Simulation Methods for Modeling Amorphous Polymers
M. Kotelyanskii 1 Center for Molecular and Engineering Thermodynamics Dept. of Chemical Engineering, University of Delaware, Newark, Delaware 19716, USA
Abstract
During recent years, computer simulation has developed as a powerful research tool to study polymer materials properties. Simulation provides a molecular-level description of the amorphous material, necessary to understand its structure and to formulate the mechanisms of material behavior. Molecular simulation of bulk polymers has become a reality and has been applied to study properties important to polymer applications, such as glass transition temperature, small molecule sorption and diffusion, plastic and elastic deformation, and molecular mobility mechanisms. Methods designed to simulate the amorphous polymers structure are discussed in the following review.
1
Current address: Polymer Science Program, Department of Materials Sci-
ence, Pennsilvania State University, University Park, PA 16802, USA. e-mail
Preprint submitted to Elsevier Preprint
28 March 1997
Compared with the simple molecular liquid, the complex molecular architecture and disordered structure lacking lattice symmetry make amorphous polymers structurally incredibly complicated. At the same time, the ability to predict properties of new possible polymers, given their chemical structure, can provide new leads and significantly reduce the effort of a trial and error search for new materials, saving a large amount of experimental work. Structural and conformational dynamics in the glass and in the melt have been studied by many authors. Polymers with repeat units of different complexity, ranging from polyethylene and simple vinyl polymers,1–3 to polystyrene,4–7 polycarbonates8–10 and more complicated polyimides,11 and aromatic polybenzoxazoles,12 have been simulated. Atomistic simulation techniques have also been used to study the thermodynamical properties of polymer melts and the glass transition in polymers.1, 13, 14 The diffusion and sorption of small molecules in polymer glasses,15 phase equilibria for systems containing chain molecules,16 and the spectra of small chromophores embedded in a glassy polymer matrix.17 More examples can be found in recent reviews.18–21
Basic Principles
A short description of the basic principles, necessary to understand the topic of this review, will be given below. More comprehensive explanation of the molecular simulation methodology can be found elsewhere.22
[email protected], FAX (814) 865-2917
2
All molecular simulations are based on the representation of the system potential energy as a sum of a number of contributions (Fig. 1) (1) The chemical bond contribution Ub , describes the energy associated with the deviation of the distance l between the chemically bonded atoms from the equilibrium bond length, l0 . It is usually represented as a quadratic function of the distance: Ub = Kb (l − l0 )2 . (2) The bond angle energy, Ua , associated with the deviation of the angle ξ, formed by two adjacent bonds, from its equilibrium value ξ0 . This term is usually a quadratic function of the deviation: Ua = Ka (ξ − ξ0 )2 . (3) The torsional energy, Utor is a function of a torsional angle φ describing the rotation around a chemical bond. This is usually expressed as Utor = Kφ [1 + cos k(φ − φ0 )]. (4) The non-bonded interaction Unb includes the interaction between chemically non-bonded atoms of the same molecule and interactions between molecules. This energy depends on the distance between two atoms. The non-bonded interaction is strongly repulsive at short distances and attractive at long, and is usually represented as a sum of the Lennard-Jones and Coulomb potentials, Unb = (A/r n − B/r m ) + qi qj /r, where r is the distance between atoms i and j, qi and qj are their partial charges, A, and B - are the constant parameters, and powers n and m are usually equal to 12 and 6 or 9 and 6, respectively. The functional form of the above equations and their coefficients (Kb , l0 , Kφ , k, φ0 , A, B,m,n, qi ,qj ) for different atom types constitute the ‘force3
field’ used to build the polymer model. Usually the parameters are chosen by the fit to some known experimental data and/or the energies of different molecular conformations obtained from quantum chemical calculations.23–27 The reliability of the molecular simulation results is largely determined by the reliability of the chosen ‘force-field’. Polymer properties are obtained from simulations as the thermodynamic averages. The thermodynamic internal energy or pressure, for instance, can be calculated from the average values of the energy or virial,22 that are the functions of the atoms coordinates or velocities in a particular configuration. In order for these averages to represent thermodynamic values measured experimentally, configurations should be sampled according to the probability distribution of a proper statistical ensemble.28 Different simulation methods essentially represent different ways of sampling system configurations with a required probability density. Monte Carlo (MC) methods and Molecular Dynamics (MD) are two major simulation techniques. MD methods follow the atomic motions by numerically integrating the equations given by Newton’s second law. At each step, forces are calculated as a function of the atomic coordinates, specified by the ‘forcefield’, and the atomic velocities and coordinates at the next step are obtained iteratively, as functions of time. Time steps are typically of the order of a femtosecond, and a typical MD simulation covers system evolution on the nanosecond timescale. Information about the atomic motions can be used to study diffusion and relaxation processes occurring in the system. Averages 4
over the trajectory, calculated in the MD simulation, provide thermodynamic values and structural characteristics.22 In the classic Metropolis Monte Carlo (MC) method,29 each new trial configuration is obtained from the current one by randomly displacing one or more atoms. If the potential energy of a new trial configuration Unew is lower, than the old energy Uold , the new configuration is always accepted; otherwise, it is accepted with a probability, p = exp[−(Unew − Uold )/kB T ]. It can be shown that the probability distribution, corresponding to the constant volume and constant temperature ensemble is reproduced. In the constant pressure ensemble, the volume is also allowed to vary, and the acceptance probability is modified as p = exp −[Unew − Uold − P (Vnew − Vold ]/kB T where P is pressure. Thermodynamic properties are calculated as averages over the sampled configurations. If the time scale can be assigned to the Monte Carlo step based on reactions rate theory, for instance, or based on comparison to the MD simulation, ‘ some features of the system dynamics can also be reproduced using MC.15 As soon as it can be proven that system configurations are sampled according to the proper distribution, there is no restriction on the way the new trial configuration is generated. This property is used to develop new, more sophisticated Monte Carlo methods for polymer systems.30–32 The computer time required for the MD and MC simulations is usually proportional to the system volume or to the number of atoms. With the current computer facilities a typical model has a size of about 20 − 40 ˚ Aand contains several hundreds or thousands of atoms. To simulate the properties of the 5
bulk material rather than of a cluster of about a thousand atoms, periodic boundary conditions are usually applied.22 A simulation cell containing system atoms is periodically replicated in all directions, thus atoms next to its border interact with the periodic images of the atoms from the opposite side of the simulation cell.22 In molecular simulations, as in any statistical study, better estimates of the averages are obtained when more uncorrelated configurations are sampled. This means simulating system evolution for longer times in MD or making more steps in MC simulations, and presents the main challenge of polymer simulations. Owing to their chain structure, the relaxation spectrum of a polymer system spans several orders of magnitude.33 Thus simulations of polymers usually require much longer MD or MC runs than simulations of low molecular weight liquids.
Generating Atomistic Models
Simulation of glasses, formed by small molecules or amorphous metal alloys, is usually done by simulating the system cooling from an equilibrated liquid state. As the temperature decreases, the change in the slope of the specific volume – temperature dependence, freezing of the diffusion mobility and viscosity increase can be observed. Temperature at which this occurs is associated with the glass transition temperature. This approach has been used to study the glass transition in simplified models of different polymers.1, 14 6
Simulation of a simple liquid can start from some artificially prepared, sometimes ordered, configuration.22 Because of the short relaxation times it does not take long for the system to ‘forget’ it, so that the calculated properties of the liquid are independent of this initial configuration. It has been observed, however, that the obtained polymer structures and simulation results are very sensitive to the way initial configurations are prepared.34, 35 This is a natural consequence of the long relaxation times peculiar to the polymer molecules. The longest relaxation time of the polymer chain increases as a squared chain length and can reach a tenth of a second.36 Thus, starting from the ordered configuration, a simulation is doomed to fail or to become extremely computationally expensive for a polymer model with a realistic molecular weight. It would require too long for the chains to relax to a random coil conformation, in which they exist in a melt or glass, from the ordered starting configuration. The problem could be overcome by either using more efficient ways of sampling amorphous polymer configurations than straightforward MD, or by finding an efficient way to generate the uncorrelated representative amorphous polymer structures. The former possibility can be realized by the recently developed sophisticated Monte Carlo techniques, based on cutting and re-growing pieces of the polymer chain.16, 30–32 These methods have been successful for the united atom model, but their performance seems to degrade significantly for more realistic models. They were recently reviewed elsewhere,,19, 21, 30, 37 and in the following we concentrate more on the latter approach, based on the generation of disordered structures. 7
Packing polymer chain with the required density and statistical properties.
The ability of a polymer chain in the melt or glass to fold randomly according to its conformational properties and at the same time to find its way through the matrix of other chains seemed for a long time to be too complicated to be reproduced in computer simulations. The first attempt to generate an atomistically detailed glassy polymer structure was published by Theodorou and Suter in 1985.2 The method is based on ‘Flory’s Hypothesis’, stating that chain conformation in the polymer melt and glass obeys Gaussian statistics and that the probability of different conformations occurring can be described by the Rotational Isomeric State (RIS) model.38 In the RIS model the potential energy of the chain is given by the sum of the interactions between the adjacent monomers. Each torsion assumes a value corresponding to one of the local potential energy minima that, in turn define the bond’s rotational isomeric states of the bond. The polymer chain is first generated by a Monte Carlo procedure, following the RIS model: the rotational isomeric state of each added segment is chosen with a probability defined by its conformational energy. This generated chain, contained inside the cell with the periodic boundary conditions, is illustrated in Fig. 2. Whenever the growing chain contour leaves the cell, it re-enters from the opposite side. The cell dimensions define the density of the system. The next step is a potential energy minimization, using the folded configuration as an ‘initial guess’. If successful, the algorithm results in the structure of the
8
polymer, packed with the required density and with the proper conformational statistics. Typically, the ‘quality’ of the obtained structures is judged by their cohesive energies and their Hildebrand solubility parameters. However, the RIS model does not account for the interactions between monomers separated by more than one bond. Neglecting the long-range interactions allows the monomers that are not the nearest neighbors along the chain to overlap. When such overlaps occur, the steep repulsive Lennard-Jones potential leads to high energies (1012 − 1015 kJmole−1 ) and to enormous forces exerted on the overlapping atoms. To overcome this problem the initial guess generation procedure has been changed, so that the long-range interactions are taken into account. When the new monomer is added to the chain, conditional probabilities of the rotational isomeric states are modified to favor the conformations with lower non-bonded energy. This improved scheme produced more homogeneous structures with lower energy (106 − 108 kJmole−1 ), but the overlaps could not be completely avoided. The energy of the initial guess was still high, and minimization was not always successful in removing the overlaps, leading to a physically impossible representation of the polymer glass. Only a relatively small fraction of the ‘initial guess’ configurations can be minimized, but as their generation is fairly simple and computationally inexpensive, it is possible to generate structures with reasonable energies by repeating the procedure again and again. The method works well for polymers with relatively simple repeat units like polypropylene2 and polyvinylchloride.3 However, more complex polymers, like polycarbonate8 and polystyrene6 re9
quired more effort. Complex multi-stage procedures of switching non-bonded interactions on and off, reducing the Van der Waals radii of the atoms, and changing the phenylene ring geometry were necessary to generate structures with reasonable cohesive energies. This methodology is widely used today. It is implemented in the popular commercial simulation software packages distributed by Molecular Simulations Inc..25 The structures of various polymers, generated by the Theodorou and Suter (TS) method, have then been used to study the monomer mobility, plastic and elastic deformation, small molecule sorption and diffusion through amorphous polymers, and other properties.8, 15, 17 Successful application of the method to the polyimide which has a repeat unit even more complex, than polycarbonate, has been published recently by Zhang and Mattice.11 Another approach to grow a chain inside the periodic cell was explored by Clarke and co-workers.35 Each successive repeat unit is added in a conformation decided by the Monte Carlo-like procedure based on its energy, but, unlike in TS method, the dihedral angle can vary continuously. The non-bonded interactions with all atoms present in the system are taken into account. The new configuration is accepted with a probability equal to exp(−∆Φ/kB T ), where ∆Φ is the energy increment due to the addition of the repeat unit. Unlike the TS method, in which chain growth always proceeds forwards, this algorithm can step back. If several successive trials do not lead to the successful addition of a new segment, the chain is cut back by one segment, and the whole procedure repeated again. This method, however, turned out to be practical 10
only for systems with relatively low density. At higher densities, corresponding to the real polymer melt, chain growth becomes deadlocked into a situation where adding a new monomer is impossible due to the absence of free space around the last added monomer. Clarke and co-workers found that it is more efficient to grow the chain inside the periodic box, taking into account only the non-bonded interactions between the close chain segments necessary to reproduce correct torsional angle statistics. Authors called this chain configuration, which is allowed to overlap with itself ‘phantom chain’. Again, this ‘phantom chain’ configuration has atomic overlaps that then have to be relaxed. The authors use molecular dynamics with a modified Lennard-Jones potential instead of the energy minimization. This relaxation was successfully performed for the case of a simple polyethylene melt by introducing softened LennardJones potentials. Vasudevan and McGrath have used a similar procedure to build the atomistic models of amorphous aromatic polybenzoxazoles.12 Notice, however, that when applied to polymers with complex repeat units containing rings, neglecting the long-range interactions may lead to the physically impossible configurations with the backbone bonds passing through the rings, or to the interwining of rings, belonging to different monomers. To overcome the deadlocking, Gupta et. al.39 suggested combining the successive chain growth with the system equilibration by moving atoms present in the system with a Metropolis MC simulation before the next segment is added to the chain. The method was applied to a model of a polymer chain, consisting of the hard spheres, connected by chemical bonds. Bond lengths and 11
angles were fixed, and the only intramolecular contribution to the energy was due to the torsional potentials. The authors succeeded in equilibrating systems of chains of up to 25 monomers. The densities studied were lower than those typically observed in a low-temperature melt close to the glass transition or to the melting point. A different approach to the problem was used by Gusev et. al.40 and by Mondello et. al 4 to prepare the atomistic models of polycarbonate and polystyrene. Instead of trying to pack chains into the small periodic box, the polymer chain is first built inside a cell large enough to accommodate the entire unfolded ‘parent chain’. The box size is then gradually reduced by applying pressure to the simulated system. Approximately 70 ps of the constant-pressure MD were sufficient to compress a polycarbonate chain from 0.01gcm−3 to its experimental density of 1.2gcm−3 .40 The simulation of polystyrene4 started from the entirely stretched chain and then the box size was gradually reduced, forcing the chain to fold to accommodate to the experimental density. The simulation time necessary for the chain folding is determined by the longest relaxation time of the polymer chain. This time increases as the square of the chain length,36 which raises doubts about the efficiency of the method for longer polymer chains.
Simulating ‘real’ polymerization
When the already prepared polymer chains are folded into the periodic simulation cell, or when they are grown at a density close to the experimental density of the polymer melt or glass, the problem of atomic overlaps reveals 12
itself either through the high energy and large forces acting on the overlapping atom pairs, or through the lack of free space necessary to add another monomer to the growing chain. Is it possible to avoid the overlaps from the very beginning? The obvious solution would be to follow the chain growth as it occurs in the ‘real’ world: to create the liquid monomer, which is free of the interatomic overlaps, and then to start introducing chemical bonds between monomer molecules. The straightforward simulation of the polymerization could be an interesting study in its own rights, but the problem is that in real systems it takes much longer than nanoseconds, it is hard to achieve 100% conversion, and it always results in a mixture of molecules with different chain lengths. An attempt to follow this route was recently reported by Gao.41 The system studied is a simplified ‘polybead’ model of a polymer. It consists of LennardJones spheres connected by harmonic links, with the equilibrium bond length equal to the Van der Waals radius of the monomer. The algorithm starts from the dense monomer liquid. During the MD run, bonds are introduced between the monomers separated by a distance smaller than some critical value. The process resembles bulk polymerization and leads to similar results: it is hard to obtain a monodisperse system, especially for the chain length of more than 100 monomers. Owing to the diffusion limitations, a substantial part of the monomers (about 20%) remained unconnected. This ‘sol’ fraction had to be removed, and the rest of the system had to be compressed to the desired density. A similar but more simple approach was used by Rigby and Roe14 13
to prepare the initial configurations for their study of the glass transition in the simplified united atom polyethylene model. The random distribution of monomers in space was generated, and bonds were introduced between the monomers located close to each other, to provide the required chain length and number of molecules. The system is then relaxed starting with the reduced bond and valence angle potential parameters Ka and Kb , which were gradually increased to their final values. We would expect more problems when applying this approach to the atomistic polymer model as, in this case, repeat units, containing several atoms, should not only get close, but they also have to approach each other in the orientation favorable to form a bond. A different version of the ‘polymerization’ process was proposed by Khare et. al..5 The ‘monomer’ liquid in this approach is ‘polymerized’ by creating a connectivity path, passing through each monomer only once. This is accomplished by a solution of the well-known ‘traveling salesman’ problem: given a set of points to find the shortest path that goes through each point only once. The bonds created by this method do not have equilibrium length, and bond angles are far from their equilibrium values. Potential energy minimization and an annealing MD run at higher temperature were performed to improve the chain geometry. The energy minimization was not difficult to perform as, in the absence of the monomer overlaps. The major contribution to the energy comes from the quadratic bond and bond angle potentials, which cause fewer numerical problems during the minimization than the steep repulsive Lennard-Jones potentials acting between the overlapping atoms. The method was applied to 14
generate the structures of atactic polystyrene at the experimental density.
Mapping atomistic model on the lattice
The approaches described above, based on the ‘polymerization’ of the monomer liquid, do avoid the severe interatomic overlaps, but do not guarantee the Gaussian chain statistics with a correct persistence length, and they do not provide any opportunity to define the monomer sequence and chain tacticity a priori. A method to generate dense amorphous polymer structures that avoids severe atom overlaps, and at the same time allows definition of the monomer sequence and tacticity, has been introduced.7 The idea is to map the atomistic model of the polymer chain on the trajectory of the self-avoiding random walk, generated on the periodic lattice. Statistical properties of the random walk can be chosen so that it creates the Gaussian chain with the desired persistence length and fills the simulation box with the size corresponding to the density of the amorphous polymer. Similar to the TS and other methods, it relies on ‘Flory’s hypothesis’, and requires the values of the polymer density and persistence length as input parameters. The self-avoiding random walk is generated on the completely occupied cubic lattice by the algorithm proposed by Pakula.42 This is a Monte Carlo method designed to sample conformations of the non-intersecting polymer chains on the completely occupied cubic lattice by propagating kinks and chain ends. The lattice chains obey Gaussian statistics with the mean squared distance between the segments < ∆r~k 2 >, 15
separated by k bonds along the chain, proportional to k. < ∆r~k 2 >= A0 ka2 . 1/2
The chain persistence length A0 a can be adjusted by biasing the acceptance probability against the chain bending. Sampling chain conformations on the lattice is computationally less expensive than off-lattice, and each Monte Carlo move rearranges large fragments of the polymer chain. Typically several thousand Monte Carlo cycles (each cycle corresponds to the N 3 Monte Carlo steps on the lattice, containing N 3 nodes) are necessary to generate uncorrelated lattice chain configurations. 1/2
The required value of the chain persistence length A0 a, the lattice spacing a, and the number of the repeat units, mapped on one lattice site ν, are defined so that the mean squared end-to-end distances and the densities of the lattice and atomistic chains will be the same. If D0 is the squared persistence length and v0 is an experimental value of the volume per one repeat unit of the simulated polymer, two conditions A0 a2 = nD0 , and v0 n = a3 have to be satisfied. This gives two equations to choose three parameters a, A0 , and ν. The choice is not unique, and it is always possible to find a reasonable set of parameter values. For atactic polystyrene,7 the self-avoiding chain with A0 = 1.5 and a = 4.374 ˚ Awas used, and the repeat unit was split in two parts (nu = 0.5). Figure 3 illustrates the mapping process. Two types of ’building blocks’, each containing one of the two parts of the repeat unit, have been prepared. At the figure a fragment of the lattice (four lattice nodes), filled with two polystyrene monomers (four ‘buildong blocks’) is shown. The dashed line
16
shows the lattice chain path, which defines the sequence in which the ‘building blocks’ are placed and later connected by the chemical bonds. It is also possible to orient the blocks in such a way that atoms to be connected by the chemical bonds between the units (aliphatic carbons in the case of polystyrene) are not screened by the substituents (phenylene rings). The tacticity of the chain can be specified by applying the symmetry transformations to the atoms in the ‘building block’ containing chiral carbon. All repeat unit atoms are contained inside the blocks so that when placed on the lattice sites there are no overlaps between the atoms. With all these precautions further potential energy minimization did not require much effort.7 To assure that the periodicity imposed by the initial lattice configuration is not present in the final structure, a several picosecond MD annealing run is required. Polystyrene structures of various sizes with periodic cell length of 21.75 ˚ Ato 39.15 ˚ Ahave been prepared by this method.7 The method proved to be robust and efficient. Four 21.75 ˚ Apolystyrene structures, containing a 62-mer chain were prepared in several hours on the IBMRISC/590 workstation and it took about the same time to generate the large 364-mer structure (39.15 ˚ Acube). The conformational statistics and X-ray scattering patterns calculated for these structures agree well with the previous simulation and experimental results.
17
The Future
The future of computer simulations is often related to the fast increase of available computer power. Only 5-10 years ago most of the molecular simulation studies were performed on mainframes, while nowadays we have the same or even greater computer resources on the desktop. This power does not only make it possible to do the work, which required weeks of calculations not so long ago, in minutes, but it also provides the possibility to address different, larger scale problems, or to drop some simplifications, which were formerly necessary to make the simulations possible. For example, we see more and more studies using the atomistically detailed models. It also allows for increasing the size of the model, which on the one hand allows better investigation of the effect of the model size on the simulation results, and on the other, allow the phenomena, involving space scales of several nanometers to be addressed, as, for instance, plastic deformation in the polymer glass.8 However, despite these advances in computer power, development and application of the molecular simulations for more realistic polymer models is impossible without developing new techniques. The relaxation times of chain molecules in the melt can be as long as a tenth of a second,36 and to study the equilibrium properties of the polymer melt with just straightforward MD, as for the low molecular weight liquids, would require computer resources that are still far beyond those available today and in the near future. Recently proposed Monte Carlo methods, based on the chain rearrangements in the dense
18
system, can be a feasible alternative.16, 31, 32, 37 Another alternative, based on the ’time coarse-graining’ is based on the idea of extracting and following the slow mobility modes of the system.30, 43 The molecular model generation methods described in this paper rely on ‘Flory’s hypothesis’ and require the polymer conformation characteristics (persistence length or probabilities of different rotational isomeric states) as an input parameter. They are usually obtained assuming that the chain conformation characteristics in the melt or glass are the same as for the single isolated chain. However, this is not always valid; it has been found for polystyrene,(see Ref.6 for example), that the equilibrium fractions of different conformers in the amorphous state tend to be different from the RIS predictions. Thus it would be desirable to design the methods avoiding these assumptions. Computer simulation of polymers will certainly profit from the developments in the field of molecular simulation in general, such as: the recently proposed methods to study the phase equilibria in dense systems,44–46 advanced methods to treat the long-range interactions,47 new algorithms capable of taking advantage of new computer architectures,48 and others.
Acknowledgments
The author is grateful to N.J.Wagner and M.E.Paulaitis for their collaboration and encouragement in this research. Financial support from the Dow Chemical Company and the Delaware Research Partnership Fund during my stay at the 19
University of Delaware is gratefully acknowledged.
20
References
[1] R.H. Boyd.
Glass transition temperarures from molecular dynamics
simulations. Trends in polymer Scinece, 4:12–17, 1996. [2] D.N. Theodorou and U.W. Suter. Detailed molecular structure of a vinyl polymer glass. Macromolecules, 18:1467–1478, 1985. [3] P. Ludovice and U.W. Suter. Detailed molecular structure of a polar vinyl polymer glass. In J.Bicerano, editor, Computational Modeling of Polymers, page 401. Marcel Dekker, New York, 1992. [4] M. M.Mondello, H.J. Yang, H. Furuya, and R.J. Roe. Molecular dynamics simulation of atactic polystyrene. 1.comparison with x-ray scattering data. Macromolecules, 27:2566–2574, 1994. [5] R. Khare, M.E. Paulaitis, and S.R. Lustig. Generation of glass structures for molecular simulation of polymers containing large monomer units; application to polystyrene. Macromolecules, 26:7203, 1993. [6] R.F. Rapold and U.W. Suter. Static atomistic modelling of the structure and ring dynamics of bulk amorphous polystyrene. Macromol.Theory Simul., 3:19– 43, 1994. [7] M.J. Kotelyanskii, N.J. Wagner, and M.E. Paulaitis.
Building large
amorphous polymer structures: Atomistic simulation of glassy polystyrene. Macromolecules, 29:8497–8506, 1996. [8] M. Hutnik, F.T. Gentile, P.J. Ludovice, U.W. Suter, and A.S. Argon. Macromolecules, 24:5962, 1991.
21
[9] F.T. Gentile, S. Arizzi, U.W. Suter, and P.J. Ludovice.
Structural
characterization o polycarbonates for membrane applications by atomic level simulations. Ind.Eng.Chem.Res., 34:4193–4201, 1995. [10] C.F. Fan, T. Cagin, Z.M. Chen, and K.A. Smith.
Molecular modeling
of polycarbonate. 1. force field, static structure and mechanical properties. Macromolecules, 27:2383–2391, 1994. [11] R. Zhang and W.L. Mattice.
Atomistic modeling of a new thermoplastic
polyimide in the amorphous state: Structure and energetics. Macromolecules, 28:7454–7460, 1995. [12] V.J. Vasudevan and J.E. McGrath. Atomistic modeling of amorphous aromatic polybenzoxazoles. Macromolecules, 29:637–645, 1996. [13] G.D. Smith, D.Y. Yoon, W. Zhu, and M.D. Ediger. Comparison of equilibrium and dynamics properties of polymethylene melts of n − c44 h90 chains from simulations and experiments. Macromolecules, 27:5563–5569, 1994. [14] D. Rigby and R.J. Roe. Macromolecules, 22:2259. [15] A.A. Gusev, F. Mueller-Plathe, W.F. van Gunsteren, and U.W. Suter. Dynamics of small molecules in bulk polymers. Adv in Poly. Sci., 116:209– 246, 1994. [16] J.J. dePablo, M. Laso, and U.W. Suter. J.Chem.Phys., 96:2395, 1992. [17] E. Leontidis, U.W. Suter, M. Schultz, H.P. Luehti, A. Renn, and U.P. Wild. The mechanism of spectral shift and inhomogenieous broadening of aromatic chromophore in a polymer glass. J.Am.Chem.Soc., 117:7493–7507, 1995.
22
[18] J. Bicerano, editor. Computational Modeling of Polymers. Marcel Dekker, Inc., New York Basel Hong Kong, 1992. [19] E.A. Colbourn, editor. Computer Simulation of Polymers. Longman Scientific and Technical, 1994. [20] R.J. Roe, editor. Computer Simulation of Polymers. Prentice Hall, 1991. [21] L. Monnerie and U.W. Suter, editors.
Atomistic Modeling of Physical
Properties, volume 116 of Advances in Polymer Science. Springer-Verlag, Berlin Heidelberg New York London Paris Tokyo Hong Kong Barcelona Budapest, 1994. [22] M.P.Allen and D.J.Tildesley.
Computer Simulation of Liquids. Clarendon
Press, Oxford, 1993. [23] S.J. Weiner, P.A. Kollman, D.A. Case, U.C. Singh, C. Ghio, G. Alagona, S. Profeta, and P. Weiner. A new force-field for molecular mechanical simulation of nucleic acids and proteins. J.Am.Chem.Soc., 106:765–784, 1984. [24] B.R. Brooks, R.E. Bruccoleri, B.D. Olafson, D.J. States, S. Swaminathan, and M. Karplus. Charmm: A program for macromolecular energy, minimization, and dynamics calculations. J.of. Comp.Chem., 4:187–217, 1983. [25] Biosym. Insight II User Guide, version 2.3.0. Biosym Technologies, SanDiego, 1993. [26] G.D. Smith, R.L. Jaffe, and D.Y. Yoon.
Conformational characteristics
poly(tetrafluoroethylene) chains based upon ab initio electronic structure calculations on model molecu les. Macromolecules, 27:3166–3173, 1994.
23
[27] W.J. Welsh and J.E. Mark. Application of quantum mechanical techniques to rigid rod polymers. In J.Bicerano, editor, Computational Modeling of Polymers, pages 581–624. Marcel Dekker, New York, 1992. [28] McQuarrie. Statistical Thermodynamics. Harper & Row, New York, 1973. [29] N. Metropolis, A.W. Rosenbluth, M.N. Rosenbluth, A.H. Teller, and E. Teller. J.Chem.Phys., 21:1087, 1953. [30] E. Leontidis, B.M. Forrest, A.H. Widmann, and U.W. Suter.
Monte
carlo algorithms for the atomistic simulation of condensed polymer phases. J.Chem.Soc.Faraday Trans, 91:2355–2368, 1995. [31] L.R. Dodd, T.D. Boone, and D.N. Theodorou. Molecular Physics, 78:961, 1993. [32] J.I. Siepmann and D. Frenkel. Mol.Phys., 75:59, 1992. [33] J. Baschnagel and K. Binder. Monte-carlo simulation of the glass-transition in polymeric systems - recent developments. Phil.Mag.B, 71:583–596, 1995. [34] N.F.A. van der Vegt, W.J. Briels, M. Wessling, and H. Strathmann. J.Chem.Phys., 105:8849, 1996. [35] J.I. McKechnie, D. Brown, and J.H.R. Clarke. Macromolecules, 25:1562, 1992. [36] M.Doi S.F.Edwards. The Theory of Polymer Dynamics.
Clarendon Press
Oxford, 1994. [37] E. Leontidis, J.J. dePablo, M. Laso, and U.W. Suter. A critical evaluation of novel algorithms for the off-lattice monte carlo simulation of condensed polymer phases. Advances in Polymer Science, 116:283–318, 1994.
24
[38] P.J.Flory. Statistical Mechanics of Chain Molecules. Hanser Publishers, 1989.
[39] S. Gupta, G.B. Westermann-Clark, and I. Bitsanis.
A continuous monte
carlo method for growth and equilibration of polymer chains at high densities. J.Chem.Phys., 98:634–637, 1993.
[40] A.A. Gusev, M.M. Zender, and U.W. Suter. Elasticity of solid polymars as a result of thermal motions. Macromolecules, 27:615–616, 1994.
[41] J. Gao.
An efficient method of generating dense polymer model melts by
computer simulation. J.Chem.Phys., 102:1074–1077, 1995.
[42] T. Pakula. Cooperative relaxations in condensed macromolecular systems. 1. a model for computer simulation. Macromol., 20:679–682, 1987.
[43] M.J. Kotelyanskii and U.W. Suter.
Dynamic monte-carlo simulation of
polycarbonate. Polym. Prepr., Am. Chem. Soc., Div. Polym. Chem., 33:663– 664, 1992.
[44] A.Z. Panagiotopoulos. Mol.Phys., 61:813, 1987.
[45] M. Laso, J.J. dePablo, and U.W. Suter. J.Chem.Phys., 97:2817, 1992.
[46] M.J. Kotelyanskii and R. Hentschke.
Gibbs-ensemble molecular dynamics:
Liquid-gas equilibria for lennard-jones spheres and n-hexane.
Molecular
Simulations, 17:95–112, 1996.
[47] S.R. Lustig, S. Rastogi, and N. Wagner. Telescoping fast multipole methods using chebyshev economization. J.of Comp.Phys., 122:317–322, 1995.
25
[48] D. Brown, J.H.R. Clarke, M. Okuda, and T. Yamazaki.
A domain
decomposition parallel processing algorithm for molecular dynamics simulations of polymers. Comp.Phys.Comm., 83:1–13, 1994.
26
U
U
U
NB
b
a
U
tor
Fig. 1. Various types of interactions between the polymer chain atoms. Bond stretching, Ub ; bond angle, Ua ; torsional potential, Utor ; and the non-bonded interactions, UNB .
27
15’ 11 00 00 11 00 11 00 11 00 11 00 3 11 00 11 00 11 00 11 000 111 00 11 000 111 000 111 00 11 000 111 000 111 000 111 000 111 000 111 000 111 000 111 000 111 000 111 000 111 000 111 00 11 000 111 000 111 00 11 000 111 000 111 00 11 00 11 000 111 00 11 00 11 00 11 00 11 000 111 00 11 00 11 00 11 00 11 000 111 00 11 11 00 11 00 11 11 00 00 00 11 00 11 00 11 00 11 00 11 00 11 00 11 00 11 00 11 000 111 00 11 000 111 00 11 00 11 00 11 000 111 000 111 00 11 000 111 00 11 00 11 000 111 000 111 00 11 000 111 00 11 000 111 000 111 000 111 000 111 00 11 000 111 000 111 000 111 000 111 000 111 000 111 000 111 000 111 000 111 000 111 15 000 111 000 111 000 111 000 111 000 111 000 111
Fig. 2. Polymer chain, folded into the periodic cell and its periodic images. ‘Primary’ box is shown in bold line, and monomers within it are are shown as filled circles. Whenever, the chain contour leaves the ‘primary’ box, it reenters from the opposite side. Non-bonded interactions are calculated using the periodic boundary conditions. Interaction between a pair of monomers 3 and 15 (dashed arrow line) is calculated as the interaction between their closest periodic images 3 and 15’ (bold arrow line).
28
Fig. 3. A 3-bond fragment of the lattice chain path (dashed line) ‘decorated’ with polystyrene atoms. Building blocks, containing polystyrene atoms are placed on the lattice sites. All repeat unit atoms are contained inside the block, to prevent the overlaps between the atoms. Blocks are oriented in such a way, that aliphatic carbons are not screened by the phenylene rings.
29
