Critical unmixing of polymer solutions

Chemical Potential of a Flexible Polymer Liquid in a Coarse-Grained Representation

While the excess chemical potential is the key quantity in determining phase diagrams, its direct computation for high-density liquids of long polymer chains has posed a significant challenge. Computationally, the excess chemical potential is calculated using the Widom insertion method, which involves monitoring the change in internal energy as one incrementally introduces individual molecules into the liquid. However, when dealing with dense polymer liquids, inserting long chains requires generating trial configurations with a bias that favors those at low energy on a unit-by-unit basis: a procedure that becomes more challenging as the number of units increases. Thus, calculating the excess chemical potential of dense polymer liquids using this method becomes computationally intractable as the chain length exceeds N ≥ 30. Here, we adopt a coarse-grained model derived from the integral equation theory for which inserting long polymer chains becomes feasible. The integral equation theory of coarse graining (IECG) represents a polymer as a sphere or a collection of blobs interacting through a soft potential. We employ the IECG approach to compute the excess chemical potential using Widom's method for polymer chains of increasing lengths, extending up to N = 720 monomers, and at densities reaching up to ρ = 0.767 g/cm3. From a fundamental perspective, we demonstrate that the excess chemical potentials remain nearly constant across various levels of coarse graining, offering valuable insights into the consistency of this type of procedure. Ultimately, we argue that current Monte Carlo algorithms, originally designed for atomistic simulations, such as configurational bias Monte Carlo (CBMC) methods, can significantly benefit from the integration of the IECG approach, thereby enhancing their performance in the study of phase diagrams of polymer liquids.

Exact Enumeration Approach to Estimate the Theta Temperature of Interacting Self-Avoiding Walks on the Simple Cubic Lattice

We compute the exact root-mean-square end-to-end distance of the interacting self-avoiding walk (ISAW) up to 27 steps on the simple cubic lattice. These data are used to construct a fixed point equation to estimate the theta temperature of the collapse transition of the ISAW. With the Bulirsch-Stoer extrapolation method, we obtain accurate results that can be compared with large-scale long-chain simulations. The free parameter ω in extrapolation is precisely determined using a parity property of the ISAW. The systematic improvement of this approach is feasible by adopting the combination of exact enumeration and multicanonical simulations.

Efficient configurational-bias Monte-Carlo simulations of chain molecules with "swarms" of trial configurations

The pruned-enriched Rosenbluth method (PERM) is a popular and powerful Monte-Carlo technique for sampling flexible chain polymers of substantial length. In its original form, however, the method cannot be applied in Markov-chain Monte-Carlo schemes, which has rendered PERM unsuited for systems that consist of many chains. The current work builds on the configurational-bias Monte-Carlo (CBMC) method. The growth of a large set of trial configurations in each move is governed by simultaneous pruning and enrichment events, which tend to replace configurations with a low statistical weight by clones of stronger configurations. In simulations of dense brushes of flexible chains, a gain in efficiency of at least three orders of magnitude is observed with respect to CBMC and one order of magnitude with respect to recoil-growth approaches. Moreover, meaningful statistics can be collected from all trial configurations through the so-called "waste-recycling" Monte Carlo scheme.

Exact partition function zeros of a polymer on a simple cubic lattice

We study conformational transitions of a polymer on a simple-cubic lattice by calculating the zeros of the exact partition function, up to chain length 24. In the complex temperature plane, two loci of the partition function zeros are found for longer chains, suggesting the existence of both the coil-globule collapse transition and the melting-freezing transition. The locus corresponding to coil-globule transition clearly approaches the real axis as the chain length increases, and the transition temperature could be estimated by finite-size scaling. The form of the logarithmic correction to the scaling of the partition function zeros could also be obtained. The other locus does not show clear scaling behavior, but a supplementary analysis of the specific heat reveals a first-order-like pseudotransition.

Tracking gas-liquid coexistence in fluids of charged soft dumbbells

The existence of gas-liquid coexistence in dipolar fluids with no other contribution to attractive interaction than dipole-dipole interaction is a basic and open question in the theory of fluids. Recent Monte Carlo work by Camp and co-workers indicates that a fluid of charged hard dumbbells does exhibit gas-liquid (g-l) coexistence. This system has the potential to answer the above fundamental question because the charge-to-charge separation, d , on the dumbbells may be reduced to, at least in principle, yield the dipolar fluid limit. Using the molecular-dynamics technique we present simulation results for the g-l critical point of charged soft dumbbells at fixed dipole moment as function of d . We do find a g-l critical point at finite temperature even at the smallest d value (10;{-4}) . Reversible aggregation appears to play less a role than in related model systems as d becomes small. Consequently attempts to interpret the simulation results using either an extension of Flory's lattice theory for polymer systems, which includes reversible assembly of monomers into chains, or the defect model for reversible networks proposed by Tlusty and Safran are not successful. The overall best qualitative interpretation of the critical parameters is obtained by considering the dumbbells as dipoles immersed in a continuum dielectric.

Freezing and collapse of flexible polymers on regular lattices in three dimensions

We analyze the crystallization and collapse transition of a simple model for flexible polymer chains on simple-cubic and face-centered-cubic lattices by means of sophisticated chain-growth methods. In contrast to the bond-fluctuation polymer model in certain parameter ranges, where these two conformational transitions were found to merge in the thermodynamic limit, we conclude from our results that the two transitions remain well separated in the limit of infinite chain lengths. The reason for this qualitatively distinct behavior is presumably due to the ultrashort attractive interaction range in the lattice models considered here.

Simulation and theory of flexible equilibrium polymers under poor solvent conditions

Grand canonical Monte Carlo simulation and simple statistical thermodynamic theory are used to model the aggregation and phase separation of systems of reversibly polymerizing monomers, capable of forming chains with or without the ability to cyclize into rings, with isotropic square-well attractions between nonbonded pairs of monomers. The general trend observed in simulation of chain-only systems, as predicted in a number of published theoretical works, is that the critical temperature for phase separation increases and the critical monomer density decreases with rising polymer bond strength. Introduction of the equilibrium between chains and rings into the theory lowers the predicted critical temperature and increases the predicted critical density. While the chain-only theories predict a vanishing critical density in the limit of complete polymerization, when ring formation is taken into account the predicted critical density in the same limit approaches the density of the onset of the ring-chain transition. The theoretically predicted effect of cyclization on chemical potential is in good qualitative agreement with a subset of simulation results in which chain-only systems were compared with equilibrium mixtures of rings and chains. The influence of attractions on the aggregation number and radius of gyration of chains and rings observed in simulations is also discussed.

Equilibrium polymerization and gas-liquid critical behavior in the Stockmayer fluid

We develop a simple theory explaining the dependence of the gas-liquid critical point in the Stockmayer fluid on dipole strength. The theory is based on the Flory-Huggins lattice description for polymer systems in conjunction with a transfer matrix model for isolated chains of reversibly assembled dipolar particles. We find that the shift of the critical point as a function of dipole strength, which originally was found in computer simulation, strongly resembles the critical point shift as a function of chain length in ordinary linear polymer systems. In particular, the decrease of the critical density with increasing dipole strength is a consequence of the existence of reversible chains near criticality. In addition we report simulation results for gas-liquid critical points well above the limiting dipole strength found previously.

Competition of mesoscales and crossover to theta-point tricriticality in near-critical polymer solutions

The approach to asymptotic critical behavior in polymer solutions is governed by a competition between the correlation length of critical fluctuations diverging at the critical point of phase separation and an additional mesoscopic length scale, the radius of gyration. In this paper we present a theory for crossover between two universal regimes: a regime with Ising (fluctuation-induced) asymptotic critical behavior, where the correlation length prevails, and a mean-field tricritical regime with theta-point behavior controlled by the mesoscopic polymer chain. The theory yields a universal scaled description of existing experimental phase-equilibria data and is in excellent agreement with our light-scattering experiments on polystyrene solutions in cyclohexane with polymer molecular weights ranging from 2 x 10(5) up to 11.4 x 10(6). The experiments demonstrate unambiguously that crossover to theta-point tricriticality is controlled by a competition of the two mesoscales. The critical amplitudes deduced from our experiments depend on the polymer molecular weight as predicted by de Gennes [Phys. Lett. 26A, 313 (1968)]. Experimental evidence for the presence of logarithmic corrections to mean-field tricritical theta-point behavior in the molecular-weight dependence of the critical parameters is also presented.

Thermodynamics and partitioning of homopolymers into a slit-A grand canonical Monte Carlo simulation study

Grand canonical ensemble Monte Carlo simulation (GCMC) combined with the histogram reweighting technique was used to study the thermodynamic equilibrium of a homopolymer solution between a bulk and a slit pore. GCMC gives the partition coefficients that agree with those from canonical ensemble Monte Carlo simulations in a twin box, and it also gives results that are not accessible through the regular canonical ensemble simulation such as the osmotic pressure of the solution. In a bulk polymer solution, the calculated osmotic pressure agrees very well with the scaling theory predictions both for the athermal polymer solution and the theta solution. However, one cannot obtain the osmotic pressure of the confined solution in the same way since the osmotic pressure of the confined solution is anisotropic. The chemical potentials in GCMC simulations were found to differ by a translational term from the chemical potentials obtained from canonical ensemble Monte Carlo simulations with the chain insertion method. This confirms the equilibrium condition of a polymer solution partition between the bulk and a slit pore: the chemical potentials of the polymer chain including the translational term are equal at equilibrium. The histogram reweighting method enables us to obtain the partition coefficients in the whole range of concentrations based on a limited set of simulations. Those predicted bulk-pore partition coefficient data enable us to perform further theoretical analysis. Scaling predictions of the partition coefficient at different regimes were given and were confirmed by the simulation data.

Simulation of phase transitions in fluids

This review provides a discussion of recent techniques for simulation of phase equilibria of complex fluids. Monte Carlo methods are emphasized over molecular dynamics methods. We describe recent developments, such as the use of expanded-ensemble, tempering, or histogram reweighting techniques. Our discussion of such developments is aimed at a general audience and is intended to provide an overview of the main advantages and limitations of each particular technique. References are provided to allow interested readers to identify and trace back most recent applications of a particular simulation technique. We conclude with general guidelines regarding selection of suitable simulation methods for particular problems and systems of interest.

Influence of solvent quality on polymer solutions: A Monte Carlo study of bulk and interfacial properties

The effect of solvent quality on dilute and semidilute regimes of polymers in solution is studied by means of Monte Carlo simulations. The equation of state, adsorption near a hard wall, wall-polymer surface tension, and effective depletion potential are all calculated as a function of concentration and solvent quality. We find important differences between polymers in good and theta solvents. In the dilute regime, the physical properties for polymers in a theta solvent closely resemble those of ideal polymers. In the semidilute regime, however, significant differences are found.

Influence of solvent quality on effective pair potentials between polymers in solution

Solutions of interacting linear polymers are mapped onto a system of "soft" spherical particles interacting via an effective pair potential. This coarse-graining reduces the individual monomer-level description to a problem involving only the center of mass (c.m.) of the polymer coils. The effective pair potentials are derived by inverting the c.m. pair distribution function, generated in Monte Carlo simulations, using the hypernetted chain closure. The method, previously devised for the self-avoiding walk model of polymers in good solvent, is extended to the case of polymers in solvents of variable quality by adding a finite nearest-neighbor monomer-monomer attraction to the previous model and varying the temperature. The resulting effective pair potential is found to depend strongly on temperature and polymer concentration. At low concentration the effective interaction becomes increasingly attractive as the temperature decreases, eventually violating thermodynamic stability criteria. However, as polymer concentration is increased at fixed temperature, the effective interaction reverts to mostly repulsive behavior. These issues help to illustrate some fundamental difficulties encountered when coarse-graining complex systems via effective pair potentials.

Flory-Huggins theory for athermal mixtures of hard spheres and larger flexible polymers

A simple analytic theory for mixtures of hard spheres and larger polymers with excluded volume interactions is developed. The mixture is shown to exhibit extensive immiscibility. For large polymers with strong excluded volume interactions, the density of monomers at the critical point for demixing decreases as one over the square root of the length of the polymer, while the density of spheres tends to a constant. This is very different from the behavior of mixtures of hard spheres and ideal polymers, these mixtures, although even less miscible than those with polymers with excluded volume interactions, have a much higher polymer density at the critical point of demixing. The theory applies to the complete range of mixtures of spheres with flexible polymers, from those with strong excluded volume interactions to ideal polymers.

Competition of mesoscales and crossover to tricriticality in polymer solutions

We show that the approach to asymptotic fluctuation-induced critical behavior in polymer solutions is governed by a competition between a correlation length diverging at the critical point and an additional mesoscopic length-scale, the radius of gyration. Accurate light scattering experiments on polystyrene solutions in cyclohexane with polymer molecular weights ranging from 200 000 up to 11.4x10(6) clearly demonstrate a crossover between two universal regimes: a regime with Ising asymptotic critical behavior, where the correlation length prevails, and a regime with tricritical theta-point behavior determined by a mesoscopic polymer-chain length.

Exactness of the annealed and the replica symmetric approximations for random heteropolymers

We study a heteropolymer model with random contact interactions introduced some time ago as a simplified model for proteins. The model consists of self-avoiding walks on the simple cubic lattice, with contact interactions between nearest-neighbor pairs. For each pair, the interaction energy is an independent Gaussian variable with mean value B and variance Delta(2). For this model the annealed approximation is expected to become exact for low disorder, at sufficiently high dimension and in the thermodynamic limit. We show that corrections to the annealed approximation in the three-dimensional high-temperature phase are small, but do not vanish in the thermodynamic limit, and are in good agreement with our replica symmetric calculations. Such corrections derive from the fact that the overlap between two typical chains is nonzero. We explain why previous authors had come to the opposite conclusion, and discuss consequences for the thermodynamics of the model. Numerical results were obtained by simulating chains of length N<or=1400 by means of the recent PERM algorithm, in the coil and molten globular phases, well above the freezing temperature.