Polymer Mean Spherical Approximation Research Articles

Phase coexistence in the hard-sphere Yukawa chain fluid with chain length polydispersity: Dimer thermodynamic perturbation theory

An extension of the dimer version of Wertheim's thermodynamic perturbation theory is proposed and used to treat polydisperse mixture of the hard-sphere Yukawa chain fluid with chain length polydispersity. The structure and thermodynamic properties of the reference system, represented by multicomponent mixture of the Yukawa hard-sphere dimers, are described using polymer mean spherical approximation. Explicit analytical expressions for the Helmholtz free energy, chemical potential, and pressure in terms of the two chain length distribution function moments are derived. The theory is used to calculate the full liquid-gas phase diagram, including critical binodal, cloud and shadow curves, and distribution functions of the coexisting phases. Effects of fractionation in terms of the distribution function and its first and second moments are studied. Predictions of the theory for these effects are in qualitative agreement with the corresponding experimental predictions, obtained recently for the polydisperse mixture of polymers in a single solvent. In particular, both theory and experiment predict that longer chain polymers equilibrate to the liquid phase while shorter chain polymers are predominantly encountered in the gas phase.

Phase coexistence in a polydisperse charged hard-sphere fluid: Polymer mean spherical approximation

We have reconsidered the phase behavior of a polydisperse mixture of charged hard spheres (CHSs) introducing the concept of minimal size neutral clusters. We thus take into account ionic association effects observed in charged systems close to the phase boundary where the properties of the system are dominated by the presence of neutral clusters while the amount of free ions or charged clusters is negligible. With this concept we clearly pass beyond the simple level of the mean spherical approximation (MSA) that we have presented in our recent study of a polydisperse mixture of CHS [Yu. V. Kalyuzhnyi, G. Kahl, and P. T. Cummings, J. Chem. Phys. 120, 10133 (2004)]. Restricting ourselves to a 1:1 and possibly size-asymmetric model we treat the resulting polydisperse mixture of neutral, polar dimers within the framework of the polymer MSA, i.e., a concept that--similar as the MSA--readily can be generalized from the case of a mixture with a finite number of components to the polydisperse case: again, the model belongs to the class of truncatable free-energy models so that we can map the formally infinitely many coexistence equations onto a finite set of coupled, nonlinear equations in the generalized moments of the distribution function that characterizes the system. This allows us to determine the full phase diagram (in terms of binodals as well as cloud and shadow curves), we can study fractionation effects on the level of the distribution functions of the coexisting daughter phases, and we propose estimates on how the location of the critical point might vary in a polydisperse mixture with an increasing size asymmetry and polydispersity.

Equation of state and liquid-vapor equilibria of one- and two-Yukawa hard-sphere chain fluids: theory and simulation.

The accuracy of several theories for the thermodynamic properties of the Yukawa hard-sphere chain fluid are studied. In particular, we consider the polymer mean spherical approximation (PMSA), the dimer version of thermodynamic perturbation theory (TPTD), and the statistical associating fluid theory for potentials of variable attractive range (SAFT-VR). Since the original version of SAFT-VR for Yukawa fluids is restricted to the case of one-Yukawa tail, we have extended SAFT-VR to treat chain fluids with two-Yukawa tails. The predictions of these theories are compared with Monte Carlo (MC) simulation data for the pressure and phase behavior of the chain fluid of different length with one- and two-Yukawa tails. We find that overall the PMSA and TPTD give more accurate predictions than SAFT-VR, and that the PMSA is slightly more accurate than TPTD.

Yukawa sticky m-point model of associating fluid

The product-reactant Ornstein–Zernike approach, supplemented by the ideal network approximation, is formulated for the Yukawa sticky m-point (YSmP) model of associating fluid. The model is represented by the multicomponent mixture of the Yukawa hard spheres with m sticky points randomly located on the surface of each hard sphere. Extensions of the regular integral equation closures, which include polymer Percus–Yevick, polymer hypernetted chain and polymer mean spherical approximations, are presented. An analytical solution of the polymer mean spherical approximation is derived and closed form analytical expressions for the structure (contact value of the radial distribution function, structure factor) and thermodynamic (internal energy) properties of the YSmP model are obtained. Due to generality and flexibility of the model it can be used to study the properties of a number of different associating fluids, including water and aqueous solutions. By way of illustration liquid–gas phase diagrams for the model with m=0, 1, 2, 3, 4 are presented and discussed. Predictions of the theory for the liquid–gas phase diagram of the YS4P model with the parameters similar to those assumed in the frames of the statistical associating fluid theory to mimic water are in reasonably good agreement with the corresponding experimental data for water.

Structural and thermodynamic properties of a multicomponent freely jointed hard sphere multi-Yukawa chain fluid

The product-reactant Ornstein-Zernike approach, represented by the polymer mean-spherical approximation (PMSA), is utilized to describe the structure and thermodynamic properties of the fluid of Yukawa hard sphere chain molecules. An analytical solution of the PMSA for the most general case of the multicomponent freely jointed hard sphere multi-Yukawa chain fluid is presented. As in the case of the regular MSA for the hard sphere Yukawa fluid, the problem is reduced to the solution of a set of nonlinear algebraic equations in the general case, and to a single equation in the case of the factorizable Yukawa potential coefficients. Closed form analytical expressions are presented for the contact values of the monomer-monomer radial distribution function, structure factors, internal energy, Helmholtz free energy, chemical potentials and pressure in terms of the quantities, which follows directly from the PMSA solution. By way of illustration, several different versions of the hard sphere Yukawa chain model are considered, represented by one-Yukawa chains of length m, where m = 2, 4, 8, 16. To validate the accuracy of the present theory, Monte Carlo simulations were carried out and the results are compared systematically with the theoretical results for the structure and thermodynamic properties of the system at hand. In general it is found that the theory performs very well, thus providing an analytical route to the equilibrium properties of a well defined model for chain fluids.

Erratum: “Multicomponent mixture of charged hard-sphere chain molecules in the polymer mean-spherical approximation” [J. Chem. Phys. 115, 540 (2001)

Abstract

Thermodynamic properties of freely-jointed hard-sphere multi-Yukawa chain fluids: theory and simulation

The hard-sphere Yukawa chain (HSYC) fluid is a simple model for chain fluids and polymers in which each molecule is composed of freely-jointed tangent spheres. The spheres each have a hard core and an attractive interaction, written as a single Yukawa potential or as a sum of Yukawa potentials. We have calculated the phase equilibria and PVT behaviour of the HSYC fluid both theoretically and by Monte Carlo simulation. The theoretical approach is based on the analytic solution of the polymer mean spherical approximation for the HSYC fluid. By comparing the theory and simulation we are able to evaluate the accuracy of the theory. We find that in general the theory performs very well, thus, it provides an analytic route to an equation of state for a well-defined model of chain fluids.

Multicomponent mixture of charged hard-sphere chain molecules in the polymer mean-spherical approximation

The analytical solution of the recently proposed ideal chain polymer mean-spherical approximation [Kalyuzhnyi, Mol. Phys. 94, 735 (1998)] is presented for the multicomponent mixture of charged hard-sphere linear chain flexible molecules. The solution applies to any mixture of chain molecules with arbitrary distribution of the charge and size of the beads along the molecular backbone. Closed form analytical expressions for the internal energy, Helmholtz free energy, chemical potentials, and pressure are derived. By way of illustration thermodynamical properties of several versions of the fluid of charged chain molecules of different length, including the molecules with uniform, diblock, and alternating distribution of the charge, are studied. Theoretical predictions are in reasonable agreement with available computer simulation predictions. We also present the liquid–gas phase diagrams for systems with diblock and alternating distribution of the charge.

Thermodynamics of the polymer mean-spherical ideal chain approximation for a fluid of linear chain molecules

A mean-spherical approximation (MSA)-like theory is proposed for the site-site molecular fluid represented by hard sphere totally flexible linear chain molecules with a long range site site pair potential of arbitrary type outside the hard core region. The theory is based on the complete association limit of the polymer mean-spherical approximation (PMSA) supplemented by the so-called ideal chain approximation, and belongs to the class of the site-site integral equation theories that are ‘proper’ in the sense of Chandler and coworkers. Both approximations have been used recently in the frame of the multidensity integral equation theory for associating fluids. In the simplest case of chain length 2 the present approach reduces to the Chandler-Silbey-Ladanyi (CSL) equation closed by the corresponding MSA closure conditions. The Hoye-Stell scheme of calculating thermodynamic properties is extended and expressions for the Helmholtz free energy, chemical potential and pressure are derived. The theory is illust...

Thermodynamics of the polymer mean-spherical ideal chain approximation for a fluid of linear chain molecules

A mean-spherical approximation (MSA)-like theory is proposed for the site-site molecular fluid represented by hard sphere totally flexible linear chain molecules with a long range site site pair potential of arbitrary type outside the hard core region. The theory is based on the complete association limit of the polymer mean-spherical approximation (PMSA) supplemented by the so-called ideal chain approximation, and belongs to the class of the site-site integral equation theories that are ‘proper’ in the sense of Chandler and coworkers. Both approximations have been used recently in the frame of the multidensity integral equation theory for associating fluids. In the simplest case of chain length 2 the present approach reduces to the Chandler-Silbey-Ladanyi (CSL) equation closed by the corresponding MSA closure conditions. The Hoye-Stell scheme of calculating thermodynamic properties is extended and expressions for the Helmholtz free energy, chemical potential and pressure are derived. The theory is illustrated by its application in the case of charged hard dumbbells, for which an analytical solution of the CSL-MSA is available. Analytical expressions for the Helmholtz free energy, chemical potential and pressure in terms of only one unknown parameter, which satisfies a nonlinear algebraic equation, are derived and used to calculate the liquid-gas phase diagram. Theoretical predictions for the coexistence curve are compared with the corresponding computer simulation predictions for the phase diagrams of the charged hard dumbbell fluid and fluid represented by the restricted primitive model (RPM) of electrolytes. In the latter case the present theory gives more accurate results than those of Zhou, Y., Yeh, S., and Stell, G., 1995, J. chem. Phys., 102, 5785, and are of the same order of accuracy as the version of Fisher-Levin theory which consistently takes into account the hard core contribution (Guillot, B., and Guissani, Y., 1996, Molec. Phys., 87, 37).

Solution of the polymer mean spherical approximation for the totally flexible sticky two-point electrolyte model

The general solution of the polymer MSA (mean spherical approximation) for an arbitrary mixture of charged hard spheres with multiple bonding obtained recently by Blum, Holovko and Protsykevytch [J. Stat. Phys,] is applied to the case of totally flexible linear polymer chains. The approximation is an extension of the associative MSA proposed by M.Holovko and Yu.Kalyuzhnyi, [Mol.Phys., 73, 1145(1991)] to study the effects of association in ionic systems. The model is defined by adding charges to the multicomponent version of the totally-flexible sticky two-point model of associating monomers introduced by Wertheim.