Preface
This book is a direct result of project PMILS (Polymer Molecular Modelling at Integrated Length/time Scales) -...
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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
^^
^ ^ ^ 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^'"^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
82
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
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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-
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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
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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 ^^^
M+6Mi""'> + c
wc.
A^Q
•znra NC.
vA^Q ,
V ^ 2
,(w+l) 172
j
W 2 + 4 (w) < ^ I+ ^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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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.
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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
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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
—
^
P 'C
0
=•
^^
420 400 0
0.2
0.4
0.6
x„
0.8
1
V, Wathelet et al.
120
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.
Estimation of critical parameters from quantum mechanics
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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:
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-;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
•"'
r
'
1
'
1"
"
I
'
-
"^^sJ^V
OWroyd^ 1 Nr=ioo H NF=1000 NF=»10000lH
-/
1
1
1
1
1
1
(
I
I
I
Hi)
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
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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
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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^^
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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
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the isochromatic fringe patterns measured experimentally. In a plane problem, the isochromatic lines correspond to the stress state according to^^
(4'=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.
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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.
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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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T.J. Sheldon et al.
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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TJ. Sheldon et al
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
T.J. Sheldon et al.
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 Lix
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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T.J. Sheldon etal
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.
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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
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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^=(
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.
198
M.Lasoetal.
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
Monomer solubility and diffusion
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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
-
—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
206
P. Gestoso andN.Ch. Karayiannis
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]
* , . , = ^ ^
0.6
\
0.4 0.2
1
'
1
200
'
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 •
\
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Oa N, Exp. 0 , Exp. N,
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i
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Q
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J J
i
f
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i • 1
5 •
1 200
1
400
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r
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600
BOO
••••
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—
• O . W U -1
-5.25 J
1•
1
— 1 — — r ^ 1—
1
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• J
^ ^.75 J
J
Q.
\
o ^.00 J •^
Karayiannis
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o J
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•
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• •
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o o o
-7.251.5
o
•
1
r-
1
o o 1
—,— —1
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
-
1
T
'
1
»— 1 — 1 — •J- - 1 -
| -
f
r
» -r
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»
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5
J
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-5.5-
•J
• 0. 0
N. -6.0- - T — T ""—, , 1.6 1.8 2.0 2.2
0 - I — ' — r — -1
2.4
2.6
1
2.8
r—1
3.0
1
1 1 1
3.2
1000/T(K')
Figure 12. Same as in Fig. 11 but for diffusion coefficient, D.
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
N.
0
1
'
\
1
'
«
J
1.4 J
]
1 1.2 J
1
\ \
J
^ 1.0 J * *^ o ?
"l
j
J
0.8
J
s ^ 0.6 J
J
i
\
0.40.2- ' — 1 — C142
\
1-
1 < 1 SBC_11x12_1«x1 SBC_9x14_lx2
SBC_7x1«_«x5
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
1
1
'1
1
1
11
r
-T
11
o
N,
A
2
IA
2.0-
S* 1.6-
}
N
E o
I
*
"o 1.2O
0.8-
i 5
0.40.0- 1 1 0142
,
•
§]
I
^
'•
1
SB C_11x12.10x1 SBC_Sx14_8x2
\
'
SBC_7x1i_6x5
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
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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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Multiscale Modelling of Polymer Properties M. Laso, E.A. Perpete (Editors) © 2006 Elsevier B.V. All rights reserved.
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
242
C Kiparissides et al.
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
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
9)
5316.608
NHCONH
30645.46
1 C03
I
5316.608
I 35843.27
17423.53 1
266
M. Hostrup et al.
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,; mC(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
Computer aided polymer design using group contribution techniques
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
M. Hostrup et al
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
270
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 k
cn
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cn
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cn
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ir»
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cn
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cn
r^ «o OS cn
f^ «o OS
sd
(Js ^
cn
on
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Cl
4 barrer and selectivity > 4.5
Computer aided polymer design using group contribution techniques
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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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,
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.
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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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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
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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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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
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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
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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
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r r=150.5°C
^.^ S 80^
"
's.
/ 1
/
^
/
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f
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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
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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
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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
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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
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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 " ^ ^ ^ ' ^
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2
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(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.
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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%
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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.
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Design ofpolyolefin reactor mixtures
fir 1 J
s il^2.5^
_ _ _ _ _ _ n—n ' ^ 15%/
20% nC^;;
1
/ ; / 10%] /,'^^5?d
^1.5J 2 1 •50.5
/
-
• '•
- ^
•
" 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
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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 '= 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
340
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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
Atomistic molecular dynamics simulations
341
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
342
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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
Atomistic molecular dynamics simulations
343
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.
344
N.Ch. Karayiannis et al.
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
;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 denotes the mean square displacement (msd) of the chain center-of-mass from its initial position after time t. Figure 10 presents -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, , 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 / (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 / (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
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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.
REFERENCES [1] A. Hakiki, R. N. Young and T. C. B. McLeish, Macromolecules 29 (1996) 3639. [2] N. Hadjichristidis, M. Xenidou, H. latrou, M. Pitsikalis, Y. Poulos, A. Avgeropoulos, S. Sioula, S. Paraskeva and G. Velis, Macromolecules 33 (2000) 2424. [3] 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. [4] P. D. Doerpinghaus and D. G. Baird, Macromolecules 35 (2002) 10087. [5] L. A. Archer and S. K. Varshney, Macromolecules 31 (1998) 6348. [6] M. T. Islam, J. Juliani, L. A. Archer and S. K. Varshney, Macromolecules 34 (2001) 6438. [7] L. J. Fetters, D. J. Lohse, S. T. Milner and W. W. Graessley, Macromolecules 32 (1999) 6847. [8] J. F. Vega, A. Munoz-Escalona, A. Sntamaria, M. E. Mufioz and P. Lafuente, Macromolecules 29 (1996) 960.
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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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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
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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-
Hierarchical approach to flow calculations for polymeric liquid crystals
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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.
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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
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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:
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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)) ) 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^--§
(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 .
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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 .
Hierarchical approach to flow calculations for polymeric liquid crystals
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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
Hierarchical approach to flow calculations for polymeric liquid crystals
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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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REFERENCES 1. P.G. de Gennes, J. P., The Physics of Liquid Crystals Clarendon Press 1995. 2. Chandrasekhar, S., Liquid Crystals Cambridge University Press Cambridge, 1992. 3. Larson, R. G.; Ottinger, H. C , Effect of Molecular Elasticity on out-of-Plane Orientations in Shearing Flows of Liquid-Crystalline Polymers. Macromolecules 1991, 24, (23), 6270-6282. 4. M. Doi, S. F. E., The Theory of Polymer Dynamics Clarendon Press: Oxford. 5. Marrucci, G.; Maffettone, P. L., Description of the Liquid-Crystalline Phase of Rodlike Polymers at High Shear Rates. Macromolecules 1989,22, (10), 4076-4082. 6. Grosso, M.; Maffettone, P. L.; Halin, P.; Keunings, R.; Legat, V., Flow of nematic polymers in eccentric cylinder geometry: influence of closure approximations. Journal of Non-Newtonian Fluid Mechanics 2000,94, (2-3), 119-134. 7. R.B. Bird, R. C. A., O. Hassager, C.F. Curtiss Dynamics of Polymeric Liquids: Kinetic Theory Vol 2. John Wiley & Sons Inc New York, 1987. 8. Grosso, M.; Keunings, R.; Crescitelli, S.; Maffettone, P. L., Prediction of chaotic dynamics in sheared liquid crystalline polymers. Physical Review Letters 2001, 86, (14), 3184-3187. 9. Knuth, D., The Art of Computer Programming: Vol 1-3. Addison Wesley New York. 10. R.G. Owens, T. N. P., Computational Rheology Imperial College Press London, p 436 11. Suen, J. K.; Nayak, R.; Armstrong, R. C ; Brown, R. A., A wavelet-Galerkin method for simulating the Doi model with orientation-dependent rotational diffiisivity. Journal of Non-Newtonian Fluid Mechanics 2003,114, (2-3), 197-228. 12. Suen, J. K. C ; Joo, Y. L.; Armstrong, R. C , Molecular orientation effects in viscoelasticity. Annual Review of Fluid Mechanics 2002, 34,417-444. 13. Dupuis, A.; Yeomans, J. M., Lattice Boltzmann modelling of droplets on chemically heterogeneous surfaces. Future Generation Computer Systems 2004,20, (6), 993-1001. 14. Denniston, C ; Marenduzzo, D.; Orlandini, E.; Yeomans, J. M., Lattice Boltzmann algorithm for three-dimensional liquid-crystal hydrodynamics. Philosophical Transactions of the Royal Society of London Series a-Mathematical Physical and Engineering Sciences 2004, 362, (1821), 1745-1754. 15. Marenduzzo, D.; Dupuis, A.; Yeomans, J. M.; Orlandini,E.,Lattice Boltzmann simulations of cholesteric liquid crystals: Permeative flows, doubly twisted textures and cubic blue phases. Molecular Crystals and Liquid Crystals 2005,435, 845-858. 16. Antony N. Beris, B. J. E., Thermodynamics of Flowing Systems: With Internal Microstructure Oxford University Press Inc, USA 1994. 17. Wang, J. M.; Wolf, R. M.; Caldwell, J. W.; Kolhnan, P. A.; Case, D. A., Development and testing of a general amber force field (vol 25, pgl 157,2004). Journal of Computational Chemistry 2005,26, (1), 114-114. 18. Wang, J. M.; Wolf, R. M.; Caldwell, J. W.; Kolhnan, P. A.; Case, D. A., Development and testing of a general amber force field. Journal of Computational Chemistry 2004,25, (9), 1157-1174.
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19. Luo, R.; Wang, J. M.; Kollman, P. A., Development of a more accurate amber united-atom force field for protein folding and large-scale biomolecular simulations. Abstracts of Papers of the American Chemical Society 2002,224, U470-U471. 20. Ryu, J.; Park, R.; Kim, D. S., Connolly surface on an atomic structure via Voronoi diagram of atoms. Journal of Computer Science and Technology 2006,21, (2), 255-260. 21. T.E. Daubert, R. P. D., (Editors) Physical and Thermodynamic Properties of Pure Chemicals. Taylor & Francis 1992. 22. M. P. Allen, D. J. T., Computer Simulation of Liquids. 1989. 23. Leach, A., Molecular Modelling: Principles and Applications. Prentice Hall: 2001; p 720 24. Daan Frenkel, B. S., Understanding Molecular Simulation: From Algorithms to Applications Academic Press London, 2001. 25. Lukasheva, N. V.; Niemela, S.; Neelov, L M.; Darinskii, A. A.; Sundholm, F., Helix conformations in poly(alkyl isocyanate) chains. Polymer Science Series A 2003, 45, (2), 194-199. 26. Lukasheva, N. V.; Niemela, S.; Neelov, I. M.; Darinskii, A. A.; Sundhohn, F.; Cook, R., Conformational variability of helix sense reversals in poly(methyl isocyanate). Polymer 2002,43, (4), 1527-1532. 27. Lukasheva, N. V.; Niemela, S.; Neelov, I. M.; Darinskii, A. A.; Sundholm, F.; Cook, R., Computer modeling of helical conformations and helix sense reversals in poly(alkyl isocyanates): New types of reversals. Abstracts of Papers of the American Chemical Society 2001,222, U358-U358. 28. Lukasheva, N. V.; Niemela, S.; Neelov, L M.; Darinskii, A. A.; Sundhohn, F.; Cook, R., New conformations and new types of helix sense reversals and defects in the chains of nonchiral poly(alkyl isocyanates). Macromolecular Symposia 1999,146,251257. 29. Shmueli, U.; Traub, W.; RosenhecK, Structure of Poly(N-Butyl Isocyanate). Journal of Polymer Science Part a'2-Polymer Physics 1969,7, (3PA2), 515-&. 30. Lifson, S.; Felder, C. E.; Green, M. M., Helical Conformations, Internal Motion and Helix Sense Reversal in Polyisocyanates, and the Preferred Helix Sense of an Optically-Active Polyisocyanate. Macromolecules 1992,25, (16), 4142-4148. 31. Tonelli, A. E., Conformational Characteristics of Poly(Normal-Alkyl Isocyanates). Macromolecules 1974, 7, (5), 628-631. 32. Bur, A. J.; Roberts, D. E., Rodlike and Ramdom-Coil Behavior of Poly(NButyl Isocyanate) in Dilute Solution. Journal of Chemical Physics 1969, 51, (1), 406-&. 33. Yang, L. J.; Luo, R., AMBER united-atom force field. Biophysical Journal 2005, 88,(1), 512A-512A. 34. Cross, C. W.; Fung, B. M., Molecular dynamics simulations forcyanobiphenyl liquid crystals. Molecular Crystals and Liquid Crystals Science and Technology Section a-Molecular Crystals and Liquid Crystals 1995,262, 507-524. 35. Cross, C. W.; Fung, B. M., Tricritical Points of Smectic-a to Nematic PhaseTransitions for Binary-Liquid Crystal Mixtures Containing Cyanobiphenyls. Liquid Crystals 1995,19, (6), 863-869. 36. Ottinger, H. C , Beyond Equilibrium Thermodynamics John Wiley & Sons Inc Chichester, 2005.
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37. Boulougouris, G. C ; Economou, I. G.; Theodorou, D. N., Calculation of the chemical potential of chain molecules using the staged particle deletion scheme. Journal of Chemical Physics imi, 115, (17), 8231-8237. 38. Laso, M.; Ottinger, H. C , Calculation of viscoelastic flow using molecular models: The CONNFFESSIT approach. Journal of Non-Newtonian Fluid Mechanics 1993,47,1-20. 39. Ramirez, J.; Laso, M., Micro-macro simulations of three-dimensional plane contraction flow. Modelling and Simulation in Materials Science and Engineering 2004,12, (6), 1293-1306. 40. Grande, E.; Laso, M.; Picasso, M., Calculation of variable-topology free surface flows using CONNFFESSIT. Journal of Non-Newtonian Fluid Mechanics 2003, 113,(2-3), 127-145. 41. Cormenzana, J.; Ledda, A.; Laso, M.; Debbaut, B., Calculation of free surface flows using CONNFFESSIT. Journal ofRheology 2001,45, (1), 237-258. 42. Laso, M.; Picasso, M.; Ottinger, H. C , 2-D time-dependent viscoelastic flow calculations using CONNFFESSIT. Aiche Journal 1997,43, (4), 877-892. 43. Van Heel, A. P. G.; Hulsen, M. A.; Van Den Brule, B. H. A. A., Simulation of the Doi-Edwards model in complex flow. Journal ofRheology 1999,43, (5), 12391260. 44. Hulsen, M. A.; Van Heel, A. P. G.; Van Den Brule, B. H. A. A., Simulation of viscoelastic flows using Brownian configuration fields. Journal of Non-Newtonian Fluid Mechanics 1997,70, (1-2), 79-101. 45. Ottinger, H. C ; vandenBrule, B.; Hulsen, M. A., Brownian configuration fields and variance reduced CONNFFESSIT. Journal of Non-Newtonian Fluid Mechanics 1997,70, (3), 255-261. 46. Soare, M. A.; Picu, R. C ; Tichy, J.; Lu, T. M.; Wang, G. C , Fluid transport through nanochannels using nanoelectromechanical actuators. Journal of Intelligent Material Systems and Structures 2006,17, (3), 231-238. 47. Ammar, A.; Ryckelynck, D.; Chinesta, F.; Keunings, R., On the reduction of kinetic theory models related to finitely extensible dumbbells. Journal of NonNewtonian Fluid Mechanics 2006y 134, (1-3), 136-147. 48. Ryckelynck, D.; Hermanns, L.; Chinesta, F.; Alarcon, E., An efficient 'a priori' model reduction for boundary element models. Engineering Analysis with Boundary Elements 2005,29, (8), 796-801. 49. Ryckelynck, D., A priori hyperreduction method: an adaptive approach. Journal of Computational Physics 2005,202, (1), 346-366. 50. Yvonnet, J.; Ryckelynck, D.; Lorong, P.; Chinesta, F., A new extension of the natural element method for non-convex and discontinuous problems: the constrained natural element method (C-NEM). International Journal for Numerical Methods in Engineering 20M, 60, (8), 1451-1474. 51. Ryckelynck, D., An a priori model reduction method for thermomechanical problems. Comptes Rendus Mecanique 2002, 330, (7), 499-505. 52. Lipschutz, M. M., Differential Geometry. McGraw-Hill: 1969. 53. Maier, W.; Saupe, A., Eine Einfache Molekular-Statistische Theorie Der Nematischen Kristallinflussigen Phase . 1. Zeitschrift Fur Naturforschung Part aAstrophysikPhysik Und Physikalische Chemie 1959,14, (10), 882-889.
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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