H. P. Deutsch

Interdiffusion and self‐diffusion in polymer mixtures: A Monte Carlo study

A lattice model for dense polymer solutions and polymer mixtures in three dimensions is presented, aiming to develop a model suitable for efficient computer simulation on vector processors, with a qualitatively realistic local dynamics. It is shown that the bond fluctuation algorithm for a suitable set of allowed bond vectors has the property that due to the excluded volume constraint no crossing of bonds by local motions can occur, and entanglement restrictions thus are fully taken into account. For athermal binary (AB) symmetrical polymer mixtures, the dependence of both self‐diffusion coefficient and interdiffusion coefficient on polymer density is obtained, simulating a thin film geometry where a film of polymer A is coated with a film of polymer B. For one density, the dependence of the interdiffusion coefficient on an attractive energy between unlike monomers is also studied. For weak attraction an enhancement of interdiffusion proportional to this energy occurs. For strong attraction, however, a rather immobile tightly bound AB layer forms in the interface which hampers further unmixing.

Equation of state for athermal lattice chains in a 3d fluctuating bond model

A generalization of the well‐known Flory and Flory–Huggins mean‐field approximations to the equation of state is derived for a three‐dimensional lattice model in which a monomer occupies an entire unit cell, and many bond lengths and bond angles are possible. By measuring the probability for particles to be in contact with the walls of the system, the pressure is determined via computer simulation over the full density range from dilute solution to dense melt. The results are used to test the mean‐field predictions. Comparing the equation of state of the present model to those of conventional lattice models and of hard‐sphere chains in continuous space, it is seen that our method approximates the continuum limit far better than single site lattice models. Also the large n des Cloizeaux scaling behavior is approached more rapidly.

Evidence against the integral equation theory of polymer blends

Monte Carlo simulations are presented for the bond fluctuation model of symmetric polymer mixtures in 3 dimensions, using a polymer density = 0.5 corresponding in that model to dense melts, and chain lengths N in the range 16 ≤ N ≤ 256. Combining multiple-histogram and finite-size scaling techniques, we obtain clear evidence for a linear dependence of the critical temperature on chain length, kBTc/e ≈ 2.15N + 1.35. For our model, the prediction Tc ∝ √N of the integral equation theory of Schweizer and Curro is clearly ruled out, at least for the range of chain lengths available. We also obtain data for the effective renormalized Flory-Huggins parameter χeff of our model, strongly supporting a strictly linear dependence of χeff on the chain length.

Monte Carlo Methods For First Order Phase Transitions: Some Recent Progress

This brief review discusses methods to locate and characterize first order phase transitions, paying particular attention to finite size effects. In the first part, the order parameter probability distribution and its fourth-order cumulant is discussed for thermally driven first-order transitions (the 3-state Potts model in d=3 dimensions is treated as an example). First-order transitions are characterized by a minimum of the cumulant, which gets very deep for large enough systems. In the second part, we discuss how to locate first order phase boundaries ending in a critical point in a large parameter space. As an example, the study of the unmixing transition of asymmetric polymer mixtures by a combination of histogram techniques and finite size scaling is described.As a final problem, we discuss the shift of the gas-liquid condensation in thin-film geometry confined between two parallel plates due to boundary fields (“capillary condensation”). Being interested in temperatures far below bulk criticality (e. g. near the wetting transition), special thermodynamic integration techniques are the method of choice, rather than the use of finite sizes scaling to map out the (asymmetric) phase diagram.

Crossover Phenomena and Finite-Size Scaling Analysis of Numerical Simulations

The implications of the scaling theory for the crossover from one universality class of critical phenomena to another one are worked out for finite-size scaling analyses of numerical simulation data. Near a multicritical point finite-size scaling critical amplitudes have a singular dependence on the distance from the multicritical point. Attention is also paid to the crossover to Landau-like behaviour when the interaction range diverges, and as an application Monte Carlo results for polymer mixtures are discussed.

First‐ and second‐order phase transitions in asymmetric polymer mixtures

The critical properties of dense asymmetric binary polymer mixtures are studied by grand canonical simulations within the framework of the three‐dimensional bond fluctuation lattice model. The monomers interact with each other via a potential ranging over the entire first peak of the pair distribution. An asymmetry is realized by giving the ratio of interactions λ≡eAA/eBB between monomers of the A species and of the B species a value different from 1. Using multiple histogram extrapolation and finite size scaling techniques for the data analysis, the two‐phase region, which is a line of first‐order transitions driven by the chemical potential difference, and the critical point are determined for a mixture of chains with 32 monomers each and various asymmetries up to λ=5. At a critical potential difference Δμc unmixing occurs below a critical temperature Tc. It is found that the quantities Δμc/(1−λ)e and 4kBTc/(3+λ)e are both independent of the asymmetry, consistent with the prediction of the Flory theory....

Monte carlo studies of phase transitions in polymer blends and block copolymer melts

SummaryThe unmixing transition of both symmetrical polymer blends AB (i.e. chain lengthsNA=NB=N) and asymmetrical ones (NB/NA=2,3) is studied by large-scale Monte Carlo simulations of the bond fluctuation model. Combination of semi-grand-canonical simulation techniques, «histogram reweighting» and finitesize scaling allows an accurate location of the coexistence curve in the critical region. The variation of the critical temperature with chain length (N) is studied and compared to theoretical predictions. For the symmetrical case, use of chain lengths up toN=512 allows a rough estimation of crossover scaling functions for the crossover from Ising to mean-field exponents. The order-disorder transitions in melts of both symmetric (compositionf=NA/(NA+NB)=1/2) and asymmetric (f=3/4) block copolymers is studied for very short chains (16≤N≤60). The interplay between structure and chain configuration is emphasized. Qualitative evidence for «dumbell formation» of chains and vacancy enrichment in A-B-interfaces and near hard walls is presented.

Monte Carlo studies of polymer interdiffusion and spinodal decomposition: A review

Abstract Putting a layer of polymer A on top of a layer of polymer B, the broadening of the interfacial profile is observed in the framework of a lattice model (‘bond fluctuation method’). The interdiffusion constant is studied as a function of chain length, vacancy concentration, and interaction energy between unlike monomers, and a comparison with pertinent theoretical predictions is made. A lattice model where polymers are represented as self-avoiding walks on a simple cubic lattice is used to model ‘spinodal decomposition’, i.e. phase separation by ‘uphill diffusion’ in the unstable part of the phase diagram of a polymer mixture. For chain lengths N ≤ 32, the linearized Cahn-like theory describes the Monte Carlo results only qualitatively. A slowing down due to chain collapse is found for deep quenches and attractive interactions.

Phase transitions in polymeric systems: A challenge for Monte Carlo simulation

Polymers are more difficult to simulate than small molecule systems, due to the large size of random polymer coils (and their slow relaxation, that is observed when dynamic simulation algorithms are used). However, variation of the chain length N of a flexible polymer chain provides a very useful additional control parameter, allowing stringent tests of theories, and new physical phenomena may emerge. As an example of these concepts, critical phenomena in polymer mixtures are described. It is shown that unmixing of symmetrical mixtures ( N A = N B = N ) is described by an equation for the critical temperature T c ( N ) = aN + b rather than T c ∝ N as claimed by some theories. While for finite N the critical behavior is Ising-like, for N → ∞ it becomes mean-field like, and this crossover creates interesting problems for the finite size scaling analysis of Monte Carlo data. Special problems occur also for asymetric mixtures (semi-grandcanonical algorithms involve chain splitting and fusion, and finite size scaling analyses must take “field mixing” into account). Finally, studies of interfaces between coexisting phases require huge systems (> 10 million lattice sizes), but can be handled on parallel computers. Less well understood, however, are ordering phenomena in block copolymers: type and wavelength of the resulting mesophases change with temperature and composition, and since the ordering is typically incommensurate with the lattice linear dimension, no simple finite-size behavior emerges.