperpolarizability. The static polarizability and hyperpolarizabilities are, of course, equivalent, to second- and successively higher-order energy derivatives with respect to the field. 4.4.4. 2n + l rule for (hyper)polarizabilities It is straightforward to calculate the polarization through order n from Eq. (70) using solutions of the perturbation equations through order n-1. This is known

25

26

D. Jacquemin et al

as the "iterative" procedure. There is a more efficient "noniterative" procedure, however, that yields the polarization through order 2n + l using CO's only through order n. Explicit noniterative formulas for the polarization through third-order (i.e. up to and including the second hyperpolarizability) are given in Eqs. (25H28)ofRef. [11]. 4.5. Finite field treatment of the CO-TDHF equation 4.5.1. Motivation The perturbation theory solution of the CO-TDHF equation that has been presented is based on a power series expansion about zero field. Of course, the electric field of interest may be so strong that such an expansion is nonconvergent. In that event the treatment of Sec.4.3 would be inappropriate. There are also other reasons for considering a finite field treatment. The classic one is the relative ease of implementing not only Hartree-Fock but higher level methods such as MoUer-Plesset perturbation theory or coupled cluster theory. Although such an approach to higher level CO field-dependent properties seems quite feasible we are not aware of any calculations that have been reported as yet. There is another rationale for the finite field approach, which is particularly important with regard to nonlinear optical (NLO) properties of polymers. It is now widely recognized that the so-called vibrational (actually, vibronic) contributions to such properties can often rival or exceed the pure electronic contribution [44]. When the electrical and/or vibrational anharmonicity is large the preferred method of calculation relies on a nuclear relaxation treatment wherein the key step is a geometry optimization in the presence of a static field. For this purpose we need non-perturbative solutions of the CO-TDHF equation in the static limit, i.e. the finite field CO (FF-CO) equation. Whereas the finite field case is a straightforward extension of the field-free problem for ordinary molecules that is not true for infinite stereoregular polymers. Thus, FF-CO is a work in progress [45]. An appropriate methodology has been formulated, as described below, but it has yet to be adequately tested or implemented in an ab initio framework. 4.5.2. Numerical evaluation of dC/dk and phase factors In the FF-CO equation there is a field-dependent intercellular charge transfer term that contains the operator d/dk. Since this term is not treated perturbatively we no longer have the simplification where the derivative that appears in an equation of a given order is obtained from derivatives already found from the solution of lower-order equations. Similarly, dC/dk can no

Calculation of Hartree-Fock energy derivatives in polymers

27

longer be obtained analytically because differentiation of the intercellular charge transfer term produces the second derivative of C with respect to^, which is unknown. Hence, numerical differentiation must be utilized instead. The CO phase factors could still be determined analytically. For the case of finite fields, however, a numerical approach seems more convenient since it dovetails with the required numerical differentiation to find dC/dk. In general there will be a contribution to this derivative due to a non-zero value of the integer m in Eq.(71). One can determine m, and thereby eliminate the unwanted contribution, by calculating the permanent dipole moment (cf. Sec.4.4.1) at each point along the geometry optimization pathway. This procedure was employed successfully in Ref. [43] at the field-free equilibrium geometry in order to obtain infrared intensities. The most straightforward numerical method for obtaining the phase factors is by means of a least squares fit to ensure that the orbital eigenvectors at successive k points are as similar to one another as possible [45]. This needs to be done self-consistently starting at ^ = 0, where the eigenvector is real, proceeding to k^Jtja and, then, across the zone boundary back to the starting point. This last step is equivalent to going from k = -jt/a to k = 0. Trial calculations using semiempirical Hamiltonians indicate that subsequent numerical differentiation using a central k point and a sufficient number of nearest neighbors will give accurate dC/dk (checked analytically for the fieldfree case) even for a relatively small total number of k points in the Brillouin zone. Keeping the total number of k points small is an important consideration as pointed out in the next sub-section. 4.5.3. Self-consistent solution of the FF-CO equation. Since the LCAO coefficients depend upon k implicitly, rather than explicitly, it is most convenient to treat the intercellular charge transfer self-consistently in combination with the two-electron coulomb and exchange contributions. In order to do so we take advantage of the normalization condition to write: dCjdk = [{dCIdk)C'S]C ;

(75)

Then, given a guess for (JC/Jik)C*S (here C* is the hermitian conjugate of C) from the previous iteration, this term may be included on the Ihs of the FF-CO equation and treated self-consistently in the same manner as the electron repulsion terms. The convergence behavior of the self-consistent-field (SCF) solution depends upon the magnitude of the field and the total number of k points, A^^. In fact, convergence problems are encountered when Nj^aE^ becomes comparable to the bandgap. If Nj^a is regarded as the equivalent length of an

28

D. Jacquemin et al.

Open chain, this is also more or less where open chain calculations experience difficulty. It is, therefore, advantageous to keep A^^ as small as possible in the manner described in Sec.4.5.2. Preliminary results, based on model semiempirical calculations [45], indicate that the above formulation will successfully allow for finite field geometry optimization. Further testing is necessary to establish the general validity of this approach with the ultimate goal being an ab initio computer code. 5. Conclusion In this chapter, we have described recent developments aiming at computing via one-shot calculations the physical and chemical properties of stereroregular polymers. In the last few years, several contributions have appeared in this crystal-orbital field and it is now possible to determine theoretically, in an almost routine fashion, properties like band structures, density of states, groundstate geometries, vibrational frequencies, elastic moduli, infra-red intensities, static and dynamic polarizabilities and hyperpolarizabilities, ... This demonstrates that ab intio quantum chemistry can be regarded as a valuable initial step in the framework of multidisciplinary integrative investigation of macromolecules. Acknowledgments. D.J. and E.P. thank the Belgian National Fund for Scientific Research for their research associate positions. This work is the results of several fruitful collaborations and discusssions. We especially thank Prof J.M. Andre, Prof Y. Aoki, Prof. D.M. Bishop, Dr. B. Champagne, Dr. Y. Dong, Dr. F.L. Gu, and Prof. M. Springborg (alphabetic order). References. 1. J. M. Andre, J. Delhalle, J. L. Bredas, Quantum Chemistry-Aided Design of Organic Polymers for Molecular Electronics (World Scientific Publishing Company, London, 1991). 2. J. Ladik, Quantum Theory of Polymers as Solids (Plenum, New York, 1988). 3. J. M. Andre, Computational quantum chemistry on polymer chains: aspects of the last half century. Chapter 36 of ^'Theory and Applications of Computational Chemistry: The First Fort Years^\ Edited by C. Dykstra (Elsevier B.V.). B. Champagne, Ab initio Polymer Quantum Theory, Chapter 1 of'Molecular simulation Methods for Predicting Polymer Properties"', Edited by V. Galiatsatos (John Wiley & Sons, New York, 2005). 4. K. Jug, T. Bredow, J. Comput. Chem. 25 (2004) 1551. 5. D. Jacquemin, Mise au point de methodes de chimie quantique pour optimiser la geometric despolymeres stereoreguliers, PhD thesis, (FUNDP, Namur, 1998)

Calculation of Hartree-Fock energy derivatives in polymers

29

6. D. Jacquemin, B. Champagne, J. M. Andre, E. Deumens, Y. Ohm, J. Comput. Chem. 23 (2002)1430. 7. D. Jacquemin, B. Champagne, J. M. Andre, J. Chem. Phys. I l l (1999) 5306. 8. D. Jacquemin, B. Champagne, J. M. Andre, J. Chem. Phys. I l l (1999) 5324. 9. B. Kirtman, F. L. Gu, D. M. Bishop, J. Chem. Phys. 113 (2000) 1294. 10. D. Jacquemin, B. Champagne, Int. J. Quantum. Chem. 80 (2000) 863. 11. D. M. Bishop, F, L. Gu, B. Kirtman, / Chem. Phys. 114 (2001) 7633. 12. F. L. Gu, D. M. Bishop, B. Kirtman, J. Chem. Phys. 115 (2001) 10548. 13. D. Jacquemin, B. Champagne, Int. J. Quantum. Chem. 85 (2001) 539. 14. B. Kirtman, B. Champagne, F. L. Gu, D. M. Bishop, Int. J. Quantum. Chem. 90 (2002) 709. 15. D. Jacquemin, J. M. Andre, B. Champagne, / Chem. Phys. 118 (2003) 373. 16. D. Jacquemin, J. M. Andre, B. Champagne, J. Chem. Phys. 118 (2003) 3956. 17. B. Champagne, D. Jacquemin, F. L. Gu, Y. Aoki, D. M. Bishop, B. Kirtman, Chem. Phys. Lett. 373 (2003) 539. 18. H. Teramae, Theor. Chim. Acta 94 (1996) 11. 19. M. J. S. Dewar, Y. Yamaguchi, S. H. Suck, Chem. Phys. Lett. 51 (1977) 175; Chem. Phys. 43(1979)145. 20. H. Teramae, T. Yamabe, C. Satoko, A. Imamura, Chem. Phys. Lett. 101 (1983) 149. H. Teramae, T. Yamabe, A. Imamura, J. Chem. Phys. 81 (1984) 3564. 21. S. Hirata, S. Iwata, / Chem. Phys. 107 (1997) 10075; J. Chem. Phys. 108 (1998) 7901; J.. Phys. Chem. A 102 (1998) 8426. 22. K. N. Kudin, G. E. Scuseria, Phys. Rev. B 61 (2000) 16640. K. N. Kudin, G. E. Scuseria, H. B. Schlegel, J. Chem. Phys. 114 (2001) 2919. 23. B. Civalleri, Ph. D'Arco, R. Orlando, V. R. Saunders, T. Dovesi, Chem. Phys. Lett. 348 (2001) 131. K. Doll, V. R. Saunders, N. M. Harrison, Int. J. Quantum Chem. 82 (2001) 1. K. Doll, Comput. Phys. Commun. 137 (2001) 74. 24. S. Hirata, S. Iwata, J. Mol. Struct. (THEOCEM) 451 (1998) 121. 25. J. Q. Sun, R. J. Bartlett, J. Chem. Phys. 109 (1998) 4209. 26. R.D. King-Smith and D. Vanderbilt, Phys. Rev. B 63 (1993) 1651; R. Resta, Rev. Mod Phys. 66 (1990) 899. 27. R.W. Nunes and X. Gonze, Phys. Rev. B 63 (2001) 155107. 28. M.V. Berry, Proc. Roy. Soc. London, Ser. A 392 (1984) 45. 29. I. Souza, J. Iniquez and D. Vanderbilt, Phys. Rev. Lett. 89 (2002) 117602. 30. S.P. Kama and M. Dupuis, J. Comput. Chem. 12 (1991) 487. 31. M. Springborg, B. Kirtman and Y. Dong, Chem. Phys. Lett. 396 (2004) 404. 32. B. Kirtman, Int. J. Quantum Chem. 43 (1992) 147. 33. P. Otto, F.L. Gu and J. Ladik, J. Chem. Phys. 110 (1999) 2717. 34. B. Champagne, E.A. Perpete, S.J.A. van Gisbergen, E.J. Baerends, J.G. Snijders, C. Soubra-Ghaoui, K. Robins and B. Kirtman, J. Chem. Phys. 109 (1998) 10489; erratum: 110 (1999)11664. 35. B. Champagne, E.A. Perpete, D. Jacquemin, S.J.A. van Gisbergen, E.J. Baerends, C. Soubra-Ghaoui, K.A. Robins and B. Kirtman, J. Phys. Chem. A 104 (2000) 4755. 36. P.R.T. Schipper, O.V. Gritsenko, E.J. Baerends, J.G. Snijders, B. Champagne and B. Kirtman, Phys. Rev. Lett. 83 (1999) 694. 37. P. Mori-Sanchez, Q. Wu and W. Yang, J. Chem. Phys. 119 (2003) 11001. 38. F.A. Bulat, A. Toro-Labbe, B. Champagne, B. Kirtman and W. Yang, J. Chem. Phys. 123 (2005) 109401 39. S. Kummel, L. Kronik and J.P. Perdew, Phys. Rev. Lett. 93 (2004) 213002. 40. M. Kamiya, H. Sekino, T. Tsuneda and K. Hirao, J. Chem. Phys. Ill (2005) 234111. 41. T. Yanai, D.P. Tew and N.C. Handy, Chem. Phys. Lett. 393 (2004) 51.

30

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42. E.I. Blount in Solid State Physics, edited by F. Seitz and D. Tumbull (Academic, New York, 1962), Vol. 13, p. 305. 43. D. Jacquemin, J-M. Andre, and B. Champagne, 118,3956 (2003) 44. See, for example, M. Torrent-Sucarrat, M. Sola, M. Duran, J.M. Luis and B.Kirtman, J. Chem. Phys. 120 (2004) 6346 and references cited therein. 45. M. Springborg and B. Kirtman, work in progress.

Multiscale Modelling of Polymer Properties M. Laso, E.A. Perpete (Editors) © 2006 Elsevier B.V. All rights reserved.

31

Chapter 2

Advanced Monte Carlo Methods for the atomistic simulation of polymers with a linear or a non-linear molecular architecture Nikos Ch. Karayiannis,^'^ Vlasis G. Mavrantzas^'' ^Department of Chemical Engineering, University ofPatras, Patras 26504, Greece ^Institute ofofCh Chemical Engineering and High Temperature Chemical Processes, Patras 26504, Greece

I. Introduction Molecular simulation methods constitute a reliable tool for the prediction of molecular structure and the understanding of the mechanisms and phenomena at the atomistic level that determine the macroscopic properties of polymers. Molecular dynamics (MD) and Monte Carlo (MC) are two such methods that can provide accurate predictions of the thermodynamic, mechanical, permeability, electrical, optical and other properties of polymeric materials. Monte Carlo, in particular, has developed to a powerful tool for simulating the properties of chain molecules, because of its capability to accelerate system equilibration through the implementation of cleverly-designed moves that do not require the system to follow the natural trajectory. This happens because MD simulations of long-chain molecules are plagued by the problem of long relaxation times, namely the fast increase in the longest relaxation time with increasing chain length [1]. Even for relatively-short chain length systems, the longest relaxation time can considerably exceed the time spans that can be simulated with the available computational resources. Despite its inability to effectively sample configurational space, MD remains the method of choice primarily because of its unique capability to provide realtime information about the system evolution and its dynamics [2,3,4], especially if multiple time steps and code parallelization are used to accelerate algorithm

32

N.Ch. Karayiannis and V.G. Mavrantzas

execution [5, 6]. But if one is interested in the static properties of the system, then MC offers an excellent alterative through the design of clever moves tailored to enhance the efficiency with which configuration space can be sampled. MC algorithms can also be combined with MD in the context of coarse-grained or hierarchical modeling methodologies by providing, for example, the initial configuration from which the MD simulation can be started. Particularly effective among the proposed MC algorithms are the most recently developed ones based on chain connectivity altering moves, which offer the opportunity of thoroughly equilibrating the polymeric system at all relevant length scales. It is the purpose of this Chapter to review the most important of these moves by presenting how they are implemented in simulations of polyethylene (PE) or polyemethylene-like systems, depending on the molecular architecture of the constituent chain molecules. Section II describes in brief the type of molecular models commonly adopted in these simulations. In Sections III and IV we describe how connectivity altering MC moves are implemented in simulations of a number of model PE systems of either a linear or a non-linear molecular architecture. Finally, Section V summarizes the main conclusions and discusses further applications of this new family of moves.

II. Molecular Model - Amorphous Cell The work presented in this Chapter refers to the application of the MC method to simulations of linear and non-linear PE systems based on a united atom model description. United-atom models for PE differ from explicit-atom ones in that the carbon atom and its adjacent hydrogens in a CH3, CH2 or CH unit are considered as a single, spherically symmetric site, with tremendous savings in terms of the total number of atoms present in the simulation box. While this seems to be an oversimplification of the atomistic description, a number of quite accurate united-atom force fields have been proposed over the years for linear and branched PE providing reliable estimates of their volumetric and conformational properties [7,8]. According to the most popular of them, bond lengths are kept constant at a value of / approximately equal to 1.54A while bond angles obey the Van der Ploeg and Berendsen bending harmonic potential [9]. As far as the torsional potential is concerned, dihedrals of the kind CHx-CH2-CH2-CHx (jc = 1, 2 or 3) are assumed to be governed by Toxvaerd's nine-term, sum-of-cosines formula [10], while dihedral angles around CH-CH2 bonds (branched systems) are sampled according to the potential function proposed by Nath and Khare [8]. Non-bonded intramolecular interactions between sites separated by more than three bonds and all intermolecular interactions are governed by a 12-6 Lennard-Jones potential (fi-j) of the form,

33

Advanced Monte Carlo methods /

I^L.j=4f

\12

^a^'

(1)

where r is the distance of the two interacting sites, and e and a the characteristic well depth and collision diameter, respectively, of the potential. Standard Lorentz-Berthelot combining rules are employed for the description of the nonbonded interactions between sites of different kind. More details about the parameters employed in a united-atom model for PE can be found in Chapter 4 of the present book. As far as the total number of interacting sites present in the simulation box is concerned, this ranges from 10^ to 2x10"^ for runs that are executed serially, with typical box dimensions on the order of 30 to 80A. For parallel runs (executed on a cluster of CPU's), these numbers can be considerably different (higher) by up to two orders of magnitude in some cases. Initial configurations are usually provided by the three-stage, constant-density Molecular Mechanics (MM) technique (the amorphous cell method) of Theodorou and Suter [11], subject to a set of appropriate (periodic) boundary conditions depending on the presence or not of interfaces. III. Monte Carlo Algorithms for Linear Polymers III.l. Simple Monte Carlo Moves Monte Carlo (MC), a stochastic method introduced in the early fifties by Metropolis et al. [12], samples the configurational space of a molecular system according to the probability density function of the statistical ensemble in which the simulation is realized. Sampling takes place through a succession of trial moves designed to generate a sequence of random states so that by the end of the simulation each state has occurred with the appropriate probability. An attempted move (i.e., a transition) from an old to a new state is accepted with a probability equal to the ratio of the occurrence probabilities of the new and old states, P^Jipld-^new), which is readily expressed as the Boltzmann factor of their total potential energy difference, AFtot, [12]:

^acc ( ^ ^ ^ " ^ ^ ^ ^ ) = ^ ^ ^

exp

,1

(2)

34

N.Ch. Karayiannis and V.G. Mavrantzas

where k^ is Boltzmann's constant. For macromolecular systems, elementary MC moves consist of displacements, rotations and librations of single atoms (or atomistic units), by taking into account the constraints imposed by the geometry and the details of the molecular architecture of the chain; then the value of AFjot is calculated to decide whether the move should be accepted or rejected according to the Metropolis criterion. According to the united-atom potential model for PE described in Section II, AFtot should be calculated as the sum of bending (AFbend), torsional (AFtor) and non-bonded (Lennard-Jones) (AFL.J) contributions: AF,, =P^,(new)-F,.(oW) = AF^<,+AF.„+AFL.j

(3)

By randomly choosing states from a given distribution, Metropolis' criterion allows sampling to be concentrated in the regions of space that make important contributions to the configurational integral. Consequently, MC is more efficient than conventional MD in simulating dense or complex systems, since energy barriers obstructing structural rearrangements are bypassed. Early MC simulations on macromolecular systems were based on a set of simple moves designed so as to mimic the slithering motion of a chain in a real fluid, such as (see Figure 1) "reptation", "end-mer rotation", and (c) "internal libration" (or flip). "Reptation" [13], for example, proceeds by moving the head of the chain to a new position, displacing all other ones along the chain, and leaving vacant the tail position. Such a move is supposed to mimic the reptation dynamics exhibited by long chains in entangled polymer melts according to the tube model of de Gennes [14] and Doi-Edwards [1].

35

Advanced Monte Carlo methods

(a)

(hi

Figure 1. Schematic representation of simple MC moves suitable for the simulation of PE chains in the united-atom description: (a) reptation, (b) end-mer rotation, and (c) internal libration (flip).

In the "internal libration" (or flip) move [15], an inner mer is picked at random and rotated by an angle whose value is chosen uniformly in an interval of approximately 20-30° around the axis formed by the two mers flanking it on either side; in a flip move, a total of four torsion and two bending angles are altered. MC moves involving the geometric reconstruction not of a single but of a group of atoms or mers either at the end (configurational bias, CB) [16-18]) or at the interior (concerted rotation, ConRot) [19,20]) of a randomly selected chain are also possible (see Figure 2). In CB, for example, an end segment of the chain (containing several atoms or mers) is removed and the missing part is re-constructed bond-by-bond and angle-by-angle in an energetically biased way to so as to avoid overlaps with the unmoved intra- and intermolecular neighboring sites. Among other applications, such a method is widely used in the context of the Gibbs ensemble MC method [21] for simulating the phase equilibria properties of chain molecules. ConRot, on the other hand, is initiated (see Figure 2) by driving atoms / andy (separated by four bonds along the chain) away from their initial positions through a small change in the torsion angles of the bonds to which they are attached. Then, trimer (4, z'b, Q formed by the

36

N.Ch. Karayiannis and V.G. Mavrantzas

atoms lying between / andy is excised from the system. The move is completed by connecting atoms / and 7 through a new trimer bridge (a bridge which has a new position and orientation). In a ConRot move, five bending and eight torsion angles are altered.

Figure 2. Schematic representation of: (a) the configurational-bias (CB), and (b) the concerted rotation (ConRot) MC moves as they are used in simulations of PE chains in the united-atom description.

The geometric problem addressed in a ConRot move (the first member of the family of moves known today as "variable connectivity" moves) for the trimer bridging construction is mathematically formulated as follows: "Given two dimers (pairs) in space, connect them with a trimer, such that the resulting heptamer has prescribed bond lengths and angles". Such a problem was originally solved by Pant and Theodorou [22], then by Mavrantzas et al. [23] through the psi-function formulation, and later by Wu and Deem [24] who casted it as a problem of solving a sixteenth degree polynomial analogous to an inverse kinematics problem involving serial chain manipulators in robotics [25]. The geometric solution requires the transformation from Cartesian to constraint variables (bond length and angles), thus it involves a number of Jacobians relating differential volume elements between the two coordinate systems whose determinants (for the forward and reverse problems, respectively) should be correctly accounted for in the Metropolis criterion [19,22,23].

Advanced Monte Carlo methods

37

MC algorithms based on combinations of the aforementioned moves have managed to successfully simulate PE meh systems characterized by small to moderate chain lengths. By trying to mimic the true dynamical aspects of the physical system under study (i.e. librations, rotations and reptations of simple atoms or entire segments), however, they fail when employed in simulations of polymer systems whose chain length exceeds roughly Cioo. A partial solution to this problem was given through the introduction of other localized moves, such as the various extensions and generalizations of the original CB [26-28] method, the internal conflgurational-bias (ICB) [29], the parallel rotation (ParRot) [30], and the wormhole (WH) [31] moves. The efficient equilibration of the dense phases of truly long (entangled) polymers continued to remain a challenge in the polymer simulation community. It was addressed only in the last decade with the introduction and successful implementation of the family of the so called chain connectivity altering or variable connectivity MC moves involving, in general, more than one chains. IIL2. Chain Connectivity Altering Monte Carlo Moves IIL2.1. End-Bridging Monte Carlo Variable connectivity moves, initially implemented for lattice simulations [32-34], proceed by appropriately excising material from two different chains and then rejoining them in a completely different way; this results in two new chains with totally different configurations than the original ones. End bridging (EB) was the first such move to be implemented by Pant and Theodorou [22] in simulations of atomistically detailed macromolecular systems in continuum space. Originally employed for the long-range equilibration of PE oligomers, EB can be coarsely described as the intermolecular analog of the ConRot move (see Fig. 2). Its schematic representation is given in Fig. 3. The move starts by randomly selecting endmer / of chain ich, which is allowed to "attack" internal mery of chain ycA by having trimer (j\,jh,jc) adjacent to it being cut off. The move is completed by connecting atoms / andy through a new trimer bridge (/V,7b'»7V) by solving the corresponding geometric problem numerically. In principle, for any given pair of "predator-prey" mers / and 7, there are two different ways with which EB can be executed depending on which of the two trimers adjacent toy is removed. As shown in Fig. 3, the move involves the transposition of three mers (atoms), a number which is comparable to the number of atoms moved in a CB or a ConRot move; the structural changes induced on the two chains participating in it are so drastic, however, that the resuhing configurations are totally different

38

N.Ch. Karayiannis and V.G. Mavrantzas

from the initial ones. As a side effect, the move effects changes also in the molecular length of the two chains: at the end of the move, the size ofich has increased while that of jch has decreased. Consequently, EB introduces polydispersity in the system, rendering it inapplicable for the simulation of strictly monodisperse systems. To control the molecular length distribution, the move is realized in a semigrand canonical statistical ensemble (Wat^ch^rfi*) where the following variables are kept constant: the total number of atoms («at) and chains (A/ch), the temperature (7), the pressure (P), and the spectrum of relative chemical potentials fi* of all chain species present in the system except two, which are taken as reference species [22]. Expressions for (i* that generate the most important chain length distributions have been derived by Pant and Theodorou [22] for bulk systems, and by Daoulas et al. [35] and Escobedo [36] for chains in systems exhibiting anisotropy and/or heterogeneity (such as at the interface with a substrate). (a) BEFORE EB

(b) AFTER EB

Figure 3. Schematic representation of the end-bridging (EB) move as it is employed in simulations of PE chain systems in the united-atom description. Left: initial configuration. The red arrow indicates that atom / of chain ich will attack atomy of chain yc//; then, trimer (j\,jb,jc) will be excised from the system. Right: chain configurations after the application of the move. Atoms / andy are connected through the new trimer bridge (J^JhJc)-

Advanced Monte Carlo methods

39

Mavrantzas et al. [23] used EB to successfully simulate a number of polydisperse PE melts characterized by a uniform distribution of chain lengths in the closed interval [A/av(l-A), iVav(l+A)], where A/av is the average chain length and A the half-width of the chain-length distribution function reduced by N^y, with A^av ranging from Cyg up to C500. In the limit of long chains, A is directly related to the polydispersity index / through 7=1+(A^ / 3). Algorithm implementation was facilitated by the maintenance and continuous update of a number of special lists, such as the lists of possible "predator-prey" pairs, the list of their allowed combinations, etc [23]. Given a chain end-mer, any internal atom belonging to a different chain is a potential candidate for EB as long as it fulfills two conditions: First, it is within the bridgeable distance from the corresponding end-mer; by definition, this distance is the all-trans distance (<^trans) Spanned by five successive atoms [(itrans = 4/cos(0niax/2)] where / is the carbon-carbon bond length, and ^max the maximum value in the distribution of the supplement of the CH2-CH2-CH;c (x = 2 or 3) bond angle. Second, the attempted EB move results in chain lengths whose values are within the lower and upper bounds of the distribution of chain lengths as defined by the prescribed polydispersity index and system average chain length. EB is initiated by randomly picking an end-mer /, one prey-mery out of its ^EB(0 potential candidates, and the direction (relative position of the excised trimer) of the bridging. The solution of the geometric problem provides a number (which is always even) of solutions for the trimer (/V,7V,7V)- In order to increase the acceptance rate of the move, the resulting configurations are subjected to a number of screening procedures to discard energetically unfavorable states. The first is a torsional energy screening: If the energy of the altered torsion angles is greater than their energy in the initial state by more than a prescribed value (AFtor), then the candidate solution is removed from the set. The second screening disregards configurations that lead to atom-atom overlaps (i.e., to pairs of non-bonded atoms whose distance is smaller than the value of a pre-assigned hard-sphere diameter,
40

N.Ch. Karayiannis and V.G. Mavrantzas

exp PseiecXold-^new) = -

Z

KA^^)\ kj

(4)

s(old-^new) S(OU /=1

exp -

The move is completed by considering also the reverse problem (new—^old). According to Fig. 3(b), this involves allowing the newly-formed end (J2) of jch to attack atomy by having its adjacent trimer {ja\jb\jc} being cut off, and by constructing a new trimer bridge {/^, y^, jc}. Clearly, the set of admissible solutions of the inverse problem should include the coordinates of atoms ja, jt and 7c- Microscopic reversibility requires subjecting all solutions of the reverse problem to the same screening procedures as was done for the solutions of the forward problem. The probability PseiectO^^^-^old) of selecting the initial state as the solution of the reverse problem is therefore defined as

exp

Ps.i.ct(^e^^old) = -

kJ

(5)

s{new-^old)

exp After the successful completion of the forward and reverse problems, the move is accepted or rejected with a probability according to the following modified Metropolis criterion V(new) W(new -> old)P^^^^ (new -^ old) J (new) exp P^^ipld -» new) = min1,

(6) W(old -> new) P^^i^(old -^ new) J(oW)exp Vjold)

KJ

where V(new) and V{old) are the sum of the torsional (Ftor) and Lennard-Jones (FL.J) energies of the final and initial configurations, respectively, J(riew) and J(old) the corresponding determinants of the geometric transformations involved in the move, and W(new^^old), and W{old-^new) the selection probabilities for the forward and reverse problems, respectively:

Advanced Monte Carlo methods

W(old^new)

=

^-

41

, W(new^old)

=-

^—r—

(7)

Simple scaling arguments put forward by Mavrantzas et al. [23], which were confirmed by subsequent atomistic EBMC simulations with a number of PE systems, showed that the performance of EB (as well as of other members of this class of chain connectivity altering moves) in equilibrating the long-range characteristics of the system should increase with increasing chain length, rendering the corresponding MC algorithm a unique tool for equilibrating longchain PE melts. In fact, the maximum relaxation (in CPU units) Tmax needed to equilibrate PE through an EBMC simulation scales with average chain length A^av as iVav"^ (which should be compared to the iVav^ "* scaling of the longest relaxation time in dynamic simulations of entangled systems). Thanks to this remarkable feature, Mavrantzas et al. [37, 38] were able to successfully sample thousands of fully relaxed equilibrium and/or oriented configurations of longchain PE systems of chain length up to C500. Uhlherr et al. [39] proposed a generalization of the existing EB algorithm, called "directed end-bridging" (DEB), where bridging constructions are not limited to just trimers but to sequences of mers of arbitrary length. This is accomplished by combining the original trimer bridging construction with an appropriate number of configurational bias (CB) regrowth steps. In the intramolecular analog of DEB, termed "directed internal bridging" (DIB), similar modifications are implemented in the original ConRot algorithm to increase its operating range. DEB and DIB moves were further integrated with domain decomposition techniques [40] to form a parallel simulation code that made possible the equilibration of a truly long Ceooo PE melt system containing more than 10^ interacting sites in the simulation box [41]. Except from PE, end-bridging and its variants have been successfully employed also in simulations of other chemically simple polymers, such as cis1,4 polyisoprene [42,43], cis-1,4 and 1,2 polybutadiene [44], and polypropylene [45]. The interested reader is referred to the recent review article by Theodorou [46] for a more detailed description of the simulated systems. 7/7.2.2. Double-Bridging Monte Carlo Despite its tremendous efficiency in equilibrating the long-range conformational features of long-chain, entangled polymer melts, EB suffers from some shortcomings associated with the fact that its operation relies on the presence of chain ends:

42

N.Ch. Karayiannis and V.G. Mavrantzas

(a) A finite (non-zero) degree of polydispersity is absolutely necessary for the move to function, rendering it inapplicable for the simulation of strictly monodisperse systems. (b) By construction, EB is highly dependent on the presence of chain ends. If topological constraints limit the relative population of free chain-ends (as, for example, in the case of terminally grafted macromolecules [47]), its efficiency drops. Even worse is the case where no free ends are found along the main chain backbone (due to its specific molecular architecture). Thus, EB cannot be used to simulate, for example, cyclic or H-shaped structures or infinitely-long polymers. (c) Its performance drops considerably as the stiffness of the chain increases or in the presence of chain orientation. Most of these disadvantages can be removed by introducing a new set of chain connectivity altering moves, the "double bridging" (DB) algorithms, which rely on the construction of two (and not one) trimer bridges among two pairs of properly selected atoms along the backbones of two different chains [48,49]. DB algorithms constitute practically a generalization of the EB move and circumvent most of its limitations, since they are not confined to chain ends. Fig. 4 presents a schematic representation of the double bridging (DB) move as it is employed in simulations of linear PE chains in the united atom description. Fig. 4(a) depicts the original configurations of the two chains ich and jch involved in the move. The move is initiated by letting internal mer / of ich "attack" (as indicated by the red arrow) internal mery of jch, followed by the removal of trimer (j\,jhjc) adjacent toy. In parallel, mer 72 ofjch "attacks" mer /2 of ich (as indicated by the cyan arrow) by having trimer (4, ih, Q connecting mers / and /2 being cut off. The move is completed by bridging the two temporarily formed pairs of ends (ij) and (J2, ii) with trimers (js!,jh\Jc) and (/V, ib\ h% respectively. As shown in Fig. 4, the resulting configurations of chains ich' and jck are strikingly different from the original ones.

Advanced Monte Carlo methods

43

(a) BEFORE DB

Figure 4. Schematic representation of the double bridging (DB) move as it is employed in simulations of linear PE chains in the united atom description. Left: local configuration of the two chains involved in the move prior to DB. The move is initiated through the attacks shown by the red and cyan arrows causing trimers (j\,jh,jc) and (/'a, k, Q to be excised from the system. Right: chain configurations after the DB move. Pairs (ij) and (/2, ii) are now bridged with trimers (/V, jhj'c) and (/a, /V, /c'), respectively.

Fig. 4 depicts just one of the possible chain combinations with which DB can be realized. In general, given two chains (ichjch) and a pair of bridgeable intermolecular neighbors (/, j \ there are four different ways with which the move can be realized depending on the relative directions of the excised trimers; these are shown in Fig. 5. For this specific example, one (out of the four combinations) preserves monodispersity, since the molecular lengths of the two chains participating in the move remain the same. By excluding the other three combinations, DB can therefore be used to simulate strictly monodisperse PE samples, thus overcoming one of the major shortcomings of the EB algorithm. In general, DB can be employed in atomistic simulations characterized by any distribution of chain lengths [48].

44

N.Ch. Karayiannis and V.G. Mavrantzas

Figure 5. Schematic representation of the possible configurations that can resuh from the four different bridging patterns or chain re-combinations through which a DB move can be realized (the example refers to the original configuration of PE chains discussed in Fig. 4(a)).

The methodology used to implement DB in simulations of PE systems, albeit more complex due to the double bridging construction, is very similar to that used to implement EB. At the beginning of a DBMC simulation, special lists are created storing all quartets of mers that can initiate a DB move obeying the following conditions: First, the distance between the mers of the chosen

Advanced Monte Carlo methods

45

pairs of mers ((/,y), (ji, ij)) should be smaller than the all-trans distance (Jtrans)Second, at least one out of the four possible DB combinations should lead to new chain configurations with molecular lengths that respect the prescribed distribution of chain lengths. In the case of a strictly monodisperse system, the latter criterion requires that for a given internal atom / in ich and a selected DB combination, there is only one candidate neighbor j in chain jch with which atom / can be bridged. Clearly, the restrictions imposed when monodisperse samples are simulated reduce significantly the set of quartets that can initiate DB. DB starts by randomly selecting an internal atom / of a chain which has at least one intermolecular neighbor j available for DB. From all available combinations, one is selected at random; simultaneously, atoms 7*2 and /2 of the secondary bridging are also identified. Appropriate triplets (/a, y'b, ^c) and (4, 4, /c) are deleted from the system, followed by the construction of the primary (/V, 7V9 yV) and the secondary (zV, ih, /V) trimer bridges. If no solution is found for any of the two geometric problems then the move is automatically rejected. Also, as in the EB move, torsional energy and hard-sphere overlap screenings can be applied to each bridge separately reducing the available solutions to a smaller set. A third screening disregards solutions based on hard-sphere overlaps among atoms of the two trimers with themselves. The two remaining individual groups are finally merged together to define distinct pairs of viable solutions. The low acceptance rate of DB necessitates selecting the attempted forward transition by weighting the available solutions with the Bohzmann factor of (not only the torsional energy as is done in the case of the EB move, but of) the sum of the torsional and L-J energies. As far as the reverse problem is concerned, this is initiated by letting atom 7*2 "attack" atomy by cutting off trimer {j^jhjc)\ simutaneously atom / "attacks" atom ii by removing trimer (/V, /V, ^V)- Finally, the attempted DB move is accepted or rejected with a probability according to a modified Metropolis criterion in the spirit of Eq. 6, where now J(new) and J{old) denote the products of the determinants of the two Jacobians of transformation for the two bridges, and W[old-^new)) and ^F(new—>old) the selection probabilities in the forward and reverse transitions, respectively; according to Eq. (7) above, the latter are inversely proportional to the number of DB candidate neighbors of atoms / and 7*2 in the old (A'DB(O) and new (A^DB(/2)) states, respectively [49]. By definition, EB [22,23] and DB [48,49] operate by rebridging pairs (EB) or quartets (DB) of mers belonging to different chains. Equally well, however, they can be designed to operate by rebridging mers belonging to the same chain (a chain "attacking" itself); the moves that result from such single or double self chain connectivity alterations are discussed in the next Section of this chapter.

46

N.Ch. Karayiannis and V.G. Mavrantzas

III.2.3. Intramolecular Chain Connectivity Altering Variants The first chain connectivity altering MC move of intramolecular origin was the "intramolecular double rebridging" (IDR). In IDR (see Fig. 6), four mers along the backbone of the same chain are reconnected through a double trimer bridging. The move is initiated by randomly selecting internal mer /, which "attacks" an intramolecular neighbor 7 lying within its all-trans distance for trimer bridging. To maintain chain connectivity, the primary bridging should be followed by a secondary bridging. In general, given two internal atoms along the chain backbone, IDR can be realized in two different ways (see patterns a and b in Fig. 6). The method used to implement IDR in a MC simulation as well as the formulation of its acceptance criterion are identical to the ones described in Sec. 111.2,2 for the implementation of DB. The second intramolecular chain-connectitivy altering move to be introduced in simulations of PE systems was the "intramolecular end bridging" (lEB); lEB constitutes the intramolecular version of EB. As shown in Fig. 7, lEB is initiated by randomly selecting end /, which "attacks" an internal mer7 of the same chain. Then a trimer next to merj is excised and a new trimer bridge is constructed to maintain chain connectivity. In general, in the IDR and lEB moves, the maximum number of the allowed bridging combinations is the one half of those allowed in the DB and EB moves, respectively; consequently, the different chain configurations that can result are also decreased by one half.

Advanced Monte Carlo methods

47

BEFORE IDR

AFTER IDR

a

Figure 6. Schematic representation of the intramolecular double rebridging (IDR) move as it is employed in the simulation of PE chains in the united atom description. Top: Initial configuration of the chain before the move is initiated (by letting mer / attack mery, as indicated by the yellow arrow). The two possible combinations {a, b) through which IDR can proceed are indicated by the arrows; the corresponding trimers to be excised are shown in red and green, respectively. Bottom: Trial configurations of the chain after attacks a and b have been completed. Also marked are mers /2 and72 connected through the secondary bridging.

48

N.Ch. Karayiannis and V,G. Mavrantzas

(a) BEFORE lEB

(b) AFTER ffiB

Figure 7. Schematic representation of the intramolecular end bridging (lEB) move, as it is employed in simulations of PE systems in the united atom description. Left: Chain configuration prior to lEB. The move is initiated with end-mer / attacking internal mery of the chain (as it is indicated by the green arrow). The trimer shown in green is to be excised from the system. Right: Trial configuration of the chain after lEB. Mers / andy are connected through the trimer bridge shown in green, rendering mery2 the new chain-end.

The intramolecular nature of the IDR and lEB algorithms renders them suitable for the atomistic simulation of macromolecular systems independently of their chain length distributions. Compared to IDR, lEB has the advantage of effecting changes also in the magnitude and orientation of the chain end-to-end vector; on the other hand, lEB cannot operate if no true chain ends are present in the system. IIL2.4. Applications of Chain Connectivity Altering Moves In the last few years, algorithms based on double bridging constructions have been employed quite extensively in simulations with a variety of model polymeric systems, either atomistic or coarser-grained ones [50,51]. Coarsegrained models, in particular, where the polymer is represented as a string of beads connected by springs governed by a finite extensible nonlinear elastic potential (FENE) [52] have been quite successful in simulations of polymer-

Advanced Monte Carlo methods

49

nanoparticle mixtures [53-55], and of end-tethered [56] or adsorbed [57] polymers. MC simulations built around the DB algorithm have also served as a means for providing equilibrated structures of bead-spring model systems for the subsequent topological analysis of entanglements [58-60] based on the concept of primitive paths [1]. DB and IDR MC simulations of atomistically detailed models, on the other hand, have also been carried out but so far they have been limited to chemically simple polymers, such as PE. In general, because of their complexity, chain connectivity altering MC schemes are difficult to implement for chemically more complex chain systems (bearing, for example, aromatic moieties or repeat units consisting of many, chemically different kinds of atoms or groups of atoms). MC simulations based on DB&IDR combinations have thus been restricted to simulations of entangled, linear monodisperse [48], poly disperse [49] and bidisperse [61] PE systems, ranging in molecular length from C24 up to Ciooo- Mention should also be made of the more recent works of Wick et al. [62, 63] who combined DB with EB to study the structural properties of pure poly(ethylene glycol) (PEG) and poly(ethylene oxide) dimethyl ether (PEODME) and to investigate the phase equilibria properties of homopolymers and block copolymers based on poly(ethylene oxide) (PEO) with CO2, and of Daoulas et al. [64] who employed DB in a MC simulation study of the structural and conformational properties of PE melts adsorbed on a graphite substrate, and compared against the predictions of a brute-force MD simulation of the same systems for the density and the loop, tail and train conformations of the corresponding adsorbed layer, as a function of chain length. In the remaining of this Chapter, we will focus on applications of chain connectivity altering MC moves to simulations of a number of polydisperse and strictly monodisperse, entangled PE melts at r = 450K and P = latm. Typical combinations of simple and more advanced moves utilized in these simulations are reported in Table I; also listed in Table I are the attempt probabilities for each move and their acceptance rates. In practice, around 45% of the attempted transitions encompasses trials with the more elementary moves (e.g., internal librations, end-mer rotations, reptations and ConRot's) including also a small percentage of volumefluctuations(vol. flue.) for sampling the density of the system, while the rest (55%) is invested to trials with the more complex (i.e., the chain connectivity altering) moves. The more local moves, which are accepted with a higher probability, are very useful in equilibrating mainly aspects of the local structure of the simulated system. It seems that they also act as "catalysts" for the good performance and operation of the more complex ones whose acceptance rate is typically one or two orders of magnitude lower. The more complex moves, however, by inducing structural changes on the level of the end-to-end vector of the simulated chains, are the only ones with which the

50

N.Ch. Karayiannis and V.G. Mavrantzas

long-range features of the simulated long-chain PE systems can be efficiently sampled within the limits of the available CPU time. As analyzed in Refs. 49 and 61, the performance of a chain connectivity altering move is governed by the following factors: a) its acceptance rate, b) the "shuttling effect" that annihilates successively accepted moves taking place between the same (initial and final) states in the system but in opposite directions, and c) the extent of the configurational changes effected by the move. By attempting to displace six atoms and modify twelve torsion angles within a single step, DB is in general less frequently accepted (by a factor of 50) than EB. Shuttling, on the other hand, can be reduced if the lists of the allowed chain connectivity altering combinations contain as many as possible different groups (pairs or quartets) of candidate predator-prey atoms. It is also highly desirable that these lists are frequently refreshed as the result of an accepted move. For example, in EB simulations of polydisperse PE systems, when reptation is intentionally omitted from the mix of moves, the performance of EB is reduced dramatically [23] not because its acceptance rate drops (in contrast, this remains constant irrespective of whether reptation is present or not) but due to the inefficient way with which the EB lists are refreshed. Such a synergy works also the other way around: if EB is intentionally left out from the cocktail of MC moves, then it is reptation whose performance decreases due to shuttling. As far as polydispersity effects are concerned, it should be mentioned that this has a tremendous effect on the performance of the two moves. In fact, as the molecular weight distribution gets more and more narrow, the number of possible combinations through which EB or DB can be completed is reduced; as a result, their performance deteriorates. For very low degrees of polydispersity (approaching unity), in particular, EB practically ceases to function, while DB continues to operate but less efficiently compared to polydisperses systems [49,61]. Table I. Typical mix of moves employed in the simulation of linear PE systems (either polydisperse or monodisperse). Also shown are their corresponding attempt probabilities and acceptance rates at T = 45 OK and P = 1 atm. Move Internal Libration End-mer Rotation Reptation ConRot EB lEB DB IDR Volume fluctuation

Monodisperse Attempt probability (%) 5 3 15 20 0 15 30 10 2

Polydisperse Attempt probability (%) 5 3 15 20 19 8 20 8 2

Acceptance rate (%) 78 20 9 10 0.1 0.1 0.002 0.002 10

Advanced Monte Carlo methods

51

Of course, in contrast to EB and DB moves, the performance of their intramolecular variants (lEB, and IDR) remains unaffected by the prescribed degree of system polydispersity, since it does not rely on the presence or not of intermolecular neighbors (nor on how often or efficiently are they updated). Equally important to the performance of the moves is the extent or the length-scale of the induced configurational change. In EB, the reconstructed trimer connects an internal atom with an end-mer belonging to a different chain (Fig. 3). Thus, for narrow MW distributions, where only small changes in the molecular lengths of the participating chains around the mean value are allowed, the "predator" end-mer is restricted to "attack" prey-atoms in other chains lying too close to their end points; thus, the length scale of the induced configurational change is significantly decreased. Such restrictions are absent in DB, in which the length scale of the attempted conformational transitions remains unaffected by the details of the MW distribution [61]. The superiority of the MC scheme based on chain connectivity ahering moves to conventional MD methods in equilibrating the long-length scale features of PE systems is shown in Fig. 8. The figure presents the decay of the orientational autocorrelation function of the chain end-to-end unit vector, , as a function of CPU time as obtained from atomistic simulations with a number of strictly monodisperse linear PE melts at T = 450K and P = latm. All reported CPU times correspond to single runs on Intel Xeon workstations at 2.8GHz. Also shown in the same figure are the corresponding curves obtained from: a) a MC simulation with a monodisperse C500 system in the absence of any chain connectivity altering moves (labeled as "no CCAM") and b) an NPT MD simulation with a monodisperse Ciooo sample, using the LAMMPS (Large-scale Atomic/Molecular Massively Parallel Simulator) software [65,66] executed in parallel on 16 CPUs using rRESPA (a multiple time step algorithm [67,68]), with the small and large time steps set equal to 1 and 5fs, respectively. The faster drops to zero the faster the system looses the memory of its initial configuration, i.e., the faster the orientational relaxation of the system. Fig. 8 clearly suggests that as the molecular length increases, the equilibration of PE's long-range features is accelerated [48,49]. As a result, MC can efficiently equilibrate also the longlength scale characteristics of a Ciooo PE melt, something which is impossible with MD [65,66].

52

N.Ch. Karayiannis and V.G. Mavrantzas

C

noCCAM

500

«. * •

C,,,,NPTMD 1000

CPU Time (10's) Figure 8. Decay of the orientational autocorrelation function of the end-to-end unit vector, , as a function of CPU time as obtained from MC simulations of monodisperse linear PE melt systems at r = 450K and P = latm. Also shown are the corresponding curves from a) MC simulation of the C500 system in the absence of chain connectivity altering moves (labeled as "no CCAM") and b) NPT MD simulation of the Ciooo system executed in parallel on 16 CPUs. All CPU times correspond to Intel Xeon workstations at 2.8GHz.

Fig. 9 shows MC simulation predictions for the dependence of density, p, on molecular length for a number of monodisperse linear PE samples at T = 450K and P = latm, and how they compare with the experimental data of Dee et al. [69]. The simulation predictions in Fig. 9 can be fitted quite accurately with a hyperbolic function of the form

p{N^J,P) =

(8)

where pa,(T, P) denotes the density value at infinite chain length and ao{T, P) is a dimensionless constant describing the rate with which p increases with A^^v, at the temperature and pressure conditions of the simulation. The best-fit values for the two parameters are: poo(450K, latm) = 0.775g/cm^ and ^o(450K, latm) ^ 3.20. The experimentally reported [48] density value for infinitely high PE at

53

Advanced Monte Carlo methods

r=450K is Pexp = 0.766g/cm^, i.e., in excellent agreement with the simulation findings. 0.78

1—•—r

i

• MC simulation O experimental data — hyperbolic fit (simulation)

0.67

I

0

I

I

I

I

I

I

I

I

I

I

I

I

I

I

I

I

I

I

I

100 200 300 400 500 600 700 800 900 1000 Molecular Length

Figure 9. MC predictions for the dependence of PE melt density on molecular length at T = 450K, P = latm. Also shown are the corresponding experimental data from Ref 69. Simulation findings have beenfittedwith the hyperbolic function of Eq. 8.

Fig. 10 presents additional results from the present MC simulations for the chain mean squared end-to-end distance, , and how it compares with the value corresponding to six times the chain mean squared radius-of-gyration, 6, as a function of PE chain length (at T = 450K and P = latm). In the long chain-length limit (e.g., for PE melts longer than about C142), it is found that, to an excellent approximation, I 6 [23]; this also agrees with Flory's random-coil hypothesis [70].

N.Ch. Karayiannis and V,G. Mavrantzas

54 1

1

10* "3

1

1

1 1 1 If

•



0

6 g

V

0

10'- 1

10^

1

1

1

1 1 1 11 1

1

D

j

0

J

o •

10'-

o •

1

•

1

1

1

1 1 1 1 1

1

10^ Molecular Length

1

1

1

1 1 1 1 1

1

10'

Figure 10. Mean squared chain end-to-end distance, , and mean squared chain radius-ofgyration multiplied by a factor of six, 6, in logarithmic coordinates as a function of chain molecular length. Predictions from the present MC simulations with a number of linear monodisperse PE melt systems at r = 450K and P = latm.

Based on the predicted chain length dependence of , one can extract also the chain length dependence of the chain characteristic ratio, CN.

Q = (N-l)l^

(9)

From the dependence of CM on the reciprocal number of C-C bonds along a linear PE chain, 1/(JV-1), one can further get an estimate of the characteristic ratio of a PE chain at infinite chain length, Coo. Based on the simulation results of Fig. 10, for PE at r = 450K: Coo=8.27 ±0.15 [49], in good agreement with the corresponding experimental value of 7.8 [71,72]. Information about the structural features of the generated PE structures at the atomic level can be provided by the (total) radial distribution function (RDF), g(r), whose Fourier transform determines the X-ray difft-action pattern of the system. Figure 11 presents the static structure factor, S(k% of a monodisperse

55

Advanced Monte Carlo methods

Ciooo PE system as obtained from the present MC simulations at r = 450K and P = latm [48] and as measured experimentally [73] through X-ray diffraction experiments at similar conditions {T = 430K and P = latm).

2.0

i

— simnlttioiitesnlts O X-tay difftaction data

-1.0

10

12

14

k(A"') Figure 11. Static structure factor, S{k), of linear monodisperse Ciooo PE melt system as obtained from MC simulations at T = 450K and P = latm. Also shown (open cycles) is the X-ray diffraction pattern of PE as calculated experimentally at r = 430K (after Ref 73).

The agreement between simulated [48] and experimentally measured [73] X-ray patterns is excellent over the entire range of wavevectors k accessed, demonstrating the capability of the proposed MC algorithm to efficiently equilibrate the short- and long-length scale structural and conformational features of linear PE. In Fig. 11, the first peak at i = 1.35A"* is attributed primarily to intermolecular correlations, while the peaks at wavevectors beyond 4A'^ reflect mainly contributions from intramolecular correlations.

56

N.Ch. Karayiannis and V.G. Mavrantzas

rv, Monte Carlo Algorithnis for Non-Linear Polymers Non-linear polymers bearing infrequently-spaced long chain branches along their backbone (long-chain branched polymers, LCB) find important technological and industrial applications due to their greater processability and improved rheological properties (such as strain hardening in extensional flows) over linear counterparts [74,75]. In the literature, the special rheology of these polymers [76-78] is explained on the basis of the tube or reptation model for entangled systems, such as the proposed pom-pom model for H-shaped polymers. An H-polymer consists of a single backbone ("crossbar") from the two ends of which a number q of branches emerge; H-structures constitute the simplest pom-pom model with q = 2. The detailed atomistic simulation of LCB polymers, on the other hand, has been restricted to rather short chain-length systems, such as the star, H-shaped and comb-shaped isomers of the linear Cioo PE system [79]. Extending these simulations to longer LCB systems can be tremendously facilitated by the design and implementation of the very efficient chain connectivity altering MC moves discussed above [80]. For the simplest model of a LCB structure the H-shaped PE system, such an approach is discussed in the remainder of this Chapter. In general, to thermally equilibrate LCB PE structures with a non-dynamic method, one should suitably adapt the DB and IDR moves discussed in Section III in order to account for the four arms that emerge from the ends of the main backbone. Figure 12 shows a schematic of such a generalization for the DB move operating on two H-shaped PE chains in the united-atom description. In fact, the move can be applied not only between the main backbones of the two chains but also between their dangling arms (branches). As far as how the double bridging is realized, the procedure followed is practically identical to that discussed before for linear PE molecules except from algorithmic (mainly bookkeeping) details associated with the presence of the two junction points.

Advanced Monte Carlo methods

57

Figure 12. Schematic representation of the double bridging (DB) move applied between two Hshaped macromolecules. Top: initial configurations before the move. Bottom: trial configurations after the application of DB on (a), (b) main backbones and (c) arms.

Figure 13 depicts how the IDR and lEB moves (the latter also termed "Hshaped branch rebridging", H-BR) [80, 81]) can be applied for H-polymers. Fig. 13(a), for example, shows how IDR can be designed in order to operate on the arms that emerge from the same junction point in the H-molecule. [Of course, IDR can also operate on a single arm or backbone, as discussed above for linear molecules]. H-BR is a specially designed move since it effects displacements of the branch point. This is achieved by excising one of the three trimers adjacent to it and reconnecting the branch point with another mer either deeper in the backbone (in which case the length of the backbone is reduced and that of the arm increases) or deeper in the arm (in which case, the length of the backbone increases and that of the arm decreases). Of course, such a move cannot be included in the simulation if the molecular lengths of the backbone and of the four individual arms should be kept monodisperse.

58

N.Ch. Karayiannis and V.G. Mavrantzas

Figure 13. Schematic representation of: (a) the intramolecular double rebringing (IDR) move when it is applied between adjacent arms and (b) the H-shaped branch rebridging (H-BR) move as it is applied on a branch point of an H-shaped molecule.

To render the MC algorithm for the simulation of H-polymers ergodic, additional moves need to be designed and implemented in order to enhance the efficiency with which polymer configurations in the neighborhood of the branch point are sampled. To this direction, besides H-BR, a new move has also been introduced: it is the so called "double concerted rotation" (d-ConRot) [80] move. As shown in Fig. 14, d-ConRot is triggered by the librations (flips) of one branch point (/) and of two neighboring mers {U, 7*4) by small changes in their torsion angles (usually chosen uniformly in the interval [-20°, +20°]). Then, trimers (/i, 12, h) and (/i, J2573) are excised, and the move is completed by connecting pairs (/, i^ and (ij^ through the trimer bridges (/V, /V» ^'3') and (/V, 72*5 j^X respectively. In the d-CONROT move, a total of nine atoms (centered around the branch point) are displaced, while fifteen bending angles and seventeen torsion angles are altered. Together with more conventional ones (see Figs. 1-2), the moves presented in Figs. 12-14 lead to the development of a robust MC algorithm for the rapid equilibration of LCB (H-shaped) polymers. Table II summarizes the attempt probabilities and acceptance rates of all these moves in a typical MC simulation run of an H-shaped PE polymer.

Advanced Monte Carlo methods

\

59

'#—.

Figure 14. Schematic representation of the double concerted rotation (d-ConRot) move. Top: initial configuration of the topology around the branch point. Middle: Branch point (/) and two randomly selected neighbors (/4,y4) lying four bonds apart from / are rotated around their initial positions; then, trimers (i\, /2, 13) and (JUJ29J3) are excised. Bottom: final configuration after dConRot. Pairs (/, /4) and (/,74) have been rebridged through the construction of trimers (/V, ii, /V) and {juJiJ^X respectively.

60

N.Ch. Karayiannis and V.G. Mavrantzas

Table II. Typical mix of moves used in the simulation of a polydisperse H-shaped PE system. Also shown are the corresponding attempt probability and acceptance rate for each move {T = 450K,P=latm). Move Internal Libration End-mer Rotation Intermolecular Reptation ConRot d-ConRot H-BR DB IDR Volume Fluctuation

Attempt probability (%) 5 3 12 16 4 6 33 20 2

Acceptance rate (%) 78 20 10 10 0.033 0.003 0.002 0.002 10

In the next paragraphs we will demonstrate the efficiency of the MC scheme, composed of the moves listed in Table II, in equilibrating a number of H-shaped PE melts in atomistic detail. In all simulations, a certain degree of polydispersity was allowed separately for the main backbone and arms, by assuming a uniform distribution of molecular lengths so that / = 1.053 for both the arms and the backbone, unless otherwise stated. The corresponding systems are denoted as H_x_j^ where x and y stand for the average number of carbon atoms per backbone and arm, respectively. Several polydisperse systems have been simulated, ranging from H_48_24 (MW = 2018g/mol) to H_300__50 (MW = 7002g/mol) and to H_400_70 (MW = 9522g/mol) [6,84]. Figure 15 presents the decay of the orientational autocorrelation function of the unit vector directed from one branch point to the other, /, as a function of CPU time for the H_128_24, H_128_48 and H_300_50 systems, as obtainedfromthe MC simulations at r = 450K and P = latm. Also shown in the same figure is the corresponding / curve for the H_300_50 system as obtained from a 4^s-long NPT MD simulation [6] executed in parallel [65,66] on 16 CPUs incorporating the rRESPA algorithm [67,68]. The rate with which the / curves of Fig. 15 drop to zero is a measure of how fast the main backbone of the H-molecule "forgets" its initial configuration (i.e., of how fast the backbone relaxes regarding its longrange orientational characteristics). The superiority of the MC algorithm based on the proposed chain connectivity ahering moves to the MD method in equilibrating the long-range characteristics of an H-shaped polymer is clear: MC is by at least two orders of magnitude faster than MD [6]. In addition, the rate of equilibration of the main backbone in the MC simulation is not affected by the value of arm molecular length (in contrast to the real dynamics [6] of

61

Advanced Monte Carlo methods

these systems or the corresponding analytical predictions of the pom-pom model [76]). li) r k

t N 1 <«k

1

0^

I

0.6

'

f

'

1

'

i

1

'

1

1

"•"--^

-]

1\ \ \''' ^''

MC,HJ28_24 — MC,H_128_48 — MC,H_300_50 - - MD, H_300_50

r1 \\\\\ \'^ • 1

1r1

n!o

1 ^

1 \^

02 1h

'

"-^^

1 S 0-4h V

'

\

J

^•

\

X

\

-j

^'*'

' *^ ' •

'*^'«'*^-.* **»••.•. N^ ^****
0.5

"1

LO

J

1.5

2.0

2.5

1.0

CPU Time (lO'^s) Figure 15. Decay of the orientational autocorrelation function of the unit vector directed from one branch point to the other, /, as a function of CPU time from MC simulations on H-shaped PE meh systems {T= 450K, P = latm). Also shown is the corresponding curve of the H_300_50 system as obtained from NPT MD simulation executed in parallel on 16 CPUs.

The high efficiency with which the proposed MC algorithm equilibrates Hshaped polymers allows one to obtain reliable predictions of their structural and volumetric properties [80]. For example, it is found that the radius of gyration of the H-polymer is smaller than for its linear counterpart characterized by the same total molecular weight. The individual average dimensions of the backbone (or arms), on the other hand, are equal to those of their linear analogues having the same length as the backbone (or arm) [80]. Also, for all studied molecular lengths, no differences were detected between the densities of an H-shaped polymer and of its linear counterpart of the same MW. How efficiently the new MC method equilibrates structural features in Hpolymers is further documented in Fig. 16 where the initial configuration of an H_400_70 PE melt is compared against that at the end stages of the simulation (i.e., after the MC has ran for about 10^ steps). The figure demonstrates the unique capability of the algorithm to relax the system from all unrealistically

62

N.Ch. Karayiannis and V.G. Mavrantzas

large clusters of free volume which would lead to an (unphysically) inhomogeneous distribution of its mass [80].

(a)

(b)

Figure 16. Atomistic snapshot of the H_400_70 PE meh system {T= 450K, P = latm), subject to periodic boundary conditions, (a) before and (b) after the application of 10^ steps with the proposed chain connectivity altering MC algorithm. Shown in blue and green are the atoms belonging to the backbone and arms, respectively, of a randomly selected H-shaped molecule.

V. Conclusions In summary, the introduction of the novel class of chain connectivity altering moves built around the double bridging (DB) algorithm [48,49] allows for the robust equilibration of long-chain, linear PE systems, independently of the form of their chain length distribution function. By further incorporating moves that can efficiently sample the configuration space around branch points, the new algorithms have opened up the way toward the simulation of branched polymer architectures, such as the H-shaped PE molecules. In general [82], MC algorithms based on chain connectivity altering moves are capable of inducing system equilibration at all length scales within modest computational time and resources, thus providing reliable estimates of its thermodynamic, structural and conformational properties, as a function of the chain average molecular length and the details of its molecular architecture. Of course, being a non-dynamic method, MC cannot provide any information about the system dynamics and its time-dependent properties. However, the new MC algorithms can be extremely useful even in dynamic simulations, since they

Advanced Monte Carlo methods

63

can provide the initial configurations from which the successive long MD simulations [6,82] can be started. A similar approach can be used to obtain estimates of the barrier properties (see, for example, Chapter 2 in Part 2 of the present book) of the polymer and of its topological properties (e.g., entanglements [83]). Currently, MC algorithms based on the proposed chain connectivity altering moves are limited to rather chemically simple macromolecules (such as, polyolefms, polydiens and polyethers). As mentioned above, however, they can be extremely useful also in simulations of more complex polymer compounds by equilibrating the polymethylene-like models onto which these complex polymers are mapped through coarse-graining. They can also be very useful in the atomistic simulation of other macromolecular systems (see, for example, Figure 17), such as: of dense brushes of terminally grafted macromolecules, of cyclic, star or short-chain branched polymers, of oriented or rigid systems, of dendritic architectures and their blends, and of the self-assembled monolayers that are formed when end-functionalized polymers are adsorbed on a solid substrate (e.g., of short alkanethiol molecules adsorbed on the Au( 1,1,1) surface, the so called R-SH/Au( 1,1,1) systems [84]).

64

N.Ch. Karayiannis and V.G. Mavrantzas

Figure 17. Potential applications of the proposed chain connectivity altering MC moves to atomistic simulations of various polymeric systems including, among others, brushes of terminally grafted (left top), cyclic (right top), and short-chain branched polymers (bottom).

Acknowledgments The authors are indebted to Prof. Doros Theodorou for introducing them to the filed of molecular simulations (in particular, to the design of MC moves for the simulation of chain molecules). Vlasis Mavrantzas is indebted to Prof. HansChristian Ottinger for his interest in the atomistic simulation of H-shaped polymers and the prediction of their rheological properties, which led first to the development of the double-bridging (DB) algorithm and then to the rest of the simulation work on non-linear polymer architectures. Stimulating discussions with Prof. Manuel Laso (University of Madrid), Dr. Martin Kroger (ETH,

Advanced Monte Carlo methods

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Zurich), Dr. Alfred Uhlherr (CSIRO, Melbourne), Drs. Joey Storer and Jaap den Doelder (Dow Benelux), Dr. Kostas Daoulas and Ageliki Giannousaki (University of Patras) are deeply appreciated. Financial support from Dow Benelux B. V. is greatly appreciated. The European Commission is thanked for valuable financial support through the GROWTH PMILS project. Finally, the home organizations (FORTH-ICE/HT and University of Patras) are thanked for the allocation of computational time.

REFERENCES [1] M. Doi and S. F. Edwards, The Theory of Polymer Dynamics, Clarendon Press, Oxford (1986). [2] K. Binder, Monte Carlo and Molecular Dynamics Simulations in Polymer Science, Oxford University Press, Oxford (1985). [3] M. P. Allen and D. J. Tildesley, Computer Simulation of Liquids, Clarendon Press, Oxford (1987). [4] M. Kotelyanski and D. N. Theodorou, Eds., Simulation Methods for Polymers, Marcel Dekker, New York (2004). [5] M. Tuckerman, B. J. Berne and G. J. Martyna, J. Chem. Phys. 97 (1992) 1990. [6] N. C. Karayiannis and V. G. Mavrantzas, Macromolecules 38 (2005) 8583. [7] M. G. Martin and J. I. Siepmann, J. Phys. Chem. B 102 (1998) 2569. [8] S. K. Nath and R. J. Khare, J. Chem. Phys. 115 (2001) 10837. [9] P. Van der Ploeg and H. J. C. Berendsen, J. Chem. Phys. 76 (1982) 3271. [10] S. Toxvaerd, J. Chem. Phys. 107 (1997) 5197. [11] D. N. Theodorou and U. W. Suter, Macromolecules 18 (1985) 1467. [12] N. Metropolis, A. W. Rosenbluth, M. N. Rosenbluth, A. H. Teller and E. Teller, J. Chem. Phys. 21 (1953) 1087. [13] M. Vacatello, G. Avitabile, P. Corradini and A. Tuzi, J. Chem. Phys. 73 (1980) 548. [14] P. G. deGennes, J. Chem. Phys. 55 (1971) 572. [15] V. G. Mavrantzas and D. N. Theodorou, Macromolecules 31 (1999) 6310. [16] J. I. Siepmann, Mol. Phys. 70 (1990) 1145. [17] J. I. Siepmann and D. Frenkel, Mol. Phys. 96 (1992) 2395. [18] J. J. de Pablo, M. Laso and U. W. Suter, J. Chem. Phys. 96 (1992) 2395. [19] L. R. Dodd, T. D. Boone and D. N. Theodorou, Mol. Phys. 78 (1993) 961. [20] E. Leontidis, J. J. de Pablo, M. Laso and U. W. Suter, Adv. Polym. Sci. 116 (1994) 283. [21] A. Z. Panagiotopoulos, in Supercritical Fluids - Fundamentals for Application, Eds. E. Kiran and J. M. H. Leveh Sengers, Kluwer Academic Publishers, Dordrecht (1994). [22] P. V. K. Pant and D. N. Theodorou, Macromolecules 28 (1995) 7224. [23] V. G. Mavrantzas, T. D. Boone, E. Zervopoulou and D. N. Theodorou, Macromolecules 32 (1999)5072. [24] M. G. Wu and M. W. Deem, J. Chem. Phys. 111 (1999) 6625. [25] D. N. Theodorou, Mol. Phys. 102 (2004) 147. [26] F. A. Escobedo and J. J. de Pablo, J. Chem. Phys. 102 (1995) 2636 [27] M. G. Martin and J. I. Siepmann, J. Phys. Chem. B 103 (1999) 4508. [28] C. D. Wick and J. I. Siepmann, Macromolecules 33 (2000) 7207. [29] A. Uhlherr, Macromolecules 33 (2000) 1351. [30] S. Santos, U. W. Suter, M. Muller and J. Nievergelt, J. Chem. Phys. 114 (2001) 9772.

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[64] K.Ch. Daoulas, V.A. Harmandaris, V.G. Mavrantzas, Macromolecules 38 (2005), 5780. [65] S. J. Plimpton, J. Comput. Phys. 117 (1995) 1. [66] Large-Scale Atomic/Molecular Massively Parallel Simulator (LAMMPS) software (version LAMMPS 2001 (Fortran 90)) distributed by Dr. S. Plimpton at Sandia National Laboratories, US. [67] M. Tuckerman, B. J. Berne, G. J. Martyna, J. Chem. Phys. 97 (1992) 1990. [68] G. J. Martyna, M. Tuckerman, D. J. Tobias, M. L. Klein, Mol. Phys. 87 (1996) 1117. [69] G. T. Dee, T. Ougizawa and D. J. Walsh, Polymer 33 (1992) 3462. [70] P. J. Flory, Statistical Mechanics of Chain Molecules, Wiley, New York (1969). [71] J. C. Horton, G. L. Squires, A. T. Boothroyd, L. J. Fetters, R. J. Rennie, C. J. Glinka and R. A. Robinson, Macromolecules 22 (1989) 681. [72] L. J. Fetters, W. W. Graessley, R. Krishnamoorti and D. J. Lohse, Macromolecules 30 (1997)4973. [73] K. J. Honnell, J. D. McCoy, J. G. Curro, K. S. Schweizer, A. H. Narten and A. Habenschuss, J. Chem. Phys. 94 (1991) 4659. [74] G. Bishko, T. C. B. McLeish, O. G. Harlen and R. G. Larson, Phys. Rev. Lett. 79 (1997) 2352. [75] D. J. Lohse, S. T. Milner, L. J. Fetters, M. Xenidou, N. Hadjichristidis, R. A. Mendelson, C. A. Garcia-Franco and M. K. Lyon, Macromolecules 35 (2002) 3066. [76] T. C. B. McLeish and R. G. Larson, J. Rheol. 42 (1998) 81. [77] T. C. B. McLeish, J. Allgaier, D. K. Bick, G. Bishko, P. Biswas, R. Blackwell, B. Blottiere, N. Clarke, B. Gibbs, D. J. Groves, A. Hakiki, R. K. Heenan, J. M. Johnson, R. Kant, D. J. Read and R. N. Young, Macromolecules 32 (1999) 6734. [78] H. C. Ottmger, Rheol. Acta 40 (2001) 317. [79] A. Jabbarzadeh, J.D. Atkinson and R.I. Tanner, Macromolecules 36 (2003), 5020. [80] N. C. Karayiannis, A. E. Giannousaki and V. G. Mavrantzas, J. Chem. Phys. 118 (2003) 2451. [81] L. D. Peristeras, I. G. Economou and D. N. Theodorou, Macromolecules 38 (2005) 386. [82] V. G. Mavrantzas, in Handbook of Materials Modeling. Volume 1: Methods and Models, Ed. Yip, Springer, The Netherlands (2005). [83] M. Kroger, Comput. Phys. Commun. 168 (2005) 209. [84] O. Alexiadis, K.Ch. Daoulas, and V.G. Mavrantzas, J. Am. Chem. Soc. (2005), submitted.

Multiscale Modelling of Polymer Properties M. Laso, E.A. Perpete (Editors) © 2006 Elsevier B.V. All rights reserved.

69

Chapter 3

A method for the systematic estimation of parameters for a stochastic reptation model Bernardino Pereira Lo,^ Andrew J. Haslam/ Claire S. Adjiman,^* Manuel Laso^ ^ Centre for Process Systems Engineering, Department of Chemical Engineering, South Kensington Campus, Imperial College London, London SW7 2AZ, UK * Laboratory ofNon-Metallic Materials, Department of Chemical Engineering, ETSII/UPM, Jose Gutierrez Abascal, 2, 28006 Madrid, Spain

Abstract A parameter estimation algorithm for the thermodynamically consistent reptation model (Ottinger, 1999; Fang et al, 2000), which is based on stochastic differential equations, is proposed. The problem is formulated using the maximum likelihood (MLE) objective function, and a modified LevenbergMarquardt algorithm is developed for its solution. Stochastic sensitivity equations are derived and used in order to obtain reliable parameter estimates. The issue of computational efficiency is addressed by varying the number of ensembles used in the integration of model based on the proximity of the current iterate to the optimal solution, as quantified by the magnitude of the trust region radius. The algorithm is applied to data for a sample of LDPE, to estimate mesoscale polymer properties from measurements of shear viscosity. 1. Introduction The understanding and the prediction of the viscoelastic properties of polymers is of immense practical importance in connection with the flow simulation of various industrial and natural processes such as plastic manufacturing, processing of foods, movement of biological fluids, injection moulding and

* Author to whom correspondence should be addressed; email: [email protected].

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fibre spinning. As a result, the development of molecular models describing polymer dynamics has been a great challenge for many years. One of the first molecular models was elaborated by Rouse (1953) to describe the dynamics of polymers in dilute solutions. However, in the case of polymer melts for which the interaction of entangled polymer chains is important, significant progress was made only when Doi and Edwards (1978a, 1978b, 1978c, 1979) postulated that each polymer chain is essentially confined to a tube imposed by other polymer chains. This was subsequently developed into the classical reptation model in which the path of motion of each polymer chain is defined by its primitive path and random orientation at chain ends. The reptation model has been the basis for many later sophisticated models of polymer rheology. A significant improvement was made when Ottinger proposed a "general equation for the nonequilibrium reversibleirreversible coupling" (GENERIC) formalism for nonequilibrium thermodynamics (Ottinger, 1999). By using this formalism and the theories of stochastic differential equations (SDEs), a thermodynamically consistent reptation (TCR) model for polymer melts was developed and implemented (Fang et ai, 2000). The model incorporates the essence of double reptation (des Cloizeaux, 1988), convective constraint release (Marrucci, 1996; lanniruberto and Marrucci 1996), chain stretching (Marrucci and Grizzuti, 1988), and it removes the assumption of independent alignment (Doi and Edwards, 1986). From mesoscale parameters which are related to both the dynamic properties and the architecture of polymer chains, the model predicts transient viscometric functions (viscosity and normal stress differences) for different shear and extensional flow conditions. The model has been shown to capture qualitatively and, in most cases, quantitatively the experimentally observed rheological behaviour in a large number of flow situations (Fang et ai, 2000). Examples include the transient overshoots of shear stress in the startup of steady shear flow at low shear rates, and the transient overshoots of the first normal stress difference at moderate shear rates. As mentioned previously, the TCR model is a model based on SDEs. The general form of an SDE is: dX,=M(t.X,;0)dt-^C7(t,X,;e)dW,

(1)

where t is time, Xt is the state variable of interest, // and cr are the drift and diffusion term respectively, 0 is a vector of model parameters and Wt is a Gaussian N(0,At^^^) noise term (a Wiener process). A key issue in using the stochastic TCR model for practical applications is the estimation of model parameters. Parameter estimation for SDE systems is a non-trivial task. Firstly, because of the existence of the noise term, it is difficult to obtain closed-form solutions for most SDEs of practical importance. Numerical methods such as

A method for the systematic estimation of parameters for a stochastic reptation model

71

the Euler and Taylor schemes are required to integrate SDEs (Kloeden and Platen, 1992) and, because of the diffusion term, a large distribution of trajectories of the state variables must be simulated to obtain reliable results, leading to high computational costs. The stochastic nature of the model also makes it difficult to obtain reliable gradients for the identification of the optimal parameters. The objective of this work is to develop a gradient-based parameter estimation algorithm for the TCR model, to obtain reliable parameters at reasonable computational cost. Once such parameters are available, it becomes possible to • understand the properties of a given polymer, • predict the viscoelastic properties of the polymer at different deformation rates: from a set of parameters predicted by the algorithm for a polymer at a specific deformation type and rate, the rheological behaviour of the polymer under different deformation conditions can be estimated. Furthermore, a parameter estimation algorithm opens the way for the design of polymers with specific rheological behaviour. By inputting the desired rheological behaviour, instead of experimental data, the algorithm can predict the mesoscale properties of the polymer required to achieve this behaviour. This chapter is organised as follows. In Section 2, the equations of the stochastic TCR model are outlined. In Section 3, the design of the parameter estimation algorithm is described, and in Section 4, the application of the algorithm to a sample of low density polyethylene (LDPE) is demonstrated. 2. TCR stochastic reptation model The formulation and implementation of the stochastic TCR model of polymer rheology are described in Fang et aL (2000). The parameters of the model are related to the dynamic properties as well as the architecture of polymer chains. They are: • the plateau modulus, G V this is the plateau value of the shear-relaxation modulus, and it characterises the transition of the dynamics of polymer chain motion from Rouse vibration at short time scales to reptation at long time scales. • the reptation time, r^: this is a characteristic relaxation time for polymer chains to diffuse away from an imaginary confining tube imposed by surrounding polymer chains. • the number of entanglement segments per polymer chain, Z: this is defined by Z = M/M'e, where M is the molecular weight of the chain and M'e is the average molecular weight between entanglement points along one chain. • the maximum stretching ratio, Amax'" this is the ratio of the contour length of a fully stretched polymer chain, Lmax, to the length when it is in its equilibrium state, LQ,

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The structural state variables in the model are the stretching ratio of the chain contour length >l(r) = L/LQ, and the configurational distribution function fiu,s,r), L is the polymer contour length; u is the unit vector describing the random orientation of a polymer chain; s is the position label of a tube segment along the chain where s = 0 and s = 1 correspond to the chain ends; r is the space position vector. The first part of the model involves the computation of the chain stretching ratio A. The time-evolution equation for A is DA Dt

(1)

— A convect i >'t dissip

where D/Dt is the material time derivative, and the total stretching rate X is split into convective and dissipative contributions. The convective contribution is defined as = XK\T

(2)

where K is the transpose of the velocity gradient tensor; T is the symmetric second-moment orientation tensor defined by

r= \ ds\d^uf{u,s,r)uu .

(3)

The dissipative contribution is defined as Adissip =

T^(^

(4)

~ 1)

r. 3Z where TS is the Rouse time defined as r^ = r^/3Z. c{A) is the effective spring coefficient

„n>)_3ZAL(^-H)

(5)

The second part of the model involves the computation of the stress tensor by updating the stochastic processes of ii and s, which are defined as:

y

f Ku,-2Du,

dUf =

\u.'I

J

:dt + 42D1 -

dW,

(6)

A method for the systematic estimation ofparameters for a stochastic reptation model

dSj

=Stotdt-{-—J—c

73

(7)

where Stot ——r

1

S

(8)

A/dissip 9

-y

W, and the three components of W, are independent Wiener processes. D is the orientation diffusion coefficient, \

(

D=

o

1

O,

o ^dissip

Yj\

Idissip

0-,—:—rl\

(9)

where H(x) is the Heaviside step function. The S[ term represents double reptation and the ^ term represents the convective constraint release mechanism. In this work Si = S^ = l/A. Although the above equations are independent stochastic processes, the boundary conditions on s result in a coupling effect between these two processes. When St reaches the boundaries 0 or 1, M, is replaced by a random unit vector. Suitable correction dynamics for unobserved absorptions at the boundary were also included. The numerical scheme used to solve the stochastic processes is described in Fang et al (2000). The transient viscometric functions (viscosity and normal stress differences) under shear and extensional flow conditions are then obtained by computing the stress tensor. The stress tensor consists of two contributions, r = T\ + ^. T\ is the original Doi-Edwards contribution T,(r) = ^Zripk^T \

\uuf{u,s,r)d^uds

(10)

and ^2 is a contribution associated with chain stretching T2 (r) = c{X)X{)i - l)n^k^T j ^

juuf(u,s,r)d^uds

(11)

where rip is the number density of polymers, ks is the Boltzmann constant, and T is the absolute temperature. Using the Doi-Edwards result for the plateau modulus

Gl=-ZnkJ, the stress tensor is simplified to

(12)

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rir) = 5G:[I + ^ ^ ^ > ^ j i; \uufiu,s,r)d'uds.

(13)

The above integrals involving the configuration distribution function / are evaluated as an ensemble average of the dyadic product uu at time t: £ ^uuf{u,s,r)d^uds = {u^u).

(14)

The shear viscosity 7] is then given by 7]^T,Jy

(15)

where y is the shear rate. The calculated expectation of the viscometric functions varies with the size of the simulation. In the TCR model implementation used in this work, the size of simulation is defined by: • Nensem, the ensemble size defined as the number of stochastic processes Ut and St being generated for the computation of the expectation of these processes. • Nblock, the number of sub-simulation blocks. In each sub-simulation, Nensem stochastic processes w, and St are first generated. The expectation of the viscometric functions are then calculated for each sub-simulation and, when all the Nblock sub-simulations are completed, the expectations of the viscometric functions are obtained as averages of the expected values over the set of sub-simulations. In this work, the number of blocks is set to 10. Extensive computational trials for different values of Nensem have been carried out. The predicted viscosities for different ensemble sizes have been compared to those at a reference ensemble size {Nensem = 1,000,000), and the accuracy for a given ensemble size, dNemsem has been defined as ^ X. {Nensem) - X. {Nensem = 7, OOP, OOP) ^' ^ ' step

XXNensem = l,PPP,PPP)

(16)

where Xj is the predicted viscosity (shear or extensional) at time-step /, and Nstep is the number of time-steps. The accuracy of the predicted viscosities also depends on the deformation rate, as viscosities are obtained by dividing the appropriate components of the stress tensor (computed from the stochastic processes) by the deformation rate, and the deformation rate affects the noisiness of the model predictions. The accuracy ^at shear rate of 1.0 s"^ and ensemble size of 100,000 is approximately 0.5%. We have found that the rate of convergence of the predicted viscosities to the values at Nemsen = 10^ is of the order 0{Nensem^'\

A method for the systematic estimation of parameters for a stochastic reptation model

75

3. Methodology and algorithm 3.1. Problem formulation The objective of the parameter estimation problem is to obtain reHable parameter values which minimise the difference between the model predictions and the experimental data. In this work, the maximum-likelihood (MLE) objective function is used:

r NE

NMi

/=1

j=\

0 = — ln(2;r)+—mm 2 2 i

^^^^,^^{x:;'{t)^x;^'\t^e)f \\

(17)

^ij

where NE is the number of experiments performed and NMi is the number of measurements in the i^ experiment. / is time and 6 is the vector of model parameters. Xy denotes the values of state variables of interest (viscosity or normal stress difference) for the/^ measurement in the f experiment, where the superscript "exp" denotes experimental data and "model" denotes model prediction. Gy is the standard deviation of the/^ experimental data point in the f^ experiment. Provided that a sufficiently large quantity of experimental data is available, the MLE estimation method possesses good statistical qualities such as consistency (as the sample size gets larger, the estimates converge to the true values of the parameters), efficiency (for large samples, it produces the most precise estimates when compared to other methods) and lack of bias (for large samples, one expects to get the true value of the parameters on average). However, the stochastic nature of model used poses some challenges in terms of the efficiency and reliability of the solution algorithm. Although derivative-free methods such as the Nelder-Mead simplex method (Nelder and Mead, 1965) and genetic algorithms such as the one proposed by Alcock and Burrage (2004) are quite popular for estimating the parameters of SDEs, a gradient-based method may lead to a more efficient search for reliable model parameters. 3.2. Proposed optimisation method In this work, the solution is obtained using a modified Levenberg-Marquardt algorithm which is specifically designed to solve least-squares problems, while accounting for the stochastic nature the model. A detailed description of the Levenberg-Marquardt algorithm is given by More (1977). The Levenberg-Marquardt algorithm identifies optimal

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parameters by a trust region method. At the current iterate, a trust region radius, A, is computed to search for the next iterate. The size of A determines the size of the step to the next iterate. When the iterate resuhs in a significant reduction of the magnitude of the MLE objective function, the size of A is increased. When there is only a small or no reduction of the value of the objective function, the size of A is decreased. Reliable gradients are required for the successful identification of optimal parameters using such an approach, and the issue of gradient calculation is addressed by deriving the stochastic sensitivity equations for the TCR model. The first step is to obtain the time derivatives of the sensitivities of the chain stretching ratio A with respect to the parameters 6 by differentiating Equation (1): D

dX\

dX convect

O Adissip

/ i o\

The sensitivities of the convective and dissipative contributions on the righthand side of Equation (18) are derived in turn by differentiating Equations (2) to (5) using the chain rule. An augmented system of stochastic sensitivity equations is then derived for the computation of the sensitivities of the random orientation unit vector u. By integrating the augmented SDE system and computing the sensitivities of the dyadic products MM, the sensitivities of the viscometric functions with respect to the model parameters are obtained. Note that the sensitivities of the viscometric functions with respect to the plateau modulus Gf^N can be obtained simply by dividing the viscometric functions by the plateau modulus. The coupling effect arising from Equations (6) and (7) results in a discontinuity in the integration of the analytical gradients. Whenever a random unit vector Ut is generated as a resuh of St leaving the interval [0,1] during the course of a simulation, the gradients are not calculated. The expectation is then calculated by averaging over Nensem - Noobs, where Noobs is the number of times St leaves the interval [0,1]. This is possible because of the relative rarity of 5, leaving [0,1]. Figure 1 shows a comparison of analytical gradients derived using the sensitivity equations and numerical gradients derived using the finite-difference method, for different step sizes h. The gradients shown are the sensitivity of transient extensional viscosity //" with respect to reptation time z;/. The parameters used are G^N = 20,000 Pa, r^ = 320 5, Z = 20 and A^ax = 18, with an ensemble size of 100,000. The numerical gradients are obtained using the central-difference formula which has a truncation error of order h^:

A method for the systematic estimation of parameters for a stochastic reptation model

2«

11

(19)

1.00E+06 1

5.00E+05 H

-5.00E+05

-1.00E+06

-1.50E+06

Figure 1: Numerical gradients of transient extensional viscosity with respect to reptation time for step sizes h = OA and 0.5, compared with analytical gradients from stochastic sensitivity equations

The numerical gradients are noisier than the analytical gradients. The numerical gradients with A = 0.1 appear unstable as the small step size amplifies the noise of the model predictions, and results in a gradient with the wrong sign for part of the trajectory. For the larger step size h = 0.5, even though the trajectory follows the same shape as the analytical gradient, the same accuracy cannot be achieved. Moreover, the analytical gradient calculations, which involve one integration of the augmented SDE, require about 40% less computational time than the numerical gradient calculations which involve two integrations of the SDE model. The analytical gradients appear more reliable, less noisy and faster to compute. The issue of computational efficiency is addressed by varying the number of ensembles used in the integration of the model from iteration to iteration. The key idea is that, when the parameter values are some distance away from the optimal solution, the search direction can be obtained reliably from relatively small and noisy simulations. As the algorithm approaches the optimal solution, the number of ensembles used in the integration of the model is increased to improve the accuracy of the predictions of the state variables and

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the reliability of the gradients. As a result the ensemble size at each iterate is calculated by an inverse function of J. In this work the function used is Mensem = 5000||V/^^|| A'^

(20)

where V / ^ is the transpose of the gradient of the objective function and ||-|| denotes the Euclidean norm. Convergence is achieved when

j||V/^4'<£

(21)

where ^is the optimality tolerance. In this work, ^is set at 0.005. The ability of the algorithm to identify true parameter values has been demonstrated in Pereira Lo et aL (2006), where pseudo-experimental data were generated by the TCR model with known parameter values at different extensional rates, and the algorithm was then tested for its ability to identify the true parameter values; results showed that the algorithm is able to estimate parameters that are close to their true values, and that the quality of fit is very high. 4. Application to LDPE In this section, the application of the algorithm to a sample of low density polyethylene (LDPE) is demonstrated (Kraft, 1996). The simulations were performed on a hyperthreaded Pentium 4 3.4GHz computer, with the RHEL 3 operating system. Because in the TCR model the reptation time and the number of entanglement segments are related by ts = ^^/3Z, it is not possible to estimate the two parameters together. As a result, Z is fixed at 10 in this work. Two optimisations are performed. First, the parameters are identified based on a set of shear viscosity data at a fixed shear rate of 1.0 s'^ Second, parameters are identified using data sets at four different shear rates (0.1 s'\ 0.3 s'\ 1.0 s"\ and 10.0 s"^). In both cases, the optimal parameter estimates and the CPU time are reported, together with overlay plots. The results for the single rate case are shown in Table 1 and Figure 2. The shape of the model-predicted trajectory agrees qualitatively with the experimental trajectories. The narrow overshoot of the model prediction is consistent with the findings described in Fang et al. (2000). Shear viscosity is insensitive to Amax, as demonstrated in Figure 3 by the small magnitude of the gradient of the transient shear viscosity with respect to A^ax, compared to other gradients (e.g. see Figure 1). As a result, optimising the parameter Amax has little impact on the estimated values of the model parameters and on the quality of fit.

A method for the systematic estimation of parameters for a stochastic reptation model

79

The finding that shear viscosity is insensitive to Xmax is also confirmed by Schweizer et al (2004) and Fang et al (2004). Table!: Parameters estimates, value of objective function and CPU time for <1LDPE sample at a single shear rate of 1.0 s'^ G%(Pa) 29270

r^(s) 2.303

Obj. Func. 10.752

^Tnax

4.859

CPU time (s) 28587

1.25

5"H{}{}{j5jj

iiiiiiii^i^tfwttmriTi 0.75

0.25

10

•

experimental data

—

model prediction

15

20

25

time / s Figure 2: Overlay plot for a LDPE sample at shear rate = 1.0 s"^ using the parameters reported in Table 1.

The results using data from 4 different shear rates are shown in Table 2 and Figure 4. The parameters estimated in this case are different from those when a single shear rate is used. Quantitatively, the fits are better at the higher shear rate of 10.0 s"* than at other shear rates. As expected, the discrepancy between the predictions and the experimental data is higher when the estimation is carried out for multiple shear rates than for a single shear rate. The quality of the predictions is consistent with findings for other polymers using the same model (e.g., Schweizer et al, 2004). The quality of the fits could be improved by using a multimodal model, in which the viscosity is obtained as the wei^ted average of viscosities predicted using different sets of parameters; each parameter set represents a different mode. This can be justified because real polymers are polydisperse, and it thus may not be possible to characterise the behaviour of a given polymer sample with a single mode. Verbeeten et al (2001) show that high quality fits to

80

B. Pereira Lo et al.

10

15

20

25

time / s Figure 3: Gradient of the transient shear viscosity with respect to the maximum stretching ratio Xmax^ computed using the parameters listed in Table 1 at a shear rate of 1.0 s'\

the same LDPE data as that used in this work can be obtained by using an extended Pom-Pom model in muhimodal form. From a statistical point of view, one should note that each additional mode introduces four new parameters to be manipulated: CfNy Td, Amax and a weighting factor. In practice this could be reduced to only three since we have shown that the shear viscosity is insensitive to Zmax, which can therefore be fixed rather than estimated. Increasing the number of modes in this way will clearly result in significant improvements in the prediction errors and overlay plots. However, it will also lead to a decrease in the confidence in the parameter estimates (i.e. an increase in the size of the 95% confidence intervals). Over-parameterisation of the model can make its use away from the experimental conditions unreliable. It is thus important to tradeoff the accuracy of the model against other statistical properties of the model. This issue is not considered here and only unimodal models are used: in a future publication, we will discuss the calculation of confidence intervals for the parameters, and their use in developing muhi-modal models. Improvements to the original TCR model can also resuh in higher quality fits. A modification of the TCR model has been proposed recently by incorporating convective conformation renewal due to flow-induced lengthening of tube segments (Fang et al, 2004). It has been shown to give a more correct representation of the behaviour of steady-state viscometric functions.

A method for the systematic estimation of parameters for a stochastic reptation model

81

Table 2: Parameters estimates, value of objective function and CPU time for a LDPE sample based on data at four different shear rates (0.1 s'^ 0.3 s ^ 1.0 s ^ and 10.0 s'*). G%(Pa) 61505

^(s) 0.733

^Tnax

5.627

Obj. Func. 266.97

CPU time (s) 70709

1.5 T

r= 1.0 s-''

A ——f

2

exper>rr»ntat c > ^ model pr^ction

4

20

experimental data model prediction

12

time / s

time / s

A —

18

24

time / s

30

10

20

30

40

50

time/s

Figure 4: Overlay plots for a LDPE sample at 4 different shear rates using parameters reported in Table 2.

5. Conclusians In this work, a parameter estimation algorithm for the stochastic TCR model has been developed and implemented. The biggest challenges associated with the design of such an algorithm are the compiitation of reliable gradients, and the high computational cost of integrating SDEs by numerical methods. The algorithm is based on a modified Levenberg-Marquardt algorithm, in which the number of ensembles used in the integration of the model is varied to improve computational performance. The gradients required for the successful identification of the parameters are derived from stochastic sensitivity equations. The application of the algorithm to estimate the plateau modulus, the

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B. Pereira Lo et ah

reptation time and the maximum stretching ratio for a sample of LDPE has been demonstrated. References Alcock J. and Burrage K., 2004, "A genetic estimation algorithm for parameters of stochastic ordinary differential equations", Computational Statistics & Data Analysis 47, 255 des Cloizeaux J., 1988, "Double reptation vs simple reptation in polymer melts", Europhys. Lett. 5,437 Doi M. and Edwards S. F., 1978a, "Dynamics of concentrated polymer systems. Part 1. Brownian motion in the equilibrium state", J. Chem. Soc, Faraday Trans 2 74,1789 Doi M. and Edwards S. F., 1978b, "Dynamics of concentrated polymer systems. Part 2. Molecular motion under flow", J. Chem. Soc, Faraday Trans 2 74,1802 Doi M. and Edwards S. F., 1978c, "Dynamics of concentrated polymer systems. Part 3. The constitutive equation", J. Chem. Soc, Faraday Trans 2 74,1818 Doi M. and Edwards S. F., 1979, "Dynamics of concentrated polymer systems. Part 1. Rheological properties", J. Chem. Soc, Faraday Trans 2 75,38 Doi M. and Edwards S. F., 1986, "The Theory of Polymer Dynamics", Clarendon, Oxford Fang J., Kroger M. and Ottinger H. C, 2000, "A thermodynamically admissible reptation model for fast flows of entangled polymers II: Model predictions for shear and extensional flows", J. Rheol. 44,1293 Fang J., Lozinski A. and Owens R. G., 2004, "Towards more realistic kinetic models for concentrated solutions and melts", J. Non-Newtonian Fluid Mech. 122, 79 lanniruberto G. and Marrucci G., 1996, "On compatibility of the Cox-Merz rule with the model of Doi and Edwards", J. Non-Newtonian Fluid Mech. 65, 241 Kraft M., 1996, "Untersuchungen zur scherinduzierten rheologischen Anisotropic von verschiedenen Polyethylen-Schmelzen", PhD thesis. Dissertation ETH Zurich Nr. 11417 Kloden P. E. and Platen E., 1992, "Numerical solution of stochastic differential equations". Springer, New York Marrucci G., 1996, "Dynamics of entanglements: A nonlinear model consistent with the Cox-Merz rule", J. Non-Newtonian Fluid Mech. 62,279 Marrucci G. and Grizzuti N., 1988, "Fast flow of concentrated polymers: Predictions of the tube model on chain stretching", Gazz. Chim. Ital. 118,179 More J. J., 1977, "The Levenberg-Marquardt algorithm: implementation and theory", in Watson G. A. (Ed.), Numerical Analysis, Lecture Notes in Mathematics 630, SpringVerlag, 105 Nelder J. A. and Mead R., 1965, "A simplex method for fiinction minimisation". The Computer Journal 7, 308 Ottinger H. C, 1989, "Computer simulation of reptation theories. I. Doi-Edwards and Curtiss-Bird models". J. Rheol. 43,1461 Ottinger H. C , 1999, "A thermodynamically admissible reptation model for fast flows of entangled polymer". J. Rheol. 43,1461

A method for the systematic estimation ofparameters for a stochastic reptation model Pereira Lo B., Haslam A. J. and Adjiman C. S., 2006, "Parameter estimation of stochastic differential equations: algorithm and application to polymer melt rheology", in Proceedings of the 16* European Symposium on Computer Aided Process Engineering, Garmisch-Partenkirchen, Germany, July 9-13,2006 Rouse P.E., 1953, "A theory of the linear viscoelastic properties of dilute solutions of coiling polymers", J. Chem. Phys. 21,1272 Schweizer T., van Meerveld J. and Ottinger H. C, 2004, "Nonlinear shear rheology of polystyrene melt with narrow molecular weight distribution - Experiment and theory". J. Rheol. 48, 1345 Verbeeten W. M. H., Peters G. W. M. and Baaijens F. P. T., 2001, "Differential constitutive equations for polymer melts: The extended Pom-Pom model". J. Rheol. 45, 823

83

Multiscale Modelling of Polymer Properties M. Laso, E.A. Perpete (Editors) © 2006 Elsevier B.V. All rights reserved.

85

Chapter 4

Implicit micro-macro methods in viscoelastic flow calculations for polymeric fluids M. Laso,^ J. Ramirez^ ^ Dept. of Chemical Engineering, ETSII, UPM, Jose Gutierrez Abascal, 2, E-28006 Madrid, Spain ^ Applied Mathematics, University of Leeds, Leeds LS2 9JT, UK

1. Introduction Since their introduction more than a decade ago, micro-macro methods (CONNFFESSIT [1], Brownian Fields [2], LPM [3]) have been appHed to a wide variety of viscoelastic flow calculation problems. Although not competitive against continuum mechanical approaches in terms of raw computational speed, they offer a useful alternative when complex polymer dynamics models have to be used for which no closed-form constitutive equation exist or when fluctuations are relevant. Micro-macro methods combine continuum-mechanical discretization techniques and stochastic models of polymer dynamics. This merger of scale-separated techniques has proved its usefulness in a wide variety of cases and its range of applicability is expanding continually. In addition, improvements over the original scheme are appearing with increasing frequency. New sophisticated techniques make it now possible to tackle a wide range of non-newtonian fluid mechanical problems including improved variance reduction [4] , Eulerian treatment of integral constitutive equations(CE's) [5] , free surfaces ([6], [7]), adaptive configurations fields [8]. As a consequence of this intense activity, there is a growing body of evidence supporting a general equivalence between micro-macro methods and continuum-mechanical methods, that can be loosely formulated as ''what can be done by continuum-mechanical methods should also be feasible in a micro-

86

M. Laso and J. Ramirez

macro approach'. Furthermore, since micro-macro methods bypass the limitations imposed on purely continuummechanical methods by the need of a closed, analytical CE's, they have as a side effect contributed to the recent increase in activity in the development of advanced dynamic and microstructural models and algorithms ([9], [10], [11], [12], [13], etc). There are however several aspects in which this correspondence has not been completely fulfilled yet. A salient one is the implicit formulation of time marching schemes. Except for the work of Somasi and Khomami [14], all previous time-dependent micro-macro calculations have been performed by means of explicit schemes: information flows in microlmacro direction via the stress tensor, which acts as right hand side "body force" in the momentum equation. Information flows in the macro-micro direction through the velocity field (and its gradient), in which the internal and external degrees of freedom of the molecular model evolve. This decoupled, unidirectional flow of information is characteristic of explicit methods and is the cause of numerical instability when the time step exceeds a threshold value. In some cases, the stabilityinduced time step size can be orders of magnitude smaller than required by the accuracy criterion. Whereas implicit time integration is routinely employed in continuum-mechanical approaches to viscoelastic flow cal culations, micro-macro methods have remained explicit except for the work of Khomami and co-workers [14], [15], who took an important step forward with their development of self-consistent semi-implicit algorithms for micro-macro simulations based on the proven concept of operator-splitting [15] . With their proposed algorithm full convergence within a time step is attained. Hence, their method has similar properties (i.e., temporal stability and accuracy) as fully implicit techniques. Unlike in continuum approaches, time-dependent micro-macro calculations have been performed up to now almost exclusively using a simple explicit time marching algorithm, the exception being the pioneering work by Somasi and Khomami (J. Non-Newt. Fluid Mech. 93, 339-362(2000)) whose iteration of a semi-implicit algorithm is equivalent to a fully implicit formulation. The use of explicit time integration puts explicit micro-macro methods at a disadvantage, since they lack the desirable stability of fully implicit methods. The limitation of using explicit time marching schemes often leads to reduced computational efficiency, since unnecessarily small time steps must be taken to ensure numerical stability.

Implicit micro-macro methods in viscoelastic flow calculations for polymeric fluids

87

The present contribution introduces a practical way to treat micro-macro time integration implicitly and thus extend the range of validity of the equivalence between micro-macro and continuum-mechanical methods. We show in a rather general way that micro-macro methods can indeed be treated in a fully implicit fashion, thus putting them on the same footing, as far as stability of time integration is concerned, as continuum approaches. 2. Implicit formulation of micro-macro methods. Governing equations The standard numerical solution of the equations describing the isothermal flow of an incompressible viscoleastic fluid [16] involves a discretization of the set of coupled partial differential equations expressing linear momentum and mass conservation equations: (1) V-5 = 0 where 7t = p 8 - h T

(2)

augmented by a differential or integral Constitutive Equation (CE) which expresses the macroscopic stress as a functional of the metric of the flow. In micro-macro methods, the CE is replaced by a pair of equations expressing i) the microscopic dynamics (in which the macroscopic field variables appear as given):

^iy^jj=^

(3)

and ii) the rule to obtain the macroscopic extra stress tensor "^ from given microscopic variables: -^=•(^..5;)

(5)

In the previous equations, u is the fluid velocity, n the total stress tensor and 7^, Y^ lists of macroscopic field variables (such as velocity, stress, pressure) and microscopic variables (such as orientation unit vector, dumbbell connector, etc. depending on the molecular model) respectively (the subscripts M and m refer to macroscopic and microscopic levels throughout). While Eq. (3) is

88

M. Laso and J. Ramirez

typically a stochastic differential equation, Eq. (4) involves an ensemble average (or in steady state calculations the upper convected derivative of such an average [17]). A discretization scheme of Eqs. (1), (3) and (4) leads to a set of coupled equations: first order initial value differential equations for macroscopic variables (w,p,etc.) and first order stochastic differential equations for the microscopic variables (Q,s, etc.). An explicit time-marching procedure treats this set of equations by splitting it into its macro- and microscopic subsets. The macroscopic subset is solved by a suitable standard method for given (i.e. right hand side) extra stress, while the microscopic subset is solved "molecule by molecule" for given velocity and velocity gradient fields. In CONNFFESSIT and LPM the macroscopic subset requires solving sparse systems of equations and the microscopic subset reduces to a series of straightforward evaluations of the discretized version of the polymer dynamics. The latter, although dominating in terms of computation due to the large number of molecules, is in general simple and linear in the number of molecules (stress calculators) since they are non-interacting, even for molecules residing in the same element. In the area of multi-particle, interacting dynamic models very little has been done . In most cases, interacting molecules are dealt with in a codeforming simulation box under periodic boundary conditions, which require the use of especial techniques [18] leading to full sub-blocks at the element level in the Jacobian. As a consequence, all the following material will deal with polymer dynamic models in which there is no explicit interaction among molecules, although dense, multi-chain molecular environments like in reptation and double reptation theories are of course included in a mean-field fashion. In the Brownian Configuration Fields (BCF) approach, each of the fields requires the solution of a PDE, again using sparse matrix techniques. Although time stepping a single configuration field is obviously more expensive than time stepping a"molecule", BCF's are greatly advantageous from the point of view of variance reduction, avoidance of tracking and spatial resolution. In typical calculations, several hundreds of fields in BCF vs. hundreds of thousands of molecules in CONNFFESSIT need to be considered. Note that while BCF nodal values for a given field are interrelated, different fields remain non-interacting.

Implicit micro-macro methods in viscoelastic flow calculations for polymeric fluids

In an implicit approach, time-stepping the discretized formulation from '

89

to

^ makes it necessary to solve a nonlinear set of equations involving the discretized representations of macroscopic fields (e.g. nodal values of velocity, pressure, etc.) and of individual trajectories (in configuration space) of the microscopic variables (e.g. components of connector or unit direction vectors for all molecules, etc.). The unknowns to be solved for are then:

This nonlinear system of equations can be solved by a number of methods. Newton iteration is among the most widely used on account of its rapid convergence and greater robustness with respect to other methods like Picard's. Newton's method requires, however, at every time step, repeated solution of a linear system of equations obtained from the Jacobian of the nonlinear equation set. The linearized system can be represented formally as:

where the right hand side b^"^ contains nodal values and microscopic degrees of freedom at the time level n. The sheer size of Eq. (5) for micro-macro methods makes a direct attack (even using sparse matrix techniques) impractical. An alternative is clearly needed. If we denote by 8^ the discretization of the flow domain Q, with discretization parameter h, the number of elements by NE = |^/,|, the number of macroscopic field variables by A^^, some numbered ordering of the set of elements by e,G €^J = l,..,NE,

the number of molecules in element /by A^C. and the

number of internal degrees of freedom of a molecule by N.^^^, the size of A is NE

of 0((Nj^ X NE H- N^j^f X ^ NC^ Y ) which, for an average-sized problem, can easily reach 0(10^x10'). The coefficient matrix can be divided naturally in four blocks:

90

M. Laso and J. Ramirez

A = where

Mm A

(6)

A.mm J the

upper-left

and

lower-right

blocks

are

square

of

size

NE

OdNj^xNEf)

and 0((N,^^fX^NCf)and

the remaining blocks are

rectangular of matching dimensions. The subindices of each block refer to their purely micro, purely macro or mixed character. The structure of A is obtained by straightforward replacement of stress values by the corresponding expression derived from the rule (4). The analogous Jacobian coefficient matrix A ^ for the purely macroscopic problem can be written as : X,. A ^ A . =

A, A

pu

pp

Au

Ap

Au

Ap

Ae

Ar

pe

pT

Ae Ae

(7)

^TC )

where rows are associated with shape functions and columns with degrees of freedom and subindices in submatrix names refer to velocity (w), pressure ( p ) , stress ( r ) and possibly auxiliary macroscopic variables coming from stabilizing or other numerical schemes (e). The Jacobian A in (6) is obtained from (7) by replacing •

•

•

in each of the submatrices of A^ AM in which T appears only once and as a first subindex, the derivatives of the microscopic dynamics (3) with respect to the variables indicated by the second subindex of the submatrix. in each of the submatrices of A^ in which r appears only once and as a second subindex, the derivatives of discretized conservation equations (in which the stress has been replaced by the rule (4)) with respect to stress in the submatrix A^ the derivatives of the microscopic dynamics (3) with respect to the internal degrees of freedom.

Implicit micro-macro methods in viscoelastic flow calculations for polymeric fluids

91

So that A has the structure: A

"^uY^

up pu

pp

pe

ep

A =

V

""pYm

eY„

^Y„,u

A„,

Y Y

•

•

•

(8)

-y

where, depending on the specific formulation, some of the blocks may be empty. The upper-left 3 x 3 block matrix in (8) is A^^the other diagonal square block is A^^and the remaining two rectangular blocks are readily identified with A^^andA^^. Thus, the block A^^^ expresses dependencies among macroscopic degrees of freedom (e.g. velocity, pressure, etc. nodal values) and the A^^ block expresses dependencies among microscopic degrees of freedom (e.g. individual components of dumbbell connectors). The off-diagonal block A^^ expresses the dependence of microscopic on macroscopic variables (e.g. terms containing Ar=(Vw)^ in the discretized stochastic differential equations). The off-diagonal block A^^ expresses the dependence of macroscopic on microscopic variables (e.g. terms containing polymer extra stress in the discretized conservation equations). 3. Reduction of system size Whereas the A^^ block has a general sparse structure and is of the usual size in macroscopic methods, the A^^ block is much larger, since there are many more microscopic than macroscopic degrees of freedom. Its structure is however simple. This simplicity can be exploited in the following way in order to reduce the size of the total micro-macro system back to that of a purely macroscopic approach. Schur's complement allows the linearized system AY^''^^^ = b^^Ho be replaced in a first stage by:

92

M. Laso and J, Ramirez

\^MM

+ A / A^^ ) Y^

—( A ^ ^

^Mm^mm^mM

)^m

"~

where b ^ and b^ are dense subvectors corresponding to the macroscopic and microscopic parts respectively of the RHS dense vector b and where the usual slash notation has been used for Schur's complement. This system is of the same size as the upper-left 3x3 block in (8). It involves the degrees of freedom for velocity, pressure and possibly stabilization auxiliary variables. Once the macroscopic degrees of freedom have been obtained, the remaining microscopic d.o.f. are obtained from:

Yr^ = A-l(bL"'-A„^rr^X

(9)

NE

which involves 0(N.^^y x ^ NC^) unknowns. For this reduction to be successful, i.e. for the computational work required to solve the modified system: (A

- A

A"^ A

W^"-'^^ = h^"> - A

A"^ b^''^

HO)

to be comparable to that in a purely macroscopic approach, some requirements have to be fulfilled: •

•

the calculation of the inverse of (or, equivalently, setting up and solving a system with coefficient matrix given by) A~J^^ and of A^^ A;|„ A^^ must be feasible with moderate effort. A ^ - A^^ A;;„ A ^ must remain sparse.

•

additional fill-in in A^^ - A^^ A^|„ A^^ should not be excessive.

•

Xf^ - ^Mm^mm^mM ^^^^ ^^ numerically well behaved.

Using the standard definition of the structure of a matrix: Srrwcr(M) = { ( / , y ) | M , ^ 0 } ,

Implicit micro-macro methods in viscoelastic flow calculations for polymeric fluids

93

the block A^_ has one of two alternative structures: mm

Struct{X^J

=\

Struct{X„®l,^)

(B)

^^^^

where I„ In is the identity nxn matrix, iV^is the number of nodal points at which stress is defined and Nj. is the number of fields in a Brownian Configuration Fields calculation. Alternative (A) corresponds to CONNFFESSIT and Lagrangian Particle Method, whereas (B) corresponds to Brownian Configuration Fields. Requirement (1) can be satisfied by choosing a proper ordering. In case (A), ordering the microscopic unknowns element-wise and then "molecule-wise" within an element (i.e. so that all molecules of a given element and all degrees of freedom of a given molecule are contiguous) produces an A^^ that is block diagonal, with as many blocks as there are molecules in the calculation and with each block of size N^^^^ x N^^^^ (e.g. 3 x 3 for a three-dimensional FENE dumbbell model). As a consequence of this very simple structure of A^^ it is possible in most instances to obtain the block diagonal A~|„ analytically. In case (B), if a field-wise ordering (i.e. all microscopic variables associated with a field are adjacent) is chosen, a block diagonal structure for A~|„ is also obtained. In this case, the blocks on the diagonal are sparse, the number of blocks being equal to the number of fields in the calculation. The structure of each block is inherited from the connectivity of the mesh where the BCF equations are solved, symbolically expressed in (1 IB) by Struct{A^). In this case, the calculation of A^^A~)„A^^must be done by sparse matrix techniques. In spite of the size of A^^, this calculation proceeds in essentially independent stages, i.e. there is never any need to attack a problem of size NE

0{{Nf^ X A^^ -h N^^^j- X ^ NC^ y ) directly. This is a natural consequence of the fact that, at every time step, Brownian Fields evolve independently of each

94

M. Laso and /. Ramirez

Other. As a consequence, the calculation of ^s4rn^~mm^mM Proceeds in a"fleldby-field" fashion reminiscent of the element-by-element approach [19]. In addition, both in (11 A) and (IIB), system size reduction is ideally suited to parallel hardware architecture [20]. Requirements (2) and (3) involve the structure of the matrix resulting from preand postmultiplication of the previous block diagonal A~)^ by rectangular blocks A^^ and A^^ which have structures: Struct{A^^)

=

Struct{augment{A^^,A^^)®l^^®l^^^^\

Struct{A^J

= Struct{augment{A^^,A^^)

® I^^ ® h,,^^)

The tall sparse submatrix A^^ contains vertical segments of nonzero entries. The segments of vertically consecutive nonzero entries correspond to all the molecules in a given element and which contain the same velocity degrees of freedom in their dynamic equation. The wide sparse submatrix A^^ contains horizontal segments of nonzero entries. The segments of horizontally consecutive filled entries correspond to all the molecules that contribute to the stress in the same element. Note that in general A^^ ^ A^^. Upper bounds for the amount of fill-in and for the maximum additional bandwidth generated by building Schur's complement can be given: Lemma: Aj^^A'^^A^^ is sparse, its bandwidth is at most the same as the bandwidth of

^^^ and the additionalfill-inis bounded by

\Struct{\^„A-:^A„^)\

where ^^^

<

NExAiGf

(12)

is the maximum vertex degree of the undirected graph

corresponding to the domain discretization ^. Proo/If the submatrix A^^ is ordered element-wise, A has the block arrowhead structure:

Implicit micro-macro methods in viscoelasticflowcalculations for polymeric fluids 95 J2

A"-

"^Km mm J2

A = A''

Ap 0

..

^r • .. 0

.

0 0

mm

where r = l,...NE, i.e. the superindex runs over the element label or BCF label. A block Schur's complement yields: NE

A ^MM

-A

A"^A

^Mm^mm^mM

=A ^MM

-YA^

_,

(A^

/^^Mm\^mm) k=\

r A^ ^mM

(13) Due to the structure of A^^ and A^^, each of the blocks in the previous sum can contain at most A(G)^ non-zero elements, i.e. A^^ A~|„ A^^ is sparse. Furthermore, the location of the non-zero elements in each term of the sum satisfies:

5/r«c?(AL(Al)"'At^)c{/|(AL) . ;tO,y/-}®{/|(At^)^^ ^0,y/-} i.e., the maximum bandwidth of the additional fill-in cannot exceed the bandwidth of A ^ , and each of the A^£ blocks A ^ ^ ( A ^ ^ )

A^^

can

contribute at most A(G)^to the additional fill-in. Furthermore, by way of Theorem 3.1 of [21] the graph G'of A ^ - A^^ A;)„ A^^ is also the graph of a well-shaped mesh, i.e. it is a bounded-degree subgraph of some overlap graph [22] and G'satisfies the A^£^^ theorem, which implies the existence of an 0(A^£^) bound on the arithmetic work of a Cholesky decomposition via generalized nested dissection. The bound in the previous Lemma is very conservative. Because of the general way in which the size reduction problem is formulated, it is difficult to find a tighter upper bound on additional fill-in. In specific examples (see Section 4

96

M. Laso and J. Ramirez

below) the amount of additional fiU-in rarely exceeds 10-20%. Requirement (4) is hard to deal with in general, since the numerical conditioning of the reduced system depends in a very specific way on the molecular model. It is however possible to obtain estimates of the condition number for specific cases. 4. Numerical examples In this Section we make more specific the rather formal principles presented in the previous Sections by means of i) a simple one-dimensional problem the reduction of which can be treated entirely analytically in a CONNFFESSIT formulation and of ii) an illustration of additional fill-in in a general twodimensional BCF setting. For the start-up of plane Couette flow of a dumbbell model, mass conservation is satisfied automatically by solutions of the type

The system obtained by a simple discretization of the momentum conservation equation and a Hookean dumbbell constitutive equation can be written as:

^!l^^+awf^^^+^::;^^+c[r7^ -r;:;]=u^;^,+Au^p+u%

(MJ

together with Qin^l) _ ^^n^l) . Qin^l) J J)^^^ + 1 gC^^D^/ = Q^"^ + VAT^T

(15)

for each of the dumbbells and T^"^'^ =^nkT(Q^''''^Q^"''^),

(16)

for each of the elements. The same symbols and dimensionless variables as in [1] have been used. Once the rule for stress calculation (16) has been inserted in the discretized momentum equation (14), the system of equations to be solved at every implicit step, written in full, is:

Implicit micro-macro methods in viscoelastic flow calculations for polymeric fluids NC,

aux4"^»- •'+bur'^+c

•"Lf^ra272 NC.

^ r

V"'^2 J

97

NQ

yNC,

I^r'^a J

1>2

M, + 4 < ' + M r , 1

Z>M<""'^+a«f"'>+6Mi""'> + c

wc.

A^Q

•znra NC.

vA^Q ,

V ^ 2

,(w+l) 172

j

W 2 + 4 (w) < ^ I+ ^
S «'s

Wr'J+flfwV/r^ +c *7S^£'-l

2y2

K-^^^NE

NE12 ^ J JNEU

^*

^f^l^l

J

\ '^NE2\^NE2\

NC^ JNE-\J\>C\J2

NC

^ '^NE22>CNE22 )

^NE\2

^^ V ^ « '''^NE\2

Where fC^ ^ is the y-th component of the velocity gradient tensor in element k, f^j^ represents the A:-th component of the spring force of the 7-th dumbbell in the i-th element (for example the linear spring law, third term in Eq. 15), Q.jj^ is the A:-th component of the dumbbell connector vector of the 7-th dumbbell in the /-th element, where: . ,^ At*De a = 4 + 12 , * \ 2 — ,

(hjRe

with dimensionless variables:

, . . At*De 6 = 1-6 , * \ 2; — ,

{h'YRe

, At c = 6 ah Re

d^At'De

98

M. Laso and J. Ramirez

nl

fJs

L

[LIU)

The blocks of the Jacobian A of the previous system of equations are then:

A

a b 0 0

b a b 0

0 b a b

0 ... 0^ 0 ... 0 b ... 0 a ... 0 b

V0

0 0 0 6

The A j ^ is block diagonal, with one 3 x 3 block per dumbbell. The entries of the block for the j-th dumbbell of the /-th element are:

"

— D e A / , iki

2 aaijl

where k and / = !,..., N^^^j- and N^^^^ is often 3. No summation over /, y, A: or / takes place in the previous expression. The tall submatrix Ay„^ has, at the row corresponding to the y-th dumbbell of the /-th element, only two non-zero entries located at columns /and z + 1 of values: -DeM and

a*-

r Q,- 1)7'

^

Implicit micro-macro methods in viscoelasticflowcalculations for polymeric fluids

99

-DeM respectively. A:and l = l,..,,N.^^y, summation over / and no summation over /, j \ k takes place in the previous expressions. The wide submatrix A^^ has, at the column corresponding to the y-th dumbbell of the /-th element, only two non-zero entries located at rows / and / + 1 of values: A

±c

+fijA

kl

where A: = 1,..., N.^^^-, the plus sign corresponds to the entry at row / the minus sign to the entry at row / + 1 and no summation over i^j^k takes place in the previous expression. The reduction of the Jacobian A +A/A =A - A A~^ A can be carried out analytically:

a+k NC2

^MM + A / A^^ —

a+k

NCj

i(e:r) +i(cl

where the superscript r labels the iteration within the Newton-Raphson loop and

100

M. Laso and J. Ramirez

GAtDe k =NC.a(h*) Re{l + At/2) for i =

l'",NE,

The full (micro-macro) right-hand side:

Y(")

=

changes to a modified macro right hand side:

which allows the velocity variables to be obtained in a first stage. Once the velocities are available, solving for the microscopic degrees of freedom (connector components) is done on a"molecule-by molecule" basis just as in an explicit approach. The following table shows the results of a numerical test case taken from [1]. Fully implicit CONNFFESSIT and BCF micro-macro schemes were solved using a Picard-type solver and a Newton solver incorporating the Schur complement size-reduction scheme for the Jacobian. CONNFFESSIT (80000 dumbbells, 40 nodes)

1

velocity error norm

t

At

Explicit

Implicit (Newton)

Implicit (Picard)

0.1

0.001

1.2e-05

1.3e-05

1.3e-05

0.1

0.01

2.2e-05

3.5e-4

3.5e-4

0.1

0.1

diverges

1.3e-2

Picard diverges

0.4

0.001

5.1e-06

5.3e-05

5.3e-05

0.4

0.01

3.3e-05

2.8e-05

2.8e-05

0.4

0.1

diverges

7.9e-04

Picard diverges

Implicit micro-macro methods in viscoelastic flow calculations for polymeric fluids

101

As expected, the fully implicit schemes shows increased stability and Newton's method has a wider range of convergence than Picard's. The implicit BCF calculation with the Newton solver was convergent for all values of At. Typical results for the velocity profiles at / = 0.01, ^ = 0.1and ^ = 0.4s are presented in Fig. 1. At very short times, the oscillations caused by the singularity at ^ = 0 s are less pronounced in the implicit calculation (Fig. 2).

O D ^

Oldroyd-BCE Ar=0.001 Ar=0.01 Ar=o.i

r = o.i

Figure 1. Velocity profiles for start-up of Couette flow for implicit and explicit algorithms and different time steps.

The next table presents a comparison in CPU time for the explicit and the implicit (both Picard and Newton) calculations and a fixed convergence criterion of 10"^ change.

M. Laso and J. Ramirez

102

Ratio of CPU times for implicit vs. explicit micro-macro calculations CONNFFESSIT

Brownian Fields

impl. Picard/expl.

impl. Newton/expl.

impl. Picard/expl.

impl. Newton/expl.

j 0.001

3.5

2.4

2.7

10.8

0.01

4.9

3.1

5.3

14.1

0.1

expl. diverges

expl. diverges

Picard diverges

29.2

^t

For this example, an implicit CONNFFESSIT step was only two to three times slower than an explicit one, whereas the implicit BCF step was between 10 and 30 times slower. This large difference is due to the availability of a fully analytical expression for the size-reduced system in CONNFFESSIT. Computing the additional terms in the Jacobian and in the RHS thus amounts to cummulating sums, which does not require a great deal of additional computation. The case of BCF is different in this respect, since an analytical expression for the reduced system is not available. The solution time reflects the considerable CPU overhead required by sparse matrix manipulation and system solution. This implicit-explicit CPU-time ratio for a BCF micro-macro calculation is comparable to the implicit-explicit CPU-time ratio for macroscopic methods.

Implicit micro-macro methods in viscoelastic flow calculations for polymeric fluids 0.4 — ^ ~ [

1 y

1

1 1 1 1 1

1

1

1

1

1

1

1

1

1

1 1

103

1 '

0J5

—

03 — h 0.25 h

— explicit. A/=0.001 •--• inqjlicit (Newton), A^ = 0.001

~ y 0.2 h 0.15 -h

-

i^>-

^-^—^—(^ 0.1 — h r^^ 0.05 h 1 1 , 0 -0.02 -0.01

"

— -—~=-^

1 , 1 , 0 0.01

1 , 0.02

1 . 1 . 1 1 1 . 1 1 1 . 1 1 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1 U

Figure 2. Velocity profiles for start-up of Couette flow for implicit and explicit algorithms at very short times. Oscillations in velocity profile are less pronounced for the implicit scheme.

In the previous simple one-dimensional CONNFFESSIT illustration, the sizereduced Jacobian retains its tridiagonal structure, additional fill-in is zero and the upper bound estimate on additional fill-in is maximally pessimistic. In 2D and 3D problems, the Jacobian will have a general sparse structure, but in CONNFFESSIT and LPM, element-wise and molecule-wise ordering of the A^^ matrix will guarantee the feasibility of building Schur's complement analytically. In BCF, the structure of A^^will not be "small, dense"-block diagonal but "large, sparse"-block diagonal. Building Schur's complement will involve sparse techniques but at the block level exclusively. The final example illustrates system size reduction and fill-in in a 2D setting for BCF. The discretization of the equations of conservation on a 2 x 2 2D

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M. Laso and J. Ramirez

quadrangular mesh using Ql^ Q^, g|* elements for velocity, pressure and stress respectively and only two BC fields (we have chosen these unrealistically low number of elements and of fields in order to have clear enough plots of the sparsity pattern of the A matrix). In addition, the stress variables are retained in the formulation, i.e., they are not eliminated from the set of macro variables. It can be proved easily that in this case the additional fill-in is confined to lie within a rectangular region of size A^^ x A^^ as can be observed in Fig. 3a and b. Fill-in of the A^^ block is 31.6% and increases to 39.7% after size reduction, i.e. by a factor of 1.26.

Figure 3a. Sparsity pattern of Jacobian for micro-macro BCF formulation. The blocks corresponding to macroscopic and microscopic unknowns are marked by lines; all nodal values for each type of variables are adjacent; the succesion of variables is as labeled on the right.

Implicit micro-macro methods in viscoelastic flow calculations for polymeric fluids

105

Figure 4b. Sparsity pattern of Jacobian for the same example after reduction by field-wise Schur's complement. Additional fill-in is marked by circles.

5. Conclusions By means of a size reduction technique, fully implicit micro-macro calculations are feasible without incurring excessive computational cost (relative to explicit micro-macro calculations). The very large nonlinear system of equations to be solved at every time step in an implicit micro-macro calculation can be reduced using Schur's complement to a system having the same size as in a purely macroscopic formulation. The necessary matrix manipulations can be done either analytically or using sparse matrix techniques but always exclusively at the block level. The need to deal with the complete micro-macro coefficient matrix (which would be unfeasible even with sparse matrix methods) can be avoided entirely. The size reduction process eliminates the microscopic degrees of freedom but leads to additional fill-in with respect to the purely macroscopic formulation. Additional fill-in is however minor and the numerical conditioning of the reduced system does not deteriorate with respect to its macroscopic counterpart.

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M. Laso and J. Ramirez

6. Acknowledgements The authors gratefully acknowledge financial support from the E.U. through contract Ref. G5RD-CT-2002-00720 and NMP3-CT-2005-016375, and partial support by CICYT grant MAT 1999-0972.

REFERENCES [I]

M. Laso, H. Ottinger, Calculation of viscoelastic flow using molecular models, J. NonNewtonian Fluid Mech. 47 (1993) 1-20. [2] A. V. H. M. Hulsen, B. van den Brule, Simulation of viscoelastic flows using brownian configuration fields, J. Non-Nev^onian Fluid Mech. 70 (1997) 79-101. [3] R. K. P. Halin, V. Legat, The Lagrangian particle method for macroscopic and micromacro viscoelastic flow computations, J. Non-Newtonian Fluid Mech. 79 (1998) 387-403. [4] J. Bonvin, M. Picasso, Variance Reduction Methods for CONNFFESSIT-like Simulations, J. Non-Newtonian Fluid Mech. 84 (1999) 191-215. [5] M. H. A.P.G. van Heel, B. van den Brule, Simulation of the Doi-Edwards model in complex flow, J. Rheol. 43 (1999) 1239-1260. [6] E. Grande, M. Laso, M. Picasso, Calculation of variable-topology free surface flows using CONNFFESSIT, J. Non-Newtonian Fluid Mech., J. Non-Newtonian Fluid Mech., 113, 127-145 (2003). [7] J. Cormenzana, A. Ledda, M. Laso, B. Debbaut, Calculation of free surface flows using CONNFFESSIT, J. Rheology 45 (1) (2001) 237-258. [8] P. Gigras, B. Khomami, Adaptive configuration fields: a new multiscale simulation technique for reptation-based models with a stochastic strain measure and local variations of life span distribution, J. Non-Newtonian Fluid Mech. 108 (2002) 99-122. [9] H. Ottinger, A thermodynamically admissible reptation model for fast flows of entangled polymers, J. Rheol. 43 (1999) 1461-1493. [10] G. lanniruberto, G. Marrucci, A multi-mode CCR model for entangled polymers with chain stretch, J. Non-Newtonian Fluid Mech. 102 (2002) 383-395. [II] P. Waperom, R. Keunings, G. lanniruberto. Prediction of rheometrical and complex flows of entangled linear polymers using the DCR model with chain stretching, J. Rheol. 47 (2003) 247-265. [12] J. N. J. Schieber, S. Gupta, A full-chain, temporary network model with sliplinks, chainlengthfluctuations,chain connectivity and chain stretching, J. Rheol. 47 (2003) 213-233. [13] N. W. M. Somasi, B. Khomami, E. Shaqfeh, Brownian dynamics simulations of bead-rod and bead-spring chains:numerical algorithms and coarse- graining issues, J. NonNewtonian Fluid Mech. 108 (2002) 227-255. [14] M. Somasi, B. Khomami, Linear stability and dynamics of viscoelastic flows using timedependent stochastic simulations techniques, J. Non-Newtonian Fluid Mech. 93 (2000) 339-362. [15] M. Somasi, B. Khomami, A new approach for studying the hydrodynamic stability of fluids with microstructure, Phys. Fluids 13 (2001) 1811-1814. [16] R. Owens, T. Phillips, Computational Rheology, Imperial College Press, London, 2002. [17] P. Doyle, E. Shaqfeh, A. Gast, Dynamic simulation of freely drainingflexiblepolymers in steady linear flow, J. Fluid Mech. 334 (1997) 251-291.

Implicit micro-macro methods in viscoelastic flow calculations for polymeric fluids

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[18] M. Laso, Calculation of non-newtonian flow of colloidal dispersions: finite elements and brownian dynamics, J. Comput-Aided Mater. 1 (1993) 85-96. [19] R. S. Crouch, T. Bennett, Efficient EBE treatment of the dynamic far-field in non-linear FE soil-structure interaction analyses, European Congress on Computational Methods in Applied Sciences and Engineering 43, 344-356. [20] D. Zois, Parallel processing techniques for FE analysis: System solution, Computers and Structures 28 (2) (1988) 261-274. [21] M. R. J. Maryska, M. Tuma, Schur complement systems in the mixed-hybridfiniteelement approximation of the potential fluid flow problem, SIAM J. Sci. Comput. 22 (2000) 704723. [22] M. Gondran, M. Minoux, Graphs and Algorithms, John Wiley & Sons, New York, 1984. [23] R. Bird, C. Curtiss, R. Armstrong, O. Hassager, Dynamics of polymeric liquids, vol.2, John Wiley & Sons, New-York, 1987. [24] H. Ottinger, Stochastic processes in polymeric fluids. Springer-Verlag, Berlin, 1996.

Multiscale Modelling of Polymer Properties M. Laso, E.A. Perpete (Editors) © 2006 Elsevier B.V. All rights reserved.

109

Chapter 5

Estimation of Critical Parameters from Quantum Mechanics Valerie Wathelet, Marie-Claude Andre, Michele Fontaine Laboratoire de Chimie Theorique Appliquee, 61 rue de Bruxelles, 5000 Namur, Belgium 1.

Introduction

The improvement of algorithms and the tremendous increase in available computer resources already made quantum chemistry (QM) successful for the treatment of many industrial problems [1]. The design of new chemical processes requires the knowledge of accurate thermodynamic data for all the substances involved. The experimental determination of such data is timeconsuming and expensive. The alternatives based on purely empirical methods sometimes lack of reliability. For example, these methods are usually unable to discriminate between isomers. First order group contribution methods often fail in the estimation of oligomer and polymer properties. As long as it is possible to connect the required material properties to molecular scale quantities, quantum chemistry appears to be of great predictive interest to improve such materials and properties. The purpose of this work is to demonstrate how quantum mechanics methods are able to determine the characteristic parameters of wellknown equations of state. This approach is necessary when physical properties are missing, and it also allow to avoid expensive experiments. Consequently, we aim at generating critical parameters (i.e. temperature, pressure and volume) by using QM methods, illustrating our recent progress in the attempt of combining computational chemistry with qualitative structure - properties relationships. Basically, we develop a strategy for calculating the critical parameters for pure compounds by using muhiple linear regressions based on quantities called "descriptors" that we estimate by QM calculations. The calculated critical values are compared with available experimental data. Description of mixtures requires convenient mixing rules, and the prediction of

110

V.Watheletetal

critical parameters for binary systems is completed by the Redlich's hyperbolic interpolation. 2.

Methodology

For the estimation of critical parameters, Tc, Pc and Vc of pure compounds, group or bond contribution methods have largely been used. In these methods, the properties of a molecule are evaluated by performing a summation of the elementary contributions times the occurrence in the molecule. An alternative to group-contribution methods is offered by the Quantitative Structure Properties - Relationship model (QSPR) [2,3]. The construction of a QSPR scheme requires experimental values of the properties of interest for a set of compounds, for which numerical descriptors, taking into account the topological and electronic properties of each compound are calculated. The descriptors are then used to generate predictive models using multiple linear regression analysis or computational neural networks. In this work, the multiple linear regression (MLR) [4-6] based on the numerical technique of least-square fitting, is applied to develop a relationship between one or more explanatory variables (descriptors) and a response variable (the property of interest) by fitting a linear equation to observed data : X = ^0 + Px^n + Pi^a + -Pp^^p for i = 1,2, ... n

(I)

To test the significance of a regression curve, the total sum of squares (TSS) is split into two components, the model sum of squares (MSS) and the residual sum of squares (RSS). If the fitted curve goes through all the original data points, the MSS is equal to the TSS and the RSS is zero. A way to measure the adequacy of the MLR is the square of the sample correlation (R^), called the determination coefficient and calculated as R = 1-(MSS/TSS). This parameter is close to 1 when the RSS (differences between experimental and computed values) is small. The multiple correlation coefficient can be spuriously large if there are large number of predictor variables (p) and a small number of observations (n). In order to check if the regression is meaningful, the so-called analysis of variance table (ANOVA) is set up, where the mean squares MMS and RMS are respectively obtained by reporting the MSS to the number of independent variables (p) and by dividing RSS by (n-p-1), where n is the number of observed data. If the MMS/RMS ratio is significantly large, the regression is meaningful. Formally, it consists on testing the null hypothesis Ho : pi = 0 against its alternative Hi : Pi ^ 0. If pi is non-zero, the null-hypothesis may be rejected and the regression is significant. The confidence limits for the regression parameters Pi measure the adequacy of each independent variable in the model. The ratio between Pi and the associated error can be compared to

Estimation of critical parameters from quantum mechanics

111

tabulated critical values for which the probability (P-value) to reject the null hypothesis has been determined. The probability to observe a regression coefficient by chance can therefore been assessed. This treatment allows to eliminate step by step the less or non significative independent variables. In the present study, MLR has been performed with the Statgraphics Plus 5.1. program [7]. Except for the molecular mass, the descriptors were estimated by ab initio calculations based on optimized geometry for each molecule. A study was performed to determine both the appropriate method and basis set to estimate the critical parameters. We have selected three common computational schemes, Hartree-Fock theory (RHF), B3LYP issued from density functional theory (DPT), and the second-order MoUer-Plesset perturbation theory (MP2). RHF is fundamental to much of electronic structure theory [8] and provides a good starting point for more elaborate theoretical methods. MP, which is an application of the perturbation theory of Rayleigh-Schrodinger, goes beyond the RHF method in attempting to treat the electron correlation. In DFT [9], one uses a general expression for the exchange-correlation functional, which includes terms accounting for both the exchange energy and the electron correlation. In addition to pure DFT methods, hybrid methods exist in which the exchange functional is a linear combination of the Hartree-Fock exchange and a functional based upon the density and possibly the density gradient. The molecular orbitals within a molecule are represented by a linear combination of atomic orbitals (LCAO). Larger basis sets induce fewer constraints on electrons and more accurate molecular orbitals, but they correspondingly require much computational resources. In this work, we have used two basis sets, the STO-3G minimal basis set, and 6-31G(d) a mediumsize basis set. Calculations were performed with Gaussian 98 [10] program on a PC Pentium III. For binary mixtures, Redlich and co-workers [11-13] have proposed an interpolation method to estimate the critical temperature and pressure as a function of the composition, by means of logarithmic-hyperbolic equations. For a binary mixture, the critical temperature evolution as a function of the composition is given by : fin r; - In 77 - y), In r, = jf, in T; + x^ In T^ +y

—r

fin r; - In T^ - V] j

x, x. .

Q)

(^In7;^-ln77-^j^,-^^ln7;^-lnr,^-^jx, where Ti^ is the critical temperature for pure component i. The composition of the mixture is given by the mole fraction Xj. When the critical temperature and

112

V. Wathelet et ai

pressure are plotted versus Xi, the terms ti and pi are the limiting slopes of the critical temperature and pressure, respectively, when Xi = 1 or Xj = 0. For the temperature, this equation corresponds to :

^^T^

/j = lim| K^xJc

,. ({dP/dx,)l,/RT-(d'P/dVdx,] T ^ (d^p/dTdvl

= lim

(3)

•^^i-^i

The estimation of the derivatives used in the estimation terms of ti and pi are based on the Redlich-Kwong equation of state :

V-nb

T"^V{y + nb)

3.

Critical parameter estimation for molecules

3. L

Computational details

3.1.1. The data set Critical temperature, pressure and volume are obtained for a set of 135 compounds at room temperature. The size of the molecules is ranging from 10 to 30 atoms corresponding to a molecular mass of 40 to 160 daltons. The functional groups amongst the data set include alcanes, alcenes, aromatics, alcohols, esters, ketones, amines, carboxylic acids, acetates, phenols, ethers, nitriles. The range of values for the critical parameters is wide : the critical temperatures spread from 393 to 782 K, the critical pressures range from 21 to 66 bar, and the critical volumes are bracketed by 139 and 624 ml/mole. 3.1.2. Selected descriptors In order to determine an empirical equation for the estimation of the critical parameters, we have selected six descriptors. The molecular volume (Vm) can be related to the critical volume of the compounds. The Vm is defined as the volume inside a envelop corresponding to a density of 10"^^ electrons/bohr^ density. The charged surface may also content information on the potential. Positively and negatively charged surfaces (S^ and S") are obtained by the analysis of the electrostatic potential on the molecular surface. The dispersion forces (London) contribute to the attractive potential and their intensity partially depends on the polarisability (a) and the ionization potential (IP). The polarisability is computed by analytically determining the derivative of the

Estimation of critical parameters from quantum mechanics

113

LCAO coefficients with respect to the external electric field. The ionization potential is evaluated via the Koopman's theorem, whereas the molecular mass (MM) is the last descriptor. 3.1.3. Quantum mechanics methods Amongst all the selected QM methods (i.e. RHF / STO-3G, RHF / 6-31G(d), B3LYP / 6-31G(d), MP2 / 6-31G(d)), the determination coefficient (R^) does not dramatically change : maximum 4 %. The ANOVA tables show a statistically significant relationship between the variables, at a 99% confidence level since the MMS/RMS ratio is very large. The degree of confidence granted to each independent variable, from 90% or higher level, makes the difference between the several models. Indeed, the extension of the atomic basis set, from STO-3G to 6-31G(d), improves the quality of the description, and the confidence level is better for a higher level of QM approximation. In the QSPR approach, the major effect of the electronic correlation consists of improving the statistic inference due to the enhanced accuracy in the estimation of the descriptors. The best results are obtained by MP2/6-31G(d), for which the needed CPU are important for the biggest molecules. So, the best compromise between computation cost and statistical quality appears to be the B3LYP functional with the 6-31G(d) atomic basis set. 3.2.

Molecular critical parameters estimation

3.2.1. Critical Volume The critical molar volume (Vc) is clearly related to the molecular volume (Vm). For instance, in the van der Waals formalism, it is fixed to three times the value of the covolume. The simple linear regression for the MP2/6-31G(d) model corresponds to : F, = - 1 8 . 4 8 + 4.05 F^

(5)

The determination coefficient indicates that the model explains more than 96 % of the variability in Vc. The standard error of the estimate shows the standard deviation of the residuals to be around 16 ml/mole. The mean absolute error (MAE) of 11 ml/mole is the average value of the residuals which corresponds to less than 5 % of the average value of the critical volume of the complete data base. Figure 1 shows the MP2/6-31G(d) Vc versus the experimental values, and confirms that the model is of high quality throughout the range of the critical volume values.

V. Wathelet et al.

114

o E 0)

£ O

>

Observed Critical Voiume (mi/mole)

Figure 1 : Comparison of the experimental values of critical volume vs. the calculated values using the MP2/6-31G(d) model.

3.2.2. Critical Pressure An acucrate description of the critical pressure (Pc in bar) requires five descriptors, which are the polarizability (a in A^), the molecular mass (MM in daltons), the inverse of molecular volume (Vm in ml/mole), the ionisation potential (IP in eV) and a combination of the positive and negative surfaces (Z = \S''-S'\/{s^+S~)in A^). These descriptors explain 80 % of the variability of the critical pressure. For the MP2/6-31G(d) model, the relationship is: P, = 2 7 . 6 - 0 . 1 4 3 a + 0.140 MM +

2.1810'

-0.100/P-11.2Z

(6)

The standard error is less than 4 bars or 8 % of the mean critical pressure. Except for the RHF/STO-3G model, the regression coefficient confidence levels are close to 99 %. The calculated critical pressure versus experimental critical values [MP2/6-31G(d)] are depicted in Figure 2. The plot shows that the model is of good quality, especially at low and middle range pressure. Structural descriptors, like positive and negative surfaces play an important role in the critical pressure. They mimic the accessibility of the heteroatoms and the relative molecular orientation, that play an essential role in intermolecular interactions. Polarisability and ionisation potential are related to the dispersion phenomena. The MM and the inverse of the molecular volume are related to the molecular density.

Estimation of critical parameters from quantum mechanics

115

(0

O """20

30

40

50

60

70

80

Observed Critical Pressure (bar)

Figure 2 : Comparison of the experimental values of critical pressure vs. the calculated values using MP2/6-3 lG(d) model.

3.2.3. Critical Temperature The estimation of the critical temperature (Tc) required three descriptors : the polarizability (a), the molecular density (MMA^m), and the total surface (S" +S^). This model explains nearly 70 % of the variability in Tc. For the MP2/63 lG(d) model, the relationship (in Kelvin) is: MM MM / \ 7 ; = 169+ 4.19 6^+ 2 7 3 ^ - ^ - 0 . 2 9 3 ( 5 - + 5 " )

(7)

The standard error deviation is 42 K, that corresponds to 7.5 % of the average critical temperature of the complete database. At the RHF 6-31G(d) level, the confidence level for the total surface is 95 %. When correlation is taken into account, like with B3LYP or MP2, the confidence level is close to 99% for every descriptor. The plot of calculated versus observed critical temperature is shown in Figure 3. The important role played by the dispersion forces in the transition phenomena is expressed by the significant polarisability contribution to the critical temperature.

116

V. Wathelet et al

Observed Critical Temperature (K)

Figure 3 : Comparison of the experimental values of critical temperature vs. the calculated values using MP2/6-3 lG((i) model.

3,3,

Validity on test molecules

In order to test the validity of the regression model, an external set is built up. Ten molecules belonging to several chemical species (alcanes, alcenes, aromatics, alcohols, amines and esters) have been selected. We have calculated the critical parameters of these compounds with the MLR method and compared them to those obtained with several contribution methods : Joback which is a first-order group-contribution method, Marrero-Pardillo that considers the contributions of interactions between bonding groups, and Marrero-Gani that integrates a group-contribution method at three levels, i.e. monofuctional, polyfunctional and larger site effect. For each critical parameter, the average relative errors (ARE) is defined as

I

Y'Est

yExp

The calculated ARE for the selected methods are presented in Table 1.

(5)

Estimation of critical parameters from quantum mechanics

Method

Vc

Pc

Joback

1.7

11

Marrero-Pardillo

1.5

5

Marrero-Gani

1.5

3

QSPR-MLR

1.5

5

111 Tc

Table 1 : Average relative errors (in %) for external databasefromthe critical parameter calculated by different methods (QSPR-MLR or group/bond contribution methods).

The critical volume is the easiest value to consistently determine. The relative error is always less than 5 percents. The average relative error for the test database is inferior to 2 %. The volume is a simple additive property directly proportional to the size of the molecule. It can consequently be awaited that a first-order method is sufficiently accurate. For the critical pressure, the relative average error is 5 % by QSPR-MLR and Marrero-Pardillo's methods. The maximum error for both methods is equal to 17 % (Figure 4). The Marrero-Gani method is somewhat better, the error is only 3 %. With the Joback's method, the relative average error and the maximum relative error are of 11.2 % and 40 % respectively. At the same time, the QSPR-MLR method yields to a relative error for the critical temperature of only 2 %. This is inferior to the Joback's error and similar to the Marrero-Pardillo's and Marrero-Gani's accuracies. • Joback M Marrero-Pardillo

S Marrero-Gani

I MLR

25 20 15 10 f$^

5 0

rfcfca , HmM , I M a (D

0 c

c

—

^


— 2

^

s

$*

-^ 2

^

^

tit

^


A

^ 9

^

">.

LU

c

o O

J

CO

E o

J.

0 Q-

Q CO JO CM

>. 0

E


c CO Q. O

>s

Q.

0 C CO

o O

E JO

0 C 0 CO X

0 0

I

I-

4

o

Figure 4 : Relative errors (in %) for critical pressure using different methods (QSPR-MLR or group/bond contribution methods) for the external database.

118

4.

V. Wathelet et al

Critical parameters estimation for polyethylene

In order to extrapolate the polyethylene critical values, the same statistical treatment is applied to a specific base dedicated to «-alkanes containing from 2 to 18 carbon atoms. We have already shown that the critical volume is directly related to the molecular volume. The average error on critical volume values is around 16 ml/mole. We also obtain an excellent agreement between calculated and experimental critical pressure values with a determination coefficient and correlation coefficient values close to 100%. The model used to estimate the critical pressure of hydrocarbons, reveals the importance of taking into account the dispersion forces, by using polarisabilities as pertinent descriptors. Calculated temperature values for «-alkanes are also correctly estimated by the QSPR-MLR method. Once more, the determination coefficient is up to 99%; the average error on critical temperature values is only 7 K. Paying attention to the coefficient probability confirms the relevance of the properties used as descriptors in this model, i.e. the polarisability and the ratio of the molecular mass to the molecular volume. Similarly than for critical pressure, these observations indicate that both the dispersion forces and the molecular density play an important role in the determination of hydrocarbon critical temperature. The comparison of the efficiency of the QSPR-MLR approaches based on the use of a general compound base, or on a specific family base (as the n-alkanes base for example), indicates that the estimation of critical temperature and pressure is really improved in the second way. In order to extrapolate the critical properties of long polyethylene oligomers, the critical temperatures and pressures for the n-alcane series are plotted as a function of the chain length. Table 2 summarizes the critical parameters obtained with different methods : group or bond contribution methods such as Joback, Marrero-Pardillo, Marrero-Gani and QSPR-MLR as well as an extrapolation of the corresponding experimental values. Tc(K)

Pc (bar)

833

10

Linear evolution

7

Marrero-Pardillo

877

8

Marrero-Gani

824

12

QSPR-MLR

792

10

Experimental Joback

Table 2 : Extrapolation of critical pressure and temperature values to the limit of the polyethylene [B3LYP/6-31G(d)] from different methods.

Estimation of critical parameters from quantum mechanics

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The discrepancies between calculated and experimental values are about 5% for critical temperature and 2% for critical pressure, as tabulated experimental values for tetracosane (C24H50) yield to a critical temperature of 800 K and a critical pressure value of 8.7 bar [14]. These two observations demonstrate a really good behaviour of QSPR-MLR approach in describing the critical point for longer compounds. Furthermore, the Joback contribution group method is unable to estimate the critical temperature value of such extended systems. This method provides continuous increasing values of Tc with the carbon chain length, and is unable to reproduce the observed experimentally limit value. This approximation is not relevant in the treatment of large oligomers. Other contribution methods lead to correct critical values with slightly larger gaps with experimental parameters. The conclusion that Marrero et al methods correctly describe the critical temperature evolution contrary to Joback method is due to the fact that the equations used to evaluate Tc and Pc have a denominator including the number of carbon atoms. Therefore they can reproduce the limit in Pc and Tc, which is not the case for Joback. 5.

Application to Binary Mixture

Figures 5 and 6 show critical temperatures and pressures for binary systems of butane-hexane and C02-pentane as a function of the mixture composition. Circles and triangles denote experimental critical temperatures and pressures [7,13], respectively. Except for CO2, critical parameters estimated with QSPRMLR model are used. Solid lines and dashed lines are calculated critical temperatures and pressures from the Redlich's logarithmic-hyperbolic interpolation functions. Experimental and calculated results are in good agreement for both mixtures, and the decrease in critical temperatures and the maximum in critical pressures are well reproduced as well.

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Figure 5 : Critical temperature and pressure for the butane-hexane mixture. Circles : experimental critical temperatures, triangles : experimental critical pressures. Solid lines and dashed lines are calculated critical temperature and pressurefromthe Redlich's logarithmic-hyperbolic interpolation functions.

Figure 6 : Critical temperature and pressure for the C02-pentane mixture. Circles : experimental critical temperatures, triangles : experimental critical pressures. Solid lines and dashed lines are calculated critical temperature and pressurefromthe Redlich's logarithmic-hyperbolic interpolation ftmctions.

6.

Conclusions

The QSPR method combined with multiple linear regression is an efficient and elegant way to estimate from scratch the critical parameters of a wide range of compounds. Calculated critical parameters for pure compounds including extrapolation to the limit of extended systems are both in agreement with experimental data and values obtained by group contribution methods, though, in some cases, these contribution methods are unable to correctly describe the saturation behaviour of critical temperature. In order to obtain the critical parameters for binary mixture, the Redlich's logarithmic-hyperbolic interpolation functions have been used. The great advantage of this method is that only critical parameters of pure compounds are required. Two mixtures have been studied butane-hexane, C02-pentane. Calculated mixture critical values are in very good agreement with experimental data, illustrating that the combination of empirical rules and QM is possible and often valuable when carefully performed.

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Acknowledgements The authors warmly thank Prof. Jean-Marie Andre for his support and Drs. Eric Perpete and Denis Jacquemin for their invaluable assistance all along this work. The calculations have been performed on the Interuniversity Scientific Computing Facility (ISCF), installed at the Facuhes Universitaires NotreDame de la Paix (Namur, Belgium), for which the authors gratefully acknowledge the financial support of the FNRS-FRFC and the "Loterie Nationale" for the convention number 2.4578.02, and of the FUNDP. Bibliography 1. 2. 3. 4. 5. 6. 7. 8. 9. 10.

11. 12. 13. 14.

M. Fermeglia, S. Priel, G. Longo, Chem. Biochem. Eng. Q. 17 (2003) 19-29 S. Grigoras, J. Comput. Chem. 11 (1990) 493-510 L.M.Egolf, M.D. Wessel, P.C. Jurs, J. Chem. Inf. Comput. Sci. 34 (1994) 947-956 P. Dagnelie, "Statistique theorique et appliquee - Tome I - Statistique descriptive et bases de rinference statistique", Paris et Bruxelles, De Boeck et Larcier (1998) P. Dagnelie, "Statistique theorique et appliquee - Tome II - Inference Statistique a une et deux dimensions", Paris et Bruxelles, De Boeck et Larcier (1998) J.H. Pollard, "A Handbook of Numerical and Statistical Techniques", Cambridge University Press (1979) Statgraphics 5 Plus A. Szabo and N.S. Ostlund, "Modem Quantum Chemistry", Dover NY (1996) R.G. Parr, ''Density-functional theory of atoms and molecules", Oxford university press NY (1989) Gaussian 98, Revision B.l, M.J. Frisch, G.W. Trucks, H.B. Schlegel, P.M.W. Gill, B.G. Johnson, M.A. Robb, J.R. Cheeseman, T. Keith, G.A. Petersson, J.A. Montgomery, K. Raghavachari, M.A. Al-Laham, V.G. Zakrzewski, J.V. Ortiz, J.B. Foresman, J. Cioslowski, B.B. Stefanov, A. Nanayakkara, M. Challacombe, C.Y. Peng, P.Y. Ayala, W. Chen, M.W. Wong, J.L. Andres, E.S. Replogle, R. Gomperts, R.L. Martin, D.J. Fox, J.S. Binkley, D.J. Defrees, J. Baker, J.P. Stewart, M. Head-Gordon, C. Gonzalez and J.A. Pople (Gaussian Inc. Pittsburgh, PA, 1995). J. Jiang, J.M. Prausnitz, Fluid Phase Equilibria 169 (2000) 127-147 O. Redlich and T. Kister, J. Chem. Phys. 36 (1962) 2002 F.J. Ackerman, O. Redlich, J. Chem. Phys. 38 (1963) 2740 BE Pauling, J.M Prauznits, J.P. O'Connell, "The Properties of Gases and Liquids", New York, McGraw-Hill (2001)

Multiscale Modelling of Polymer Properties M. Laso, E.A. Perpete (Editors) © 2006 Elsevier B.V. All rights reserved.

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Chapter 6

Micro-macro calculations of 3D viscoelastic flow Jorge Ramirez/'^ Manuel Laso,^ ^Applied Mathematics, University of Leeds, Leeds LS2 9JT, UK ^E.T.S.L Industrials, Jose Gutierrez Abascal,28006 Madrid, Spain

I. Introduction Most industrial polymer processes (extrusion, injection molding, fiber spinning, etc.) involve complex geometries which are generally three dimensional or have 3D effects like recirculation flows or complex vortices. In these cases, a simplified two dimensional model is not enough to capture all the relevant effects that are observed in the actual experiments. Most benchmark problems proposed in the computational rheology literature are 2D, essentially because they are easier to formulate and cheaper to solve, but the computational power of CPUs and the memory capacity of computers has increased so much during the last decade, combined with their decreased cost, so that it has become possible to tackle more realistic viscoelastic problems in 3D geometries^ ^ Although no 3D benchmark problems have been established yet, which is an important point if we want to compare the robustness and accuracy of the different solvers and methods available, the contraction flow is the case that mostfi*equentlyappears in the literature. Up to now, all 3D viscoelastic simulations presented have been based on a purely macroscopic description, using a closed-form constitutive equation to complement the conservation equations of mass and momentum. There are, however, situations in which a more detailed level of description is needed in order to fully understand the observed behavior of a particular polymer in a given complex flow. It is in these cases where the so-called micro-macro techniques can be of great help. In this approach, the conservation equations are supplemented by a molecular model from the kinetic theory^ to describe the rheology of the fluid. Molecules are represented by a coarse-grained model and

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their configurations are updated using Brownian dynamics simulations. The viscoelastic contribution to the stress tensor is obtained from the configurations of the individual molecules, as an average over the configurational space. This method requires much more computational power than traditional macroscopic methods, but it avoids the need of a closure approximation, and complex flows of molecular models without a macroscopic closed-form equivalent equation, such as FENE dumbbells or reptation models, can be calculated. In addition, information about the conformation of the molecules, as detailed as the corresponding molecular model employed, is readily available from these calculations. The original contribution to the combined use of molecular models in complex flows was called CONNFFESSIT (Calculation of Non-Newtonian Flows: Finite Elements and Stochastic Simulation Technique)^. It presented some computational disadvantages, such as the spatial fluctuations of the stress and the non-optimal spatial resolution, and has been improved during the last decade, mainly with two promising techniques: the Brownian configuration fields* and the backward-tracking Lagrangian particle method^. Micro-macro methods have been applied successfully in many 2D viscoelastic problems^^^"*, but we are not aware of any 3D computation using molecular models. In this work we want to show that it is possible with current available computational power to run typical 3D viscoelastic flow simulations using micro-macro methods. We will see that the well known 2D micro-macro techniques can be readily extended to three dimensions, and will present results of 3D 4:1:4 contraction flow using dumbbell models. The stress will not be placed on the resolution of these results. We realize that micro-macro methods cannot compete in terms of mesh refinement and spatial resolution with standard macroscopic methods, like finite volumes or finite elements. The importance of this work relies on the use of molecular models to solve complex 3D viscoelastic flow problems. In section 2 the equations that govern the problem are introduced. In section 3 the numerical and computational schemes are described and in section 4 the results for the steady-state creeping 4:1:4 contraction flow are presented. Finally, in section 5 some conclusions are extracted and some hints are presented as for the possible future directions along this line of research. 2. Governing equations The equations governing the conservation of momentum and mass for an incompressible, isothermal steady-state creeping viscoelastic flow are:

Micro-macro calculations of 3D viscoelastic flow

125

-;7,V.(Vw + Vw^) + V / 7 - V r = 0

(1)

where u is the velocity of the fluid, p is the pressure, r is the extra stress due to the polymer, and rj^ is the viscosity of the solvent. The extra stress tensor T can be obtained by means of a macroscopic constitutive equation or, as in the present case, using a molecular model. In this work, we have chosen the archetypal molecular model, the dumbbell^, which is the most frequently used model in benchmark tests. In this microscopic model, the conformation of molecules is represented by two massless beads connected by a spring, which corresponds to the molecule end-to-end vector Q. The evolution of the molecules is expressed as a diffusion equation for the probability distribution function \if[Q\r,t); but here, as it is common in micro-macro methods, we use an equivalence between diffusion equations and stochastic differential equations^^ which allows us to express the evolution of individual molecules as

dQ = {Vu)'.Q-^F[Q]

dt+

4k T

-

(2)

where ^ is the friction coefficient of the beads, kg is the Boltzmann constant, T is the absolute temperature, F{Q) is the intramolecular force (the force exerted by the connector spring) and ^ is a standard Wiener process. The extra stress T can be obtained by means of a Kramers expression T=

n{QF{Q))-nkJI

(3)

where n is the number density of dumbbells in the fluid. If the connector spring is Hookean, i.e. f{Q) = HQ, with //the spring constant, it can be easily shown that the molecular model is equivalent to an Oldroyd-B constitutive equation. If a slightly more complicated and realistic form is chosen for the intramolecular force by adding finite extensibility, such as the FENE model.

= HQ

(4)

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with QQ being the maximum allowed length of the connector, the molecular model does not have a macroscopic closed-form counterpart. Dumbbell models are appropriate to represent solutions that are sufficiently diluted such that the individual molecules do not interact with one another. In the remainder of the article, the connector spring is Hookean and the length of the connector vector of the dumbbells is scaled with yJkgT/H . 3. Numerical scheme We consider the steady-state, 3D 4:1 plane contraction flow in the computational domain Q. represented schematically in figure 1. We are interested mainly in the effects upstream of the contraction, and therefore the selected outlet channel length is short (Lo = 8 Dout)- In this complex flow problem, the fluid is exposed to shearing and extensional effects, and the contraction comer is a geometrical singularity. All the boundaries are walls, except for the inflow F/^ and outflow Tout sections. We have selected a width h of the same order of magnitude as the height of the entrance, in order to better capture 3D effects. Due to the symmetry of the domain, we only need to consider a quarter of it in the calculations.

D,

h

^

z

Figure 1: Domain for the contraction flow problem. Du, = 2, Du/Dout = 4, h = Du,. Li = 2 Du, and Lo — 8 Dout-

We are interested in the steady-state solution of equations (1-3). The time domain [0,/] is divided in Nr steps. Equations (1-3) will be integrated using a

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flow

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time-stepping scheme until convergence for the velocity and stress fields is accomplished. For the spatial discretisation of the system of equations in Q, we use the finite element method^^ with the help of the publicly available dealll library^^. The computational scheme is split in three steps.

Problem 1 (Generalized Stokes) The conservation equations (1) are solved by considering the extra stress field r" as a known volume force. The stress r" is calculated with equation (3) using the value of the molecular conformations Q" at the previous time step. The resuhing generalized Stokes problem reads: knowing r" e T, find [u"^\p""')e\5xV

such that

;7,(vr^'+(vr*^)\Vv)-(p''^\V.v)-(v.«''*\^) = (r",Vv)

(5)

where (,) is the standard scalar product in the domain Q, for all (v,^)€ UxP . In the present work, we use cuboid finite elements, and the approximating functional spaces are continuous piecewise triquadratic polynomials Q2 for the velocity space U and continuous piecewise trilinear functions Qi for the pressure space P and the extra stress space T. The following boundary conditions are prescribed for the velocity field: • No-slip at the walls. • Symmetry of the velocity field at the two symmetry planes (y=0 and 2=0).

•

Outflow conditions at r„„, ( u";' = wf^ = 0)

At the inflow F/^, a fully developed velocity profile w,„ is imposed. For the calculation of this profile, a steady-state Stokes problem is solved at Tm for a long rectilinear channel subjected to a constant pressure gradient, such as to obtain a given flow rate V . This velocity profile is known to be valid as a steady-state solution for the flow of a Newtonian or an Oldroyd-B fluid in a long channel. 3.LProblem2(BCF) The molecular configurations are represented by continuous, Eulerian fields^. Nf Brownian configuration fields (BCFs) are initialized from the equilibrium

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distribution of the chosen dumbbell model, with the same value for each individual field over the whole computational domain. Due to the advection term that appears in the Eulerian formulation of the stochastic differential equation for the configuration fields, we use the streamline upwind/Petrov Galerkin (SUPG) method^^ to stabilize the numerical scheme. Knowing the velocity field at step «+/, each BCF is updated. The finite element problem to solve for each Brownian field is: knowing w"*' G U , find Q"*' G Q such that

for all ^G Q, /=1, Nf, where ASUPG is a number of the order of the element size in the flow direction and A = f/4// is the relaxation time of the dumbbells. Note that equation (6) is the weak formulation of equation (2) after scaling the length of the dumbbells to make them non-dimensional and after adding a transport term. The elasticity of the flow is given by the Deborah number DQ = Z{U^)/D^^ , where (w,) is the average velocity in the outlet channel. The time discretization of the equation is implicit and the approximating functional space Q for the molecular configurations is the space of continuous piecewise trilinear polynomials (Qi). We need to prescribe boundary conditions at the inflow section T/^ for each of the configuration fields. At the boundary T/^, we solve Nj. transient 2D configuration field problems Q^^, having the same initial values as the BCFs Q. in the domain Q., subjected to the same random noise AlV. as these Q, and evolving under the prescribed inflow velocity profile ii^. The transient state of each one of these 2D fields Q^j is used as a Dirichlet boundary condition for the transient calculation of the corresponding 3D configuration field Q. An important computational aspect in micro-macro simulations is the reduction of the statistical error in the stochastic simulation. In this work, we use a simple method for variance reduction. We calculate the evolution of an ensemble of dumbbells Q in quiescent conditions, having the same initial state and being subjected to the same random noise AW. as the configuration fields Q.

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3.2, Problem 3 (Extra stress tensor) Knowing the value of the molecular configuration fields Q"^^ and the fields in quiescent conditions Q, the extra stress tensor is found by projection: find T^^^sT such that

(r-,a) =(nkj^t^ U^X^

(7)

for all aeT , Note again that equation (7) is the variational form of equation (3) after scaling the length of the dumbbells and substituting the identity tensor by the equilibrium stress tensor obtained as the average over the ensemble of molecules in quiescent conditions.

Figure 2: Computational mesh used in the calculations (not to scale). The origin of coordinates is set in the centre of the inflow section.

The computational mesh is depicted infigure2. The origin of coordinates is set at the center of the inflow section T/^. There are two planes of symmetry aty=0 and z=0 and, apart from the inflow jc=0 and outflow x=8, the rest of boundaries are walls. Due to the computational cost of the solution of a high number Ny of configuration fields at each time step, the level of refinement of the mesh is moderate (3879 elements). The system matrix of the generalized Stokes

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problem (Problem 1) is independent of time and is assembled only once at the beginning of the simulation. It is solved at each time step using a conjugate gradient algorithm and an incomplete LU decomposition as a preconditioner. The system matrix of the finite element problem 2 is independent of the field Q considered, and is thus assembled only once per time step. For each of the Nf configuration fields, we only need to assemble the right hand side vector of the problem and solve it. To be able to simulate a large number Nf of Brownian fields, this part of the problem has been parallelized in a cluster of A^c nodes (PCs) using MPI*^. Problem 1 is solved at one node only (the so called 'master node'), and the resulting velocity field is transferred to every other node in the cluster. Each particular node has its own copy of Nf/Nc configuration fields and handles their initialization and their evolution independently of the others. For the solution of the resulting algebraic system of each BCF (Problem 2), we use a preconditioned conjugate gradient algorithm. The mass matrix arising in the determination of the extra stress (Problem 3) is assembled only once at the beginning of the simulation by one of the nodes (master). Each node assembles its corresponding part of the right hand side of the problem, and transfers it to the master node who assembles the total right hand side and solves equations (7). 3.3. Preliminary convergence tests A series of calculations has been done in order to test the convergence of the presented method. The test case chosen is the startup of shear flow in a long 3D channel. The polymer is modeled by Hookean dumbbells with relaxation time A = 1 and viscosity 77^ = Xnk^T = 1, and the solvent viscosity is 7, = 1. The domain is a straight channel along the JC coordinate, with square section of side 1 (y,ze [0,1] X [0,1]). The lower plane >F=0 is held fixed, while the upper plane y='\ moves with velocity Ux=\, resulting in a shear rate y = l . The boundaries z=0 and z=\ are considered as symmetry boundaries and there is no flux of stress through them. The length of the channel is set such that the polymer, for the given shear rate, attains the steady state before reaching the outflow boundary. This equivalent ID problem for the Oldroyd-B constitutive equation can be solved analytically, and the solution can be compared to what is obtained with the proposed numerical scheme.

Micro-macro calculations of 3D viscoelastic flow

131 r

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'

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'

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OWroyd^ 1 Nr=ioo H NF=1000 NF=»10000lH

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Figure 3: Startup of steady shear flow (shear rate y = 1) for Hookean dumbbells (relaxation time X ^ 1, viscosity iip "" 1), and different number of Brownian fields Nf,-. (a) Shear viscosity as a function of time, (b) First normal stress coefficient as a function of time.

In figure 3, the results for the transient viscosity of the polymer, calculated as rip=t^lY, and the transient first normal stress coefficient, ¥\ ={^xx~^yy)/y^'

^® presented. The smooth line represents the analytical

solution of the Oldroyd-B constitutive equation, and the different rather noisy lines represent the numerical value calculated by the 3D micro-macro method at the middle point on the outflow boundary (y=z=0.5), for different number of Brownian fields, Nf = 100, 1000, 10000. It can be observed that the numerical results converge to the analytical solution with the increasing value of A^/.. A calculation of the transient error in the values of rj^^ and y/^^ obtained from the numerical solution indicates an error that scales with A^^"^^^, as expected for a statistical average. 4. Results and discussion Two runs of Hookean dumbbells in a Newtonian solvent at a fixed flow rate F = 0.2 have been carried out, with relaxation times A = 0.1 (De=0.36) and A = 0.2 (De=0.72). The polymer viscosity is AnkBT = OMS and the ratio of solvent to polymer viscosity is 1/9. The chosen number of BCFs is 7V^=1000 and the number of parallel nodes is Nc=20, The time step is A^ = A/10 and the simulations have been run to a total time of t = 30>1. After analysis of the transient values of the stress tensor, it has been observed that the steady-state is completely attained at r = lOA. From that moment on, solutions are saved each At = Z and are used later to calculate averages. The total number of degrees of freedom of the discretized problem is about 14.5 million (14.4 million

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accounting only for molecular degrees offreedom).The total simulation time is about 50 hours on a cluster of Pentium II PCs at 550 MHz. Resulting fields from both simulations do not differ much, and only results from the second simulation (De=0.72) are shown in this section. 4.1. Velocity field

0.75

Figure 4: Some groups of streamlines for the Hookean dumbbells fluid through the plane 4 : 1 three-dimensional contraction (De ^ 0.72). Streamlines in the symmetry plane (z ^ 0, ), in a parallel plane in the middle of the domain (z ^ 0.5, ) and in a parallel plane closer to the side wall (z = 0.99, ). Starting points for the trajectories have vertical coordinates ranging from >^-0toj-0.95.

The resulting velocity field, averaged over 20 steps after the steady-state is attained, shows some noise. In figures 4-6 some representative particle trajectories of the flow under study are shown in different views. In figure (4), three groups of streamlines started at the inflow boundary and having different coordinate z are depicted. The first group is in the symmetry plane (z=0). All trajectories belonging to this group remain plane until they exit the domain. The observed vortex attachment length is slightly smaller than other 2D plane contraction calculations reported in the literature^°'^°, which is reasonable because the 3D geometry affects the kinematics of the flow. This result suggests that the De numbers of 3D simulations need to be scaled appropriately in order

Micro-macro calculations of 3D viscoelastic flow

133

to compare the results at the symmetry plane z=0 with those obtained in similar plane 2D calculations. For the simulated Deborah numbers, from 0 to 0.72, the trend of the symmetry plane vortex is to reduce its size, in agreement with previous reported results. A second group of streamlines having z=0.5 shows evidence of the 3D nature of the flow. Particles of fluid are not able to find their way through the contraction and are displaced to the side wall by the fluid that is moving closer to the center of the channel. These particles are slowed down and find their way out of the domain at a position closer to the side wall z=l. It must be reminded that most of the fluid is transported around the symmetry axis, which represents the center of the channel. As there is no place in the central part of the contraction channel to allow for the passage of so much fluid, it tries to find its way by moving towards the side wall, pushing the rest of the liquid. The third group of streamlines, started at a position much closer to the side wall, at z=0.99, take much more time to exit the domain. They almost remain plane, but mostly because they are so close to the side wall that they cannot be displaced any more. If the streamlines in figure (4) are studied with more detail, it can also be observed that some of those in the symmetry plane seem to end at the contraction wall. This is an artifact of the numerical simulation, due to the coarse mesh used in the calculations and to the noise of stochastic origin. The resolution of the streamlines in the areas of very low velocity is thus low. (a)

(b)

0

O.S z

1

Figure 5: Side view (a) and top view (b) of some trajectories of particles started at lines parallel to the upper wall of the domain {y=\). The trajectories are started at>' = 0.5 ( ) and at j = 0.9 (— ) and z ranges from 0 to 0.9.

In figure 5, two different groups of streamlines started at the inflow at different y are shown in two different views. In the side view (see figure 5(a)), the trajectories do not show any particular behaviour, except for the slight

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oscillations that can be seen near the outflow boundary, which are of stochastic origin. The outflow boundary conditions have been imposed to the velocity too soon after the contraction, and this affects the dymamics of the dumbbells, which in turn make the velocity field more noisy. A longer outlet channel will probably reduce these oscillations. In the top view (see figure 5(b)), it is even more evident how the molecules turn aside from a planar trajectory. The closer the particles are to the upper wall (y=l), the more they are pushed away to the side wall by the fluid moving in the vicinity of the symmetry axis x of the domain. The closer the trajectories are to the side wall (z=l), the weaker is the deviation from the planar trajectory. Those particles in the vicinity of the salient comer follow even more complicated streamlines, as it can be seen in figure (6). They move along helical trajectories from the symmetry plane (z=0) to the side wall (2=1). This effect observed in the micro-macro simulations is in qualitative agreement with previously published results21,22

0.75 h

0.25

Figure 6: Perspective view of some complex three-dimensional helical streamlines appearing close to the salient comer.

4.2. Extra stress field

135

Micro-macro calculations of 3D viscoelastic flow

Txy

-2.50

^2,00

'ISO

'f.OO

-0.500

0.000

Figure 7: Contours of the component Xxy of the extra-stress tensor on some planes (z =- 0, z ^ 0.99) of the 4 : 1 three-dimensional contraction.

Some of the components of the extra-stress tensor are shown in figures (7) and (8). In figure (7), the component r^ of the stress tensor is represented at the two different planes of the domain (z=0 and z=0.99). The largest values of the stress component r^ occur at the contraction channel, close to the upper wall, where the flow is dominated by shear flow (the largest component of the strain rate tensor is y^). But it is evident, from the figure, that the values of T^ are different at the symmetry plane (z=0) and close to the side wall (z=0.99). It is clear that the no-slip condition prescribed at the side wall affects y^ in the surrounding region of the domain, reducing its value significantly. Nevertheless, the value of the components of the velocity gradient d^u^ and d^u^ is not negligible at the walls, which makes the shear stress grow along the contraction channel. In general, the values of r^ at the symmetry plane z=0 are higher than those obtained in the corresponding 2D simulation at a similar De numbei^^

136

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(a)

Figure 8: (a) Stress tensor component TXX- (b) First normal stress difference Nj ^ TXX - Xyy . The planes selected to represent the stress components are the same as in the previous figure.

The same is true for the normal stress r^x» which is represented in figure 8 (a). On the symmetry axis x, around the contraction, the flow is purely extensional and the component of the stress r^x increases steeply when the flow traverses the contraction along this line. The effect is not easily observable because the molecular relaxation time is short compared with the characteristic time of the flow, and this makes the Qx component of the molecules relax very fast after the contraction, and so r^x relaxes also. The effect is also masked because the stress component z^x is affected by the large shear along the walls in the contraction channel (Yxy along the upper wall (y=0.25) and a non-negligible value of y^ along the side wall (z=l)). The shear flow increases the value of Txx at the walls, as it can be seen in the figure, masking the increase in the same normal component due to the extensional flow. This is a well known feature of contraction flows, that they are not able to stretch the molecules by extension, but mostly by shear. The three-dimensional character of the flow can be better perceived by considering the first normal stress difference N\. The effect of the side walls can be observed in the value of N\ in figure 8(b). The contour plot representation is very similar to figure 8(a) in the symmetry plane, but in the plane closer to the side wall, the difference in the value of N\ is emphasized by the fact that the stress component r^ at the side wall, in the contraction channel, has a very small value. From the stress component r^^ and the first normal stress difference N\, we can plot the approximate isochromatic isolines, which are frequently compared to

Micro-macro calculations of 3D viscoelastic flow

137

the isochromatic fringe patterns measured experimentally. In a plane problem, the isochromatic lines correspond to the stress state according to^^

(4<^Ar,^f=^,

^ = 1,2,3,.

(8)

where ^^ is the light wavelength, C is the stress optical coefficient and h is the width of the channel.

Figure 9: Detail of the calculated isochromatic isolines according to equation (8), in a plane perpendicular to the z direction, close to the contraction, at positions 2 = 0 and z = 0.99.

In figure 9, as there are no experimental data to compare with, we have represented the fictitious isolines of equation (8), considering ^jhC = 1, in two different planes. The three-dimensional effect is even more evident. While in the symmetry plane, the isolines close to the contraction are slightly butterfly shaped, in the parallel plane closer to the wall (at z=0.99, see figure 9(b)), the structure is completely different. The no-slip conditions at the lateral walls reduce both the shear rate {yx component) and the extensional flow {pcx component), affecting the shape of the isolines. A real measured birefringence pattern would integrate the whole domain, so it is important to take into account the 3D geometry when comparing the simulations with the isochromatic lines obtainedfromthe experiments. 4.3. Molecular configurations One of the advantages of micro-macro simulations over traditional methods is that the configurations of the molecules can be collected all over the domain. For ease of visualization, we have represented some populations of molecules, as clouds of points, at three selected representative positions on the symmetry plane z=0 (figure 10). Each point in the population diagrams represents the configuration of a single dumbbell in terms of the coordinates of the connector

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vector Q. Only the Qx and Qy components are represented in the populations. The third component of the connector vector, Q^, is permanently at equilibrium in that plane. The population of molecules at the area close to the salient comer (figure 10 (a)), with spherical symmetry of the components Qx and Qy, is nearly at equilibrium. The low shear rates and zero extension rate to which the molecules are subjected in that area allow them to relax and recover the equilibrium state. When the molecules approach the reentrant comer, they experience some shear rate in the yx component, which tends to stretch and orient them, as can be seen in the second population depicted in figure 10 (b). After the molecules traverse the contraction, they are subjected to a strong shear flow (large value of y^), and hence they stretch and orient in the direction shown in the third distribution of points represented in figure 10 (c). The same plots could be represented on the symmetry plane >'=0. Components Qy and Qz would exchange their roles on this second plane, but the extension of the Qx component of the connector vector of the molecules and the orientation of the molecules on the xz plane, at the contraction and along the contraction channel, would be lower because the contraction ratio along the z direction is 1:1.

0.25

t

1

Figure 10: Molecular configurations (Qx and Qy components) at three selected positions on the symmetry plane of the 4 :1 contraction. From left to right: (a) salient comer, (b) re-entrant comer, and (c) contraction channel.

139

Micro-macro calculations of 3D viscoelastic flow

(a)

(b)

Figure 11: Perspective view(a) and top view(b) of an ensemble of steady-state trajectories initiated at the same point.

One of the evident effects of the stochastic part of the micro-macro methods in the solution is the addition of statistical noise to the macroscopic fields u,p and r . As an illustrative example, we have plotted in figure 11 an ensemble of trajectories which have been started at the same points in the inflow boundary, at different time steps starting at / = lOX and at time intervals of At = X. The problem is considered to be at the steady-state at / > lOX, and each one of the streamlines (calculated by integrating the velocity field) is representative of the steady-state velocity field of the problem. Due to the finite number of Brownian fields used to represent the molecules, the stress field presents temporal fluctuations which affect the velocity field through the momentum conservation equation (1). The result is a statistical representation of the velocity field in terms of an average value (whose streamlines are represented in figure 4) and a statistical error bar. In figure 11, we can get an idea of the shape of this error bars. The streamlines should be considered as statistical tubes surrounding the trajectories obtained at different time steps after the steady-state. In order to increase the level of certainty about the trajectories, the radius of those tubes should be reduced by increasing the number of molecules present in the problem or by the use of a more refined variance reduction method^"*'^^. 5. Conclusions Micro-macro simulations have been run for the first time for the calculation of the flow through a three dimensional plane 4:1 contraction. The computational method is essentially the same as for lower dimensional simulations, but special care must be taken of memory and simulation time due to the size of the

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problem. Although the level of refinement of the finite element mesh and the degree of elasticity of the flow are moderate, results are in good agreement with previous literature. Additionally, information about the state of the molecules can be gathered from these simulations. Work is being done to extend the simulations to more elaborated problems and models. It would be interesting to see how micro-macro methods compare with the experiments in 3D situations. Also, more realistic molecular models should be used if we want to simulate concentrated solutions or melts. The Brownian fields method^ is no longer appropriate for molecular models that don't admit variance reduction methods, like the reptation model proposed in^^. In those cases, the backwards tracking Lagrangian particle method^ seems to be the more appropriate. References 1. Rasmussen H K 1999 J. Non-Newton. Fluid Meek 84 217 2. Bogaerds A C B, Verbeeten W M H, Peters G W M and Baaijens F P T 1999 Comput. Methods Appl Meek Eng. 180 413 3. Mitsoulis E 1999 Comput. Methods Appl. Meeh. Eng. 180 333 4. Schoonen J F M, Swartjes F H M, PetersGWM, Baaijens F P T and Meijer H E H 1998 J. Non-Newton. Fluid Meeh. 79 529 5. Xue S C, Phan-Thien N and Tanner R11999 J. Non-Newton. Fluid Meeh. 87 337 6. Bird R B, Curtiss C F, Armstrong R C and Hassager O 1987 Dynamies ofpolymerie liquids Kinetie Theory vol 2 (New York: Wiley) 7. Laso M and Ottinger H C 1993 J. Non-Newton. Fluid Meeh. 47 1 8. Hulsen M A, van Heel A P G and van den Brule B H A A 1997 J. Non-Newton. Fluid Meeh. 70 79 9. Wapperom P, Keunings R and Legat V 20007. Non-Newton. Fluid Meeh. 91 273 10. Feigl K, Laso M and Ottinger H C 1995 Maeromoleeules 28 3261 11. Hua C C and Schieber J D 1998 J. Rheol. 42 477 12. van Heel A P G, Hulsen M A and van den Brule B H A A 1999 J. Rheol. 43 1239 13. Cormenzana J, Ledda A, Laso M and Debbaut B 2001 J. Rheol. 45 237 14. Fan X J, Phan-Thien N and Zheng R 1999 J. Non-Newton. Fluid Meeh. 84 257 15. Ottinger H C 1996 Stoehastie Proeesses in Pofymerie Fluids (Berlin: Springer) 16. Crochet M J, Davies A R and Walters K 1984 Numerieal Simulation of Non-Newtonian Flow (New York: Elsevier) 17. Bangerth W, Hartmann R and Kanschat G deal.U Differential Equations Analysis Library Teehnieal Referenee IWR, http://www.dealii.org 18. Brooks A N and Hughes T J R 1982 Comput. Methods Appl. Meeh. Eng. 32 199 19. GroppW, Lusk E and Skjellum A 1999 Using MPI—Portable Parallel Programming with the Message-Passing Interface 2nd edn (London: MIT Press) 20. Phillips T N and Williams A J 1999 J. Non-Newton. Fluid Meeh. 87 215 21. Mompean G and Deville M 1997 J. Non-Newton. Fluid Meeh. 72 253 22. Mompean G and Deville M 2002 J. Non-Newton. Fluid Meeh. 103 271 23. Baaijens F P T, Selen S H A, Baaijens H P W, Peters G W M and Meijer H E H 1997 J. Non-Newton. Fluid Meeh. 68 173

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24. Melchior M and Ottinger H C 1996 J. Chem. Phys. 105 3316 25. Bonvin J and Picasso M 1999 J. Non-Newton. Fluid Meek 84 191 26. Fang J, KrOger M and Ottinger H C 2000 J. Rheol 44 1293

Multiscale Modelling of Polymer Properties M. Laso, E.A. Perpete (Editors) © 2006 Elsevier B.V. All rights reserved.

143

Chapter 7

The derivation of size parameters for the SAFT-VR equation of state from quantum mechanical calculations T. J. Sheldon", B. Giner'^ C. S. Adjiman", A. GaHndo', G. Jackson\ D. Jacquemin^, V. Wathelef, E. A. Perpete^ ^ Centre for Process Systems Engineering, Dept. of Chemical Engineering, Imperial College London, South Kensington Campus, London SW7 2AZ, UK * Departamento de Quimica Organica-Quimica Fisica, Facultad de Ciencias, Universidad de Zaragoza, 50009 Zaragoza Spain. ^Laboratoire de Chimie Theorique Appliquee, F.U.N.D.P., Rue de Bruxelles, 61, 5000 Namur, Belgium

1. Introduction Equations of state play an important role in the development of efficient production processes. During the course of a process simulation or optimisation, thousands of evaluations of thermophysical properties under different operating conditions are required. Computationally inexpensive models such as equations of state, which can be used over a wide range of temperatures, pressures and compositions, are essential for this activity. For complex systems such as polymers and hydrogen-bonding compounds, molecular equations of state such as SAFT (statistical associating fluid theory) and its variants [1,2,3] are particularly well suited. However, such equations require a minimum of three parameters for each pure compound to be modelled, and additional parameters for mixtures. Their use is thus dependent upon the availability of reliable parameter values for the compounds of interest. SAFT parameters for pure components are usually obtained using phase equilibrium data. In order to obtain statistically significant values of the

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parameters, a large number of experimental data points are required. Furthermore, it is desirable to include different types of data, such as saturated vapour pressures and saturated liquid densities to allow the resolution of different parameters: the density is known to be most sensitive to size parameters, while the vapour pressure depends most strongly on energy parameters. Even when a large and varied data set is available, some of the parameter values remain difficult to determine with precision. In SAFT-like equations, this is the case of the chain-length (aspect ratio) parameter, m, which is often fixed a priori based on physical arguments, rather than fitted to experimental data [e.g.,4,5]. When using SAFT with variable range (SAFTVR), the identification of the range parameter can be made more reliable by including additional data such as the speed of sound [6]. Finally, when association is present, it can be difficult to partition the attractive interactions between the dispersive and associating terms and data such as the fraction of bonded molecules can be useful [7]. In practice, there are few compounds for which extensive experimental data are available. It is therefore desirable to look for alternative approaches which provide reliable parameter values while reducing the dependence on experimental data. In their recent review of challenges in thermodynamics, Arlt et aL [8] identify the combined use of quantum mechanics and equations of state as one of the key developments for thermodynamic modelling. Several efforts have been made in this direction, since the pioneering work of Wolbach and Sandler [9]. They proposed a mapping to derive a relationship between the association parameters for the original SAFT equation [1,2] using enthalpy, entropy and heat capacity changes for dimerisation reactions, calculated using Hartree-Fock or density functional theory ab initio quantum mechanics. They also derived a relationship between the two SAFT size parameters and the molar volume calculated using quantum mechanical calculations. This procedure thus avoids having to fit two of the SAFT parameters to experimental data. The remaining parameters were fitted to VLE data (vapour pressures and liquid densities). This approach was tested on water, methanol and three acids, giving good agreement with experimental data. The SAFT parameters thus derived were also used successfully to model mixtures, using mixture parameters fitted to experimental data [10]. Yarrison and Chapman [11] used the same association parameters in other variants of the SAFT equation of state, and obtained good results, particularly with the Hartree-Fock based parameters. In related work Fermeglia and Pricl [12] derived parameters for a reformulated version of the Perturbed Hard Sphere Chain Theory (PHSCT) equation of state [13] from semi-empirical quantum mechanical calculations and molecular mechanics. They derived the surface and volume parameters by applying a Connolly surface algorithm to molecular geometries generated using the AMI

SAFT- VR parameters from quantum mechanics

145

semi-empirical method. The energy parameter was mapped onto a ratio of energies calculated from molecular dynamics (MD) simulations in the NPT ensemble. The approach was successfully applied to chlorofluorohydrocarbons. Fermeglia and Pricl [14] also proposed a method based entirely on MD simulations, in which they obtained parameters for the PHSCT equation and for the Sanchez-Lacombe equation [15] to model the PVT behaviour of four polymers. In a later study they tackled the prediction of mixture behaviour by deriving binary interaction parameters from MD simulations or by fitting these parameters to activity coefficient data [16] predicted by COSMO-RS, which is itself entirely based on quantum mechanical calculations [17]. For the systems studied (mostly alcohols and aromatics), they found that, of the two approaches, the binary interaction parameters derived from MD simulations gave the best results. However, they also found that COSMO-RS usually gave comparable or better results. This, and the reliance of the results of any MD-based approach on the availability of a good intermolecular potential model, suggests that methodologies based on quantum mechanics may offer a more broadly applicable route to equation of state parameters. In this work, we focus on deriving the size parameters of the SAFT equation of state ab initio, making sure that the approach is applicable to different types of compounds. In the next section, we present briefly some background information on the SAFT-VR equation of state [18,19] used here. We then present the methodology for the derivation of the parameters. Finally, we apply it to a test family of pure compounds and comment on the representation of the VLE thus obtained. 2. The SAFT-VR Equation of State 2.1. General background Over the last 50 years there has been a great deal of effort in developing equations of state that can be used to describe the thermodynamics and bulk phase equilibria of fluids and fluid mixtures. Analytical theories can now provide a quantitative description of bulk fluids comprising molecules with complex interactions, such as associating systems, amphiphiles, polymers, and electrolytes (see the excellent reviews in the collection by Sengers et al [20]). A successful modem equation of state for complex fluids is the statistical associating fluid theory (SAFT) [1,2] The SAFT approach has been rapidly superseding better-established empirical equations of state. The basis of the SAFT description is Wertheim's first-order perturbation theory (TPTl) for associating systems [21,22,23,24]. In its numerous incarnations the SAFT approach has been shown to be very versatile in describing the fluid phase

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behavior of systems ranging from small strongly associating molecules such as water, to long-chain alkanes, polymers, and electrolytes (see the comprehensive reviews by Miiller and Gubbins [3,25]). The most recent versions of the SAFT EOS include the soft-SAFT [26,27], the variable-range SAFT-VR [18,19], and the perturbed-chain PC-SAFT [28,29] and simplified PC-SAFT [30,31] descriptions. These approaches differ in the way the attractive intersegment interactions are treated and in the choice of reference fluid. The soft-SAFT approach is based on segments interacting through a Lennard-Jones potential [32] with a hard-sphere fluid as the reference in the perturbation theory. In the PC-SAFT EOS a hard-sphere chain fluid is used as the reference instead of a hard-sphere fluid. The attractive term is empirically modified to fit the experimental vapour pressures and saturated densities of the «-alkanes, which leads to an excellent description of the thermodyamics of chain molecules. The SAFT-VR equation of state describes a fluid of associating chain molecules with the segments of the chain interacting through attractive interactions of variable range (square-well, Sutherland, Yukawa, and Lennard-Jones potentials have all been examined); a reference system of attracting monomers is used to build up the chain and the associative contributions to thefi*eeenergy following the TPTl theory of Wertheim. The main advantage of the SAFT-VR description is that one explicitly includes the range of the intermolecular potential, which allows for the nonconformal nature of the interactions between molecules to be taken into account. The variable range is particularly useful in describing polar molecules such as refrigerants or polyelectrolytes. 2.2. The SAFT'VR molecular model and parameters In the SAFT-VR equation of state, any given molecule is represented as a chain of tangentially bonded spherical segments of equal diameter, with association sites if necessary. The dispersive interactions between segments can be represented by different potentials with variable range. The square well potential is used here, as it has been shown to be both versatile and accurate through application to a large set of compounds (the references have been collected recently in a paper by Gloor et al. [33]). In this model, the simplest compounds are represented by four parameters, illustrated in Figure 1: m, the chain length (aspect ratio); tr, the segment diameter; s, the depth of the attractive well, and X which characterises the range of the attractive interactions.

Ul

SAFT-VR parameters from quantum mechanics

u(r)

0

o

Xa

Figure 1. SAFT-VR model for a non-associating compound, consisting of w segments of diameter cr, interacting via a square-well potential u(i) (shown on the right), where r denotes the intersegment distance. The potential u(r) has a depth s and a range Xa

When association is modelled, several additional parameters are introduced. The type of association model must first be specified by choosing the number of site types and, within each site type, the number of sites. These determine what kinds of aggregates (e.g., dimers, chains) can form. Then, for each pair of site types, two parameters must be set: the strength of the interaction, ^^, and its range r"^. Usually, association is only allowed to take place between sites of different type. A typical model molecule with two sites of different type is shown in Figure 2. ^^(rss)

0 fss

r

cr

re G

Figure 2. SAFT-VR model of an associating compound consisting of three segments of equal diameter cr interacting via a square-well potential and with two types of site (one site of each type). The sites interact through a square-well potential li^^ir^s) as shown on the right, where r^s is the intersite distance. The potential ii^^(rss) has a depth d^^ and a range TCG

148

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3. From Quantum Mechanics to the SAFT-VR m and a Parameters 3.1. Overview of methodology In this work, we obtain size parameters for SAFT-VR (m and d) from quantum mechanical information on the dimensions of the molecule. This requires a mapping to translate the complex QM representation of a molecule into the bonded-spheres representation of SAFT-VR. For this purpose, we treat every molecule as a spherocylinder of diameter crand with aspect ratio m (Figure 3). The choice of an enveloping spherocylinder is based on the observation that it is the convex hull of a linear chain of bonded spheres, and therefore a good approximation of the effective volume of such a chain, and that its volume and aspect ratio are readily calculated even for a non-integer number of spheres. The aspect ratio of our convex model (a hard cylinder of diameter G and length L capped by two hemispherical caps of diameter o) is Llc&\, One can obtain a relationship between the aspect ratio of the hard spherocylinder (L/cH-1) and the aspect ratio of a linear chain of m spherical segments rigorously when the segments are bonded tangentially (i.e., when m is an integer): L/o+l = m. In the case of chains of fused hard-sphere segments the aspect ratio of the enveloping convex hard spherocylinder could be retaled to a non-interger value of m through the corresponding expressions for the second virial coefficient [34,35,36,37]; we choose not to use these more complex (albeit more rigorous) mapping functions here, and opt for the simpler linear dependence of the aspect ratio with the chain length m as in the case of the system of tangent spheres. Several quantities could be calculated from quantum mechanics to obtain the equivalent spherocylinder dimensions. For instance, one could use the volume and surface of a molecule, or its volume and aspect ratio. Molecular surfaces are particularly difficult to compute consistently, and we have therefore opted for volume and aspect ratio. We then apply the following strategy 1. Using QM, calculate the volume FgMof a molecule, and the dimensions of the smallest box containing the molecule Li
(1)

SAFT-VR parameters from quantum mechanics

149

(2)

3. Obtain the remaining SAFT-VR parameters by fitting to experimental data, such as saturated vapour pressures and liquid densities.

c

.^

.K

^K

.>x

moFigure 3: A spherocyUnder with diameter crand aspect ratio or number of spheres m = 4.24

In the remainder of this section, we detail steps 1 and 3 of the methodology. 3.2. Quantum mechanical calculations 3.2.1. General methodology In the last decade, the determination of molecular wavefunctions has become more and more sophisticated with the increasing availability of CPU resources. This often requires the calculation of determinants built from orbitals that are usually obtained from linear combinations of atomic basis functions. In the Hartree-Fock (HF) framework, increasingly large and flexible basis sets, from the minimal STO-3G to the Pople series (including polarised and diffuse functions) or Dunning's correlation-consistent suite, are bringing our capabilities closer to the theoretical HF limit. This results in more and more accurate results, provided linear dependencies are not an issue. The wavefunction completely characterises the system, but does not immediately provide any of its physical and chemical properties, nor the features of the electronic distribution. The application of appropriate operators is needed to obtain these observables [38]. In the statistical interpretation of this treatment, the square of the modulus of the wavefunction is associated with a probability. The multiplication by the number of electrons in the system, followed by the integration over the spin coordinate and all but one of the space coordinates, yields the probability density p(f) of finding an electron (regardless of its spin) in a given infinitesimal volume at position r. In terms of a basis of A'atomic functions, this can be written as

150

p(r) = 2'ZimjMr)fj(r)

T.J. Sheldon etal

(3)

where ^ r ) denotes the wavefunction at position r, (f denotes the complex conjugate of ^, and the elements of the density matrix D are given by the product of the Linear Combination of Atomic Orbitals (LCAO) coefficients C over the doubly occupied molecular levels: As a result, given a threshold on the electronic density, the molecular volume can be obtained from the LCAO coefficients by a straightforward spatial integration. Care must be taken to perform a sufficiently accurate integration. For instance, the number of sampling points is important if a Monte-Carlo integration technique is employed. If a classical Newtonian integration is performed, the resolution of the underlying grid must be chosen appropriately. We use the latter technique because it immediately returns the dimensions of the smallest box enclosing the molecule, in addition to the volumetric figures. 3.2.2. Calculation of volume and aspect ratio The GaussianOS [39] suite of programs is used to compute the electronic density over a three-dimensional cartesian grid, using the lOP feature of Gaussian03 which gives full control over the number of points and step size along each direction. The box used must be sufficiently large to inscribe the molecule. Considering each of the grid points as the centre of a small cube, a cube is said to be occupied if the electronic density at the corresponding grid point is greater than a given threshold. The total molecular volume can easily be recovered by a discrete summation of the infinitesimal volumes of the occupied cubes. The longest distance between two occupied cubes gives the longest dimension of the molecule. The smallest cartesian box containing all occupied cubes gives the other two dimensions. 3.2.3. Tuning of QMparameters to n-alkanes In order to produce reliable parameters to be transferred into the SAFT methodology, ethane and n-octane are used as a benchmark to adjust the different computational parameters involved in the QM approach. For different choices of parameters, the QM information is used in the overall methodology as described in Section 3.1 and the calculated vapour-liquid equilibrium is then compared to experimental data. A detailed study of the basis set and level of approximation is also undertaken. To obtain a good balance between CPU requirements and accuracy, the 6-311+G* basis set, a Pople type split-valence basis set with the addition of one set of diffuse functions and one set of polarization functions, is selected in the Restricted Hartee-Fock (RHF) formalism. The threshold on the molecular electronic density is fixed to 10'^^

SAFT- VR parameters from quantum mechanics

151

|^|/bohr^. This level of approximation is used for all of the compounds studied, regardless of their size, shape or electronic structure. 3.3. Fitting to experimental data In step 3 of the methodology, we gather vapour-liquid equilibrium data for the pure compound of interest. To obtain consistent results, we base our work on saturated vapour pressures {P^ and liquid densities (//) as a function of temperature. Equations of state do not usually represent both the subcritical and critical regions well with a single set of parameters. Here, we focus on the subcritical region, and use data at temperatures below 90% of the critical temperature. We use a relative least-squares objective function, giving equal weight to every data point. We thus solve the following problem (4)

mm — e N

y

exp,/

where N is the total number of experimental data points; Np is the number of vapour pressure data points; Np is the number of liquid density data points; 6 is the vector of parameters to be estimated (s, A, and association parameters if sites are present); P\Ti\6) is the saturated vapour pressure at temperature T, calculated using SAFT-VR, with m and eras obtained in step 2; f^(Ti;0) is the corresponding saturated liquid density; and the subscript "exp,/" denotes the measured value at temperature T/. To facilitate comparison with other SAFT models of the same compounds, the quality of fit is also assessed a posteriori by calculating the average absolute percentage error, AAPE, given by AAPE = ±f^\^^'^^'^^PI X100% ^

'=^

(5)

P/,exp

4. Computational Results A diverse set of compounds for which data are readily available is selected to test the proposed methodology. It includes w-alkanes of varying size, to ensure that non-sphericity is captured well, some small compounds covering a range of polarities (N2, CO, CO2), basic aromatic and cyclic compounds (benzene and cyclohexane), some associating compounds and highly polar compounds, including water and two refrigerants (trifluoromethane, also known as R23, and 1,1,1,2-tetrafluoroethane, also known as R134a). The methodology is applied to

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each compound and the resuhs are compared with "standard" SAFT-VR models obtained by fitting to experimental data only. All experimental data are obtained from DECHEMA's DETHERM accessed through the UK Chemical Database Service [40]. In the case of CO, benzene, cyclohexane and the refrigerants, the standard models are obtained by fitting all parameters to the data. In the case of «-alkanes and water, the aspect ratio m is fixed based on physical arguments, as has been done in the past (e.g., [5]) and all other parameters are fitted to experimental data. Similarly, the value of m used for N2 and CO2 is consistent with their bond lengths and Pauli radii. The structural information generated from the QM calculations is given in Table 1. In principle one should recover a TABLE 1. Dimensions (Ly, L2, L3) of the minimal Cartesian box containing a given molecule and molecular volume VQM computed at the RHF/6-3111+G* level with a 10'^^ |e|/bohr^ electronic density threshold. I, (A)

L,(A)

Ls(A)

VQM(A')

CH4

3.75

3.75

193

27.0

C2H6

3.85

4.49

5.19

44.1

C3H8

3.93

4.33

6.45

61.0

C4H10

3.90

4.42

7.77

78.1

C5H12

3.82

4.47

8.94

95.7

CeHn

3,94

4.42

10.37

112.1

C7H16

3.84

4.42

11.52

129.1

CgHig

3.83

4.43

12.81

146.1

C10H22

3.91

4.54

15.39

180.1

C12H26

3.95

4.48

17.94

214.2

C14H30

3.84

4.52

20.42

248.2

N2

3.03

3.03

4.12

23.8

CO

3.01

3.01

4.25

24.1

CeHe

3.45

6.26

6.91

83.4

C6H12

5.18

6.42

6.87

101.0

CO2

2.97

2.97

5.03

30.7

H2O

3.10

3.10

3.40

18.2

R23

3.91

4.62

4.88

39.7

R134a

4.81

4.90

6.18

60.9

Molecule

box of equal dimensions (cube) in the case of molecules with tetrahedral symmetry such as methane. It is gratifying that even though no assumption

SAFT-VR parameters from quantum mechanics

153

about the molecular symmetry is made in our general approach, we obtain an aspect ratio of close to 1 for methane. The value of the objective function (Equation (4)) and the average absolute percentage error (Equation (5)) are reported in Table 2 for the QM/SAFT and standard models. The SAFT-VR parameters obtained using QM information are presented in Table 3, and the standard model parameters are presented in Table 4. TABLE 2. Objective function value, average absolute percentage errors (AAPE) for vapour pressure and liquid density data used in fitting, and average absolute percentage errors for all available data points. QM-based model AAPE fitted points

Standard model

AAPE all points pa

AAPE fitted points

AAPE all points

Obj.

pa

CH4

0.0018

0.72

6.13

0.90

8.55

0.0165

1.17

3.84

0.99

8.54

C2H6

0.0047

5.05

2.46

5.95

14.97

0.0058

4.78

1.50

5.95

15.52

C3H8

0.0055

6.11

0.97

7.19

7.10

0.0058

5.84

1.90

6.74

7.20

C4H10

0.0071

5.50

2.14

6.19

5.36

0.0108

7.11

3.52

7.12

6.11

C5H12

0.0044

4.96

0.91

6.12

3.35

0.0048

4.92

2.10

5.81

4.08

QHH

0.0014

2.54

0.53

3.61

1.09

0.0033

4.65

1.95

4.69

2.36

C7H16

0.0017

2.96

1.72

3.98

6.54

0.0027

3.76

1.83

4.21

6.67

CgHig

0.0012

2.65

0.60

3.49

12.04

0.0028

4.79

0.84

4.89

12.94

C10H22

0.0022

3.42

0.48

3.73

1.77

0.0038

5.11

1.26

5.15

2.36

C12H26

0.0034

3.54

—

3.75

—

0.0053

5.86

—

5.83

—

C14H30

0.0012

2.53

—

2.53

—

0.0103

8.95

—

8.95

—

N2

0.0003

0.64

1.46

0.69

4.35

0.0003

0.46

1.52

0.47

3.54

CO

0.0014

1.83

3.10

2.00

5.44

0.0001

0.73

0.41

0.82

2.26

C6H6

0.0060

2.04

7.36

3.35

7.36

0.0001

0.57

0.68

0.57

0.68

^6^12

0.0020

1.54

3.89

2.32

3.79

0.0003

0.39

1.30

0.47

1.33

CO2

0.0024

1.03

4.53

1.86

9.46

0.0003

0.61

1.18

1.17

10.94

H2O

0.0005

1.10

1.45

1.12

7.30

0.0002

0.46

0.91

1.63

5.85

R23

0.0052

3.54

5.19

5.40

5.11

0.0011

0.95

0.91

0.97

2.83

R134a

0.0048

2.13

6.29

2.81

5.93

0.0262

0.46

0.48

0.51

1.56

Molecule

/

/

Obj.

pa

/

pa

//

154

T.J. Sheldon et al.

TABLE 3. SAFT-VR parameters obtained with the proposed QM/SAFT methodology. The model for water is a four-site model, that for R23 is a three-site model and that for Rl34a is a two-site model. In all cases, two types of sites are used.

m

a/A

(€/k)/K

I

(^^/k)/K

HB c

CH4

1.0480

3.6503

161.71

1.4516

—

—

C2H6

1.3481

3.8135

249.42

1.4201

—

—

CsHg

1.6412

3.9057

267.63

1.4445

—

—

C4H10

1.9923

3.9152

257.15

1.4974

—

C5H12

2.3403

3.9235

264.37

1.5060

—

C6H14

2.6320

3.9610

282.77

1.4877

—

C7H16

3.0000

3.9513

251.08

1.5611

—

—

CgHis

3.3447

3.9440

256.32

1.5576

—

—

C10H22

3.9361

3.9927

264.08

1.5592

—

C12H26

4.5418

4.0167

259.18

1.5798

—

—

C14H30

5.3177

3.9875

240.61

1.6200

—

—

N2

1.3597

3.0907

76.13

1.5866

—

—

CO

1.4120

3.0527

76.81

1.6055

—

C6H6

2.0029

3.9917

348.32

1.4840

—

—

C6H12

1.3263

5.0595

548.92

1.3377

—

—

CO2

1.6936

3.0630

231.00

1.4189

—

—

H2O

1.0968

3.1194

740.53

1.2760

366.87

0.7752

R23

1.2481

308.24

4.0801

1.2231

235.39

1.0000

R134a

1.2848

402.74

4.6368

1.2473

295.09

1.0000

Molecule

For all the compounds considered, the QM-based models yield good fits as measured by the value of the objective function. For the w-alkanes, the objective function is slightly lower (i.e., a closer representation of the experimental data is obtained) with the new proposed approach than the standard model. Although it should be noted that in the derivation of the standard model, m has been fixed before optimising the remaining parameters. The values of w in both approaches are very similar, and it is remarkable that the QM-based approach performs so well given that a further degree of freedom, a, has been removed from the parameter estimation problem. Overall, the QM results are of good quality, with average deviations on the vapour pressure ranging from 0.69% to 7.19%, and those on the liquid density ranging from 1.09% to 14.97%. The largest discrepancies between the performances of the two types of models are

SAFT- VR parameters from quantum mechanics

155

observed for the density. This can be expected as the density is most sensitive to the size parameters, and therefore largely determined by fixing the values of m and <x For compounds where good quality density data are available, one could consider deriving m only from quantum mechanics, and allowing cr to be optimised together with the energy parameters in order to obtain an improved fit. TABLE 4. Standard SAFT-VR parameters providing an optimal description of the vapour-liquid equilibria without QM information. When no source is indicated, the standard model is obtained as part of this work. The same numbers of sites are used as in Table 2.

Molecule

m

dk

s/k/K

X

^^IklYi

r"^

Source

c

CH4

1.0000

3.6847

167.30

1.4479

—

—

[5]

C2H6

1.3333

3.8115

249.19

1.4233

—

—

[5]

C3H8

1.6667

3.8899

260.91

1.4537

—

—

[5]

C4H10

2.0000

3.9332

259.56

1.4922

—

—

[5]

C5H12

2.3333

3.9430

264.37

1.5060

—

—

[5]

C6H14

2.6667

3.9396

251.66

1.5492

—

—

[5]

C7H16

3.0000

3.9567

253.28

1.5574

—

—

[5]

CgHig

3.3333

3.9455

249.52

1.5751

—

—

[5]

C10H22

4.0000

3.9675

247.08

1.5925

—

—

[5]

C12H26

4.6667

3.9663

243.03

1.6101

—

—

[5]

C14H30

5.3333

3.9745

249.74

1.6023

—

—

[5]

N2

1.3000

3.1940

84.53

1.5340

—

—

[5]

CO

1.5190

3.0042

73.95

1.6004

—

—

CeUe

2.7604

3.3473

193.62

1.7391

—

—

C6H12

3.0015

3.4364

167.67

1.7919

—

—

CO2

2.0000

2.7864

179.27

1.5157

—

—

[41]

H2O

1.0000

3.0333

300.43

1.7183

1336.85

0.6843

[7]

R23

2.4589

131.90

2.8305

1.6344

669.50

0.5516

R134a

3.1161

136.66

2.9487

1.6863

1280.44

0.5340

A closer examination of the «-alkane parameters shown in Tables 3 and 4 reveals similar trends in their variation with carbon number for the standard method and for the new approach using QM information. In Figure 4, the linear dependence of m, mX, ms and mc^ on carbon number is illustrated. The QM behaviour is seen to follow closely that of the standard model.

156

T.J. Sheldon et al.

0

5

10

15

0

5

•

5

1000 ^

1000

O

1200 ^

15

Number of carbon atoms

Number of carbon atoms 14UU •

10

ft

800 -

•«

600 400 -

# 9

200 - (» 0 -

0

5

10

Number of carbon atoms

15

0

5

10

15

Number of carbon atoms

Figure 4. Behaviour of the SAFT-VR parameters as a function of the number of carbon atoms for the Ai-alkane series. The open circles denote the standard model of [5] and the squares denote the QM-based models derived in this work.

The chain length parameter, w, is usually the most difficult parameter to estimate and it has been fixed rather than optimised in the standard models for N2, H2O and CO2. Similar values are computed by the QM approach for N2 and H2O. However, for CO2, a lower value of m of 1.6936 is found in the QM approach. Altliough this is compensated to some extent by a larger value of cr, this is reflected in a relatively poor representation of saturated liquid densities in the fitting range. A much improved fit can however be obtained by setting m = 1.6936 and allowing crto be optimised. In this case, the optimal parameters are o-= 2.9918 A, s/k = 219.75 K, yl.=1.4481. This yields average deviations of 0.23% for vapour pressure and 1.15% for liquid density within the fitting range, and 0.46% for vapour pressure and 7.12% for liquid density for all data points.

SAFT- VR parameters from quantum mechanics

157

As can be seen from Table 2, this constitutes a small improvement over the standard model. For the remaining compounds, the standard model was obtained by estimating all the parameters including m from experimental data. The chain length parameter found by the QM-based approach tends to be smaller those of the standard models. The spherocylindrical model adopted here is less well suited to molecules such as benzene and cyclohexane, and this is reflected in larger density deviations for the QM-based model. As for CO2, much improved performance can be achieved by using the m value calculated by the proposed methodology, but allowing cr to vary. In the case of the refrigerants, in addition to differences in the size parameters, there is a marked variation in the energy parameters. 5. Concluding remarks A methodology to obtain two of the parameters in the SAFT-VR equation from ab initio computations has been proposed. The chain length parameter, w, and the segment diameter, a, for SAFT-VR are derived by mapping molecular dimensions calculated via the Restricted Hartree-Fock (HF) formalism onto a spherocylinder. The dimensions used are the molecular volume, calculated by integrating the electronic density, and the smallest and largest dimensions of a box containing the molecule. There are two adjustable parameters in these calculations: the electronic density threshold, and the grid size. Values for these parameters were chosen by tuning to data for ethane and octane. The molecular volume and the molecular aspect ratio as obtained from the HF calculation are then used to determine the SAFT parametes m and a . Once m and a have been computed, the remaining SAFT-VR parameters are estimated using experimental VLB data. Application of this approach to several members of the w-alkane series (up to C14) and to other representative molecules has yielded promising results. The quality of the representation of VLB provided by the QM-based models is generally comparable to that obtained with standard SAFT-VR models, although larger deviations from experimental data are sometimes observed for density, particularly the cyclic and aromatic compounds tested. For the nalkanes series, the QM-based parameters follow the expected trends with respect to carbon number. The proposed approach is useful in several ways. Firstly, it allows a physical value for the chain length parameter m to be established. This is valuable because this parameter is difficult to estimate reliably using VLB data only. Secondly, the proposed approach leads to a reduction in the number of parameters which must be estimated by numerical optimisation, and therefore

158

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an increase in the statistical significance of those parameter values, especially when experimental data are scarce. Finally, by combining the proposed method with ab initio techniques to derive the other SAFT parameters [e.g. 9], this work provides a stepping stone towards data-free methods to model the phase behaviour of new compounds. References. 1. W.G. Chapman, K.E. Gubbins, G. Jackson and M. Radosz, Fluid Phase Eq. 52 (1989) 31. 2. W.G. Chapman, K.E. Gubbins, G. Jackson and M. Radosz, Ind. Eng. Chem. Res. 29 (1990) 1709. 3. E.A. Muller and K.E. Gubbins, Ind. Eng. Chem. Res. 40 (2001) 2193. 4. C. McCabe and G. Jackson, Phys. Chem. Chem. Phys. 1 (1999) 2057. 5. P. Paricaud, A. Galindo and G. Jackson, Ind. Eng. Chem. Res. 43 (2004) 6871. 6. T. Lafitte, D. Bessieres, M.M. Piileiro, J.-L. Daridon, J. Chem. Phys. 124 (2006) 024509. 7. G. Clark, A. J. Haslam, A. Galindo, and G. Jackson, Mol. Phys., in preparation (2006). 8. W. Arlt, O. Spuhl, A. Klamt, Chem. Eng. Proc. 43 (2004) 221. 9. J.P. Wolbach and S.I. Sandler, Ind. Eng. Chem. Res. 36 (1997) 4041. 10. J.P. Wolbach and S.I. Sandler, Int. J. Thermophysics 18 (1997) 1001. 11. M. Yarrison and W.G. Chapman, Fluid Phase Eq. 226 (2004) 195. 12. M. Fermeglia and S. Pricl, Fluid Phase Eq. 166 (1999) 21. 13. M. Fermeglia, A. Bertucco, D. Patrizio, Chem. Eng. Sci. 52 (1997) 1517. 14. I.e. Sanchez and R.H. Lacombe, J. Phys. Chem. 80 (1976) 2352. 15. M. Fermeglia and S. Pricl, AIChE J. 45 (1999) 2619. 16. M. Fermeglia and S. Pricl, AIChE J. 47 (2001) 2371. 17. A. Klamt, J. Phys. Chem. 99 (1995) 2224. 18. A. Gil-Villegas, A. Galindo, P.J. Whitehead, S.J. Mills, G. Jackson and A.N. Burgess, J. Chem. Phys. 106(1997)4168. 19. A. Galindo, L. A. Davies, A. Gil-Villegas, and G. Jackson, Mol. Phys. 93 (1998) 241. 20. J. V. Senger, R. F. Kayser, C. J. Peter, and H. J. White, Jr., Equations of State for Fluids and Fluid Mixtures (Elsevier, Amsterdam, 2000), Vols. 1 and 2. 21. M. S. Wertheim, J. Stat. Phys. 35 (1984) 19. 22. M. S. Wertheim, J. Stat. Phys. 35 (1984) 35. 23. M. S. Wertheim, J. Stat. Phys. 42 (1986) 459. 24. M. S. Wertheim, J. Stat. Phys. 42 (1986) 477 25. E. A. Muller, and K. E. Gubbins, in Equations of State for Fluids and Fluid Mixtures, edited by J. V. Sengers, R. F. Kayser, C. J. Peters, and H. J. White, Jr., Elsevier, Amsterdam, 2000, Vol. 2. 26. F. J. Bias, and L. F. Vega, Mol. Phys. 92 (1997) 135. 27. F. J. Bias, and L. F. Vega, Ind. Eng. Chem. Res. 37 (1998) 660. 28. J. Gross, and G. Sadowski, Ind. Eng. Chem. Res. 40 (2001) 1244. 29. J. Gross, and G. Sadowski, Ind. Eng. Chem. Res. 41 (2002) 1084. 30. N. von Solms, M. L. Michelsen, and G. M. Kontogeorgis, Ind. Eng. Chem.Res., 42 (2003) 1098. 31.1. Kouskoumvekaki, N. von Solms, M. L. Michelsen, G. M. Kontogeorgis, Fluid Phase Equilibr.215(2004)71. 32. J. K. Johnson, and K. E. Gubbins, Mol. Phys. 77 (1992) 1033. 33. G. J. Gloor, G. Jackson, F. J. Bias, E. Martin del Rio, and E. de Miguel, J. Chem. Phys. 121 (2004)12740.

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T. Boublik, Mol. Phys. 68 (1989) 191. T. Boublik, C. Vega, and M. Diaz-Pefta, J. Chem. Phys. 93 (1990) 730. M. D. Amos and G. Jackson, J. Chem. Phys. 96 (1992) 4604. D. C. Williamson and G. Jackson, Mol. Phys. 86 (1995) 819. A. Szabo and N.S. Ostlund, Modem Quantum Chemistry: Introduction to Advanced Electronic Structure Theory, Dover Publications, 1996. 39. M. J. Frisch, G. W. Trucks, H. B. Schlegel, G. E. Scuseria, M. A. Robb, J. R. Cheeseman, J. A. Montgomery Jr.T. Vreven, K. N. Kudin, J. C. Burant, J. M. Millam, S. S. Iyengar, J. Tomasi, V. Barone, B. Mennucci, M. Cossi, G. Scalmani, N. Rega, G. A. Petersson, H. Nakatsuji, M. Hada, M. Ehara, K. Toyota, R. Fukuda, J. Hasegawa, M. Ishida, T. Nakajima, Y. Honda, O. Kitao, H. Nakai, M. Klene, X. Li, J. E. Knox, H. P. Hratchian, J. B. Cross, V. Bakken, C. Adamo, J. Jaramillo, R. Gomperts, R. E. Stratmann, 0. Yazyev, A. J. Austin, R. Cammi, C. Pomelli, J. W. Ochterski, P. Y. Ayala, K. Morokuma, G. A. Voth, P. Salvador, J. J. Dannenberg, V. G. Zakrzewski, S. Dapprich, A. D. Daniels, M. C. Strain, O. Farkas, D. K. Malick, A. D. Rabuck, K. Raghavachari, J. B. Foresman, J. V. Ortiz, Q. Cui, A. G. Baboul, S. Clifford, J. Cioslowski, B. B. Stefanov, G. Liu, A. Liashenko, P. Piskorz, I. Komaromi, R. L. Martin, D. J. Fox, T. Keith, M. A. Al-Laham, C. Y. Peng, A. Nanayakkara, M. Challacombe, P. M. W. Gill, B. Johnson,W. Chen, M. W. Wong, C. Gonzalez, J. A. Pople, Gaussian 03, revision B.04; Gaussian, Inc.: Wallingford, CT, 2004. 40. D.A. Fletcher, R.F. McMeeking, D. Parkin, J. Chem. Inf. Comput. Sci. 36 (1996) 746. 41. F.J. Bias and A. Galindo, Fluid Phase Equilibr. 194 (2002) 501.

Multiscale Modelling of Polymer Properties M. Laso, E.A. Perpete (Editors) © 2006 Elsevier B.V. All rights reserved.

161

Chapter 8

Implicit Viscoelastic Calculations using Brownian Configuration Fields Jorge Ramirez,^'^ Manuel Laso,^ ^Applied Mathematics, University of Leeds, Leeds LS2 9JT, UK ^E.T.S.L Industrials, Jose Gutierrez Abascal,28006 Madrid, Spain

1. Introduction For the numerical simulation of complex viscoelastic flows, two different possibilities are available: a purely continuum mechanical approach and the micro-macro or C O N N F F E S S I T ' methods, which are the combination of continuum-mechanical discretization techniques for the conservation equations and kinetic theory models^ for polymer dynamics. Although they are not competitive against the continuum-mechanical approach in terms of computational efficiency, micro-macro methods are necessary when a closed form constitutive equation is not available. Closure approximations are most of the times favorable because they offer huge savings in computational effort, at the price of missing some aspects of the underlying physics. On the other hand, micro-macro methods conform correctly to the physics of the molecular model, but their computational cost is higher and the results typically show statistical fluctuations. However, these methods have proved to be very useful in a wide range of problems including different molecular models^"^, and geometries like free surfaces^'^ or three-dimensional conditions^. Additionally, improvements based on the original idea^ continue to appear periodically in the literature^"^^ extending the scope of application and the efficiency of micro-macro methods. An alternative approach to the micro-macro conception, that can also be used in complex flows, is the solution of the Fokker-Planck equation instead of the equivalent Stochastic Differential Equation^^. Recently, this approach has been improved considerably with a very efficient and accurate method^^ that in many situations can perform better than CONNFFESSIT. The only limit of that

162

/. Ramirez andM. Laso

method is the case of molecular models for which the configurational space has a high number of dimensions. In such context, it is still more efficient and useful to implement the micro-macro idea based on the particular realization of ensembles of molecules and the solution of stochastic differential equations. An important feature of complex viscoelastic flows, specially those appearing in most industrial processes, is that they are time dependent in nature. It is increasingly important to have efficient, accurate and robust time-dependent methods to simulate industrial type flows. The stability of the numerical scheme is another aspect that must be taken carefully into account. It is well known that implicit time dependent discretizations, apart from being more stable than explicit schemes, frequently allow to get reliable results at an extended region in the parameter space. On the other hand, implicit time integration is more demanding in terms of the computer resources required. Previously, micro-macro calculations have been done almost exclusively by means of a time marching explicit, uncoupled numerical scheme, lacking the desirable stability of implicit methods. In the numerical solution of time dependent partial differential equations that include advection terms, an explicit discretization in time must fulfill some kind of Courant-Friedrich-Lewy (CFL)like condition to ensure the stability of the numerical scheme. This imposes a very severe restriction on the size step allowed in the calculation, especially with increasing mesh refinement. However, a fiilly implicit time discretization is unconditionally stable with respect to the time step size, and the latter is restricted only by the physics of the problem or the accuracy requirements. Recently, Somasi and Khomami^"* developed a self-consistent semi-implicit algorithm based on the 0-method and a Picard-like iteration to attain full convergence at each time step. Their method has similar properties as a fully implicit implementation in terms of accuracy and self-consistency, but it still misses the desired stability, concerning the size of the time step, due to the explicit sub-steps that are part of the 9 -method. In our previous paper^^ we introduced a practical way to treat the implicit time integration of problems arising from the micro-macro concept. Our method is based on the size reduction of the linear system of equations resulting from the discretization of the problem, using a Schur's complement approach. In that work, the theoretical basis of the method was outlined, and some results and numerical analysis for a simple one-dimensional problem, for which a fully analytical treatment was possible, were presented.

Implicit viscoelastic calculations using brownian configuration fields

163

In the present article, we implement and extend the ideas presented in our previous work^^ for the simulation of complex flows using the Brownian Configuration Fields (BCF)^ approach, hi the present, more general, implementation, the analytical treatment of the problem is not practical because the cost of construction of the Schur complement is unacceptably high. Therefore, alternative special techniques must be implemented to deal with the very large size of the resulting linear system. In section 2 the governing equations and the numerical scheme for the implicit Brownian Fields of complex flows are presented, along with the proposed new size reduction method. In section 3, the method is applied to some benchmark tests using bead-spring molecular models and the resuhs are compared with the analytical solutions, when available, or the numerical solutions of the corresponding macroscopic constitutive equations; the convergence and performance of the method are discussed as well. Finally, in section 4, a summary of the results and plans for future work are presented. 2. Model and methods 2.1. Governing equations The equations describing the isothermal flow of an incompressible viscoelastic fluid with no body forces, are the standard momentum and mass conservation equations*^; in non-dimensional form, these equations read: Su

dt

^

^

^

(1) V.w=0

where « is the fluid velocity, p is the hydrostatic pressure and x is the extra stress tensor. Variables are made non-dimensional by picking a characteristic velocity Vand a characteristic length L from the particular problem at hand, and performing the following substitutions: for the velocity (H* =Vu), length {r* = LP ), time {t* =tL/V), pressure (p* = prjV/L) and extra stress (r* =TT]V/L), In these expressions, letters labelled with * denote dimensional variables, whereas unlabelled letters indicate their non-dimensional counterparts; rj is the total viscosity of the fluid, a is the ratio of solvent viscosity rjs to total viscosity ;/, and the Reynolds number is defined as KQ=pVL/r], with p being the fluid density.

164

/. Ramirez and M. Laso

This set of partial differential equations (PDE) must be supplemented by a differential or integral constitutive equation (CE), which relates the macroscopic extra stress to the history of the flow. In the micro-macro approach, the CE is replaced by (i) an equation expressing the dynamics of a mesoscopic kinetic theory molecular model as a function of the macroscopic velocity field (typically a stochastic differential equation, SDE), and (ii) another equation expressing the extra stress tensor r as a function of the conformation of the molecules as given by the molecular model used (typically an ensemble average). To illustrate this concept, we introduce the non-dimensional equations for the micro-macro formulation of the Hookean dumbbell molecular model^ (due to its simplicity and because it will be used later in this work), using the Brownian Fields^ representation:

.-^((ee)-«)=o In equation (2), the connector vectors Q., representing the end-to-end vector of the molecules, is rescaled with the equilibrium average connector length yjkgT/H , where kg is Boltzmann's constant, T is the temperature and H is the elastic constant of the connector spring. The relaxation time of the dumbbells is 'k=C/4H, C, being the friction coefficient of the beads of the dumbbell, and the non-dimensional Weissenberg number, which measures the degree of elasticity of the flow, is defined as We=A,F/I. f^ is a standard vector of independent Wiener processes and 8 is the identity tensor. The angle brackets ( ) represent an ensemble average over a population of N representative dumbbells (/ = l...iV). To obtain better statistics, a large N is required; typically, in the case of BCF, N is of the order of 10^ Dumbbell models are appropriate to represent solutions that are sufficiently diluted such that the individual molecules do not interact with one another. It can be easily shown^^ that Hookean dumbbells correspond to the molecular description of an Oldroyd-B viscoelastic fluid. 2,2. Numerical scheme The set of equations (1) and (2) are discretized using a mixed DEVSS-G/SUPG finite element method^^"*^. To increase the statistical efficiency of the simulation, we reduced the variance of the stress with the help of the control variates method^^. More precisely, an ensemble of Hookean dumbbells starting

Implicit viscoelastic calculations using brownian configuration fields

165

from the same initial conditions as the BCF and evolving under quiescent conditions, is used to approximate the equilibrium nondimensional conformation tensor \QQ) which, at equilibrium, must be equal to 8. The temporal discretization of the equations is done using the implicit (backward) Euler method for the momentum conservation equation, and a fully implicit Euler-Maruyama method for the stochastic differential equations. Using this scheme, both the order of convergence of the PDE and the weak order of convergence of the SDE coincide. The proposed numerical scheme reads: ' ^-w^+ Re(w - " • ~V)w +"V« - - -;?8 -(1 - - -a)G + T," Vv - • =—(b, Vv)^

(3)

-{V-u,q)

(4)

=0

(l-a)(G-VM,E) = 0

(5)

A/

2We

X =

h 1^

\uy

^j (7)

Q- + -^Q-

= Q-^" + j — A P T

(8)

where G = VM is the discrete finite element interpolant of the velocity gradient and /z is a measure of the size of each finite element along the direction of the flow, (x, y) and (jc, y)r are the standard scalar products of jc and y in the domain Q and on the boundary F, respectively, and b is the traction vector on the boundary F. All terms are evaluated at the current time step («+l), except those labelled with «, which are evaluated at the previous time step. The approximating functional spaces are continuous biquadratic polynomials (Q2) for the velocity, and continuous bilinear polynomials (Qi) for the pressure, gradient of velocity, stress tensor and connector spring.

/. Ramirez and M. Laso

166

After discretization, the fiilly implicit formulation leads to a system of nonlinear algebraic equations. We selected the Newton-Raphson (NR) method for the solution of the problem, due to its superior convergence rate. It is important to know that, for typical micro-macro flow problems and medium mesh refinement (for instance, we can consider some hundreds of finite elements and some thousands of BCF), the number of degrees of freedom can easily get very large (on the order of millions). Hence, we have a non-linear system of equations with some millions of unknowns. In the NR scheme, the non-linear system of equations is linearized and the resulting algebraic system J 5x = b, where J is the Jacobian, 8x is the vector of corrections of the unknowns and b is the vector of residuals, has to be solved several times at each time step until convergence is fulfilled. If the equations and unknowns of the discretized versions of J, 8x and b are ordered according to the variables, they show the following block structure (the lines inside the matrices and vectors are drawn to assist in the descriptions below): uu

J^

^

Jo, 0

0 0 0

JQ,U JQ,U

J =

Jofi

KQ.-

8X^=(<5A:J,

J'^ 0 «^GG

0 0 0

^

0 0

0

•

0

• .. • ..

0

•

0

'•~.

0

~~

•^.a.

• ••

-^.a

•

"^^QN

0

• ..

0

•

0

..

0

•

0

0

0

Jr.

0

•^ac

0

0

^Qfi

0

•^aa 0

0

0

0

0

0

0

0 0

Qfi

•^OvG

SXp \SXf^

b^=(*. Mba

jk_

Sx^ \^^Q

*. |*a *e

"^QiQi

^^Q

0

'"

^1

0 0~1

• QNQS

^^o

^.)

(9)

0

*^aa • ..

•

'"

*^^o )

y

^^^^

(11)

It can be noted that the Jacobian has a general arrowhead block structure. Each non-empty sub-block of the Jacobian is at the same time a sparse matrix with the sparsity pattern given by the underlying connectivity of the finite element

Implicit viscoelastic calculations using brownian configuration fields

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mesh. Also, due to the fact that molecular fields do not interact with each other, as is the case in many molecular models from the kinetic theory, the sub-block J (lower-right part of the Jacobian, also the largest part of the Jacobian due to the large number of BCF) has an advantageous block diagonal structure. 2.3. Size reduction method Due to the large size of the resulting linear system that needs to be solved at each loop of the NR method, it is impractical, even using iterative methods, to solve the complete system of equations. Instead, we recur to a size reduction algorithm that takes advantage of the sparse block structure of the Jacobian in equation (9). The idea is to reduce the problem to the size of a typical macroscopic CFD or Navier-Stokes problem (upper-left part of the Jacobian), for which a large number of efficient numerical techniques are available. We do such reduction by performing a block Gauss elimination step of all variables except w and p. The block elimination procedure is equivalent to the following substitution of variables, operating at the block level:

Sx^=r:if^i^5xA V /=l

(12)

J

After the substitutions are performed, we are left with a modified Navier-Stokes problem that can be expressed with the following simplified notation: '

ttu

(8x-

(13)

0 . 5x . \

pJ

where only the sub-blocks labelled with an asterisk are modified from those in the original Jacobian of equation (9). The modified blocks have the following expressions: (14)

*S = h - -^SG-^GG^G - -^uzK

b.+

twM

•^QG-^GG^G-^a)

(15)

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To build the modified block of the residual fe! we need to solve A^+2 linear systems. The cost of this operation is approximately equal to the cost of solving one step of the same micro-macro problem if it is put in an explicit, uncoupled way. This needs to be done at the beginning of each NR loop. However, the construction of the modified block of the Jacobian j ! - would need the inversion of N^l matrices, which is completely out of question due to the memory and CPU requirements of such operation. Instead, we propose to solve the modified Navier-Stokes problem by using a preconditioned GMRES method. This iterative method is based on matrix-vector muhiplications, and it can be altered so it can use the modified version of the Jacobian at each iteration. To perform a matrix-vector multiplication using the modified Jacobian block J*--, we need to solve A/H-2 linear systems (and, eventually, to assemble the 4A/^ sub-blocks J^^ , J=., J^^ and J^^ if the memory is limited). In order to reduce the numerical work, it is essential to reduce the number of iterations of the solver. With iterative solvers like GMRES, this is typically accomplished by the application of an appropriate preconditioner. In the present problem, we have the additional complication that the matrix that we want to solve is never available in memory, and it is hence very difficult to estimate a good preconditioner. In our previous work^^, we showed that, for a simple onedimensional case, the condition of the Jacobian corresponding to the purely macroscopic part of the problem is not deteriorated after the size reduction procedure. Assuming that this property of the method still holds in two and three dimensional problems, we decided to rely on a preconditioner that only considers the original Navier-Stokes sub-blocks of the Jacobian, before any reduction algorithm is applied. We selected a block preconditioner that has been proposed recently^ ^ whose convergence is only weakly dependent on the Reynolds number and does not depend on the mesh size. The proposed solution algorithm comprises the following three steps, that must be performed at each NR loop: 1. Preparation of the modified residual block b*-. 2. Solution of the modified Navier-Stokes problem to obtain Sx^ and Sxp^ using a preconditioned GMRES method. The solver applies a special matrix-vector product that considers the modified Jacobian sub-block

^„3. Back substitution to obtain SXQ, ^x^and Sx^ .

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Steps (1) and (3), as well as each iteration of the GMRES method in step (2), implies the solution ofN-^2 linear systems. At the end of each NR loop, we are left with the updated values of the fields i?, /?, G, x and Q . If the solution has converged, we move on to the next time step; if not, another NR loop is performed. It must be noted that, due to the sparse block structure of the part of the Jacobian referring to the interaction between microscopic variables, J^^, all operations involving BCF can be done on a per field basis. This permits to adapt the algorithm very easily to a parallel hardware environment. 3. Results and discussion To test our new method, we solved a selected number of benchmark problems, most of them having analytical solution. The problems were chosen carefiilly to address the issues of convergence, efficiency or computational cost, and applicability of the proposed algorithm. In the following subsections, the results of applications of the method to problems of increasing difficulty are presented. 3.1. Numerical test: startup of 2D Couetteflow ofHookean dumbbells

ky

V W^

'^^Z ^^y

X

D Figure 1. Channel domain where tests and applications are solved.

First, we examined the performance of the method in a typical benchmark case: the start up of two-dimensional plane creeping Couette flow of a solution of Hookean dumbbells (domain and boundary conditions schematically depicted in Figure 1). There is no slip at the plates and periodic boundary conditions are applied to the fields w , G, i and each BCF g . The dimensionless length of the flow channel is D=2n, Unless otherwise specified, the conditions of the

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simulations were the following: mesh of 10x40 finite elements, We=l, zl/=0.1 and a=0.5$. / "vy \ xy

"\J-.y'^^

,

Np = 500 Np=1000 Np = 2000 N^ = 4000 Oldroyd-B

4000

10 Fig. 2. Start up of shear stress xxy in plane creeping Couette flow (We = 1, a = 0.5, At = 0.1). The inset shows the error for three different BCF ensemble sizes, along with the fit to a function proportional to 1/VA^.

We checked the convergence of the method for different BCF ensemble sizes. In Figure 2, the transient growth of the shear stress for different N is presented. In the insert of the same figure, it can be observed that the error, computed as yjcrlN {G being the variance of the solution at the steady state), has the expected convergence for a statistical average. We also checked the convergence of the error for different time step sizes At, The expected weak order of convergence 1 in the value of the stress was verified. Regarding the computational cost of the implicit algorithm, compared to the explicit version, we observed the following: • Only 2 NR loops were needed at each time step (the condition for convergence was set to a maximum of 10'^ in any single value of the residual vector b or vector of corrections 6x). This was somehow an

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•

111

expected result, because we are solving a problem that is only weakly non-linear. After the initial time step, only 1 GMRES iteration is needed to converge (10"*^ tolerance in the residual of the modified Stokes problem)*. The number of GMRES iterations does not depend on A^, mesh refinement. At or We.

The last two points mean that we have an explicit time integration method which is linear in the number of molecules or BCF (something that is evident in the explicit case but is not clear in the implicit, non-linear, coupled case), and whose computational cost is only twice the cost of the explicit, uncoupled algorithm. The explanation of this behavior is simple: the flow in this particular case is homogeneous in space, and the conformational state of the molecules does not affect the value of the hydrodynamic variables u and p. The solver converges very easily because, after the first time step, the hydrodynamic fields essentially do not change. 3.2. Application 1: startup of plane pressure flow ofHookean dumbbells The same geometry (see Figure 1) was used to solve the start up of plane pressure flow. Parabolic velocity profiles were imposed at the inflow and outflow boundaries, no slip boundary conditions were applied at the upper plate, and G, r and Q^ were subjected to periodic boundary conditions. In this particular problem, the Weissenberg number is defined as We = A(F)/Z, where ( F ) is the average velocity over the width of the channel. Note that, although the velocity field remains homogeneous at all times (except for the typical fluctuations arising from the averages of the stochastic variables), the pressure field depends on the value of the extra stress and is, thus, not homogeneous in space and time. Several runs were performed for different values of At and A^, and the same correct convergence as in the previous paragraph was observed at any point in space and time. The comparisons were done with the analytical solution of the equivalent Oldroyd-B problem, also available in this case (we do not present transient evolution of the results because they look very similar to Figure 2).

* To have an idea of the computational requirements of this problem, around 200 seconds are needed for each preconditioned GMRES iteration (1000 BCF) on an INTEL® Xeon™ CPU at 2.80 GHz.

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60

N^ I 40

20

20

J 500

150

N^ 1000

(c)

0

150 h

100

100

50

50

0.2

0.4

At

0.5

0

llh ^O

W

O.

0

OAf = We/10 • Ar=0.1

We

10

Fig. 3. Average number of GMRES iterations (Ni) at each time step, as a function of (a) the number of BCF N, (b) the finite element mesh size \/h, (c) the time step At and (d) the Weissenberg number We (A^= 100, ^^ = 0.1, /z = 1, We = 1 and a = 0.5, unless specified). The straight lines are linearfittingsto the points and are represented only as a guide to the eye.

More important is the analysis of the computational cost of the implicit method in the present, non-homogeneous case. Again, only two NR loops were needed to converge at each time step. In Figure 3, a more detailed analysis of the computational cost is presented, in terms of the average over the complete time dependent simulation (200 time steps) of the total number of GMRES iterations (Ni) needed to attain convergence at each time step (we added the GMRES iterations of the two NR loops). In Figure 3(a), it can be observed that Nj decreases with the number of molecules or fields N. An increase of N reduces the size of the fluctuations in the value of the extra stress, which affects the efficiency of GMRES, making it easier for the solver to find the solution. In terms of CPU time, the cost is sublinear with respect to N, which is better than in the explicit case. In (b), we can remark that performance of the solver is basically not affected by the fineness of the mesh, measured in terms of h, the element size*. This is a result that is in agreement with the predicted performance of the preconditioner, and supports our assumption that the * h=l is considered to be the base case (10x40 finite elements).

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condition of the macroscopic (Stokes) block of the linear system is not appreciably deteriorated by the size reduction algorithm. In (c), however, it can be seen that the size of the time step At seems to affect the performance of the solver. This can be explained by considering that larger time steps imply larger changes of the transient values of the unknowns at each time step, and it is therefore more difficult for the solver to find the converged solution. In (d), Nj is represented against We in two different situations: using the same time step in absolute terms {At = 0.1), and using the same time step relative to We (zl/=We/10). In the first case, the computational requirements of the simulation don't grow; in the second case, though, it can be seen that Nj grows rather fast with We. Again, the larger changes in the unknown fields could be an explanation for this trend. However, we believe that, for large We, the condition of the Stokes block of the problem might be altered by the size reduction method. To increase the efficiency in these more demanding problems, it is necessary to search for a better preconditioner that considers all the blocks of the Jacobian. Further steps in the investigation of this point are being taken. 3.3. Application 2: linear stability analysis of the Couette flow of Hookean dumbbells A much more demanding problem is the analysis of the linear stability of the steady-state Couette flow of a solution of Hookean dumbbells, using the micromacro approach. In this study, we followed an approach presented recently^"*. The idea is to integrate the time evolution of random or prescribed imposed disturbances, linearized around a particular steady-state solution, using the finite element technique. The long time range behaviour of the norm of the disturbance should be governed by the most dangerous eigenvalue. This procedure was introduced in a previous work by Sureshkumar and coworkers^^, and the same principles presented in that article hold here, except for the fact that the steady-state in micro-macro simulations is actually unsteady^ due to the fluctuations produced by the finite size of the ensemble of molecules. In this case, both the steady-state and perturbation evolve in time and must be integrated. The governing equations (1) and (2), in the case Re=0 and with the additional DEVSS-G equation, are linearized around the steady state. For the sake of completeness, we include here the set of evolution equations for the perturbation fields:

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/. Ramirez andM. Laso -V'M' + Vp' + (l-a)V-G'-V-T' = 0 V-M' = 0

(l-a)(G'-VM') = 0

We iVtr^

(16)

' 2We

The same discretization techniques described in section 2.2 are applied here. The equations for plane creeping Couette flow of Hookean dumbbells are evolved until a steady-state is attained (a simulation length of around 10 times the relaxation time of the dumbbells is enough for that purpose). At that point, an initial, periodic and random, perturbation is prescribed for each of the perturbed BCF Q.. Then, both the base flow and the perturbed variables are integrated in time, and the growth or decay of the disturbance for any variable v is monitored using the following norm:

P^{t) = slt{<.^<2+where v, j is the /

+ <.)

(17)

component of the variable v (scalar, vector or tensor)

evaluated at the /* node, and Nn is the total number of nodes of the discretization. The main difference between this approach and the previous one^"* is that, due to the fully implicit discretization in time, we need to prescribe the disturbance in the molecular variables and not in the velocity field. Moreover, we are allowed to use larger time steps than in previous works. This is a clear advantage of implicit methods over explicit or semi-implicit algorithms, specially for large We and very fine meshes. Although the size of the time step affects the accuracy of the solution, a calculation using large time steps can help to reach faster the regime where we want to extract the value of the most dangerous eigenvalues. In our simulations, we always used a time step At = We /lO.

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We =10 We = 5 We = 2 We=l

0.01

0.0001 h

20

40

60

80

100

t

Fig. 4. Evolution of the normalized norm of the disturbance of the stress for Hookean dumbbells at We = 1, 2, 5 and 10 with 2000 trajectories.

In Figure 4, the evolution of the normalized norm of the disturbance of the extra stress tensor in the linear stability analysis of creeping plane Couette flow of Hookean dumbbells is presented. As expected for the unconditionally stable Couette flow, the stress norm shows an exponential decay. The long time behavior of this norm can be used to evaluate the real part of the most dangerous eigenvalue, by fitting the norm at long times to a function of the form f{t) = Qxp(^at + b), a being the sought eigenvalue. In Table 1, the eigenvalues computed from Figure 4 are presented, along with the results from a previous micro-macro semi-implicit algorithm, and from the solution of the generalized eigenvalue problem. The results are in good agreement with previously published results. If Figure 4 is compared with Figure 7 in the work by Somasi and Khomami^"*, it can be observed that, although both result in similar long time dynamics (same slope of the logarithmic decay), the short time dynamics seem to differ. Apart from the influence of the non-normal interactions of the eigenmodes on the dynamics of the system, we believe that the discrepancies at short times may be due to actual deviations in the dynamics of the perturbations of the molecules. In the previous semi-implicit approach, the initial perturbations of the dumbbells originated as a reaction to a prescribed perturbation of the velocity field. In the present fully implicit work, the perturbation must be prescribed directly at the molecular level, and therefore the

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evolution of the norms are different. We believe that the present approach is more realistic, although this is not reflected in a higher accuracy in the eigenvalues of Table 1. The lower accuracy of our results is due to the significantly larger value of the time step.

We 1.0 2.0 5.0 10.0

Hookean A -0.72 -0.37 -0.15 -0.061

Hookean B -0.85 -0.46 -0.17 -0.079

OLD-B (GEVP) -1.02 to-0.98 -0.52 to -0.48 -0.22 to-0.18 -0.073

Table 1: Most dangerous eigenvalues for Hookean dumbbells. Column A shows the results from the present work, and column B the resultsfromreference 14, GEVP values beingfromreference 22.

3.4. Application 3: startup of creeping plane Couetteflow of Rouse chains As we mentioned in Section 1, in the case of molecular models for which the configurational space has a high number of dimensions, we believe it is still necessary and useful to implement the micro-macro idea based on the particular realization of ensembles of molecules and the solution of SDEs. In this context, it is important to check the performance and applicability of the implicit algorithm presented in this paper in the case of high-dimensional molecular models. The archetypal molecular model with a high-dimensional configurational space is the Rouse model. In addition, this molecular model, expressed in normal coordinates, is fully equivalent to a multi-modal Oldroyd-B constitutive equation and analytical solutions for particular examples are readily available.

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111

100

50

N s Fig. 5. Computational cost for Rouse chains. Hours on an InteKD Xeon^^ CPU at 2.80 GHz, as a function of the number of segments in the chain Ns (N = 1000, At = 0.1, h = 1, We = 1 and a = 0.5).

We tested the convergence and efficiency of the implicit micro-macro method with Rouse chains in the same benchmark problem as with Hookean dumbbells. The domain and boundary conditions are equivalent to those in section 3.1. We obtained the same degree of convergence in all points of space and time as in that case. More important is the check of the computational cost. In an explicit, uncoupled simulation, it is clear that the cost is linear in the number of springs in the chain. In a fully implicit formulation, this behavior is not so evident because, although the spring force of the Rouse model is linear, the coupling between micro and macro variables is not linear and it is not clear that the CPU time should scale linearly with the size of the problem. However, as Figure 5 shows, the CPU time needed for this test case is proportional to the number of segments in the Rouse chains. Again, only two NR loops were needed, and one GMRES iteration per NR loop. This is a consequence of the homogeneous value in space of the hydrodynamic fields. More GMRES iterations are expected in more complicated problems.

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4. Conclusions An implementation of the implicit time integration of micro-macro problems based on the Brownian Configuration Fields for complex flows was presented for the first time. A novel size reduction method was presented to deal with the large size of the linear systems under consideration. The algorithm allows an independent treatment for each molecular field and is well suited to parallel hardware architecture. The method was tested on well known benchmark problems for viscoelastic flows, and the results compare very favorably with obtainable analytical expressions. This equivalence is highly encouraging. Although not competitive with other micro-micro approaches based on explicit or semi-implicit uncoupled formulations in simple problems, we believe that the proposed algorithm can help to extend the range of convergence and the quality of the results in more demanding cases at higher We numbers. An extensive comparison, in terms of efficiency, stability and accuracy, between the different methods to solve time dependent problems based on the micro-macro idea is of paramount importance in order to select which features of each algorithm are interesting and, perhaps, design a better numerical scheme based on the combination of them. It would also be interesting to check if the proposed implicit algorithm allows to extend the limit of accessible We numbers in well known benchmark problems solved within the micro-macro framework. An additional important issue is the search for an optimal preconditioner for the implicit micro-macro calculations. As the Jacobian matrix is never available in memory, it is rather difficult to estimate how the preconditioner should be. In the present case, we rely on preconditioners that are known to perform very well in the Stokes problem. We believe that there may exist other possibilities that could help to significantly increase the efficiency of our method. Two potential improvements could be the extension of the block preconditioner used in the present paper or the use of multigrid solvers. Finally, we believe that the same ideas presented in this paper can be applied in the case of the simulation of viscoelastic flows based on the constitutive equation approach. The behavior of most polymeric materials used in the industry cannot be correctly approximated by means of a single mode constitutive equation. The most frequently applied approach is to represent the rheological behavior of the fluid by using several uncoupled modes of a certain CE. The simulation of complex flows of muhimodal viscoelastic fluids represents, conceptually, a similar numerical problem to the micro-macro approach. It is of tremendous importance to find numerically stable and accurate algorithms to the resolution of such problems, which also lead to very large

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fields

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linear systems. The most often used method to tackle this kind of problems is the Discontinuous Galerkin method^^ which does a locally implicit treatment of the equations at the level of the element, but deals explicitly with the flux terms at the boundaries. This method also allows the reduction of the problem to the size of a typical Stokes calculation, but it is known to have convergence problems in time-dependent simulations with low order time integration schemes. It is important to create algorithms which are appropriate to the solution of very large problems like those arising from very fine discretizations or three dimensional geometries, specially in the case of multi-modal constitutive equations. Also, after the recent introduction of the logconformation method^"* which solves the high Weissenberg number problem, it becomes even more important to search for efficient schemes and solvers for large problems. We think that algebraic size reduction methods, combined with an appropriate preconditioning technique, could be a possible option. References 1. M. Laso, H.C. Ottinger, Calculation of viscoelastic flow using molecular models — the CONNFFESSIT approach, J. Non-Newtonian Fluid Meek 47 (1993) 1-20. 2. R.B. Bird, C.F. Curtiss, R.C. Armstrong, 0 . Hassager, Dynamics of polymeric liquids, vol. 2, Kinetic Theory, John Wiley & Sons, New York, 1987. 3. A.P.G. van Heel, M.A. Hulsen, B.H.A.A. van den Brule, Simulation of the Doi-Edwards model in complex flow, 1 Rheol 43 (5) (1999) 1239-1260. 4. X.-J. Fan, N. Phan-Thien, R. Zheng, Simulation of fibre suspension flows by the Brownian configuration field method, J. Non-Newtonian Fluid Meek 84 (1999) 257-274. 5. J. Cormenzana, A. Ledda, M. Laso, B. Debbaut, Calculation of free surface flows using CONNFFESSIT, J. Rheol 45 (1) (2001) 237-258. 6. E. Grande, M. Laso, M. Picasso, Calculation of variable-topology free surface flows using CONNFFESSIT, 1 Non-Newtonian Fluid Meek 113 (2003) 127-145. 7. J. Ramirez, M. Laso, Micro-macro simulations of three-dimensional plane contraction flow. Modelling Simul Mater. Sci. Eng. 12 (2004) 1293-1306. 8. M.A. Hulsen, A.P.G. van Heel, B.H.A.A. van den Brule, Simulation of viscoelastic flows using Brownian configuration fields, J. Non-Newtonian Fluid Meek 70 (1997) 79-101. 9. P. Halin, G. Lielens, R. Keunings, V. Legat, The Lagrangian particle method for macroscopic and micro-macro viscoelastic flow computations, J. Non-Newtonian Fluid Meek 79 (\999)3Sl-403. 10. P.G. Gigras, B. Khomami, Adaptive configuration fields: a new multiscale simulation technique for reptation-based models with a stochastic strain measure and local variations of life span distribution, J. Non-Newtonian Fluid Meek 108 (2002) 99-122. 11. D. Tran-Canh, T. Tran-Cong, Computation of viscoelastic flow using neural networks and stochasfic simulation, Korea-Aust. Rheol. J. 14 (2002) 161-174. 12. X. Fan, Molecular models and flow calculations: II. Simulation of steady planar flow. Acta Meek Sin. 5 (1989) 216-226. 13. A. Lozinski, C. Chauviere, J. Fang, R.G. Owens, A Fokker-Planck simulation of fast flows of melts and concentrated polymer solutions in complex geometries, J. Rheol. 47 (2003) 535-561.

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14. M. Somasi, B. Khomami, Computation linear stability and dynamics of viscoelastic flows using time-dependent stochastic simulation techniques, J. Non-Newtonian Fluid Mech. 93 (2000)339-362. 15. M. Laso, J. Ramirez, M. Picasso, Implicit micro-macro methods, J. Non-Newtonian Fluid Mech. 122(2004)215-226. 16. R.B. Bird, R.C. Armstrong, 0. Hassager, Dynamics of polymeric liquids, vol. 1, Fluid Mechanics, John Wiley & Sons, New York, 1987. 17. R. Owens, T. Phillips, Computational Rheology, Imperial College Press, London, 2002. 18. R. Guenette, M. Fortin, A new mixed finite element method for computing viscoelastic flows, J. Non-Newtonian Fluid Mech. 60 (1995) 27-52. 19. M.J. Szady, T.R. Salamon, A.W. Liu, D.E. Bomside, R.C. Armstrong, R.A. Brown, A new mixed finite element method for viscoelastic flows governed by differential constitutive equations, J. Non-Newtonian Fluid Mech. 59 (1995) 215-243. 20. J. Bonvin, M. Picasso, Variance reduction methods for CONNFFESSIT-like simulations, J. Non-Newtonian Fluid Mech. 84 (1999) 191-215. 21. D. Kay, D. Loghin, A. Wathen, A preconditioner for the steady-state Navier-Stokes equations, SIAMJ. Sci. Comput. 24 (1) (2002) 237-256. 22. R. Sureshkumar, M.D. Smith, R.C. Armstrong, R.A. Brown, Linear stability and dynamics of viscoelastic flows using time-dependent numerical simulations, J. Non-Newtonian Fluid A/ec/i. 82(1999)57-104. 23. F.P.T. Baaijens, S.H.A. Selen, H.P.W. Baaijens, G.W.M. Peters, H.E.H. Meijer, Viscoelastic flow past a confined cylinder of a low density polyethylene melt, J. NonNewtonian Fluid Mech. 68 (1997) 173-203. 24. M.A. Hulsen, R. Fattal, R. Kupferman, Flow of viscoelastic fluids past a cylinder at high Weissenberg number: stabilized simulations using matrix logarithms, J. Non-Newtonian Fluid Mech. 127 (2005) 27-39.

Multiscale Modelling of Polymer Properties M. Laso, E.A. Perpete (Editors) © 2006 Elsevier B.V. Allrightsreserved.

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Chapter 1

Monomer Solubility and Diffusion in Confined Polyethylene by Mapping Atomistic Trajectories onto the Macroscopic Diffusion Equation M. Laso,^^ N. Jimeno^, C. Aleman^ ^ Dept. of Chemical Engineering, ETSII, UPM, Jose Gutierrez Abascal, 2, E-28006 Madrid, Spain ''Institutfur Polymere,ETHZurich, CH-8093 Zurich, Switzerland ^ Departament d'Enginyeria Quimica, Facultat de Quimica, Universitat de Barcelona, Mart i Franques 1,Barcelona E-0802, Spain

I. Introduction Modeling heterogeneous supported catalysts for polyeolefin synthesis is a challenging area in which great advances have been made in the last few years ([!]» [2]). Recent models have achieved a considerable degree of realism through the incorporation of a wide range of transport and kinetic phenomena in a consistent way ([3], [4]). The increasing detail of such meso- and microscopic models makes it necesary to deal with thermodynamic and transport phenomena at very small spatial scales. Although the complexity (fractal geometries, coexistence and multicomponent equilibrium and diffusion of monomer, oligomers and polymer, actual kinetics on active sites, etc.) of existing supported catalysts puts them out of reach of atomistically detailed simulations, the very small time and length scales at which diffusion, phase equilibria and reaction (incorporation of monomers to growing polymer chains) take place pose some fundamental questions as to the applicability of macroscopic continuum descriptions of these phenomena.

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The present investigation attempts to quantitatively answer the question of the applicabihty of macroscopic descriptions to i) solubility (phase equilibrium) and ii) diffusion in very confined geometries. To that end, atomistic modeling techniques are applied to an idealized pore in which a polymer chain (polyethylene, PE) and monomer (ethylene, Et) are present. Although modeling the reaction (i.e. chain growth at the expense of monomer) at an active site in the pore could be treated on a semiempirical basis [5], it has been deliberately left out in order to focus on the thermodynamic and transport aspects of the problem. The main goal of the present work was therefore to determine monomer availability at the reaction site as controlled both by diffusivity and solubility.

II. Model systems for atomistic ethylene/poly-ethylene structures confined in a pore In all work reported in the following, the pore was modeled as a cylindrical cavity of total depth H + Djl consisting of a cylinder of diameter D and height H capped by a semisphere of diameter D . Two pore sizes were considered: Z) = 3 nm and Z) = 1.5 nm (see Figure 1). In both cases, / / « 5 nm (average over the Monte Carlo run; see below). These dimensions are thought to be representative of typical structures of supported catalysts where active catalyst sites are located [6]. The model system consisted of a single molecule of linear polyethylene of length 1246 backbone carbon atoms in the pore with Z) = 3 nm and 311 backbone carbon atoms in the pore with Z) = 1.5 nm, while the number of ethylene molecules fluctuated in the Monte Carlo (MC) and was fixed in the Molecular Dynamics (MD) calculations reported below. Polyethylene was modeled using a united-atom representation for both methylene and methyl end groups, nonbonded interactions being described by a Lennard-Jones potential:

f^\' V!f'{r) = Ae

r^V (1)

with r.j being the scalar minimum image distance between sites / andy. Along a PE chain, all pairs of sites separated by more than three bonds along the chain and all intermolecular sites interact via the Lennard-Jones pair potential.

Monomer solubility and diffusion

185

Potential tails are cut at 1.45 Cp^ and brought smoothly down to zero at 2.33 (7p^ using a quintic spline, BuDcPE

Figure 1. Schematic of pore. A single chain of PE is grown in the pore. The bulk is represented by a separate cubic simulation box under three-dimensional boundary conditions.

In the generation of the starting structure, bond lengths were kept constant, whereas bond angles were assumed to fluctuate around an equilibrium angle OQ of 112^ subject to the Van der Ploeg and Berendsen bending potential [7] of the form:

^w..W = ^^.(^-^o)'

(2)

with K^ = 482,23 kJ/mol, Associated with each dihedral angle (/> was also a torsional potential of the form [8]: Korsion (f)

= ^0 + ^1 ^^^ ^ + C2 COS^ ^ + C3 COS^ ^ + C4 COs"^ ^ + +C5 COS^ ^

(3)

with Co = 9.28, Ci = 12.16, c^ = -13.123, c^ = -3.06, c^ = 26.25, all of them in kJ/mol. In Molecular Dynamics calculations, C-C bonds were constrained by a standard harmonic potential [9], Ethylene was also represented in the united atom mode following [10]. Pore walls were represented by a Catlow potential frequently used in zeolite modeling [11].

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III. Model systems and techniques IILl. Solubility of ethylene in polyethylene structures confined in a pore Solubility is the first of the two key factors influencing permeability of ethylene in PE. Its direct measurement in nanopores is still beyond the reach of available experimental techniques. It is therefore natural that current modeling work assumes the macroscopic value of Et/PE solubility to be valid at all scales down to the finest structures of the catalyst. Molecular modeling techniques, in particular Monte Carlo, offer a way to computationally test the validity of this assumption. To this end, we performed Gibbs ensemble simulations ([12], [13]), in which thermodynamic equilibrium was established between a pure ethylene box and the PE-filled pore depicted in Figure 1 at 353 K and 10 bar. The first type of moves in a Gibbs ensemble simulation, namely NVT moves, were achieved by simple Metropolis moves for ethylene (both in the pure ethylene box and in the pore). Efficient configurational sampling for the polymer in NVT moves was much more challenging, since neither Continuum Configurational Bias [14] nor End-Bridging Monte Carlo [15] were effective in the restricted space of the pore. The Extended Concerted Rotation (ECROT) of Leontidis et al, [16] however proved to be very efficient at performing local rearrangements of polymer segments. ECROT moves were supplemented with a combination of single angle Metropolis, reptation and flip moves. Although not as efficient as ECROT, these moves are known to act as a "lubricant" in MC calculations and improve overall sampling when used in conjuntion with larger scale moves like ECROT or End-Bridging. The overall strategy for moves consisted thus in 85% ECROT, 5% single angle Metropolis, 5% reptation and 5% flip moves. Another main hurdle in the simulation of such highly constrained systems was the generation of the starting configuration for the MC run. The very recent [17] and highly efficient [18] initial guess generator Polypack was used to densely pack the PE chain in the pore. Polypack is one of the most flexible and efficient polymer structure generators and is based on a geometric optimization approach. The polymer chain packing problem is cast as a geometric optimization taks which is then solved by heuristic search algorithms. It can generate dense packings of long chains for virtually any polymer structure, no matter how complex its architecture.

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The ability of Polypack to produce acceptable, dense initial structures of long chains in the very confined space of a pore is very remarkable. (C1246 in the Z) = 3 nm pore and C311 in the D = 1.5 nm pore). Although these initial structures had high intramolecular non-bonded energies due to a few overlaps, these overlaps were not severe and were rapidly eliminated by the ECROT algorithm during the equilibration phase (5x10^ steps). After equilibrating the confined PE structures, a full Gibbs ensemble simulation was initiated, in which cycles of 1000 NVT moves (configurational sampling), 1 NpT move (volume fluctuation) and 10000 NjuT moves (ethylene exchange between boxes) were executed. For the small pore, 3.3x10^ cycles and for the large pore 5.8x10^ cycles were performed. Volume fluctuations in the ethylene box were achieved in the usual way (scaling of cubic box edge). Coupled volume fluctuations in the pore were carried out by changing H and leaving D constant, which is consistent with the assumption that the pore is an undeformable cavity and volume fluctuations can only take place in the axial direction. This procedure is tantamount to placing a movable lid or piston over the pore. Volumefluctuationscorrespond to displacements of this piston. During NpT moves, the lid was endowed with the same repulsive potential as the pore walls. In view of the small size of the ethylene molecule, simple brute-force insertions of ethylene molecules into both boxes (pure Et box and PE-filled pore) were used for particle exchange and resulted in sufficiently high acceptance ratio 1.8% and 3.4% for the small and large box, respectively. A block analysis showed that the efficiency of the suite of MC moves was amply sufficient to generate decorrelated PE structures after every 800 cycles in the small pore and every 2200 cycles in the large pore. All computations were carried out in a simple parallel fashion on individual processors of a 24-CPU Beowulf Linux cluster composed of inexpensive Pentium III processors. The complete MC simulation of the small system ( D = 1.5 nm pore), including the equilibration phase, took 26 hours, whereas the D = 3 nm pore required 510 hours (accumulated wall clock over all processors). Figures 2 and 3 are graphical representations of typical configurations produced during the MC calculation. They give an idea of the smallness of the pore and of the high degree of confinement of the ethylene/polyethylene system.

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Figure 2. Side and top views of C1246 in large pore (£) = 3 nm, i f « 5 nm)

As a consequence of the strong confining effect exerted by the pore walls, high-energy bond torsions appear at isolated locations within the PE. This local energy penalty is compensated for by the global increase in packing efficiency. An aspect worth further investigation is the effect such strained chain conformations have on polymerization reaction kinetics. Descending the spatial modeling scale, the next logical step would be the study by means of quantum chemistry methods of changes in reaction velocity induced by unusual chain conformations in the immediate vicinity of a catalytic active site. Such a QC study would take as starting point the (classically described) chain conformations generated in the present work and would aim at detecting differences in catalytic activity induced by strained chain conformations.

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Monomer solubility and diffusion

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Figure 3. Side and top views of Cjn in small pore ( D = 1.5 nm, / / « 5 nm) III 2. Diffusivity of ethylene in polyethylene structures confined in a pore Diffusivity being a dynamic or transport property, it is natural to use a dynamic method such as a MD to investigate it. Fully decorrelated configurations from the MC calculations (previous paragraphs) were used as initial configurations [18] for MD simulations. We employed standard MD using the velocity Verlet method [19] with A/ = 0.45 fs. Periodic rescaling was used to maintain temperature at the set value [20]. In addition to the PE chain, a number of ethylene molecules were also present in the system. The number of ethylene molecules was taken to be the nearest integer to the average number of ethylene molecules present in the corresponding MC calculation. Independent MD runs were started from individual MC configurations and executed on different processors. For each initial structure, trajectories for all ehtylene molecules were stored. Results were subsequently pooled over all structures and trajectories in order to reduce statistical uncertainty in the diffusivity.

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TV. Results IV. 1, Solubility of ethylene in polyethylene confined in a pore The main results of the Gibbs ensemble calculations are density and composition of both phases. The average density in the pore was found to be 738 ±0.2 kg/m^ (one standard deviation of the mean) with typical fluctuations of 40kg/m^ (one standard deviation) for the large pore and 671 ±0.3 kg/m^ with typical fluctuations of 35 kg/m^ (one standard deviation) for the small pore. An independent calculation of the same molecular model of PE in the bulk (three-dimensional periodic boundary conditions) yielded 801 ±0.2 kg/m^ for pure PE [18]. The differences between pure bulk and confined systems are therefore highly significant and are a consequence of the confinement: although quite flexible, PE chains have a certain rigidity and therefore feel the presence of the pore walls when the pore size approaches the order of magnitude of the chain persistence length. The effect is more marked in the smaller pore, since the diameter of the pore is barely sufficient to allow the PE chain to turn around within it. As a matter of fact, the PE chain adopts conformations with high torsional and non-bonded potential energy. These conformations are rich in gauche defects and have a correspondingly higher proportion of "hairpin" turns than chains in the bulk. Unlike in the work of Baschnagel et al [21], the density profile of monomeric residues of the PE chain across the pore does not display any obvious maximum in the center of the pore. The PE chain fills the pore quite uniformly in the axial direction as well. On the other hand, the inability of PE to fill the pore at the same density as in the bulk implies that additional volume must be available for the ethylene solute compared with a bulk system. This effect is also evident in the relatively high acceptance ratios for molecule insertions in the Gibbs ensemble calculation. Solubility of ethylene was found to be 0.012 ±0.001 gEt/gPE in the large pore and 0.019 ±0.001 gEt/gPE in the small pore, which is higher than experimental values, than EOS calculations and also higher than bulk simulations at the same conditions (0.004 ±0.0003 gEt/gPE). It therefore seems that ethylene solubility is noticeably enhanced in confined geometries. Since the ethylene molecule is much smaller than the pore diameter, solubility is enhanced through an energetic mechanism (greater available volume) and is not hindered by an entropic mechanism, as could be the case in very small pores of a size comparable to that of the ethylene molecule. The consequence is an

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overall enrichment effect with respect to the bulk. Enhanced ethylene concentration close to active sites will almost certainly have a positive influence on polymerization reaction rate, although this point warrants a careful investigation. Equilibrium thermodynamic properties of highly confined PE are therefore markedly different from those of bulk ("macroscopic") PE. The more so, the smaller the pore diameter. Although the system studied is an idealization of real PE confined in real catalyst pores, the results obtained are highly suggestive that at the very small scale, where monomer incorporation to the chain takes place, PE bears little resemblance to the bulk material. IV. 2. Diffusivity of ethylene in polyethylene confined in a pore Analyzing the MD runs in confined geometry in order to obtain the diffusivity of ethylene in PE is a rather difficult task for two reasons. First, the slow progression of ethylene molecules through the PE matrix makes it necessary to run extremely long calculations. Secondly, ethylene diffusion takes place in a restricted volume and therefore cannot achieve a full threedimensional Einsteinian diffusion regime due the presence of walls. Current sophisticated methods for the evaluation of diffusivity ([22], [23]) are unfortunately not applicable in confined geometries, since they rely on the diffusion through an infinite domain (periodically infinite as e.g. imposed through periodic boundary conditions). Here we have followed an alternative approach based on tagging diffusant molecules and monitoring their positions (and through averaging, their concentration) as they follow deterministic trajectories through the fiilly elastic PE matrix. In this way, concentration of ethylene through the pore can be mapped at different times. Thanks to the axial symmetry, data can also be accumulated by projecting on the radial-axial plane, with a significant improvement in the quality of the statistics. Nevertheless the method is subject to strong statistical noise. The results obtained below were obtained from a set of 146 independent MD trajectories, each of them 1.1x10^ steps long. Typically, around a dozen molecules of ethylene diffused in each structure. Unlike for small penetrants [22], full polymer chain flexibility was found to be an absolute requirement for ethylene to diffuse.

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There is a subtle point worth discussing in the evaluation of the concentration profiles. One of our goals is to answer the question of whether diffusion in a confined polymer obeys classical macroscopic laws. In an periodically infinite simulation box, the determination of the diffusivity according to [22], although expensive in terms of computation, is rather straightforward, since that method is actually using the well known solution of a diffusion problem in an infinite 3D domain. The situation in a finite domain like the pore is quite different, since no analytical solution with which to compare the evolution of the concentration is available. Answering the question posed above requires that we compare the time dependent concentration field obtained from the MD calculation with time dependent concentration fields obtained by solving the macroscopic diffusion equation in the same confined domain and with a given value of the diffusivity. The answer to the question is positive (i.e. the behavior is in agreement with macroscopic conservation and constitutive laws) if it is possible to find a numerical value for the diffusivity for which the solution of the continuum diffusion equation matches the results of the MD calculations. According to this mapping strategy, we have solved the unsteady diffusion equation in the axisymmetric geometry defined in Figure 1 using finite elements techniques. Different values of the diffusivity were tried until an optimum match to the concentration field from the MD calculation was obtained. For the large pore (Figs. 4 and 6), the solution of the unsteady diffusion equation with D^^p^ = 7.2x10"^ mVs is found to match the MD results within the statistical uncertainty of the latter (small closed contour lines in Figure 4 are due to statistical noise). Even close to the bottom of the pore, where the diffusion problem is more strongly two dimensional and the contour lines have the greatest curvature, the agreement is complete. We can therefore conclude that the diffusive behaviour of ethylene in a PE-filled pore of about 3 nm diameter is well described by the macroscopic diffusion equation. On the other hand, the value of D^^p^ is higher than in bulk PE, most probably as a consequence of the lower density of PE in the pore and the associated enhanced available volume for diffusion.

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Monomer solubility and diffusion

Figure 4. Contour plot of ethylene concentration in the large pore on a plane that contains the pore axis and at ^ = 10 s. Isolines correspond to equally spaced concentrationvalues. Solid lines are results of MD calculation. Dahsed lines represent the numerical solution of the macroscopic diffusion equation (cylindrical coordinates) in the pore for D^^ p^ = 7.2 X10

mVs (best fit).

Figure 5. Contour plot of ethylene concentration in the large pore on a plane that contains the pore axis and at / = 5 X10 s. Isolines correspond to equally spaced concentrationvalues. Solid lines are results of MD calculation. Dahsed lines represent the numerical solution of the macroscopic diffusion equation (cylindrical coordinates) in the pore for Dg^ p^ = 1 2 . 5 x 1 0 m^/s (best fit).

For the small pore the best match between MD and macroscopic calculations was found for a value of D^^p^ = 12.5x10"^ mVs. The quaUty of this optimum match, however, is poor (Figure 5 and Figure 7). Ahhough it is possible to match the MD results in specific regions of the pore (e.g. close to the pore entry in Figure 5, where the isolines overlap), it is not possible to simultaneously extend the match to the whole spatial and temporal domains. Ethylene

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concentrations deeper in the pore are systematically higher than what the macroscopic diffusion equation predicts. _

S

0.0100

,

0.0075

B c o

\\

r

— K

>

1

«

1—n

Large pore

i

7^\

V

1

«

diffusion equation J MT) 1

t = 200 ps

^L

V

V

o

1

1—

X >^ ^%

\

1

1

Lv W^ V \

C

— 1

,

= 500 ps ^ /

N

0.0050

\ j

V

1 r = 1000 psJ

*-

I —.

/

Xs,

0.0025

x\

V ^ ^ ^^^^^^'^'*^*. j

[ / = 50 p s ' v s . L

1

1

1

1

^"n^

I'^'^VuTj

z(lO-^m) Figure 6. Ethylene concentration profiles along the pore axis for the large pore at four different times. Solid lines are results of Nffi) calculation. Dashed lines are numerical solution of the macroscopic diffusion equation (cylindrical coordinates) in the pore for D^^ p^ = 7 . 2 x 1 0 m^/s (best fit).

Monomer solubility and diffusion

m a.

0.020

197

t

fe) w c o

§ o c o o

0.015

0.010

>

0 . 0 0 5 L.

-imi

m

z(lO-^m) Figure 7. Ethylene concentration profiles along the pore axis for the small pore at four different times. Solid lines are results ofMD calculation. Dashed lines are numerical solution of the macroscopic diffusion equation (cylindrical coordinates) in the pore for D^Et,PE

= 12.5x10"'

m^/s (best fit).

This discrepancy is more pronounced at longer times (Figure 7). As we have seen, the 1.5 nm pore is already small enough to severely constrain the conformation of the PE chain and prevent its efficient filling of the pore. As a consequence, ethylene diffuses faster than in the bulk and in a way that cannot be reconciled with the macroscopic diffusion equation with a constant value of ^Et,PE • ^^ ^s ^f course possible to maintain the macroscopic formalism by assuming a phenomenological, position dependent D^^p^, This would of course come at the price that there is no a priori macroscopic way of predicting that dependence. As far as ethylene diffusivity is concerned, the 1.5 nm pore seems to be truly microscopic and the macroscopic description breaks down.

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V. Conclusions The atomistic investigation of solubility and diffusivity of ethylene in a nanopore filled with polyethylene by means of MC and MD techniques strongly suggests that thermodynamic and transport properties of a confined system differ markedly from their counterparts in the bulk. Polyethylene density in the pore is significantly lower than in the bulk as a consequence of the restricted space available. Hairpin chain turns and bonds in gauche state occur with higher probability than in the bulk. These high-energy conformations are required for the PE chain to be able to fit in the available pore volume. Solubility of ethylene is predicted to be enhanced due to the lower density of the PE matrix and the ensuing greater available volume for insertion. The same mechanism is responsible for the enhanced value of the diffusivity as compared with the bulk value. Ethylene transport is seen to obey the macroscopic diffusion equation in the large pore. In the small pore however, the behavior of ethylene is quantitatively and qualitatively different from behavior in the bulk and cannot be described by the macroscopic diffusion equation with constant diffusivity. As far as ethylene transport in PE is concerned, a 1.5nm pore behaves as a microsocopic object, whereas a pore of 3 nm diameter can already be described by standard macroscopic conservation and constitutive laws. Molecular modeling techniques offer a way to investigate the former type of system. The observed deviations in solubility and diffiisivity of ethylene in polyethylene confined in a nanopore are such that they would increase the amount of monomer available for reaction (incorporation into a growing polyethylene chain) should the pore contain an active catalytic site. This observation may be of some relevance for the design and modeling of more efficient supported catalysts. Acknowledgment The authors would like to acknowledge the very fruitfiil vigorous discussions with all partners of the PMILS project. support by the EC through contracts G5RD-CT-2002-00720 2005-016375, and partial support by CICYT grant MAT greatfiiUy acknowledged.

interaction and Major financial and NMP3-CT1999-0972 are

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REFERENCES [1] Ferrero, M.; Chiovetta, M. Catalysts Fragmentation During Propylene Polymerization: Part I. The effects of Grain Size and Structure. Pol. Eng. Sci. 1987,27,1436. [2] Ferrero, M.; Chiovetta, M. Catalysts Fragmentation During Propylene Polymerization: Part II. Microparticle Diffusion and Reaction Effects. Pol. Eng. Sci. 1987,27,1447. [3] Hutchinson, R.; Chen, C; Ray, W. Polymerization of Olefins through Heterogeneous Catalysis. X. Modeling of Particle Growth and Morphology. J. Appl. Pol. Sci. 2992,44, 1389. [4] Estenoz, D.; Chiovetta, M. Olefin Polymerization using Supported Metallocene Catalysts: A Process Representation Scheme and Mathematical Model. J. Appl. Pol. Sci. 2001, 85, 285. [5] Leach, A.R. Molecular Modeling. Principles and Applications, Longman, 1996. [6] Dickson, R.M.; Norris, D.J.; Tzeng Y.-L.; Moemer W.E. Three-Dimensional Imaging of Single Molecules Solvated in Pores of Poly(acrylamide) Gels. Science 1996,274,966. [7] Van der Ploeg, P.; Berendsen H. J. C. Molecular dynamics simulation of a bilayer membrane, J. Chem. Phys. 1982,76, 3271 [8] Ryckaert, J. P.; Bellemans, A. Molecular dynamics of liquid n-butane near its boiling point. Chem. Phys. Lett. 1975, 30,123 [9] Brooks III, C.K.; Karplus M.; Pettitt, B.M. Proteins: a theoretical perspective of dynamics, structure and thermodynamics, in Advances in Chemical Physics, Vol. LXXI, John Wiley & Sons, 1988 [10] Cornell, W.D.; Cieplak P.; BaylyC.L; Gould I.R.; Merz Jr. K.M.; Ferguson D.M.; Spellmeyer D.C.; Fox T.; Caldwell J.W.; KoUman P.A. A Second Generation Force Field for the Simulation of Proteins, Nucleic Acids, and Organic Molecules J. Am. Chem. Soc. 1995,117,5179. [11] Raj N.; Sastre G.; Catlow C.R.A. Diffusion of Octane in Silicalite: A Molecular Dynamics Study, J. Phys. Chem. B 1999,103, 11007. [12] De Pablo, J.J.; Laso, M.; Suter, U.W. Estimation of the chemical potential of chain molecules by simulation. 1992 J. Chem. Phys. 96,6157. [13] Laso, M.; De Pablo, J.J.; Suter, U.W. Simulation of phase equilibria for chain molecules, J. Chem. Phys. 1992, 97, 2817 [14] Rosenbluth, M.N.; Rosenbluth, A.W. Equations of state calculations by fast com-'puting machines. J. Chem. Phys 1953,21, 1087; Siepmann, J.I.; Frenkel, D. Configurational Bias Monte Carlo: a new sampling scheme for flexible chains. Mol. Phys. 1992, 59; de Pablo, J.J.; Laso, M.; Suter, U.W. Simulation of Polyethylene Above and Below the Melting Point. J. Chem. Phys. 1992, 96, 2395. [15] Mavrantzas, V. G.; Boone, T. D.; Zervopoulou, E.; Theodorou, D. N. End-Bridging Monte Carlo: An Ultrafast Algorithm for Atomistic Simulation of Condensed Phases of Long Polymer Chains. Macromolecules 1999, 32, 5072 [16] Leontidis, E.; De Pablo, J.J.; Laso, M.; Suter, U.W. A Critical Evaluation of Novel Algorithms for the Off-Lattice Monte Carlo Simulation of Condensed Polymer Phases, Adv. Polym. Sci. 116,283, "Atomistic Modelling of Physical Properties", Chapter VIII, Springer Verlag (1994) [17] MuUer, M.; Nievergelt, J.; Santos, S.; Suter, U.W. A novel geometric embedding algorithm for efficiently generating dense polymer structures J. Chem, Phys. 2001 114,9764 [18] Garcia Pascua, O., Ahumada, O., Laso, M., Miiller. M. The effect of the initial guess generator on molecular mechanics calculations. Molecular Simulation 2003 29(3), 187

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[19] Swope, W,C.; Anderson, H.C.; Berens, P.H.; Wilson, K.R. A Computer Simulation Method for the Calculation of Equilibrium Constants for the Formation of Physical Clusters of Molecules: Application to Small Water Clusters. J. Chem. Phys. 1982, 76,637 [20] Allen, M. P.; Tildesley D.J. "Computer Simulation of Liquids", Oxford University Press (1987) [21] Baschnagel, J.; Mischler C; Binder K. Dynamics of confined polymer melts: Recent Monte Carlo simulation results. Joumal de Physique IV. Proceedings of the International Workshop on Dynamics in Confinement, Eds.: B. Frick, R. Zom, H. Buttner, Vol. 10, Pr 7, May (2000) [22] Gusev A.A.; MuUer-Plathe F.; van Gunsteren W.F.; Suter U.W. Dynamics of Small Molecules in Bulk Polymers. Special volume on "Atomistic Modeling of Physical Properties of Polymers" of Adv. Polym. Sci. 1994,116, 273. [23] Gusev A. A.; Suter U.W. A Model for Transport of Diatomic Molecules through Elastic Solids. J. Comput.-Aided Mater. Des. 1993,1,63.

Multiscale Modelling of Polymer Properties M. Laso, E.A. Perpete (Editors) © 2006 Elsevier B.V. All rights reserved.

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Chapter 2

Detailed Atomistic Simulation of the Barrier Properties of Linear and Short-Chain Branched Polyethylene Melts Through a Hierarchical Modeling Approach Patricia Gestoso,^'^ Nikos Ch. Karayiannis"^'^ ''Accelrys Ltd., 334 Cambridge Science Park, Cambridge CB4 OWN, UK ^Rhodia Recherches, Centre de Recherches de Lyon, Saint-Fons Cedex 69192, France ^Department of Chemical Engineering, University ofPatras, Patras 26504, Greece '^Institute of Chemical Engineering and High Temperature Chemical Processes, Patras 26504, Greece

I. Introduction Sorption and diffusion of small molecules in polymers play an influential role in industrial applications. Many technologically important processes rely on the design of macromolecular materials with tailored barrier characteristics, with the permeability coefficient {P) being the key factor determining the quality of barrier end-products. Among others, gas permeation through polymers is critical to the technology of membranes for industrial gas separation and purification, the production of packaging materials, coatings, biosensors and drug implants and the removal of residual monomer or solvents from the products of polymerization. For example, a polymeric membrane designed for use in food and beverage packaging has to be practically impermeable to certain gases like oxygen (O2). In a similar fashion, storage tanks should efficiently contain gases, and thus exhibit very low permeability coefficients. Finally, in gas separation and purification processes the design goal is the development of a barrier material selective to one molecular species.

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To successfully confront the challenge of designing polymeric membranes with tailored barrier characteristics, a connection must be established between macromolecular morphology and chemical composition with the barrier properties of the end-product material. Towards this direction, one may resort to direct experimental measurements consisting of two main steps: first, the novel membrane is synthesized and characterized regarding the morphological features, and then its barrier properties are measured and tested under a wide variety of conditions [1,2]. In industrial practice, the material morphology is usually dictated by the manufacturing conditions and the thermodynamic properties of the material throughout its processing history. While experimental studies represent undoubtedly the most reliable and direct way to determine the ability of a novel polymer to act as an effective barrier structure, the execution of a large series of trial-and-error synthetic processes and successive permeability measurements may be proven, in practice, too expensive and/or time-consuming. Li parallel to direct calculations, large collections of experimental data over a wide range of systems and conditions may serve as the basis for the fabrication of phenomenological correlations [3]. From the theoretical point of view, sorption and diffusion in and through macromolecular systems can be explained and predicted based mainly on the concept of free volume (FV) [4-6], the dual-mode transport model [7-10] and molecular theories [11-14]. In practice, no plastic material is perfectly impermeable: Sorption and diffusion of light gases is possible, to certain extend, in any polymeric matrix as the packing of chains and their constituent atoms, even in the case of very dense polymers, leaves holes or micro voids [15], the sum of which forms the free (or unoccupied) volume (FV). The cavities of free volume can accommodate gas molecules depending on the size and shape of the penetrants relative to their own volume and shape, and the polymer-penetrant interactions in relation to the penetrant-penetrant and polymer-polymer interactions. The subset of sorption sites where the molecules can reside comprises the accessible volume (AV) of the polymer, depending on both the host matrix and the penetrant. Finally, computer simulations and modeling can play an important role in the establishment of the dependence of barrier properties on structure by elucidating, at the atomistic level, all the relevant mechanisms controlling the whole transport process. Computer-generated polymer structures built on detailed molecular models can be subjected to novel simulation techniques that allow, within modest computational time and resources, the accurate prediction of permeability under a wide range of conditions without the usually high cost of experiments or without resorting to simplifications and assumptions invoked by most theoretical studies. In recent years, the continuous increase in computational power, the affordable cost of powerful workstations and the

203

Atomistic simulation of the barrier properties

development and implementation of novel simulation techniques have drawn the attention of the polymer community, setting computer-aided design of materials in par with the well-established experimental and theoretical approaches. Li the present chapter we report our recent work concerning the detailed atomistic simulation, through a hierarchical modeling approach, of the sorption and diffusion of small gas penetrant molecules at infinite dilution (low concentration) through purely amorphous polyethylene (PE) systems characterized by different molecular architectures and molecular weights under a variety of temperature conditions. Section II presents a brief description of the molecular mechanisms and the physical aspects of gas permeation in polymers. Section III deals with the description of the methods adopted in the simulation of sorption and diffusion in rubber and glassy polymers. Results of our recent work are presented in Section IV. Finally, main conclusions and future plans are analyzed in Section V. II. Molecular mechanism of low-concentration gas transport in polymers A coarse schematic representation of the low-concentration gas permeation in a polymeric membrane is given in Fig. 1.

y |iiiiiiiiiiiiB

-<

X

Po

•

Figure 1. Coarse schematic representation of the low-concentration permeation of penetrant molecule in polymeric membrane of thickness / as a result of the pressure difference Ap.

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P. Gestoso andN.Ch. Karayiannis

Permeation can be envisioned as a three-step process where the penetrant molecule a) is absorbed at the surface of the inlet side of the membrane, b) diffuses through the polymer matrix and c) is desorbed for the outlet surface of the membrane [16,17]. At equilibrium, penetrant concentration, c, is related to the pressure, /?, by the isothermal relation c = S{c)p

(1)

where S{c) is the solubility coefficient. At infinite dilution (low concentration of the penetrant molecule) Eq. 1 reduces to a form of Henry's law and solubility is practically independent of concentration (or pressure). Accordingly, penetrant concentration in the inlet and outlet surfaces of the polymeric membrane (Fig. 1) is given by

^in=*^An '^out=^/^out

(2)

Transport process across the membrane can be quantified by the permeation rate (flux), y, according to Pick's first law

J =-Z)^ dx

(3)

where D is the diffusion coefficient. The rate of change in the concentration profile is given by

^ =± Z , ^ d/ dx V doc)

(4)

If the diffusion coefficient, D, is independent of the position in the sample, x, (i.e. the medium is homogenous [18]), then Eq. 4 corresponds to Pick's second law

At

dx^

Atomistic simulation of the barrier properties

205

At steady state, Eq. 5 is reduced to dc ^ d^c ^ dc ^ ^ — = 0 => —r- = 0 => — = constant

dt

dx

... (6)

dx

Based on Eq. 2 and the steady-state conditions in Eq. 6, the permeation rate (Eq. 3) can be rewritten as

/

I

I

The low-concentration permeabiUty coefficient, P, is given as the product of diffusion and solubility coefficients P =D5

(8)

where the permeability coefficient, P, is related to permeation flux, J, through

P='-^ JAp

(9)

Based on Eq. 8, the low-concentration permeation of a penetrant molecule in a polymer matrix, expressed through the permeability coefficient (P), can be considered as the combined effect of kinetic (diffusion) and thermodynamic (sorption) factors, quantified by the diffusion (Z)) and solubility (5) coefficients, respectively. Solubility of a penetrant in a polymeric membrane depends mainly on the nature and magnitude of the polymer-penetrant interactions relative to the polymer-polymer and penetrant-penetrant interactions. In addition, clusters of free volume (sorption sites) should exist within the macromolecule, large enough to accommodate the penetrant molecules [19]. Penetrant diffusion is mainly controlled by the size and shape of the penetrant, the magnitude of the potential interactions with the host matrix and the size, shape, distribution and connectivity of the free-volume cavities. In the melt state, at high temperatures (well above the glass transition temperature Tg), polymer segments undergo significant thermal fluctuations leading to enhanced

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chain mobility. Consequently, the size, shape, distribution and connectivity of the sorption sites are continuously rearranged and the penetrant is "carried along" by density fluctuations caused by the thermal motion of the surrounding polymer chains [19]. In a glassy polymer {T < Tg) the corresponding transport mechanism is considerably different: glassy configurations are trapped in local minima of the potential energy and conformational transitions between different energy minima are prohibited by exceptionally high energy barriers. As the mobihty of the polymer segments is significantly reduced, the distribution and connectivity of the network of sorption sites remains practically unaltered within the time frame of the transport process, undergoing only minor fluctuations. The work of Takeuchi [20] revealed that the penetrant molecule spends most of its time trapped within a formed cavity of free volume and only infrequently it performs jumps from one cluster to a neighboring one through channels which are formed instantaneously via fluctuations in regions of lower density or enhanced molecular mobility. Consequently, penetrant diffusion in glassy polymers is orders of magnitude slower than in high-temperature melts. III. Simulation methods to study permeation in polymers The most straightforward way to study the transport behavior of a small molecular weight penetrants in a polymer melt is to employ Molecular Dynamics (MD) simulations [21] (see also chapter II.6 of the book), following the motion of the penetrant for long times when the hydrodynamic limit is reached and Fickian diffusion is established. Diffusion coefficient, Z), is readily calculated by invoking the Einstein equation

D = \mi\

([rp(O-r/0tf)] 6/

(10)

where the brackets, <>, denote averaging over all trajectories and [rp(0-rp(0)]^ is the mean square displacement (MSD) of the penetrant at time t. In the most common implementation, MD simulations track the dynamics and motion of both the polymer matrix and the penetrant molecules, which either coexist from the very beginning of the simulation (i.e. the chain segments are originally grown and relaxed in the presence of penetrants) or the penetrants are inserted a posteriori in the amorphous cell in an energetically-biased way to avoid overlaps with the existing polymer segments [22-38].

Atomistic simulation of the barrier properties

207

The main advantage of the MD method is that, by providing direct and detailed information about system dynamics, it practically constitutes a computational experiment, as the assumptions and simplifications invoked in its implementation are only related to the applied potential force field (molecular model). In parallel, MD's major limitation emanates as well from its working pattern: even in a state-of-the-art implementation executed on parallel supercomputers, thousands of CPU hours are required to study the real dynamics and transport behavior of large penetrant molecules in complex macromolecular environments. Even worse, at temperatures below Tg, penetrant mobility in glassy structures is too slow to be tracked by conventional MD simulations within reasonable computational time. An excellent alternative is the Transition State Theory (TST) as introduced by Arrizi, Gusev and Suter [39-42] for the computational study of gas transport in polymeric glasses. Li the limit of low concentration, TST is applied on detailed atomistic systems and evolves in a coarser level of representation as a succession of infrequent events. For each transition, the "reaction trajectory" leading from a local energy minimum to another through a saddle point in configuration space is tracked, and the transition rate constant is evaluated. Li the Gusev-Suter implementation [40-42], TST is initiated as a 3-D grid of fine resolution (typically 2-3 A) is laid on the amorphous cell consisting of the equilibrated polymer chains, covering every part of its volume. Next, a spherical probe with dimension identical to the one of the penetrant molecule (represented as a united-atom site) is inserted in every point (x, y, z) of the 3-D grid and the potential energy £'ins(jc, y, z) is calculated as the non-bonded interactions between the probe and the polymer segments. Thus, based on the calculated value of ^ms(^, y, z), the simulation cell is divided into clusters of accessible volume (regions of low energy, containing the local minima of the energy) and domains characterized by high energy (excluded volume regimes). If there are no potential interactions between the dissolved species (i.e. infinite dilution) the excess chemical potential, //ex, according to the Widom method, is given by

//„=/?nn

exp

^ E. ^ ms

(11)

where R is the universal gas constant, T is the temperature, k^ the Bohzmann constant and the brackets, <>, denote averaging over all the grid points and probe insertions. Additionally, the low-concentration solubility coefficient, S, is related to the excess chemical potential, /iex, through

208

P. Gestoso andN.Ch. Karayiannis

'5 = exp(-|^J

(12)

Each energy local minimum is associated with a sorption "microstate" in the configuration space of the penetrant. Adjacent microstates (i.e. sorption sites) are separated by high-energy surfaces. The penetrant molecule spends most of its time "trapped" in the microstates and only infi*equently hops from one microstate to a neighboring one. Penetrant diffusion can be envisioned as a succession of rare transitions between adjacent microstates. An elementary transition from microstate / to microstate 7 can be described by a characteristic first-order rate constant, /:,_;. For temperatures low enough, for which the polymer matrix can be considered frozen, the rate constant for the transition from microstate / to microstatej is given by [39-42]

* , . , = ^ ^


where h is Plank's constant, k^ is the Boltzmann constant and g/, Qy are the partition fimctions of the molecule in the microstate / and on the high-energy surface (separating microstates / and7), respectively. The partitionfimction,Qy on the separating surface is given by [39^2]

Qij =

7 1 ^ — \ p(x,y,z)dA

(14)

where m is the mass of the dissolved molecule, p(jc, y, z) is the Boltzmann probability density function for finding the inserted atom at point (x, y, z) and dL4 is an elementary surface area on the separating surface between / and 7, whose total area is denoted by Ay. Similarly, the partition function of the molecule in microstate / is given by [39-42]

Q,=[^^T

lp(x,y,z)jy

(15)

Atomistic simulation of the barrier properties

209

where dF is an elementary volume of the microstate i whose total volume is denoted by Vt. Finally, the probability p{x, y, z) of finding the inserted atom at point (jc, y, z) is given by

p(x,y,z)ocQxp

Einsix.y.z)^

(16)

kj

Li the original formulation of TST for glassy systems the polymer matrix was considered as completely frozen, leading to significant underestimation of the solubility and diffusivity coefficients. In the modified approach, the small thermal fluctuations of the constituent atoms are taken into account under the simplification that the matrix segments execute independent "elastic" motions, tethered to their equilibrium positions through harmonic springs. These displacements around the equilibrium positions follow a Gaussian probability density function [39-42] \

VY

fr(Ar,, Ar2...., Ar^) oc exp

-E^2(Ar,^)

(17)

where N is the total number of polymer atoms in the simulation box and Ar^t = fk - is the displacement of atom k from its equilibrium position, . In the simplest version, an average value independent of atom type, 3A^, is assigned to the corresponding MSD , A^ being termed as "smearing factor". In the modified version of TST that incorporates the elastic motion of the polymer segments the corresponding density function ^(x,;^, z) is given by

p(x.y.z)«Jexpl-5M|^

A

^(Ar„...,Ar^)c/V,...jV^

(18)

where W(Ari,...,ArAf) is given by Eq. 17. Knowledge of the probability density function allows the calculation of the transition rate constants through Eqs. ISIS. At equilibrium, the average residence time, r, in a microstate /, is given by [19,39-42,45]

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P. Gestoso andN.Ch.

Karayiannis

/

where the index / runs over all sorption sites adjacent to the reference microstate /. Consequently, the corresponding probability for the transition from microstate / to the adjacent microstate7,/7i_j, is given by

"-''it.

<^"'

Based on the information about the distributions of rate constants and lengths governing the transitions, the spatial connectivity and topology of the microstates and the mobility of the dissolved molecule (as provided from Eqs. 13,19,20), a kinetic Monte Carlo (kMC) simulation is performed to generate long stochastic trajectories, each consisting of a large succession of transitions of penetrant walkers hopping in the 3-D coarse-grained lattice [19,39-42,45,46]. The low-concentration diffusion coefficient, D, can be calculated from the Einstein equation (Eq. 10) when Fickian diffiision is established. Early TST simulations [39-42] demonstrated that for small times scales the motion of the penetrant is not diffusive. Listead, an "anomalous diffiision" behavior is observed, where the mean square displacement of the penetrant, [rp(0-rp(0)]^. rp(0-rp( scales with time, t, as [Y^{tyr^{0)f ©c f with « < 1. Similar conclusions were drawn by the early MD works of Miiller-Plathe et al. for oxygen (O2) and helium (He) diffusion in amorphous polyisobutylene (PIB) [22], Pant and Boyd for methane (CH4) in polyethylene (PE) [24] and Chassapis et al. for O2 in PE [28]. Experimentally, short-time anomalous diffiision behavior has been identified through NMR measurements during diffiision of penetrants in zeolite systems by Karger et al. [47] and in molecular sieves by Ylihautala et al. [48]. An interesting observation in the original Gusev-Suter TST simulations [41] is that the crossover from anomalous to Fickian diffiision occurs when the MSD of the penetrant walkers is commensurate to the dimensions of the amorphous cell, suggesting that Fickian diffusion is artificially established in the simulation by the applied periodic boundary conditions. This finding has been confirmed by a very recent TST approach for O2 diffiision in glassy configurations of poly(ethylene terephthalate) (PET) and poly(ethylene isophthalate) (PEI) [49]. To capture correctly the duration of the anomalous behavior one should repeat TST simulation on simulation boxes with progressively larger dimensions. On

Atomistic simulation of the barrier properties

211

the other hand, as indicated in the work by Karayiannis et al. [45], the diffusion coefficient of penetrant walkers, for a given lattice with prescribed topology and connectivity, depends solely on the distribution of rate constants governing the transitions between microstates. Short-range spatial heterogeneity affects the duration of the anomalous behavior but has no effect on the estimated diffusivity. The rather coarse representation of the thermal motion of the polymer atoms as expressed by the smearing factor (same for all atomic species) limits the apphcability of TST [39-42] to relatively simple and small gas molecules (He, H2, O2, N2 and marginally CH4), whose sorption in the polymer matrix does not affect the thermal motion of the surrounding chains. Large molecules, for example CO2, may force the polymer atoms to undergo significant local relaxation in order to accommodate their presence, rendering the smearing factor of Eq. 17 obsolete to effectively capture the local dynamics of the chains. In addition, the simplified harmonic form of the smearing factor in the original TST approach is not adequate to describe more complex segmental relaxation mechanisms like phenyl ring vibrations and flips that may have a profound impact on the "dynamic flexibility" of the system [49]. Solution to this central shortcoming of the original TST can be provided by allowing the local relaxation of the polymer environment around the dissolved molecule through appropriate implementation of the multidimensional Transition State Theory, as proposed by Greenfield and Theodorou [50-54]. To this date, TST has been successfully applied on various penetrantpolymer systems providing reliable and accurate predictions for the solubility, (5) diffusivity (D) and permeability (P) coefficients (the latter being indirectly calculated from Eq. 8) of light gases (He, H2, O2, N2 and CH4) in glassy polymers. For larger and non-spherical penetrants, like CO2, TST fails to capture the correct sorption and transport behavior [40-42,49,55-68]. As the TST approach departs from the detailed atomistic level and resorts to a coarser level of description and because of the invoked simplifications related with the simplistic representation of the segmental relaxation of the polymer host, the approach should be validated against available data either from experimental studies or MD simulations. Critical preconditions of utmost importance for the successful prediction of the barrier properties, using either TST or MD, are a) the generation of realistic atomistic configurations of the polymer systems, fully equilibrated at all length scales, b) the calculation of the barrier properties as statistical averages over as many as possible totally uncorrelated computer-generated structures and c) the apphcation of a very accurate and, if possible, CPU inexpensive force field. As analyzed extensively in chapter 1.2 of the book, for chemically simple macromolecular systems (for example polyolefms, polydiens and polyethers)

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P, Gestoso and N. Ch. Karayiannis

full-scale equilibration is provided by the application of stochastic Monte Carlo algorithnis built around advanced chain-connectivity altering moves [69-74]. In the case of more complex macromolecular systems bearing aromatic units and bulky inflexible and polar groups along the backbone (for example polyesters and polyimides), equilibration is achieved by a procedure consisting of a large succession of short MD runs in the canonical (NVT) or isothermal-isobaric (NPT) ensembles, corresponding practically to a series of annealings, coolings, compressions and decompressions at various conditions of temperature and pressure [55-65]. hi the next section we will present results about the low-concentration solubility, diffusivity and permeability of oxygen and nitrogen in linear and short-branched polyethylene melts as obtained from application of a hierarchical modeling methodology integrating simulation techniques applied on different time and length scales. IV. Multiscale Modeling of the Barrier Properties of Polyethylene Systems IV. 1. Summary of the Hierarchical Approach The adopted hierarchical methodology is divided into three phases: 1. Application of chain-connectivity altering MC moves [69-74] for the rapid and robust equilibration of purely amorphous PE samples in the atomistic level of description employing a united-atom representation. 2. Conversion of selected representative and uncorrected configurations of the simulated systems to an explicit (all-atom) representation by appropriately adding the corresponding hydrogen atoms. Application of short NVT MD runs accompanied by static energy minimizations to afford additional local relaxation due to the addition of hydrogen atoms. 3. AppHcation of the Gusev-Suter Transition State Theory (TST) [3942] to calculate the barrier properties of the computer-generated PE samples at conditions of infinite dilution (low concentration). IV.2. Systems Studied All simulated linear polyethylene samples are denoted as "C^r" where N is the total number of carbon atoms per chain. The short-chain branched (SCB) analogs are referred to as "SCB_(A^br+l)^A^freq_^br^Cb", where A^br is the number of branches distributed regularly along the main (linear backbone), Cb is the

Atomistic simulation of the barrier properties

213

branch length (i.e. the number of carbon atoms of each branch) and TVfreq is the branching frequency (i.e. the number of carbon atoms along the main backbone between two successive branch points). More details about the notation and the molecular characteristics of the SCB structures can be found in chapter II.6. Regarding the linear PE systems, at all studied temperatures a small degree of polydispersity was allowed to accelerate the equilibration phase performed through MC techniques, which is the most time-consuming stage of the whole scheme. For the short-chain branched systems all molecular characteristics were kept strictly monodisperse (branch length, Cb, branch frequency Nf^^, and number of branches, iVbr) to avoid adding extra structural parameters, such as heterogeneity in the branching frequency and in the number of branches per chain. In addition, all SCB samples bear exactly the same number of carbon atoms (corresponding to a total molecular weight of 1990g/mol, i.e. linear C142 PE as an analog) so as to attribute any detectable differences in the barrier properties solely on molecular characteristics. Table I summarizes the details for all simulated PE systems. Initial linear structures were provided by the three-stage constant-density energy minimization technique of Theodorou and Suter [75,76], which generates configurations trapped at a local energy minimum with reaUstic bonded geometry (bond length, bending angles and torsions). Initial dimensions of the cubic amorphous cell were selected so as to correspond to density values that followed the quantitative trend as a function of molecular length reported from our previous atomistic MC simulations on similar PE structures [71]. The initial configuration for each SCB system was generated through the commercial software package Materials Studio (version 3.0) [77] by Accelrys Inc at identical density values as their linear counterparts of the same molecular weight. All simulations of the equilibration phase were executed with a molecular model (potential force field) adopting the united-atom (UA) representation where each CH3, CH2 and CH unit is treated as a single spherical interacting site (pseudoatom). In previous atomistic simulations the molecular model, a hybrid combination of accurate force fields existing in the literature [78-80], has provided excellent estimates of the volumetric [71], conformational [72] and dynamic [81] properties of PE samples. More details about the employed potential force field regarding the mathematical expressions and the corresponding parameters describing all bonded and non-bonded interactions can be found in chapter 11.6 of the book.

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Karayiannis

Table I. Details of the Simulated PE Systems.

no. of chains 40 C78 22 Cl42 6 C500 3 ClOOO SCB 11 xi2; 10x1 22 22 SCB 9x 14 8x2 22 SCB 7x 16 6x5 22 SCB 5x:22 4x8 System

no. of sites 3120 3124 3000 3000 3124 3124 3124 3124

Total MW (g/mol) 1094 1990 7002 14002 1990 1990 1990 1990

Polydispersity Index 1.083 1.083/1.000 1.053 1.053 1.000 1.000 1.000 1.000

IV. 3, Step 1: Equilibration of samples in united-atom representation All linear monodisperse or polydisperse PE systems were subjected to exhaustive MC simulations based on chain-connectivity altering moves. As described in chapter 1.2 in the case of polydisperse samples, molecular weight distribution is controlled by casting the MC simulations in the semi-grand statistical ensemble through the use of an appropriately defined spectrum of chemical potentials [69,70]. Chain lengths of polydisperse systems were sampled uniformly in the closed interval [iVav(l - A), A^av(l + A)], where TVav is the number-average chain length and A denotes the half-width of the chainlength distribution reduced by A^av The polydispersity index, /, (reported in Table I for all PE samples) is related to A through / = 1 + A^ / 3. The simulation of polydisperse linear systems was carried out with the following mix of moves: internal libration (flip) (5%) [82], end-mer rotation (3%), intermolecular reptation (15%) [69,73], concerted rotation (20%) [83], end bridging (15%) [69,70], double bridging (20%) [71-73], intramolecular double rebridging (10%) [71-73], intramolecular end bridging (10%) and volume fluctuation (2%). To compare against the SCB analogs, a single strictly monodisperse C142 PE sample was simulated using a reduced set of moves (all algorithms that introduce polydispersity were excluded): internal libration (5%), end-mer rotation (3%), reptation (15%), concerted rotation (20%)), double bridging (35%), intramolecular double rebridging (10%), intramolecular endbridging (20%) and volumefluctuation(2%). The complex molecular architecture of SCB systems required the application of a more advanced MC scheme consisting of novel local or chainconnectivity altering moves [73]. Since all branching parameters were held fixed, direct application of chain-connectivity altering algorithms was not possible at this stage. For this reason all initial SCB configurations were

215

Atomistic simulation of the barrier properties

subjected to very long NPT MD simulations for a total duration of 100 and 400ns at T = 450 and 300K, respectively. All MD simulations were conducted using the LAMMPS (large-scale atomic/molecular massively parallel simulator) software (version LAMMPS 2001) [84,85]. Temperature and pressure fluctuations were controlled through the Nose-Hoover thermostat [86,87] and Andersen barostat [88], respectively. For the integration of the equations of motion, the multiple time step algorithm rRESPA [89,90] was employed with the small and large steps set equal to 1 and 5fs, respectively. More details about the dynamics of the SCB PE melt configurations can be found in chapter II.6 of the book.

5000

T"

instantaneous value -running average value selected frames

4500 4000 3500 3000 V 25002000 1500 1000

-»—r 1

10 MC Steps (10')

Figure 2. Instantaneous and running average values of the end-to-end distance, , as a function of MC steps for polydisperse, linear C142 PE system at 7 = 450K and P = latm. Filled cycles denote representative and statistically uncorrelated frames from the MC trajectory taken for successive TST calculations.

216

P. Gestoso andN.Ch. Karayiannis

instantaneous value -running average value selected frames

0.79 H

0.78 H

0.74

I

0

1

I

I

2

I

I

3

I

4

I

I

5

I

I

I

6

I

7

I

I

8

I

I

9

I

I

10

MC steps (10')

Figure 3. Same as in Fig. 2 but for the density.

At the end of the MC (or MD) simulations, a set of representative and statistically uncorrelated frames was carefully selected from the whole trajectory with dimensions (the radius of gyration for SCB, and the end-to-end distance for linear melts) and density similar to the average values obtained from sampling over the equilibrated part of the trajectory. Figs 2 and 3 present the instantaneous and running average values of the mean squared end-to-end distance, , and density, respectively, as a function of MC steps as obtained from simulations on linear polydisperse C142 system at r = 450K and P = latm. In both figures filled cycles denote representative and statistically uncorrelated frames from the MC trajectory selected for successive TST calculations. This procedure was repeated over all MC trajectories and the number of selected frames increased as the temperature decreased. Similar procedure for extracting configurations for successive studies of the barrier properties was adopted in the case of the NPT simulations of SCB analogs. Figure 4 presents the dependence of density on average chain length, A/av (or equivalently molecular weight) as obtained from MC simulations on linear polydisperse PE samples at T = 300 and 450K. Also shown are available experimental volumetric data at the same conditions of temperature and pressure as the simulations [91,92]. At room temperature {T = 300K) PE is a semi-crystalline material and the density value reported by Cubero et al. [92] is corrected so as to correspond to a purely amorphous sample. It is evident that at both temperatures the agreement between the density as calculated from MC

Atomistic simulation of the barrier properties

211

simulations and as reported experimentally is excellent, the relative error being less than 1%. As reported in a previous work [49], the precise calculation of density is a critical prerequisite for the successful and accurate prediction of the barrier properties. Even for relatively small errors of about 2-3%, one may need to impose the density to match the corresponding experimentally known values; otherwise consecutive TST simulations may fail to capture the qualitative and quantitative trends for sorption and diffusion [49]. Fig. 5 depicts the variation of density with temperature as obtained from MC simulations on a polydisperse, linear Cyg system. As shown by the linear fit in Fig. 5 it can be concluded that density increases hnearly with temperature in the whole range of appUed temperatures. Divergence from linearity at the lowest studied temperature (T= 250K) can be attributed to the onset of transition to the glassy state. Recent MC simulations suggest a glass transition temperature in the range of 190-200K for the linear C78 amorphous sample. At high enough temperatures, in the melt regime, MC is capable of equilibrating the volumetric and conformational characteristics of PE systems within two CPU hours. 0.88-1

1

1

1

1

1 —— 1 _

1

1

—1

1

1

0.860.84-

•

-^ 0.82-

• \

£ 3

A

0.80 -

g 0.78-

i ]

a> 0

0.760

0.74n 77 -

(3

• 0

simulation 300K simulation 450K 1

200

•

1

400

D

#

experimental 300K experimental 450K

'

]

-

1

600

800

1000

Molecular Length

Figure 4. Dependence of density on average molecular length as obtainedfromMC simulations on linear polydisperse PE systems at r = 300 and 450K (in both cases P = latm).

218

P. Gestoso andN.Ch.

0.62 200

250

300

350

400

450

500

550

Karayiannis

600

Temperature (K) Figure 5. Density as a function of temperature as obtainedfromMC simulations on a linear polydisperse C78 PE system at P = latm. The solid line corresponds to the best linear fit on simulation data.

The effect of molecular architecture (short-chain branching) on density is depicted in Fig. 6 as obtained from NPT MD simulations on PE systems bearing a total of 142 carbon atoms at T = 300K and P = latm. Highest density is recorded for the SCB configuration bearing the shortest (and most frequently distributed) branches (Ci), whereas for longer branches (C2-C6) density slightly decreases with increasing branch length. It can be safely concluded that shortchain branching has a very minor effect on density, at least in the range of molecular weight and temperature reported here: The relative difference between the higher and lower density values lies within the statistical error of the simulation method, about 1%. Volumetric trends of Fig. 6 are in excellent qualitative agreement with available experimental data [93]. The density values corresponding to higher temperatures {T = 450K) for molten PE either in a purely linear pattern or bearing symmetrically attached short branches along the main backbone are presented in chapter IL6 (see for example Fig. 7 and the corresponding discussion).

219

Atomistic simulation of the barrier properties

0.88

0.86 H E

c

o 0.84 H

0.82 C142

SBC 11x12 10x1

SBC 9x14 8x2

SBC_7x16_6x5

SBC 5x22 4x8

Molecular Architecture Figure 6. Dependence of density on molecular architecture as obtainedfromNPT MD simulations on short-chain branched (SCB) and linear PE systems at 7 = 300K and P = latm.

IV.4. Step 2: Conversion to explicit-atom configurations - Local relaxation Early MD simulation studies recognized the important role of the detailed representation of the constituent polymer atoms in the calculation of sorption and diffusion coefficients of penetrant molecules. According to the work of MuUer-Plathe et al. [22,94], united-atom (UA) description artificially raises transport rates by almost two orders of magnitude, even if the adopted UA model provides the correct density. An explicit-atom (EA) model packs more efficiently polymer segments of the same configuration, thus leading to a notably different distribution of free volume even if the density of the UA and EA systems is identical [17,22,94]. In the first stage of the hierarchical approach we adopted the UA description for the full-scale equilibration of the simulated samples. Besides the wellestablished asset of providing excellent predictions regarding density (Fig. 4) and intra- and intermolecular packing [71,72], the incorporation of the present UA model resulted in a tremendous saving in computational time and resources for system equilibration (a factor of three when compared against EA models), which is the time-limiting stage of the whole methodology. The critical

220

P. Gestoso andN.Ch. Karayiannis

requirement of a detailed (explicit-atom) representation of the polymer matrix in the final phase of the TST appUcation requires the conversion of the exiting UA configurations into EA structures by appropriately adding a posteriori the corresponding hydrogen atoms [95]. The resuhing EA structures were fiirther adapted to be compatible with the supported file formats of the commercial software Insightll (version 400P+) from Accelrys Inc. [96] employed for the successive TST calculations. The inclusion of hydrogen atoms and the resultant transformation from UA to EA configurations may cause energetic overlaps between added hydrogens and existing polymer segments, leading to unrealistic local packing. Therefore, a supplementary cycle of local equilibration was adopted to remove all energetically unfavorable overlaps and to impose a realistic distribution of free volume along the chains. The protocol used for the additional relaxation consisted of static structure optimizations (using an energy minimization Molecular Mechanics (MM) algorithm [75,76]) accompanied by short NVT MD simulations. The duration of each MD run was set at 50ps, the time integration step was equal to Ifs and the thermostat by Berendsen et al. [97] was employed to dump all temperature fluctuations. Casting of the short MD runs in the canonical (NVT) ensemble was decided on the basis that all simulated PE systems, as obtained from the application of the UA model during the equilibration stage, were characterized by densities in excellent agreement with available experimental data. In all simulations of EA samples all bonded and non-bonded potential interactions were described by the COMPASS [98,99] (Condensed-phase Optimized Molecular Potentials for Atomistic Simulation Studies) force field which is widely considered one of the most accurate, allpurpose EA potentials. Preliminary executions of the proposed hierarchical scheme on selected linear PE configurations using either COMPASS or PCFF [100,101] (Polymer-Consistent Force Field) suggested that the former provided predictions for the diffusion and solubility coefficients of oxygen and nitrogen molecules in better agreement with available experimental data. In addition, preUminary simulations indicated that a total of 3 successive combinations of MM and NVT MD runs were adequate to provide local relaxation as determined from the evohition of the total potential energy of the system and of the inter(g(r)) and intramolecular (w(r)) carbon-carbon distribution fiinctions. After the completion of the third MM/MD cycle, the potential energy was seen to reach a constant value, whereas no appreciable difference was detected in the structural features in the atomic level as evidenced from the pair distribution functions. IV, 5. Step 3: Application of Transition State Theory Gusev-Suter TST was employed, as implemented in the gsnet and gsdif subroutines of the Insightll software [96], to calculate the barrier properties of

221

Atomistic simulation of the barrier properties

PE matrices equilibrated in the first two phases of the present methodology. gsnet calculates the free energy surface of the system, thus separates the volume of the amorphous cell into sorption sites (characterized by low energy) and regimes of high-energy (excluded volume). Additionally, the connectivity of microstates is determined as well as the distribution of the first-order rate constants governing the transitions between microstates (sorption sites). Initially, a 3-D grid was laid on the amorphous cell covering the entire volume. A limitation in the maximum number of points in the 3-D grid in conjunction with the large size of the equilibrated PE configurations ranging from 40 up to 45A, notably larger than the cell dimensions encountered in typical a all-atom TST (25-30A), restricted the grid resolution to 0.5A, which is somewhat coarser than the typical value of 0.3 used in previous TST works [49]. Extensive test simulations with different grid resolutions ensured that the grid spacing of 0.5A did not affect the successive prediction of solubility, diffusivity and permeability. The penetrant molecule was represented as a united-atom, spherical site which interacts with the atoms of the polymer matrix through short-range forces characterized by a 9-6 potential (t/vdw) of the form 9

a vdW (ry) = £ 2

(

\6"]

-3 G

(21)

where ry is the distance between the penetrant / and the polymer atom 7 and e and a are the well depth and the collision diameter of the COMPASS force field. For various penetrant molecules the corresponding 6: and a parameters are reported in Table II.

Table II. Well-depth, e, and collision diameter, a, for various gas penetrant molecules as implemented in Gusev-Suter TST. Penetrant

e (kcal/mol)

^(A)

O2 N2 CO2

0.2344 0.1889 0.4500

3.46 3.70 4.00

222

P. Gestoso andN.Ch. Karayiannis

The cutoff distance for the non-bonded interactions was equal to 9.5A, as test runs with larger values revealed no dependence of the calculated barrier properties on the truncation radius. The smearing factor (Eq. 17) was calculated through a self-consistent scheme with the two relevant parameters being the most probable residence time, tp, of the penetrant in the microstates and the smearing factor. Liitially, an arbitrary value was assigned for the smearing factor and the distribution of the residence times in microstates was computed using Eq. 19. Knowledge of the distribution allowed the identification of Tp and a new value was attributed to the smearing factor based on the mean square displacement (MSD) of the polymer atoms from their equilibrium positions for time equal to Tp. To extract the corresponding MSD, an additional 50ps-long NVT MD simulation was executed after the completion of the relaxation cycle. Convergence occurred when the relative error between two successive values of the smearing factor was smaller than a prescribed threshold equal to 2.5%. Based on the adopted self-consistent scheme, the calculated smearing factor depends not only on the segmental mobility of the polymer host and the applied conditions (temperature) but also on the penetrant (as revealed by the inclusion of the most probable residence time). By determining the free energy surface of the system, identifying of all sorption sites and calculating the corresponding smearing factor (a single common value for all different species of the polymer host) through the gsnet subroutine, the low-concentration solubility coefficient, iS, can be readily calculated through Eqs. 11-12. Based on the information about the connectivity and distribution of microstates, the rate constant distribution and the smearing factor, the lowconcentration diffusion coefficient, D, was extracted by employing the gsdif subroutine of Insightll. Kinetic Monte Carlo (kMC) simulations, applied on the coarse network of microstates, provided the trajectories of the penetrant walkers as a function of time. The diffusion coefficient, Z), was calculated in the longtime limit where Fickian diffusion is established through the Einstein equation (Eq. 10). The total duration of the kMC was 10"*s and the mean square displacement (MSD) was averaged over the independent trajectories of 1000 walkers. Finally, the low-concentration permeability coefficient, P, was calculated indirectly as the product DS (Eq. 8). All reported results about the barrier properties (S, D and P) of PE to O2 and N2 came as statistical averages over either four {T> 500K), five (500K >T> 350K) or eight (350K >T> 250K) uncorrelated structures for each simulated system.

Atomistic simulation of the barrier properties

223

IV. 6. TST-based results IV.6.1. Comparison with experimental data A series of TST calculations were conducted at T = 300K and P = latm to validate the accuracy of the proposed multiscale scheme and its applicability on three different gas penetrant molecules: oxygen (O2), nitrogen (N2) and carbon dioxide (CO2), wilh the latter being widely considered [62,63,65,68] as too large (and anisotropic in shape) to be effectively modeled by the present implementation of TST. As akeady mentioned, at standard conditions polyethylene is a semi-crystalline polymer, with a non negligible crystalline volume fraction, v^, especially in the case of linear (HDPE) polyethylene samples. On the other hand, computer-generated PE macrostructures are completely amorphous. To compare TST-based results with available experimental data one should "correct" the latter so as to correspond to ideal purely amorphous samples. Pant and Boyd [24] proposed the following relationship between the experimentally measured diffusion coefficient, Dexp, and the "corrected" one, Z)'exp characterizing non-crystalline PE systems

D

3 D =-—^^ "' 2 ( l - v ^ )

(22)

Similar correction can be applied on the solubility coefficient

<^^>

^•-F^

where iSexp is the experimentally measured solubility coefficient in the semicrystalline material and ^yexp is the "corrected" value corresponding to an ideal non-crystalline sample. Consequently, the permeability coefficient "corrected" for crystallinity, Fexp is given by . exp

.

e.pe.p

2D

S

3(l_vJ(l-v)

2 P 3(l-v)^

^ ^

224

P. Gestoso andN.Ch. Karayiannis

where Pexp is the experimental value of the permeability coefficient of the actual semi-crystalline PE material. All reported experimental data were corrected for crystaUinity (through Eqs. 22-24) so as to correspond to purely amorphous samples allowing the direct comparison with TST-based predictions. Table III lists the low-concentration S, D and P coefficients as obtained from application of the hierarchical scheme to the polydisperse, linear Ciooo system {T = 300K, P = latm) and the corresponding experimental data for a linear C26000 system {T = 298K, v^ = 0.43) as reported by Michaels and Bixler [100] (experimental solubility data were calculated indirectly through S = P / D based on the D and P values of Ref 100). A close inspection of the data in Table III suggests that present TST simulations provide low-concentration solubility coefficients, S, which are in very good agreement with the experimental data for the whole range of gas penetrants, even for CO2. Past simulation works concluded that solubility coefficients, especially for large penetrant molecules (CH4 and CO2) are strongly and artificially affected by system size effects and, whenever small simulation boxes were used, predicted S values departed considerably from experimental values [68,101,102]. It appears that the present TST analysis on relatively large boxes (40-45A) rectifies related issues, although for a definite statement a more systematic approach is necessary using progressively smaller simulation cells. Diffusivity values are also in good agreement with experimental findings as long as the size of the penetrant molecule is small. For example, the best agreement is observed for oxygen (relative error of around 10%), while N2 diffusion rate is underestimated by the proposed TST approach within half order of magnitude. In contrast to the small gases, TST predicts CO2 diffusivity with a larger error: the simulated value is one order of magnitude smaller than the experimental one. It is clear that in its present formulation Gusev-Suter TST is not able to capture quantitatively the CO2 transport behavior due to its large and non-spherical size and to possible alternations of the surrounding polymer segments imposed by its presence that can not be correctly represented by the smearing factor. Finally, permeability coefficients of O2 and N2 (calculated indirectly as the product DS) were overand underestimated, respectively by similar factors in the range of half order of magnitude. Given the accurate TST-based predictions of sorption and diffusion of O2 and N2 through linear PE samples at standard conditions, in the next sections we will examine the effect of molecular weight (MW), temperature (7) and molecular architecture (short-chain branching, SCB) on the barrier properties of amorphous PE to these gases.

Atomistic simulation of the barrier properties

225

Table III. Simulation and experimental results for solubility, diffusion and permeability coefficients for O2, N2 and CO2 in purely amorphous linear PE matrices (T= 300K).

Penetrant O2 N2 CO2

S lO-^cm^(STP)/ cm^Pa) Simul. Exp. 1.14 ±0.08 0.418 ±0.08 3.20 ±0.3

0.828 0.400 4.47

D 10-'cm^ / s Exp. Simul. 13.2 ±1.2 4.84 ±0.8 0.41 ±0.1

12.1 8.42 9.79

P \0'^^ cm^STP) cm/ cm^ Pa s Exp. Simul. 15.0 ±0.6 2.02 ±0.09 1.31 ±0.1

10.0 3.37 43.7

IV,6.2. Effect of Molecular Weight Figures 7 and 8 present the solubility coefficient, S, of O2 and N2 as a function of molecular length for hnear PE systems as obtained from TST calculations at T = 450 and 300K. Regardless of the temperature, TST results suggest that 8(02) > S(N2) in the whole range of simulated chain lengths, in agreement with experimental trends [100]. The results also agree with semiempirical predictions based on the correlation function proposed by Teplyakov and Meares [103] according to which log(5) scales linearly with the effective well-depth parameter of the penetrant molecule (£(02) > e(^2) as seen in Table II). A qualitative difference is observed in the dependence of S on MW as a function of the applied temperature: at standard conditions {T = BOOK), S appears to depend weakly on molecular length and a maximum appears at A^av = 500. On the other hand, at higher temperature {T = 450K), solubility decreases hyperbolically with MW in a fashion reminiscent to the increment of density as a function of molecular length (see Fig. 4). As density increases, the magnitude and distribution of free volume decreases, consequently the number and size of the sorption sites that can accommodate the penetrant molecules are also reduced. Based on the results shown in Figs. 7 and 8 it can be concluded that MW has a profound effect on solubility at higher temperatures but at lower its effect diminishes.

P. Gestoso and N. Ch. Karayiannis

226

Molecular Length Figure 7. Low-concentration solubility coefficient, S, of O2 and N2 as a function of molecular length as obtainedfi-omTST calculations on polydisperse linear PE samples at 7 = 450K. 1.6

1

1

1

»

1—

•

1

'

1 —

»

1

Exp. O2

J O 1.4-1

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Exp. N2

\

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it

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ii> 0.6

\

0.4 0.2

1

'

1

200

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400

1

1

600

800

•

1

'

1000

Molecular Length Figure 8. Same as in Fig. 7 but at 7 = 300K. Also shown are available experimental data after Ref 100.

227

Atomistic simulation of the barrier properties 1.5 1.4 H

m 0.

1.3

o #

N. Exp. N.

1.2 ^11-1 o

Q

{

i

{

0.9-1 0.8

J

0.7 0.6 200

400

600

800

1000

Molecular Length

Figure 9. Low-concentration diffusion coefficient, Z), of O2 and N2 as a function of molecular length as obtained from TST calculations on polydisperse linear PE samples at 7 = 450K. Also shown is available experimental data for N2 diffusivity after Ref [104]. 4.5-1

1

1

11

1

1

— 1

'

4.0-

-r• 0

3.5-

n •

\

—r—

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i

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1 200

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400

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1000

Molecular Length Figure 10. Same as in Fig. 9 but at 7 = 300K. Also shown are available experimental data from Ref 100.

228

P, Gestoso andN.Ch. Karayiannis

Figs. 9 and 10 depict the simulated diffusion coefficient, Z), for O2 and N2 as a function of molecular length at T = 450 and 300K, respectively. As already mentioned in the previous section, simulated difiusivities at room temperature are in very good agreement with experimental data [100]. Even at the higher temperature {T= 450K), the predicted diffusivity of N2 compares very well with the corresponding experimental measurements reported by Yoshiyuki [104] with the relative error being less than 10%. Based on this resuh it is interesting to notice that Gusev-Suter TST, even if it invokes simplifications and assumptions that have to be validated (in contrast to MD) and departs from the atomistic level by resorting to a coarser representation, is able to capture quantitatively the transport behavior of small gas penetrants in polymers, even in the melt state. Qualitatively, the transport behavior of both penetrants, O2 and N2, exhibits the same dependence on the average molecular weight of the polymer matrix at different temperatures: the kinetically-driven transport process is strongly correlated to the amount and distribution of free volume (and consequently density) and the reduction of the latter has a profound effect on the penetrant diffusion rate. TST-based results further suggest that between the two polymeric samples (C500 and Ciooo) gas diffusion is faster in the matrix with the higher segmental mobility (i.e. C500) revealing the effect of the dynamical flexibility of the polymer matrix on penetrant diffusivity [81]. Obviously, as shown in Figs. 9 and 10, gas diffusivity is also strongly affected by the size of the penetrant molecule: the larger the molecule, the slower its transport in the polymer, in agreement with the correlation function proposed by Teplyakov and Meares, valid over a very wide range of polymer/penetrant systems [103]. According to with this correlation, log(Z)) drops linearly with (f, where d is the effective diameter of the penetrant. Consequently, D{0^ > D(H^ for the whole range of simulated chain lengths and applied temperatures.

Table IV. Low-concentration permeability coefficients of O2 and N2 in linear polydisperse PE systems as obtainedfiromTST simulations at r = 450 and 300K. P(10-^^cm^(STP) cm / cm^ Pa s) PE System

O2(r=300K)

N2(r=300K)

O2(r=450K)

N2(r=450K)

C78

3.85 ±0.12

0.657 ± 0.20

25.0 ±2.0

9.70 ±1.0

Cl42

3.05 ±0.10

0.457 ± 0.15

19.7 ±2.0

7.69 ± 0.8

C5OO

1.82 ±0.08

0.259 ±0.10

17.1 ±1.6

6.42 ±0.7

Ciooo

1.50 ±0.06

0.202 ± 0.09

15.8±1.8

5.83 ±0.7

Atomistic simulation of the barrier properties

229

The low-concentration permeability coefficient, P, for O2 and N2 in linear PE characterized by various molecular lengths as obtained from TST simulations at T = 450 and 300K is summarized in Table IV. The decrement in permeability with increasing chain length is mainly kinetically driven (decrease in difflisivity as shown in Figs. 9 and 10) attributed primarily to the reduction in the free volume and secondarily to the reduced segmental mobility of the polymer atoms (dynamic flexibility of the polymer host). Perm-selectivity of a polymeric membrane, a,/,, quantifying the separation efficiency of the material, of critical importance in gas separation processes, can be defined as the ratio of the low-concentration permeability coefficients of the two gases i andy P(i) ^ D{i) S{i)

Pij)

DU)S(j)

where dyj and Si/j are the diffusivity and solubility selectivities of the polymer host to the pair of gases, respectively. Based on the permeability coefficients exhibited in Table IV, at room temperature ao2/N2 = 7.4, which suggests that permeation of O2 in amorphous linear PE is preferred to N2 by a ratio of 7.4, which compares reasonably well with the available experimental value of ao2/N2 = 3.0 [100]. This preferential permeation of O2 against N2 in PE steams equivalently from kinetic (diffusion selectivity) and thermodynamic (sorption selectivity) factors.

IV.6.3. Effect of Temperature Fig. 11 presents the dependence of the logarithm of solubility coefficient, log(5), for O2 and N2 on reciprocal temperature as obtained from TST calculations on a polydisperse linear C78 system. Penetrant solubihty decreases exponentially with increasing temperature as the gas molecule (O2 or N2) experiences more difficulties to dissolve in the polymer matrix as the temperature increases. On the other hand, as shown in Fig. 12, diffusivity decreases with decreasing temperature. TST-based diffusion coefficients of both O2 and N2 in Unear PE follow a distinctly non-Arrhenius temperature dependence, in agreement with similar observations for methane diffusion in PE from atomistic MD simulations of Pant and Boyd [24]. Table V summarizes selectivity ratios for solubility, soim, diffusivity J02/N2 and permeability aoimi in linear Cyg as obtained from TST simulations. Based on the selectivity data listed in Table V, it is evident that temperature decrease favors the preferential sorption and diffusion of O2 against N2 in amorphous PE.

230

P. Gestoso andN.Ch.

T—

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2.0

2.5

r-

1

3.0

1

1

3.5

4.0

1000/T(K')

Figure 11. Logarithm of solubility coefficient, log(5), of O2 and N2 as a function of the inverse of temperature as obtained from TST calculations on linear C78 system. "0.\t

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3.4

231

Atomistic simulation of the barrier properties

Table V. Solubility 5O2/N2> diffusivity doimi and permeabilityflfo2/N2selectivities of linear C78 PE to O2 and N2 as obtained from TST calculations at various temperatures. Temperature

(K)

^011^2

^02/N2

250

2.98

300

2.73

2.71

7.44

350

2.44

1.64

3.99

400

2.28

1.37

3.11

450

2.14

1.27

2.70

500

2.00

1.20

2.41

550

1.91

1.12

2.14

600

1.82

1.10

2.01

IV.6.4. Effect of Molecular Architecture (Short-Chain Branching) The effect of molecular architecture was studied by comparing the barrier properties of short-chain branched (SCB) PE systems against the linear analogs of the same total molecular weight (C142). The molecular characteristics for all simulated SCB samples are given in Table 11. NPT MD simulations of the equilibration phase revealed that short-chain branching has a rather minimal effect on density as depicted in Fig. 6, in agreement with reported experimental findings [93] in the purely amorphous melt phase. Figs. 13 and 14 present the dependence of solubility coefficient, S, on short-chain branching at r = 450 and 300K, respectively, hi general, short-chain branching produces an increment in gas solubility to a small extend with a maximum being observed for the nonlinear structure with branches bearing 5 carbon atoms (C5). The maximum difference in all cases between TST-based values of 5 does not exceed the range of 20-30%, consequently SCB has a rather small effect on solubility of gas molecules in PE.

232

P. Gestoso andN.Ch.

T C142

Karayiannis

-I 1 1 i r SBC_S)c14_tx2 SBC_7x1«_«x5 SBC_Sx22_4x«

SBC_11ic12_1dx1

Molecular Architecture

Figure 13. Effect of short-chain branching (SCB) on low-concentration solubility coefficient, S, as obtainedfi"omTST simulations on PE systems at r = 450K. I.O -1

1

•

1.6 J 1 •

0.

J

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0

1

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1

'

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J

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0.40.2- ' — 1 — C142

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Molecular Architecture

Figure 14. Same as in Fig. 13 but at 7 = 300K.

1 ' SBC_5x22_4xi

233

Atomistic simulation of the barrier properties

T 1 1 1 r SBC_11x12_10x1 SBC_Sx14_»x2 SBC_7x16_«x5

SBC_53t22^4x«

Molecular Architecture Figure 15. Effect of short-chain branching (SCB) on low-concentration diffusion coefficient, D, as obtained from TST simulations on PE systems at r = 450K. ^.O '

1

2.4-

i

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1

'1

1

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\

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Molecular Architecture Figure 16. Same as in Fig. 15 but at 7= 300K.

1

1

'

SBC_5x22_4x«

234

P. Gestoso andN.Ch.

Karayiannis

Figs. 15 and 16 present the dependence of diffusion coefficient on molecular architecture as obtained from TST simulations at T = 450 and 300K, respectively. In contrast to solubility, transport rates of penetrant molecules in short-chain branched systems appear to be somewhat smaller than those in linear counterparts characterized by the same molecular weight. Given that density is not significantly affected by the addition of branches (as shown in Fig. 6), an explanation can be given in terms of slower mobility of the polymer segments in SCB structures, especially in the vicinity of branch points as analyzed in chapter 11.6 of the book. The decrement in diffusion rates of the gas molecules, as a result of the short-chain branching, is more pronounced at lower temperatures. Low-concentration permeability coefficients, P, of O2 and N2 in PE matrices characterized by various molecular architectures are reported in Table VI. Based on the relative differences of the TST-predicted permeabilities for different structures it is evident that short-chain branching has a minimal effect on the barrier characteristics of purely amorphous PE systems. Accordingly, differences encountered between end-product LLDPE and HDPE materials could be attributed to the different degrees of crystallinity.

Table VL Low-concentration permeability coefficients of O2 and N2 in monodisperse short-chain branched (SCB) and linear PE systems as obtainedfromTST simulations at r = 450 and 300K. All systems are characterized by MW = 1990g/mol. P(10-^^cm^(STP) cm / cm^ Pa s) PE System

O2(r=300K)

N2(r=300K)

O2(r=450K)

N2(r=450K)

Cl42

2.59 ±0.10

0.382 ± 0.06

20.3 ±2.0

7.90 ± 0.8

SCB_llxl2_10xl

1.59±0.15

0.232 ± 0.08

19.4 ±2.5

7.64 ±0.9

SCB_9xl4_8x2

2.17 ±0.12

0.315 ±0.02

21.6±1.6

8.39 ±0.7

SCB_7xl6_6x5

2.80 ±0.15

0.359 ±0.06

20.9 ±1.8

8.13 ±0.7

SCB_5x22_4x8

2.39 ±0.10

0.381 ±0.05

21.7 ±3.0

8.66 ±0.6

Atomistic simulation of the barrier properties

235

V. Conclusions - Future Plans We have presented a study of the barrier properties of purely amorphous polyethylene samples to oxygen and nitrogen gases at infinite dilution as obtained from a hierarchical, multiscale modeling approach integrating state-ofthe-art simulation techniques. In the first stage, atomistic MC simulations built around chain-connectivity altering moves [69-74, an analysis of this technique is given in chapter 1.2 of the book] provide, within modest computational time, vast trajectories of representative and uncorrected confiigurations of the studied systems. Properly selected structures from the MC-generated trajectories with representative long-range characteristics, atomic packing and volumetric properties serve as excellent configurations for successive simulation studies based on Transition State Theory (TST) [39-42], as implemented in Insightll commercial software package [96]. Regarding computational time, the proposed hierarchical scheme, by integrating atomistic MC and Gusev-Suter TST (as implemented in Insightll), outperforms any conventional approaches by many orders of magnitude, especially for the modeling of the barrier properties of truly long, entangled systems (i.e. Ciooo PE) at low temperatures. Because of its simplifying assumptions, TST at its present formulation can be applied to small spherical gases at low concentration of the penetrant molecule. TST-based results revealed very good agreement between the calculated solubility, diffusivity and permeability coefficients of O2 and N2 in PE and available experimental data for a wide range of conditions. Sorption and diffusion are strongly correlated with the amount of free volume and the segmental mobility of the polymer atoms as identified by the effect of molecular weight on the barrier properties. Solubility decreases with increasing temperature as the penetrant molecule exhibits difficulties in condensing in the polymer matrix, while diffusivity shows a distinctly non-Arrhenius behavior as a function of temperature. Short-chain branching has a rather small effect on the barrier properties if the comparison is made between linear and branched structures bearing exactly the same molecular weight. Solubility in short-chain branched samples is to some extent favored when compared to linear analogs, whereas the opposite trend is observed regarding diffusivity. Finally, based on TST results, perm-selectivity of PE favors the permeation of O2 against N2 by a factor of around 7, in reasonable agreement with experimental trends which suggest a ratio of 3. As revealed from present findings, preferential permeation of O2 steams equivalently from kinetic (diffusivity selectivity) and thermodynamic (solubility selectivity) factors. Further work concerns the modeling of the barrier properties of glassy PE samples (at temperature in the vicinity and below Tg) and the expansion of the

236

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Karayiannis

proposed multiscale methodology to other chemically simple polymeric systems. Application of the present technique to chemically complex macromolecules requires the adaptation of a coarse-graining strategy where the atomistic structure of the polymer is mapped into a coarse-grained representation involving fewer degrees of freedom [105-109]. Li parallel, current efforts focus on the generalization of the TST approach to handle large, non-spherical gas molecules (for example CO2) by allowing the polymer matrix to undergo structural relaxation to accommodate the presence of penetrants.

Acknowledgments P. Gestoso acknowledges Rhodia Recherches for permission to publish this work. N. Ch. Karayiannis expresses his gratitude to Prof. Doros Theodorou (University of Athens) for his guidance in the modeling of diffusion in polymers. Stimulating discussions with Prof. Manuel Laso (University of Madrid), Prof V. Mavrantzas (University of Patras), Prof. R. Gani and V. Soni (Technical University of Denmark), Dr. James Wescott (Accehys Ltd.), Dr. Nikolas Zacharopoulos and Niki Vergadou (National Research Center "Demokritos") and all partners of the PMILS project are deeply appreciated. This work was financially supported by the "Polymer Molecular Modeling at Integrated Length/time Scales" (PMILS) European Community project (Contract No G5RD-CT-2002-00720).

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P. Gestoso andN.Ch.

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M. L. Greenfield and D. N. Theodorou, Macromolecules 31 (1998) 7068. M. L. Greenfield and D. N. Theodorou, Macromolecules 34 (2001) 8541. D. Hofmann, J. Ulbrich, D. Fritsch and D. Paul, Polymer 37 (1996) 4773. D. Hofmann, L.Fritz, J. Ulbrich and D.Paul, Polymer 38 (1997)6145. L. Fritz and D. Hofinann, Polymer 38 (1997) 1035. L. Fritz and D. Hofinann, Polymer 39 (1998) 2531. D. Hofmann, L. Fritz and D. Paul, J. Membrane Sci. 144 (1998) 145. D. Hofmann, L. Fritz, J. Ulbrich, C. Schepers and M. Bohning, Macromol. Theor. Simul. 9 (2000) 293. [61] D. Hofmann, L. Fritz, J. Ulbrich and D. Paul, Comput. Theor. Polym. Sci. 10 (2000) 419. [62] M. Heuchel and D. Hofinann, Desalination 144 (2002) 67. [63] D. Hofmann, M. Heuchel, Y. Yampolskii, V. Khotimskii and V. Shantarovich, Macromolecules 35 (2002) 2129. [64] D. Hofinann, M. Entrialgo-Castano, A. Lerbret, M. Heuchel and Y. Yampolskii, Macromolecules 36 (2003) 8528. [65] M. Heuchel, D. Hofinann and P. Pullumbi, Macromolecules 37 (2004) 201. [66] J. R. Fried and P. Ren, Comput. Theor. Polym. Sci. 10 (2000) 447. [67] M. Lopez-Gonzalez, E. Saiz, J. Guzman and E. Riande, J. Chem. Phys. 115 (2001) 6728. [68] E. Kucukpinar, P. Doruker, Polymer 44 (2003) 3607. [69] P. V. K. Pant and D. N. Theodorou, Macromolecules 28 (1995) 7224. [70] V. G. Mavrantzas, T. D. Boone, E. Zervopoulou and D. N. Theodorou, Macromolecules 32 (1999)5072. [71] N. Ch. Karayiannis, V. G. Mavrantzas and D. N. Theodorou, Phys. Rev. Lett. 88 (2002) 105503. [72] N.. Ch. Karayiannis, A. E. Giannousaki, V. G. Mavrantzas and D. N. Theodorou, J. Chem. Phys. 117(2002)5465. [73] N. Ch. Karayiannis, A. E. Giannousaki and V. G. Mavrantzas, J. Chem. Phys. 118 (2003) 2451. [74] D. N. Theodorou, Bridging Time Scales: Molecular Simulations for the Next Decade, Eds. P. Nielaba, M. Mareschal and G. Ciccotti, Springer-Verlag, Berlin, 2002. [75] D. N. Theodorou and U. W. Suter, Macromolecules 18 (1985) 1467. [76] D. N. Theodorou and U. W. Suter, Macromolecules 19 (1986) 139. [77] Commercial simulation software Materials Studio (version 3.0) by Accelrys Inc. http://www.accelrys.com/products/mstudio/. [78] M. G. Martin and J. L Siepmann, J. Chem. Phys. B 102 (1998) 2569. [79] S. K. Nath and R. J. Khare, J. Chem. Phys. 115 (2001) 10837. [80] S. Toxvaerd, J. Chem. Phys. 107 (1997) 5197. [81] N. C. Karayiannis and V. G. Mavrantzas, Macromolecules 38 (2005) 8583. [82] V. G. Mavrantzas and D. N. Theodorou, J. Chem. Phys. 31 (1998) 6310. [83] L. R. Dodd, T. D. Boone and D. N. Theodorou, Mol. Phys. 78 (1993) 961. [84] S. J. Plimpton, J. Comput. Phys. 117 (1995) 1. [85] LAMMPS software distributed by Dr. S. Plimpton at Sandia National Laboratories, US. All equilibration NPT MD simulations were carried out using version LAMMPS 2001 (Fortran 90). [86] S Nose, Mol. Phys. 52 (1984) 255. [87] W. G. Hoover, Phys. Rev. A 31 (1985) 1695. [88] H. C. Andersen, J. Comput. Phys. 52 (1983) 24. [89] M. E. Tuckerman, B. J. Berne and G. J. Martyna, J. Chem. Phys. 97 (1992) 1990. [90] G. J. Martyna, M. E. Tuckerman, D. J. Tobias and M. L. Kein, Mol. Phys. 87 (1996) 1117. [91] G. T. Dee, T. Ougizawa and D. J. Walsh, Polymer 33 (1992) 3462. [92] D. Cubero, N. Quirke and D. F. Coker, J. Chem. Phys. 119 (2003) 2669.

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[93] J. L. Lundberg, J. Polym. Sci. Part A 2 (1964) 3925. [94] F. Muller-Plathe, S. C. Rogers and W. F. van Gunsteren, Macromolecules 25 (1992) 6722. [95] O. Ahumada, D. N. Theodorou, A. Triolo, V. Arrighi, C. Karatasos and J. P. Ryckaert, Macromolecules 35 (2002) 7110. [96] Commercial simulation software Insightll (version 400P+) by Accelrys Inc. http://www.accelrys.com/products/insight/. [97] H. J. C. Berendsen, J. P. M. Postma, W. F. van Gunsteren, A. Di Nola and J. R. Haak, J. Chem.Phys. 81(1984)3684. [98] H. Sun, P. Ren and J. R. Fried, Comput. Theor. Polym. Sci. 8 (1998) 229. [99] H. Sun, J. Phys. Chem. B 102 (1998) 7338. [100] A. S. Michaels and H. J. Bixler, J. Polym. Sci. 50 (1961) 413. [101] T. R. Cuthbert, J. J. Wagner, M. E. Paulitis, G. Murgia and B. D. Aguanno, Macromolecules 32 (1999) 5017. [102] T. R. Cuthbert, N. J. Wagner and M. E. Paulaitis, Macromolecules 30 (1997) 3058. [103] V. Teplyakov and M. Meares, Gas Sep. Purif. 4 (1990) 66. [104] Y. Sato, K. Fujiwara, T. Takikawa, Sumamo, S. Takishima and H. Masuoka, Fluid Phase Equilb. 162(1999)261. [105] W. Tschop, K. Kremer, J. Batoulis, T. Burger and O. Hahn, Acta Polym. 49 (1998) 61. [106] W. Tschop, K. Kremer, O. Hahn, J. Batoulis and T. Burger 49 (1998) 75. [107] T. Aoyagi, F. Sawa, T. Shoji, H. Fukunaga, J. Takimoto and M. Doi, Comput. Phys. Commun. 145 (2002) 267. [108] F. Muller-Plathe, Chem. Phys. Chem. 3 (2002) 754. [109] N. Zacharopoulos, N. Vergadou and D. N. Theodorou, J. Chem. Phys. 122 (2005) 244111.

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241

Chapter 3

From polyethylene rheology curves to molecular weight distributions Costas Kipa^issides^ Prokopis Pladis*, 0ivind Moen^ ^Laboratory of Polymer Reaction Engineering, CPERI, P.O.Box 361, 57001 ThermiThessaloniki, Greece ^ Borealis AS, 3960 Stathelle, Norway 1. Introduction to the Problem It has been shown that Molecular Weight Distributions can be determined from linear viscoelastic melt properties (shear storage modulus G' (CD) and the stress relaxation modulus G (t). A method for the determination of Molecular Weight Distributions from viscosity-shear rate data would have the following two advantages: 1. A piston rheometer can be utilized to perform the necessary measurements. These rheometers are more convenient in use and are more robust instruments than the mechanical spectrometers that are used to measure G' (co) and G(t). 2. In order to use the G' (o) and G (t) methods it is essential that the Plateau modulus (G^ ) must be known. This is a very difficult, and sometimes impossible, quantity to measure. A knowledge of the Plateau modulus ( G ^ ) is not required for the viscosity method that is going to be described next. Two main approaches have appeared in the pertinent literature: The Malkin and Teishev approach is based on a concept that at high shear rates, high molecular weight components of a polydisperse mixture act as filler in a viscous medium. Although they have calculated the MWD from viscosity data they concluded that the flow curve is not unique for a specific MWD. They demonstrated this by using a mathematically formulated MWD. However the precision of their

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method was said to be limited due to a) sparse data used and b) deviation due to curve fitting. Bersted and Siee used a similar basic idea, but developed a different mathematical approach to the problem to show how the viscous effects of high molecular weight components are truncated at higher deformation rates. They used this method to determine the flow curve from a MWD and also to solve the inverse problem. However they concluded that it is impractical to determine the MWD from the flow curve because the high rates that are essential for the determination are usually experimentally inaccessible. This is a common problem for all polydisperse polymers and of course LDPE. Mavridis and Shroff have shown that for a HDPE whose MWD extends from 10^ to 10^ in molecular weight, the rheological accessible range was half of that needed to calculate the MWD. Tuminello applied the Bersted and Slee approach to determine the MWD of broad polystyrene. Nonlinear least squares techniques were used to extrapolate to the high deformation rate behaviour thus overcoming the problem. He accurately predicted the shape and breadth of the MWD. Thus, he concluded that this approach was promising. 2. Mathematical Formulation of the Problem 2.1. Malkin and Teishev Approach It has been shown that the monodisperse polymer melts flow almost like Newtonian liquids up to the critical shear stress Os (i.e. the viscosity has a value r|o that does not change with the shear rate, y ). At a > GS shear flow becomes impossible and linear polymers behave as cured rubbers. The chemical intermolecular link function is played by relaxation entanglements with a lifetime longer than r|o /as. The higher the Molecular Weight of a polymer the higher the characteristic relaxation time r|o /Os, and the lower the limiting shear rate y = (TIO /GS)^ up to which the flow is still possible. The critical shear stress as is a constant for each polymer homologous series and depends only slightly upon the chemical nature of a polymer. In contrast mixtures of monodisperse fractions behave just like a polydisperse commercial polymer melt. This experimental fact directly points to the correlation between the flow curve and the MWD of a polymer.

From polyethylene rheology curves to molecular weight distributions

243

The flow curve of a monodisperse polymer melt can be written as: ^0

•

at


or

at

a>G^

or

y
>

^(r) = ^ •

r>rs

(1)

where r|o is the Initial Newtonian viscosity and YS = (r|o/as)"^ is the critical shear rate depending on the Molecular weight, M. It is known that

(2) and thus ys=(S) Usually a is taken to be 3.4 Equation (2) is completely valid for monodisperse polymers only. For polydisperse polymers Weight Average Molecular Weight must be used instead ofM. For a binary mixture of some polydisperse polymer with Mi and M2 Molecular Weights (Ml > M2) one can write: (4) The flow curve of the binary mixture is expressed as AX Y<

rj(r) =

K,M," K,(w,M,+w,M,r (5)

244

At

C. Kiparissides et al.

'—r
K,M,"

'•

K,M,"

n(r) = w,{c7jrf"+w2Kl'"M2 (6) At

y>-

T]{y) =

ajy (7)

This approach can be generalized for continuous MWD M(r)

n(y) =

\ (K,M,''f"fiM)dM + {aJyy"'

j f{M)dM M(r)

(8) where f(M) is the weight MWD function. The first term reflects the polymer fractions that can still flow at some shear rate and the second term is related to the high MW fractions. The value of M(Y) depends on shear rate:

M{y) = {cTjK,yf"

^^^

Thus for a given MWD it is possible to calculate theflowcurve. For the inverse problem the following dimensionless variables are introduced: •\ila

(10) X^iylYsf^

(11)

m = M I Mw

(12)

From polyethylene rheology curves to molecular weight distributions

245

Equation (9) is rewritten in the following form: MX

f

l^"^

r = Jm/(m)c/m+ —

\ f{m)dm (13)

A double differentiation of this equation in accordance with the Leibniz rule is given according to the original paper of Malkin and Teishev. Note that Tuminello (1991) reaches a final result for the same equation with the right part divided by a factor of 2 which, however, is not correct. /

X.

dX^

\IX=m

\dX,

(14)

Equation (14) is the exact solution of the problem. MWD is calculated by the derivatives of the flow curve. 2,2. The Bersted and Slee Method In their approach, Bersted and Slee made the following assumptions for a monodisperse component of a polydisperse mixture: 1. Like the previous method, they assumed that the relaxation time spectrum was truncated at high shear rates. However the form of this truncation was written in a different way ^(rJ=

fH(T)dT (15)

where H(T) is the relaxation time spectrum and Xm is the maximum allowable relaxation time at a specific shear rate Ym . 2. The relationship between x^ and Ym is Xm = 1 / Ym • Since x^ ^(Mmf'^ then it can be written as :

Yn,

(16)

Mm is the Molecular weight of the component with terminal relaxation time x„

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C Kiparissides et al.

3. The relaxation time distribution from a specific component will include the distributions of all lower molecular weight components. 4. The distribution of relaxation times for a monodisperse polymer was assumed to be the same as that of a monodisperse component of the same molecular weight in a polydisperse mixture: No coupling effects between chains of different molecular weight were assumed. The weight fraction of chains with MW >Mni is written: ^1/3.4

.

,1/3.4

Urn

E ^/ =

i=m+\

^..1-K

(17)

The weight fraction of each monodisperse component Wm by numerically differentiating adjacent digitized values from the flow curve as shown in the following expression: /

\l/3.4

vl/3.4 \l/3.4

/

M^^M.

m-\

/

\ i1/3.4 /i.4

/

M..,-M^

vl/3.4 \

(18)

2.3, The Tuminello methodology Tuminello used the Bersted and Slee approach but made some useful modifications in order to carry out the lack of experimental data at high shear rates. They used the known fact that the plot of Ti/r|o vs. log(co) give a sigmoid curve with high and low rate limits at 0 and 1, respectively. Thus, fitting a sigmoid function (such as hyperbolic tangent) to the available low rate data would give reasonable extrapolations to high rates if there is no unusual shape to the distribution at low MW. Since a) the left side of Equation (17) is equivalent to unity minus the cumulative MWD and b) Equation (16) expressed the MW in terms of an equivalent frequency , Equation (17) can be rewritten to solve for the cumulative weight fraction: W^ {cumulative) = 1 (19) The differentiation of the cumulative distribution would yield the differential MWD.

From polyethylene rheology curves to molecular weight distributions

241

The basic steps in his methodology are: • Smooth the reduced viscosity plot of [ (rj^ IK^ )^'^'^ vs. {K^Q)^ ^-1/3.4 j ^^jj^g ^ cubic spline least squares fitting procedure. • Differentiate the smoothed version of the reduced viscosity plot. • Fit a function to the cumulative Molecular weight data. The following hyperbolic tangent function was used for this curve fitting. Its sigmoid shape allows fitting to the existing data and reasonable extrapolation to the known limits of 0 and 1 at low and high MW.

^ ( 3 ) , | - 4 { l + tanh[^,(^^C,)]} '='

^

(20)

where 0 < F(Z) < 1, IAi=l, 0 < Ai< 1 n is an integer, when n=l a symmetrical sigmoid curve is described. Asymmetric or bimodal distribution requires n to be greater than 1. Z = log(MW) • Determination of the MWD by differentiating F(Z) • The analytical differentiation of F(Z) gives

4F(Z)] ^ ^ AB, stch' [B,(Z + C,)] d{Z)

i=i

2

pjj

that is the Molecular Weight Distribution • Calculation of MW averages and Polydispersity Index • The Number and Weight Average Molecular Weights are calculated from the cumulative curve using the values between 0.001 and 0.999 3. Polydispersity Estimation As described above the calculation of the entire Molecular Weight Distribution requires experimental data from a large range of shear rates. This is not possible for many polymers for very high shear rates and for several, such as LDPE, for very low shear rates. However it is still possible to calculate the polydispersity index from these data. This can be achieved either by using the previous described methodology proposed by Tuminello or by using the calibrating curve method of Malkin and Teishev. The first method was described earlier and in the following section the second method is going to be discussed.

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C. Kiparissides et al

The dimensionless equation (13) includes in the integral terms the MWD equation f(m) 7 = J mf(m)dm + (^] ] f(m)dm 0 y^Jvx

(13)

One can then use some known analytical functions for the MWD containing two "free" parameters. Varying these parameters one can calculate Y(X) dependencies, correlate the Polydispersity Index values with some characteristic points and construct a universal calibration curve. Comparing the calibration curve with experimental data (which were transformed into dimensionless) it is possible to estimate the Polydispersity Index. The distribution curves that can be used depend on the problem. This is due to the known fact that some distributions are symmetrical and, thus, are suitable for monodisperse samples and others are asymmetrical and suitable for polydisperse polymers. For LDPE grades the use of Wesslau (Log-Normal) distribution and the Beasly MWD seems reasonable. The Wesslau distribution is written as /(M) =

1 yprSM

exp

(InM-//)' (22)

and in dimensionless form

fim) =

exp

\n(nm) 2\nn

(23)

where m= M/M^ and n is the polydispersity index. The parameters depend on average molecular weights are given ju = —\n(MwMn) (24) n = ==^ = exp((J^ / 2) Mn

(25)

From polyethylene rheology curves to molecular weight distributions

249

The Beasly distribution is

and written in dimensionless form is 2n(n - l)m

firn) = where

s{\-P) Mw Mn

(28)

2(1-yS) 1-2)5

Qg^

Equations (23) and (27) used in equation (13) determine the set of flow curves Y(X) for different values of the polydispersity index. For the transformation of the experimental data the r|o and YS must be known for the polymer. Since Ys = is /r|o and for LDPE TS=3X10^ Pa, the only unknown in the problem is r|o. The knowledge of this value is needed for an accurate estimation of the polydispersity index which can be done by comparing the reduced experimental data with the calibration curves extracted by Equations (13), (23) and (27) for different values of polydispersity. The reduction of the experimental data is done by the following equations r

,

-|l/3.4

4p=U(r)exp/^0 L -I X

=(y exp

V/ exp

/yf" is/

(30) (31)

250

C. Kiparissides et al

4. Calculation of the Relaxation time spectra 4.1. Definitions The dependence of linear viscoelastic data is usually discussed in terms of the relaxation spectrum H(T), where H(T) is the relaxation strength of the material at relaxation time i. All other linear viscoelastic properties can be derived by the relaxation spectrum: Storage Modulus:

J--

' 'l + icDzf

(32)

Loss Modulus: G\co)=

rH(T)

(COT)

^ ^

,d]xiT (33)

Complex Modulus:

G\(o) = 4G'{(of+G\(of

(34)

Complex Viscosity: 7 (fi>) = (O

(35)

Dynamic Viscosity:

7 («) = (O

(36)

Loss Tangent: tan<5(
(37)

The previous equations can be used in the calculation of other rheological properties:

From polyethylene rheology curves to molecular weight distributions

251

Plateau Modulus: Gl = ^HiT)d\nr

=

^^G\co) (38)

Zero-shear Viscosity: 7]Q= I / / ( r ) r J i n r = lim

rj\co) (39)

Compliance:

(40) where co is the frequency in rad/s 4.2. Calculation of the Relaxation time spectra: The procedure to calculate the relaxation time spectrum, for a given MWD, using the four assumptions made by Bersted and Slee: • The relaxation time spectrum is truncated at high shear rates; Xm is the maximum allowable relaxation time at a specific shear rate Ym . • The relationship between Xm and ym is Xm = 1 / Ym . • The relaxation time distribution from a specific component will include the distributions of all lower molecular weight components. • No coupling effects between chains of different molecular weight were assumed. The steps of the procedure are: 1. For every molecular weight fraction Mi, calculate the effective average Mw* from the following equation: m-\

M for Mc = Mi

m

(41)

252

C. Kiparissides et al.

2. Calculate the relaxation time from (42) 3. Calculate the relaxation strength

H(T) =

Kc^2iM^r^-''

(43)

Note that rio(M) = KiM" 4. Repeat steps 1-3 until the complete relaxation spectrum is completed. The dynamic module are calculated:

(44)

(45) 4.3. Temperature dependence of the rheological properties The basic assumption is that the relaxation spectra derived from data at different temperatures can be made to superimpose by vertical (along the relaxation strength axis) and horizontal (along the time axis) shifts. That is bj,H(T/aj.,T)

= H(TJo)

(46)

where a j and br are the horizontal and vertical shift factors respectively, T and To are the temperature and reference temperature respectively, aj and br are defined as a J = exp

R

1

1

r + 273

r„+273

A' (47)

From polyethylene rheology curves to molecular weight distributions

bj. = e x p

R

1

1

r + 273

7;+273

253

(48)

The horizontal shift factor reflects the temperature dependence of the relaxation time while the vertical shift factor reflects the temperature dependence of the modulus: (49)

GliT) = Gl{T,)/b,

(50)

The temperature dependence of the rheological properties is

(51) = bj^

(52) The two shifting factors can be calculated independently from dynamic data by using: • The loss tangent to calculate ax • The complex modulus to calculate bx From other types of linear relaxation modulus data as well as steady-shear viscosity data, the two activation energies can be calculated, but not independently. The steady-shear data can be superimposed by the same factors. Thus (53) where a (Y,T) is the shear stress at y shear rate. The two shift factors (activation energies) can be calculated by steady-shear data by using numerical methods as nonlinear-least squares: logCa^ y) (54)

254

C. Kiparissides et al.

where m is the order of the fitting polynomial and Ck is the coefficients of the k-term of the fitting polynomial. However by this method the estimated activation energies are statistically correlated. Thus the most correct way to calculate the activation energies is with the dynamic data. Usually in the pertinent literature the vertical shift factor is ignored and not taken into consideration. That means that bj = 1 and Ey = 0 . This is true for most of the polymer, but when the vertical shift is neglected in the case of long chain branched LDPE then anomalous results are observed as reported in literature: the computed horizontal activation energy is stress depended. Values of the activation energies reported in literature for different LDPE samples are: EH: 14.62-16.47 Kcal/mole Ev : 0.97 - 2.44 Kcal /mole The need for a vertical shift in the case of LDPE not required for other polyethylene shows a direct connection between the shift factor and long chain branching. However the relationship between the activation energy and the amount of LCB is depending and on other parameters (i.e. LCB distribution and branch length) which complicate the problem. The fact that LDPE requires a vertical shift can be explained, as suggested by Graessley, by two possible ways. 1. The concept of molecular weight between entanglements 2. The suppression of reptation in branched polyethylene Both mechanisms influence the temperature dependence of the rheological properties, but the effect of each one is not clear yet.

From polyethylene rheology curves to molecular weight distributions

255

References Cooper A.R.,1989, "Recent Advances in Molecular Weight Determination" Polymer Engineering andScience, 29(1), 2 Hieber C.A. and Chiang H.H.,1992, ''Shear-Rate-Dependence Modelling of Polymer Melt Viscosity", Polymer Engineering and Science, 32(14), 931 Kalyon D.M, Chiou Y.-N., Kovenklioglou S. and Bouaffar A., 1994, ''High Pressure Polymerization of Ethylene and Rheological Behavior of Polyethylene Product", Polymer Engineering and Science, 34(10), 804 Malkin A.Y. and Teishev A.E., 1991, "Flow Curve - Molecular Weight Distribution: Is the Solution of the Inverse Problem Possible?", Polymer Engineering and Science, 31(22), 1590 Mavridis H. and Shroff R., 1992, "Temperature Dependence of Polyolefm Melt Rheology", Polymer Engineering and Science, 32(23), 1778 Mavridis H. and Shroff R., 1993, "Appraisal of a Molecular Weight Distribution to Rheology Conversion Scheme for Linear Polyethylenes", Journal of Applied Polymer Science, 49, 299 Ritzau G., Ram A. and Izrailov L.,1989, "Effect of Shear Modification on the Rheological Behavior of Two Low-Density Polyethylene (LDPE) Grades", Polymer Engineering and Science, 29(4), 214 Tuminello W.H. and Cudre-Mauroux N., 1991, "Determining Molecular Weight Distributions From Viscosity Vesrus Shear Rate Flow Curves", Polymer Engineering and Science, 31(20), 1496

Multiscale Modelling of Polymer Properties M. Laso, E.A. Perpete (Editors) © 2006 Elsevier B.V. All rights reserved.

257

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Computer Aided Polymer Design Using Group Contribution Techniques Martin Hostrup^, Peter.M. Harper^, 0ivind Moen^, Nuria Muro-Sune*^, Vipasha Soni^, Jens Abildskov*^, Rafiqul Gani^* ^Rambell Danmark A/S, 2820 Virum, Denmark ^Borealis AS, 3960 Stathelle, Norway ^CAPEC, Department of Chemical Engineering, Technical University of Denmark, 2800 Lyngby, Denmark Corresponding author

1. INTRODUCTION This chapter highlights certain aspects of polymer design and analysis through the use of systematic computer aided techniques. Different polymer design and analysis, starting with the simple problem of design of polymer repeat units based on a set of pure polymer (repeat unit) properties and ending with complex problems involving design of polymer blends plus additives to satisfy a set of functional properties of the polymer-based chemical product, are highlighted. Solution of all these problems require the evaluation of different sets of end-use properties related to the polymer (depending on the micro-structure as well as macroscopic structure of the polymer). In the design of polymer repeat units, only end-user properties based on the macroscopic structure is sufficient, while in more complex problems involving polymer blends, transport properties or configuration of repeat units, microscopic structural properties also play an important role. Also, as the search space for the polymer design problem is potentially very large, computer aided techniques that can reliably reduce the search space and identify a feasible set of alternatives within which the optimal solution may lie, is an option worth investigating.

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M. Hostrup et al

The main features of any computer aided polymer design technique are methods for quick and reliable prediction of target properties, algorithm for generation of polymer repeat unit structures and their distribution over a chain length and an selection engine for screening of alternatives. The methods for prediction of target properties, connect the other two items. Group contribution (GC) based methods, are simple, have acceptable accuracy for preliminary design calculations, are predictive in nature, and therefore, ideally suited for computer aided polymer design techniques. Their use, however, depends on the availability of sufficient number of groups and their contributions towards specific properties. Since the typical groups (known as first-order groups) are not able to include all structural information, GC^ methods, where additional structural information is provided through higher-order groups may be used. The higher-order groups can be identified through molecular modelling and/or computational chemistry. This chapter presents a methodology for computer aided polymer design (CAPD); describes a suite of GC-based models for prediction of polymer properties; describes the main features of a prototype CAPD software for design and analysis of polymer based products; and, illustrates the application of the CAPD methodology through 3 different application examples. The application examples also highlight the need and importance of the corresponding property models and the application of the CAPD software. 2. METHODOLOGY FOR CAPD Given a set of desirable end-use properties and functions of a polymer (or polymer-based product), determine the structure of the polymer repeat unit and/or its distribution (such as branched or linear) over a specified chain-length that best matches the desired (target) properties and fimctions. The simplest polymer design problem is the design of the polymer repeat unit matching a desired set of pure polymer properties. A slightly more complex design problem is to find the polymers matching a desired set of polymer and polymer solution properties. A more complex design problem may involve the same set of properties as above but involving, as well, the configuration of the repeat units (for example, the arrangement of polymer repeat units in branched configurations). Finally, even more complex design problems involving the design of polymer-based products may be formulated as given, polymer(s) and fillers (atoms, molecules), find structures that match desired behaviour of the product (such as the controlled release of drugs, pesticides, food or aroma

Computer aided polymer design using group contribution techniques

259

products from a polymer-based microcapsule). In this case, the design problem may also involve the identification of chemicals that when added to the polymer system, improves the function (performance) of the final product. Properties such as the diffusion or permiability of a chemical (active ingredient) through the polymer based microcapsule plays a very important role. The polymer design problems defined above may be reformulated as a generic CAPD problem, as follows: Given a set of building blocks and a specified set of target properties, determine the structure of the polymer rrpeat unit and/or its distribution and chain length that matches these properties. In this respect, the CAPD problem is the reverse problem of property prediction where given the identity of the molecule and/or the molecular structure, a set of target properties are calculated. CAPD maybe performed at various levels of size and complexity, depending on the polymer repeat unit structure and its distribution. For example, design of polymers and polymer-based products are usually based on properties estimated from macroscopic structural information. In the design of structured products such as (non-linear) polymers, the structural differences are observed by employing meso- and/or microscopic representation of the polymer repeat unit structure and/or its distribution. Therefore, the property models and the polymer structural representation differ according to the type of polymer based product being designed. Computer aided polymer blend or polymer solution design problems can be defined as. Given a polymer, a set of chemicals and a specified set of property constraints, determine the optimal polymer blend and/or polymer solution. Here, we do not know which chemicals to use in the final product and in what amount they should be present but we know the molecular structures of the candidate chemicals and of the polymer repeat unit. The design of formulated (polymer-based) products and polymer blends are typical examples of mixture design. Here, a chemical is added to an active ingredient (representing a mixture or blend) in order to enhance one or more specified functional properties of the original product. For example, the performance of a polymer based product may be improved by changing its viscosity through the addition of a chemical and forming a polymer blend or solution.

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The fundamental objective of CAPD, therefore, is to identify a polymer repeat unit (and/or its distribution) or a polymer blend or a polymer solution having specific (desired) properties. This can be achived by identifying the end-use properties defining the product performance and then controlling the end-use properties by manipulating the structural (also micro-structural) properties. The structures of the polymer repeat units can be represented using appropriate descriptors (groups, atom connectivities, bond energies, etc.). This means that the property evaluation methods must also be based on these descriptors. The most common approach in CAPD is to generate chemically feasible polymer repeat unit structures from a set of descriptors (represented by groups or fragments) and to test them by estimating their desired (target) properties. The properties are estimated by using some kind of fragment-based methodology, where the contributions for a specific target property of each fragment present in the structure (polymer or chemical compound) are added to determine the final property value. The set of feasible polymer repeat units are identified as those that match the property targets, given as a series of constraints with upper and/or lower bounds of property values. The optimal compound is identified from the set of feasible polymer repeat units through a problem specific selection criteria or objective function. The principal differences between the various methods of solutions (that could be applied to CAPD) are how the various steps are performed, the type of descriptors used and how the necessary property values are obtained. A discussion on the various types of solution approaches can be found in Achenie et al. (2003). In this chapter, the generate-and-test approach is used.

3. GENERATE-AND-TEST APPROACH Based on the above, the following multi-level generate-and-test approach for CAPD has been developed. The CAPD problem is decomposed into four subproblems starting from level 1 and ending at level 4. • •

•

Level 1 - problem definition: Define the desirable end-use properties in terms of upper & lower bounds and goal values. Level 2 - generate-and-test (repeat units): Identify the polymer repeat units that fall within the upper and lower bounds of the desirable properties and functions of the polymer. If polymer repeat unit structure is already known, by-pass this level. Level 3 - generate-and-test (repeat unit distribution): If branched configuration of polymer repeat units are desired, for the list of feasible

Computer aided polymer design using group contribution techniques

•

261

polymer repeat units, identify the polymer structure in terms of branching information that comes closest to satisfying the goal values. Otherwise, go to setp 4. Level 4 - final selection: Detailed analysis of the identified polymer structures and final selection.

To solve the sub-problems at each level, a suite of property modelling tools are needed. Note that purely predictive methods such as the group contribution based property estimation models are needed for levels 2 and 3. Section 4 presents a class of group contribution based models that have been used in this chapter. A short description of the main issues and needs are given below for each level. Level 1: At this level, a good understanding of the integration across scales is needed because the target values for the end-use properties and functional properties may not be available and in many cases may need to be generated. Also, the controlling of the end-use properties through the manipulation of the macroscopic and/or microscopic structure needs to be investigated. In design and analysis problems involving identification of the product and verification of its performance (such as controlled release of an active ingredient through a polymeric microcapsule), ability to correlate microscopic and macroscopic parameters to the end-use and functional properties is very important. Finally, because fast and simple models need to be used, the properties calculated from the micro and/or meso scales need to macroscopic structural parameters for quick estimated of the target end-use properties. Level 2: At this level the property calculations are mainly at the macroscopic level, but for a wide range of properties - also covering pure polymer as well as polymer solution properties. Therefore, models such as group contribution^"^ (that is extended group contribution) for pure polymer properties and GC-based models for polymer solution properties become very useful. Note that at this stage, the properties are calculated at the polymer repeat unit level. Level 3: At this level, property models need to integrate across scales so that structure of a selected polymer repeat unit can be evaluated. The group contribution^^"^ methods can be used since they incorporate the

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results from the smaller scales into them as additional structural contributions to the property being estimated. Level 4: At this level, all property modelling techniques would be useful since the final selection should be made with the most reliable models. In this way, the work in levels 1-3 are principally to reduce the work in level 4, which can be time consuming and expensive. 4. GC-BASED PROPERTY MODELS In this section, GC-based methods for prediction of pure polymer properties as well as polymer solution properties used for various types of polymer design problems are presented. The criteria for their selection have been that they must be predictive in nature, have a wide application range, have acceptable accuracy, and, they must be flexible enough to fit a computer aided system for polymer design. The properties include pure polymer properties (the glass transition temperature, viscosity, etc.) and polymer solution properties (solubility, partition coefficient, diffusion coefficient) at the polymer repeat unit. In addition, properties such as the radiation of gyration, density and solubility of gases in polyethylene (PE) has been modelled as a function of parameters defining the arrangement of the PE repeat units in branched configurations. Properties are classified as Low properties (which require low computational expense and can be modelled through group contribution or topological indices techniques) and High properties (which can only be computed through computationally demanding simulations ranging from the nanoscale to the mesoscale). For the Low properties, property models that can handle a wide range of polymers should be available. The polymers will be characterized in terms of repeat units (for the average end-use properties) as well as higher level structural information (length of side chains, frequency of side chains, etc.).

4.L Pure Polymer Properties A list of the pure polymer properties together with their model reference is given in Table 1. The property models for the glass transition temperature is presented in detail in this chapter as this model has been further developed by the authors.

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Table 1: List of GC-based property models for pure polymer properties Property

Model Reference

Remarks

Density, p

Van Krevelen

GC-based

Molar volume, v

Elbro et al.

GC-based, function of temperature

Glass transition temperature. To

ProPred (Gani);

GC-based;

Van Krevelen

GC-based

Dielectric constant, 8

Van Krevelen

GC-based

Viscosity, r|

Yinghua et al.

GC-based

Zero shear viscosity, r|zs

Van Krevelen

Secondary property

Water absorption, Ws

Van Krevelen

GC-based

1 N2 permability, PN2

Van Krevelen

GC-based

1 O2 permability, P02

Van Krevelen

= N2 permability *3.8

Van Krevelen

= N2 permability *24

CO2 permability, Pco2

|

4,1.1, Glass Transition Temperature The glass transition temperature (TQ) of a polymer repeat unit is calculated from the contributions of the functional groups representing the polymer repeat unit structure. TG = (567.128 + IniNi + ImjMj )IM^

(1)

In Eq. 1, «/ is the number of times first-order group / appears in the polymer repeat unit, /Wy is the number of times second-order group j appears in the polymer repeat unit, Ni and Mj are the regressed contributions for groups / and7, respectively; and M^ is the average molecular weight of the polymer. The model has been developed by correlating 388 data points from more than 200 different polymer repeat units. The correlation statistics is given in Table 2 while the group contribution values are given in Tables 3a (first-order groups) and 3b (second-order groups).

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Table 2: Correlation statistics for the GC-based model for glass transition temperatures of polymer repeat units. Number of

Average absolute

Standard

data points

error(K)

deviation (K)

12

14.7

18.4

Poly[ureas]

13

11.2

13.3

Poly[di-esters]

7

7.4

8.3

1 Carbohydrates

8

14.5

16.9

1 Poly[styrenes]

21

20.6

26.6

1 Poly[acrylates]

21

9.8

12.6

1 Poly[carbonates]

13

9.4

12.7

1 Nylons

13

10.9

13.9

Total*

388

13

18

Polymer family

1 Poly[urethanes]

1

Total includes polymers not listed under "polymer family"

4.L2. Other properties and GC^ methods For all the other properties, the models as given in the cited references (see Table 1) have been used. The only difference is that the polymer repeat unit is described in terms of groups from a master table (which are also used as the building blocks for representing the polymer repeat units). The master table also includes the corresponding groups needed for each property model. In this way, once a polymer repeat unit structure is generated or specified, all property models for which the corresponding groups and their contributions are available, can be applied to predict the required properties. If a group or its contributionss are missing, they can be created through an atom connectivity index method (Gani et al. 2005) that does not require the use of additional experimental data.

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Table 3a: First-order groups and their contributions for the TG model (see Eq. 1) First-order group CH3

Tc-contribution

Tc-contribution

-2890.47

First-order group CCl

CH2

2686.096

CHC12

45063.10

CH

11553.63

CC13

29077.06

C

23193.65

ACCl

18471.71

CH2=CH

-1412.39

ACN02

30207.04

CH=CH

2880.945

I

42042.92

ACH

2588.087

Br

22354.37

AC ACCH3 •

10811.23

ACF

11174.33

10538.63

CF3

15163.85

ACCH2

10619.01

CF2

12577.07

ACCH

20839.63

CF

26110.23

OH

2198.939

11291.85

CH3C00

13939.45

COO Si

CH2COO

14887.94

SiO

16773.31

CH30

-303.827

C0NHCH2

29683.86

CH20

4994.031

C2H502

12960.73

CH-0 CH2CN

18763.45

CH3S

3825.832

COOH

837.6483 16128.90

CH2S F

10562.46 205.6617

CH2C1

21728.91

O

1385.750

CHCl

12648.13

NHCOO

30554.09

CH2 (Cn > 9)

5316.608

NHCONH

30645.46

1 C03

I

5316.608

I 35843.27

17423.53 1

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Table 3b. Second-order groups and their contributions for the Tg model (see Eq. 1) Second-order group

TG-

1 Second-order group

TG-

1

contribution |

contribution |

(CH3)2CH-

1544.60

6533.23

(CH3)3C-

2583.30

>C-COO-(C)„,

-1146.48

-CH(CH3)CH(CH3)-

-3987.02

(CH2)n,; m<9 styrenes

4879.86

-CH(CH3)C(CH3)2-

10537.95

-C(CH3)2C(CH3)25-membered ring

14394.35 -9128.05

-AC-NH-COO-C-

6-membered ring

-2603.92

-AC-NH-CO-NHCH2-

^cyclic^m? n i - ^ l

6536.25

(CH2V; methacrylate

m<9 -828.72

CH3COOCH; CH3COOC

-1076.56

(CH2V acrylamine

m<9 -1204.60

ACCOO >CH-OH

Carbihydrate-ring

AC-O-CHm

758.85 -92.10 1556.90

AC-Br

-1001.47

1 AC-I

12533.86

>SiO(CH3)2 >C(CH3)2 AC-0-CO-C -Cn-O-CO-Cm-CO-0-

>CH-OOC-(C)„

3561.68

AC-C„-0-CH™

1589.49

>CH-COO-(C)„,

-2337.18

-CF2-CF-

10644.27

Cn-CO-NH-(C)™

-7205.38

-C-CH(CF2-)-

10945.61

AC-CH(CH3)-AC

-10065.45

-0-AC-AC-O-

14313.05

AC-C(CH3)2-AC

-9517.57

CH-C0-0-(CH2)„,

-642.09

>C(CH3)COO(CH2)„

2143.94

-0-AC-AC-CH3

5280.42

AC-C-AC

1949.86

AC-CH2-AC

-4710.36

1 AC-CH-AC

1 -5120.25

1 AC-C03-AC

1 -C-NH-COO-C-

2485.52

-CH2-NH-CO-NHCH2-

-10113.48 913.09 6307.60

19976.35 -1613.43 -5306.30 -3309.66

-3109.14

1

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267

According to the atom connectivity index method, any group can be represented by atom connectivities and if the corresponding atom connectivity contributions for the target properties are available, the contribution of the new group (or an existing group with missing contribution) can be directly predicted through the following equation.

where, Y is the pure component property to estimate; Ai is the number of ithatom occurring in the molecular structure; v^O is the zeroth-order (atom) connectivity index; vxl is the first-order (bond) connectivity index; a, is the contribution of atom /; b and c are adjustable parameters; and, disa constant. Adding this feature to any GC-based method converts it to a GC^ method with a very wide application range without the need for additional experimental data. 4.2. Properties of Polymer Solutions - Partition Coefficients In this section, GC-based models for prediction of polymer solution properties (partition coefficients) is presented. The first attempt has been to select and implement a model for the prediction of the thermodynamic partition coefficients (Kp/sow) through infinite dilution activity coefficient calculations (Eq. 2).

^PISOIV

""

^00

(3)

The challenge here is to use a simple model that is predictive and which can be extended for application to a wide range of complex chemicals. The selected model is the "GC-Flory Equation of State" by Bogdanic et al. (1994), which is a simple activity coefficient model based on a group contribution approach, with an existing parameter table that provides accurate and predictive results for solvent-polymer systems. This model is based on a modified form of the Flory equation of state [Flory et al. (1964)], which has been converted into a groupcontribution form by Chen et al. (1990). The original Chen et al. (1990) model was revised and simplified together with a new parameter table provided by

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268

Bogdanic et al. (1994). This revised model has been used in this chapter as the starting point. The model equations are given in Appendix A. The GC-Flory EoS is a predictive model in the sense that only the molecular structure (represented by groups), the temperature and the composition of the system need to be provided in order to estimate the activity coefficient, as illustrated in Figure 1. One advantage of the GC-Flory EOS over some of the other GC-models [Entropic-FV, Kontogeorgis et al. (1993) and UNIFAC-FV, Oishi and Prausnitz (1978)] is that they need accurate data of density for the solvent as well as the polymer at the system temperature. The parameters of the GC-Flory EOS (constitutive model) are both pure component (Ci) and group interaction parameters (Smn and 8mm)-

Molecule structure

—¥

Groups

Parameters

^ —¥•

GC-Flory model

A

*

Activity coefHcient

T,wi | l « Q, = Ln Cl'^-'^ + Ln Q/' + Ln Qf"^ |

Figure 1. Procedure of activity coefficient calculation through GC-Flory EoS model. 4,2. L Testing of the GC-Flory EOS In order to verify the performance of the GC-Flory EOS with the original group parameters, the calculated values of partition coefficients for some test systems were compared. In Table 4a the experimental values [Hao et al. (1992)] of activity coefficients at infinite dilution (fi^^j 2) of two chemicals in a polymer (Poly(n-butyl methacrylate), PBMA) are compared with the ones calculated with the GC-Flory EoS. It can be noted that good agreement has been obtained even though the model parameters were not regressed or tuned with these data. In Table 4b, the results involving three chemicals (having more complex molecular structures than the solvents in Table 4a) in Poly(ethylene-co-vinyl acetate), EVA, are given. These examples were selected so that the available parameter table of the GC-Flory EoS model could be used and the predicted property values could be compared with experimental data reported by Pitt et al. (1988). Although there are some differences between the experimental and calculated values, it should be noted that this has been a pure prediction, that is, without any adjustment of the original parameters. This means that through a

Computer aided polymer design using group contribution techniques

269

modelling framework for generating new groups and parameters, the application range of this model and its quantitative accuracy can be further extended. Table 4a. Comparison of experimental and calculated activity coefficients at infinite dilution, in weight-basis (Q"*j 2) Experimental

Calculated

^00

Comp. 1

Comp. 2

T^K)

" u

"1.2

n-pentane

PBMA(91000)

343.2

11.0

11.65

Chlorobenzene

PBMA (73500)

393.2

3.18

3.38

Table 4b. Comparison of experimental and calculated drug partition coefficients AI

between polymer and water (K /^), at 298 K Calculated

Experimental AI

Al

Complex chemical (AI)

Polymer

Androstenedione

EVA

2.61

2.182

Testosterone

EVA

2.66

2.217

Progesterone

EVA

3.01

3.210

4.2.2. Extension of GC'Flory EoS The next step has been to systematically extend the group parameter table so that a large range of chemicals and polymers can be handled. This was necessary because the available parameter table [Bogdanic et al. (1994)] did not have the groups to describe the large and/or complex chemicals as well as the parameters to estimate the needed properties. An extension of the model has been made [Muro-Sune et al. (2005a)] and the procedure employed for parameter estimation is highlighted in Figure 2. The data required for estimation of the group parameters included low molecular weight pure component thermal expansivities (a) and enthalpies of vaporization (AHyap) together with the corresponding VLE data. As illustrated in Figure 2, for a new group n, the volume and surface area parameters (/?, Q) are given together with the pure

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M. Hostrup et al.

component thermal expansivity, a{T), of the solvent as a function of temperature. With these data, the GC-Flory model was used to generate the heat of vaporization (AHyap) of the solvent as a function of temperature and the activity coefficients (Qi), for guessed values of the model parameters (the Cparameters and the interactions, £„„ and €„„,)' The calculated values were then checked against experimental values and the procedure was repeated until a minimum of the objective function was obtained. AHvap,eip(T)

Qie xp

a(T)

C(T) y

"N ^ Rn, Qn

GC-Hory model

S2i,cak AHvap,calc(T) T

T

Objective function

^

Sim

Sum

Figure 2. Scheme for the group parameter estimation procedure (Rn and Qn are the hard core volume and surface area of new group n, respectively). More details on the methodology for parameter estimation together with the extended group parameter table have been presented elsewhere [ Muro-Sune et al. (2005a)]. The following new groups have been added: aC-O, aC-OH and Cl(C=C). The important point about introducing new groups is to find solventsolute mixtures that are smaller than the new chemical and polymer, respectively, but have the same groups needed to represent the new chemical and the polymer. More details on the methodology for parameter estimation together with the extended group parameter table can be found in [Muro-Sune et al. (2005a). The following new groups have been added: aC-O, aC-OH and C1-(C=C). The important point about introducing new groups was to find solvent-solute mixtures that represented the complex chemicals of interest (in this case, those commonly used as active ingredients in drugs, pesticides, food, etc.) and polymer, respectively, but where the same groups are needed to represent the chemical and the polymer structures (Muro-Sune et al. 2005b). Figures 3a and 3b illustrate the performance of the new groups when compared to the corresponding experimental data. In Figure 3 a, the performance of the new groups (aCO, aCOH and C1-C=C) and their regressed group interaction parameters covering 59 systems and datapoints is highlighted. In figure 3b, the improvement in the performance of the GC-Flory EOS due to the introduction

Computer aided polymer design using group contribution techniques

111

and use of the aCO group as opposed to CH30 group in the aromatic chemical compound is highlighted.

Figure 3a. Comparison of calculated and experimental data for polymer systems involving aCO, aCOH and CL-C=C groups [Muro-Sune et al. (2005b)]. 2.3

-7

^

2.2 2.1 2

I 1.9

j
i+ .-'++

1

•H-++

1.6

f

/W>

1.5

J X-"^ . a

b

y''

.-'

y

B 1-8 1.7 o 0*

1.4 -^ 1.4

\ ^=-^ 1.5

1.6

T

1.7

+ (a) old groups 0 (b) new groups 1

r

1.8 1.9 InQ exp

\

1

1

2.1

2.2

2.3

Figure 3b. Comparison of calculated and experimental data for polymer systems involving groups (new) aCO & (old) CH30 [Muro-Sune et al. (2005a)].

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M. Hostrup et ah

4.3. Properties of Polymer Solutions - Diffusion Coefficients The selected model in this section is based on the free-volume theory of diffusion [Vrentas & Duda (1977)]. Extensive work has been presented on the free-volume theory as well as attempts to obtain a purely predictive model [Zielinski & Duda (1992)] for solvents in polymers. The objective here is to apply and test the same theory for the diffusion of the complex chemicals in polymers. The details on how to use the theory in a predictive manner have been presented elsewhere [Zielinski & Duda (1992)] and only the model equations needed for property estimation are given here (see Appendix B). As the original model did not have the necessary parameters for the chemicals of interest, they had to be first estimated according to the procedure suggested by Zielinski & Duda (1992). In this case, the parameter defined as the ratio of molar volumes of solvent and polymer jumping units (§), needs a special mention. In the case where the solvent is "small" this parameter is easily obtained through Eq. B.6, given that the volume of the solvent jumping unit (Vlj) corresponds to its molar volume at 0 K (VIO) and the volume of the polymer jumping unit can be calculated through available correlations (Eqs. B.7-B.8). On the other hand, when the molecule is "large" (as is the case of most AI compounds), Eq. B.6 is not valid and thus the estimation of this parameter is not straightforward. In these cases, two alternatives are possible depending on: (a) if experimental data is available or (b) if no experimental data is available. Application of the first alternative is illustrated for Permethrin (a pesticide molecule) for which experimental data of diffusion coefficient in Polypropylene as a function of temperature is available. The modelling results are presented in Figure 4 and could be considered satisfactory given that a very limited number of diffusion data was available. From this estimation of the ^ parameter the volume of the Permethrin jumping unit can be back-calculated. This is in fact a property of the complex chemical, which is independent of the polymer and can therefore be used to calculate the ^ parameter for this same chemical in any other polymer. That is, once this ^ parameter has been estimated, it can be used to predict the diffusion coefficient for this chemical in all other polymers without the need for additional experimental data. This feature has been illustrated by Muro-Sune et al. (2005b) through modelling of controlled release of the chemical through a polymeric membrane microcapsule, where the predicted diffusion coeeficient was used.

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273

For the second alternative where no experimental data was available, an extension of the free-volume theory for large molecules [Vrentas et al. (1996)] was used. This alternative is based on accounting for the effects of the molecular shape (asymmetry) of the compound and this feature has also been illustrated by Muro-Sune et al (2005b) through a case study where molecular modelling has been used to generate the necessary information for the estimation of the ^ parameter. Again, the estimated diffusion coefficient has been used to predict the controlled release of the chemical from a polymeric membrane based microcapsule. l.OE-10 1 l.OE-11 l.OE-12 ^

l.OE-13

"a Q

A. ^*A

l.OE-14 •A

l.OE-15 l.OE-16 H l.OE-17 200

250

300

350

400

T(K)

Figure 4. Diffusion coefficient of Permethrin in Polypropylene. Comparison of estimated ( — ) and experimental data (D). 4.4. Polymer Properties as a Function of Repeat-Unit Configuration

Data for radius of gyration, density and self diffiisivity of polyethylene (PE) was generated by ICE-FORTH (Patras, Greece) as a function of chain length as well as branching parameters, as shown in Fig. 5. These data were then analyzed and a GC^ model was developed to represent the data. In this way, the molecular modelling parameters and data were converted to accurate

274

M. Hostrup et al.

engineering models (GC+) for the end-use properties. The developed models, are given in Tables 5-7.

A-l

Br_Ax]B_CA-l )xC B : Branching Frequency (in CH2 units) (in this example 10) A : Number of Linear Segments along the backbone (in this example 8) A-1^^ Number of Branches (in this example 7) C : Branching Length (in CH2 units) (in this example 6) [br^ch point is counted as atom of the main backbone]

Figure 5. Branching parameters for branched polyethylene

Table 5. Model details for radius of gyration for repeat unit distribution Radius of gyration (Rp) Independent variables: C, n CH2, n CH Dependent variable: Radius of gyration RG = -38.72593 + 3.15745*n_CH2 - 2.47186*n_CH + 5.563*C Adj R-Square Standard Deviation R-square 0.9919 0.9897 25.0629 Table 6. Model details for density for repeat unit distribution Density (p) j Independent variables: C, n CH2, n CH Dependent variable: Density p = 0.75105 + 4.26E-05*n_CH2 + 3.78E-04*n CH+1.39E-04*C Standard Deviation Adj R-Square R-square 0.0030 0.5511 0.6858

!

Computer aided polymer design using group contribution techniques

275

Table 7. Model details for self diffusivity for repeat unit distribution SelfDiffusivity(Do) Independent variables: C, n CH2, n CH Dependent variable: Self Diffusivity Do = 13.1882 - 0.04785*n CH2 + 3.81E-05*(n CH2)' - 0.22388*n CH 0.21599*C Adj R-Square Standard Deviation R-square 0.8901 0.9389 0.8305

Model details for pemeability of Oxygen and Nitrogen in Polyethylene p

o.

^ - In I ^^-^- " ^^"^^ 17.66e -13 with R^ = 0.9989

\Mle6

pN, ^ - In I ^^-^-" ^^'^^ 11.67^-12 with/?'= 0.9984 1.212e7 J^carbon IS the number of carbons in the chain of PE.

5. SOFTWARE FOR CAPD Figures 6a-6c provide a schematic overview of the computer program developed on the basis of the multi-level CAPD algorithm described in section 3 and the property models described in section 4. Note that the current version of the computer program is only able to design and analyze polymer repeat units based on the pure polymer properties listed in Table 1. For other CAPD examples (involving polymer solution properties and/or polymer branching parameters), stand-alone programs have been developed, which have not yet been integrated to the CAPD software.

M. Hostrup et al.

276

START

I

Read data files (groups + contributions)

DISK

z For NoGrp := ng_nnin to ng„max

Reset local storage

DATA: GroupVector[1 ..nDiffGr]

i Call: RecursivePolyDesign(NoGrp, GrpType=1)

I Next NoGrp Ref to extern, function.

C STOP ^

Figure 6a. Algorithm for linear polymers

277

Computer aided polymer design using group contribution techniques

\

NO

(Groups to be I considered) \

p\ GrpType:=GrpType + 1

GroupVectorJGrpType] = n oo^ur

NO

YES

Call: AddLlnRepeatUnit(GroupVector)

iiiURN

Figure 6b. RecursiveLinearPolymerDesign

278

M. Hostrup et al.

GroupVector

Call PredictProperties(GroupVector)

Call CheckConstraints(Properties)

PropPredict

CheckConstraints

Discard GroupVector

Figure 6c. AddLinearRepeatUnit

5.7. CAPD Software An MS-Excel based CAPD software that incorporates the generate-and-test approach described in section 3 has been developed. The CAPD software consists of 3 main tools that incorporates levels 2-3 of the methodology, as highlighted below. A user-interface takes care of level 1, while the detailed analysis for level 4 needs to be done with other software or database, if available.

Computer aided polymer design using group contribution techniques

279

1. Structure generation tool: Generate all combinations of groups for the repeat unit (see Fig. 7a of how groups are combined to form fragments of a repeat unit). The combinatorial algorithm (of recursive nature) is currently capable of generating description of repeat units for linear polymers. In the current form, the algorithm is not capable of determining the order of occurrence of the groups in the repeat unit.

- CH2 -CO-

"

^

-C(=0)0-0-CONH-

|l:(-CH2-CHOH-CH2-) 2:(-CH2-0-CH2-) X:

Figure 7a. Combinations of groups (box on left hand side) to form repeat unit fragments (box on right hand side). 2. Property prediction tool: Predict the polymer properties using the available group contribution methods. See section 4 for further details. 3. Screening tool: Screen the generated repeat unit structures with respect to their properties to identify the feasible set; order the feasible set according to a specified criteria. Based on the above software architecture, CAPD consists of the following two separate modules (as shown in Fig. 7b): • •

A calculation engine (Design-DLL) written in C++ for property estimation, structure generation and screening A user Interface in MS-Excel that also handles the communication with the calculation engine.

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280

COM interface layer

Figure 7b. Overall structure of the MS-Excel based CAPD software.

5.2. Use of the CAPD Software The use of the CAPD software is quite straightforward. The user needs to specify the number of groups that one repeat unit should be constructed of; the identities of the that should be used as building blocks; the upper and lower bounds for the target properties (constraints). On execution and successfiil completion, the results are shown in terms of a table of feasible polymer repeat unit structures and their corresponding calculated property values. 6. APPLICATION EXAMPLES 6.1. Example 1: Pure Polymer Design Polyethylene and polypropylene is used for coating paperboard to increase the barrier properties. This is especially popular for packaging dairy products in the Nordic countries. To protect the products from degrading oxygen, oxygen should be kept away from the dairy product. To protect the paperboard fi*om being waterlogged as little water as possible should be absorbed on the coating and thence getting into contact with the board.

Computer aided polymer design using group contribution techniques

281

The MS-Excel based software has been used to screen the types of polyolefin for this application example. A minimum permeability for oxygen and carbon dioxide were specified as shown in the input to the routine in Table 8. Allowing up to 4 different groups in the repeat unit and sorting the screened feasible alternatives according to water absorbance gave the results shown in Table 9. Table 8. Input to the CAPD software for Example 1. Consider group in eat unit Repeat unit size Min no. Q r o u p s l H ^ ^ ^

CH2 CO COO 0 CONH CHOH CHCI CCI2 CHF CF2 CHCN CHCH3 C(CH3)2 -ACCHAC -CH=CH-CH=C(CH3)-CH=C(CI)-Si(CH3)2-

Design properties Min 1 Max Densityl Glass transition temperature Water absorption Dielectric constant N2 permability 02 permability C02 permability Zero shear viscosity (r|o) f%^^^

Extra input required: Temperature Ave. Molecular Mass

•K

Bg/mol

Target |

11^^ llll^^ HH

polymer

BHHH^'^^s*^^

l i ^ S H n cm2/s*Pa •HHH ^ P P ^ N * s / m 2 (=10 Poise)

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283

The results suggest polypropylene as the one that absorbs the least amount of water while polyethylene absorbs the most. Varying mixtures of polyethylene and polypropylene gives intermediate water absorbance. It can also be seen that the glass transition temperature of the polymer increases the more polypropylene is added. This also means that coating with polypropylene requires higher processing temperatures for the coating process than using polyethylene. 6.2. Example 2: Polymer design for membrane based gas separations Membrane gas separators are separation units which split a given gas stream into two product gas streams, a high pressure retentate stream and a low pressure permeate stream. The membrane provides a selective mass transfer layer. Due to difference in chemical potential the species permeate through the membrane material at different rates. Polymeric membranes are normally asymmetric in shape, i.e., a thin active membrane layer is laminated on a highly porous support layer giving mechanical support against the large pressure gradients applied in practice. Pressurized feed gas is normally fed to the shell side and the components permeate at different rates to the fibre bore. The retentate gas that is depleted in fast permeating components is withdrawn at essentially the feed pressure. In this work a non-porous membrane is being considered. The case study of air separation is illustrated in this work. Oxygen of more than 30% purity (for combustion) and more than 40% purity (for medical use) are required. 6.2.1. Mass transfer in gas separation membranes The one-dimensional flux, N^k^ of component k through a nonporous polymer membrane is given by Pick's first law (assuming constant diffusivity). Inserting the phase equilibrium relation describing the solution equilibrium, the equation becomes: do

D ,

,

(DS\,

.

(3) The product of diffiisivity. A, and solubility, 5/, is called permeability. Permeability is the key property that controld mass transfer in this separation

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M. Hostrup et al.

process. Permeability was calculated by mainly two ways in this work. One by using group contribution method based on free volume theory by Vrentas et.al. (1996). Second approach is to get polymer properties as a function of microscopic structural paramters by molecular modeling [Karayiannis et. al.(2003)]. In general, a hierarchical modeling approach of three steps is followed: 1) Monte Carlo (MC) simulations to extract information about the density, radius of gyration and static structure factor (i.e. local packing of the polymer atoms); 2) Molecular Dynamics (MD) simulations to extract information about the diffusivity of the polymer chains and characteristic relaxation times; 3) Transition State Theory (TST) to extract information about the free volume, solubility, diffusivity and permeability of small gas molecules to a polymer matrix. In this work, the structural, volumetric and dynamic properties of a rather simple polymer, those of polyethylene (PE), as a function of its molecular architecture, defined by branch length and branch frequency are used. Sufficient data about the equilibrium radius of gyration of linear and branched PE, the longest relaxation time, and the chain center-of-mass self-diffusion coefficient have been obtained in this case through a state-of-the-art Monte Carlo simulation algorithm. By analyzing these data using well established group contribution methods, closed-form analytical expressions have been developed capable of relating these properties to features of molecular structure and conformation of the polymer. For the case of PE considered here, the permeability of oxygen and nitrogen has been related to the molecular length (number of carbon atoms) of the main chain backbone.

6.2.2. Problem Definition The reverse problem to be solved requires the following information: Feed: 200 mole/s air; TF = 298.150C; PF = 1 bar Permeate: Desired Separation: XO2 = 50%; O2 recovery = 0.0038, with recovery = moles of O2 in permeate/moles of O2 in feed Based on the above information, it was found that the polymer that would do the desired separation should have the following permeabilities: PO2 > 4 barrer and selectivity > 4.5

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285

6.2.3. Results - Polymer repeat unit configuration 20 polymers from literature; 13 polymers from group contribution method and 9 polymers generated through molecular modeling were selected and plotted as shown in Fig. 8 (selectivity vs. permeability). The horizontal and vertical lines show the property targets and all the polymers in the shaded region match the target properties. Hence, the validation step was performed only for these polymers (shown in Table 10). It can be seen that the separation tasks are achieved for the polymeric membranes.

-1

1

r-

*#

M 6 QL

a? 5h 1! >

4

m

3h

%* * m

-1

-U

0

0.5

1

1.5

2

2.5

S

$.5

4

LogPo2

Figure 8: All polymers on property target plots (polymers on the shaded part satisfy the target permiability)

The best polymers that appear on the upper right hand side of the shaded part (see Fig. 8) are polymers 8-10 in Table 10.

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286

Table 10: Validation for selected polymers Number

Polymer

% Purity

Recovery

Method

1

6FDA-6FpDA

54.24

1.40E-02

Literature

2

6FBPA/TERE

50.68

1.04E-02

Literature

3

HFPC

56.91

4.66E-03

GC

4

TMHFPC

55.95

7.10E-03

GC

5

TMHFPSF

56.86

4.81E-03

GC

6

TBHFPC

56.67

1.09E-03

GC

7

PE-78

59.02

5.19E-03

MM

8

PE-142

62.27

3.46E-02

MM

9

PE-500

63.77

2.09E-02

MM

PE-1000

65.69

1.73E-02

MM

1 10

6.3. Example 3: Verification of diffusion coefficient This case study involves the application of the free-volume theory in a purely predictive manner for the estimation of the diffusion coefficient without use of any experimental diffusivity data and the predicted values are analyzed through the controlled release of a drug from a polymeric microcapsule. The case study concerns the drug Codeine [CAS Number 76-57-3] that is encapsulated, together with a carrier (an ion exchange resin), within a microcapsules made of polyurea. The polymer wall is formed by water promoted polyreaction of the monomer Methylene diphenyl diisocyanate (MDI, [CAS Number 101-68-8]), that can give a cross-linked polymer. The controlled release has been predicted for a fixed composition of the microcapsules under different scenarios defined by the parameters given in Table 11a. Each scenario has a different membrane thickness (defined through a different monomer-MDl to resinate ratio). As some of the data needed by the model was not available, their values have been assumed (marked in italics in Table 11a). For example, the dimensions (and the size distribution) of the capsules were not available and therefore, values have been assumed (using as basis dimensions of commercial microcapsules) which were then used to

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calculate the wall thickness for each of the experiments. With respect to the needed properties of the system, summarized in Table lib, the diflftisivity has been predicted with the developed model while the values for the partition coefficient (Km/r) and for the membrane-donor coefficient (Km/d), were available. The value of the partition coefficients accounts for the respective solubility of the encapsulated compound with the polymer and the release medium, and the donor medium, respectively. Since these do not change from one experiment to another, they were kept constant. Table 11a. Summary of the input data required for the mathematical release model. Variable MDI/resinate h(m) Max. radius (m) Min. radius (m) Mean radius (m) Standard Deviation Radius Step Vb (m3) t(s) Cd,initial (g/m3) Vd (m3)

Scenario 1 0.1 2.8610-9 32910-9 2910-9 12910-9 3-10-8 110-8 40010-6 12600 358.44-103 0.485-10-6

Scenario 2 0.25 6.72-10-9 329-10-9 29-10-9 129-10-9 310-8 110-8 40010-6 12600 324.72-103 0.536-10-6

Scenario 3 0.5 12.23-10-9 329-10-9 29-10-9 129-10-9 3-10-8 1-10-8 400-10-6 12600 280.697-103 0.620-10-6

Scenario 4 1.0 20.8510-9 32910-9 29-10-9 129-10-9 3-10-8 1-10-8 400-10-6 12600 220.825-103 0.788-10-6

Table 1 lb. Properties required by the controlled release model Variable D (m2/s) Km/r Km/d

All scenarios 1.027-10-19 2.67 0.11

Remarks Estimated Available Available

The value of the diffusion coefficient of Codeine in polyurea is estimated, as mentioned above, in a completely predictive manner through the extended free volume model. In order to perform the free volume theory based calculations, some parameters related to the polymer viscosity with respect to temperature (the WLF parameters - see appendix B, Eqs. B.5 to B.7) are required. For the polymer of interest, that is, polyurea, these parameters (or experimental data to

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estimate them) are not available. Therefore, the parameters corresponding to a polymer that is assumed to have a similar behaviour, a polyurethane, has been used. The value for the diffusion coefficient is estimated at the temperature for which the release experiments are reported, that is 309.15 K. The estimated parameters for the free volume theory based model (Eq. B.l plus associated equations for the parameters) are summarized in Table 12, and the estimated value for the diffusion coefficient is given in Table 1 lb. Table 5. Free volume theory estimated parameters (pure prediction) for Codeine in Polyurea (modelled as Polyurethane). Compound

V(OK)xlO* Kli/yxlO-'' (m3/mol) (m3/gK) 214.5 1.362

Codeine Polyurea 148.8 (Polyurethane)

0.323

K2i-Tgi (K) -132.75

DO Stheory (m2/s) 5.142-10-'' 2.26

-181.4

-

-

Finally the release of Codeine from the polyurea microcapsules is calculated using a controlled release model [Muro-Sune et al (2005b)] for the four scenarios listed in Table 1 la and the simulated release behaviours are compared with experimental data in Figure 9. The performance of the release model with the predicted diffusion coefficient is very good in all the cases, except for the scenario with the smallest thickness (Scenario 1). This can be attributed to different degrees of cross-linking of the polymer that would affect the value of the diffusion coefficient. In the mentioned case (Scenario 1) the amount of monomer forming the wall is smaller than in the others, the degree of crosslinking being reduced and thus increasing the value of the diffusion coefficient. This is shown in Figure 10 where a higher value for the diffusion coefficient is used (D = 1.5910-19 m2/s) and the release data are accurately described.

Computer aided polymer design using group contribution techniques

0

50

0

50

289

200 250 150 t(min) Figure 9. Comparison of experimental and estimated Codeine release values as a function of time (Km/d=0.11). Experimental data: (n) Scenario 1, (A) Scenario 2, (x) Scenario 3, (o) Scenario 4. Predicted release: Scenario 1, Scenario 2, Scenario 3, — Scenario 4. 100

250 200 150 t(min) Figure 10. Comparison of experimental and estimated Codeine release values as a function of time (Km/d=0.11). Experimental data: (n) Scenario 1, (A) Scenario 2, (x) Scenario 3, (o) Scenario 4. Predicted release: Scenario 1, Scenario 2, Scenario 3, — Scenario 4. Plus: Scenario 1 with D = 1.59-10-19 m2/s. 100

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7. CONCLUSIONS

This chapter has highhghted the issues related to computer aided polymer design and analysis. For computer aided techniques to be applicable, it is necessary to have available, an appropriate suite of property models that use the same structural parameters as those used to generate the structures of the polymer product. The property models need to have a wide range of application and they need to be simple, yet predictive with acceptable accuracy. Also, it should be possible to correlate the target end-use properties with macroscopic and microscopic structural parameters defining the polymer or polymer based product. Once the property models have been validated, the CAPD algorithm and software (if available) can help to identify a feasible set of promising alternatives within which the optimal product is likely to be found. In this way, the computer aided techniques help to reduce the time and resources spent on finding and developing a polymer product. However, as the example of the design of polymer repeat unit distribution at various chain lenghts have pointed out, data relating end-use property to the microscopic structure, in most cases, is not available. Therefore, a considerable effort is needed not only to generate the data but also to use the data to develop GC^ methods that can be used for polymer design and analysis. One alternative is to use molecular modelling and/or computational chemistry to generate these data. This, however, needs time and the pseudo-experimental data also needs to be verified. Current and future work is concentrating on further developing the CAPD software together with more GC^ polymer property models.

List of Symbols

A

Surface area through which diffusion takes place (m^)

C

Temperature dependent molecular external degrees of freedom parameter WLF parameter of component j WLF parameter of component j (K) Polymer-solvent binary mutual diffusion coefficient (m^/s) Self-diffiision coefficient of the solvent (m^/s) Constant pre-exponential factor (m^/s) Apparent diffusion coefficient (m^/s)

Cij ^^^ C2j^^^ D Di Do Dapp

Computer aided polymer design using group contribution techniques

E

291

T

Energy (per mol) that a molecule needs to overcome attractive forces which constrain it to its neighbours (cal/mol) Free-volume parameter of component j (m^/g K ) Free-volume parameter of component j (K) Partition coefficient of the AI between the donor and the polymer m e m b r a n e Partition coefficient of the A I between the polymer m e m b r a n e and the release medium Partition coefficient between polymer and solvent M a s s (g) Molecular weight of component j (g/mol) Molecular weight of the polymer j u m p i n g unit (g/mol) N u m b e r of points to evaluate the function Total number of moles of the system (kmol) N u m b e r of moles of component i (kmol) N u m b e r of particles Pressure of the system (Pa) Surface area of component i Surface area o f group n (normalized V a n der Waals surface area, U N I F A C ) G a s constant Hard-core volume of group n (normalized V a n der Waals volume, U N I F A C ) Temperature of the system (K)

To Tgj V

Reference temperature (298.15 K ) Glass transition temperature of component j ( K ) V o l u m e (m^)

Vd Vr V Vi Vi*

Donor volume (m^) Receiver volume (m^) Molar volume of the system (m^/kmol), in GC-Flory E o S Molar volume o f component i, in GC-Flory E o S Molar hard-core volume of component i (m^/kmol), in G C Flory E o S Reduced volume of the system Reduced volume o f pure component i

Kij K2j Km/d Km/r Kp/soiv M Mj M2J nf n ni Np P qi Qn R Rn

V V Vj* Vi^ Vij

Specific critical hole free volume required for a j u m p (m^/g), for component j , in free volume theory Molar volume o f a solvent at 0 K (mVmol) Molar volume of a solvent j u m p i n g unit (m^/mol)

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Molar volume of a polymer jumping unit (mVmol) Weight fraction of component i Molar fraction of component i Coordination number (z=10)

Wi Xi

z

Greek letters a

p X AH vap AM ASji Af

Thermal expansivity (K"^) Polymer specific proportionality constant Flory-Huggins polymer-solvent interaction parameter Enthalpy of vaporization (J/kmol) Mass change (g) Interaction energy parameter (J/kmol of contact sites) Interaction energies between unlike groups n and m (J/kmol of interaction sites) Interaction energies between groups Interaction energies between like groups (J/kmol of interaction

eji(v),Sji

sites)

(V,)

Energy interaction parameters

y

v„'-(i) a (pi

at aiiattr comb

at

Volume fraction of component j Overlap factor accounting for shared free volume Surface area fraction of component i Microcapsule mean radius (m) Number of groups n in component i Standard deviation Segment volume fraction of component i Activity coefficient (weight basis) of component i Infinite dilution activity coefficient (weight basis) of component i Attractive contribution to the activity coefficient Combinatorial contribution to the activity coefficient Free-volume contribution to the activity coefficient Ratio of molar volumes for the solvent and the polymer jumping units

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Subscripts calc exp ij m, n max min P r solv

Calculated (value) Experimental (value) Component i and j respectively Group of type m, n Maximum Minimum Polymer Release medium Solvent

Superscripts attr comb fV CXD

Attractive Combinatorial Free-volume Infinite dilution

References Achenie, L. E. K, Gani, R., Venkatasubramanian, V., Computer Aided Molecular Design: Theory & Practice^ CACE-12, Elsevier, The Netherlands, 2003. Bogdanic, G. & Fredenslund, A. (1994). Revision of the Group-contribution Flory equation of state for phase equilibria calculations in mixtures with polymers. 1. Prediction of Vapor-Liquid equilibria for polymer solutions. Industrial & Engineering Chemistry Research, 33, 1331-1340. Chen,F., Fredenslund, A. & Rasmussen, P. (1990). Group-contribution Flory equation of state for Vapor-Liquid equilibria in mixtures with polymers. Industrial & Engineering Chemistry Research, 29, 875-882. Elbro, H. S., Fredenslund, Aa. & Rasmussen, P. (1990). Macromolecules, 23, 4707. Flory, P. J., Orwoll, R.A. & Vrij, A. (1964). Statistical thermodynamics of chain molecule liquids. I. An equation of state for normal paraffin hydrocarbons. J. Am. Chem. Soc, 86, 3507. Gani, R. (2002). ICAS Documentation, PEC02-14, CAPEC Internal Report, Technical University of Denmark, lyngby, Denmark.

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Gani, R., Harper, P. M. & Hostrup, M. (2005). Automatic creation of missing groups through connectivity index for pure-component property prediction. Industrial Engineering and Chjemistry Research^ 44, 7262. Hao, W., Elbro, H.S., Alessi, P. (1992) Polymer solution data collection. Part 2+3 (vol. 14), DECHEMA Chemistry data series. Karayiannis, N. Ch., Giannousaki, E. A. & Mavrantzas, V. G. (2003). Journal of Chemical Physics, 118(6), 2451-2454. Kontogeorgis, G.M., Fredenslund, A. & Tassios, D.P. (1993). Simple activity coefficient model for the prediction of solvent activities in polymer solutions. Industrial & Engineering Chemistry Research, 32, 362. Muro-Sune, N., Gani, R., Bell, G., Shirley, I. (2005a). Predictive property models for use in design of controlled release of pesticides. Fluid Phase Equilibria 22S'229,127-133. Muro-Sune, N., Gani, R., Bell, G., Shirley, I., (2005b). Model-based computeraided design for controlled release of pesticides. Computers and Chemical Engineering, 30, 28-41. Oishi, T. & Prausnitz, J.M. (1978). Estimation of solvent activities in polymer solutions using a groupcontribution method. Industrial & Engineering Chemistry Research, 17, 333. Pitt, C.G., Bao Y.T., Andrady, A.L. & Samuel, P.N.K. (1988).Thecorrelation of polymer-water and octanol-water partition coefficients: estimation of drug solubilities in polymers. Int. J. of Pharmaceutics, 45,1-11. Van Krevelen, D. W. Properties of Polymers. Elsevier: Amsterdam, 1990. Vrentas, J.S. & Duda, J.L. (1977). Diffusion in polymer-solvent systems: II. A predictive theory for the dependence of diffusion coefficients on temperature, concentration and molecular weight. J. of Polymer Science: Part B: Poly. Phys., 15,417. Vrentas, J.S., Vrentas, CM., Faridi, N. (1996). Effect of solvent size on solvent self-diffusion in polymer-solvent systems. Macromolecules 29, 3272-3276. Yinghua, L., Peisheng, M. & Ping, L. (2002). Estimation of liquid viscosity of pure compounds at different temperatures by a corresponding-states group contribution method. Fluid Phase Equilibria, 198, 123-130. Zielinski, J.M. & Duda, J.L. (1992). Predicting polymer-solvent diffusion coefficients using free-volume theory. AIChEJ., 38, 3,405-415.

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Appendix A: GC-Flory EOS The model equations are listed below without further explanations, these being out of the scope of this paper. For more details see Bogdanic et al. (1994). The activity coefficient (on weight basis) for compoinent i is given by, > InQ =lnQ''""''+lnD^+lnQ \Comb

(A.1)

Where, each of the terms on the right side of Eq. A.l is given by. \Comb lnQf°'"''=ln-^ + l-^

(A.2)

Vf.

^~"3_0

InQf =3(1 + C,)ln

-1/3

v"^-l 1

V

lnQr=-z^, —[^,(v)-^„(v,)]+l-lnX^;exp(-A^,//fr)-^

(A.3)

S^.^-PI^I^^ ^ RT (A.4)

With the following definitions for each of the variables occuring in Eqs. A.2 A.4,


^,=

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H^'j^j

(A.5)

(A.6)

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296

(A.7)

v;=(1.448)(15.17)XvA n

(A.8) j

(A.9) n

(A. 10) where, R. TOM

^ Jl

1 ^m

/

m

/

lYi

T,n

n

(A.11) ZJK

(A.12)

^n

n

where.

,(0 _ v ^ a

(A.13)

^i

(A.14) A£j,=Sj,-£,=£jXv)-£„{v)

(A.15)

where.

V

V

(A.16)

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297

Finally, the last term on the right hand side of Eq. A4 is obtained from the following Equation of state.

P =

\

nRT

E'^'^iv)

(A.17)

+

where the energy parameter E** is defined as,

E"-='£^zq,n,\

j;^e,exp{-A£jRT)

^-3/?ln|^lX«,

dC. d(\/T)

(A.18)

Appendix B. Free-volume theory based model for diffusion coefficient A summary of the main equations for the fi-ee-volume theory based model for diffusion coefficient is presented below (for more details the reader is referred to Zielinski & Duda (1992)). The polymer(2)-solvent(l) binary mutual diffusion coefficient (D) is expressed as, D = D,(l-^,)'(l-2j^,)

(B.l)

where Di is the self-diffusion coefficient of the solvent, and is calculated as.

-(wX+w,^v;)

-E]

D, = Z)o exp — exp RT

^w,^(/:,-r,.+7')+w,^(^,-7;,+r) (B.2)

for

Tg
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M. Hostrup et al.

At high temperatures (T > Tg +150 °K), diffusion is no longer free-volume limited and energy effects become dominant. The parameters in Eq. B.2 can be calculated from the following equations, '^

j

- = 2.303C,7^pWLF ^2J

(B.3)

(B.4)

f^2J

4=

(B.5) ^2J

Ifth e solvent is "ismall"', the riight hand side of Eq. B.5 is represnted by,

4=

K _ MX

r

M,

^2J

= 0.6224 *r^2(K-)-86.95

(B.6)

(B.7)

(B.8)

Vrentas et al. (1996) proposed (Eq. B.9) a modification of Eq. (B.5) for calculation of the parameter ^, to be used with the free-volume theory based model for the diffusion and the self-diffusion coefficients (Eqs. (B.l) and (B.2)), respectively.

^-T7-¥

(B.9)

The parameter \|/, which previously was set equal to 1, must now different and calculated through Eq. (B.IO), which includes the effect of the asymmetry of the molecule through the so-called aspect ratio (B/A).

Computer aided polymer design using group contribution techniques

^-uv.,,y,,('-^>B)

299

(«•'«>

The final expression for the ^ parameter is therefore obtained as,

With this modified model, the molecules still jump as single units but ^ ^ ^L because the average hole free volumes of the solvent and the polymer are different (due to the asymmetry of the molecule).

Multiscale Modelling of Polymer Properties M. Laso, E.A. Perpete (Editors) © 2006 Elsevier B.V. All rights reserved.

301

Chapter 5

Design of polyolefin reactor mixtures Andrew J. Haslam^, 0ivind Moen^, Claire S. Adjiman^ Amparo Galindo^ and George Jacksona " Centre for Process Systems Engineering, Dept. of Chemical Engineering, Imperial College London, South Kensington Campus, London SW7 2AZ, UK ^Borealis AS, 3960 Stathelle, Norway 1. Introduction 1.1. Economic issues Polyolefins were among the first plastics produced, with roots back to 1933 when ICI made the first grams of polyethylene. Polyethylene was the material that made RADAR a practical instrument by offering good electrical insulation, enabling instrumentation for warning of attacks. It had a deciding influence on the outcome of World War II. The main uses of polyolefins today are in packaging, giving a substantial environmental benefit by reducing the weight of goods to be transported and thereby reducing the amount of hydrocarbons burnt by lorries. Without polyolefins it would also be difficult to imagine today's supermarket where food and goods are packed in clear plastics, giving the customer the opportunity to inspect goods prior to purchase. A number of production processes have been developed over the years making the production and sale of polyolefins a very competitive business. The market is extremely volatile and difficult to predict as shown in Figure 1. The price of polyethylene and polypropylene varies by a factor of two over time. The margin between olefins and polyolefins varies even more, by nearly a factor of four. One method by which producers are able to survive financially is by producing speciality plastics, as these have added value. As can be seen also in the figure, the prices for speciality products (compounds and rota-moulding products) are higher than those of the commodity polyolefins. However a major

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A.J. Haslam et ai

polyolefin producer cannot produce speciality products only; it must also offer larger quantities of materials fulfilling the general needs of the customers. 1 Price trend (Norwegian Krone / kg)

^ Compounds „^^^

^ HOPE

v^

-,

jJ*^^

Black Rota Moulding

i

«{ /(

^

Poiyolefins

LDPE Ethylene

Propylene

Olefins

1 i i li i 1 i 1 1 i 1 1 1 1986 1987 1988 1989 1990 1991 1992 1993 1994 1995 1996 1997 1998 1999 2000 2001 2002 2003 2004

Figure 1. The difference in the price of polyolefin and the raw-material olefin varies considerably with time, making it a challenge to operate a profitable polyolefin production business. Manufacture of higher-priced speciality products (compounds and rota-moulding products) is only a partial solution.

The key to financial success in the production of bulk poiyolefins lies in the continual improvement of design and operation, which can be achieved through a better understanding of the physical phenomena that underpin the production process. In particular, this work will show that a knowledge of the thermodynamic behaviour of the olefin and polyolefin can result in significant improvements in productivity. 1.2. Industrial methods of polyethylene production The polyethylene (PE) produced back in 1933 was made with the aid of a highpressure free-radical reaction giving a product referred to as Low Density PolyEthylene (LDPE). In the 1950s and 1960s, other polymerisation processes, based on the use of catalysts, were developed. These catalytic-aided processes are still being developed and have led to more-efficient ways of polymerising olefins. Loop reactors containing slurry of liquid olefin/polymer have become popular; the reactor walls function as large heat-removal areas, enabling a high degree of polymerisation without overheating the reactants. Other efficient reactor types are gas-phase fluidised-bed reactors, in this case removing

303

Design ofpolyolefin reactor mixtures

polymerisation heat via a large through-flow of olefin, again enabling a high degree of polymerisation. Conversion in gas-phase reactors (GPRs) was increased during later decades by the introduction of so-called condensed-mode

The Borstar Process Diluent

•" i Ethylene Comonomer Hydrogen

Ethylene Comonomer Hydrogen

Prepoly reactor to develop particle morpholocly • slurry polymerisation in propane •

/)«6.5MPar=50..70^C

• f = 1 0 . , SOmin

r-^ )t^

\

Loop reactor

Gas Phase reactor

to produce low molecular tail • slurry polymerisation in supercritical propane • />«6.5MPa, r = 7 0 . . 9 5 « C • / = 0.5..2h

to produce high molecular tall • gas phase polymerisation • p^2.0..

2.5 MPa, T= 70-90 ^C

• f=1 ..3h

• Lifetime of polymer

Figure 2: The Borstar configuration of loop reactors and gas-phase reactors offers an efficient way of polymerising olefins both for specialty as well as commodity polyolefins.

operation, in which condensed gas is introduced into the reactor to aid in the heat removal process, thereby boosting the production rate considerably. The Bostar process of Borealis, illustrated schematically in Figure 2, is a combination of these two types of reactors. A puzzling observation during polyethylene production has been the increase in the ethene polymerization rate when co-monomers are added to GPRs. This is often referred to as the "co-monomer effect" [ 1 ^ ] . It occurs in single GPRs as well as in GPRs preceded by a loop reactor. A number of different explanations have been proposed, few enabling any sort of quantitative prediction of the increased polymerisation rate. In the case of hex-1-ene comonomer, Banaszak et al [4] attributed the increase in large part to the increase in ethene solubility in PE brought about by the presence of the hexene. Preliminary calculations carried out at Borealis using commercially available thermodynamic modelling software (Multiflash® [5],) suggested that such an increase in absorption could be accounted for solely on thermodynamic grounds. (Full details of this preliminary calculation are provided in an Appendix at the end of this chapter.) Given the economic issues in polyolefin

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production described earlier, this observation in particular provided strong motivation for a more thorough thermodynamic investigation of the GPR process. This case study is the result of that investigation. 2. The Design Problem 2.1. Statement of the design problem The task at hand is to design a reactor mixture that will offer increased yield of polymer, ultimately for use in the BORSTAR process. Li particular, using knowledge of the thermodynamics of the system, we seek to understand the comonomer effect and to use this understanding to design a bench-reactor experiment that will increase the yield of polymer in GPRs, without necessarily increasing the amount of co-monomer present. 2.2. Techniques of solution In the past, polymer producers have tackled this type of problem using intuition arising from many years of hands-on experience with the reactors, combined with a knowledge of the necessary chemical ingredients for the polymerisation reaction, to design bench- or pilot-reactor experiments; improvements in reactor mixtures or conditions were based on the results of these experiments. Clearly, a more systematic approach is desirable and this has led to academic interest in modelling various aspects of the reactor processes. Among these aspects is the thermodynamic modelling of the phase equilibria of the polymer + gas mixtures inside the reactors. Polymer-gas systems exhibit rich phase behaviour, with regions of vapour-liquid equilibrium (VLE) and liquid-liquid equilibrium (LLE) characterised by upper and/or lower critical solution temperatures. The phase behaviour of multi-component mixtures of small-molecule gases in PE is still poorly understood. This general class of problem is often treated as an adsorption problem; it is common to study co-adsorption with models such as that of Brunauer-Emmett-Teller (BET) [6] in which adsorption is assumed to involve a non-additive physical process (there are no relationships describing how two or more gases adsorb together). The term adsorption can however be misleading, because the gas molecules usually diffuse through the amorphous polymer, so that the terms absorption or solubility are more appropriate. The techniques for thermodynamic modelling of such polymer-gas systems are reviewed in references 7 and 8. A current state-of-the-art technique

Design ofpolyolefin reactor mixtures

305

is the statistical associating-fluid theory (SAFT), which is based on a continuum approach for associating and chain molecules developed by Wertheim [9]. The more recent versions of the SAFT EOS include the variable-range SAFTVR [10,11], and the perturbed-chain PC-SAFT [12,13] descriptions. Since the initial application of SAFT to polymer systems by Huang and Radosz [14] the success of SAFT in the treatment of polymer phase equilibria has become firmly established; in particular, SAFT-VR [15-18] and PC-SAFT [18-24] have been successfully employed in the treatment of such systems. 2.3. Constraints on the design problem There are both practical and thermodynamic constraints on the problem. The first practical constraint is the high demand for bench time in an industrial environment. In this context, modelling can help to reduce the number of experiments needed in two ways. First, it increases our understanding of the key variables and how they affect the process. Second, it allows combined experiments to be designed, in which the influence of several variables is explored in a manner which provides maximum information. Modelling can thus be used to keep the number of experiments to a minimum. Heavy investment on plant infrastructure over a period of years means that it would be financially disadvantageous to make significant changes to the polymerisation-reaction conditions of temperature and pressure from those used in existing plant. Consequently, experiments must be performed at a temperature in the region of 80°C, and pressure in the region of 2 MPa. Furthermore, the process itself becomes a constraint on the problem in the following way. Under these conditions, the reactants (olefins) are present in gaseous form while polymer produced will be liquid. In other words, the system is designed to operate with a two-phase vapour-liquid equilibrium (VLE), therefore ultimately we must seek a system in which gases and PE exist in such a two-phase VLE if it is to be of value for use in the BORSTAR process. (Although the reactor may be operated in condensed mode, this does not mean that the prevailing phase equilibrium is liquid-liquid. Rather, it means that gaseous reactants are introduced to the reactor in condensed form so that the vaporisation of these gases can be used to assist in using up the excess heat from the polymerisation reaction.) The VLE constraint described in the preceding paragraph not only applies to the final system chosen, but also to any bench experiment that is carried out. This is a consequence of the shortage of bench time in the industrial context. In the event that altered conditions in an experiment give rise to gas condensation in the reactor and hence liquid-liquid equilibrium (LLE), not only does the experiment fail, but the bench reactor itself becomes "gummed up" and is

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therefore out of commission for further experiments until it has been cleaned. Therefore it is a very important constraint in designing an experiment that gas condensation be avoided at all times. 3. Strategy employed in this case study The crucial issue in the GPR process is the constraint that the reactants (olefins) are present in gaseous form while the catalyst, the site at which the reaction takes place, resides in the liquid (polymer) phase. Thus the olefins must be absorbed into the polymer in order for the reaction to proceed; in principle, the greater the absorption of olefin, the greater the yield of polymer. Hence, we seek to increase the yield via such an increase in olefin absorption. The method involves a combination of thermodynamic modelling and experiment, in multiple stages. Thermodynamic modelling is used in the first three stages to examine various aspects of polymer + gas phase equilibria, to build up an understanding of the factors influencing the absorption of the individual gases; the fourth stage consists of an experimental olefin polymerisation. The tool employed in the modelling is the SAFT-VR equation of state; the purecomponent phase equilibria of all the gases present in the GPR have been treated previously using this approach. [10,17,25] The stages in the procedure are summarised as follows: 1: Model the binary single-gas + PE phase equilibria to provide a reference for Stage 2. STAGE

STAGE 2: Model ternary (two gases + PE) systems to study the effect of the presence of one gas on the absorption of another ("co-absorption"). STAGE 3: Armed with insight from the study of ternaries, when modelling full GPR (multicomponent) systems, mixtures can be tailored to manipulate the predicted absorptions of the important gases (olefins). In this third stage, candidate bench-reactor-experiment mixtures and conditions are identified.

4: Carry out bench-reactor experiments based on the findings from Stage 3. In these bench experiments measure any actual increase in yield of polymer obtained using tailored mixtures. STAGE

By comparing experimental results from Stage 4 with predicted absorptions from Stage 3, further modelling may be carried out to design a refined bench experiment. In principle, given available bench time. Stages 3 and 4 can be iterated to obtain the reactor mixture providing optimum yield of polymer.

Design ofpolyolefin reactor mixtures

307

4. Thermodynamic modelling: applying the SAFT EOS to polymer systems 4.1. Theory 4.1.1. Assumptions We first note briefly the assumptions underlying the modelling described in this chapter. Under reactor conditions relevant to this study, PE would not be fully amorphous since typical reactor temperatures lie beneath the glass-transition temperature. The first important assumption is that we may neglect the presence of polymer crystallite. Partial crystallinity of the PE may affect the absorption of gas in various ways. Crystalline zones may link the chain molecules together to form a polymer network, [26,27] which could inhibit the swelling of the PE associated with absorption of gas. In this case the solubility of gas would be lower than in the corresponding fully amorphous PE sample in which the molecules are free to expand. Alternatively, the light molecules could depress the "freezing point" of the crystallites for purely coUigative reasons and this would enhance the solubility of the gas in the sample. In this work we neglect the possibility of swelling or cyroscopic effects as these are secondary factors in describing how crystallinity affects gas absorption. Here, as is usual [28-32], we assume that gases are not absorbed into the crystallite, which is considered to behave as a barrier to the diffusion of gas molecules in the sample. This assumption may be too crude - for example, if the gas molecules are small (such as H2) they may penetrate into the crystalline lattice. On the other hand, in very crystalline polymers, amorphous regions may be occluded inside crystallites, rendering them inaccessible to gas absorption. However, under the conditions considered in this study the assumptions of zero absorption in the crystalline region of the polymer and full gas accessibility in the amorphous region are felt to be reasonable The proportion of the PE that is crystalline is given by the degree of crystallinity, Wcrys (which is determined from experimental measurements), and so under these assumptions the effect of crystallinity on gas absorption simply involves a linear scaling (by (1 -Wciys)) and thus affects the thermodynamics in a trivial way (We note in passing that a simple method was proposed in Reference 17 for the calculation of the crystallinity of PE as a function of temperature, again taking no account of the cryoscopic effect of the presence of gas molecule in the PE sample, which would decrease the mehing point and thereby the crystallinity. [33-35]) The second important assumption is that we may neglect kinetic effects associated with the diffusion of gas through the polymer matrix. In this study we model the equilibrium thermodynamics of the gas + polymer system, and thus implicitly assume that the reactor can come to thermodynamic equilibrium;

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A.J. Haslam et al.

it is possible that in practice the diffusion of gas through the polymer will be too slow to allow this equilibrium to be reached. The final assumption that we note here is made as a practical aid in the calculation of polymer + gas VLE. Due to the extremely small mole fractions of polymer present in the gas phase, numerical difficulties may be encountered in solving the phase equilibrium. Consequently, the calculation is facilitated by assuming that no polymer is present in the gas phase. We have tested this assumption for the case of the lowest-molecular-weight polymer considered (for which the assumption is the least applicable) and found the results to be indistinguishable from those where co-existence is solved allowing polymer to be present in the gas phase. (This will not be true for the liquid-liquid region of the phase diagram; no such assumption is made in this case.) 4.1.2. SAFT-VR The reader can find full details concerning the SAFT-VR EOS in references 10 and 11; here we provide only the details necessary in the context of this chapter. The theory is based on a continuum approach for chain molecules developed by Wertheim, [9] in which afirst-orderthermodynamic perturbation theory is used to determine the thermodynamic properties of associating fluids. [36-39] In the limit of infinitely strong association a simple expression is obtained for the EOS of a fully flexible chain fluid, [40,41] which is naturally derived in terms of the free energy. Correspondingly, the SAFT EOS also is generally written down in terms of the free energy, as a function of the size of the chain and the parameters describing the intermolecular potential model; thermodynamic properties are obtained as the appropriate derivatives. For example the negative of the volume derivative yields the EOS expressed in the traditional form, in terms of the pressure of the fluid. Pure Substances In the SAFT-VR approach the molecules are modelled asflexiblechains formed from m spherical segments. Each segment in a chain has the same diameter a, but segments belonging to different species can have different diameters. The dispersive interactions between the segments can be modelled using any standard attractive pair potential of depth € and variable range A; in this work the square-well potential is used. The four parameters, e, Z, a and m thus characterise a substance and once these are known, the thermodynamic properties of the substance may be calculated. For short alkanes, the parameter m, which is related to the aspect ratio of the chain molecule, is given by [42] /w = l + ( C - l ) / 3

(1)

Design ofpolyolefin reactor mixtures

309

where C is the number of carbons in the chain. Thus, for example, /w = 2 for butane. The remaining three parameters are obtained by fitting to experimental vapour-pressure data and saturated liquid densities. PE is treated as a long alkane. For most PEs the necessary experimental data are not available to fit the SAFT-VR parameters. However, it has been demonstrated [16,17] that linear relationships can be used to relate the parameters to the chain length. Equation (1) can be recast in terms of the polymer molecular weight {MW in g mol'*), yielding m = 0.02376 MW+ 0.6188

(2)

and the remaining parameters are obtained from the correlations [17] mA = 0.04024 MW + 0.657

(3)

ma' =\,532\2MW

+ 30.753

(4)

— = 5A65ilMW +194.263

(5)

ntP

where ks is Boltzmann's constant. In the limit of high MW (in practice, MW> 10000 g mol"*) these expressions approach limiting forms: m = 0m376MW; C7= 4.01 A; e/ kg = 230.04 K and >l= 1.694. [17] For non-alkanes, pure-component parameters are generally obtained by ensuring the optimal description of experimental vapour-pressures and saturated-liquid densities (although work, described elsewhere in this book, is now underway to obtain parameters using quantum mechanics). Mixtures The usual combining rules for mixtures are used to evaluate the unlike size Otj and energy ^y interaction parameters: cx, = ^ ey=(}-ky)^.

;

(17) (18)

Here ky is a binary parameter that can be adjusted to reproduce the experimental data of binary mixtures. The case kij = 0 corresponds to the Berthelot combining rule. For those interactions that require a correction {i.e., ky ^ 0) only one value is used for the entire range of temperature and pressure. The need for a non-zero value of kij might be expected when the mixture comprises two molecules whose bonding is chemically of different nature (e.g., (5p^-hybridised) alkanes

310

A.J. Haslam et al.

TABLE 1. SAFT-VR square-well pure-component parameters substance propane n-butane n-pentane ethene propene but-1-ene nitrogen hydrogen

MW[gmor^] 44.10 58.12 72.15 28.05 42.08 56.11 28.01 2.0

m 1.6667 2 2.3333 1.3333 1.6667 2 1.3 1.0

^[A]

elks[K\

3.8899 3.9332 3.9430 3.6627 3.7839 3.7706 3.1940 3.1053

260.91 259.56 264.37 222.17 259.80 228.49 84.53 37.018

k 1.4537 1.4922 1.5060 1.4432 1.4465 1.5564 1.5340 1.8000

with (5p^-hybridised) alkenes). A combining rule is also required for the range parameter. A simple arithmetic combining rule is used for the range Xtj of the cross interaction square-well potential:

"

jj

In Table 1 we provide the SAFT-VR parameters used for the pure-component gases considered in this study. 4.2. Results of SAFT-VR modelling

4.2.1. Binary (gas + PE) systems Whereas experimental data are very scarce for ternary (two gases + PE) and higher mixtures, experimental binary data are available for most of the gases relevant to the GPR process. By first comparing experimental with predicted absorptions for these representative binary systems, we are able not only to confirm that the SAFT-VR approach satisfactorily captures the VLE of PE + small-molecule gas systems, but also (where necessary) to obtain the binary interaction parameters ky that will be used in later calculations for multicomponent mixtures. In Figure 3 we show the VLE of six binary systems; here, and frequently thereafter, we abbreviate the gases as follows: ethene (C2=), propane (C3), nitrogen (N2), propene (€3=), w-butane {nC^X but-1-ene (wC4=), and «-pentane (wCs). In general, SAFT-VR captures the VLE of these systems well. In Figure 3((3) we present two isotherms for the binary system of «-pentane + LDPE (MW = 76000 g mol'*), at T= 150.5 °C, and 201 °C. The theory captures the lower-temperature isotherm well; the experimental points [43] lie mostly between the two predicted absorption curves. No binary

311

Design ofpolyolefin reactor mixtures

20-1

lUU-

r r=150.5°C

^.^ S 80^

"

's.

/ 1

/

^

/

1

f

„•

go-

/ / /•

O) >» 40-

£t 3

o ^0-

o

/ 1 /r=2oix 1 y^ 1 0 y^ 1

u ^

C»10-

^ -

X

VK^K^' = UM

1 (g) nCg in (76 kg mol-^) LDPE || 2

1

3

4

T= 170*^

^

/m

C/)

(I

m :

y ^ 0

1

2

1 <^> ^a ^^ ^^^^ ^ ^^^^^ ^^ 1 3

4

1

6

7

p/MPa

8

10

12

14

16

18 20

p /MPa Figure 3: Experimental gas-absorption isotherms in amorphous PE (symbols) are modelled using SAFT-VR (continuous curves). All non-zero binary-interaction parameters (ky) are indicated on the graphs; these values are used throughout this work.

binary interaction parameter was needed in the calculation (i.e., ^/, = 0), as would be expected in the case of interactions between alkanes. The curves tend towards an asymptote, which corresponds to the saturation (vapour) pressure

312

A.J. Haslam et al.

(Psat) of pure «-pentane. As this figure confirms, SAFT-VR reproduces this value accurately, and can successfully describe the absorption in PE of a gas with a low/?saf The second experimental isotherm at 7 = 201 °C is more poorly represented by the SAFT prediction, except at low pressure. The disagreement of the prediction with the experimental data at -- 3 to 4 MPa is due to the near proximity of the temperature of the isotherm to the critical temperature of pure «-pentane (197 °C); it arises because analytical equations of state cannot, by themselves, accurately capture the behaviour of real systems in the proximity of the critical point. In the case of «-pentane, SAFT-VR over-estimates the saturation pressure of the pure fluid by ~ 0.5 MPa; [25] this corresponds closely to the separation of the asymptotes of the experimental points and SAFT curve in Figure 3(a). This is a useful illustration of the effect of near criticality of the absorbing gas. In this particular case, critical-region errors will not concern us further since in our later mixture calculations, with temperature constrained to be ~ 80°C (see Section 2.3), pentane will not be at near-critical conditions. Before describing Figure 3(i) through (e) we note that the experimental absorption data represented in these graphs, corresponding to binary systems of PE + propane, ethene, propene and but-1-ene (respectively), were all obtained using the same polymer and, mostly, under the same conditions. [44,45] Such data are ideal for our study since the binary-interaction parameters that will be used in predictions for higher-component mixtures are obtained by fitting to experimental binary absorption data. Clearly this reduces risk of inconsistency, providing greater confidence in the (relative) values of the binary-interaction parameters thus obtained. We therefore choose to use these data even though the PE in question is of rather low molecular weight (1940 g mol'^). In Figure 3(6) we present the predicted absorption of propane in linear low-density PE (LLDPE) (MW= 1940 g mol"^). The curve determined with SAFT beautifully represents the experimental data [44]. A non-zero ky is used here (A:/, = 0.02). The need for a non-zero ktj is surprising, given that the interaction again represented an alkane-"alkane" interaction. Although the value is small, this nevertheless appears slightly inconsistent with the zero ky used for pentane + PE (Figure 3(a)). The need for a non-zero k^ here is another manifestation of being within the influence of the critical region of the lightgas, in which SAFT-VR fails to capture accurately the behaviour of the fluid. In this region, one has two options: either re-adjust the pure-component parameters to fit the critical region at the detriment of the sub-critical states, or adjust the binary-interaction {ktj) parameters leaving the pure-component parameters unaltered. The latter option was chosen here in order to be consistent with the way we treat the other components, however it should be noted that the ky value obtained is thus specific to this study. (Such a procedure is not necessary for «pentane (Figure 3(a)) because the temperature of more interest is that of the

Design ofpolyolefin reactor mixtures

313

lower-temperature isotherm, which is sufficiently below the critical point predicted with the theory.) In Figure 3(c) we show the SAFT-VR description of the experimental absorptions [44] of ethene in (MW = 1940 g mol"^) LLDPE at temperatures of 140 °C, 170 °C and 200 °C. Since these temperatures are all above the critical temperature of ethene, saturation is not an issue and the absorption plots are almost linear. Not surprisingly, a binary-interaction parameter is needed: kij = 0.057. We note particularly the adequacy of the description at the lower values of pressure; as will be seen, we will later be concerned mostly with pressures under 5 MPa. In Figure 3(d) we show predictions of the absorption of propene in LLDPE (MW= 1940 g mol*), again at temperatures of 140°C, 170°C, and 200 °C. A binary-interaction parameter (A:/, = 0.028) is required to fit the experimental data [44], Again we note the excellent description at the lower values of pressure. In Figure 3(e) we show predictions of the absorption of but1-ene in LLDPE at temperatures of 155 °C, 165 °C, 195 °C and 220 °C. A binary-interaction parameter (A:/, = 0.02) is required to fit the experimental data [45]. Compared with €2= + LLDPE and €3= + LLDPE, we find the trend of kij decreasing with increasing molecular size for small alkenes; this is physically sensible since the effect of the double bond is expected to decrease with increasing chain length. When one is at temperatures above the 3-phase line and above the upper critical end point (UCEP) for the mixture, there is a continuous fluid-fluid equilibrium going from the pure-polymer VLE to the critical point in the mixture. [8] At the lowest pressures the behaviour corresponds to that of a vapour phase consisting of gas molecules with almost no polymer present (in fact we imposed the composition of the polymer to be zero in these states), and a liquid phase consisting of gas absorbed in polymer. As the pressure is increased and more of the absorbed molecules accumulate in the polymer liquid phase, the densities of the two co-existing phases (one fluid rich in the polymer and the other rich in the light molecule ("gas")) become comparable and liquid like. As a consequence, the behaviour goes continuously from a VL-type equilibrium to a LL-type equilibrium (an example of the continuity of the gas and liquid states). When the system is at a temperature below that of the predicted upper critical end point (UCEP) two separate regions are seen: a VLE region and an LLE region separated by a three-phase line. (Note that in PE + light-molecule binary systems the UCEP is virtually indistinguishable from the critical point of the small molecule.) At temperatures below the LCEP no liquid-liquid separation is seen, and there is just one gas-liquid region. Across the range of temperatures considered in Figure 3(e) for but-1-ene + LLDPE, the nature of these transitions is nicely illustrated. The temperatures

314

A.J. Haslam et al.

155 °C and 165 °C are predicted to be below the UCEP (and above the LCEP): at each temperature we see a curve with a marked discontinuity in the slope, which corresponds to the VLLE triple-point pressure. For r=195°C and 220 °C the system is predicted to be above the UCEP and a continuous curve is seen, albeit with a change in the curvature. It is gratifying tofinda similar small change of curvature in the experimental data [45] at the highest temperature. Of the systems depicted in Figure 3 we show the VL-LL behaviour only for «C4= + PE, as experimental LL data points are available for this system. The LLE has already been discussed for the nCs + PE system depicted in Figure 3(a) [17]; at r = 201 °C the SAFT-VR calculation indicates that the LL branch meets the VL branch at /? - 3.95 MPa and pentane solubility -- 131g / lOOg PE. There will be no discernible three-phase region for the other mixtures, since the temperature in each case is well in excess of the critical temperature of the gas. In Figure 3(/) we show SAFT-VR predictions of the absorption of nitrogen in high-density PE (HOPE) (MW = 111000 g mol'^). The behaviour for the absorption isotherm of N2 in PE, at the temperature of interest, is qualitatively similar to that seen for ethene: an almost linear dependence of absorption with pressure (Henry's Law [17]). A large ky is needed to obtain a good representation of the experimental data [46] of the mixture, which is to be expected for this gas (due to its non-negligible quadrupole). The k^ value is comparable and consistent with the range of -- 10-18% reported by GarciaSanchez et al [47] for PC-SAFT studies of N2 + long-hydrocarbon mixtures; these authors concluded that the binary interaction parameter for N2 + PE is an increasing function of the polymer MW. In the foregoing section we have confirmed the suitability of SAFT-VR to study gas + PE systems, and (where necessary) obtained the kij values required to describe all of the gas + PE interactions that will be relevant in the remaining discussions. To understand and predict the simultaneous absorption of more than one gas in a given polymer in a gas-phase reactor, we will consider multicomponent systems consisting of some or all of N2, €2=, C3, nC^, nC^=, nCs with a PE that is characteristic of a typical industrial process. Since there are no suitable experimental data available we have to choose a prototype reference polymer. For PE above a molecular weight of -^ 10000 gmol^ the VLE of gas + PE is relatively insensitive to the MW[17]; for our reference PE we choose a slightly larger MW of 12000 gmol* as a representative value for the gas-phase process. The first task is to compare the individual binary VLE of these gases in the reference PE at a suitable temperature within the range of temperatures representative of industrial gas-phase polymerisation. We choose to work at T=SO °C; unless otherwise indicated, all subsequent calculations will be carried out at this temperature.

Design ofpolyolefin

315

reactor mixtures

4.2.2. Reference Binary Systems JJ300-

n C5

Absorption In (MW-12000gmol-^)PE || atrB80°C II

Q.

0)250;;;2ooO) ^150-

Zioo^ 500- Mril^M^BMipSiMH^HMqMMHqMMM|MMi^^

Figure 4: SAFT-VR-calculated gas-absorption isotherms in amorphous PE for binary mixtures of propane, but-1-ene, «-butane and «-pentane + the reference PE (MW = 12000 g mol"^)

In Figure 4 we present, on the same axis, the binary VLE predictions for C3, «C4=, WC4 and nCs in our reference PE (MW = 12000 g mol ) at the reference temperature of 80°C. Neither the absorption curve for C2= nor that for N2 are shown on this figure, these gases are supercritical at r = 8 0 ° C , and the (approximately linear) curves are almost indistinguishable from the horizontal axis at this scale. We note that at 80 °C the polymer is expected to be semicrystalline. As discussed in Section 4.1.1, the degree of crystallinity affects gas absorption since a semi-crystalline sample has less amorphous polymer in which to absorb gas, and also since there may be effects related to the inhibition by the crystal matrix of the polymer swelling that occurs whengas is absorbed. However, for the reasons outlined in the Introduction we neglect the crystallinity here and throughout this study - nevertheless it should be kept in mind that for accurate predictions of experimental absorptions crystallinity would need to be considered. (We note that using the method of Paricaud et al [17], the crystallinity of the reference PE is estimated to be Wciys = 0.253 at r=80°C.) As will become clear from later discussion, it is important to know approximately at what pressure the LLE curve intersects the absorption curve {i.e. the position of the three-phase line); specifically, for the less volatile gases, we need to be sure that this intersection does not occur on the shallow part of the curve (at low values of gas absorption), as this would mean the appearance of additional liquid phases during the process. Unfortunately, except at high temperature, the calculation of the LLE in polymeric systems is demanding (for numerical reasons). One method employed is to first calculate the LLE at high

316

A.J. Haslam et al.

temperature (where it is numerically less demanding), and then to use this information to assist in calculating the LLE at lower temperatures. From the calculations illustrated in Figure 3(e), it can be seen that with decreasing temperature the intersection of the LLE with the absorption curve occurs at decreasing pressure and increasing values of absorption (as the region of liquidliquid coexistence decreases in size). For the systems illustrated in Figure 4, the SAFT-VR predictions indicate that in the case of propane, the intersection occurs at a pressure of ~ 3.4 MPa, and absorption - 90 g / lOOg PE. In the case of but-1-ene, we have not calculated the position of the intersection at r = 80°C: at the higher temperature of 100°C the intersection predicted by SAFT-VR already occurs at an absorption of 140 g / lOOg PE; the intersection at r = 80°C will be at a higher absorption value (- 200 g / lOOg PE) and thus certainly on the steep part of the absorption curve. Similarly, calculations at higher temperatures show that the intersection in the case of «-butane will also be high on the steep part of the absorption curve, while in the case of pentane it appears that the region of LLE may have disappeared completely at 80°C (at 145°C, the intersection occurs at an absorption - 600 g / lOOg PE). Thus we can indeed be sure that the interception does not occur on the shallow part of the absorption curve for any of these mixtures at T-' 80 °C. 4.2.3. The Ternary System^ PE+nC4+ N2 : Enhancement and Inhibition of Absorption The terms co-absorption, co-solvency and anti-solvency are commonly used to describe the absorption of two or more gases in polymer. As we have already seen (e.g.. Figure 3(e)) the phase behaviour can be fairly complex in gas + PE systems with regions of both VLE and LLE. To avoid confusion, we prefer to use the expressions "enhanced absorption" and "inhibited absorption" to refer to the relative increase or decrease in the absorption of a gas-phase component in the polymer-rich liquid phase. It is clear from the preceding sections that, to a degree of approximation, the lighter the gas, the less it absorbs in the PE. However, we know that two light gases will mix with each other readily. We have also seen that the absorption of a gas in PE rises sharply as the saturation pressure of the purecomponent gas is approached, i.e., as the gas becomes "liquid like". If we consider these two points together in relation to a mixture of two gases and PE, we might ask whether a less-volatile gas might help a more-volatile gas to absorb in the polymer, as the partial pressure of the less-volatile gas approaches its saturation pressure. Correspondingly, will a more-volatile gas inhibit absorption of a less-volatile gas? But-1-ene is one of the least volatile of the gases typically present in reactors as the co-monomer for the PE, while nitrogen

Design ofpolyolefin reactor mixtures

317

(Jf150 Q. 0)120

\

|(«)100%iiC^

1 nc,= / -

XI 30^

/'/MPa Figure 5: Predicted gas absorptions in PE of the ternary mixture of but-1-ene + nitrogen + PE (MW = 12000 g mol"^) at r = 80 °C, for a range of vapour compositions; note that (c) and (d) differ only in the scale on the vertical axis. The dotted line represents the gas saturation pressure.

which is also usually present, is highly volatile, and at reactor conditions is supercritical (it is non-condensable under these conditions and saturation never occurs), so a ternary system of nC4= + N2 + PE is ideal to search for any such enhancement / inhibition-of-absorption effect. In Figure 5 we present SAFT calculations of absorption of ^€4= and N2 in our reference PE (MW = 12000 g mol'*), for mixtures varying from the binary nC4= + PE in (a) through ternaries with increasing (vapour) mole fraction of N2 (b-e) to the binary N2 + PE in (/). Note the change of vertical scale after (c); (c)

318

A.J. Haslam et al.

and {d) correspond to the same mixture, but the change of vertical scale is made to highlight first the absorption of «C4= and then that of N2. From Figure 5 one can see that as «C4= in the vapour is replaced by N2, the calculated absorption of «C4= decreases (this is not surprising - there is less of it to be absorbed). However, it is also clear that as N2 in the vapour is replaced by nC^=, the absorption of N2 is increased, even though there is less N2 to be absorbed. This counter-intuitive result suggests that there may indeed be some enhancement / inhibition-of-absorption effect. Qualitatively similar trends were observed in other ternary systems containing the reference PE together with any two of the other reactor gases. We can eliminate the effect of the reduction or increase in the quantity of one or other gas and isolate the effect of the second gas on absorption of the first by defining the adjusted solubility, SADJ such that for a gas, / S'ADJ,/

= (absorption of gas /) / (vapour mol fraction of gas /)

(20)

This measure approximately accounts for changing amounts of each gas present (it would be exact for an ideal mixture, in which the partial pressure of each gas is well defined). In Figure 6 we present adjusted solubilities for selected ternary mixtures using SAFT-VR. In Figure 6(a) we summarise the results presented in Figure 5 in terms of the adjusted solubility, representing the reference PE + nC/[= + N2 ternary system. The adjusted solubility of N2 in PE is seen to rise with increasing mole fraction of ^€4= (rising in each case to a maximum at a pressure corresponding to the saturation pressure of the gas mixture). At the same time, the adjusted solubility of «C4= decreases with increasing mole fraction of N2 in the mixture (note the different vertical scales in the two graphs). This indicates that, thermodynamically at least, «C4= strongly enhances the absorption of N2, and that N2 strongly inhibits the absorption of «C4=. In Figure 6{b) we show the adjusted solubilities for the ternary system PE + «C4= + C3. Similar trends to those in {a) are seen, showing that «C4= enhances the absorption of C3, while C3 inhibits the absorption of ^€4=. The calculations for the ternary PE + C2= + C3 are represented in Figure 6(c), from which it can be seen that C3 enhances the absorption of C2= and that C2= inhibits the absorption of C3. In summary we have shown in Figure 6 that, thermodynamically, an enhancement / inhibition of absorption is indeed expected. In conjunction with Figure 4, this figure suggests a correlation between the saturation pressure, /?sat» of the gas-mixture and this absorption enhancement / inhibition effect: a lessvolatile (low-/?sat) gas enhances the absorption of a more-volatile (high-;?sat) gas, while the latter inhibits the absorption of the former.

319

Design ofpolyolefin reactor mixtures 4-1 LU

a. O)

A

Increasing m frac nC^^ ^^^

o o 2-1

j

1^

1

i \

^

1

,1 /

\IA

^^^"^n^^^^^ T ^ ^ ^ ^ ^ T " ^ ^ ^ ' ^

u-1

1

2

150-

(a)

1 Adjusted sol*y Ng 1;

100%nC4=

lU Q. 120-1 oj

75% nC^^ § 9050% iiC4= II 32.5% nC4= 3 ®H 20% nC^^ Sg^ 0% nC4=

^

0-1

3

jD/MPa S 120-

p/MPa Figure 6: Adjusted solubility, 5ADJ, defined as absorption I vapour mole fraction, of gas in the reference PE (12000 g mol"^), calculated for three ternary mixtures at r = 80°C: {a) nitrogen + but-1-ene + PE (see also Figure 5); {b) propane + but-1-ene + PE; (c) ethene + propane + PE. •S'ADJ highlights the enhancement of absorption of a more-volatile gas resulting from the presence of a less-volatile gas, and the corresponding inhibition of absorption of the former resuking from the presence of the latter.

We now turn our attention to the mixtures of more than three components to investigate how this effect might be utilised to engineer an increased alkene absorption in PE for typical reactor conditions.

320

A.J. Haslam et al.

4.2.4. Multicomponent Systems and Enhancement / Inhibition of Absorption: identification of candidate experimental mixtures This aspect of the thermodynamic modelling constitutes Stage 3 of the strategy laid out in Section 3 of this chapter; the task here is to identify candidate experimental reactor mixtures for which increased yield of polymer may be expected. From the previous section, it is clear that a highly volatile gas, such as nitrogen, is expected to inhibit the absorption of less-volatile alkenes in PE in a ternary mixture. Should this be true also in a multicomponent mixture then the implications are quite profound for the GPR process, where nitrogen is typically present in large quantities as a so-called "inert", or "diluent" gas; it is used initially to degas and then to pressurise the reactor. Indeed, it is typically present in larger quantities than any other gas in the mixture; for example, a typical copolymerisation mixture might comprise 20 mole % ethene, 10% butene, 20% propane (inert) and 50% nitrogen (inert). Consequently, an obvious approach to designing a reactor mixture with greater alkene absorption in the polymer is to replace nitrogen with a less-volatile diluent gas, which should also be chemically inactive in the reaction. The most obvious candidate gases are alkanes, such as butane or pentane. The least-volatile alkane that is gaseous at the reactor conditions is pentane, which is therefore our chosen gas (hexane could remain as liquid in the reactor and hence is not suitable). Ideally, to test our prediction of enhanced absorption, one would carry out two absorption experiments with a "dead" polymer (end product): a reference experiment with, for example, the typical gas mixture described above, followed by an experiment in which nitrogen is partially or completely replaced by pentane. However, as indicated in Section 2.3, due to practical limitations on industrial bench time, it is not possible to perform an experiment solely to test the predictions of the modelling. Instead, a suitable bench experiment was chosen with which this experiment could be combined. A co-polymerisationreaction experiment in which nitrogen constituted the only diluent gas was available as a reference; in such an experiment the yield of polymer is measured, but no direct measurement of gaseous absorption is obtained. 4.2.4.1. Calculations for experimental reactor mixture Unfortunately, due to the nature of the experimental set-up, the global composition is in the reactor is known (albeit approximately), but the vapourphase composition is not. Since this composition is required as input for the thermodynamic modelling, it must first be estimated. At the start of an experiment but-1-ene is introduced by weight and the other gases are introduced according to a measured overall pressure (see Section 5); the vapour-phase composition is then kept approximately constant via a carefully monitored, continuous feed of the alkenes. The volume of the reactor is known, together

Design ofpolyolefin reactor mixtures

321

with an estimated weight of polymer, of estimated density; thus an approximate value for the total volume occupied by the vapour can be calculated. The reactor pressure/? =2 MPa and temperature r = 85 °C. The gas mixture in this experiment comprised nitrogen, but-1-ene, ethene, and a trace amount of hydrogen. A VLE flash calculation, constrained by a material balance, was performed using the SAFT-VR EOS. The overall vapourphase composition was thereby estimated as 67.00 mole % nitrogen, 23.61% ethene, 9.35% but-1-ene and 0.04% hydrogen. Since almost all of the input values to this calculation carry uncertainty, there is accordingly considerable uncertainty in the composition obtained. 4,2,4.2. Enhancement/inhibition of absorption: effect of replacing nonreacting nitrogen. As has already been described, it is desirable to obtain the highest absorptions of the reacting gases (€2= and ^€4=) in the liquid, PE-rich phase, since by so doing the yield of PE in the polymerisation reaction would increase. In Figure 7 we show the effect of partial or complete replacement of N2 in the reactor mixture with «-pentane. In {a) we show the calculation for the reference experiment. Within the range of pressure investigated, the gas absorptions are all approximately linear; there is no suggestion of gas condensation. In (6), instead of 67 mole % N2, the vapour contains 57% N2 and 10% nC^, There is now clear curvature in each gas absorption plot; off the horizontal scale it is predicted that this mixture will saturate (condensation will occur) at just under 4.6 MPa. (Henceforth we shall refer to the pressure at which the second liquid phase, rich in the predominantly light components, appears as the saturation pressure of the mixture.) In order to increase the yield of polymer, the saturation pressure must be brought close to reactor pressure (2 MPa). In (c) the vapour phase contains 15% nCs and 52% N2; this mixture is predicted to saturate at just under 3 MPa. When compared with {a) the gas absorptions at 2 MPa have all increased - even that of the nitrogen. Finally, in (d\ instead of 67% N2, the vapour phase contains 20% nCs and 47% N2. With this amount of «-pentane, the predicted saturation pressure has been reduced to just in excess of2 MPa; this represents the maximum increase in absorption that can therefore be achieved without the system exhibiting LLE - one of the constraints on the problem discussed in Section 2.3. In Figure 8, we present a summary of the alkene absorptions in PE from the calculations represented by Figure 7, together with those obtained in a further calculation with 62 vapour mol % N2 and 5% wCs. In (a) we show the absorptions of €2=, while alongside in {b) we present those of nC^=, At typical reactor pressure of -- 2 MPa, these calculations indicate that alkene absorption in PE for the original mixture (with 67% N2) may be increased by up to -- 500%

322

A J. Haslam et al.

liJ «^5

l(^)10%iiC5;57%N2

/

sJ =2

.

nC2=/K

J

31H 0 0.5 1 1.5 2 2.5 3 3.5 4

/7/MPa

5o' F ' ^ T ^ i "

1111111111 | i 11| 1

C1 0.5 1 1.5 2 2.5 3 3.5 4

/>/MPa

1 (r) 15% nC^; 52% N^

s]

/

1

• nc = .^ •^ 0)3H • ^

/1 \

0 4^

;"^5

'

•IH • ' 2 '\ / 3 1 i • /'

nC2= /

Jy^

i

1

h

1

/

^ N2

«ol01y0.5: ^ ^1 ^1.5^ 2- "2.5 /i/MPa

1

3 3.5 4

0 0.5 1 1.5 2 2.5 3 3.5 4

/^/MPa

Figure 7: Compared with the reference polymerisation experiment, depicted in (a), partial replacement of nitrogen by «-pentane lowers the predicted the gas-mixture saturation pressure, resuhing in increased absorption of all gases at reactor pressure (2 MPa). In (d), the theoretical limit is reached as the saturation pressure (the asymptote of the asborption curves) lies just above reactor pressure; further replacement would result in condensation in the reactor (LLE). Vapourphase composition: 23.61 mol% ethene, 9.35% but-1-ene, nitrogen and «-pentane as indicated, 0.04% hydrogen (the absorption of hydrogen is too small to be seen at this scale). T=S5 °C.

by judicious substitution of N2. This would be a truly remarkable result in the context of the polymerisation reaction if it could be achieved in practice, however, as will be discussed later, this figure should be treated with caution. It is noticeable that the two plots of Figure 8 appear very similar, except in the scale on the vertical axis. Thus it can be seen that not only does SAFT-VR predict that substantial increases in absorption of both alkenes can be engineered by replacing N2 with «-pentane as indicated, but that this can also be achieved in such a way that the re/a//ve increases in the absorptions of the two alkenes are very similar. This may be important since the nature and morphology (structure, in terms of frequency of chain branching, density etc.) of the polymer that is synthesised in the GPR process is determined by the relative proportions of the two alkenes in the reacting mixture; only very slight alterations (if any) in the gas-mixture composition would be required in order to produce the same polymer.

323

Design ofpolyolefin reactor mixtures

fir 1 J

s il^2.5^

_ _ _ _ _ _ n—n ' ^ 15%/

20% nC^;;

1

/ ; / 10%] /,'^^5?d

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-

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- ^

•

" 1 ^ - ^

CO oJ .-r-7***^. .,....,. . ....... 1 0 0.5 1 1.5 2 2.5 /7/MPa

0

0.5

1 1.5 2 /7/MPa

Figure 8: Summary of predicted alkene absorptions in PE at r = 85 °C upon partial replacement of nitrogen with «-pentane in our reference polymerisation experiment (see also Figure 7). The dotted lines highlight reactor pressure (2 MPa) and corresponding absorption values. The vapour mole % of pentane is indicated for each curve.

From the systems that have been studied it now only remains to select a mixture to use in the bench experiment. Before doing so, it is important to set these predicted enormous increases in alkene absorption into context. There are two important issues to consider. The first of these is that while the qualitative trends are clear, the predictions cannot be considered quantitative. Gas absorption is very sensitive to the gas-mixture saturation pressure; this, in turn, is very sensitive to the vapour-phase composition, which has been estimated in rather crude fashion. The interpretation of these predictions is thus that by direct replacement of nitrogen with w-pentane in the vapour phase, the saturation pressure can indeed be reduced to a value close to the reactor pressure (thereby substantially increasing alkene absorption), but that there is uncertainty in the level of replacement required to do so. The other important issue is that although it is straightforward in the calculation simply to replace x% N2 in the vapour with jc% A2C5 while keeping the remainder of the mixture unaltered, it is not straightforward to do so in the reactor, where the global composition is controlled rather than the vapour-phase composition. Since a given weight of but-1-ene is initially introduced to the reactor, if more of this weight is absorbed into the polymer phase - as is predicted when pentane is included - then less remains in the vapour phase. The gas-phase composition estimated earlier is therefore no longer the same when «pentane has replaced some nitrogen. As a consequence, the predicted absorptions are also no longer valid. The vapour-phase composition must be recalculated for the selected mixture, and correspondingly new absorption calculations must be made. Again, the difference in calculated absorption will correspond to a different amount of but-1-ene remaining for the vapour phase, so an iterative set of self-consistent calculations of vapour-phase composition

324

A.J. Haslam et al.

and gas absorption is needed. With this in mind, the absorptions in Figures 7 and 8 represent predictions for an experiment in which the vapour-phase composition is controlled. This is nevertheless an important result, demonstrating that equal relative increases in absorptions of both alkenes may be obtained. We note that it would be possible to conduct such an experiment by adjusting the initial weight of butene in the experiment so that the concentration of butene in the vapour (with nitrogen and «-pentane) after partitioning matched that of the reference experiment (with just nitrogen). However, at this stage the purpose is to confirm the theoretical prediction that nitrogen substitution will result in an increase in polymerisation activity; had the initial weight of butene been increased then it would be necessary to prove that the increase was in no way due to the presence of extra butene. 4.2.4.3. Candidate reactor mixture for bench experiment, and absorption prediction. The best choice of mixture for the bench reactor experiment will be one for which there is only a negligible chance that the system will exhibit LLE, but yet will be close enough to saturation so that the increase in absorption will be large enough to affect the polymerisation activity. Unfortunately these two requirements are somewhat contradictory. At this point, the more-important consideration is the avoidance of LLE, therefore the mixture with 20 mole % nCs was rejected in favour of that with 15% nCs and 52% N2. For this mixture, a revised estimate of the vapour-phase composition (in the presence of npentane) and corresponding gas absorptions were obtained. The revised estimate of vapour-phase composition comprised 6.9% but-1-ene, 52% nitrogen, 15% pentane, 26.05% ethene and 0.05% hydrogen; the previous estimate included 9.35% but-1-ene and 23.61% ethene. In Figure 9 we present the predicted absorptions of gas in the PE for the chosen candidate experiment. In (a), for ease of comparison, we show again the original calculation (Figure 7(c)) and alongside in {b) we show the revised predictions. It is clear that the predicted saturation pressure is higher in (fe) than in {a): - 3 . 1 MPa, compared with -'2.9MPa. The reason for this is that the proportion of butene in the vapour phase has been reduced, while that of the more-volatile ethene has been increased. Accordingly, the gas absorptions in PE at reactor pressure (2 MPa) are slightly lower, with the exception of that of ethene. The alkene absorptions are summarised in Figure 10. In Figure 10(a) the predicted absorption of ethene is shown, for the reference experiment (no npentane), for the original prediction (9.35% nC^=\ see also Figure 8(a)) and for the revised prediction (6.9% «C4=). Here it is evident that at 2 MPa, there is virtually no difference in the predicted absorption of ethene of the original and revised calculations (in both cases, predicted absorption is increased by <- 100%

Design ofpolyolefin reactor mixtures

325

^6 111 i < ^ t ^ % C 4 « ; 23J1%C^|| II 1 ' * >4 T-

/'

1 h^' / 1 • nC,=/

h11' 1!

o

/' / / ' / ^ "^2=/

f / ^ — ^

^

0 0.5 1 1.5 2 2.5 3 3.5 4

0 0.5 1 1.5 2 2.5 3 3.5 4

/p/MPa

/r/MPa

Figure 9: Global composition, not vapour-phase composition is controlled in the experiments; when nitrogen is replaced by /i-pentane in the reference experiment the other gases repartition between the liquid and vapour phases, affecting gas absorptions in the PE. Absorptions are shown for 15 vapour mole % «-pentane with 52% nitrogen. In (a) (also Figure 7(c)) the repartitioning is not taken into account; in {b) the repartitioning results in less but-1-ene content in tiie vapour phase, resulting in an increase in the calculated gas-mixture saturation pressure, compared with (a), and a corresponding decrease in gas absorptions at reactor pressure (2 MPa). Vapour-phase composition: 6.9% but-1-ene, 52% nitrogen, 15% «-pentane, 26.05% ethene and 0.05% hydrogen.

compared with the reference experiment); the increase in the proportion of ethene in the vapour phase offsets the reduction in the absorption-enhancement effect due to the smaller proportion of butene. However, in (Z?) it is apparent that the revised calculation predicts a smaller increase in absorption of butene, compared to the reference experiment, than the original calculation (albeit still an increase): the revised prediction is of an increase of about 50% atp = 2 MPa, compared with an increase of about 100% in the original calculation. In summary, a reactor experiment is suggested in which the nitrogen content of the vapour is reduced from 67 mole % to 52% by substitution with «-pentane. With this level of nitrogen substitution, based on the assumptions laid out in Section 4.1.1, we would expect an overall increase in olefin (ethene and butene) absorption of between 50% and 100%. As mentioned earlier, gas absorption is not measured directly in the experiment. Instead, the polymerisation activity is measured - this is equivalent to the yield. Assuming that there are no masstransfer limitations, this means that an increase in activity of 50% to 100% could be achieved. However, we again emphasise that due to the approximate manner in which the vapour-phase composition was estimated, the predicted increase in activity should be considered semi-quantitative rather than fully quantitative.

326

A.J. Haslam et al.

J |<^) Solubility C^ 1

with 9.35%

i with 6.9%

1

1.5

2

plUPa

2.5

\

/ \ / / / ^ / / /^

J

:>>v,;^^^^ ^

A

.^.-'^T-^

0

0.5

1

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1

pentaneA 2

2.5

3

/^/MPa

Figure 10: Predicted alkene absorptions in PE for the reference experiment with no pentane (dashed curve), and the substitution experiment with 15 vapour mole % «-pentane and 52% nitrogen. The continuous curve (9.35% but-1-ene) represents the prediction without taking into account repartitioning of the other gases between the vapour and liquid phases; the dot-dashed curve (6.9% but-1-ene) represents the prediction when the repartitioning is included. At reactor pressure (dotted line), the predicted ethene absorption is the same in each case, however the predicted increase in but-1-ene absorption is lower when repartitioning is taken into account.

5. Experimental confirmation To test the prediction of the theory that incorporating heavier alkanes in the gas mixture actually increases the polymerisation activity in gas-phase polymerisation of ethene, experiments in bench-scale reactors were performed, based on the thermodynamic modelling described in Section 4. The experiments were conducted in a 2-litre and a 5.3-litre reactor. The first stage of the twostage polymerisation experiments was to carry out a slurry polymerisation similar to the first stage of the Borstar process. This coated the catalyst and produced about 300 g of polymer. After this the reactor was evacuated, nitrogen purged through the polymer powder to get rid of the solvent from the slurry polymerisation, and the temperature set to the reaction value of r = 85°C. The appropriate weight of but-1-ene was added (10.1 g in the case of the 2-litre reactor and proportionally 26.8 g for the 5.3-litre reactor). In the substitution experiment, the reactor was then filled with nitrogen to 10.4 bar (1.04 MPa) and 3 bar (0.3 MPa) of «-pentane; in the reference experiment it was filled to 13.4 bar (1.34 MPa) of nitrogen. Ethene was then used to get the polymerisation pressure of 20 bar (2 MPa). The results from the polymerisation experiments are given in Table 2. The alkenes were supplied by a continuous feed throughout the reaction, carefully monitored to ensure that their concentrations remained approximately constant. Since the reaction is continuous, the yield of polymer is measured in terms of the polymerisation activity, defined as the mass of polymer produced per mass of catalyst per hour.

327

Design ofpolyolefin reactor mixtures Table 2: Results from polymerisation with and without nitrogen replacement by w-pentane. Experiment 2 litre reactor but-1 -ene co-monomer Inert: 13.4 bar Nitrogen 5.3 litre reactor but-1 -ene co-monomer Inert: 13.4 bar Nitrogen 5.3 litre reactor but-1-ene co-monomer Inert: 3 bar Pentane, 10.4 bar Nitrogen

Activity [gPEgCAT^hr-^]

MW [gmol-^]

Percentage change

1908

12000

Reference

1929

10000

Reference

2547

11000

32 % increase

The reference experiment was performed twice to ensure repeatability.

Comparing the polymerisation in the presence of «-pentane with the reference in which no «-pentane was present shows that the activity of the polymerisation in the presence of pentane was about 30% higher than with nitrogen only. This increase in activity is consistent with the predictions set out in the previous section, although the magnitude of the experimentally observed increase is a little lower than predicted. This could be due to the neglect of kinetic effects with respect to the diffusion of the alkenes through the polymer matrix to reach the catalyst, but could also be due to inaccuracies in the estimation of the vapour-phase composition. Such inaccuracies could result in the actual gas-mixture saturation pressure being higher than predicted, which would in turn result in a lower activity than predicted. The PEs produced in each experiment appear to be equivalent, although detailed analysis has not been carried out. For example, the molecular weight of the polymer produced in the substitution experiment is equal to the average of the molecular weights obtained in the two reference experiments. From the theoretical predictions, one might have expected a PE of different morphology, due to the difference in the relative increase of absorption of ethene and but-1ene (see Figure 10). This may in fact be an indication that the mixture is some distance from LLE, which would be consistent with the relatively small increase in activity (compared with the possible increase suggested by the theory) and would support the earlier interpretation that the actual saturation pressure was higher than that predicted. For example, consider the absorption curves in Figure 10(a) and (ft): the difference between the relative increase in predicted ethene absorption and but-1-ene absorption increases with pressure, and will be at a maximum at the saturation pressure. Conversely, at a pressure far from the saturation pressure the difference in the relative increases in absorption of the

328

A.J. Haslam et al.

two alkenes is quite small, so that the differences in the PE synthesised would also be small. In view of the foregoing discussion, it should now be safe to perform an experiment in which even more nitrogen is replaced by «-pentane. However refined modelling should first be done in the light of the experimental resuhs to choose an appropriate level of substitution. In such an experiment, we can now confidently expect an even larger increase in activity. Moreover, now that the effect has been demonstrated, if desired, the initial weight of but-1-ene added can be adjusted to maintain an equal alkene concentration in the vapour phase after partitioning. 6. Discussion and Conclusions In this case study, we have successfully designed a GPR mixture that gives a significant increase in the yield of polythene compared to the conventional mixture, with no requirement for a change in the reactor conditions. Furthermore, potential still exists for further large increases in yield. The absorptions of the gases in a multi-component mixture with amorphous PE are predicted to be at their highest at the saturation pressure of the gaseous mixture. Engineering enhanced absorption of one or more of the light gases present (such as ethene and butene) may therefore be achieved by controlling the pressure at which this saturation occurs. Reactor mixtures generally contain non-reacting diluent gases ("inerts"), such as nitrogen, to pressurise the reactor. Replacing nitrogen in the reactor mixture by, e.g., npentane or a similar alkane lowers the saturation pressure of the mixture, thereby increasing the absorption of all the gases in the mixture, resulting in an increased yield of polymer. This trend is predicted using thermodynamic modelling with SAFT -VR, and a substantial increase in yield is confirmed by experiment, though the increase was slightly less than suggested by the theory. This difference could be due to the neglect of mass-transfer effects; the alkenes must diffuse through the liquid polymer to reach the catalyst, and this was not considered in the calculations. It could also be due to uncertainties in the composition of the gas phase in the reactor. The gas-mixture saturation pressure is sensitive to this and so, in turn, is the enhancement of absorption of all gases. In the case of a co-polymerisation of but-1-ene and ethene, if nitrogen is replaced by pentane in such a way that the proportion of the other gases in the vapour phase remains constant, similar increases in absorption of both ethene and but-1-ene are predicted, suggesting that the polymer produced will be equivalent in terms of branch structure and density. The enhancement of absorption effect offers an attractive explanation of the co-monomer effect, in which the presence of higher molecular-weight

Design ofpolyolefin reactor mixtures

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alkenes in co-polymerisation is known to increase yield compared with equivalent homo-polymerisation. Higher alkenes are less volatile than ethene and replacing some ethene in the reaction mixture with a higher alkene will lower the gas-mixture saturation pressure, in just the same way as replacing nitrogen with pentane, thereby increasing the absorption of all the gases present. Further modelling and experiment is still required to exploit the enhancement of alkene absorption to its full potential. Now that the enhancement effect has been demonstrated, careful experimentation using gasphase analysis is needed to provide more-accurate information of gas-phase partitioning in the reactor. At the current level of modelling, the gas-phase composition is crudely estimated, and the estimate used as input to the absorption calculation. Using instead the accurate experimental composition, absorption calculations could be carried out that would be more representative of the actual experimental set-up. Up to now the predictions from the modelling were used cautiously, due to the risk of condensation (LLE) in the reactor. However, with this extra experimental input, results from future modelling could be used with much greater confidence and a new reactor mixture designed that takes greater advantage of the effect of enhanced absorption, further increasing the yield of polymer. Appendix: Preliminary PC-SAFT calculation for test-case GPR mixture As described in the Introduction, in the past, a puzzling observation during polyethylene production was the increase in the ethene polymerization rate when co-monomers are included in the GPR mixtures - the so-called "comonomer effect". [ 1 ^ ] Prior to the investigation described in this chapter, preliminary calculations were carried out at Borealis using commercially available PC-SAFT software: Multiflash® [5] from Infochem Computer Services Ltd., London. The parameters used are supplied by Infochem and also taken from Reference 19; these are shown in Table Al. In PC-SAFT, three parameters are used to describe each substance: size parameters a and m and an energy parameter e, (PC-SAFT lacks the extra parameter X of SAFT-VR, which is most useful in treating systems with polar interactions, and in extending treatment to deal with charged systems.) Note that although the same symbols are employed, these e, <jand m are not the same as those used in SAFT-VR. Details of PC-SAFT may be found in References 12 and 13. There are other subtle differences between PC-SAFT and SAFT-VR (the interested reader is directed to Reference 18 for a more-detailed discussion) but in the context of this work, they may be assumed to be broadly equivalent techniques.

330

A J. Haslam et al. PC- SAFT pure-component parameters substance

MW [g mol"^]

m

^[A]

ethene but-1-ene nitrogen PE

28.05 56.11 28.01 46883

1.593 2.2864 1.2053 12330

3.445 3.6431 3.313 4.0217

e/ksm 176.47 222 90.96 252

As a test case, a mixture was studied comprising 100 moles of ethene, 40 moles of a mixture of nitrogen and but-1-ene in varying proportions, and 0.11 moles of amorphous PE (giving a total of 140.11 moles). Calculations were carried out at r = 80°C and /? = 20 bar (2 MPa). The but-1-ene content of the but-1-ene-nitrogen mixture was varied from zero to 100 %. In Figure Al the but-1-ene and ethene content in the PE is plotted against the mole fraction of but-1-ene of the whole 140-mole mixture.

0.05

0.1

0.15

0.2

mole fraction butene Figure Al. The amount of ethene and but-1-ene in the PE phase as a function of the fraction but1-ene present, calculated using PC-SAFT.

It is seen that the ethene content in the PE phase is enhanced with the presence of but-1-ene even when the driving ethene partial-pressure remains the same. It should also be noted that the but-1-ene content is also enhanced above what can be explained by the increasing driving pressure. Such an enhancement will result in a large increase in the concentration of polymerising monomers in the polyethylene encapsulating the active catalyst. This explains the increased polymerisation rate, as the rate depends directly on the concentration of the monomer in contact with the catalyst. The calculation of increased absorption relates only to the amorphous part of the polyethylene; it is assumed that no

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monomer is absorbed in the crystalline part. Estimates of the fraction of amorphous and crystalline polyethylene can be based on measurements of the polyethylene density. This is very well described in the work by Paricaud et aL [17,25], to which the interested reader is referred. The above calculations showed promising leads to explain the co-monomer effect and provided motivation for a further study based on a SAFT-based thermodynamic description. References. 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20. 21. 22. 23. 24. 25. 26. 27. 28. 29.

G.C. Han-Adebekun, W.H. Ray, J. Appl. Polym. Sci. 65 (1997) 1037. T.S. Wester and M. Ystenes, Macromol. Chem. Phys. 198 (1997) 1623. P. Kumkaew, L. Wu, P. Praserthdam and S.E. Wanke, Polymer 44 (2003) 4791. B.J. Banaszak, D. Lo, T. Widya, W.H. Ray and J.J. de Pablo, Macromolecules 37 (2004) 9139. Multiflash, Infochem Computer Services Ltd, London, UK, www.infochemuk.com S. Brunauer, P. H. Emmett and E. Teller, J. Am. Chem. Soc. 60 (1938) 309. S. M. Lambert, Y. Song and J.M. Prausnitz, "Equations of State for Polymer Systems", in: "Equations of State for Fluids and Fluid Mixtures"; Elsevier, Amsterdam, 2000; Vol. 2. P. Paricaud, A. Galindo and G. Jackson, Mol. Phys. 101 (2003) 2575. M.S. Wertheim, J. Chem. Phys. 87 (1987) 7323. A. Gil-Villegas, A. Galindo, P.J. Whitehead, S.J. Mills, G. Jackson and A.N. Burgess, J. Chem. Phys. 106 (1997) 4168. A. Galindo, L.A. Davies, A. Gil-Villegas and G. Jackson, Mol. Phys. 93 (1998) 241. J. Gross and G. Sadowski, Ind. Eng. Chem. Res. 40 (2001) 1244. J. Gross and G. Sadowski, Ind. Eng. Chem. Res. 41 (2002) 1084. S. H. Huang and M. Radosz, Ind. Eng. Chem. Res. 29 (1990) 2284. C. McCabe and G. Jackson, Phys. Chem. Chem. Phys. 1 (1999) 2057. C. McCabe, A. Galindo, M.N. Garcia-Lisbona and G. Jackson, Ind. Eng. Chem. Res. 40 (2001)3835. P. Paricaud, A. Galindo and G. Jackson, Ind. Eng. Chem. Res. 43 (2004) 6871. A.J. Haslam, N. von Solms, C.S. Adjiman, A. Galindo, G. Jackson, P. Paricaud, M.L. Michelsen and G.M. Kontogeorgios, Fluid Phase Equilib. (submitted, October 2005). F. Tumakaka, J. Gross and G. Sadowski, Fluid Phase Equilib. 194 (2002) 541. LA. Kouskoumvekaki, N. von Solms, T. Lindvig, M.L. Michelsen and G.M. Kontogeorgis, Ind. Eng. Chem. Res. 43 (2004) 2830. LA. Kouskoumvekaki, G.J.P. Krooshof, M.L. Michelsen and G.M. Kontogeorgis, Ind. Eng. Chem. Res. 43 (2004) 826. N. von Sobns, LA. Kouskoumvekaki, T. Lindvig, M.L. Michelsen and G.M. Kontogeorgis, Fluid Phase Equilib. 222 (2004) 87. T. Lindvig, M.L. Michelsen and G.M. Kontogeorgis, Ind. Eng. Chem. Res. 43 (2004) 1125. N. von Solms, M.L. Michelsen and G.M. Kontogeorgis, Ind. Eng. Chem. Res. 44 (2005) 3330. P. Paricaud, PhD. Thesis: "Understanding the fluid phase behaviour of polymer systems with the SAFT theory". Imperial College London, 2002. C. E. Rogers, V. Stannett and M. Szwarc, J. Phys. Chem. 63 (1959) 1406. A.S. Michaels and R.W. Hausslein, J. Polym. Sci., Part C: Polym. Symp. 10 (1965) 61. S.J. Moore and S.E. Wanke, Chem. Eng. Sci. 56 (2001) 4121. A.S. Michaels and R.B. Parker, J. Polym. Sci. 41 (1959) 53.

332 30. 31. 32. 33. 34. 35. 36. 37. 38. 39. 40. 41. 42. 43. 44. 45. 46. 47.

A J. Haslam et al. T.F. McKenna, Eur. Polym. J. 34 (1998) 1255. S.K. Nath and J.J. de Pablo, J. Phys. Chem. B 103 (1999) 3539. B. Flaconneche, J. Martin and M.H. Klopffer, Oil Gas Sci. Technol. 56 (3) (2001) 261. C. Pan and M. Radosz, Ind. Eng. Chem. Res. 38 (1999) 2842. A. Ghosh and W.G. Chapman, Ind. Eng. Chem. Res. 41 (2002) 5529. H. Adidharma and M. Radosz, Ind. Eng. Chem. Res. 41 (2002) 1774. M.S. Wertheim, J. Stat. Phys. 35 (1984) 19. M.S. Wertheim, J. Stat. Phys. 35 (1984) 35. M.S. Wertheim, J. Stat. Phys. 42 (1986) 459. M.S. Wertheim, J. Stat. Phys. 42 (1986) 477. M.S. Wertheim, J. Chem. Phys. 87 (1987) 7323. W.G. Chapman, G. Jackson and K.E. Gubbins, Mol. Phys. 65 (1988) 1057. G. Jackson and K.E. Gubbins, Pure Appl. Chem. 61 (1989) 1021. R.K. Surana, R.P. Danner, A.B. de Haan and N. Beckers, Fluid Phase Equilib. 139 (1997) 361. T. Heuer, G.-P. Peuschel, M. Ratzsch and C. Wohlfarth, Acta Polymerica 40 (1989) 272. C. Wohlfarth, U. Finck, R. Schultz and T. Heuer, Die Angewandte Makromol. Chemie 198 (1991)91. Y. Sato, K. Fujiwara, T. Takikawa, Sumamo, S. Takishima and H. Masuoka, Fluid Phase Equilib. 162(1999)261. F. Garcia-Sanchez, G. Eloisa-Jimenez, G. Silver-Oliver and R. Vazquez-Roman, Fluid Phase Equilib. 217 (2004) 241.

Multiscale Modelling of Polymer Properties M. Laso, E.A. Perpete (Editors) © 2006 Elsevier B.V. All rights reserved.

333

Chapter 6

Atomistic Molecular Dynamics simulation of shortchain branched polyethylene melts Nikos Ch. Karayiannis/'^ Vlasis G. Mavrantzas^'^, Dimitrios Mouratides"^, Elias Chiotellis*^, Costas Kiparissides^ ^Department of Chemical Engineering, University ofPatras, Patras 26504, Greece ^Institute of Chemical Engineering and High Temperature Chemical Processes, Patras 26504, Greece ^Department of Chemical Engineering and Chemical Process Engineering Research Institute, Aristotle University ofThessaloniki, Thessaloniki 54124, Greece 1. Introduction Based on the molecular architecture of the constituent chains polyethylene (PE) is usually classified as HDPE (High Density PE), LLDPE (Linear Low Density PE), and LDPE (Low Density PE). HDPE consists of linear chains resulting from the catalytic reaction of pure ethylene monomers, LLDPE consists of chains bearing short branches sparsely distributed along their main backbone and LDPE consists of chains with densely packed branches of variable length that may carry additional arms thus creating a complex, dendritic-like structure. Recently, LDPE and other well-defined polymers with several kinds of molecular architecture (mainly of a long-chain branched structure, LCB) have been synthesized by metallocene and other single-site catalysts through anionic living polymerization techniques by employing organolithium initiators [1-6]. In general, dangling branches are considered as "long" if [3] their length exceeds the characteristic molecular length between entanglements, A/g, which is around C70 for PE [7]. Metallocene-catalyzed HDPE are characterized by both low polydispersity and total absence of branching when compared against more conventional PE samples leading to different rheological properties [8]. LLDPE samples bearing short-chain branches have been synthesized through copolymerization of ethylene with a-olefins using mainly Ziegler-Natta catalysts [9-11]. Fig. 1

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depicts simple sketches of single chains of a) HDPE, b) LDPE and c) LLDPE samples. (a)

(b)

(c)

Figure 1. Schematic representation of the molecular architecture of (a) high density polyethylene (HDPE), (b) low density polyethylene (LDPE) and (c) linear low density polyethylene (LLDPE) chains.

As a result of their different molecular characteristics, end-product PE materials exhibit different physical and rheological properties. For example, LDPE is more ductile than HDPE but it presents less tensile strength and hardness, while LDPE is inferior to LLDPE in terms of tensile strength and impact puncture resistance but considerably superior in terms of processability [3]. While recent advances in polymer synthesis result in LCB polymers with a well-defined structure, in the case of LLDPE the situation is more complicated [12-14]: the attached branches, which consist commonly of one up to six methylene units, are not distributed uniformly in regular intervals along the main backbone [9,10]. For such highly heterogeneous non-linear samples, the complete experimental study and precise characterization of the effect of the exact length, frequency and distribution of branches on the physical and rheological properties of LLDPE is almost impossible. Thus, the study of the viscoelastic properties of linear (HDPE) and LCB (LDPE) polymers, which are of paramount importance for industrial process, has drawn the majority of interest in the polymer research community. Today, shear thinning seems to be a universal phenomenon common to all PE macrostructures (HDPE, LDPE and LLDPE), but in extensional flows, LCB is found to lead to extreme strain hardening. Form the theoretical point of view, key aspects of the dynamics and rheology of entangled monodisperse linear melts have been successfully described by the reptation theory of Doi-Edwards [15] based on the original concept of the tube model by de Gennes [16,17]. Additional mechanisms contributing to the rheology of linear macromolecules are further captured through the concepts of contour-length fluctuation (CLF) [16,18], constraint release (CR) [16,19] and convective constraint release (CCR) [20]. Building on the tube model and the concept of dynamic dilution (DD) [21], Bishko et al. [22] and McLeish and

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335

Larson [23] formulated the "pom-pom" molecular theory for the rheology of LCB polymers with the H-shaped being the simplest model pom-pom molecule. While a lot of progress has been made regarding the theoretical coupling of structure-property relationship for HDPE and LDPE, the corresponding behavior of LLDPE macromolecules bearing irregularly distributed branches which are too small to entangle with surrounding chains (or branches), can not be interpreted on the basis of a general theoretical framework. Serving as an alternative approach, molecular simulations can play an important role in this direction, by elucidating the molecular-level mechanisms that govern the macroscopic physical and rheological behavior of these polymers, thus improving our understanding of the correlation between chain architecture (linear versus branched) and macroscopic properties. Unlike the difficulties encountered in the experimental characterization of the highly heterogeneous LLDPE materials, computational modeling can generate short-chain branched (SCB) samples with a well-defined molecular architecture, characterized by a regular distribution of branches along the main backbone; the corresponding simulation findings can therefore serve as a guide for the formulation of more accurate molecular theories and rheological models. In this chapter we review our most recent work in the atomistic simulation of the dynamic properties of model short-chain branched PE systems in the melt state. Section II presents the force field usually employed to describe the potential interactions in branched PE systems, on the basis of a united-atom model representation. Section III gives an overview of the Molecular Dynamics (MD) method, highlighting also results of past simulations with a number of linear and non-linear PE samples. Results from our most recent efforts with some well-defined entangled SCB configurations are presented in Sec. IV. Section V, finally, summarizes the main conclusions and discusses future plans.

2. Potential Force Field Atomistic simulations of chain-like molecules can in general be carried out using either a detailed molecular model in which all atoms are represented explicitly, or a less detailed one in which entire groups of atoms are lumped into single quasiatomic entities. For polyethylene, the former (explicit-atom, EA) description requires treating hydrogen (H) and carbon (C) atoms separately as individual sites [24], while the latter (united-atom, UA) allows considering each CH3, CH2 and CH unit as a single, united pseudoatom [25]. United-atom models result in enormous savings in terms of the total number of interacting sites present in the system; however, they are considerably less accurate than explicit-atom ones. To improve accuracy over the UA representation, anisotropic united-atom (AUA) models, in which the centers of the non-bonded

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interactions are moved slightly away from the carbon centers of the united pseudoatoms, have also been proposed [26,27]. All results presented in the next sections of this chapter have been obtained from molecular dynamics (MD) simulations carried out with a united-atom model, which is a hybrid combination of the TraPPE (Transferable Potentials for Phase Equilibria) and NERD models proposed by Martin and Siepmann [28] and Nath et al. [29], respectively, for the description of the potential (bonded and non-bonded) interactions between CH3, CH2, CH and C units in hydrocarbons. To improve the predictive capability of the model, the values of some of its potential function parameters have been borrowed from the AUA description of Toxvaerd [30] and the extended NERD model of Nath and Khare [31] for branched hydrocarbons. Such a model has already been employed in past simulations of a number of entangled linear [32,33] and LCB (H-shaped) [34,35] PE melts over a wide range of chain lengths, providing excellent estimates for their volumetric, thermodynamic and conformational properties. In all SCB PE systems studied here, only one branch emanates from each junction point, which means that there exist no bare carbon atoms (C) in the system. Thus, the various pseudoatoms along a SCB chain will be denoted here as CH^ where x = 1 for the united atoms at the branch points, x=3 for the united atoms at the free ends of the chain (both in the main backbone and the short arms), and x = 2ior the rest of atoms along the chain or the arms of the studied polymers. According to the united-atom model employed in this study: • bond lengths, /, are taken to fluctuate harmonically around a mean value of k = 1.54A according to the following bond stretching potential (Fstretch) [29,31]

where ^B is Boltzmann's constant and ki = 96500 K A"^. • bond angles, 6, are governed by the Van der Ploeg and Berendsen harmonic bending potential (Fbend) [36]

Y^Jk,{9-e,f

(2)

where k^ = 62500 K rad' and ^o adopts the values of 114° and 109.47° for the bending angles around the CH2 and CH pseudoatoms, respectively

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dihedral angles around the CH2-CH2 bond are sampled using the following nine-term, sum-of-cosines torsional potential [30]

£ ^

= ic,cos'W

(3)

/=0

while those around the CH-CHx bonds (x = 2 or 3) are sampled using the following expression for the torsional potential [31]

(4) 7=1

• All pairs (jj) of intramolecular neighbors separated by more than three bonds along a chain and all intermolecular ones interact through a standard 12-6 Lennard-Jones potential (FL.J)

V^.Mj) = ^eij

u

(5)

\^v J where ry stands for the distance between the interacting sites and £/, and Oy for the well depth and collision diameter of the potential, respectively. To describe nonbonded interactions between sites of different kind, standard Lorentz-Berthelot combining rules are used cT,+cr, ij

y i i j j

^

IJ

(6)

An example of the different interactions and of the potential functions governing them is shown in Figure 2. What is presented in the figure is examples of two-body (bond stretching), three-body (bond bending), four-body (torsional), as well as longer-range (intra- and inter-molecular) nonbonded interactions governing the interacting sites belonging to two segments along two different SCB chains in the system.

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intermolecular non-bonded

bond bending

torsional

Figure 2. Schematic representation of polymer segments belonging to different chains in the united-atom description of the present study. Typical examples of 1-2 (bond stretching), 1-3 (bond bending), 1-4 (torsional) and non-bonded (inter- and intramolecular) interactions are indicated by the corresponding arrows.

The mathematical expressions and the values of their parameters for all kinds of potential interactions invoked by the UA model of the present simulation study of linear and SCB PE melt configurations are summarized in Table I.

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Table I. Potential functions and the values of their parameters used to describe all inter- and intramolecular interactions in the present atomistic MD simulations of linear and SCB PE systems.

type of interaction potential form

and

type of pseudoatom

bond stretching

parameter values

/O=1.54A,^/ = 96500KA-2

Fstretch(0/^B=l/2^,(/-/o)' bond bending

CHjf-CH2~CHy

fe = 62500Krad^ ^0=114°

(jc,>'=I,2,3) CH_;f-CH-CHy

torsional

(;(,>'= 1,2, 3)

ke = 62500K rad ^ OQ = 109.47°

Co = lOOlK, ci = 2130K, C2 = -303K, C3 = -3612K, C4 = 227K, cs = 1966K,

(x,>'=l,2,3)

ce = -4489K, Cj = -1736K, cg = 2817K co=1416.3K,ci = 398.3K,

CHjf-CH-CH2-C'riy

fc>;=l,2,3) Nonbonded

CH^r

VUr^ = 4£y[(V^.j)^2-(ay/ry/]

{X = 2, 3)

C2 = 139.12K,C3 = -901.2K, ^l=+l,^2 = -l,^3=+l fCHx = 46K, acHx = 3.95A

CH fCH = 39.7K, acH = 3.85A

Initial configurations for each one of the simulated systems were created in amorphous cells subject to periodic boundary conditions in all dimensions using the three-stage, constant-energy minimization technique of Theodorou and Suter [37] as implemented in the Materials Studio (version 2.1) software package of Accelrys Inc. [38]. These initial structures were subsequently subjected to an exhaustive pre-equilibration run via a long NPT MD simulation at the desired temperature (r= 450K) and pressure (P = latm) conditions. Alternatively, initial configurations for the present atomistic MD studies could be provided through a more robust and faster approach using a Monte Carlo (MC) method similar to that discussed in Chapter 2 of this book. As explained in more detail there, at the heart of such a MC algorithm is a set of advanced chain connectivity altering moves [32-35,39,40], which has made possible the robust and efficient thermal equilibration of a number of linear, and

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LCB PE systems. Extending the algorithm to the case of the SCB PE systems of the present study is already underway. III. Molecular Dynamics Molecular Dynamics (MD) offers a direct way of simulating a classical many-body system [41-44] and obtaining information about its structural and dynamic properties, throu^ the time integration of a set of coupled differential equations expressing Newton's law of motion for every atom (or united group) / in the simulation box containing a total of A/^ atoms (or united groups)

' W / ^ = ^=-V/(rpr2,...,r^_pr^), « / = - ^

(7)

In Eq. (7), / is the time, W/, U/ and r/ the mass, velocity and position of the /th atom in the simulation box, F, the force exerted on the /th atom, and F(ri,r2,...,r;v-75i*A^) the function of the total potential energy of the system. Based on the discussion of section II for the united-atom nature of the molecular model employed in the present simulation study, the latter is the sum of bond stretching, bond-angle bending, torsional angle and (intra- and inter-molecular) nonbonded contributions. Consequently, the total force on atom / can also be written down as the sum of two-body, three-body, four-body and (intra- and inter-molecular) non-bonded contributions, as follows: -V/(r„r2,...,r^_pr^) = X^i-2(»'p»'^) j

J

k

nonbonded

MD proceeds by integrating in time the set of Newton's equations of motion (see Eq. (7)) for all atoms in the system with appropriate initial conditions for all atomic positions and velocities. As mentioned above, initial positions can be provided by building an amorphous cell subject to periodic boundary

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conditions, followed by an energy minimization step [37,38] (and, if possible, an exhaustive pre-equilibration run with a suitable MC algorithm [3235,39,40]). Initial velocities, on the other hand, are assigned randomly to each atom from a Maxwell-Boltzmann distribution of velocities at the desired temperature. Direct numerical integration of Newton's equations (Eq. 7) leads to the conservation of the total energy of the system, casting MD simulations in the microcanonical (constant energy - constant volume, NVE) statistical ensemble. In practice, more valuable information arises from simulations in different ensembles where temperature (canonical ensemble, NVT) and/or pressure (isothermal-isobaric, NPT) are kept constant. This is achieved by introducing additional degrees of freedom (e.g., a heat reservoir or a pressure piston) to couple the system with external variables and dump out deviations from the desired temperature and pressure values, also by appropriately modifying the equations of motion. Commonly used thermostats (thermo-coupling methods) have been proposed by Nose [45], Hoover [46] and Berendsen et al. [46], while isotropic and anisotropic deformations of the simulation cell are mostly controlled by the Andersen [47] and Parrinello-Rahman [48,49] barostats, respectively. In the case of extended ensembles, adjustable parameters are employed (termed "thermal inertia" and "piston mass" for thermostats and barostats, respectively) to control the rate of temperature (or pressure) fluctuations [42]. An important notice is that for all statistical ensembles (NVE, NVT and NPT) with appropriately defined thermostats/barostats such that a Hamiltonian can be defined, this should be a conserved quantity of the system that is its value should remain constant in the course of the MD simulation. Numerically solving Newton's equations of motion requires the definition of an integration time step, dt; this should be considerably smaller (by almost one order of magnitude) than the fastest characteristic relaxation time of the system (in our case, the time characterizing bond stretching), but large enough in order to minimize the CPU time associated with the frequent calculation of forces (which consumes most of the computational time). Typical values range from dt = 0.5 up to 2fs, depending on the potential force field and the applied conditions. Integration of the differential equations is undertaken either by a high-order, predictor-corrector numerical method as proposed by Gear [51] or by a variant (leapfrog, velocity-Verlet) of the Verlet scheme [52,53]. Novel multiple time step algorithms where the fast modes are integrated with a small time step (dt]) and the slow ones with a time step which is «/times longer than the small one (dt2 = rifdti) have also been developed and implemented, resulting in significant CPU savings. One of the most widely used explicit multiple-time step integrators for atomistic simulations is the reversible REference System

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Propagator Algorithm (rRESPA) of Tuckerman et al. [54] and Martyna et al. [55]. Specific atomistic MD simulations require constraining specific degrees of freedom, such as the bond lengths. Such rigid-bond constraints (where the bond lengths do not obey the harmonic potential of Eq. 1 but are fixed to their equilibrium length, k) can be implemented through the application of specific algorithms based on the SHAKE [56], RATTLE [57], and the Edberg-EvansMoriss [58] methods. One of the nicest features of the MD method is that its performance can be significantly improved through code parallelization and execution on sharedmemory supercomputing facilities or distributed memory clusters of workstations through a synchronous exchange of data between the available processors. Commonly employed techniques for parallelizing an MD code for molecular systems include [44] the atom decomposition (or replicated-data) method, the force decomposition method, and the domain decomposition method. For very large systems containing hundreds of thousands of interacting sites, the efficiency of the domain decomposition method can be as high as 90% [59]. Even in its state-of-the-art implementation where the rRESPA multiple time step integrator [53,54] is used, a spatial decomposition scheme is employed for parallelizing the corresponding MD code [58], and the system is exhaustively pre-equilibrated with a state-of-the-art MC algorithm [60-64], the longest time for which an atomistic MD method can track the evolution of a PE system consisting of chains bearing a couple of hundreds of interacting sites is on the order of a few microseconds (|is). Thus, the atomistic simulation of truly long LCB PE melts bearing well entangled arms and backbones is definitely out of reach with today's computational resources. Similar conclusions can be drawn for the case of long, well-entangled linear PE systems [35, 65]. To extend the dynamic studies to high PE melts (of relevance to industrial practice), one should depart from the atomistic level of description and adopt a coarser representation of the chain, such as through a FENE (finite extensible nonlinear elastic) model [66] or by treating chains as uncrossable strings of blobs [67], It is also important to mention that from the three different PE structures (HDPE, LDPE and LLDPE), the linear (HDPE) one has attracted the majority of the simulation work in contemporary literature [60-67]. Of course, the recent implementation of the chain connectivity altering MC moves for H-polymers opened up the way toward the direct simulation of LCB systems in atomistic detail [34,35]. MC and MD simulation work on LLDPE systems, on the other hand, has been limited either to small alkanes or oligomeric samples [11, 68-76] with the exception of the recent work of Jabbarzadeh et al. [77] who reported on the effect of molecular shape on the viscoelastic properties of the star, H-shaped

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and comb-shaped isomers of the linear CiooPE system based on the findings of non-equilibrium MD simulations. IV. Results IV. 1. Systems Studied All MD simulations of the present study with short-chain branched PE structures have been executed in the isothermal-isobaric (NPT) ensemble at T = 450K and P = latm, using the rRESPA method, with the small integration step selected equal to 1 fs and the large one equal to 5 fs. The MD runs were carried out with the large-scale atomic/molecular massively parallel simulator (LAMMPS) code [59,78] that can run on virtually any parallel platform. The Nose-Hoover thermostat [45,46] and the Andersen barostat [57] were implemented to control temperature and pressure fluctuations, respectively. The simulations were executed in parallel [59] on the processors of a Linux cluster consisting of 8 Intel Xeon dual workstations at 2.4GHz. To compare against the structural, volumetric and dynamic properties of linear PE melts of the same total MW (the so called "linear analogues"), two linear and strictly monodisperse PE systems with chain length equal to C142 and C3205 respectively, were also simulated under exactly the same conditions. System pre-equilibration was achieved through a long NPT MD run (of duration equal to 50 and 100ns for the two families of systems, respectively). Initial configurations for the subsequent production runs were chosen so as to be characterized by dimensions (as quantified through the radius of gyration) and density equal to their average values at the end of the pre-equilibration run. All short-chain branched (SCB) PE systems simulated in this work are defined by specifying: a) their branch length, Cb, (i.e., the number of carbon atoms per branch), b) their branching frequency, Nf^eq, (i.e., the number of carbon atoms along the main backbone between successive branch points), and c) the total number of branches per chain, A/br. Based on this definition, the notation used to describe the simulated PE structures is "SCBJA/br+l)xA/freq_A/brXCb" whcrc (iVbr+1) is the number of linear (equal in length) intervals along the main backbone separated by iVbr branches. Figure 3 presents a schematic of a single SCB chain in the united-atom representation of the present work. Based on the notation just adopted, this is a SCB_4xlO_3x4 molecule, bearing a total number of 52 (4x 10_3x4) carbon atoms, i.e., its linear analogue is a C52 PE system.

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Figure 3, Schematic representation of a short-chain branched (SCB) PE macromolecule in the united-atom representation of the present study. For this molecule: iVbr = 3, A^freq = 10 and Cb = 4, therefore, it is denoted as SCB_4xlO_3x4.

The large number of parameters {N\^, N^^, Cb) needed to precisely define the molecular architecture of SCB PE molecules, even for regularly distributed branches, complicates somewhat the analysis of their structural and dynamic properties, if we want to compare them to those of their linear counterparts. It is clear that by including even more structural parameters in the analysis, like heterogeneity in the branching frequency and/or polydispersity in the total chain length mimicking the industrial samples, their systematic study would require the execution of a large number of simulations, something which is beyond the scope of this work. For this reason, all SCB PE systems selected to be simulated here have been assumed to consist of chains which, for a given SCB_(iVbr+l)^A^freq_^br^Cb structurc, bear: (a) the same number of carbon atoms per branch, (b) the same number of carbon atoms between successive backbones along the main backbone, and (c) the same total number of carbon atoms. A detailed description of the molecular characteristics (iVbr, iVfreq^ Cb, MW) of all simulated systems (SCB and their linear analogues) is given in Table II. Table 11. Details of the Simulated PE Meh Systems ( r = 450K, P = latm)

System

no. of chains 22 SCB 11x12 10x1 22 SCB 9x14 8x2 22 SCB 7x16 6x5 22 SCB 5x22 4x8 24 SCB 12x25 11x2 24 SCB 6x50 5x4 SCB_8x35_7x6 24 22 Cl42 32 C320

no. of interacting sites 3124 3124 3124 3124 7728 7680 7728 3124 10240

Cb

A^br

N&^

1 2 5 8 2 4 6 -

10 8 6 4 11 5 7 0 0

12 14 16 22 25 50 35 -

Total MW (g/mol) 1990 1990 1990 1990 4510 4482 4510 1990 4482

Overall, the systems described in Table II can be divided in two categories based on their total chain length: the first set contains all SCB systems whose

Atomistic molecular dynamics simulations

345

total molecular length is equal to that of a C142 linear PE system, and the second set all SCB systems whose total molecular length is equal to that of a C320 linear PE system. We refer to the first set as the C142 family of systems and to the second as the C320 family of systems. IV. 2. Conformational Properties Conformational properties in a polymeric system can be analyzed in terms of chain dimension parameters, such as the mean-square end-to-end distance, , and the radius of gyration, , where the brackets denote averages over all chains in the simulation box and over all configurations. For the simulated SCB PE systems, the distance between the two ends of the main backbone is a well-defined quantity but it cannot be considered as a representative measure of its size, since it does not account for the molar mass distributed along the short (or long) dangling arms. Thus, our analysis of chain conformational properties for the simulated SCB PE systems will be restricted here to the calculation of the chain mean-square radius-of-gyration, , and its dependence on the architectural features of the system. If /Wi is the mass of the fth atom in the simulated system and R; its position vector, then the mean square radius of gyration, , is defined as

=^

(9)

Z-. /=1

where iV denotes the total number of atoms per chain and Rem the center of mass of the molecule:

^^^—.—

(10)

Z'"' 1=1

Figure 4 presents the time evolution of the running average value of in the simulated SCB 12x25_llx2, SCB_6x50_5x4, and SCB_8x35_7x6 PE

346

N.Ch. Karayiannis et al.

systems, as obtained from the present NPT MD simulations at T = 450K and P = latm. Also shown in the same figure is the corresponding curve for their monodisperse linear C320 PE analogue. After a short initial equilibration period, reaches a constant value which is seen to be: (a) practically the same (within the statistical error of the simulation data) for all SCB systems in the given family of systems (i.e., almost independently of the details of their molecular architecture), and (b) significantly smaller than the value characterizing the C320 linear analogue by almost 20%. This demonstrates undoubtedly that SCB PE melts possess a more compact structure than their linear PE counterparts. i

1

•

r

'

1

»

r

'

1000 J

<

H

900-

A V

1

/

800-

/

^*

*——— """*

\ 700- ^

600-

— •—--..-^i

'•**

J

^ 3 ^

SCBJ2x25J1x2 SCB 6x50 5x4 SCB 8x35 7x6 ,

,

200

,

p

i

400

1

600

1

^ J r-

800

1

1

1000

Time (ns) Figure 4. Time evolution of the running average value of the chain mean-square radius-ofgyration, , as obtained from the present NPT MD simulations with the C320 family of PE systems. In all cases, r = 450K and F = latm.

More quantitative conclusions about the dependence of on the details of the chain molecular architecture can be drawn from the data depicted in Figure 5 for the C142 family of systems. Again, branched structures adopt in all cases configurations that are significantly smaller than the linear systems of the same chain length. It is further observed that by reducing the branching frequency or increasing the branch length, decreases somewhat. More precisely, as more and more molecular mass is re-positioned from the main chain backbone to the branches, the mean size of the coil seems to shrink. To

347

Atomistic molecular dynamics simulations

quantify such a dependence, we can study how varies as a function of the "branch material fraction" ^b defined as the ratio of the molecular mass distributed along branches over the total mass of the molecule:

^

=

mass distributed along the branches total molecular mass

KrC, N^C, + iN^ + l)N,^

(11)

Then, it is observed that for the C142 family of SCB PE systems, decreases by around 15% for the samples characterized by ^b = 0.07, by about 23% for the samples characterized by ^b = 0.11, and by about 30% for the samples characterized by g)yr02\, compared to its value for the linear C142 system, at the same temperature and pressure conditions.

0142

SCBJ1x12_10x1

SCB_9x14_8x2

SCB_7x16_6x5

SCB_5x22_4x8

Molecular Architecture Figure 5. Dependence of on the details of PE molecular architecture. The results have been obtained from the present NPT MD simulations with the C142 family of systems, at r = 450K and P=latm.

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N.Ch. Karayiannis et al.

IV. 3. Volumetric Properties Figure 6 presents the time evolution of the running average value of the density, p, for the C320 family of the simulated PE systems (at T= 450K and P = latm). It is observed that the linear C320 system and the SCB member of its family with only 2 carbon atoms per arm are characterized by identical density values. On the other hand, the densities of the two other SCB PE systems (those bearing a little longer branches, equal to C4 and Ca, respectively) differ by less than 0.5% from the density of the linear C32onielt, which can also be considered as being within the statistical error of the simulation predictions. Exactly the same conclusions are drawn by examining the C142 family of systems, whose density plots are reported in Figure 7: the maximum difference detected in the densities of the SCB samples relative to that of the linear PE analogue hardly exceeds 1%. 0.769 0.768 ^ E

0.767-4

c

0.766-1

SCB 12x25 11x2 SCB 6x50 5x4 SCB_8x35_7x6

320

01

c

mm mm M « ^ i > - > •

0.765 J 0.764 0.763

—I—

600

800

1000

Time (ns)

Figure 6. Time evolution of the system density running average value, as obtainedfromthe present NPT MD simulations with the C320 family of PE systems, at r = 450K and P = latm.

On the basis of the information provided by Figs. 6-7, it is concluded that short-chain branching has a minimal effect on polymer density for purely amorphous PE samples. This simulation finding is supported by similar qualitative conclusions reported by Lundberg [79] on the basis of experimental

Atomistic molecular dynamics simulations

349

measurements with SCB and linear PE melts. Consequently, the density differences characterizing LLDPE (or, equivalently, SCB) and HDPE (or, equivalently, linear) end-products at room temperatures should be attributed mostly (or, almost exclusively) to the different extents of crystallinity in the two materials [HDPE materials are characterized by a higher degree of crystallinity compared to short-chain branched PE materials which are more amorphous]. In the purely amorphous melt phase, both molecular architectures are calculated to possess identical volumetric properties.

0.78

0.72-^T 0142

1

\

1

SCB_11x12J0x1

1 SCB_9x14.8x2

1

\ SCB_7x16_6x5

1

r SCB.5x22_4x8

Molecular Architecture Figure 7. Density as a function of molecular architecture for the C142 family of PE systems simulated here at r = 450K and P = latm.

IV.4. Dynamic Properties The MD simulations of the present study are particularly useful since they can provide first principles information for the effect of short-chain branching on the dynamics of molten PE. Of particular importance is the calculation of the maximum chain orientational relaxation time, TC, as quantified through the time it takes the autocorrelation function for the chain end-to-end unit vector, , to drop to zero. In general, the more rapidly drops to

N.Ch. Karayiannis et al.

350

zero, the faster the system loses the memory of its initial configuration, i.e., the faster the rate of its orientational relaxation. For the C142 and C320 families of the simulated PE samples, the corresponding -vs.-/ functions are shown in Figures 8 and 9, respectively.

Time (ns) Figure 8. Time decay of the orientational autocorrelation function of the unit vector u directed from the one end of the main chain backbone to the other, for the C142 family of PE systems. Results from the present NPT MD simulations at r = 450K and P = latm.

To make quantitative the dependence and variation of the orientational relaxation on the molecular characteristics of the systems, the simulated -vs.-/ curves can be fitted piecewise with stretched exponential (Kohlrasuch-Williams-Watts, KWW) functions of the form < u(OQi(0) >= exp

t

\fi' (12)

''KWW .

where fi and /KWW are the two fitting parameters of the KWW expression. The orientational relaxation time, TC, can then be calculated from the integral of the time autocorrelation function from / = 0 up to / -^ QO:

351

Atomistic molecular dynamics simulations oo

T, = J< u(OQi(0) > dt

(13)

and the values obtained are reported in Table III. 1

n

1

1

1

'

1

'

—1

1

1

0.8 J

H

^3,0

0.6C* 3 V

0.4-

SCBJ2x25J1x2 SCB_6x50_5x4 SCB 8x35 7x6

H

-^

\

0.2-

0.0-

•

1

100

«

'

—1

200

i"^

300

•

400

500

Time (ns) Figure 9. Same as with Fig. 8 but for the C320 family of PE systems.

It is observed that all samples within a given family of systems (the C142 and the C320 ones, here) are characterized by similar Xc values (within the statistical error of the simulation). This means that different combinations of branch length and branching frequency that correspond, however, to the same total number of carbon atoms per chain exhibit the same or very similar (orientational) relaxational behavior. That is, when the comparison for the effect of short-chain branching on the terminal relaxation is made on the basis of the same total chain length, no appreciable differences are computed. This happens because by re-positioning carbon atoms from the main backbone to the branches results, on the one hand, in a decrease of the molecular length of the main chain backbone (which causes Xc to decrease) but, on the other hand, in a drastic decrease in its dynamic flexibility [68,80] (which causes Xc to increase), since dangling branches act as effective high-energy barrier obstacles enhancing excluded volume interactions and steric hindrance for the translational and

352

N.Ch. Karayiannis et al.

rotational motion of the segments of the main backbone, especially in the vicinity of the branch points [68]. The net result is then a Xc value which is only a function of the total chain length of the molecule, and not of the relative fraction of backbone and branch material in it.

Table III. MD predictions for the chain orientational relaxation time, TC, and the self-diffusion coefficient, DQ, for the simulated C142 and C320 families of PE systems {T= 450K, P = latm). System

tc (ns)

Cl42

1.1 7.6 8.3 6.7 6.7 80 67 72 86

SCB_ll>;12_10xl SCB_9xl4_8x2 SCB_7xl6_6x5 SCB_5x22_4x8 C320

SCBJ2x25_llx2 SCB_6x50_5x4 SCB_8x35_7x6

Da (10-' cm^/s) 8.0 ± 0.6 5.2 ± 0.2 7.3 ±1.0 4.6 ±0.3 6.3 ± 0.2 1.3 ±0.1 0.9 ±0.1 1.0 ±0.1 0.8 ±0.1

Of course, when one compares different families of structures, then the computed differences are significant, following the scalings of the known molecular theories for polymeric systems (Rouse, Rouse combined with free volume, and reptation): for example, the C320 family is characterized by TC values which are one order of magnitude higher than the C142 family of systems. Further information about the dynamics of the simulated PE systems can be provided by calculating the self-diffusion coefficient, DQ, of the chain center-ofmass. For long time scales when the hydrodynamic limit is reached and the molecular transport obeys Fick's law, DQ can be calculated from Einstein's equation, according to which:

([R^CO-RGCO)]')

D. = lim-^

'-

(14)

where <[RG(t)-RG(0)]^> denotes the mean square displacement (msd) of the chain center-of-mass from its initial position after time t. Figure 10 presents <[RG(t)-RG(0)]^>-vs.-t plots as obtained from the present NPT MD simulations for the Ci42 family of PE melts. The long times scales up to which the NPT MD

353

Atomistic molecular dynamics simulations

simulations have been carried out (500 ns) has ensured that the centers-of-mass of the chains in all samples have traveled distmices which are at least 7 times longer than their dimensions (as quantified through their radius of gyration see, e.g., Fig. 5), which allows one to calculate quite reliably the corresponding self- diffusivity. 25000

20000 < A

15000H

it

10000H

0^

5000

Time (ns) Figure 10. Mean square displacement of the chain center-of-mass, <[RG(t)-RG(0)]^>, for the C142 family of PE systems, as obtained from the present NPT MD simulations at r = 45OK and P latm.

The same plots are shown again in Fig. 11 but this time in the form of <[RG(t)-RG(0)]^> / (60 graphs, whose long-time limits define the chain selfdiffusion coefficient DQ for the simulated systems. It is evident that after an initial and quite prolonged non-Fickian (or anomalous) diffusive regime, Fickian diffusion sets-in for all systems independent of their molecular architecture. The duration of the anomalous diffusion regime increases somewhat with increasing branch length, as a result of the increased heterogeneity in the system molecular architecture. Calculated values of the chain self-diffusion coefficient. Do, for all simulated systems are summarized in Table III. On the basis of information provided by the data of Table III it is concluded that the addition of short branches along a given linear backbone (keeping the total chain length constant)

354

A^. Ch. Karayiannis et ah

causes, in general, a decrease in the chain self-diffusion coefficient. Unfortunately, the relatively large statistical error associated with the calculation of Dodoes not allow us to precisely quantify this dependence, since the observed differences are in some cases within the statistical noise of the simulation data. 4.0 3.5-1 3.0 4

320

SCB 12x25 11x2 SCB 6x50 5x4 SCB_8x35_7x6

0.0

— I —

200

400

600

—I—

800

1000

Time (ns) Figure 11. Time evolution of the ratio <[RG(0-RG(0)]^> / (60, as obtained from the present NPT MD simulations with the C320 family of PE systems, at r = 450K and P - latm.

V. Conclusions We have presented results from long atomistic MD simulations for the effect of molecular architecture on the structural, volumetric and dynamic properties of purely amorphous PE melt systems bearing a well-defined number of shortchain branches frequently spaced along the main (linear) backbone. Two families of PE microstructures were studied, corresponding to a total of 142 (MW = 1990g/mol) and 320 (MW = 4482g/mol) carbon atoms per chain, respectively. As quantified by the mean chain radius of gyration, short-chain branched (SCB) PE mehs are characterized by significantly smaller dimensions than linear PE melts of the same total chain length under the same temperature and pressure conditions, due to the more symmetric arrangement of their

Atomistic molecular dynamics simulations

355

material around the chain center-of-mass as molecular mass is removed from the linear backbone and is distributed along the dangling branches. In contrast, SCB and linear PE melts of the same chain length exhibit identical (or almost identical) volumetric properties, suggesting that the differences recorded in the densities of the end products of the PE industry under the names "LLDPE" (linear low density polyethylene) and "HDPE" ( h i ^ density polyethylene) are due to their totally different degrees of crystallinity at temperatures below their melting point. The present MD simulations have further allowed us to calculate the chain center-of-mass long time diffusive behavior and its dependence on the molecular characteristics of the constituent chains: short-chain branching causes a decrease in the chain self diffusion coefficient compared to the value exhibited by the linear melt of the same total chain length by a factor which can range from 10 up to 40% depending on the molecular characteristics of the simulated system (branch length, branching frequency, and total chain length). Based on the results reviewed here it appears that short-chain branching (SCB) has a rather small effect on the equilibrium dynamics of PE melts. Further simulations of even longer (higher-MW) PE samples are definitely needed in order to clarify more precisely the effect of branch length, branching frequency and number of branches on the dynamic properties. Acknowledgements The authors are indebted to European Commission for financial support through the GROWTH PMILS project. Very fruitful discussions with Prof Manuel Laso (University of Madrid), Prof. Rafique Gani (Danish Technical University, Copenhagen), Dr. Prokopis Pladis (University of Thessaloniki) and all partners of the PMILS project are also warmly acknowledged.

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[47] H. J. C. Berendsen, J. P. M. Postma, W. F. van Gunsteren, A. Di Nola and J. R. Haak, J. Chem.Phys. 81(1984)3684. [48] H. C. Andersen, J. Chem. Phys. 72 (1980) 2384. [49] M. Parrinello and A. Rahman, Phys. Rev. Lett. 45 (1980) 1196. [50] M. Parrinello and A. Rahman, J. Appl. Phys. 52 (1981) 7182. [51] C. W. Gear, Numerical Initial Value Problems in Ordinary Differential Equations, Prentice-Hall, Englewood Cliffs, 1971. [52] L. Verlet, Phys. Rev. 159 (1967) 98. [53] L. Verlet, Phys. Rev. 165 (1967) 201. [54] M. Tuckerman, B. J. Berne and G. J. Martyna, J. Chem. Phys. 97 (1992) 1990. [55] G. J. Martyna, M. E. Tuckerman, D. J. Tobias and M. L. Klein, Mol. Phys. 87 (1996) 1117. [56] J. P. Ryckaert, G. Ciccotti and H. J. C. Berendsen, J. Comput. Phys. 23 (1977) 327. [57] H. C. Andersen, J. Comput. Phys. 52 (1983) 24. [58] R. Edberg, D. J. Evans and G. P. Morriss, J. Chem. Phys. 84 (1986) 6933. [59] S. Plimpton, J. Comput. Phys. 117 (1995) 1. [60] D. N. Theodorou, Bridging Time Scales: Molecular Simulations for the Next Decade, Eds. P. Nielaba, M. Mareschal and G. Ciccotti, Springer-Verlag, Berlin, 2002. [61] J. J. de Pablo and F. A. Escobedo, AIChE J. 48 (2002) 2716. [62] D. N. Theodorou, Mol. Phys. 102 (2004) 147. [63] W. Paul, Computational Soft Matter: From Synthetic Polymers to Proteins, Eds. N. Attig, K. Binder, H. Grubmiiller and K. Kremer, Julich NIC Series, 2004. [64] W. Paul and G. D. Smith, Rep. Prog. Phys. 67 (2004) 1117. [65] V. A. Harmandaris, V. G. Mavrantzas, D. N. Theodorou, M. Kroger, J. Ramirez, H. C. Ottinger and D. Vlassopoulos, Macromolecules 36 (2003) 1376. [66] K. Kremer and G. S. Grest, J. Chem. Phys. 92 (1990) 5057. [67] J. T. Padding and W. J. Briels, J. Chem. Phys. 117 (2002) 925. [68] M. Mondello and G. S. Grest, J. Chem. Phys. 103 (1995) 7156. [69] M. Mondello, G. S. Grest, A. R. Garcia and B. G. Silbemagel, J. Chem. Phys. 105 (1996) 5208. [70] R. Khare, J. de Pablo and A. Yethiraj, J. Chem. Phys. 107 (1997) 6956. [71] K. S. Kostov, K. F. Freed, E. B. Webb III, M. Mondello and G. S. Grest, J. Chem. Phys. 108(1998)9155. [72] L. I. Kioupis and E. J. Maginn, Chem. Eng. J. 3451 (1999) 1. [73] L. I. Kioupis and E. J. Maginn, J. Phys. Chem. B 103 (1999) 10781. [74] L. G. MacDowell, C. Vega and E. Sanz, J. Chem. Phys. 115 (2001) 6220. [75] B. Abu-Sharkh and I. A. Hussein, Polymer 43 (2002) 6333. [76] X.-b. Zhang, Z.-s. Li, Z.-y. Lu and C.-C. Sun, Polymer 43 (2003) 3223. [77] A. Jabbarzadeh, J. D. Atkinson and R. I. Tanner, Macromolecules 36 (2003) 5020. [78] Large-scale Atomic/Molecular Massively Parallel Simulator (LAMMPS) software distributed by Dr. S. Plimpton as Sandia National Laboratories, US. All MD simulations reported in the present review have been carried out using version LAMMPS 2001 (Fortran 90). [79] J. L. Lundberg, J. Polym. Sci. Part A 2 (1964) 3925. [80] N. C. Karayiannis, V. G. Mavrantzas and D. N. Theodorou, Macromolecules 37 (2004) 2978.

Multiscale Modelling of Polymer Properties M. Laso, E.A. Perpete (Editors) © 2006 Elsevier B.V. All rights reserved.

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Chapter 7

Hierarchical Approach to Flow Calculations for Polymeric Liquid Crystals M. Laso^ L.M. Muneta^ M. Miiller^, V. Alcazar'', F. Chinesta^, A. Ammar® ""ETSII, UPM, Jose Gutierrez Abascal, 2, E-28006 Madrid, Spain ^NovodeXAG, Technoparkstrasse 1, CH- 8005 Zurich, Switzerland ^ Dept. of Organic Chemistry, Universidad de Salamanca, £-37071 Salamanca, Spain ^LMSP UMR8I06CNRS-ENSAM'ESEM, 151 Boulevard deVHopital, F-75013 Paris, France ' Laboratoire de Rheologie, UMR 5520 CNRS-UJF-INPG, 1301 Rue de la Piscine, BP 53 Domaine Universitaire, 38041 Grenoble Cedex 9, France I. Introduction Liquid crystalline polymers (LCPs) are macromolecules that contain long, rigid or approximately rigid segments. Because of these rigid units, LCPs can display structural phase transitions between isotropic (disordered) and nematic (oriented) states. LCPs are classified as thermotropic or lyotropic depending on whether the structural transitions are induced by changes in temperature or in concentration, respectively^*^. Some of these materials are known to have interesting macroscopic properties, such as high modulus in the solid phase and low viscosity in the melt or in solution. They can also display a rich phase behaviour as temperature or concentrations or both are changed, even in quiescent conditions. Non-equilibrium, macroscopic flow conditions can further complicate their phase behaviour.^ Macroscopic viscoelastic flow calculations for LCPs typically start with the derivation of macroscopic, approximate equations for quantities of interest, such as order parameters. Analytical developments towards a closed constitutive equation very often necessitate the introduction of more or less ad hoc closure •

•

4-6

approximations. The difficulty in obtaining accurate closures has motivated the use of direct simulations. These take place typically at one of two levels of description:

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1. Integration of the partial differential equation (PDE) governing the temporal evolution of the orientational probability distribution function (pdf) yf{u J) fox the orientation of the molecule, as given by the unit vector w, in a coarse gramed sense, i.e. u represents the molecule by its head-to-tail connector vector. This equation is typically a non-linear Fokker-Planck (FP) equation which has to be solved in configuration space by means of suitable discretization methods. Solution techniques are basically identical to those used to integrate the accompanying macroscopic conservation equations, also expressed as PDEs. For Doi's widely used model for LCPs the FP equation for takes the form:

dY{u.t) dt

du

[(MVv-wMw:Vv)^(w,r)] +

'Tu\^'^^^

^-<->i(^:

(1)

where l//^iu,t) is the probability that a rod-like molecule has an orientation given the by the unit vector u at time ^; V v is the local velocity gradient the LCP molecule is subjected to at time t, and D^[u) is the orientation-dependent rotary diffusivity given by: 4 r

/

/

^2

Dr{li) = Dr — miu )sin(w ,u)du

(2)

IT J

where i)^ is the rotary diffusivity in a hypothetical isotropic solution of molecules at the given concentration and sin(w ,u)is the positive sine of the angle between the unit vectors u and u describing the orientation of two LCP molecules'^. V^^(u) is the excluded volume interaction potential, e.g. in the Onsager form:

^Ev {Ji) = 2cdL^ksTJi/r(u)smU,u\du^

(3)

(see Section III below). The integrals in (2) and (3) are performed over the surface of a unit sphere the radial vectors of which form the configuration (in this

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361

a case, orientation) space. The operator -r—is the gradient operator on this ou manifold:

^u

L^= — ^ - J - ' d e -"" sine dtp

S the unit second order tensor and S^ and S^ are the polar and azimuthal unit vectors in spherical coordinates^. A way to solve the rather complex PDE (1) is to factor and expand the pdf in the eigenfimctions of the Laplacian operator in spherical coordinates, i.e. in spherical harmonics^' ^:

¥(.u,t) = 'Z'Zb,„(t)Y,„(u)

(5)

/=0 m=-l even

A weak formulation of Eq. (1) is obtained by truncating the expansion at a level dictated by the desired numeral accuracy and subsequently applying a Galerkin scheme. Typically, n is of 0(10) and the resulting set of ordinary differential equations, although rather cumbersome, can be integrated numerically with moderate effort. 2. Integration of a large number (an ensemble) of individual trajectories U^(t) for the unit vector defining the orientation of single LCP molecules*. The evolution of individual molecules is described by means of a stochastic differential equation (SDE) for the time evolution of the Markovian process U_(t). If properly constructed, the evolution of an ensemble of trajectories is in exact correspondence with the evolution of the pdf as described by (1). Integration of the SDE associated with Doi's model can be accompUshed by the simple algorithm^:

Uj+mj)At + ^y..

(2DMjWfWj

=

Uj + mj)At + (2DMjWr

^J (6)

* C/(f) is used instead of w(f) to emphasize the character of stochastic process of the fonner and distinguish itfromthe latter.

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where C/^ = Ujij^t). This algorithm guarantees that the constraint C/^ = 1 is fulfilled at all times and for allfinitetime steps. It does not require application of the transverse projector operator ^-C/C/nor of the transverse gradient

.

The simplicity of this algorithm is in stark contrast with the substantial complication of the numerical schemes used to integrate (1). However, it must be borne in mind that the ensemble approach yields "noisy" results, as a consequence of the stochastic term W_j, the components of which are three random numbers sampled fi-om a Gaussian distribution of mean zero and variance unity^, and as a consequence offiniteensemble size. It can therefore be computationally very expensive to obtain high accuracy solutions, since the amplitude of noise, i.e. the error bar inherent in any average quantity computed over the ensemble, decreases with the square root of the number of trajectories. Both levels of description can be used to compute macroscopic values of an arbitrary quantity A{u):

\A(u)yr{u,t^)du' =^Y,A{Uf)

(7)

While the ensemble approach is not suited to obtain the full pdf l/^iuj) with reasonable accuracy, it is however a very powerful alternative for the calculation of moments of l//^(u,t) or of general average expressions like Eq. (7), since they can be computed satisfactorily even in configuration spaces of very high dimensionality. Throughout the previous discussion of the two main alternatives, the velocity gradient was considered to be a given, spatially constant, at most time-varying, magnitude, i.e. the velocity gradient field was assumed to be spatially homogeneous. Inhomogeneous flows are clearly important, since most real4ife flow problems belong in this class. Over the last 15 years, both of the above techniques have started to be used to solve inhomogeneousflowproblems in combination with computational fluid dynamics methods.^^ In addition, the numerical values of the parameters appearing in (1), namely the rotary diffusivity and the strength parameter of the excluded volume interaction KV(M) 9 ^^ either set to values within given ranges in parametric studies, or estimated by straightforward arguments.

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The present investigation is a modest attempt to advance the current state-of multiscale approaches to flow calculations of LCPs. Only very few complex, i.e. nonhomogeneous, flow calculations for models such as Doi's have been performed up to date^^' ^^. In some cases, the kinematics were decoupled from the molecule dynamics, i.e. although spatially non-homogeneous, the velocity field was obtained non-consistently from a simpler constitutive law, like in the work by Grosso et al.^, where Doi's equation was solved in given Newtonian kinematics in a largeeccentricity journal-bearing geometry. More recently. Lattice Boltzmann methods have also been used in conjunction with the LCP model of Beris and Edwards^^"^^. In the first part of this work, coarse-graining is applied to obtain one of the parameters in Doi's equation from a detailed atomistic model of the liquid-crystal forming polymer poly-(«-propyl isocyanate) [-CO-N(C3H7)]n (PPIC) dissolved in toluene. II. Atomistic-level (Level 1 and 2) description of poly-(/f-propyl isocyanate) (PPIC) The lowest, most detailed, level of description considered in this work is atomistic (we will refer to it as Level 1 in the following). The force field of Amber^^"^^ was used at this Level 1 to obtain the most stable polymer configuration for single chains in vacuo and then in solution. To this end, a single PPIC containing 40 structural repeat units [-CO-N(C3H7)] and terminated with capping H atoms W2is simulated infiiUexplicit detail both in vacuo and immersed in a solvent of toluene (CyHg) molecules also represented fiiUy explicitly under periodic boundary conditions. In the calculations in a solvent, the number of toluene molecules was chosen so that, after subtracting the helix volume (more precisely, the volume of the Connolly surface^^ of the helix for a probe sphere of radius 0.38 nm, representative of a toluene molecule) from the total volume of the simulation box, the density of toluene in this remaining volume matched the experimental macroscopic value at r = 298K^\ Atom naming convention, definitions of bond lengths, bond angles and torsion angles for the helix backbone and side propyl groups were as follows:

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h N

I.

N..

R

R= 7,3

Figure 1. Naming convention for atoms and bond lengths in PPIC backbone and side group

' r," H^^CS>H''

a,

"n

N

N

R

R

R

R

0,, "' Figure 2. Bond angle definition for PPIC backbone and side group

^i?\ N///L/

N \J

N

N

N

R

R

R

R

'^''^

-^t?, Z,

-i3 ^CH^

C H2

Figure 3. Torsion angle definition for PPIC backbone and side groups

Hierarchical approach to flow calculations for polymeric liquid crystals

365

The simulation box was a right-angle parallelepiped of 8.1 by 1.5 by 1.5 nm. The PPIC center of mass was placed in the geometrical center of the box, the helix axis was aligned with the long dimension of the box and it was ensured in all cases that interactions with periodic copies of the central chain were absent. Time integration was performed by means of a velocity Veriet MD algorithm^^. Temperature was set at 298 K and a Nose-Hoover thermostat^^"^"^ was used to keep temperature at this set value. Electrostatic contributions to the force field were computed by considering the helix environment as a continuum with a constant value of the dielectric constant. The values of8 = 1 and € = 2A were used, as representative of the vacuum and the nonpolar solvent. Partial electrostatic charges were found to be important for backbone atoms, where highly polar moieties reside. Side propyl branches can be considered electrostatically neutral for all practical purposes. Starting configurations for the PPIC chain were constructed following the parameters of Lukasheva et al.^^, which corresponds to a left-handed Natta-Corradini 8/3 helix. Starting from this helix conformation, the system was integrated for a total run length of 12 ps. During the run, helix conformation was monitored by following the length of the head-to-tail vector, which reacts very sensitively to conformational (torsional) changes in the polymer backbone. Side group conformation was similarly monitored by following the length the vector joining the side chain-helix attachment point with the center of the last C atom in the propyl group, i.e. the vector from backbone nitrogen atom A^^ to carbon atom C^ 3. After a short transient of approx. 3.4 ps, both the helix and the side group conformations were found to reach a steady average state with low amplitude oscillations around these most probable values. This steady state was characterized by a stable 8/3 helix, virtually undistinguishable from the starting structure (which was taken from the in vacuo runs) and by almost fully extended side propyl groups. The rapid convergence to a stable conformation very similar to that obtained in Refs.^^"^^ using the PCFF force field in the absence of solvent, and also very close to experimentally determined structures'^. References ^°"^' suggests that rigid helical conformations are indeed a robust characteristic of PPIC chains. This is also in agreement with previous work'^ and with the estimated persistence length of PPIC.

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Figure 4. Stick perspective representation of PPIC helix backbone in most probably confonnation; 8 residues (3 helix turns) and only one propyl side group is also represented. Solvent molecules not represented. Left, helix axis contained in plane of p^er. Right, helix axis perpendicular to plane of paper. Blue segments correspond to N atoms, grey to C, white to H. Observe tight winding of 8/3 helix and ahnost fully extended configuration of side propyl group.

Figure 5. Two views of the same Augment of the PPIC helix as in previous figure; left and right images are taken after performing a 90° rotation around the helix axis. Atoms are represented as spheres with van der Waals radii. Helix backbone is very effectively shielded fi-om solvent molecules by side propyl groups.

The first-level calculation just described, although incorporating a single PPIC 40mer chain, requires a significant amount of computation. Moving on to a semi-

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dilute or concentrated system using Level 1 description would require consideration of a large, multi-chain system, each of them comprising (for a degree of polymerization of 40) 520 explicit atoms plus additional explicit solvent molecules, comprising 15 atoms each. This MD calculation, although feasible with the computational resources available today; would however be impossible to extend to the time scale required to observe Einstenian rotary diffusion regime. Therefore, a direct attempt to observe typical multi-chain, cooperative behaviour and transitions in LCP's remains a major computational challenge for present-day hardware. However, the high rigidity of the helix in PPIC makes it amenable to an alternative approach where a moderate simplification (coarse-graining) of the full atomistic detail is performed. In this second stage, advantage was taken of the following observations collected during the MD run: • the PPIC backbone torsion and bending angles, and bond lengths, remained virtually unchanged during the entire run, with only very small fluctuations around their most probable values. The following table summarizes the essential helix parameters. For torsional angles, mean and standard deviation of the mean are given: («)

(r,>

UM)

fc)

n residues per turn

-161.3° ±1.3°

37.5° ±0.9°

-98.1° ±3.9°

175.8° ±5.5°

2.68

Po helix angle per residue 132°

1

T helix advance per residue .201 nm

furthermore, changes in torsional angles in side propyl changes were also very minor (see previous table) and the last (2nd) torsional angle (Zn)

^^ ^^

average very close to trans. by performing short MD bursts branching off from the main conformational trajectory and with a force field modified in its electrostatic contribution (partial charges of the side groups set to zero), it was observed that the electrostatic contributions to the force field have a very direct effect on helix stability but none whatsoever on side propyl group conformation nor on inter-chain interactions.

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Based on these observations, a simplified yet, for our purposes, faithful representation of PPIC chains was implemented (called Level 2 in the following). In the Level 2 representation: • explicit hydrogen atoms are removed and methyl (-CH3) and methylene (-CH2-) groups are described as single entities or united atoms, as is current practice in many atomistic polymer simulations. • explicit toluene molecules were similarly represented by united atoms, namely aromatic -CH=, -CH2- and -CH3 groups. United atom parameters were taken from the Amber united-atom force field^^ and from the work of Cross and • except for the deletion of C-H bonds as a consequence of the use of pseudoatoms, PPIC helix geometry was kept in its full detail, but bond lengths, bond angles and torsion angles were frozen at their most probable values, as collected during the MD run. This "freezing" of conformational degrees of freedom was performed for all bond lengths, all bond angles and all torsional angles in the helix, while the torsional angles Xi 1 ^^^ Xt 2 ^^^^ allowed to vary. The rationale behind this simplification is that, for the purpose of studying the interaction of helices which are known or have been shown to be very rigid, the only relevant feature of the internal structure of the helix is itsrigidity.Since uiterhelix interaction is mostly short-rage steric (excluded-volume) and long-range electrostatics plays no role, it suffices to keep an explicit representation of those helix atoms which reside in an outermost shell. In Level 2, helices consist of a rigid core and pendant propyl groups with torsional degrees offreedom.Helices interact with one another and with the neutral toluene molecules via the short range LJ potential. This coarse-graining makes physical sense and leads to a significant saving of computational effort. Along the polymer backbone, only two types of atoms are present: nitrogen and carbon. Hence, for a polymer consisting of n repeat units, 6w degrees of freedom were necessary to specify the precise conformation of its backbone. A possible set of such coordinates is the set of all Cartesian coordinates of its constituent atoms. Alternatively, the coordinates of one atom of the backbone considered as chain origin (3 d.o.f), the Euler angles defining the orientation of a bond in the backbone (3 d.o.f.), 2 n ~ 3 torsional angles, 2n-\ bond lengths and In-2 bond angles, in all, 6n d.o.f. By employing the assumption of rigidity of the main chain the number of degrees offreedomwas reduced byfixingthe values of 2« - 3 torsional angles and 2n-2 bond angles, i.e. a total of 4 « - 5 holonomic constraints. These constraints were imposed by a minimum triangulation scheme that ensures helix rigidity by fulfilling the following set of equalities:

Hierarchical approach to flow calculations for polymeric liquid crystals

{nq - Lq^^) ~ ^1 = 0

(« - 1 constraints)

{LN, - rj^^ ^) - ATj = 0

(« - 1 constraints)

(?:Q " Ij^,,, f-K,=0

(« - 1 constraints)

(re, - ?:i^,,)' - ^4 = 0

369

^^^

(« - 2 constraints)

toto/: 4« - 5 constraints where the K 's were obtainedfromthe single-chain MD run. Anderson's adaptation of the SHAKE algorithm^^ to velocity Verlet^^' ^"^ was used to satisfy (8).

(Lcrw)-K.=^ Figure 6. Schematic representation of backbone constraints required to impose helix rigidity.

III. Level 3: Coarse-grained molecular description via Doi*s model The task of coarse-graining or projecting a modelfroma given level to another with many fewer degrees offreedomis in general non-univocal, since it depends on the specific aspects of the material which are of interest at the coarse-grained level. In the present framework of Doi's LCP model, the entire atomistically detailed representations of PPIC (Levels 1 and 2) need to be reduced to the two scalar parameters appearing in the constitutive equation to be used for viscoelastic flow calculations, namely: 1. The strength parameter appearing in the excluded volume interaction potential F^y(M). Determining the interchain interactionfromLevel 1 and 2 MD runs is probably at the limit of feasibility today, even when resorting to major

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computational capabilities. Thennodynamic consistency requires a careful analysis of the from of the friction matrix in theframeworkof GENERIC in analogy to (8.169) of Ref ^^, which is applicable to reptation models. Strictly speaking, thermodynamic consistency also implies that the Level 3 of description should incorporate the orientation distribution fiinction ^(M,^) directly and not any strength parameter. This path was however not explored in the scope of the present work. Hence, and although not strictly self-consistent, the calculation of the effective excluded-volume potential was made according to the widely used Onsager form"^: VEV = K (H) = '^cdl^k^T ^yfiu) sin(w, u)du

(9)

where d is the rod diameter, L its length and c is the number concentration of rods in solution. Values of d = 0.79 nm and L = 6.8 nm were readily extracted from Level 1 simulations. This latter value should be compared with experimental values (=200-300 nm) of the persistence length of PPIC in nonpolar solvent solution ^' ^^' ^°. 2. the rotary diffusivity in a hypothetical isotropic solution of molecules at the given concentration, appearing in (2) This much rougher level of description will be referred to as Level 3. Although at atomistic levels 1 and 2, the nature of the interaction between polymer helices can only be energetic, via the force field and mediated by solvent molecules as well, at Level 3, interchain interaction is of entropic nature and the explicit solvent is absent. Regarding the second point above, the rotary diffusivity that characterizes the rotational Brownian motion of the head-to-tail vector w, is illustrated in the following figure:

Figure 7. Rotational Brownian motion of LCP head-to-tail vector.

Hierarchical approach to flow calculations for polymeric liquid crystals

371

For short times, in the sense that D/«l, the random motion of wean be regarded as Brownian motion of the tip of the head-to-tail vector (scaled to unity) on a two-dimensional flat surface"^. The mean square displacement of the unit vector u(t) in a time t can be written as: {{u(t)-U(0)f) = 4D^t

(10)

where D^ is the rotary diffusivity that appears in Eq. (2). Since the units of this rotary diffusivity are inverse time, this rotary diffusivity is frequently employed to make time dimensionless by introducing the relaxation time T^ = D~^ and t =t/T^=

tDj.. The same factor is also employed to render the strain rate

dimensionless: T = Yr^ = ylD^. In most studies published to date, the actual numerical value of D^ comes either i) from an experimental measurement in solution, for example via the Miesowicz viscosity, ii) from an estimation based on geometric factors and solvent viscosity, or iii) in parametric studies, numerical values are selected from intervals often chosen because striking changes in dynamical behaviour take place within them. In this work, however, we have followed a hierarchical route in which MD results from Levels 1 and 2 are used to compute D^, according to (10). Unlike in the atomistically detailed simulations at Levels 1 and 2, the extraction of the rotary diffusivity must be done in a multi-particle setting, i.e. on a system appreciably larger that that needed for the single-rod calculation. To this end, a large atomistically detailed. Level 2 system was prepared containing 57 identical PPIC molecules, each of them containing 40 residues and capped by terminal hydrogens. They were placed in a cubic simulation cell of 8 nm edge, together with 1824 toluene molecules, comprising a total number of 26471 pseudoatoms. Initial polymer helix configurations were generated in the absence of solvent molecules by uniformly sampling space for positioning helix origins, and uniformly sampling the unit sphere in order to define the orientation of the helix axis. When rod placements using this uniform sampling scheme led to unrealistic rod-rod overlaps, the latest trial rod was discarded and a new attempt performed. Since the rod volume fraction is not excessively high (see below), this simple procedure was more than adequate. Rod-rod overlaps were detected using the rod dimensions estimated at Level 1, namely rod length L = 6.8 nm and rod diameter b = 0.79 nm. From these values, the volume of individual PPIC molecules, considered as cylinders, was 3.33x10"^^m^, so that the volumefractionoccupied by the rods was 0.37 and the numeric volumetric concentration was c = l.11x10^*. This value can be compared

372

M. Laso et al.

with the volume concentration at which the isotropic phase is estimated to become unstable: -'-

^^ =1.39x10^^ TtdU

so the system under consideration was at the boundary between semi-dilute and concentrated. In terms of c , the concentration of the system was c = 0.79. Once polymer molecules were placed in the simulation cell, solvent molecules were introduced one by one using a similar brute force scheme in a first stage. As the filling procedure progressed and density increased, overlaps were progressively more frequent, since, at liquid-like densities, brute force insertion of a molecule as large as toluene had a very low probability. As total system density reached 80% of the final one /? = 1174 kg/m^, insertions of toluene molecules were complemented by a van der Walls radius staged inflation^^ which alleviated major overlaps. Initial structure preparation was finalized by energy relaxation via simple MetropoUs Monte Carlo and by subsequently performing a full-scale equilibration MD run of 10 ps duration. An integration time step of 1.9 fs was used. In order to extract the value of the rotary diffusivity in the isotropic phase, an isothermal production MD run was carried out to a total of 5 ns. During the production run, helix orientation and center of mass trajectories were stored periodically for all chains in the system. Due to the high packing density, helix (rod) mobility was severely hindered. From the two plots included in the next figure (where the 1824 toluene molecules have been omitted for clarity) it is possible to judge the considerable packing density of PPIC helices in the isotropic solution:

Hierarchical approach to flow calculations for polymeric liquid crystals

373

Figure 8. Dense PPIC in toluene system. All 57 helices have been represented by semi-transparent cylinders (dots placed on the surface of cylinders) of diameter fe = 0.79 nm and length Z = 6.8 nm. In the left figure, cube edge has been set to 12 nm for representation only and periodic boundary conditions have been suppressed to improve visibility. Intiierightfigure,PPIC rods are folded back into the simulation cell by means of periodic boundary conditions, and cell edge has been set to 8.0 nm. Solvent (toluene) molecules not represented.

Although helix center of mass difiusion had not progressed beyond the rod length, i.e. ((?:com.(0~?!c.o.m.(0)) )
The main result of this extensive

calculation is presented in the following two figures (in linear and log scales respectively), which show Einstenian diffusive behavior of the unit head-to-tail vector at long times. Although it may seem that even longer MD runs would lead to better approximations to Z)^, a key result from the GENERIC formalism of nonequilibrium thermodynamics indicates that the MD run is not only adequate, but should not be made any longer^^.

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{{m-mf) 0

time (s)

0.01

MO

((«(0-w(0))')

MO

MO MO

1 lO"^^

1 10^

MO'

time (s) Figure 9. Diffusion on unit surface of the tail-to head vector, averaged over all PPIC polymer rods in the system.

The value of the rotary diffusivity obtainedfroma least squares fit of the long-time behaviour o f / ( w ( 0 - w ( 0 ) f \ turns out to be Z)^ =12.7x10^ (±3.1) s\

Hierarchical approach to flow calculations for polymeric liquid crystals

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IV. Viscoelastic flow calculations using CONNFFESSIT for Doi*s constitutive equation At this stage, all parameters required to perform complex viscoelastic flow calculations using CONNFFESSIT^^ are available. These parameters are the rotary diffusivity, the temperature and the geometric parameters of the rods (length and diameter). Furthermore, the use of CONNFFESSIT makes it possible to use either an average rotary diffusivity (averaged over the instantaneous molecular orientation distribution) or to apply an orientation-dependent rotary diffusivity. CONNFFESSIT was used to perform a viscoelastic flow calculation in a complex geometry, namely a 3:1 (ratio of tube radii) or 9:1 (ratio of tube cross sectional areas) axisymmetrical contraction. The calculation was 3D and did not exploit the axial symmetry^. The objective of the calculation in the contraction geometry, was to solve the equations of mass and momentum conservation for the incompressible and isothermal flow of the PPIC solution in toluene, whose rheological behavior is described by Doi's model. The conservation equations for mass and momentum to be solved are: y ^ ^ + y^v Yv + V p - V . ^ + 7 7 , ( V v - f ( V v f ) = 0

^^^^

V.v = 0 The stress is seen to contain a Newtonian contribution, due to the toluene solvent, and a non-Newtonian contribution ^ arismg jfrom the presence of polymer molecules, PPIC rigid helices or rods in the present case. Unlike in purely macroscopic formulations^^, in CONNFFESSIT the polymer contribution to the rheological behavior of the complex fluid is obtained as a function of the local, instantaneous velocity gradient and of the instantaneous configurations of the members (coarse-grained molecules) of ensembles of unit vectors which obey the dynamics (6). In addition, a rule for the calculation of the stress is required, which for Doi's model consists of a Brownian contribution and an excluded volume contribution which can be written in terms of the configurational probability distribution function"^:

** the reason for perforaiing what is essentially a 2D calculation in a 3D setting is that for higher values of the strain rate, three dimensional behaviour can emerge even if the initial and boundary conditions are axisymmetric, due to out-of-plane attractors for U .

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ck^T

uu — S \-—z^d^u 3~Jdu ou

=P

=P

=P

(12)

=P

ck^T =P

i'^-l^]Tu^i

^ EV

d'u

In Eq. (12) the integral operation L. d^u is equivalent to J... smOdOdg) with integration ranges

O<0<7r,O<(p<27r^.

The basic idea of CONNFFESSIT is to use a discretization method, such as finite elements, to solve the mass and momentum conservation equations while the polymer contribution to the stress is obtained by averaging the individual contributions of a great many molecules and not from a constitutive equation^^"^^. These molecules are entrained by the fluid much as it happens in real flow situations. In addition to their macroscopic degrees of freedom (spatial coordinates), they possess internal degrees offreedom,those of w, the time evolution of which is governed by the stochastic differential equation (6). The integration domain for the conservation equations and the stochastic version of the constitutive equation was the 3:1 axisymmetric contraction flow. The flow in this geometry is essentially axisymmetric Poiseuille flow far away upstream and downstreamfromthe contraction, but has a significant extensional component close to the contraction itself At high values of the strain rate, off-plane attractors for the director lead to loss of axial symmetry. Although the calculations to be presented were performed in a fiill 3D fashion, this regime W2is not explored in the present work. The domain was discretized by means of 17453 vertices in 91834 tetrahedra. The resulting mesh is shown in the following figure.

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0.45-1

0.45

0.4 0.35 0.35

0.3

0.25 0.25 0.2

0.15

0.15 0.05

0.05 0.02

0.02

0.02 -0.02

Figure 10. Finite Element mesh for the 3:1 axisymmetric contraction. The mesh contains 17453 vertices and 91834 tetrahedra. Dimensions in 1 0

m.

The mesh is refined around the abrupt contraction, as can be observed in the cutaway 3D view of the mesh above.

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.02

-0.01

0

0.01

0.02

0.03

Figure 11. Refinement of the Finite Element mesh close to the contraction. The view on the left is a transversal section at Z = 0 . 1 5 . The section on therightcontains the symmetry axis. Dimensions in

10"'m. In spite of this local refinement, the mesh is still relatively coarse compared with some of the meshes used in macroscopic calculations. Our relatively large element size is dictated primarily by available computational resources. It is also important to keep in mind that, to ttie best of our knowledge, this is the first time a 3D CONNFFESSIT calculation is performed (3D Brownian Configuration Fields"*^"^^ had been already done by Ramirez and Laso^^). Although conceptually identical to 2D calculations, complex flow 3D problems are computationally at the edge of feasibility with present day hardware. To give an idea of the size of the numerical task at hand, the integration domain contained a total of 67108864 (2^^) "molecules", i.e. C/ processes, uniformly distributed over the 91834 tetrahedra in which the domain was partitioned. This high number of individual trajectories was required in order to keep the statistical uncertainty in the calculation of the polymer contribution to the stress acceptably low. This is a particularly important concern in Doi's model, since the fmite size of the ensemble has a "noise" effect on the determination of the instantaneous value of the rotary diffusivity according to Eq. (2) and of the stresses^. Initial director vectors for the PPIC rods ("molecules") were chosen to be isotropically distributed in the wide part of the contraction flow and homeotropically distributed in the narrow part of the channel. Boundary conditions at the inlet were parabolic velocity profile and isotropic director distribution at all locations. The inlet flowrate was set at 1.94 10"^^ mVs (193 pl/s). The characteristic

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dimensions of the contraction and the flowrate were chosen as typical of those used in recently developed devices for fluid transport using nanoelectromechanical actuators'*^. Instead of a reference time based on the rotary diffusivity, an apparatus-related reference time scale was defined as the transit time through the domain for a molecule along the cemterline and with a constant velocity set at the average inlet value, which leads to f^^^ =1.81 10^ s. Calculation times and shear rates were made dimensionless using t^^j-. The inlet radius was taken as the reference length L^^f . The dimensionless rotary diffusivity was then D^ = 0.23 (all variables in the following are understood to be dimensionaless, except where units appear explicitly). Finally a mappingfromthe Onsager expression for the excluded volume to the Maier-Saupe excluded volume function (see below) based on a simple fit of the integral of the force acting on the atomistic (Level 2) PPIC rods to the Maier-Saupe potential for a perfect parallel alligment, which resulted in a 4 dimensionless Maier-Saupe interaction potential of U = \4,5—cdL^ = 7.21 Selected individual particle trajectories are shown in the next figure. Each individual curve corresponds to the projection of the three-dimensional trajectory of one of the 2^^ molecules on the plane defined by the symmetry axis and the starting point of the trajectory on the inflow (leftmost) boundary. The effect of statistical noise on the resulting velocity field is clearly seen as wiggles in what should otherwise be smooth trajectories. This effect is however magnified in the upper figure by the use of different scale factors in the axial and radial axes. In the lower part of thefigurea different set of trajectories is depicted and both axes are identically scaled so that the length-width aspect ratio is the real one. In the upper part of the figure it can also be observed that some particle trajectories cross the symmetry axis of the domain and also present loops. Both of these effects are a consequence of the projection of 3D trajectories on the plane of the figure (although the calculation is fully 3D and does not exploit the symmetry of the domain, in this figure, axis are shown as those of cylindrical coordinates r and z).

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Figure 12. Representative individual particle trajectories. Lines are projections of 3D trajectories on the plane defined by the symmetry axis of the domain and the starting point of the trajectory. Loops and axis crossings appear for some trajectories as a consequence of the planar projection.

The field of the velocity modulus V = ^vf ^Vj+Vj^ and the pressure field are presented as a shaded surface plots in the next two figures. In both figures, two snapshots at two different times have been overlapped so that the magnitude of fluctuations due to finite ensemble size can be appreciated. Spatial fluctuations at fixed time are visible as differences in shades of gray either at the upper or at the lower half of the plot.

H0.O:

0,45

^{Wm) Figure 13. Field of velocity modulus ( F = ^ v f + V2 + V3 ) represented as shaded surface plot. Upper and lower halves correspond to two different times t = 25 and f = 3 0 , both at steady state, in order to show the magnitude of spatial and temporal fluctuations in the velocity field. Molecule trajectories from previous figure have been overlaid.

381

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0.45

^(10"'^) Figure 14. Pressure (p)

field represented as shaded surface plot. Upper and lower halves correspond

to two different times ^ = 2 5 and ^ = 3 0 , both at steady state, in order to show the magnitude of spatial and temporal fluctuations in the pressure field.

Temporalfluctuationsare responsible for differences in shading between upper and lower parts of the plot (which correspond to two different times). Finally, the director field at ^ = 10 is represented as an arrow plot in the next figure. In order to make the visualization of the full 3D field of director vectors as clear as possible, first azimuthal averaging over the entire [0,2;r] range was performed in order to obtain a 2D field, which was then smoothed by projection, using a coarse two-dimensional mesh based on quadrangles: if thefimctionalspace for u = (C/) is denoted F^;, find w G F^ such that for all seFjj we have:

'-'^-irM'-' t 7=1

where N^ is the number of trajectories used to build the azimuthal average, and the parentheses imply multiplication by the test fimction ^ and integration over the domain. For the approximation space F^ discontinuous bilinear polynomials ( Q ) were used. The initially isotropic orientational distribution evolves in the course of the flow development. At ? = 10 it has givenriseto a complex spatial organization in which axially oriented domains seem to be dominant. An m-depth analysis of the dynamic evolution of textures and domains is currently being pursued but will not be dealt with here smce the purpose of the present work is to illustrate the scale crossing methodology only and not to investigate texture evolution.

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r (10"'m)

Z(10 m) Figure 15. Director field represented at ^ = 10. The area represented is a close-up of the zone around the contraction.

V. Level 4: Orientational distribution function for Doi*s model via reduced order modelling Although powerful and of general validity, the CONNFFESSIT approach applied in the previous section is very computationally intensive, especially for 3D problems. A radically different approach to the calculation of the orientational probability distribution function ^ ( w , 0 will be employed here. It serves as an independent cross check, and also as the last and most macroscopic level (Level 4) in tiie hierarchical chain of description of liquid crystal polymers under flow. It is worth recalling that starting from a very detailed, atomistic model of individual polymeric helical chains for PPIC, here we reach the level of description of configurational distribution functions; i.e. even the concept of a highly simplified, coarse-grained stochastic model based on head-to-tail helix vector u is superseded by that of a distribution of orientations of such molecules. The discretization methodology to be applied here is based on a weak formulation of the conservation and constitutive equations, but employs the concept of model reduction^^. Its basis has been described in detail"*^* ^^ and appUed to homogeneous flows, i.e. to predicting the evolution of ^(w,^)in a given, spatially uniform velocity gradient field. The key idea behind the reduction method is the use of a

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reduced set of basis functions with wide support, such that they constitute the optimal choice for representing the solution at the desired level of detail It operates extracting automatically, and in a way completely transparent for the user, the most relevant information of the unknown solution for constructing the functional approximation to the macroscopic fields from the information just extracted. The new shape functions (the most characteristic functions related to the model solution) have a large support, i.e. they are defmed in the whole domam in an appropriate manner (see below). Thus, the number of degrees of freedom involved in the solution of the Fokker-Planck equation is very significantly reduced. The construction of those new approximation functions is done with an 'a priori' approach, which combines a basis reduction (using the Karhunen-Loeve decomposition) with a basis enrichment based on the use of some Krylov subspaces'^^'^V In order to show the capabilities of the reduction method, we have taken the kinematic results (velocity and velocity gradient fields) of the viscoelastic flow calculation obtained by means of CONNFFESSIT along a selected streamline. Therefore, in the following, the flow kinematics and the microstructure evolution are fully uncoupled, i.e. the PPIC molecules, described by means of u in Level 3 and by means of y^{u,t) at the present Level 4, will evolve in a known velocity field which is time and, implicitly, spatially dependent, the known dependence being given by the Lagrangian history of flow along a streamline selectedfi-omthe solutions to the viscoelastic flow problem computed in Section IV. To this end, the velocity gradient history entering the FP equation along a streamline that starts at r = 0.025 (which corresponds to the uppermost or lowermost streamlines plotted in the figures in Section IV) was extracted in the following way: the 3D velocity field was averaged azimuthally over the entire [O, ITT] range in order to obtain a 2D field, which was then smoothed by projection, in the same way as it was done for the director field. This reason for doing this projection was that although the calculation was entirely 3D in Cartesian coordinates, all fields remained, except for fluctuations, essentially axisymmetric. Hence, the statistical noise in the solution can be greatly reduced by averaging. For this reason, in the remainder of this Section, the variables r and z will appear m labels, subindices, etc. But it should be kept in mind that the calculation was not performed under the assumption of axial symmetry. Only after projection and smoothing were the variables r and z introduced. The evolution of the velocity gradient, given by its shear and extensional components along the streamline, is shown in the following two figures, for shear and for extension respectively.

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In both of these figures, the origin of time is assigned to the instant when the trajectory reaches the surface that separates the wide and the narrow parts of the domain, i.e. when it reaches z = 0.15. Thus, negative times correspond to flow in the wide cylindrical part of the domain, positive times to flow in the narrow part. Hence, the flow along this streamline contains both shear and elongational components in varying proportions. Roughly speaking, there is no appreciable shear until the streamline reaches the narrow portion of the contraction, whereupon shear reaches an almost constant value.

Figure 16. Temporal evolution of the shear component of the velocity gradient along a streamline entering the domain at j ; = 0.025. Negative times correspond toflowin the wide cylindrical part of the domain; positive times to flow in the narrow part (see text).

Elongation however is mostly confined to the part of the flow where the cross section drops abruptly. There is no appreciable extensional component far from this region. Notice also the rather noisy curves for both components of the strain rate tensor.

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-10.0

Figure 17. Temporal evolution of the elongational component of the velocity gradient along a streamline entering the domain aX y = 0.025. Negative times correspond to flow in the wide cylindrical part of the domain; positive times to flow in the narrow part (see text).

The rate of strain tensor to be used as the input in the calculation of the evolution of ^(w, t) will therefore have the form:

0

rJt)]

0

-4(0

0

rJO

0

f.(0

-e^it)

r=

(13)

where the time dependences come from the CONNFFESSIT calculation as tabular data and are represented in the two previous figures (r,z labels are used because azimuthal averaging has been performed; calculation and unsmoothed strain rates are of course Cartesian 3D).

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V.l. Discretization of the Fokker-Planck equation In order to solve Eq. (1) for given Vv, a natural discretization of this problem defined on the unit surface is based on the use of the spherical polar and azimuthal angles ( ^ , ^ ) : Tcos^cos^^ w = sin^cos^

(14)

sin^ where both angular coordinates are defined in: (p^O^ [0,2;r[x[05^]. However singularities are encountered at the ends of the 6 definition interval, because of the

Dr ^3 V ; 2 ^^ ^® P^'^ expression of the Fokker-Planck equation. Moreover,

term . 2

this kind of discretization requires an explicit imposition of the periodicity condition. Thus, it was chosen to work in Cartesian coordinates, where the surface of the unit sphere is approximated by a set of planar triangular facets. The nodal coordinates of the vertices of these triangles are given by: fx^ (15)

w.= \^ij

which satisfy ^jcf+>^f+zf =1, and where the natural periodicity is implicitly verified. Integrating Eq. (1) along the streamlines where the kinematics history, that is v(^), is known, that equation can be rewritten in the following form that only involves the conformation (orientation) coordinates and time:

dy/

dy/ _ a y

dy/ ^.

^ + £o(?^)i^ + £ i ( « ) ^ - A T H T + ^o(?^.^)?^ + ^ i ( « . ^ ) ^ = 0 du ~ du dt du

where y^(t,u) = l/r(x(t),t,u)

and x(t) = x(t = 0)+ \v(t')dt\

(16)

The two scalar

0

functions EQ(U), HQ(U,S) as well as the two vector fields E^(u), Kiin^^) ^ ^ ^^ easily deduced from the equation (1). We denote by Q, the domain in configuration space where equation (16) is defined, namely the two dimensional manifold spanned by the unit orientation vector w, i.e. the surface of the unit sphere.

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3 87

First, the problem is formulated in the Finite Elementframeworkusing a weighting function j / * :

-lrD,^du+

lrHo(li,S)V
= (i

The computational domain H (in configuration space) is partitioned into a collection of non-overlapping finite elements (triangular facets covering the surface of the unit sphere). A linear and continuous interpolation of the distribution fimction is then built in each triangle, form which:

r(u)=j;^N,(u)w: 3

and yr^*iu) = ^N^(u)i/r^

where ^/ and i/rf are the values at node i of ^

and ^* respectively, and N^(u) is the associated shape function which takes a unit value at the node / vanishing at the other nodal positions. Integrating equation (17) by parts and taking into account that the configuration space for u is unbounded, we obtain: LV"*^^^^ lv^%(u)Wdu+

l'V*Eliu)^du

Due to the advection-diffusion character of that equation, an appropriate stabilization of the Finite Element scheme is needed to avoid numerical instabilities induced by the convection term. Stabilization is achieved by using a SUPG formulation, which modifies the weighting functions related to the advection term —* ^ as described later. The stabilized variational formulation results in:

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3

where in each element 'y/ (w) = ^ NiQi)wf with NXu) = N^u) + ^

^^^-^

where h is the characteristic element size and J3 is the upwinding parameter given by Pe where the Peclet number Pe is given by ^_max(||£,fe)||,||afe,^)||);r 2A In order to construct EQ(u)Qnd HQ(U,S) the differential operators must be projected on the plane tangent to the unit sphere. The required gradient projection is as follows: if we consider a triangular facet defined by its three vertices: ixi,yj,zj), (x2,y2,Z2% (x3,y3,Z3) reference coordinates (^,r|) can be defined using an isoparametric geometrical interpolation:

We denote by ej and e^ two vectors defined in the plane containing the triangle, which can be expressed as:

Hierarchical approach to flow calculations for polymeric liquid crystals

/

389

(

3 X^,x,

fdx] 1

rax]

9# \dy e,=

e, =

r^dz U^J

ay \drj \dz_

r=l

9^ \

The relation between the gradient in the Cartesian coordinates and the one expressed on the manifold tangent to the unit sphere is given by^^:

3x 3y^ = Ul

^2)

r.^ ?1 ?1 r

.T". V ?1 ^2 T

V^2^1

^2^2y

y 3z

where

r=z^,i^'^)¥: i=l

and the matrix

^1 §.\

§.\ §.1

V^2^1

^2^2 J

contains the elements of the metric tensor g

corresponding to a given triangular facet. Note that the required properties of differentiability, symmetry, non-singularity and positive-definiteness are satisfied by construction in Cartesian coordinates. As it is usual in the FEMframework,after numerical integration and assembling of element contributions the following non-linear system is obtained:

The non-linearity follows from the dependence of the nematic potential on the average of the distribution fimction itself on the unit sphere. When one uses an explicit time discretization technique, the non-linear system can be linearized by considering the nematic potential calculated at the previous time step.

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V.2. Model Reduction In this section the ideas described by Ammar et al."*^ are applied to the discrete problem resulting from the discretization of the Fokker-Planck equation that governs the evolution of the configuration distribution. We consider that the probability distribution have been accurately described in the time interval [0,f^]. We assume that at time tp > t^ the reduced approximation («) basis is given by B . The /, j -component of B represents the value at node / of the

y-th

eigenvector

tp,pe[l,...P]^tp=t^,ihe

(y'G l,r^"M).

Moreover,

at

certain

times

solution is assumed properly computed and defined

by the reduced vectors a^"^. Knowing a^"^, the finite element description of l//^ at time tp results: V^ = B B

cip - We can assume that the first approximation basis

contains a single vector that corresponds to the initial probability distribution

Now, we can compute the evolution of ^in \ta,t^ J solving the explicit system:

that can be written in the reduced approximation basis as: which results in:

^[:L = ((i ^^y I ^"O'^i ^"^ M"' ( M - A r ( g + ^ ( | ^''^a^;^)])g^''^a^;^ (18) In order to conclude about the solution accuracy at time t^ we compute the residual according to:

h = g X"IA. -M-^ (M-Af (G + ^ ( | ^''y^))| ^'^^aj;^ If R]^ <8 ,WQ can put t^ = t^ and continue the solution of the evolution problem in the reduced approximation basis B ^"^ for t>t^. When |^^ > ^ we need to look for a smaller time t^ ( ^ a + " ^ = ^;?) such that LRJ <€

after a new

solution of the evolution problem in [t^Jfi]- If the residual criterion is still

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391

unsatisfied when t^ - 1 ^ becomes smaller than the time step At, then we set ^p+\ "^ K ^^^ proceed to an enrichment of the approximation basis. The new reduced approximation basis 5^"^^^ is defined by adding to the significant information extracted fi-om a^ , Vp {p
+ V), some Krylov

subspaces. The construction of B_ ^"^^^ proceeds as follows: We define the matrix Q contaming the reduced vectors a' , Vp < a and we solve the eigenvalue problem defmed by:

whose solution results in r^"^ couples i'^j^yMk)> where we assume that //j >//2---,>//^(„). We select the r^"^^^ eigenvectors ^^ related to the eigenvalues verifying //^ > 10~ / / j , and these eigenvectors define the matrix F of (^^"^x^:^''*'^). Now we can write

Obviously, the change in the reduced approxunation basis implies a change in the expression of the reduced vectors a^ , Vp. For this purpose we can write

from which it results or ^(-i) ^ ^ ( | (..i))r| (-1) j-^(| («-i))^| C')^;;), Vp Now, we can add to J ^"^^^ a number N^^ (^new " ^ in the present work) of Krylov subspaces defined at time tj^:

where the matrix K is defined by:

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The introduction of the Krylov's subspaces impHes a new change in the expression of the reduced vectors a^"^ . For this purpose we add to those N^^ new vectors components that are assumed initially null because the new approximation functions just introduced operate only for t>tp , Thus we can write:

(5rr=((«rr'0.o.o):V;' The algorithm just presented can be summarized in pseudocode as follows: tp=t„+T While

(T»At;t^

At

Compute the evolution of the reduced variables in

[t„,tg]

Update the matrix H at each iteration /^||^(^ = ^/?)||< 0-001 then:

5-=h--ta ta

"-h

h = ta + 2S --h.)||> o.oo: 5 = h-ta ta = ta

h=

ta ^

2

Karhunen-Loeve expansion in [Oj^] Basis enrichment using Krylov's subspaces related to the residual at t^

V.3. Orientational distribution function and order parameter along a streamline The reduction scheme just presented was appUed to obtain the dynamic evolution of\|/(w,0 along a streamlme starting at r = 0.025. In accordance with the boundary conditions imposed in the CONNFFESSIT calculation, the initial orientational distribution was set to isotropic on the unit sphere:

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393

\|/(tt,r = 0) = (l/47c)5(|tt|-l). Instead of the Onsager form for the interaction .53 ,

potential, Eq. (9) the Maier-Saupe form was assumed: VEviu) = ~Uk,nuu:S)

(19)

where [/ is a dimensionless interaction potential and the second order traceless symmetric orientation tensor S is defined by:

£ = {«">--§

(20)

where the averaging ( ) is done with the instantaneous value of the orientational distribution function: {Wi)= j(ww)Kw.0^w

(21)

A scalar orientational order parameter S associated with the second order tensor S is defined by:

S = J^S'4

(22)

5 is a useful measure of the average degree of orientation, since it takes the value 0 in the isotropic phase, the value of 1 in a situation where all rods are perfectly aligned and intermediate values in the nematic phase. In addition, in this Section a constant value of the rotary diffusivity D^ was employed. The use of (19) mstead of (3) for the excluded volume interaction, and of constant D^ instead of (2) for the rotary diffusivity represent two simplifications, so that no perfect agreement can be expected between the results obtained by integrating (1) along a streamline by model reduction and the CONNFFESSIT results. The simplifications however are not so drastic as to make the comparison meaningless, as will be seen below. A more sophisticated treatment by reduction, ui which the more general forms for the excluded volume interaction and for the rotary diffusivity are used is also possible and constitutes work in progress. From a numerical point of view, this can be done using a fixed point strategy in thefi-ameworkof an expUcit algorithm. For the calculation by model reduction, a mesh of 2560 nodes on the unit sphere was used. It is important to emphasize that this size of the mesh is also the size of the sparse linear system that would have to be solved at each iteration when using a semi-implicit finite element technique, in contrast with the dense but very small size of the system to be solved when using model reduction. This is a very attractive feature of model reduction and is of course a consequence of the use of very few characteristic functions (typically, a few tens) in order to represent the whole temporal evolution of the distribution function when using model reduction. In this

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particular calculation, the number of significant functions used is approximately 20 (according to the criterion of 10"^ for the ratio between the contributions of the least significative and the most significative one) The evolution of y^{u^t) along the selected streamline is presented in the following figures as a sequence of snapshots of a colour surface plot on the unit sphere.

yM

Figure 18. Surface plot of ^ ( w , f) at Z' = - 1 0 . 0 .

Figure 19. Surface plot of ^ ( M , t)dXt =

-5.Q,

Hierarchical approach to flow calculations for polymeric liquid crystals

Figure 20. Surface plot of l/r(u, 0 at f = - 0 . 9 .

y M

Figure 21. Surface plot of ^ ( w , t)ziit

= -0.35 .

395

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Figure 22. Surface plot of y/{u, f) at f = 0 . 0 .

Figure 23. Surfece plot of ^ ( M , ^ at f = 0 . 7 .

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Figure 24. Surface plot of lf/^(u, f) at ^ = 1 0 . 0 .

This series of snapshots nicely shows how starting from an isotropic distribution, Y{u,t) becomes sharply peaked along the flow direction ( z ) . In the next figure, the orientational order parameter is presented as afimctionof time.

10.0

Figure 25. Scalar order parameter as a function of time.

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Starting from the initial isotropic state, and due almost exclusively to shear flow close to the tube wall, S rapidly climbs from 0 to about 0.8 and stays at that level until the streamline approaches the sudden contraction (which happens at ^ = 0 ) . At that point, the velocity field departs from pure shear and develops an appreciable extensional component, the competition of the two deformation mechanisms initially leads to a decrease in order and then to a sequence of oscillations before settling on a final value of around 0.7 in the narrow part of the domain.

0.0

Yn 0.0

10.0

Figure 26. The scalar order parameter as a function of time. The extensional and shear components of the velocity gradient as a function of time have also been included. The abscissa axes of the three plots are properly scaled, so that the vertical dashed lines join simultaneous values of S , y and S .

The evolution of y/{u^t) and of the scalar order parameter along the streamline closely follow the history of the strain rate tensor, as can be seen in the previous figure. A note of caution is however necessary: although a sharp peak in the extensional component of the rate of strain at the contraction (f = 0 ) is a physically correct feature of the fiow field, the sharp oscillations around the peak are very

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probably an artifact caused by insufficient mesh resolution and statistically noise in the CONNFFESSIT calculation, imposed by available computational resources. Hence, the oscillations of the order parameter around ^ = 0 must to some extent be attributed to this artifact. Besides, the simplifications in rotary dififiisivity and in excluded volume made in the reduced model, make this figure of qualitative value only. VI. Conclusions Starting fi*om an atomistically detailed description of the LC-forming poly-(wpropyl isocyanate) PPIC, a sequence of coarse-graining steps have been performed in order to obtain a macroscopic description of a PPIC solution in toluene which can be employed in viscoelastic flow calculations. The two first levels of description reside at the atomistic level and differ only in a computationally convenient reduction of the number of degrees of freedom through the introduction of holonomic constramts. The jump to the Level 3 description represents a major reduction in the detail of the description, so that the 0(10^) degrees of freedom in Level 1 are reduced to only two mesoscopic parameters describing the rotary diffiisivity of PPIC helices (considered as rigid rods) and the entropic excluded volume interaction between rods via purely geometric parameters and not in a fiiUy thermodynamically consistent way. These parameters are then used in the fi-amework of Doi's model for LCP's to perform a complex viscoelastic flow calculation in a three dimensional 3:1 (9:1 cross section area ratio) cylindrical contraction using CONNFFESSIT. Finally, the use of a model reduction technique was demonstrated by integrating the Fokker-Planck equation, which controls the dynamic evolution of y/^(u,t), along a streamline for which the history of the strain rate had been calculated with CONNFFESSIT. VII. Acknowledgments The authors would like to acknowledge the very finitfiil interaction and vigorous discussions with all partners of the PMILS project. Very thorough and constructive criticism of a first version of the manuscript by Prof Hans Christian Ottinger is also greatly appreciated. Major financial support by the EC through contracts G5RDCT-2002-00720 and NMP3-CT-2005-016375, and partial support by CICYT grant MAT 1999-0972 are gratefiiUy acknowledged as well as generous allocations of CPU time on hardware at CesViMa and BSC.

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403

Subject Index Acceptance Rate 39, 45,49, 50, 58, 60 Adaptive configurations fields 85 Adjusted solubility 318 Algorithm 361, 362, 365, 393, 395,403 Amber 363, 368 Amorphous cell 32, 33, 206, 207,210, 213, 221,339,341 Amphiphile 145 Andersen Barostat 343 Anisotropic united atom (AUA) model 336 Anomalous diffiision 210 Association 144, 146, 147, 151 Assumption of independent alignment 70 Aspect ratio 144, 146, 148, 149, 150, 152, 153,157 Atom decomposition parallelization method 342 Atomistic 183, 200, 201 Atomistic Simulation 43, 48, 51, 56, 63, 34, 335, 342 Autocorrelation Function 350, 351 Azimuthal361,383,387

B Backbone 57, 58, 59, 60, 61, 62, 184, 364, 365, 366, 367, 369, 370 Barrier properties 201, 202, 211-213, 216217,220,222,224,231,235 Basis 384, 391, 392, 393 Basis function 149 Bending Harmonic Potential 32 Bending Potential 336 Benzene 151, 152, 157 Berendsen 336, 341 Bersted and Slee 242, 245 Binary-interaction parameter defined 309 evaluated for gas + PE, 310-314 Binary mixture 119 Bloch's orbitals 6, 7, 20 Bond angle bending 336, 337, 338, 339, 340

Bonded interactions 336, 337, 338, 339, 340 Bond Length 336, 342 Bond Stretching Potential 336, 337, 338, 339 Borstar process 302 Branch Material Fraction 347 Branch Length 344, 347, 352, 354, 356 Branch Point 57, 58, 59, 60, 61, 62, 336, 344, 352 Branching Frequency 344, 347, 352, 346 Brownian 371, 372, 377, 379, 405 Brownian Fields 85, 88, 93, 102 Brownian motion 371, 372

Canonical Ensemble (NVT) 341 CAPD algorithm 260, 276 CAPD methodology 258 CAPD software 175, 278 Carbon dioxide (CO2) 211, 221, 223-225, 236 Catalysts 183, 184, 199 Chain-Connectivity Altering (Algorithm / Move) 32, 37, 41,42,46, 48,49, 50, 51, 52, 56, 60, 62, 63, 64, 369, 343 Chain length 146, 148, 156, 157 Chain stretching 70 Characteristic Ratio 54 Chemical Potential 38 Closure approximations 161 CO 151, 152, 153, 154, 155 C02 151, 152, 153, 154, 155, 156, 157 Co-absorption 306 Coarse grained 360 Coarse-graining 363, 367, 369, 370,401 Co-monomer effect 303 Complex Modulus 250 Complex Viscosity 250 Compliance 251 Computer Aided Polymer Design (CAPD) 258 Concerted Rotation (move) 35, 36, 58, 59 Configuration space 360, 388, 389

404

Subject Index

Confined 184, 186, 187, 192, 193, 194, 195,199,201 Confinement 188, 193 CONNFFESSIT 376, 377, 379, 384, 385, 387, 394, 395, 401, 402,404, 405 Connolly surface 364, 403 Conservation equations 163, 360, 376, 377 Configurational Bias (move) 35, 41 CONNFFESSIT 85, 88, 93, 96,100, 102,103 Conservation equations 87, 90, 125 Constitutive equations (CE's) 85, 86, 87 Constraint Variables 36 Continuum Configurational Bias 186 Continuum-mechanical methods 85, 86, 87 Contraction 376, 377, 378, 379, 380, 383, 385,400,401,404 Contraction flow 126 Convective conformation renewal 70 Convective constraint release 70 Couette flow 96, 102, 103, 169 Coupled-Perturbed procedure 17 Courant-Friedrich-Lewy (CFL) 162 Critical parameter estimation 112 Molecules 113 Polyethylene 118 Critical pressure 114 Critical temperature 115 Critical volume 113 Crystalline 359,402 Crystalline Orbitals 6, 21 Crystallinity 223-224, 234 Crystallinity of PE 307 Cyclohexane 151, 152, 157

D dealll library 127 Deborah number 129 Degrees of freedom 368, 369, 370, 377, 384,401 Density 33, 49, 52, 53, 144, 149, 150, 151, 152, 153, 154, 155, 156, 157, 193, 195, 199,344,348,349,350 DEVSS-G 165 Diameter 146, 147, 148, 149, 157 Dihedral Angle 337 Diffusion 183, 200, 201-206, 208, 210-211, 217, 219-220, 222-225, 227-230, 233237 Diffusion coeffcient 352, 353, 354, 355

Diffrisivity 209, 211-212, 221, 224, 227229,231,235 Dimer 147 Discretization 360, 377, 384, 387, 391 Doi's model 361, 370, 376, 377, 379, 384, 401 Domain377, 379, 380, 381, 383, 384, 385, 386,387, 388, 400 Domain decomposition parallelization method 342 Double Bridging (DB) (move) 42,43, 44, 48, 56, 57, 62 Double reptation 70 Dumbbell model 93, 86, 125, 164 FENE 126 Dynamic Viscosity 250

E Edberg-Evans-Moriss constraint method, 342 Eigenvectors 393 Einstein equation 206, 210, 222 Electric Field 19 Electrolyte 145, 146 Electrostatic charges 365 Electrostatics 369 Elongation 386 Energy minimization 212-213, 220 End-Bridging 186, 200 End Bridging (EB) (move) 37, 46, 48 End-Mer Rotation (move) 34, 35, 49, 50, 60 Energy Minimization 339, 341 Enhancement / inhibition of absorption 316, 320,321,325,328 Ensemble 186, 187, 193 Entangled (melts) 334, 335, 336, 342 Entangled Polymer Melts 34,41 Entanglements segments per polymer chain 71,78 EOS 193 Equation of state 144, 145, 146 Ethylene 184, 186, 187, 188, 192, 193, 194, 195, 196, 198, 199 Euler-Maruyama 165 Exchange-correlation potential 19 Excluded volume 360, 363, 370, 377, 380, 395,401 Explicit-atom (EA) model 212, 219-220, 335

Subject Index

405

Extended Concerted Rotation 186 Extensional 377, 385, 386, 400, 401 extra stress tensor 125, 129 Kramers expression 125

FENE 342 FENE dumbbell model 93 Fick's law 204 Finite Element 378, 379, 388, 389 Finite-Field 27 Flexibility 195 Flip (move) 34, 35, 58, 188 Fluctuations 188, 193, 367, 381, 382, 385 Fokker-Planck 360, 384, 387, 391, 401 Force decomposition parallelization method 342 Force field 32, 207,211, 213, 220-221, 335, 341, 363, 365, 366, 368, 371, 403, 404 Friction matrix 370

G Galerkin361,402 Gas absorption / solubility in polyethylene binary systems 310 ternary systems 316 multicomponent systems 320 Gas-phase reactors 302 gauche 193, 199 Gaussian 150 GC^ 264 GC-based property models 262 GC-Flory EOS 267 Generate and Test Approach 260 GENERIC 70, 370, 374 Geometrical derivatives 11 GMRES 168 Gradient 12, 76, 77 Group Contribution (GC) 116, 118,262 Gyration radius 343, 345, 346, 353, 354

H Hartree-Fock 3, 5, 144, 149, 175 Helix 364, 365, 366, 367, 368, 369, 370, 372, 373, 374, 384, 403, 404 Henry's law 204

Hessian 16 Hierarchical Modeling 32, 66 High density polyethylene (HDPE) 242, 333, 334, 335, 342, 349, 355 Hierarchical modeling 201, 203,212 Holonomic constraints 369, 401 Hookean dumbbell 96 Hoover 341, 343 H-shaped 42, 56, 57, 58, 60, 61, 62

Integration Time Step 341 Intermolecular potential 145, 146 Internal Libration (move) 34, 35, 58 Intramolecular End Bridging (move) 46,48 Intramolecular Double Rebridging (move) 46,47 isochromatic isolines 137 Isothermal-Isobaric Ensemble (NPT) 339, 340, 341, 343, 346, 347, 348, 349, 351, 353, 354, 355 Isotropic 359, 360, 371, 373, 380, 383, 394, 395, 399, 400

Jacobian 36, 45, 88, 89, 90, 98, 99, 100, 102, 103, 104, 105

K Karhunen-Loa 384, 394 Kinematics 363, 385, 388 Kinetics 183,200 Kinetic Monte Carlo 210, 222 Krylov 384, 392, 393, 394 KWW function 350

Lagrangian Particle Method (LPM) 85, 88, 93 LAMMPS51,67,343,357 Laplacian operator 361 Lattice Boltzmann 363, 403 LCAO 150 LCPs 359, 360, 363 Lennard-Jonnes 32, 34, 40, 184, 185 Lennard-Jones Potential 337

406

Subject Index

Levenberg-Marquardt algorithm 75 Linear low density polyethylene (LLDPE) 333, 334, 335, 342, 349, 355 Linear stability analysis 173 Liquid-liquid equilibrium 305 log-conformation method 179 Long-Chain Branched Polymers 56 Long-range corrections 9, 13, 18 Loop reactors 302 Lorentz-Berthelot rules 337 Loss Modulus 250 Loss Tangent 250 Low density polyethylene (LDPE) 78, 242, 302, 333, 334, 335, 342, 333, 334, 335, 342, 349, 355

M Macroscopic 183, 184, 186, 194, 195, 196, 197, 198, 199 Maier-Saupe 380, 394 Malkin and Teishev 241, 242 Mapping 183, 195 Mass conservation 87, 96 Mavridis and Shroff 242 Maximum likelihood 75 Maximum stretching ratio 71, 78-80 Maxwell-Boltzmann distribution 341 Mesh 378, 379, 383, 395, 401 Message Passing Interface (MPI) 130 Methyl 368, 403 Methylene 368 Metric tensor 391 Metropolis 196 Metropolis Criterion 34, 36,40,45 Microcanonical Ensemble (NVE) 341 Micro-macro methods 85, 86, 87, 89 backward-tracking Lagrangian particle method 124 Brownian configuration fields 124, 128, 164 CONNFFESSIT 124, 161 Microscopic dynamics 87, 90 Microscopic Reversibility 40 Miesowicz viscosity 382 Minimum image 185 Mixture 143, 144, 145 Modeling 184, 186, 190, 199

Molecular Architecture 32, 34, 42, 62, 203, 214, 218-219, 224,231, 234, 333, 334, 335, 343, 344, 346, 347, 348, 349, 350, 354, 355 Molecular Dynamics (MD) 31, 32, 34, 49, 51, 52, 60, 61, 63, 335, 336, 339, 340, 341, 342, 346, 347, 348, 349, 350, 351, 353, 354, 355, 356, 365, 367, 368, 370, 372, 373, 374 Molecular Mechanics (MM) 33 Molecular Weight (MW) 39, 50, 51, 60, 61, 333, 342, 343, 344, 345, 355, 356 Molecular weight dependence 203, 213214, 216, 218,224-225, 228, 235 Molecular Simulation 31, 64 Momentum conservation 96 Monodisperse (Melt / System) 334, 343, 346 Monodispersity 43 Monomer 146, 183 Monte Carlo (MC) 31, 33, 37,41, 56, 184, 186,200,201,339,341,343 Multi-level CAPD 260 Multiple Linear Regression (MLR) 110

N n-alkane 146, 150, 151, 152, 154, 155, 156 Nematic359,391,395,402 NERD (force field) 336 Newtonian 363, 376,402,404,405 Newton's Law (Equation) 340, 341 Newton's method 89, 101 Newton-Raphson method 166 Nitrogen (N2) 151, 152, 153, 154, 155,211212,220-231,234-235 non-Arrhenius behavior 229, 235 Non-bonded Interactions 33, 335, 336, 337, 338, 339, 340 Non-equilibrium thermodynamics 374 Non-Linear Optical Properties 4, 22, 23, 25 Nose (Thermostat) 341, 343

o Oldroyd-B 126,164 Oligomer 4, 118 Onsager360,381,380,394

407

Subject Index Optimisation 157 Order parameter 394, 395, 399,400,401 Orientational 384, 394 Orientational Autocorrelation Function 51, 52, 60, 61 Orientational Relaxation 349, 350, 351, 352,353 Oxygen (O2) 201, 210-212, 220-231, 234235

Pure polymer design 280 Pure polymer properties 262

QSPRllO Quantum-Chemistry 5,111 Quantum mechanics 144, 145, 148, 155

R Packing 373 Parallelization 342 Parameter estimation 70, 71, 75, 154 Parrinello-Rahman (Barostat) 341 Partition coefficients 267 Penetrant 202-211, 219, 221-225, 228-229, 234-236 Permeability 201-202, 205,211-212, 221225,228-229,231,234-235 Phase 23, 26 Picard89, 100, 101, 102 Plateau Modulus 71, 73, 76, 90, 251 Polydispersity 38, 39, 42, 50, 51, 60 Polyethylene (PE) 118, 183, 184 199, 200, 201, 203, 210, 212-214, 216-229, 231235 Polyethylene production 302 Poly gamma Functions 10, 14, 15, 18 Poly-(n-propyl isocyanate) 366,401 Polymer 3, 143, 145, 14 Polymer Design 257, 258 Polymer design application example 280, 283, 286 Polymerisation activity 326 Polyolefmsl83,302 Polypack 187 Pore 184, 185, 186, 187, 188, 190, 192, 193, 194, 195, 196, 197, 198, 199 Potential 184, 185, 186, 187, 193, 200 PPIC 363, 364, 365, 366, 367, 368, 370, 371, 372, 373, 374, 375, 376, 380, 384, 385, 401 Pressure 38, 52, 339, 341, 343, 348, 355 Problem definition 258, 259 Process 143 Properties of polymer solutions 272 Properties as fiinction of repeat unit configurations 273 Pseudoatoms 368, 372

Radial Distribution Function 54 Radius of Gyration 54, 61, 344, 345, 347, 353,355 Rate ofstrain 387, 401 RATTLE method 342 Redlich's hyperbolic interpolation 111 Reduction 384, 385, 394, 395,401, 405 Refi-igerantl46, 151, 152, 157 Relaxation strength 252 Relaxation time 252 Relaxation time spectra 250 Repeat unit 157, 260, 262 Replicated data parallelization method 342 Reptation 187 Reptation (move) 34, 35, 37, 49, 50, 60 Reptation time 76,78 Reptation Theory 334 RHS 92, 103 Rigid-Bond Constraints 342 Rigidity 367, 369, 370 Rod 360, 371, 372, 373, 374 Rotary diffusivity 360, 363, 371, 372, 373, 376, 380, 395, 401 Rouse model 176 rRESPA method 342, 343

SAFT 145, 148, 157 SAFT-VR 308-310 Sampling 186, 187,200 Saturation pressure 311 Schur's complement 91, 92, 94, 95, 100, 103, 105 Segment 146, 147, 148, 157 Selectivity 229, 235 Self-diffiision Coefficient 353, 354 Sensitivity equations 76 SHAKE algorithm 370 SHAKE method 342

408 Shape functions 384 Shear 380, 385, 386,400 Shear viscosity 74, 78-80 Short-Chain branched (SCB) 63, 64, 201, 212-213, 218-219, 224, 231-235, 333, 335, 336, 337, 338, 339, 340, 343, 344, 345, 346, 347, 348, 349, 355, 356 Simulation 143, 145 Site 146, 147, 151, 154, 155, 184, 190, 199 Smearing Factor 209, 211, 222, 224 Smoothing 385 Solubility 204-205, 207, 209, 211-212, 220-226, 229-232, 234-235 Sorption 201-203, 205-206, 208, 210-211, 217, 219, 221-222, 224-225,229, 235 Solubility 183, 186,193,199 Spherocylinder 148, 149 Spline 185 Square well 146 Stabilization 389 Static Structure Factor 54, 55 Statistical Ensemble 33, 38 Stochastic differential equations 70, 85, 88, 91,125,164,361,377 Stokes Problem 127 Storage Modulus 250 Streamlines 385, 388 Stress 86, 87, 88, 90, 93, 94, 96,104, 376, 377, 379 Stretched Exponential Function 351 SUPG 128, 165, 389 Synthesis 183

Temperature 38, 52, 339, 341, 343, 348, 349, 355, 356 Temperature dependence 203, 206, 216218, 224-225, 228-231, 235, 252 Texture 383 Thermodynamic 144 Thermodynamically consistent reptation model 70 Theta-method 162 Time-Dependent Hartree-Fock 20, 23, 26 Toluene 363, 368, 369, 372, 373, 374, 376, 401 Torsional angles 368, 369

Subject Index Torsional Potential 32, 337, 338, 339, 340 Trajectories 183, 371, 372, 373, 379, 380, 381,382,383 Transition State Theory (TST) 207, 209212, 215-217, 220-221, 223-236 Transport 183, 184, 192, 199 Transverse gradient 362 Transverse projector 362 TraPPE (force field) 336 Tube Model 334 Tuminello 242, 246 Trimer Bridging 36,41,46 Trust region radius 76

u United-Atom (Model / Representation) 207, 212-214, 219, 221, 335, 336, 338, 340, 344 United-Atom Representation 32, 33, 34, 35, 36, 38, 56 Unit sphere 360, 372, 387, 388, 390, 391, 394, 395, 396 Upwinding 389

Vapour-liquid equilibrium 305 Vapour pressure 144, 146, 149, 151, 153, 154, 156 Variable-Connectivity Move 36, 37 Variance reduction 129, 165 Variational 389 Velocity 360, 362, 363, 365, 370, 377, 380, 381, 382, 384, 395, 396, 397, 400 Velocity gradient 360, 362, 377, 384, 385, 386, 387, 400 Verlet341,365,370 Vertices 378, 388, 390 Viscoelastic 359, 370, 376, 385,401, 404, 405 Viscoelastic flow 85, 86 VLE 144, 145, 157 Volume 144, 148, 149, 150,152, 154 Volume Fluctuation (move) 49, 50, 60

Subject Index

409

w Water 144, 146, 151, 152, 154 Wavefunction 149, 150 Widom method 207 Wiener process 70, 73 Weissenberg number 164, 171

z Zero-shear Viscosity 251 Ziegler-Natta 334

411

Postface Going through this book, the attentive reader will sense a thread sewing together the sometimes very different chapters. Indeed, our strong intention from the outset was far more than simply to assemble a collection of scientific papers into a single volume; we conceived this as the natural culmination of a three-year-long common work. In modem research, tentative group working is often encountered, though it remains a very difficult task to build up strongly cohesive teams; individuals usually underestimate the Strength of teamwork and, moreover, a shared or common benefit can be more difficult to handle than a direct personal return. The success of this project, and subsequently of this book has a lot to do with group management: every step along the usual forming, storming, norming and finally performing sequence has gained from both the required coordination level and the necessary support firom the partners in the consortium. Thus, the integration of knowledge - Wisdom, know-how and way of working - has been coupled with the integration of people and cultures from all over Europe. There is inherent Beauty in work that is successful not only scientifically, but also at such a deeply human level. We hope that in the coming years there will continue to be new opportunities for this type of effort in collective research and we sincerely wish that anyone involved will also enjoy such a successfiil and rewarding experience.

Namur and Madrid, May 2006

E.A. Perpete, M. Laso