- Email: [email protected]
Molecular modelling of polymers
596
Molecular modelling of polymers Julian- H R Clarke Significant
advances
have been made in the application
molecular
modelling
techniques
properties
of polymers.
to understanding
The highlights include the use of
detailed atomistic studies of segntental motions in amorphous interpretation of diblock
polymers
of experimental
copolymer
of
the
motions and penetrant
aimed at enhancing
data, and mesoscale
microphase
the modelling
separation.
Addresses Chemistry Department, UMIST, Manchester M60 1 CD, UK; e-mail: [email protected] Current Opinion in Solid State & Materials Science 1998, 3:596-599 Electronic identifier: 1359-0286-003-00596 0 Current Chemistry Ltd ISSN 1359-0286 Abbreviations DqD dissipative particle dynamics glass transition temperature Ts
Introduction It is now more than ten years since atomistic modelling began to make an impact in synthetic polymer science. There have been reviews of this early progress, and one of th&e appeared recently [ 11. Consolidation and diversification describe the thrust of new developments since the beginning of 1997. There have been several new atomistic studies aimed at enhancing the interpretation of experimental data in terms of structure and mechanisms, and also at analysing the motion of small penetrant gas molecules. There has also been some effort to improve the force fields used to model polymer interactions so as to reproduce more realistically the static and dynamic properties. For the first time ab initio molecular dynamics simulations have been applied to investigate the polymerization process. There have been several new and exciting developments in larger scale (mesoscopic) simulations of polymers. These are the areas that will be the focus of this review.
incredibly high quench rates (> 1010 KS-~). One therefore, expects values of T, (glass transition temperature) to be higher, transitions to be broader than observed in the laboratory and one cannot expect the same amount of densification as in the laboratory. This was recognized in a recent molecular dynamics study of poly(ethylene terephtlialate) [3]. The model was set up so that the characteristic ratio, the dipolar correlation factors and density were satisfactorily represented in the melt but it was found that it was not possible to achieve the experimental glass density below T, One manifestation of glass formation in polymers is the freezing out of conformational transitions [l]. This does not necessarily all happen at T,, however, and there is a whole body of dielectric relaxation data for instance on subglass transitions in polymers [4]. Recent molecular dynamics simulations have been carried out on amorphous polyethylene at temperatures below the experimental glass transition in order to try and characterize the character of the torsional motions contributing to the y-relaxation in amorphous polyethylene and to compare the relaxation spectrum with dielectric relaxation data [S’]. In reality this polymer shows three transitions, due to partial crystallization, glass formation in the amorphous part and a subglass y-transition. In nanosecond simulations of small samples the probability of crystallization is sufficiently small so that purely amorphous samples can be investigated. Dielectric relaxation can be observed in slightly oxidized polyethylene and the experimental loss peak map is shown in Figure 1, where the data are compared with the simulation results. Although there is hardly any overlap in the frequency regimes of simulation and experiment there is still an apparent continuity between the two sets of results. The interpretation of the molecular dynamics results is that the y-process is characterized predominantly by next nearest neighbour cooperative conformational transitions (e.g. the migration ofgau& conformations gtt-‘ttg), which are also important in the melt. They just become less frequent as the temperature is lowered.
Dynamics in polymer melts and glasses The dynamic structure factor for single-chain relaxation has been studied by molecular dynamics for polyethylene melts composed of chains with sizes less than the entanglement length, which is the regime where Rouse dynamics are expected. The simulation results show good agreement widh those from neutron spin echo spectroscopy [Z]. There have been several studies aimed at characterizing the glassy state in polymers although this is a controversial issue in molecular dynamics simulations [l]. A great deal of care must be exercised when comparing simulation results with experiment. Even if a good force field is chosen for a particular polymer which yields a good fit of, say, pVT (pressure, volume, temperature) properties in the melt, no satisfactory way has yet been found of forming glassy samples except at
The dynamics of bisphenol A polycarbonate has been studied close to the simulation glass transition temperature [6]. The frequency and activation energy of the phenylene ring-flip at 300K are close to those found by NMR. In this case conformational transitions between adjacent torsion angles appear to be strongly correlated. As in the case of polyethylene it appears that it is the probability rather than the character of these cooperative transitions which changes with temperature. Processes .occurring on or beyond 100 ns must be considered as rare events on the time scale of molecular dynamics simulations and one expects more advantage will be taken in future of special techniques that were originally
Molecular modelling of polymers
developed for examining, for example, the mechanisms of chemical reactions.[7]. The only requirement is that the physical process can be described in a straightfo~ard way. Much can also be learned from the use of approximate methods and one example of this is a recent investigation of methyl group rotation in poly(methy1 methacrylate) using a quasistatic approach [B]. Essentially the conformational change of interest is driven in steps by a forcing potential so that the energy profile along the transition path can be mapped out. So far this has only been performed on static con~gurations but there is no reason why in favourable cases it could not also be performed during molecular dynamics simulations using suitable constraints. As in the potential of mean force calculations this would also yield the free energy profile across the transition state.
Clarke
597
Figure 1
8.0 -
Solubiiity and mobility of penetrants Understanding the factors controlling solubility and mobility of gas molecules continues to be a subject of enormous practical importance with respect to both the barrier and permeation properties of polymer films. It has been shown that the diffusion of a penetrant involves occasional jumps between cavities through the opening of a ‘neck’ due to thermal motions within a glassy polymer f9]. It was found that the jump rate did not seem to correlate simply with changes in particular types of degrees of freedom but was, perhaps understandably an order of magnitude faster in the neighbourhood of chain ends. This emphasizes the importance bf using long chains in these simulations. These jump rates are rare events on the time scales of atomistic simulations and a good deal of effort has been put into estimating diffusion coefficients using multidimensional transition-state theory (which is reviewed in [ 11). One interesting new approach [lo’] is to treat the process of diffusion’as a unimolecular rearrangement and to develop semiempirical means of estimating the activation energy, frequency factor and jump length. Prescriptions are given [lo’] for calculating these parameters from bulk properties of the polymer plus penetrant determined by short time simulations. This is a promising approach which reproduces experimental trends and, with further development, could be used to interpolate and extrapolate experimental diffusion data. A method of estimating the water content of a hydrophilic polymer which employs a combination of thermodynamic integration and Widom particle insertion methods has been described [ll]. The method was tested successfully on amorphou samples of various polyamide oligomers. The results show reasonable agreement with experimental data for equilibrium water contents. Studies of the diffusion of water in mixtures of water and poly(viny1 alcohol) show that the diffusion mechanism changes from random walk in pure water to a hopping mechanism at high polymer concentration [ 121. This may lead to anomalous diffusion at length scales comparable to the thickness of the active layer of membranes.
Illustrating the continuity between molecular dynamics (MD) results and dielectric relaxation experiments for the ~transition in polyethylene. The MD results were obtained by frequency transforming the autocorrelation function for the motions of vectors bisecting the angle formed by and in the same plane of CH,-CH,-CH, triplets. The circles and squares are experimental data for either slightly oxidized (filled symbols) or lightly chlorinated (open symbols) material; triangles are MD data. Reproduced with permission from WI.
Quantum simulations
of polymers
One new area of activity which holds great promise is the use of ab i&o molecular dynamics simulations based on density functional theory. This has been used for a detailed examination of the initiation step in the polymerization of isoprene induced by ethyl lithium [13’]. The polymerization is known to proceed with a high stereoselectivity to (Z)-1,4-polyisoprene. Using simulated annealing techniques it was found that a &-isomer was energetically most favourable in the initiation step. The calculations indicate that c&tram isomerization turns out to be both rate determining and the stereoselective step. Lithium plays a crucial role by preferentially stabilizing specific structures through the formation of ‘chelated’ complexes and agostic interactions. A rather different application of molecular dynamics based on density functional theory is the calculation of Youngs modulus for crystalline polyethylene [ 141. Quantum effects on various properties of the orthorhombic phase of crystalline polyethylene have also been investigated by path integral Monte Carlo methods [ 151.
Towards more realistic force fields Realistic studies of molecular motion in polymers demand the use of satisfactory force fields. There has been a sustained effort to optimize the force fields used
598
Polymers
Figure 2
obtain estimates of the molar-mass material.
solubility
parameter
of
high-
A new united atom force field for simulations of l,l-polybutadiene based on nl/ itlitio quantum chcmistr! calculations on model motccules has been de\.etoped [IO]. ‘I’he characteristic ratio and its temperature dependence for L-1 ,+polybutadiene and ~/UK-I .~-pol~hlltadienc. and the characteristic ratio of a random copolymer of Cc and tram units, as predicted by 3 tile KIS (rotational isomeric states) model, are in reasonable agreement with cxpcrimental values. ‘l’he model also gives good agrecmcnt \vith the experimental melt density of the polymer as a function of temperature.
Mesoscopic simulations
Time evolution of the mesoscopic structure of a model AsB, block copolymer. Top left shows an unstable gyroid phase. The interval between successive pictures is 2000 time units. The top right structure shows the beginning of a transformation into a rod-like morphology, Rods are more clearly visible In the bottom left structure and, after a final 2000 time units, all the sideways broken. Reproduced with permission from [27”].
connections
are
to model saturated hydrocarbon chains, largely based on studies of short chain alkancs. ‘I’here u-e two approaches either fully atomistic or the so-called united atom model: in the latter both the carbon und hydrogen atoms of either CHz or (:H, groups are represcntcd by single interaction centres, either centred on the carbon atom (CIA model) or slightly displaced from the centre (AIlA model). ‘I’he attraction of these models is that they provide up to a factor of ten increase in simulation speed. ‘I’he AlJA model shows better agreement with experiment than the ITA model for the equations of state of walkanes up to decane [ 161. \‘arious fully atomistic fields ha1.c been compared in the calculation of vapor/liquid phase equilibria in I/-alkanes using Ylonte (Lrto simulations [17]. ITp to u-octane the L\‘illi;lms and OFIS-AA force fields yield liquid denand critical points that are in sitics, boiling temperatures. acceptable, albeit not in quantitative agrecmcnt with experiments, \vhercas the fluid-phase behavior of the \lhIFFC)-l model shohvs very large deviations. A new atomistic force field for amorphous pol\;(ethylene oxides) has been described [1X]. Simulations are able to predict densities which agree with experiment to within 1%-Z% over extended ranges of at lust ZOOK in tetnperature and 180 hlPa in pressure. Solubility parameters have brcn calculated as a function of chain length for poly(ethylene glycol) oligomers and LIX~ effectively to
of polymers
‘I’here are many important properties of polymers that involve large distance or time scales, for cxamplc, microphase separation in copolymers. In these cases the computing requirements for fully atomistic simiilations exceed what is available by several orders of magnitude. In an> case for many applications the degree of detail pro\,ided in wch simulations may be unnecessary Several straw gies have evolved to enable modelling on larger distuncc scales. l:or instance the lattice bond fluctuation nmdcl has been widely used and recently reviaved [ZO]. ‘I’herc ha\,e also been significant developments in the LISC of so-called ‘coarse grain’ continuous spacc simulation techniqiics based on the bead-spring model of ;I polymer 121 1. Another
recent approach is to LISC dissipati\,e prticlc dynamics (DPD) [ZZ’]. ‘I’h’IS introduces the concept of ‘fluid particles’ [23] which are attached by springs and again act as centrcs of mass but which each reprrscnt ;I large number of atoms. The equations of motion also inwh e frictional and random forces and arc somewhat similar to 13rou nian dynamics except that in the case of 1)I’l) linear momentum is conserved, a condition Lvhich is required to reco\ er h? drodynamic behaviour in the limit of large distance and time scales. One important difference from the bead polymer model discussed above is that these particles interact by an extremely soft repulsive force: in fact there is a tinitc probability that the particles can actually pass through each other, a feature \vhich considerabl\speeds LIP the relaxation of polymer melt structures. Although there seem to be siomc worrying inconsistencies in the method [24] it has been shown to reproduce an N’V’I’ ensemble if the fluctuationdissipation relation is satisfied [ZZ*]. Although either coarse-grain or lIPI> simulations are relatimely easy to perform the fundamental challenge is to establish 3 basis for interpreting what a coarse-grain simulation means in terms of the chemical constitution of 3 sample. A direct approach to solving this problem was IISUI in recent simulations of polycarbonate melts using the bead-spring model where bead interaction parameters were determined from an atomistic model b); ri numerical
Molecular
procedure. The effective simulation speedup was - lo3 [25,26]. In the case of DPD simulations of copolymers a more indirect approach has been adopted in which the parametrization was based on fits to the interfacial tension and the Flory-Huggins x parameters for the two components [27”]. This latter study provides interesting insight into the pathway along which a block copolymer melt finds its equilibrium structure after a temperature quench, as shown in Figure 2. renormalization
References
and recommended
reading
l
Clarke JHR: Molecular dynamics modelling of amorphous polymers. In The Physics of Glassy Polymers. Edited by Young RJ, Haward RN. London: Chapman and Hall; 1997:33-83.
2.
Paul W, Smith GD, Yoon DY, Farago B, Rathgeber S, Zirkel A, Willner L, Richter D: Chain motion in an unentangled polyethylene melt: a critical test of the rouse model by molecular dynamics simulations and neutron spin echo spectroscopy. Phys Rev Lett 1998, 80:2346-2349.
3.
4.
Hedenqvist MS, Bharadwaj R, Boyd RH: Molecular dynamics simulation of amorphous poly(ethylene terephthalate). Macromolecules 1998, 31 :1556-l 564. McCrum NG, Read BE, Williams G: Ane/as&z and Dielectric in Polymeric Solids. New York: Wiley; 1967.
Effects
5. Jin Y, Boyd RH: Subglass chain dynamics and relaxation in . polyethylene. J Chem Phys 1998, 108:9912-9923. 450 ns molecular dynamics simulations are used to characterize the y-process in amorphous polyethylene and to show reasonable agreement for relaxation times with dielectric relaxation data. The process is dominated by correlated next-nearest-neighbour conformational transitions. 6.
7.
8.
9.
599
11.
Knooo B. Suter UW: Atomisticallv modeling the chemical Dotential of s’iall ‘molecules in dense polimer micr&tructures. 2. iflater sorption by polyamides. Macromolecules 1997, 30:6114-6119.
12.
MullerPlathe F: Diffusion of water in swollen poly(vinyl alcohol) membranes studied by molecular dynamics simulation. J Membrane Sci 1998,141:147-154.
Rothlisberger U, Sprik M, Klein ML: Living polymers - ab inifio molecular dvnamics studv of the initiation steo in the polymerizatibn of isoprene induced by ethyl liihium. J Chem Sot Faraday Trans 1998,94:501-506. .Shows that ab m/t!o molecular dynamics slmulatlons based on density tunctional theory can be used directly to study the initiation step of polymerization reactions. 13. .
14.
Hageman JCL, Meier RJ, Heinemann M, deGroot RA: Young modulus of crystalline polyethylene from ab initio molecular dynamics. Macromolecules 1997, 30:5953-5957.
15.
Martonak R, Paul W, Binder K: Orthorhombic phase of crystalline polyethylene: a constant pressure path-integral Monte Carlo study. Phys Rev E 1998, 57:2425-2437.
16.
Toxvaerd S: Equation of state of alkanes. 2. J Chem Phys 1997, 107:5 197-5204.
1 7.
Chen B, Martin MG, Siepmann JI: Thermodynamic properties of the Williams, OPLS-AA, and MMFF94 all-atom force fields for normal alkanes. J Phys Chem B 1998, 102:2578-2586.
18.
Rigby D, Sun H, Eichinger BE: Computer simulations of poly(ethylene oxide): force field, PVT diagram and cyclization behaviour. Po/ym lnt 1997,44:31 l-330.
19.
Smith GD. Paul W: United atom force field for molecular dvnamics simulations of 1,4-polybutadine based on quantum chemi&y calculations on model molecules. J Phys Chem A 1998, 102:1200-l 208.
20.
Binder K, Paul W: Monte Carlo simulations of polymer dynamics: recent advances. J Polym Sci Part B-Polym Phys 1997, 35:1-31.
21.
Murat M, Kremer K: From many monomers to many polymers: soft ellipsoid model for polymer melts and mixtures. J Chem Phys 1998, 108:4340-4348.
of special interest * of outstanding interest
1.
Clarke
GrayWeale AA, Henchman RH, Gilbert RG, Greenfield M, Theodorou DN: Transition-state theory model for the diffusion coefficients of small penetrants in glassy polymers. Macromolecules 1997, 30:7296-7306. This article shows that the occasional jumps between cavities which characterize diffusion of small molecules in a polymer can be treated using transition state theory. Experimental trends can be reproduced using semi-empirical means of estimating the activation energy, frequency factor and jump length.
Papers of particular interest, published within the annual period of review, have been highlighted as: l
of polymers
10. .
Conclusions Applications of atomistic modelling to enhance the interpretation of experimental data, particularly relating to the dynamic properties of polymers continue to grow. As a result of instrumental and computational developments there is is now a convergence in the time and length scales accessible to neutron scattering and computer simulation, which should stimulate increasing opportunities for establishing close links between modelling and experiment. The other exciting development area is in mesoscale modelling of polymers which provides the ability to study phenomena on time and distance scales lo3 or more larger than those in atomistic simulations.
modelling
22. .
Groot RD, Warren PB: Dissipative particle dynamics: bridging the gap between atomistic and mesoscopic simulation. J Chem Phys 1997, 107~4423-4435. This paper describes a method of parametrizing large scale simulations effectively describing millions of atoms by firstly performing simulations of molecular fragments retaining all atomistic details to derive Floty X-parameters, then using these results as input to a dissipative particle dynamics simulation to study the formation of micelles, networks, mesophases, and so on. 23.
Espanol P: Fluid particle model. Phys Rev E 1998, 57:2930-2948.
24.
Fan CF, Cagin T, Shi W, Smith KA: Local chain dynamics of a model polycarbonate near glass transition temperature: a molecular dynamics simulation. Macromol Theory Simull997, 6:83-i 02.
Pagonabarraga I, Hagen MHJ, Frenkel D: Self-consistent dissipative particle dynamics algorithm. furophys Lett 1998,42:377-382.
25.
Tschop W, Kremer K, Batoulis J, Burger T, Hahn 0: Simulation of polymer melts. I. Coarse-graining procedure for polycarbonates. Acta Po/ym 1998,49:61-74.
Carter EA, Ciccotti G, Hynes JT, Kapral R: Constrained reaction coordinate dynamics for the simulation of rare events. Chem Phys Left 1989, 156:472-477.
26.
Tschop W, Kremer K, Hahn 0, Batoulis J, Burger T: Simulation of polymer melts. II. From coarse-grained models back to atomistic description. Acta PoIym 1998,49:75-79.
Nicholson TM, Davies GR: Modeling of methyl group rotations in PMMA. Macromolecules 1997, 30:5501-5505. Greenfield ML, Theodorou DN: Coupling of penetrant and polymer motions during small-molecule diffusion in a glassy polymer. Molec Simull997, 19:329.
27. Groot RD, Madden TJ: Dynamic simulation of diblock copolymer .. microphase separation. J Chem Phys 1998,108:87 13-8724. This demonstrates the strengths of the dissipative particle dynamics method, in this case predicting the dynamical pathway along which a block copolymer melt finds its equilibrium structure after a temperature quench.
Molecular modelling of polymers Julian- H R Clarke Significant
advances
have been made in the application
molecular
modelling
techniques
properties
of polymers.
to understanding
The highlights include the use of
detailed atomistic studies of segntental motions in amorphous interpretation of diblock
polymers
of experimental
copolymer
of
the
motions and penetrant
aimed at enhancing
data, and mesoscale
microphase
the modelling
separation.
Addresses Chemistry Department, UMIST, Manchester M60 1 CD, UK; e-mail: [email protected] Current Opinion in Solid State & Materials Science 1998, 3:596-599 Electronic identifier: 1359-0286-003-00596 0 Current Chemistry Ltd ISSN 1359-0286 Abbreviations DqD dissipative particle dynamics glass transition temperature Ts
Introduction It is now more than ten years since atomistic modelling began to make an impact in synthetic polymer science. There have been reviews of this early progress, and one of th&e appeared recently [ 11. Consolidation and diversification describe the thrust of new developments since the beginning of 1997. There have been several new atomistic studies aimed at enhancing the interpretation of experimental data in terms of structure and mechanisms, and also at analysing the motion of small penetrant gas molecules. There has also been some effort to improve the force fields used to model polymer interactions so as to reproduce more realistically the static and dynamic properties. For the first time ab initio molecular dynamics simulations have been applied to investigate the polymerization process. There have been several new and exciting developments in larger scale (mesoscopic) simulations of polymers. These are the areas that will be the focus of this review.
incredibly high quench rates (> 1010 KS-~). One therefore, expects values of T, (glass transition temperature) to be higher, transitions to be broader than observed in the laboratory and one cannot expect the same amount of densification as in the laboratory. This was recognized in a recent molecular dynamics study of poly(ethylene terephtlialate) [3]. The model was set up so that the characteristic ratio, the dipolar correlation factors and density were satisfactorily represented in the melt but it was found that it was not possible to achieve the experimental glass density below T, One manifestation of glass formation in polymers is the freezing out of conformational transitions [l]. This does not necessarily all happen at T,, however, and there is a whole body of dielectric relaxation data for instance on subglass transitions in polymers [4]. Recent molecular dynamics simulations have been carried out on amorphous polyethylene at temperatures below the experimental glass transition in order to try and characterize the character of the torsional motions contributing to the y-relaxation in amorphous polyethylene and to compare the relaxation spectrum with dielectric relaxation data [S’]. In reality this polymer shows three transitions, due to partial crystallization, glass formation in the amorphous part and a subglass y-transition. In nanosecond simulations of small samples the probability of crystallization is sufficiently small so that purely amorphous samples can be investigated. Dielectric relaxation can be observed in slightly oxidized polyethylene and the experimental loss peak map is shown in Figure 1, where the data are compared with the simulation results. Although there is hardly any overlap in the frequency regimes of simulation and experiment there is still an apparent continuity between the two sets of results. The interpretation of the molecular dynamics results is that the y-process is characterized predominantly by next nearest neighbour cooperative conformational transitions (e.g. the migration ofgau& conformations gtt-‘ttg), which are also important in the melt. They just become less frequent as the temperature is lowered.
Dynamics in polymer melts and glasses The dynamic structure factor for single-chain relaxation has been studied by molecular dynamics for polyethylene melts composed of chains with sizes less than the entanglement length, which is the regime where Rouse dynamics are expected. The simulation results show good agreement widh those from neutron spin echo spectroscopy [Z]. There have been several studies aimed at characterizing the glassy state in polymers although this is a controversial issue in molecular dynamics simulations [l]. A great deal of care must be exercised when comparing simulation results with experiment. Even if a good force field is chosen for a particular polymer which yields a good fit of, say, pVT (pressure, volume, temperature) properties in the melt, no satisfactory way has yet been found of forming glassy samples except at
The dynamics of bisphenol A polycarbonate has been studied close to the simulation glass transition temperature [6]. The frequency and activation energy of the phenylene ring-flip at 300K are close to those found by NMR. In this case conformational transitions between adjacent torsion angles appear to be strongly correlated. As in the case of polyethylene it appears that it is the probability rather than the character of these cooperative transitions which changes with temperature. Processes .occurring on or beyond 100 ns must be considered as rare events on the time scale of molecular dynamics simulations and one expects more advantage will be taken in future of special techniques that were originally
Molecular modelling of polymers
developed for examining, for example, the mechanisms of chemical reactions.[7]. The only requirement is that the physical process can be described in a straightfo~ard way. Much can also be learned from the use of approximate methods and one example of this is a recent investigation of methyl group rotation in poly(methy1 methacrylate) using a quasistatic approach [B]. Essentially the conformational change of interest is driven in steps by a forcing potential so that the energy profile along the transition path can be mapped out. So far this has only been performed on static con~gurations but there is no reason why in favourable cases it could not also be performed during molecular dynamics simulations using suitable constraints. As in the potential of mean force calculations this would also yield the free energy profile across the transition state.
Clarke
597
Figure 1
8.0 -
Solubiiity and mobility of penetrants Understanding the factors controlling solubility and mobility of gas molecules continues to be a subject of enormous practical importance with respect to both the barrier and permeation properties of polymer films. It has been shown that the diffusion of a penetrant involves occasional jumps between cavities through the opening of a ‘neck’ due to thermal motions within a glassy polymer f9]. It was found that the jump rate did not seem to correlate simply with changes in particular types of degrees of freedom but was, perhaps understandably an order of magnitude faster in the neighbourhood of chain ends. This emphasizes the importance bf using long chains in these simulations. These jump rates are rare events on the time scales of atomistic simulations and a good deal of effort has been put into estimating diffusion coefficients using multidimensional transition-state theory (which is reviewed in [ 11). One interesting new approach [lo’] is to treat the process of diffusion’as a unimolecular rearrangement and to develop semiempirical means of estimating the activation energy, frequency factor and jump length. Prescriptions are given [lo’] for calculating these parameters from bulk properties of the polymer plus penetrant determined by short time simulations. This is a promising approach which reproduces experimental trends and, with further development, could be used to interpolate and extrapolate experimental diffusion data. A method of estimating the water content of a hydrophilic polymer which employs a combination of thermodynamic integration and Widom particle insertion methods has been described [ll]. The method was tested successfully on amorphou samples of various polyamide oligomers. The results show reasonable agreement with experimental data for equilibrium water contents. Studies of the diffusion of water in mixtures of water and poly(viny1 alcohol) show that the diffusion mechanism changes from random walk in pure water to a hopping mechanism at high polymer concentration [ 121. This may lead to anomalous diffusion at length scales comparable to the thickness of the active layer of membranes.
Illustrating the continuity between molecular dynamics (MD) results and dielectric relaxation experiments for the ~transition in polyethylene. The MD results were obtained by frequency transforming the autocorrelation function for the motions of vectors bisecting the angle formed by and in the same plane of CH,-CH,-CH, triplets. The circles and squares are experimental data for either slightly oxidized (filled symbols) or lightly chlorinated (open symbols) material; triangles are MD data. Reproduced with permission from WI.
Quantum simulations
of polymers
One new area of activity which holds great promise is the use of ab i&o molecular dynamics simulations based on density functional theory. This has been used for a detailed examination of the initiation step in the polymerization of isoprene induced by ethyl lithium [13’]. The polymerization is known to proceed with a high stereoselectivity to (Z)-1,4-polyisoprene. Using simulated annealing techniques it was found that a &-isomer was energetically most favourable in the initiation step. The calculations indicate that c&tram isomerization turns out to be both rate determining and the stereoselective step. Lithium plays a crucial role by preferentially stabilizing specific structures through the formation of ‘chelated’ complexes and agostic interactions. A rather different application of molecular dynamics based on density functional theory is the calculation of Youngs modulus for crystalline polyethylene [ 141. Quantum effects on various properties of the orthorhombic phase of crystalline polyethylene have also been investigated by path integral Monte Carlo methods [ 151.
Towards more realistic force fields Realistic studies of molecular motion in polymers demand the use of satisfactory force fields. There has been a sustained effort to optimize the force fields used
598
Polymers
Figure 2
obtain estimates of the molar-mass material.
solubility
parameter
of
high-
A new united atom force field for simulations of l,l-polybutadiene based on nl/ itlitio quantum chcmistr! calculations on model motccules has been de\.etoped [IO]. ‘I’he characteristic ratio and its temperature dependence for L-1 ,+polybutadiene and ~/UK-I .~-pol~hlltadienc. and the characteristic ratio of a random copolymer of Cc and tram units, as predicted by 3 tile KIS (rotational isomeric states) model, are in reasonable agreement with cxpcrimental values. ‘l’he model also gives good agrecmcnt \vith the experimental melt density of the polymer as a function of temperature.
Mesoscopic simulations
Time evolution of the mesoscopic structure of a model AsB, block copolymer. Top left shows an unstable gyroid phase. The interval between successive pictures is 2000 time units. The top right structure shows the beginning of a transformation into a rod-like morphology, Rods are more clearly visible In the bottom left structure and, after a final 2000 time units, all the sideways broken. Reproduced with permission from [27”].
connections
are
to model saturated hydrocarbon chains, largely based on studies of short chain alkancs. ‘I’here u-e two approaches either fully atomistic or the so-called united atom model: in the latter both the carbon und hydrogen atoms of either CHz or (:H, groups are represcntcd by single interaction centres, either centred on the carbon atom (CIA model) or slightly displaced from the centre (AIlA model). ‘I’he attraction of these models is that they provide up to a factor of ten increase in simulation speed. ‘I’he AlJA model shows better agreement with experiment than the ITA model for the equations of state of walkanes up to decane [ 161. \‘arious fully atomistic fields ha1.c been compared in the calculation of vapor/liquid phase equilibria in I/-alkanes using Ylonte (Lrto simulations [17]. ITp to u-octane the L\‘illi;lms and OFIS-AA force fields yield liquid denand critical points that are in sitics, boiling temperatures. acceptable, albeit not in quantitative agrecmcnt with experiments, \vhercas the fluid-phase behavior of the \lhIFFC)-l model shohvs very large deviations. A new atomistic force field for amorphous pol\;(ethylene oxides) has been described [1X]. Simulations are able to predict densities which agree with experiment to within 1%-Z% over extended ranges of at lust ZOOK in tetnperature and 180 hlPa in pressure. Solubility parameters have brcn calculated as a function of chain length for poly(ethylene glycol) oligomers and LIX~ effectively to
of polymers
‘I’here are many important properties of polymers that involve large distance or time scales, for cxamplc, microphase separation in copolymers. In these cases the computing requirements for fully atomistic simiilations exceed what is available by several orders of magnitude. In an> case for many applications the degree of detail pro\,ided in wch simulations may be unnecessary Several straw gies have evolved to enable modelling on larger distuncc scales. l:or instance the lattice bond fluctuation nmdcl has been widely used and recently reviaved [ZO]. ‘I’herc ha\,e also been significant developments in the LISC of so-called ‘coarse grain’ continuous spacc simulation techniqiics based on the bead-spring model of ;I polymer 121 1. Another
recent approach is to LISC dissipati\,e prticlc dynamics (DPD) [ZZ’]. ‘I’h’IS introduces the concept of ‘fluid particles’ [23] which are attached by springs and again act as centrcs of mass but which each reprrscnt ;I large number of atoms. The equations of motion also inwh e frictional and random forces and arc somewhat similar to 13rou nian dynamics except that in the case of 1)I’l) linear momentum is conserved, a condition Lvhich is required to reco\ er h? drodynamic behaviour in the limit of large distance and time scales. One important difference from the bead polymer model discussed above is that these particles interact by an extremely soft repulsive force: in fact there is a tinitc probability that the particles can actually pass through each other, a feature \vhich considerabl\speeds LIP the relaxation of polymer melt structures. Although there seem to be siomc worrying inconsistencies in the method [24] it has been shown to reproduce an N’V’I’ ensemble if the fluctuationdissipation relation is satisfied [ZZ*]. Although either coarse-grain or lIPI> simulations are relatimely easy to perform the fundamental challenge is to establish 3 basis for interpreting what a coarse-grain simulation means in terms of the chemical constitution of 3 sample. A direct approach to solving this problem was IISUI in recent simulations of polycarbonate melts using the bead-spring model where bead interaction parameters were determined from an atomistic model b); ri numerical
Molecular
procedure. The effective simulation speedup was - lo3 [25,26]. In the case of DPD simulations of copolymers a more indirect approach has been adopted in which the parametrization was based on fits to the interfacial tension and the Flory-Huggins x parameters for the two components [27”]. This latter study provides interesting insight into the pathway along which a block copolymer melt finds its equilibrium structure after a temperature quench, as shown in Figure 2. renormalization
References
and recommended
reading
l
Clarke JHR: Molecular dynamics modelling of amorphous polymers. In The Physics of Glassy Polymers. Edited by Young RJ, Haward RN. London: Chapman and Hall; 1997:33-83.
2.
Paul W, Smith GD, Yoon DY, Farago B, Rathgeber S, Zirkel A, Willner L, Richter D: Chain motion in an unentangled polyethylene melt: a critical test of the rouse model by molecular dynamics simulations and neutron spin echo spectroscopy. Phys Rev Lett 1998, 80:2346-2349.
3.
4.
Hedenqvist MS, Bharadwaj R, Boyd RH: Molecular dynamics simulation of amorphous poly(ethylene terephthalate). Macromolecules 1998, 31 :1556-l 564. McCrum NG, Read BE, Williams G: Ane/as&z and Dielectric in Polymeric Solids. New York: Wiley; 1967.
Effects
5. Jin Y, Boyd RH: Subglass chain dynamics and relaxation in . polyethylene. J Chem Phys 1998, 108:9912-9923. 450 ns molecular dynamics simulations are used to characterize the y-process in amorphous polyethylene and to show reasonable agreement for relaxation times with dielectric relaxation data. The process is dominated by correlated next-nearest-neighbour conformational transitions. 6.
7.
8.
9.
599
11.
Knooo B. Suter UW: Atomisticallv modeling the chemical Dotential of s’iall ‘molecules in dense polimer micr&tructures. 2. iflater sorption by polyamides. Macromolecules 1997, 30:6114-6119.
12.
MullerPlathe F: Diffusion of water in swollen poly(vinyl alcohol) membranes studied by molecular dynamics simulation. J Membrane Sci 1998,141:147-154.
Rothlisberger U, Sprik M, Klein ML: Living polymers - ab inifio molecular dvnamics studv of the initiation steo in the polymerizatibn of isoprene induced by ethyl liihium. J Chem Sot Faraday Trans 1998,94:501-506. .Shows that ab m/t!o molecular dynamics slmulatlons based on density tunctional theory can be used directly to study the initiation step of polymerization reactions. 13. .
14.
Hageman JCL, Meier RJ, Heinemann M, deGroot RA: Young modulus of crystalline polyethylene from ab initio molecular dynamics. Macromolecules 1997, 30:5953-5957.
15.
Martonak R, Paul W, Binder K: Orthorhombic phase of crystalline polyethylene: a constant pressure path-integral Monte Carlo study. Phys Rev E 1998, 57:2425-2437.
16.
Toxvaerd S: Equation of state of alkanes. 2. J Chem Phys 1997, 107:5 197-5204.
1 7.
Chen B, Martin MG, Siepmann JI: Thermodynamic properties of the Williams, OPLS-AA, and MMFF94 all-atom force fields for normal alkanes. J Phys Chem B 1998, 102:2578-2586.
18.
Rigby D, Sun H, Eichinger BE: Computer simulations of poly(ethylene oxide): force field, PVT diagram and cyclization behaviour. Po/ym lnt 1997,44:31 l-330.
19.
Smith GD. Paul W: United atom force field for molecular dvnamics simulations of 1,4-polybutadine based on quantum chemi&y calculations on model molecules. J Phys Chem A 1998, 102:1200-l 208.
20.
Binder K, Paul W: Monte Carlo simulations of polymer dynamics: recent advances. J Polym Sci Part B-Polym Phys 1997, 35:1-31.
21.
Murat M, Kremer K: From many monomers to many polymers: soft ellipsoid model for polymer melts and mixtures. J Chem Phys 1998, 108:4340-4348.
of special interest * of outstanding interest
1.
Clarke
GrayWeale AA, Henchman RH, Gilbert RG, Greenfield M, Theodorou DN: Transition-state theory model for the diffusion coefficients of small penetrants in glassy polymers. Macromolecules 1997, 30:7296-7306. This article shows that the occasional jumps between cavities which characterize diffusion of small molecules in a polymer can be treated using transition state theory. Experimental trends can be reproduced using semi-empirical means of estimating the activation energy, frequency factor and jump length.
Papers of particular interest, published within the annual period of review, have been highlighted as: l
of polymers
10. .
Conclusions Applications of atomistic modelling to enhance the interpretation of experimental data, particularly relating to the dynamic properties of polymers continue to grow. As a result of instrumental and computational developments there is is now a convergence in the time and length scales accessible to neutron scattering and computer simulation, which should stimulate increasing opportunities for establishing close links between modelling and experiment. The other exciting development area is in mesoscale modelling of polymers which provides the ability to study phenomena on time and distance scales lo3 or more larger than those in atomistic simulations.
modelling
22. .
Groot RD, Warren PB: Dissipative particle dynamics: bridging the gap between atomistic and mesoscopic simulation. J Chem Phys 1997, 107~4423-4435. This paper describes a method of parametrizing large scale simulations effectively describing millions of atoms by firstly performing simulations of molecular fragments retaining all atomistic details to derive Floty X-parameters, then using these results as input to a dissipative particle dynamics simulation to study the formation of micelles, networks, mesophases, and so on. 23.
Espanol P: Fluid particle model. Phys Rev E 1998, 57:2930-2948.
24.
Fan CF, Cagin T, Shi W, Smith KA: Local chain dynamics of a model polycarbonate near glass transition temperature: a molecular dynamics simulation. Macromol Theory Simull997, 6:83-i 02.
Pagonabarraga I, Hagen MHJ, Frenkel D: Self-consistent dissipative particle dynamics algorithm. furophys Lett 1998,42:377-382.
25.
Tschop W, Kremer K, Batoulis J, Burger T, Hahn 0: Simulation of polymer melts. I. Coarse-graining procedure for polycarbonates. Acta Po/ym 1998,49:61-74.
Carter EA, Ciccotti G, Hynes JT, Kapral R: Constrained reaction coordinate dynamics for the simulation of rare events. Chem Phys Left 1989, 156:472-477.
26.
Tschop W, Kremer K, Hahn 0, Batoulis J, Burger T: Simulation of polymer melts. II. From coarse-grained models back to atomistic description. Acta PoIym 1998,49:75-79.
Nicholson TM, Davies GR: Modeling of methyl group rotations in PMMA. Macromolecules 1997, 30:5501-5505. Greenfield ML, Theodorou DN: Coupling of penetrant and polymer motions during small-molecule diffusion in a glassy polymer. Molec Simull997, 19:329.
27. Groot RD, Madden TJ: Dynamic simulation of diblock copolymer .. microphase separation. J Chem Phys 1998,108:87 13-8724. This demonstrates the strengths of the dissipative particle dynamics method, in this case predicting the dynamical pathway along which a block copolymer melt finds its equilibrium structure after a temperature quench.