Monte Carlo simulation of a lattice model for ternary polymer mixtures

Monte Carlo studies of symmetrical polymer mixturesAB, modelled by selfavoiding walks withN _A =N _B =N steps on a simple cubic lattice, are presented for arbitrary concentrations of vacancies _v in the range from _v =0.2 to _v =0.8 and chain lengthsN64. We obtained the phase diagrams and the equation of state for three choices of the ratio / _AB ( being the energy between monomers of the same kind, _AB being the energy between different monomers). Flory-Huggins theory provides only a qualitative understanding of these results. If the equation of state is fitted with an effective Flory-Huggins parameter _eff , the latter turns out to be strongly dependent on both concentration and temperature.Contributed paper delivered at the Tagung der Deutschen Physikalischen Gesellschaft, Fachausschuß Polymerphysik, Berlin, March 30–April 3, 1987.     

Monte Carlo simulation of phase separation of A/B/A‐B ternary mixtures

Monte Carlo simulations were used to model A/B/A‐B ternary mixtures with different AB diblock copolymer volume fractions for which both the dispersed and continuous phase volume fractions were kept constant. For concentrations of the diblock copolymer below a critical value, the domain size increment of the dispersed phase decreases linearly with the copolymer concentration. This is in agreement with the predictions of Noolandi and Hong. The dependence of the domain size as a function of the copolymer volume fraction can also be fitted by the equation of Tang and Huang. Our simulations indicate, for the first time, that the micelles form before saturation of the interface occurs. This means that the formation of the micelles is not a result of the saturation of the interface.     

Critical phenomena in polymer mixtures: Monte Carlo simulation of a lattice model

A lattice model of a symmetrical binary (AB) polymer mixture is studied, modelling the polymer chains by self-avoiding walks withN _A =N _B =N steps on a simple cubic lattice. If a pair of nearest neighbour sites is taken by different monomersAB orBA, an energy _ab is won; if the pair of sites is taken by anAA or aBB pair, an energy is won, while the energy is reduced to zero if at least one of the sites of the pair is vacant. To allow enough chain mobility, 20% of the lattice sites are vacancies. In addition to local motions of the chain segments we use a novel grand-canonical simulation technique:A chains are transformed intoB chains and vice versa, keeping the chemical potential difference fixed. The phase diagram is obtained forN=4, 8,16 and 32; the critical behaviour is analysed by finite-size scaling methods. It is shown that the critical exponents are those of the Ising model (=0.32,=0.63) rather than those of the Flory-Huggins meanfield theory (==1/2). Implications of these results for real polymers are briefly discussed.Paper presented at the Frühjahrstagung 1986, Polymerphysik, der Deutschen Physikalischen Gesellschaft, Kaiserslautern, March 12–14, 1986.     

Monte Carlo simulation of polymer mixtures: recent progress

The phase diagram of symmetrical polymer blends (A,B) confined into thin films is studied, considering both the effect of finite film thickness D and of surface forces at the confining walls that either prefer both the same species, or different species. In the case of <“>neutral<”> walls confinement enhances the compatibility of the blend. The critical temperature is depressed, the coexistence curve gets flattened (reflecting a crossover from 3‐dimensional to 2‐dimensional critical behavior). But if both walls preferentially attract species A, then also the critical composition of the blend is shifted to the A‐rich side of the phase diagram, and the coexistence curve exhibits a bulge just above the wetting transition temperature. If one wall attracts A and the other B, lateral phase separation sets in via a first order transition. Above this transition, an interface parallel to the walls is stabilized in the system.     

A Monte Carlo simulation of the inhomogeneous Lebwohl-Lasher lattice model

A Monte Carlo computer simulation of a modified Lebwohl-Lasher model is presented. The model consists of a set of interaction centres placed at the sites of a cubic lattice. The angular part of the pair potential is a second Legendre polynomial of the relative orientation between the two particles, like that of the Lebwohl-Lasher model. Each particle interacts with its six nearest neighbours with an attractive anisotropic potential differing in strength for the four horizontal and the two vertical neighbours. Various values of the in-plane to out-of-plane coupling ratio δ have been considered, i.e. δ = 0.75, 0.5, 0.1, 0.0. The latter case corresponds to the limiting situation of a two-dimensional lattice. Systems with a 1000 particles have been simulated for δ = 0.75, 0.5 and 0.1 while a sample of 3600 particles has been investigated for the two-dimensional lattice. Comparisons are made with available simulations and with mean field theory. We find that the molecular field theory predictions worsen as the effective coordination number is decreased. Energy, specific heat, second and fourth rank order parameters have been evaluated for the various models. We also present, for the first time, a way of approximately reconstructing the pair distribution, G(r₁₂, ω₁₂), using maximum entropy and second and fourth rank two particle order parameters.     

Monte Carlo simulation of polymer welding

Monte Carlo Modelling of random polymer chains, course grained onto a cubic F lattice, provides the ability to monitor the long range relaxation processes and the dynamic parameters of chains up to 400 units long. The model, described and verified by Haire et al. (Haire KR, Carver TJ, Windle AH. A Monte Carlo model for dense polymer systems and its interlocking with molecular dynamics simulation. Computational and Theoretical Polymer Science 2000; in press), is here applied to the study of molecular parameters in the vicinity of different types of surface and also to the process of polymer welding, whereby adhesion between two adjacent surfaces is achieved by the interpenetration of chains which are across the surface.The model demonstrates that a surface distorts the conformation of chains adjacent to it to give an oblate molecular envelope, that the concentration of vacant sites and chain ends increases near to the surface and that the density of points representing the centres of mass of the chains increases in the sub-surface regions. These results confirm earlier predictions and provide additional confidence in the model.Modelling of the welding process leads to the parameter intrinsic weld time, t_w, which is the time from initial perfect contact of the surfaces to the achievement of a weld within which the chain conformation is indistinguishable from the bulk. After the initial period in which the mating surfaces roughen, the welding proceeds according to the t^1/4 law predicted by reptation theory. The time to a given level of interdiffusion across the boundary is proportional to the chain length l, a comparatively weak dependence, while t_w is proportional to l³, a strong dependence. This is the same dependence on length as for the relaxation time of the chain end-to-end vectors. In fact, the agreement between the relaxation time, measured on the model of the bulk, and t_w is surprisingly close, at least for the monodisperse polymers investigated here.     

Monte Carlo simulation of polymer adsorption

We developed and employed the incremental gauge cell method to calculate the chemical potential (and thus free energies) of long, flexible homopolymer chains of Lennard-Jones beads with harmonic bonds. The free energy of these chains was calculated with respect to three external conditions: in the zero-density bulk limit, confined in a spherical pore with hard walls, and confined in a spherical pore with attractive pores, the latter case being an analog of adsorption. Using the incremental gauge cell method, we calculated the incremental chemical potential of free polymer chains before and after the globual-random coil transitions. We also found that chains confined in attractive pores exhibit behaviors typical of low temperature physisorption isotherms, such as layering followed by capillary condensation.     

Monte Carlo simulation of a polymer-analogous reaction in a polymer blend

The autocatalytic polymer-analogous reaction A → B in a blend composed of two contacting layers of compatible homopolymers A and B is studied by numerical simulation using the dynamic continuum Monte Carlo method. The evolution of the numerical density of units A and units initially belonged to the chains of homopolymer A is investigated in the course of the reaction and interdiffusion. Local characteristics of the distribution of the homopolymer with respect to its composition and blocks A and B with respect to their length are calculated at different times. The dispersions of the above distributions are appreciably higher than the corresponding dispersion of the Bernoullian copolymer of the same average composition, despite the random character of the reaction. This effect can be provided by changes in the composition of the blend on the scale of the reacting chain as well as by the diffusive mixing of the above chains. For the products of the polymer-analogous reaction, the broadening of the compositional distribution is predicted also by the theoretical model, which describes interdiffusion in the reacting system on scales that are markedly greater than the size of a polymer chain.     

Kinetic lattice Monte Carlo simulation of viscoelastic subdiffusion

We propose a kinetic Monte Carlo method for the simulation of subdiffusive random walks on a Cartesian lattice. The random walkers are subject to viscoelastic forces which we compute from their individual trajectories via the fractional Langevin equation. At every step the walkers move by one lattice unit, which makes them differ essentially from continuous time random walks, where the subdiffusive behavior is induced by random waiting. To enable computationally inexpensive simulations with n-step memories, we use an approximation of the memory and the memory kernel functions with a complexity O(log?n). Eventual discretization and approximation artifacts are compensated with numerical adjustments of the memory kernel functions. We verify with a number of analyses that this new method provides binary fractional random walks that are fully consistent with the theory of fractional Brownian motion.     

Monte Carlo simulation studies of polymer systems

Polymers molecules in solution or melt are more or less flexible and continuously change their shape and size. Thus, characteristic properties of the system fluctuate around statistical mean values which are dependent on the concentration of the solution, on the quality of the solvent used, and on the specific structure of the molecules, e.g. linear or star-branched. The most direct approach to these quantities on a molecular level are computer simulations. Due to restrictions of computer power fully atomistic simulations of macromolecules are presently still at the beginning but several arguments justify the use of simplified models. The most efficient way dealing with polymer systems are Monte Carlo simulations based on lattice chains, at least as long as static properties are of interest only. In the present paper a short introduction to the field is given and selected examples are presented in order to demonstrate the usefulness of these methods.     

A Monte Carlo lattice model for chain diffusion in dense polymer systems and its interlocking with molecular dynamics simulation

The paper reports the development of a Monte Carlo lattice model (cubic F) of polymer chains which is able to access times where the diffusion of the centre-of-mass of the chains is the dominant process, even though the chain lengths are well above that for entanglement. The volume of the model is large when compared with the volume of gyration of the individual molecules. The model incorporates an algorithm, which allows for the possibility of co-operative motions over sections of the chains and increases the time efficiency of the simulation. Both the model and the modifying algorithm have been tested against the known scaling laws.The model, for shorter chains, is ‘reverse mapped’ into full atomic detail as polyethylene and the shorter time processes simulated using molecular dynamics (MD). The MD model is tested against experimental diffusion data for polyethylene, of the same molecular weight and at the same temperature, and then used to time-calibrate the lattice model.Both the fine grained MD model and the coarse grained MC model are thus interlocked to cover a time range from the individual atomic motions of MD up to the order of a microsecond, a range of six orders of magnitude.     

Kinetic Monte Carlo modeling of chemical reactions coupled with heat transfer

In this paper, we describe two types of effective events for describing heat transfer in a kinetic Monte Carlo (KMC) simulation that may involve stochastic chemical reactions. Simulations employing these events are referred to as KMC-TBT and KMC-PHE. In KMC-TBT, heat transfer is modeled as the stochastic transfer of "thermal bits" between adjacent grid points. In KMC-PHE, heat transfer is modeled by integrating the Poisson heat equation for a short time. Either approach is capable of capturing the time dependent system behavior exactly. Both KMC-PHE and KMC-TBT are validated by simulating pure heat transfer in a rod and a square and modeling a heated desorption problem where exact numerical results are available. KMC-PHE is much faster than KMC-TBT and is used to study the endothermic desorption of a lattice gas. Interesting findings from this study are reported.     

Monte Carlo simulation of polymer analogous reaction in confined conditions: effects of ordering

The influence of energetic parameters of the interchain homo- and heterocontacts on a local ordering of Bernoullian copolymers has been studied using Monte Carlo simulations and probabilistic analysis. The results of both methods are in a good agreement. Then simple Monte Carlo procedure was employed to study the ordering in products of a polymeranalogous reaction with accelerating effect of neighboring groups. When the reaction with intra- and interchain acceleration and local ordering proceed simultaneously in confined conditions, the ordering might affect the process so that the formation of certain nano-structures (in particular, not trivial strip-like ones) is possible.     

Mesoscale simulation of polymer reaction equilibrium: combining dissipative particle dynamics with reaction ensemble Monte Carlo. I. Polydispersed polymer systems

We present a mesoscale simulation technique, called the reaction ensemble dissipative particle dynamics (RxDPD) method, for studying reaction equilibrium of polymer systems. The RxDPD method combines elements of dissipative particle dynamics (DPD) and reaction ensemble Monte Carlo (RxMC), allowing for the determination of both static and dynamical properties of a polymer system. The RxDPD method is demonstrated by considering several simple polydispersed homopolymer systems. RxDPD can be used to predict the polydispersity due to various effects, including solvents, additives, temperature, pressure, shear, and confinement. Extensions of the method to other polymer systems are straightforward, including grafted, cross-linked polymers, and block copolymers. To simulate polydispersity, the system contains full polymer chains and a single fractional polymer chain, i.e., a polymer chain with a single fractional DPD particle. The fractional particle is coupled to the system via a coupling parameter that varies between zero (no interaction between the fractional particle and the other particles in the system) and one (full interaction between the fractional particle and the other particles in the system). The time evolution of the system is governed by the DPD equations of motion, accompanied by changes in the coupling parameter. The coupling-parameter changes are either accepted with a probability derived from the grand canonical partition function or governed by an equation of motion derived from the extended Lagrangian. The coupling-parameter changes mimic forward and reverse reaction steps, as in RxMC simulations.     

Properties of simple models of polymer networks. A Monte Carlo simulation

A model polymer network was constructed from branched chains. Each chain was built on a simple cubic lattice forming a star-branched polymer consisting of f = 3 arms of equal lengths. The fragment of network under consideration consisted of 1, 2 and 3 star polymers with different topology of connections. The only potential used was excluded volume (athermal chains). The properties of the network were determined by the means of computer simulations using the classical Metropolis sampling algorithm (local micromodifications of chain conformation). The behaviour of linear chains of the same molecular weight was also studied as a state of reference. The influence of attaching the next star-branched chain to the network on its static and dynamic properties was studied. The short-time dynamic behaviour of chain fragments was determined and discussed.     

Grafted polymer layers under variable solvent conditions: A Monte Carlo simulation

Polymer chains anchored with one end at a hard wall under variable solvent conditions are investigated by Monte Carlo simulations using the bond- fluctuation model. Detail information on the structural properties are obtained above, at, and below the Θ-point and discussed in terms of the appropriate theories. In particular, the scaling of the brush thickness is formulated and verified by the simulation data. For the dynamics at the Θ-point, both the relaxation time of the chain configuration and the mean-square time displacement are studied. At temperatures distinctly below the Θ-point, we find that the layer develops considerable lateral inhomogeneity in its density, which has not been predicted by previous theories.     

A lattice Monte Carlo simulation of the FePt alloy using a first-principles renormalized four-body interaction

Although a lattice Monte Carlo method provides an effective, simple, and fast way to study thermodynamic properties of substitutional alloys, it cannot treat by itself the off-lattice effects, such as thermal vibrations and local distortions. Therefore, even if the interaction among atoms at lattice points is calculated accurately by means of first-principles calculations, the lattice Monte Carlo simulation overestimates the order-disorder phase transition temperature. In this paper, we treat this problem in the investigation of the FePt alloy, which has recently attracted considerable interest in its magnetic properties. We apply a simple version of the potential renormalization theory to determine the interaction among atoms, including partly the off-lattice effects by means of first-principles calculations. Then, we use the interaction to perform a lattice Monte Carlo simulation of the FePt alloy on a fcc lattice. From the results, we find that the transition temperature obtained after the present renormalization procedure becomes closer to the experimental value.     

Monte Carlo simulation for inhomogeneous chemical kinetics: Application to the Belousov–Zhabotinskii reaction

A Monte Carlo algorithm, capable of simulating numerically the time and space dependence of chemical concentrations in a reacting system, is presented. This method is used to study the phenomenon of trigger waves in the Oregonator model of the Belousov–Zhabotinskii reaction, including the diffusion of species X and Y in one dimension. The results show that a small disturbance in a homogeneous mixture can grow into a chemical (trigger) wave propagating in space at constant velocity. The dependence of this velocity on several factors is studied, namely, initial concentrations, the diffusion of Y, and the stoichiometry of the autocatalytic step of the model. A comparison of the Monte Carlo results with a previous simulation also is discussed.