Yukawa Closure Research Articles

A complete integral equation theory for accurate thermodynamics of chain molecules

An integral equation theory is developed for n-alkane chains ranging in carbon number from eight to infinity through an adaptation of the polymer referenced interaction site model (PRISM). Thermodynamic perturbation theory (TPT) is adapted as the basis for expressing the equation of state through the coupling parameter expansion (CPE) as an extension of the theory developed by Ramana et al. for spherical molecules. The key feature of PRISM theory is the assumption that all pair correlations for a given site type are equivalent, regardless of whether they appear at the end of the chain or in the middle. The present analysis assumes that the PRISM result represents the average over all sites in the chain. The site-site interaction energies are then implemented through a weighted average based on their frequency of occurrence. The CPE is demonstrated through the fourth order in temperature, although higher order terms are accessible in principle. An advantage of the TPT representation is that convergence need be achieved only once for each chain length and density, then the TPT coefficients can be tabulated and interpolated for future reference. This approach moderates difficulties with molecular simulation of fully equilibrated long chains and uncertainty in the estimation of higher order TPT coefficients. Step potentials are used as the basis for validating and refining the theory using published simulation results for n-alkanes through 80 carbons. We find that the first order TPT contribution is fairly accurate regardless of conformational details but the second order TPT contribution is remarkably sensitive to conformation. We can obtain accurate characterization of the second order contribution by varying the stiffness parameter of the semi-flexible chain model. Higher order TPT contributions follow from the second order result. The theory would not be “complete,” however, without accurate characterization of the TPT reference contribution, which simultaneously impacts the first order contribution. We combine the compressibility route with a Yukawa closure to accurately characterize the reference fluid thermodynamics as a function of density, chain length, and bond length. Altogether, the result is an accurate model of all contributions to chain molecular thermodynamics, including the underlying fluid structure. A sample application is demonstrated by generating the phase diagrams through 80 carbons. It is observed that the flatness near the critical region is reproduced by including the higher order TPT contributions, but the impact of higher order contributions in the critical region is diminished for 80 carbon chains.

Thermodynamics of the soft and extended soft mean spherical model

A new analytical theory is discussed that extends the range of the mean spherical approximation (MSA) to extreme situations where the asypmtotic limits are known. The MSA [J. K. Percus and G. Yevick, Phys. Rev. 110, 250 (1964); J. L. Lebowitz and J. K. Percus, Phys. Rev. 144, 251 (1966)] yields explicit thermodynamics and structure in terms of a single screening parameter [L. Blum, Mol. Phys. 30, 1559 (1975)] and gives the correct high density limiting behavior of the contact pair distribution function and thermodynamics [Y. Rosenfeld and L. Blum, J. Chem. Phys. 85, 1556 (1986), Y. Rosenfeld, J. Stat. Phys. 37, 215, (1984)] and has been very useful for practical applications. However, it misses the low density limiting behavior which is implicit in the second virial coefficient. The soft mean spherical approximation (SMSA), which was proposed over 30 years ago by Blum and Narten [L. Blum and A. H. Narten, J. Chem. Phys. 56, 5197 (1972), A. H. Narten, L. Blum, and R. H. Fowler, J. Chem. Phys. 60, 3378 (1974)], corrects this problem interpolating between the MSA and the PY. It is an amazingly accurate approximation. For a given switch distance σ sw the SMSA closure is The present communication gives a fully analytical solution of SMSA using the Yukawa closure of the Ornstein Zernike equations [L. Blum and J. A. Hernando, in: Condensed Matter Theories, edited by S. Hernandez, J. W. Clark (Nova, New York, 2001), Vol. 16, p. 411, L. Blum and M. Arias, J. Phys.: Cond. Matter 18, S(2437), ArXiv cond-mat 0602477 (2006)]. This then leads to useful extensions that are discussed in this article.

Explicit partial static structure factors of neutral Yukawa fluids

An explicit expression for the partial static structure factors for neutral Yukawa liquids is presented, starting from Blum and Ginoza's MSA solution of the Ornstein-Zernike equation for the factorizable Yukawa closure.

Analytical solution of the Yukawa closure of the Ornstein-Zernike equation IV: the general 1-component case

Analytical solution of the Yukawa closure of the Ornstein-Zernike equation IV: the general 1-component case

Analytical solution of the Yukawa closure of the Ornstein-Zernik equation IV: the general 1-component case

In previous work we have studied the solution of the Ornstein-Zernike equation with a general multiyukawa closure. Here the direct correlation function is expressed by a rapidly converging sum of M (complex) exponentials. For a simple fluid the mathematical problem of solving the Ornstein-Zernike equation is equivalent to finding the solution of a linear algebraic equation of order M. The solution for the arbitrary case is given in terms of a scaling matrix Γ. For only one component this matrix is diagonal and the general solution using the properties of M-dimensional SOM Lie group is given. In the Mean Spherical Approximation (MSA) the excess entropy is obtained and expressed as a sum of 1-dimensional integrals of algebraic functions. We remark that the general solution of the M exponents-1 component case was found in our early work (Blum, L., and Hoye, J. S.,1978, J. stat. Phys., 19, 317) in implicit form. The present explicit solution agrees completely with the early one. Other thermodynamic properties such as the energy equation of state are also obtained, explicitly for 2 and 3 exponentials. The analytical solution of the effective MSA is also obtained from the simple variational form for the Helmholtz excess free energy δA (1) where (2) where both the excess energy ΔE(Γ) and the excess entropy ΔS(Γ) are functionals of Γ, which opens interesting possibilities that are discussed elsewhere. We remark that this is a non-trivial property, which is certainly true for the MSA (Chandler, D., and Andersen, H. C., 1972, J. chem. Phys., 57, 1930). It implies cross-derivative properties for the closure equations, which have been verified in all cases.

Analytical solution of the Yukawa closure of the Ornstein—Zernike equation III: the one-component case

A new general form of the multi-Yukawa closure of the Ornstein—Zernike equation is discussed for the case of one component and two exponentials. This closure is given in terms of a diagonal scaling matrix Γ. If we are dealing with a pure fluid then the general form of the closure is where β(m) is obtained by solving a linear matrix equation where the matrix M m , depends only on z n , and Γ(nn). Explicit expessions for the excess thermodynamic functions are also found.

Analytical solution of the Yukawa closure of the Ornstein-Zernike equation III: the one-component case

Analytical solution of the Yukawa closure of the Ornstein-Zernike equation III: the one-component case

Analytical solution of the Yukawa closure of the Ornstein-Zernike equation. II. The full solution

Analytical solution of the Yukawa closure of the Ornstein-Zernike equation. II. The full solution

Analytical solution of the Yukawa closure of the Ornstein—Zernike equation. II. The full solution

New and simpler forms are presented of the closure of the analytical solution of the Ornstein—Zernike equation for the general case consisting of a sum of M Yukawa exponentials with factorizable coefficients in terms of an M × M scaling matrix γ (Blum, L., Vericat, F., and Herrera, J. N., 1992, J. statist. Phys., 66, 249) are presented. The general solution is given in terms of M(M—1) symmetry relations and M boundary conditions. The general form for the multicomponent case is obtained. For only one component the closure for n = 1, …, M is

An Extension of the Analytical Solution of the Ornstein–Zernike Equation with the Yukawa Closure

The analytical solution of the Ornstein–Zernike equation with one Yukawa closure of the factorizable-coefficient case is extended from the scalar-factorization case to the vector-factorization case. As a result, the scaling parameter is extended from a scalar quantity to a matrix quantity, and the scaling matrix \(\)

Effective Interaction Potential between Colloidal Particles in a Solution

In a solution containing microions from an added electrolyte, the effective interaction potential between colloidal particles is studied on the basis of the primitive model. The approximation is employed that the correlations of solvent-particles with microions can be neglected, and the potential is calculated with the use of the mean spherical approximation solution of the Ornstein-Zernike equation with the Yukawa closure. In the case of the hydrophobic colloid, the theory is discussed in comparison with the DLVO theory and a role of the solvent-mediated potential is emphasized.

Mathematical comment on adhesive hard sphere models for fluids

A mathematical discussion is provided on the relationship between Baxter's adhesive hard sphere fluid and the hard sphere Yukawa fluid with sufficiently strong and short range attractive Yukawa tails. It is proved that the Percus-Yevick approximate solution of the Ornstein-Zernike equation for the former fluid can be included mathematically in the generalized mean-spherical approximation solution with the Yukawa closure, and the equivalence conditions of both solutions are presented.

On the analytical solution of the Ornstein-Zernike equation with Yukawa closure

We discuss the solution of the Ornstein-Zernike equation for most general closure consisting of a sum ofM Yukawa-type exponentials. A formal solution for the factored case is bound for an arbitrary mixture of hard spheres introducing a general scaling matrix Γ of dimensions M×M. A sufficient number of equations for this matrix is obtained from symmetry considerations and the boundary condition. We discuss also restricted and semirestricted case, for which explicit solutions in terms of the scaling parameters and input parameters are found.

Monte Carlo simulations and integral equation theory for microscopic correlations in polymeric fluids

Monte Carlo simulations are performed for polymer chains modeled as pearl necklaces of freely jointed tangent hard spheres; chains composed of 20, 50, and 100 beads are studied at volume fractions ranging from 0.1 to 0.35. The mean-square end-to-end distance, the radius of gyration, and the intramolecular and intermolecular site–site distribution functions are monitored in the simulations. Various approximations for the intramolecular structure factor, ŵ(k), are tested. It is found that the ŵ(k) from the semi-flexible chain model is the most accurate. The polymer ‘‘reference interaction site model’’ (PRISM) theory of Curro and Schweizer is tested using both approximate and exact expressions for ŵ(k). It is found that, at the densities examined here, the theory is accurate for the local structure except near contact where it tends to overestimate the value of the intermolecular site–site distribution function, g(r). The polymer-RISM theory is also solved with the generalized mean spherical approximation (GMSA), which uses a Yukawa closure for the direct correlation function. The contact value of g(r), required in the GMSA, is obtained approximately, but accurately, via a perturbation expansion for a hypothetical fluid and the generalized Flory dimer equation of state. The GMSA theory results in improved predictions for g(r) when compared with the original polymer-RISM theory, but there are still some differences between theoretical predictions and simulation results near contact.

Local structure of fluids containing chain-like molecules: Polymer reference interaction site model with a Yukawa closure

The local structure of fluids composed of chain-like molecules (modeled as a pearl necklace of freely jointed hard spheres) is investigated via an integral equation approach. A Yukawa closure for the direct correlation function is used in the framework of polymer-RISM theory. This introduces two free parameters which are chosen by matching the theory’s predictions for the compressibility and the contact value of the site–site distribution function, to an equation of state and simulation data, respectively. The theory shows good agreement when compared to Monte Carlo simulations of tangent diatomics, and freely jointed 4-mers, and 8-mers. Methods for estimating the contact value of the site–site distribution function are discussed.

Analytical solution of the mean-spherical approximation for a system of hard spheres with a surface adhesion.

The solution of a model for adhesive hard spheres is obtained within the mean-spherical approximation. The adhesive part of the potential is modeled as the limit of a Yukawa tail when both the amplitude and the inverse range tend to infinity. Use is then made of the analytical solution of the Ornstein-Zernike equation with a Yukawa closure, developed by Waisman, Hoye, Stell, and Blum. The system presents a liquid-gas phase transition, and the critical exponents \ensuremath{\gamma} and \ensuremath{\delta} are those of the spherical model.

Solution of Ornstein-Zernike Equation for a Mixture of Hard Spheres with Yukawa Closure: the Case of Factorizable Coefficients

The solution of Blum and Hoye, which is the solution of Ornstein-Zernike equation for a mixture of hard spheres with Yukawa closure: c i j ( r )=Σ n K i j ( n ) exp (- z n r )/ r for r >σ i j , c i j ( r ) and σ i j being respectively the direct correlation function and the closest distance between i - and j -type spheres, has been studied in the case of factorizable coefficients, namely, K i j ( n ) = K ( n ) d i ( n ) d i ( n ) . In this case it is shown that (i) the set of algebraic equations defining the solution is remarkably simplified and, in particular, that (ii) in the single Yukawa-term case the set is solved and the solution is given by rational functions of a parameter. The latter is the generalization of the Blum's solution for a primitive model of electrolytic solutions in the sense of producing the Blum's solution as the limiting case.

Solution of the generalized mean spherical approximation for charged hard-sphere mixtures

We discuss in detail the method of solution of the GMSA for charged hard spheres of unequal sizes, briefly reported in a preceding paper. The theoretical approach is based on the use of the Wiener-Hopf factorization technique. It is shown that the solution of the Ornstein-Zernike equation with Yukawa closure can be obtained in terms of a system of coupled nonlinear equations, which can be solved numerically. Since the solution is manifold, we face the problem of how to choose the physically meaningful branch. To this aim we devise a procedure, which, starting from the well-known GMSA for equally sized charged hard spheres, builds up the solution for the present case by «perturbative» steps. The equal-size limit is then investigated in detail and close contact of our results with those obtained by other authors is established. The problem of the thermodynamic consistency of the GMSA and the usefulness of a comparison of the different ways one can impose it are also discussed.

Analytic solution of the RISM equation for symmetric diatomics with Yukawa closure

We present the site-site direct correlation function c(r) for a fluid of hard diatomic symmetric molecules obtained from Monte Carlo simulation data via the RISM integral equation. This c(r) ensures that the site-site correlation function given by the RISM equation is exact, and thus provides a basis for critically examining the usual closure for the RISM equation. As an example of an improved closure we present the analytic solution of the RISM integral equation with a Yukawa closure for c(r).

Solution of the Ornstein-Zernike equation for a multicomponent system with Yukawa closure

Solution of the Ornstein-Zernike equation for a multicomponent system with Yukawa closure