How to derive and parameterize effective potentials in colloid-polymer mixtures

Polymer chains in colloid-polymer mixtures can be coarse-grained by replacing them with single soft particles interacting via effective polymer-polymer and polymer-colloid pair potentials. Here we describe in detail how Ornstein-Zernike inversion techniques, originally developed for atomic and molecular fluids, can be generalized to complex fluids and used to derive effective potentials from computer simulations on a microscopic level. In particular, we consider polymer solutions for which we derive effective potentials between the centers of mass, and also between mid-points or end-points from simulations of self-avoiding walk polymers. In addition, we derive effective potentials for polymers near a hard wall or a hard sphere. We emphasize the importance of including both structural and thermodynamic information (through sum-rules) from the underlying simulations. In addition we develop a simple numerical scheme to optimize the parameterization of the density dependent polymer-polymer, polymer-wall and polymer-sphere potentials for dilute and semi-dilute polymer densities, thus opening up the possibility of performing large-scale simulations of colloid-polymer mixtures. The methods developed here should be applicable to a much wider range effective potentials in complex fluids.Comment: uses revtex4.cls; submitted for archival purpose

Many-body interactions and correlations in coarse-grained descriptions of polymer solutions

We calculate the two, three, four, and five-body (state independent) effective potentials between the centers of mass (CM) of self avoiding walk polymers by Monte-Carlo simulations. For full overlap, these coarse-grained n-body interactions oscillate in sign as (-1)^n, and decrease in absolute magnitude with increasing n. We find semi-quantitative agreement with a scaling theory, and use this to discuss how the coarse-grained free energy converges when expanded to arbitrary order in the many-body potentials. We also derive effective {\em density dependent} 2-body potentials which exactly reproduce the pair-correlations between the CM of the self avoiding walk polymers. The density dependence of these pair potentials can be largely understood from the effects of the {\em density independent} 3-body potential. Triplet correlations between the CM of the polymers are surprisingly well, but not exactly, described by our coarse-grained effective pair potential picture. In fact, we demonstrate that a pair-potential cannot simultaneously reproduce the two and three body correlations in a system with many-body interactions. However, the deviations that do occur in our system are very small, and can be explained by the direct influence of 3-body potentials.Comment: 11 pages, 1 table, 9 figures, RevTeX (revtex.cls

Density profiles and surface tensions of polymers near colloidal surfaces

The surface tension of interacting polymers in a good solvent is calculated theoretically and by computer simulations for a planar wall geometry and for the insertion of a single colloidal hard-sphere. This is achieved for the planar wall and for the larger spheres by an adsorption method, and for smaller spheres by a direct insertion technique. Results for the dilute and semi-dilute regimes are compared to results for ideal polymers, the Asakura-Oosawa penetrable-sphere model, and to integral equations, scaling and renormalization group theories. The largest relative changes with density are found in the dilute regime, so that theories based on non-interacting polymers rapidly break down. A recently developed ``soft colloid'' approach to polymer-colloid mixtures is shown to correctly describe the one-body insertion free-energy and the related surface tension

The Asakura-Oosawa model in the protein limit: the role of many-body interactions

We study the Asakura-Oosawa model in the "protein limit", where the penetrable sphere radius $R_{AO}$ is much greater than the hard sphere radius $R_c$. The phase behaviour and structure calculated with a full many-body treatment show important qualitative differences when compared to a description based on pair potentials alone. The overall effect of the many-body interactions is repulsive.Comment: 9 pages and 11 figures, submitted to J. Phys.: Condensed Matter, special issue "Effective many-body interactions and correlations in soft matter

Coarse-graining polymers as soft colloids

We show how to coarse grain polymers in a good solvent as single particles, interacting with density-independent or density-dependent interactions. These interactions can be between the centres of mass, the mid-points or end-points of the polymers. We also show how to extend these methods to polymers in poor solvents and mixtures of polymers. Treating polymers as soft colloids can greatly speed up the simulation of complex many-polymer systems, including polymer-colloid mixtures.Comment: to appear in Physica A, special STATPHYS 2001 edition. Content of invited talk by AA

Accurate effective pair potentials for polymer solutions

Dilute or semi-dilute solutions of non-intersecting self-avoiding walk (SAW) polymer chains are mapped onto a fluid of ``soft'' particles interacting via an effective pair potential between their centers of mass. This mapping is achieved by inverting the pair distribution function of the centers of mass of the original polymer chains, using integral equation techniques from the theory of simple fluids. The resulting effective pair potential is finite at all distances, has a range of the order of the radius of gyration, and turns out to be only moderately concentration-dependent. The dependence of the effective potential on polymer length is analyzed in an effort to extract the scaling limit. The effective potential is used to derive the osmotic equation of state, which is compared to simulation data for the full SAW segment model, and to the predictions of renormalization group calculations. A similar inversion procedure is used to derive an effective wall-polymer potential from the center of mass density profiles near the wall, obtained from simulations of the full polymer segment model. The resulting wall-polymer potential turns out to depend strongly on bulk polymer concentration when polymer-polymer correlations are taken into account, leading to a considerable enhancement of the effective repulsion with increasing concentration. The effective polymer-polymer and wall-polymer potentials are combined to calculate the depletion interaction induced by SAW polymers between two walls. The calculated depletion interaction agrees well with the ``exact'' results from much more computer-intensive direct simulation of the full polymer-segment model, and clearly illustrates the inadequacy -- in the semi-dilute regime -- of the standard Asakura-Oosawa approximation based on the assumption of non-interacting polymer coils.Comment: 18 pages, 24 figures, ReVTeX, submitted to J. Chem. Phy

Refinement of the crystal structure of caesium dichloride

CslC12, trigonal, space group R'3m with a= 5.469 (2) A, ~ = 70.67 (3) °, Z= 1. The atomic positions have been determined by least-squares refinement of counter intensities, the final R being 0.031 for 256 reflexions. The I-CI bond length is 2.548 A

Steady-state nucleation rate and flux of composite nucleus at saddle point

The steady-state nucleation rate and flux of composite nucleus at the saddle point is studied by extending the theory of binary nucleation. The Fokker-Planck equation that describes the nucleation flux is derived using the Master equation for the growth of the composite nucleus, which consists of the core of the final stable phase surrounded by a wetting layer of the intermediate metastable phase nucleated from a metastable parent phase recently evaluated by the author [J. Chem. Phys. {\bf 134}, 164508 (2011)]. The Fokker-Planck equation is similar to that used in the theory of binary nucleation, but the non-diagonal elements exist in the reaction rate matrix. First, the general solution for the steady-state nucleation rate and the direction of nucleation flux is derived. Next, this information is then used to study the nucleation of composite nucleus at the saddle point. The dependence of steady-state nucleation rate as well as the direction of nucleation flux on the reaction rate in addition to the free-energy surface is studied using a model free-energy surface. The direction of nucleation current deviates from the steepest-descent direction of the free-energy surface. The results show the importance of two reaction rate constants: one from the metastable environment to the intermediate metastable phase and the other from the metastable intermediate phase to the stable new phase. On the other hand, the gradient of the potential $\Phi$ or the Kramers crossover function (the commitment or splitting probability) is relatively insensitive to reaction rates or free-energy surface.Comment: 12 pages, 6 figures, to be published in Journal of Chemical Physic

An integral equation approach to effective interactions between polymers in solution

We use the thread model for linear chains of interacting monomers, and the ``polymer reference interaction site model'' (PRISM) formalism to determine the monomer-monomer pair correlation function $h_{mm}(r)$ for dilute and semi-dilute polymer solutions, over a range of temperatures from very high (where the chains behave as self-avoiding walks) to below the $\theta$ temperature, where phase separation sets in. An inversion procedure, based on the HNC integral equation, is used to extract the effective pair potential between ``average'' monomers on different chains. An accurate relation between $h_{mm}(r)$, $h_{cc}(r)$ [the pair correlation function between the polymer centers of mass (c.m.)], and the intramolecular form factors is then used to determine $h_{cc}(r)$, and subsequently extract the effective c.m.-c.m. pair potential $v_{cc}(r)$ by a similar inversion procedure. $v_{cc}(r)$ depends on temperature and polymer concentration, and the predicted variations are in reasonable agreement with recent simulation data, except at very high temperatures, and below the $\theta$ temperature.Comment: 13 pages, 13 figures, revtex ; revised versio

Density functional theory and demixing of binary hard rod-polymer mixtures

A density functional theory for a mixture of hard rods and polymers modeled as chains built of hard tangent spheres is proposed by combining the functional due to Yu and Wu for the polymer mixtures [J. Chem. Phys. {\bf 117}, 2368 (2002)] with the Schmidt's functional [Phys. Rev. E {\bf 63}, 50201 (2001)] for rod-sphere mixtures. As a simple application of the functional, the demixing transition into polymer-rich and rod-rich phases is examined. When the chain length increases, the phase boundary broadens and the critical packing fraction decreases. The shift of the critical point of a demixing transition is most noticeable for short chains.Comment: 4 pages,2 figures, in press, PR