Soft Mean Spherical Approximation Research Articles

Ground-state correlation energy of counterions at a charged planar wall: Gibbs-Bogoliubov lower-bound approach

Recent simulation results imply the lowering of the ground-state correlation energy per counterion at a charged planar wall, compared with that of the 2D and 3D one-component plasma systems. Our aim is to correctly evaluate the ground-state energy of strongly-coupled counterion systems by considering a quasi-2D bound state where bound counterions are confined to a layer of molecular thickness. We use a variational approach based on the Gibbs-Bogoliubov inequality for the lower-bound free energy so that the liquid-state theory can be incorporated into the formulations. The soft mean spherical approximation demonstrates that the lowered ground-state energy can be reproduced by the obtained analytical form of a quasi-2D bound state.

Soft mean spherical approximation for dusty plasma liquids: Level of accuracy and analytic expressions

The soft mean spherical approximation is employed for the study of the thermodynamics of dusty plasma liquids, the latter treated as Yukawa one-component plasmas. Within this integral theory method, the only input necessary for the calculation of the reduced excess energy stems from the solution of a single non-linear algebraic equation. Consequently, thermodynamic quantities can be routinely computed without the need to determine the pair correlation function or the structure factor. The level of accuracy of the approach is quantified after an extensive comparison with numerical simulation results. The approach is solved over a million times with input spanning the whole parameter space and reliable analytic expressions are obtained for the basic thermodynamic quantities.

Soft mean spherical approximation for dusty plasma liquids: One-component Yukawa systems with plasma shielding.

The structure and thermodynamics of strongly coupled dusty plasmas are investigated with the soft mean spherical approximation. This integral theory approach is analytically solvable for Yukawa pair interactions yielding a closed-form solution for the direct correlation function. The pair correlation function, the structure factor, and basic thermodynamic quantities are calculated for a wide range of parameters. Exact consistency between the "energy"-"virial" thermodynamic routes and approximate consistency between the "energy"-"compressibility" paths is demonstrated. Comparison with extensive molecular dynamics results is carried out and a remarkable agreement from the Coulomb limit to the strongly screened limit is revealed. The soft mean spherical approximation is concluded to be particularly well suited for the study of dusty plasma liquids, uniquely combining simplicity and accuracy.

Mercedes–Benz water molecules near hydrophobic wall: Integral equation theories vs Monte Carlo simulations

Associative version of Henderson-Abraham-Barker theory is applied for the study of Mercedes-Benz model of water near hydrophobic surface. We calculated density profiles and adsorption coefficients using Percus-Yevick and soft mean spherical associative approximations. The results are compared with Monte Carlo simulation data. It is shown that at higher temperatures both approximations satisfactory reproduce the simulation data. For lower temperatures, soft mean spherical approximation gives good agreement at low and at high densities while in at mid range densities, the prediction is only qualitative. The formation of a depletion layer between water and hydrophobic surface was also demonstrated and studied.

Determination of colloidal interaction potentials from small angle scattering data

The scattering curves of monodisperse, globular particles measured by small angle scattering are products of a form factor and of a structure factor. The form factor contains the information on particle shape and size and can often be measured for dilute samples. The structure factor contains the contributions due to particle interactions. It should be approximated by a structure factor model based on the Percus-Yevik, hypernetted chain, or soft mean spherical approximation closure relations. Combining these relations with a flexible model for the interaction potential, it is possible to determine the interaction potential from the scattering data. Scattering curves of charged spherical colloids and of spheres interacting by depletion forces are therefore simulated and evaluated, resulting in potentials that are close to the ones used for the simulation. The applicability of the approach is finally tested on a 5% solution of lysozyme at pH 4.5, which gave an interaction potential that agreed well with the one expected for such a sample.

Liquid-vapor interfaces inXY-spin fluids: An inhomogeneous anisotropic integral-equation approach

An integral-equation approach is developed to study interfacial properties of anisotropic fluids with planar spins in the presence of an external magnetic field. The approach is based on the coupled set of the Lovett-Mou-Buff-Wertheim integro-differential equation for the inhomogeneous anisotropic one-particle density and the Ornstein-Zernike equation for the orientationally dependent two-particle correlation functions. Using the proposed inhomogeneous angle-harmonics expansion formalism we show that these integral equations can be reduced to a much simpler form similar to that inherent for a system of isotropic fluids. The interfacial orientationally dependent direct correlation function can be consistently constructed by means of a nonlinear interpolation via its values obtained in the coexisting anisotropic bulk phases. A soft mean spherical approximation is employed for the closure relation. This has allowed us to solve the complicated integral equations in the situation when both spatial inhomogeneity and orientational anisotropy are present simultaneously. The approach introduced is applied to an XY fluid model with ferromagnetic spin interactions. As a result, the density-orientation and magnetization profiles at the liquid-vapor interfaces are calculated in a wide range of temperatures up to subcritical regions. The influence of the external field on the microscopic structure of the interfaces and the surface tension is also analyzed in detail.

Integral equation procedure based on tailored orthogonal functions for theXYspin fluid in an external magnetic field

The classical XY model describes particles in three-dimensional space that carry magnetic moments or spins whose motion is restricted to rotations in a plane. Introduction of an external magnetic field lying in the same plane then generates a system that is anisotropic in the azimuthal angle phi . We use numerical simulations and integral equation techniques to study this system, producing in the latter case a formalism that is identical to that of the simpler isotropic version having no external field. The basis for this simplification is a generalization Em(phi) of the ordinary exponential basis set eimphi that restores orthogonality in the presence of the external field. We display results of sample calculations obtained with two integral equation closures, reference hypernetted-chain and soft mean-spherical approximation, both coupled to the Lovett-Mou-Buff-Wertheim relation, along with results from the numerical simulations for comparison. Construction of the Em(phi) is described in an Appendix.

Liquid-vapor and liquid-liquid interfaces in Ising fluids: An integral equation approach

The microscopic structure and thermodynamic properties of liquid-vapor and liquid-liquid interfaces in Ising fluids are studied using an integral equation approach. The calculations are performed in the absence and presence of an external magnetic field by solving the corresponding set of Lovett-Mou-Buff-Wertheim integrodifferential equations for the one-particle density distribution functions. The two-particle inhomogeneous direct correlation functions are consistently constructed by nonlinear interpolation between the bulk ones. The bulk correlation functions of the coexisting phases are obtained from the Ornstein-Zernike equations with a modified soft mean spherical approximation for the closure relation. As a result, the density and magnetization profiles at liquid-vapor and liquid-liquid interfaces as well as the surface tension and adsorption coefficients are evaluated in a wide temperature range including subcritical regions. The influence of an external magnetic field on the liquid-vapor interfaces is also considered.

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.

Integral equation study of an ideal Ising mixture

We construct an integral equation scheme for magnetic binary mixtures of an ideal soft-core Ising fluid and a soft-sphere fluid by mapping the system onto an equivalent nonmagnetic ternary mixture. We apply the multicomponent Ornstein-Zernike equation together with a closure relation based on the soft mean spherical approximation and a field constraint for the Ising fluid component. Phase coexistence curves are calculated both by directly evaluating the chemical potentials via the bridge function, and by using a Maxwell-like construction which is derived in the text. Our results are compared to Monte Carlo data obtained earlier, and we find that the second method yields a much better agreement with the simulations.

XY-spin fluids in an external magnetic field: An integral equation approach

We develop an integral equation approach to study anisotropic fluids with planar spins in the presence of an external field. As a result, the integral equation calculations for these systems appear to be no more difficult than those for ordinary isotropic liquids. The method presented is applied to the investigation of phase coexistence properties of ferromagnetic XY-spin fluids in a magnetic field. The soft mean spherical approximation is used for the closure relation connecting the orientationally dependent two-particle direct and total correlation functions. The Lovett-Mou-Buff-Wertheim and Born-Green-Yvon equations are employed to describe the one-particle orientational distribution. The phase diagrams are obtained in the whole range of varying the external field for a wide class of XY-spin fluid models with various ratios of the strengths of magnetic to nonmagnetic Yukawa-like interactions. The influence of changing the screening radii of the interaction potentials is also considered. Different types of the phase diagram topology are identified. They are characterized by the existence of critical, tricritical, critical end, and triple points related to transitions between gas, liquid, and para- and ferromagnetic states, accompanied by different external field dependencies of critical temperatures and densities corresponding to the gas-liquid and liquid-liquid transitions. As is demonstrated, the integral equation approach leads to accurate predictions of the complicated phase diagram behavior which coincide well with those evaluated by the cumbersome Gibbs ensemble simulation and multiple-histogram reweighting techniques.

Ising fluids in an external magnetic field: an integral equation approach.

The phase behavior of Ising spin fluids is studied in the presence of an external magnetic field with the integral equation method. The calculations are performed on the basis of a soft mean spherical approximation using an efficient algorithm for solving the coupled set of the Ornstein-Zernike equations, the closure relations, and the external field constraint. The phase diagrams are obtained in the whole thermodynamic space including the magnetic field H for a wide class of Ising fluid models with various ratios R of the strengths of magnetic to nonmagnetic Yukawa-like interactions. The influence of varying the inverse screening lengths z(1) and z(2), corresponding to the magnetic and nonmagnetic Yukawa parts of the potential, is investigated too. It is shown that changes in R as well as in z(1) and z(2) can lead to different topologies of the phase diagrams. In particular, depending on the value of R, the critical temperature of the liquid-gas transition either decreases monotonically, behaves nonmonotonically, or increases monotonically with increasing H. The para-ferro magnetic transition is also affected by changes in R and the screening lengths. At H=0, the Ising fluid maps onto a simple model of a symmetric nonmagnetic binary mixture. For H--> infinity, it reduces to a pure nonmagnetic fluid. The results are compared with available simulations and the predictions of other theoretical methods. It is demonstrated that the mean spherical approximation appears to be more accurate compared with mean field theory, especially for systems with short ranged attraction potentials (when z(1) and z(2) are large). In the Kac limit z(1), z(2) -->+0, both approaches tend to nearly the same results.

Decay of correlations in fluids: The one-component plasma from Debye-Hückel to the asymptotic-high-density limit

The decay of structural correlations in the classical one-component plasma is analyzed by calculating the poles of the Fourier transform of the total (pairwise) correlation function $h(r)$ for two integral equation theories, the soft mean spherical approximation and the hypernetted chain (HNC). We show that for all except the largest values of the plasma coupling constant $\ensuremath{\Gamma},$ the leading-order pole contribution provides an accurate description of $h(r)$ at intermediate range, as well as the ultimate asymptotic decay. The crossover from monotonic decay at weak coupling to exponentially damped oscillatory decay at strong coupling is shown to arise from the same mechanism as that which occurs for charge correlations in binary ionic fluids. We calculate the values of $\ensuremath{\Gamma}$ at which the crossover occurs in the two theories. The role of higher-order poles and (within the HNC) other singularities in determining the intermediate range behavior of $h(r)$ for strong coupling is discussed. We investigate the properties of the solutions of the integral equations in the strong coupling, $\stackrel{\ensuremath{\rightarrow}}{\ensuremath{\Gamma}}\ensuremath{\infty},$ asymptotic high-density limit (AHDL). Pad\'e approximants are employed in order to test the validity of the scaling laws proposed for the potential energy, direct correlation function, and for the poles and their contributions to $h(r)$ in the AHDL. Our numerical results provide strong support for the validity of the theoretical predictions concerning the AHDL.

Spontaneous alignment in charged-particle systems: Fixed-length vs micellar situations

We use a recently developed version of the soft-mean-spherical approximation (SMSA) to treat spontaneous alignment in suspensions of ionic, rod-like particles. In particular our theory derives from the charge-smearing idea of Onsager in which a lower bound is established for the free energy of the interacting system. First we treat the usual case of fixed-particle lengths and establish the regimes in which the charged lines behave like hard spherocylinders (i.e., the ‘‘smearing’’ length plays the role of a hard-core diameter). In general, the nematic phase is superceded by the solid whenever the rod length is too small or too large. Then we consider explicitly the micellar situation in which the particle lengths are determined by the thermodynamics (and vice versa). Here we find the possibility of a reentrant isotropic phase. At low temperatures the nematic becomes unstable because the alignment-induced growth leads to too large a charging (‘‘self’’) energy for the micelles. We also predict an ideal solution-like behavior for the growth of micelles in sufficiently concentrated isotropic phases. Finally we comment on the differences between our charge-smearing description of spontaneous alignment in ionic systems and a collective-coordinate approach due to Deutsch and Goldenfeld.