Random Mixing Approximation Research Articles

A simple theory for interfacial properties of dilute solutions.

Recent studies suggest that cosolute mixtures may exert significant non-additive effects upon protein stability. The corresponding liquid-vapor interfaces may provide useful insight into these non-additive effects. Accordingly, in this work, we relate the interfacial properties of dilute multicomponent solutions to the interactions between solutes. We first derive a simple model for the surface excess of solutes in terms of thermodynamic observables. We then develop a lattice-based statistical mechanical perturbation theory to derive these observables from microscopic interactions. Rather than adopting a random mixing approximation, this dilute solution theory (DST) exactly treats solute-solute interactions to lowest order in perturbation theory. Although it cannot treat concentrated solutions, Monte Carlo (MC) simulations demonstrate that DST describes the interactions in dilute solutions with much greater accuracy than regular solution theory. Importantly, DST emphasizes a fundamental distinction between the "intrinsic" and "effective" preferences of solutes for interfaces. DST predicts that three classes of solutes can be distinguished by their intrinsic preference for interfaces. While the surface preference of strong depletants is relatively insensitive to interactions, the surface preference of strong surfactants can be modulated by interactions at the interface. Moreover, DST predicts that the surface preference of weak depletants and weak surfactants can be qualitatively inverted by interactions in the bulk. We also demonstrate that DST can be extended to treat surface polarization effects and to model experimental data. MC simulations validate the accuracy of DST predictions for lattice systems that correspond to molar concentrations.

Order-Disorder Transitions and Free Energies in Asymmetric Diblock Copolymers

Simulations of simple bead-spring models of asymmetric diblock copolymers are used to study the dependence of order-disorder transitions and free energies upon the invariant degree of polymerization N̅ and the fraction f of beads in the minority block. Well-tempered metadynamics is used to determine values of (χN)ODT along the lamellar-disorder and hexagonal-disorder transitions over the range 0.1875 ≤ f ≤ 0.5 for two models with different values of N̅ = 480 and 1920, where χ is an effective Flory–Huggins interaction parameter, N is the degree of polymerization, and (χN)ODT is a value of χN at the order-disorder transition (ODT). More extensive studies are performed for systems with f = 1/4, which undergo a hexagonal-disorder transition. Equivalent results for both phase boundaries and free energies are obtained for one pair of systems with different numbers of beads per chain but matched values of f = 1/4 and N̅, in agreement with the corresponding state hypothesis. Comparison of results for (χN)ODT for systems with f = 1/4 and several values for N̅ show a systematic decrease in (χN)ODT with an increase N̅, consistent with the expected approach to the self-consistent field (SCFT) prediction as N̅ → ∞. Results for the free energy per chain in the disordered and hexagonal phases of systems with f = 1/4 show that SCFT gives rather accurate predictions for the free energy in the ordered hexagonal phase but that the random-mixing approximation underlying SCFT significantly overestimates the free energy of the disordered phase.

Solubility in compressible polymers: Beyond the regular solution theory

The age-old idea of “like dissolves like” requires a notion of “likeness” that is hard to quantify for polymers. We revisit the concepts of pure component cohesive energy density c P and mutual cohesive energy density c 12 so that they can be extended to polymers. We recognize the inherent limitations of c 12 due to its very definition, which is based on the assumption of no volume of mixing (true for incompressible systems), one of the assumptions in the random mixing approximation (RMA); no such limitations are present in the identification of c P. We point out that the other severe restriction on c 12 is the use of pure components in its definition because of which c 12 is not merely controlled by mutual interactions. Another quantity c 12 SRS as a measure of mutual cohesive energy density that does not suffer from the above limitations of c 12 is introduced. It reduces to c 12 in the RMA limit. We are able to express c 12 SRS in terms of c 12 and pure component c P’s. We also revisit the concept of the internal pressure and its relationship with the conventional and the newly defined cohesive energy densities. In order to investigate volume of mixing effects, we introduce two different mixing processes in which volume of mixing remains zero. We then carry out a comprehensive reanalysis of various quantities using a recently developed recursive lattice theory that has been tested earlier and has been found to be more accurate than the conventional regular solution theory such as the Flory–Huggins theory for polymers. In the RMA limit, our recursive theory reduces to the Flory–Huggins theory or its extension for a compressible blend. Thus, it supersedes the Flory–Huggins theory. Consequently, the conclusions based on our theory are more reliable and should prove useful.

Lessons from the design of a novel atomic potential for protein folding

We investigate all-atom potentials of mean force for estimating free energies in protein folding and fold recognition. We search through the space potentials and design novel atomic potentials with a random mixing approximation and a contact-correlated Gaussian approximation of decoy states. We show that the two derived potentials are highly correlated, supporting the use of the random energy model as an accurate statistical description of protein conformational states. The novel atomic potentials perform well in a Z-score and fold decoy recognition test. Furthermore, the designed atomic potential performs slightly and significantly better than atomic potentials derived under a quasi-chemical assumption. While accounting for connectivity correlations between atom types does not improve the performance of the designed potential, we show these correlations lead to ambiguities in the distribution of energetic contributions for atoms on the same residue. Within the confines of the model then, many potentials may exist which stabilize all native folds in subtly different ways. Comparison of different protein conformations under the various atomic potentials reveals both a remarkable degree of correspondence in the estimated free energies and a remarkable degree of correspondence in the identity of the contacts types that make the dominant contributions to the estimated free energies. This consistency may be interpreted as a sign that the design procedure is extracting physically meaningful quantities.

Simulation study of the coil-globule transition of a polymer in solvent

Molecular dynamics simulations are used to study the coil-globule transition for a system composed of a bead-spring polymer immersed in an explicitly modeled solvent. Two different versions of the model are used, which are differentiated by the nature of monomer-solvent, solvent-solvent, and nonbonded monomer-monomer interactions. For each case, a model parameter lambda determines the degree of hydrophobicity of the monomers by controlling the degree of energy mismatch between the monomers and solvent particles. We consider a lambda-driven coil-globule transition at constant temperature. The simulations are used to calculate average static structure factors, which are then used to determine the scaling exponents of the system in order to determine the theta-point values lambdatheta separating the coil from the globule states. For each model we construct coil-globule phase diagrams in terms of lambda and the particle density rho. The results are analyzed in terms of a simple Flory-type theory of the collapse transition. The ratio of lambdatheta for the two models converges in the high density limit exactly to the value predicted by the theory in the random mixing approximation. Generally, the predicted values of lambdatheta are in reasonable agreement with the measured values at high rho, though the accuracy improves if the average chain size is calculated using the full probability distribution associated with the polymer-solvent free energy, rather than merely using the value obtained from the minimum of the free energy.

Analytic density-functional self-consistent-field theory of diblock copolymers near patterned surfaces.

Analytical solutions are derived for the density profiles and the free energies of compressible diblock copolymer melts (or incompressible copolymer solutions) near patterned surfaces. The density-functional self-consistent-field theory is employed along with a Gaussian chain model for bonding constraints and a random mixing approximation for nonbonded interactions. An analytical solution is rendered possible by expanding the chain distribution function around an inhomogeneous reference state with a nontrivial analytical solution, by retaining the linear terms, and by requiring consistency with the homopolymer limit. The density profiles are determined by both real and complex roots of a sixth-degree polynomial that may easily be obtained by solving a generalized eigenvalue problem. This analytical formulation enables one to efficiently explore the large nine-dimensional parameter space and can serve as a first approximation to computationally intensive studies with more detailed models. Illustrative computations are provided for uniform and patterned surfaces above the order-disorder transition. The results are consistent with the previous self-consistent-field calculations in that lamellar ordering appears near the surface above the order-disorder transition and the lamella order perpendicular or parallel to the surface depending on the commensurability between the periods of the surface pattern and the density oscillations.

Mean-field Gaussian chain theory for semidilute theta chains in a slit

Mean-field Gaussian chain theory for nondilute polymer solutions is being applied to polymer chains in the theta condition confined to a slit in a wide range of concentrations. Various existing lattice chain theories were used to obtain the effective potential for the Gaussian chain. Calculation results for the monomer density profile, the partition coefficient, and the chain dimension across the slit at various concentrations were computed and compared with the results obtained in the lattice Monte Carlo simulations. The need for a higher concentration to flatten the density profile and to cause the weak-to-strong penetration transition, compared with athermal chains, was confirmed. At the monomer–monomer interaction widely used as the theta condition on the cubic lattice, the chain cluster theory by Freed and the random mixing approximation by Huggins provide a better agreement with the simulation results than the Flory model and the Guggenheim model do. The agreement is, however, not as good as the one for athermal chains.

Monte Carlo test of the lattice cluster theory: Thermodynamic properties of binary polymer blends

The particle exchange method is used to compute the exchange chemical potential for binary mixtures of linear polymers in Monte Carlo lattice model simulations. The internal energy and the specific heat are also calculated for each thermodynamic state. Simulations are performed for nine blend compositions, with the total monomer volume fraction held fixed at φ=0.8 or φ=0.7. Both polymer species have the same polymerization indices, M=40, 50, or 100. The two species are distinguished by their van der Waals interaction strengths, ε11, ε12, and ε22. Several ratios ε22/ε11 are considered, and the simulation temperatures range from almost athermal conditions to near-critical states. The parameter ε12 is either computed as the geometrical average ε11ε22 or is assigned an independent value. Three approximate solutions to the model are compared to the simulations: Flory–Huggins theory, Guggenheim’s random mixing approximation, and the lattice cluster theory (LCT). The results confirm the higher accuracy of the LCT. However, some predictions for the specific heat are again found to be inaccurate because of the low order cutoff of the high temperature perturbative expansion.

Universal equation of state for an interacting multicomponent mixture of polymers

We present a closed form universal equation of state for an interacting multicomponent mixture of polymers of any architecture and dispersity. The equation is obtained by solving the model on a Bethe lattice and goes beyond the random mixing approximation. The latter property endows our theory with features that are consistent with real systems. The equation of state, though an approximate one, is thermodynamically consistent and is valid even in the incompressible limit. The predictions of the equation are consistent with simulations and experiments, as discussed.

Lattice theory of a multicomponent mixture of monodisperse polymers of fixed architectures

We present a lattice theory for a multicomponent mixture of p distinct polymeric species, each of a prescribed architecture but without any cycles and s monomeric species along with a solvent species, the latter playing the role of a reference species whose amount is controlled not by any activity but by the sum rule of fixed amount of material. The theory is an extension of our previous work on a binary mixture of polymers in bulk or a general mixture next to a surface. The model allows for nearest-neighbor interactions between unlike species. The chemical bondings are allowed to be between monomers (of the same species) that are nearest-neighbor. The resulting theory is obtained by solving the model on a Bethe lattice. The theory has a very simple structure and supersedes random mixing approximation to which it reduces in a special limit, the random mixing approximation limit, see text. We study the behavior of a ternary system numerically and compare it with that of a binary system. We also compare the predictions of our theory with simulations and find them to be consistent. However, our theoretical predictions are inconsistent with the conventional Flory–Huggins theory. Thus, our theory is superior to the Flory–Huggins theory.

Lattice theory of polymer solutions with endgroup effects

We present a unified lattice theory for a binary solution where endgroups are treated differently from middle groups. This is a simple example of a triblock and the present study provides a starting point for studying a general triblock system. We replace the original homogeneous lattice by a Bethe lattice of the same coordination number as the original lattice. The model is solved exactly on the Bethe lattice. The resulting solution goes beyond the random mixing approximation and provides us with an approximate theory of the model on the regular lattice. The contributions of endgroups on various thermodynamic properties of a binary solution are investigated in a quantitative way using the theory. In particular, our theory predicts that contributions to the energy are more important than to the entropy.

New statistical mechanical treatment of systems near surfaces. III. Polydisperse linear and branched polymers in an interacting solution

We apply a recently developed analytical but approximate method to study the behavior of polydisperse linear and branched polymers in a solution with interaction and near various kinds of surfaces. We consider equilibrium polydispersity controlled by a set of activities. The method allows us to account for polymer connectivity and excluded-volume effects and goes beyond the random mixing approximation. The method also allows us to probe the entire density range, which is not possible in a Monte Carlo simulation. The density profiles show oscillations for bulk densities φmb larger than some threshold value φmt, the amplitudes of those oscillations being a strong function of the solvent-monomer interaction parameter w. The behavior of linear and branched polymers is almost identical for the same bulk densities φmb, except when the solvent-monomer interaction is repulsive and the surface-monomer interaction is not attractive. In this case, the effect of branching is to increase the correlation length ξ describing the approach of a density profile to its bulk value. We also compute the surface thermodynamic functions (the entropy and the free energy). While the surface free energies are identical for linear and branched polymers, the surface entropies are very different.

New statistical mechanical treatment of systems near surfaces. II. Polydisperse linear and branched polymers in an athermal solution

We apply a recently developed analytic but approximate method to study the behavior of polydisperse linear and branched polymers in an athermal solution and near various kinds of surfaces. We consider equilibrium polydispersity controlled by a set of activities. The method allows us to account for polymer connectivity and excluded-volume effects and goes beyond the random mixing approximation. The density profiles of various kinds exhibit oscillations for bulk densities φmb larger than some threshold bulk density φmT. The origin of these oscillations is related to the decreasing branch of the recursion function, as explained in the text. The correlation length ξ related to these oscillations increases as φmb increases. On the other hand, the correlation length ξ controlling the approach of various density profiles to their respective bulk values in the range φmb<φmT increases as φmb decreases. The free energy and the entropy are uniquely determined. Various surface properties are also easily determined. We demonstrate the existence of surface polymerization transition in the case of linear chains as surface interactions are varied, even though there is no bulk transition. The method allows us to probe the entire density range which is not possible in a Monte Carlo simulation. Our computations are ultrafast by at least three orders of magnitude compared to rival methods and easy to implement. Our results agree with Monte Carlo simulations but disagree with the mean-field predictions.

New statistical mechanical treatment of systems near surfaces. I. Theory and principles

We present a new theoretical framework for a statistical mechanical and thermodynamic description of any general inhomogeneous system (not necessarily polymeric) in the presence of surfaces. The framework is an extension of a lattice theory recently developed for a homogeneous system and requires approximating the original lattice by a recursive lattice which, for simplicity, we take to be a modified tree structure (see Fig. 4), TM as described in the text. The tree is formed recursively by two basic elements, the main tree 𝒯 and the surface tree T¯. The model is solved exactly using a recursion technique. The technique allows us to account for connectivity, architecture, excluded-volume effects, interactions, etc. exactly. The resulting description goes beyond the random-mixing approximation used in most mean-field theories. We consider a general model of a multicomponent system and its exact solution on the modified tree TM provides us with an approximate theory of the inhomogeneous system on the original lattice. We provide a general discussion of the theory and principles involved. Our method produces results similar to those of Monte Carlo simulations but can even be applied to cases where Monte Carlo simulations are not possible. We also obtain surface free energy and the surface entropy that is not easily obtained in a Monte Carlo simulation. Our method is more reliable than the mean-field method of Scheutjens and Fleer, whose predictions are, in many cases, in direct contradiction with the Monte Carlo simulations. Our method is fast by at least three orders of magnitude compared to rival methods.

Small-angle neutron scattering experiments on microemulsion droplets: relation to the bending elasticity of the amphiphilic film

Small-angle neutron scattering (SANS) experiments and measurements of interfacial tension have been carried out on droplet type microemulsions in the system nonionic surfactant (C i E j )/hydrocarbon/water. The SANS spectra in the shell contrast can be very well interpreted by a model of polydisperse spherical shell structured aggregates. In addition it has been shown that a more precise description is obtained if one takes into account the diffuseness of the amphiphilic film. However, the outcome for the structural parameters does not depend on the choice of the particular model employed. For microemulsion droplets structural quantities such as radius, polydispersity index, and macroscopic interfacial tension can be related directly to the bending constants (bending modulus κ and Gaussian modulus \\ ̄ gk) of the system. The sum 2κ + \\ ̄ gk can be deduced independently from interfacial tension and polydispersity index (from SANS data in the shell contrast) and both are in good agreement. Variation of the chain length of the surfactant shows that this sum increases with increasing thickness of the amphiphilic film. In contrast, variation of the chain length of the hydrocarbon does not influence the elastic properties of the surfactant film. A dilution series also allows for a determination of 2κ + \\ ̄ gk from the concentration dependence of the deduced radii and polydispersity indices. Again good agreement is observed with the values obtained before and the most appropriate entropy of mixing term to describe such a microemulsion appears to be a random mixing approximation. Experiments both in oil-in-water (O/W) and water-in-oil (W/O) droplets indicate that 2κ + \\ ̄ gk decreases with increasing temperature (W/O at higher temperature). In general one can state that the properties of these microemulsions are well described by the elastic theory for the amphiphilic film. Structural parameters such as radius and polydispersity are related to the interfacial tension and if two of these quantities are known the third can be predicted reliably.

Effective chi in a lattice theory of mixtures.

Using general thermodynamic arguments for an incompressible lattice system, we demonstrate that the effective \ensuremath{\chi} must be a complicated function of temperature, composition, polymerization indices, and the coordination number of the lattice, even though the bare \ensuremath{\chi} is merely a parameter. We go beyond the random mixing approximation and present a lattice theory of mixtures which yields an effective \ensuremath{\chi} consistent with experiments. \textcopyright{} 1996 The American Physical Society.

A fourth-order, density-functional, random-phase approximation study of monomer segregation in phase-separated, lamellar, diblock-copolymer melts

It is shown that the random-phase approximation (RPA) for correlations in block-copolymer melts derives from the free energy functional of ideal chains augmented by a random-mixing approximation for the interactions between monomers. The derivation facilitates comparisons to other approaches to monomer segregation. The RPA functional is applied to the segregation of symmetric, lamellar, block-copolymer phases while employing the full wave-vector dependence of the third- and fourth-order vertex functions. Compared to calculations which accurately treat only the second-order vertex function, the fourth-order theory shows that the wave-vector dependence of the higher-order terms plays an important role in determining the degree of segregation of the melt

Entropy and energy of mixing in polymer solutions: simple expressions that approximate lattice cluster theory

A new expression for free energy of mixing is proposed. This equation contains a repulsive part (entropy term) and an attractive part (internal energy term). The entropy of mixing is Guggenheim's random-mixing term. The energy of mixing is a simple, exponentially-weighted, surface-fraction-averaged energy. It is shown that the attractive term is simple, yet accurate. Comparisons are made with Flory theory, Guggenheim's random-mixing approximation, Guggenheim's quasichemical theory, lattice cluster theory, Born-Green-Yvon formalism and experimental data for real systems. This equation provides good results for phase equilibrium calculations, energies of mixing and activities

A Born–Green–Yvon integral equation treatment of incompressible lattice mixtures

The case of a binary incompressible lattice mixture is treated using the Born–Green–Yvon (BGY) integral equation approach with the Kirkwood superposition approximation. Analytic expressions for ΔEmix and ΔAmix are derived without invoking the random mixing approximation. The BGY predictions for ΔEmix and ΔS(nc), the noncombinatorial contribution to the entropy of mixing, are compared with those of the lattice cluster (LC) theory of Freed and co-workers for mixtures in which at least one of the two components is a polymer.

Excess gibbs free energies of mixtures of substances containing nitrogen groups

Vapour-liquid equilibria for dimethylethylamine(1) + n-heptane or cyclohexane(2) and for N,N,N',N'-tetramethylalkanediamine(1) of general formula (CH 3) 2N-(CH 2) u -N(CH 3) 2, with u = 1,2,3, + n-heptane(2) were determined at 298.15 K by head-space gas-chromatographic analysis of the equilibrated vapour phase withdrawn directly from the equilibration apparatus. Measurements on the test mixture, ethanol + cyclohexane, were also made. Excess molar Gibbs free energies G E were obtained by a least-squares treatment of the equilibrium x,y results. The G E values are positive for all the systems and increase with the ratio of nitrogen to carbon atoms in the amine or diamine molecule. They also depend on the distance between the two N atoms in the diamines. The experimental G E values are compared with the predictions of a group contribution model in the random-mixing approximation.