Monte Carlo simulation of polymeric materials: Recent progress

Monte Carlo simulations are presented, dealing with phase diagrams of block copolymer melts and polymer blends, including the unmixing kinetics of the latter systems. The theoretical background is briefly reviewed: Ginzburg-type criteria reveal that in mixtures of long flexible polymers a “crossover” occurs from mean-field behavior (as described by Flory-Huggins theory) to nonclassical Ising-type behavior, and spinodal curves can be unusually sharp. This crossover is demonstrated by large scale simulations of the bond fluctuation model, and it is also shown that for symmetric mixtures the critical temperature scales with chain length as T_c α N. The prefactor in this relation is distinctly smaller than predicted by Flory-Huggins, but the Curro-Schweizer integral equation theory prediction T_c α √N is clearly ruled out. Tests of the Cahn theory on the initial stages of spinodal decomposition of polymer blends will also be reported. To conclude, the mesophase formation in block copolymers is discussed, and it is shown that the simulations agree very well with experiment. The pronounced chain stretching that already occurs in the disordered phase is compelling evidence against the validity of simple random phase approximation concepts for these systems. This shows how Monte Carlo simulations can assist in better understanding large classes of polymeric materials.     

Composition fluctuations in a homopolymer-diblock copolymer mixture covering the three-dimensional Ising, isotropic Lifshitz, and Brasovskiĭ classes of critical universality

The phase behavior of a three-component polymer blend consisting of a critical mixture of polybutadiene and polystyrene (PB/PS) with varying amount of a symmetric PB-PS diblock copolymer was explored with small-angle neutron scattering. Our focus were thermal composition fluctuations which we discuss in terms of mean field, three-dimensional Ising, isotropic Lifshitz, and Brasovski? classes of critical universality. Particular attention is spent to the observation of a narrow reentrant two-phase regime and double critical point in the Lifshitz critical regime as well as the Lifshitz line. Critical exponents of the isotropic Lifshitz case are proposed in spite of the demonstrated nonexistence of the isotropic Lifshitz critical point. The Ginzburg number (Gi) and Flory-Huggins parameter were determined over the whole diblock concentration range; Gi changes by three orders of magnitude, two orders of magnitude of that change over a 0.03 diblock concentration interval within the isotropic Lifshitz regime.     

Design of miscible polyolefin copolymer blends

Theoretical guidelines are established for designing miscible blends of amorphous polyolefin copolymers. On the basis of calculations for an athermal and incompressible model of copolymer melts, limits are placed on the compositions and structural differences between blend components that are consistent with thermodynamic stability of a single liquid phase. Specific cases analyzed include binary blends of random copolymers containing short branches and blends of graft polymers with long flexible branches, either periodically or randomly placed. The predictions are shown to be in good agreement with recent experimental studies of miscibility in model polyolefin copolymer blends. © 1995 John Wiley & Sons, Inc.     

Theoretical aspects of small-angle neutron scattering from multiphase polymers

A molecular theory for small-angle neutron scattering from polymer mixtures is reviewed and extended to consider multiphase polymer systems such as block copolymers and their blends with homopolymers. Methods are developed for the isolation of scattering functions for individual components in these blends. These methods rely on two contrast-matching techniques: the concept of “composition matching,” where a mixture of deuterium-labeled and protonated species is used to match the contrast of a third component; and the synthesis of “phase-matched” block copolymers, where the contrast of the block copolymer sequences are matched. Methods are discussed specifically for the isolation of single chain and single sequence scattering functions for diblock and triblock copolymers, their blends with homopolymers, and star copolymers.     

对称长链和近对称短链两嵌段聚合物混合体系相行为研究

利用自洽平均场理论(SCMFT)系统地研究了对称长链和近对称短链两嵌段聚合物混合体系在纳米尺度下的自组装行为.体系中具有较高聚合度的对称长链熔体处于层状相,聚合度较低的近对称短链熔体处于无序相,而其混合体系却随着两种成分的不同比例呈现出有序-无序相转变、有序-有序相转变及有序-无序两相共存等复杂的相行为,计算结果与近期类似体系的实验有着较好的吻合.同时与两种对称的两嵌段聚合物混合体系的计算结果进行了比较,得出这两种体系的异同之处.     

Interplay of chemical and physical factors in reacting polymer blends. Theoretical considerations

Various effects produced by copolymers in polymer blends are discussed, with an emphasis on the role of interchain interactions. Simple theoretical models are considered to study the following problems: the interplay of diffusion and macromolecular reaction in compatible and incompatible blends, the stabilizing effect of premade diblock copolymer on the system of minor phase particles in incompatible blends, the kinetics of transesterification in a homogeneous blend. The effect of diblock copolymer on the Ostwald ripening in a polymer blend is stated in more details; the possibility of narrowing the size distribution of minor phase particles is predicted.     

Monte Carlo studies of static properties of interacting lattice polymers with the cooperative-motion algorithm

In this paper we review some applications of the cooperative-motion simulation algorithm to dense polymer melts. The basic idea behind the algorithm and its implementation on a computer are discussed. Furthermore we show how to include intra-and intermolecular interactions in the simulation. Exemplary results are presented concerning on one hand the static properties of athermal and semiflexible lattice polymers and on the other hand the phase behaviour of polymer mixtures and diblock copolymers.     

New polymer systems exhibiting microphase separation transition with the formation of nano-microstructures

The concept of microphase separation was up to now widely applied mainly to the conformational transitions in block-copolymer solutions and melts. However, recently it became obvious that this concept has a much more general meaning. It was shown that microphase separation transition can be observed in random copolymers, interpenetrating polymer networks, polyelectrolyte mixtures, poor solvent polyelectrolyte solutions, ionomer solutions and melts, polymer blends and solutions with nonlocal entropy of mixing. In all these examples the emerging microdomain structures correspond to the nanometer scale, therefore the study of these effects can lead to the new ways of obtaining polymer materials with controlled nano-microstructure. In this presentation the review of our recent findings on microphase separation in some of the above-mentioned systems will be presented. 1. The problem of microphase separation in the systems containing weakly charged polyelectrolytes (polyelectrolyte mixtures and poor solvent polyelectrolyte solutions) will be considered. From the methodic point of view, it will be shown that this problem can be solved by direct minimization of the free energy, without the use of “weak segregation” or “strong segregation” assumptions which are common in the theory of block-copolymers. The final phase diagrams exhibit wide macroscopic phase separation regions, which is their main difference from the corresponding phase diagrams for block-copolymer systems. The formation of microdomains is thus coupled with macroscopic phase separation: in most of the cases microdomain structure is formed in one of the coexisting phases after macroscopic phase separation takes place [1] - [2]. 2. The formation of the multiplet structure in ionomer melts and solutions can be also considered as the microphase separation in the random copolymer system with the formation of the “micelles” (or clusters) of ionic links. The parallels with micelle formation in block-copolymer systems can be established if one considers a new “superstrong segregation regime” for block-copolymer microstructures. This regime can be indeed observed for diblock copolymers with one ionomeric and one neutral block [3]. 3. The microphase separation transition in ordinary polymer blends and solutions is also possible. The conditions for this effect are: (i) significant entropic contribution to polymer/polymer or polymer/solvent miscibility, (ii) the nonlocal character of this contribution with a high value of the nonlocality radius. It is argued that one can expect that the entropy nonlocality radius increases in the vicinity of the glass transition for the blend or polymer solutions (in the latter case solvent molecules act like “poor solvent plasticisers”). Computer simulation data supporting the theoretical prediction of microphase separation transition in these systems will be presented [4] - [5].     

The influence of polymer structure and component ratios on blend miscibility

The mean field, rigid lattice treatment as applied to polymer mixtures has been used to estimate segment-segment interaction parameters for a wide range of polymers. These parameters incorporate, without distinction, contributions from non-combinatorial entropy effects, dispersion forces and any specific interactions that operate in the polymer blend. Thus while these parameters can be used to predict successfully the nature of the phases in untested polymer blends, structural effects may also play a role in determining miscibility, and these may have to be assessed individually. Examples of structural effects are described using chlorine-containing polymers and blends of copolymers with an anhydride ring attached in two different ways to the polymer chain. The extension of binary interaction parameters to the prediction of phase behaviour in complex ternary copolymer blends and the effect on the phase behaviour of changing the component ratios in the blends, is also illustrated.     

Influencing the morphology of polymer blends through graft copolymers

Graft copolymers have a potential as compatibilizers in two-component thermoplastic polymer blends, and also as impact-modifiers in one-component thermoplastics. The compatibility of the blocks of the copolymer (i.e. the grafts and the main chain) with the chains of the matrix polymers must be adjusted carefully. Blends of various polymers, especially of polystyrene (PS) and poly(vinyl chloride) (PVC), with graft copolymers on the basis of polybutadiene are discussed. An excellent compatibilizer, for blends PS/PVC, is a block-graft copolymer, derived from a diblock copolymer of Styrene and butadiene, with grafts of cyclohexyl methacrylate monomelic units.     

Instabilities in block copolymer films induced by compressible solvents

We combine a simple lattice-gas model for fluid mixtures along with polymer mean-field theory for block copolymer melts to study the stability of thin films of diblock copolymers in the presence of compressible fluid solvents. Using a free energy analysis, the stable and unstable thicknesses of a copolymer thin film are obtained for given solvent conditions. Our results suggest that the interplay between confinement, the compressibility of the solvent, and its selectivity to polymer component can lead to significant changes on the ordering and stability of the diblock copolymer thin films. Our results are in qualitative agreement with recent experimental results.     

Organizing thermodynamic data obtained from multicomponent polymer electrolytes: Salt‐containing polymer blends and block copolymers

The objective of this review is to organize literature data on the thermodynamic properties of salt‐containing polystyrene/poly(ethylene oxide) (PS/PEO) blends and polystyrene‐b‐poly(ethylene oxide) (SEO) diblock copolymers. These systems are of interest due to their potential to serve as electrolytes in all‐solid rechargeable lithium batteries. Mean‐field theories, developed for pure polymer blends and block copolymers, are used to describe phenomenon seen in salt‐containing systems. An effective Flory–Huggins interaction parameter, χ_eff , that increases linearly with salt concentration is used to describe the effect of salt addition for both blends and block copolymers. Segregation strength, χ_effN , where N is the chain length of the homopolymers or block copolymers, is used to map phase behavior of salty systems as a function of composition. Domain spacing of salt‐containing block copolymers is normalized to account for the effect of copolymer composition using an expression obtained in the weak segregation limit. The phase behavior of salty blends, salty block copolymers, and domain spacings of the latter systems, are presented as a function of chain length, composition and salt concentration on universal plots. While the proposed framework has limitations, the universal plots should serve as a starting point for organizing data from other salt‐containing polymer mixtures. © 2019 Wiley Periodicals, Inc. J. Polym. Sci., Part B: Polym. Phys. 2019 , 57, 1177–1187     

Relationships among coarse-grained field theories of fluctuations in polymer liquids

Two closely related field-theoretic approaches have been used in previous work to construct coarse-grained theories of corrections to the random phase approximation for correlations in block copolymer melts and miscible polymer blends. The "auxiliary field" (AF) approach is based on a rigorous expression for the partition function Z of a coarse-grained model as a functional integral of an auxiliary chemical potential field. The "effective Hamiltonian" (EH) approach is instead based on an expression for Z as a functional integral of an observable order parameter field. The exact effective Hamiltonian H(eff) in the EH approach is defined as the free energy of a system with a constrained order parameter field. In practice, however, H(eff) has often been approximated by a mean-field free energy functional, yielding what we call a mean-field effective Hamiltonian (MFEH) approximation. This approximation was the starting point of both the Fredrickson-Helfand analysis of fluctuation effects in diblock copolymers and earlier work on the Ginzburg criterion in polymer blends. A more rigorous EH approach by Holyst and Vilgis used an auxiliary field representation of the exact H(eff) and allowed for Gaussian fluctuations of this field. All applications of both AF and EH approaches have thus far relied upon some form of Gaussian, or "one-loop" approximation for fluctuations of a chemical potential and/or order parameter field about a mean-field saddle-point. The one-loop EH approximation of Holyst and Vilgis and the one-loop AF theory are equivalent to one another, but not to the one-loop MFEH theory. The one-loop AF and MFEH theories are shown to yield predictions for the inverse structure factor S(-1)(q) that (in the absence of further approximations to either theory) differ by a function that is independent of the Flory-Huggins interaction parameter χ. As a result, these theories yield predictions for the peak scattering intensity that exhibit a similar χ-dependence near a spinodal. The Fredrickson-Helfand theory for the structure factor in disordered diblock copolymer melts is an asymptotic approximation to the MFEH one-loop theory that captures the dominant asymptotic behavior of very long, symmetric copolymers very near the order-disorder transition.     

On the phase behavior of blends of polymers and nematic liquid crystals

The phase behavior of mixtures of polymers and nematic liquid crystals (LC) is investigated. Two types of systems are examined. The first one deals with blends in which the polymer is made of linear chains. In this case, a systematic study of the effects of various parameters on the phase diagrams is performed. In particular, it is shown how increasing the polymer size and/or the LC molecule size increases the miscibility gap of the mixture. It also reduces the region where a single nematic phase is observed in the presence of a tiny amount of polymer. Likewise, the relative effects of the isotropic and the nematic interaction parameters on the phase diagrams are examined. The second part of this investigation deals with blends involving crosslinked polymers. Here, substantial differences are observed as compared to the case where the polymer components are made of linear chains. These differences are illustrated by showing the phase diagrams in similar conditions for both blends. Unlike the case of a linear polymer matrix, it is observed that the single nematic phase and the nematic-isotropic spinodal branches are absent from the phase diagram of crosslinked polymers. This results into significant distortions of the phase diagram. In order to highlight all these effects, examples representing hypothetical blends are considered. These examples are chosen for illustration of the results in which the choice of numerical parameters is made consistently with the existing values in the literature which makes comparison with published data possible.     

Interfacial tension of evaporating emulsion droplets containing amphiphilic block copolymers: effects of solvent and polymer composition

Evaporating droplets of volatile organic solvent containing amphiphilic block copolymers may undergo hydrodynamic instabilities that lead to dispersal of copolymer micelles into the surrounding aqueous phase. As for related phenomena in reactive polymer blends and oil/water/surfactant systems, this process has been ascribed to a nearly vanishing or transiently negative interfacial tension between the water and solvent phases induced by adsorption of copolymer to the interface. In this report, we investigate the influence of the choice of organic solvent and polymer composition for a series of polystyrene-b-poly(ethylene oxide) (PS-PEO) diblock copolymers, by in situ micropipette tensiometry on evaporating emulsion drops. These measurements suggest that the sensitivity to the organic solvent chosen reflects both differences in the bare solvent/water interfacial tension as well as the propensity of the copolymer to aggregate within the organic phase. While instabilities coincident with an approach of the interfacial tension nearly to zero were observed only for copolymers with PEO content greater than 15 wt.%, beyond this point the interfacial behavior and critical concentration needed to trigger surface instability were found to depend only weakly on copolymer composition.     

The influence of elongational flow on association rate and phase behaviour of binary polymer blends

Flow‐induced phase separation in binary blends of end‐associating polymers is studied analytically. To describe the conformational and orientational properties of a polymer chain a simple dumbbell model is applied. It is demonstrated that the association rate for formation of associated diblock copolymer‐like chains decreases with an increase of flow rate. This is due to the extra‐stretching of the associated chain compared with the two initial homopolymer chains. The decrease in the fraction of associated diblock copolymer‐like chains makes the homogeneous state less stable, so the effect of flow manifests itself in the enhancement of the segregation tendency in these kind of associated polymer blends.     

Interfaces in polymer blends

We investigate the structure and thermodynamics of interfaces in dense polymer blends using Monte Carlo (MC) simulations and self‐consistent field (SCF) calculations. For structurally symmetric blends we find quantitative agreement between the MC simulations and the SCF calculations for excess quantities of the interface (e.g., interfacial tension or enrichment of copolymers at the interface). However, a quantitative comparison between profiles across the interface in the MC simulations and the SCF calculations has to take due account of capillary waves. While the profiles in the SCF calculations correspond to intrinsic profiles of a perfectly flat interface the local interfacial position fluctuates in the MC simulations. We test this concept by extensive Monte Carlo simulations and study the cross‐over between “intrinsic” fluctuations which build up the local profile and capillary waves on long (lateral) length scales. Properties of structurally asymmetric blends are exemplified by investigating polymers of different stiffness. At high incompatibilities the interfacial width is not much larger than the persistence length of the stiffer component. In this limit we find deviations from the predictions of the Gaussian chain model: while the Gaussian chain model yields an increase of the interfacial width upon increasing the persistence length, no such increase is found in the MC simulations. Using a partial enumeration technique, however, we can account for the details of the chain architecture on all length scales in the SCF calculations and achieve good agreement with the MC simulations. In blends containing diblock copolymers we investigate the enrichment of copolymers at the interface and the concomitant reduction of the interfacial tension. At weak segregation the addition of copolymers leads to compatibilization. At high incompatibilities, the homopolymer‐rich phase can accommodate only a small fraction of copolymer before the copolymer forms a lamellar phase. The analysis of interfacial fluctuations yields an estimate for the bending rigidity of the interface. The latter quantity is important for the formation of a polymeric microemulsion at intermediate segregation and the consequences for the phase diagram are discussed.     

Design of bicontinuous polymeric microemulsions

Recent experiments suggest that thermodynamically stable, bicontinuous microemulsions can be achieved in symmetric ternary blends of two homopolymers and a diblock copolymer by formulating alloys with compositions near mean-field isotropic Lifshitz points. We argue that practical application of this design criterion may require use of homopolymers of unequal molecular weights and block copolymers of different architecture. We demonstrate the existence of, and explicitly locate, mean-field isotropic Lifshitz points in ternary blends with homopolymer molecular weight asymmetry and either AB diblock or ABA triblock copolymer architectures. These calculations considerably expand the parameter space for observing bicontinuous microemulsions and allow for more flexibility in tailoring melt rheological properties and solid-state mechanical properties. © 1997 John Wiley & Sons, Inc. J Polym Sci B: Polym Phys 35: 2775–2786, 1997     

Calculation of phase equilibria in random copolymer systems

Synthesis and application of copolymers are not seldom connected with different phase equilibria. Their precise knowledge is of importance for industrial processing as well as it is a profound basis for a better understanding of the nature and thermodynamics of such systems. As a common situation today, enough experimental information is seldom available in the necessary or desired amount, and a lot of model calculation is, therefore, more or less unavoidable to cover the desired ranges of application. Different equations-of-state as well as lattice models are discussed with respect to their applicability for calculating liquid-liquid and gas-liquid phase equilibria in copolymer solutions and blends. Examples for high-pressure phase equilibria in monomer/copolymer mixtures, liquid-liquid demixing in copolymer blends and for the isotropicnematic phase equilibrium in systems with rigid rod-like copolymers characterized by distributions of rigid and flexible chain parts are given. The effects of copolymer polydispersity are included by means of continuous thermodynamics. Literature references for original sources, earlier reviews and further applications round up this paper.