Hypernetted Chain Closure Research Articles

Efficient treatment of solvation shells in 3D molecular theory of solvation

We developed a technique to decrease memory requirements when solving the integral equations of three-dimensional (3D) molecular theory of solvation, a.k.a. 3D reference interaction site model (3D-RISM), using the modified direct inversion in the iterative subspace (MDIIS) numerical method of generalized minimal residual type. The latter provides robust convergence, in particular, for charged systems and electrolyte solutions with strong associative effects for which damped iterations do not converge. The MDIIS solver (typically, with 2 × 10 iterative vectors of argument and residual for fast convergence) treats the solute excluded volume (core), while handling the solvation shells in the 3D box with two vectors coupled with MDIIS iteratively and incorporating the electrostatic asymptotics outside the box analytically. For solvated systems from small to large macromolecules and solid-liquid interfaces, this results in 6- to 16-fold memory reduction and corresponding CPU load decrease in MDIIS. We illustrated the new technique on solvated systems of chemical and biomolecular relevance with different dimensionality, both in ambient water and aqueous electrolyte solution, by solving the 3D-RISM equations with the Kovalenko-Hirata (KH) closure, and the hypernetted chain (HNC) closure where convergent. This core-shell-asymptotics technique coupling MDIIS for the excluded volume core with iteration of the solvation shells converges as efficiently as MDIIS for the whole 3D box and yields the solvation structure and thermodynamics without loss of accuracy. Although being of benefit for solutes of any size, this memory reduction becomes critical in 3D-RISM calculations for large solvated systems, such as macromolecules in solution with ions, ligands, and other cofactors.

Assessment of Three-Dimensional Distribution Functions and Triplet Distribution Functions for Hard Spheres Calculated Using a Three-Dimensional Ornstein–Zernike Equation with Hypernetted-Chain Closure

We calculated three-dimensional (3D) distribution functions around a contact dimer composed of hard spheres immersed in a fluid composed of same-sized hard spheres calculated using a three-dimensional Ornstein–Zernike equation with hypernetted-chain closure (3D-HNC-OZ theory). The results of the 3D-HNC-OZ theory were compared with those calculated using Monte Carlo simulations. Even though the packing fraction of solvent was high, such as in ambient water, the 3D-HNC-OZ theory gave semiquantitatively reasonable results. This means that the triplet distribution function was also calculated reasonably well, although the triplet distribution functions are not explicitly included in the equations of the 3D-HNC-OZ theory. However, the accuracy depended on the configuration of the solute. Our results are discussed in a biological context, such as molecular recognition and the stability of folded proteins.

An Integral Equation Study of Reentrant Behavior in Attractive Interactions between Like-Charged Macroions Immersed in an Electrolyte Solution

The effective interaction between two like-charged macroions immersed in an electrolyte solution was studied using a Ornstein–Zernike equation with hypernetted-chain closure (HNC-OZ theory) with a particle model for the ions and solvent molecules to discuss some biological phenomena. We examined conditions under which the Coulomb interaction was greater than the thermal energy. Although the macroions repelled each other in a dilute electrolyte solution, a strong effective attraction between macroions occurred only in a medium concentration electrolyte. The effective interaction is analyzed using the superposition approximation to discuss the bond formation. When the effective attraction appears, the dense cloud of countercharged ion exists between the macroions and the cloud mediates the attraction like a covalent bond.

Application of the SRISM approach to the study of the capacitance of the double layer of a high density primitive model electrolyte

In this study the Singlet Reference Interaction Site Model (SRISM) is employed to the study of the electrode charge dependence of the capacitance of a planar electric double layer using the primitive model of the double layer for a high density electrolyte that mimics an ionic liquid. The ions are represented by charged hard spheres and the electrode is a uniformly charged flat surface. The capacitance of this model fluid is calculated with the SRISM approach with closures based on the hypernetted chain (HNC) and Kovalenko-Hirata (KH) closures and compared with simulations. As long as the magnitude of the electrode charge is not too great, the HNC closure shows the most promise. The KH results are reasonably good for a high density electrolyte but are poor when applied at low densities.

Calculation of the hydration energy of polyvalent metal ions by the RISM method

The energy of hydration of Group Ia, IIa, and IIIb metals has been calculated on the basis of the integral equation theory of molecular liquids. Calculations were performed in the RISM approximation with hypernetted chain closure, partial inclusion of the orientation effect, and semiempirical corrections for excluded volume and for the existence of hydrogen bonding with the solute molecule. The analysis of calculation data made it possible to apply a correction to the computation method used, which ensured better convergence between calculation results and experimental data.

Atomistic Description of Binary Lanthanoid Salt Solutions: A Coarse-Graining Approach

The experimental difficulties inherent to the solution chemistry of actinoids and lanthanoids have led to the use of a wide variety of models, from the microscopic to the macroscopic scale, in an attempt to represent their solution properties. Molecular dynamics (MD) simulations, with explicit solvents, have been successfully used to describe the structural characteristics, but the limits on the accessible length and time scales do not allow for an equivalent description of the macroscopic properties. In this study, we propose a multiscale approach, based on MD simulation results, to study the thermodynamic and structural properties of a series of lanthanoid-chloride aqueous solutions. An inversion procedure, based on the approximate hypernetted chain (HNC) closure and the Stillinger-Lovett sum rules for ionic liquids, is used to obtain the effective ion-ion potentials from MD-generated radial distribution functions (RDF). Implicit solvent Monte Carlo (MC) simulations are then performed to compute the osmotic coefficients of the salt solutions. This coarse-grained strategy provides accurate effective pair potentials for the lanthanoid salts, derived from an atomic model. The method presented here is an attempt to bridge the gap between MD and the thermodynamic properties of solutions that are experimentally measured.

Freezing Transition and Interaction Potential in Monolayers of Microparticles at Fluid Interfaces

The structure and the interaction potential of monolayers of charged polystyrene microparticles at fluid interfaces have been studied by optical microscopy. Microparticles of different sizes have been studied over a broad range of surface particle densities. The structural characterization is based on the analysis of images obtained by digital optical microscopy. From the experimental images, radial distribution functions, hexagonal bond order correlation functions, and temporal orientational correlation functions have been calculated for different monolayer states at both the air/water and oil/water interfaces. The interaction potential has been calculated from the structure factor using integral equations within the hypernetted chain closure relationship. For particles trapped at the oil-water interface, it was found that, upon increasing the surface coverage, a freezing transition occurs, that leads to the formation of a 2D crystalline structure. We have studied the freezing densities of particle monolayers at the oil/water interface and compared them with Monte Carlo simulation results reported by H. Löwen. In contrast, at the air-water interface, freezing is inhibited due to the formation of particle aggregates.

Effect of particle size and Debye length on order parameters of colloidal silica suspensions under confinement

Using atomic force microscopy (AFM) and small angle X-ray scattering (SAXS), we show a full comparison between structuring of nanoparticles in confinement and in bulk in order to explain the effect of confinement on characteristic lengths and the scaling law of the characteristic lengths. Three different-sized particle suspensions are used to check the generalization and the correlation between the characteristic lengths and the system parameters, like particle diameter and Debye length. The two characteristic lengths obtained from AFM force curves, the oscillatory wavelength λ, which is related to the average particle distance, and the decay length ξ, which measures how far particle correlates to obtain periodic oscillations, are in good agreement with the mean particle distance 2π/qmax and the correlation length 2/Δq in bulk, respectively, obtained from the structure peaks of SAXS diagrams. Although confinement causes layering of nanoparticles parallel to the confining surfaces, the characteristic lengths in the direction perpendicular to the confining surfaces follow the bulk behavior. The wavelength scales as ρ−1/3 with the particle number density ρ irrespective of the particle size and the ionic strength and shows a pure volume effect. Upon comparing with literature results, the λ = ρ−1/3 scaling law can be applied more generally for charged particles, as long as the repulsive interaction is sufficiently long-ranged, than the previous expression of λ = 2(R + κ−1), which only approaches the value of average particle distance under specific conditions. The decay length ξ is controlled both by the particle size and the ionic strength of the suspensions, and ξ = R + κ−1 is proposed in the paper. In addition, the interaction strength, the force amplitude and maximum scattering intensity, increases linearly with particle concentration. On the other hand, the Monte Carlo (MC) simulations and approximate hypernetted chain (HNC) closure calculation based on Derjaguin-Landau-Verwey-Overbeek (DLVO) potential are employed to study the characteristic lengths from the theoretical point of view. The experimental wavelengths are in good agreement with the theoretical counterparts and the experimental decay lengths show the same qualitative behavior as theoretical ones on the particle size and ionic strength.

Microscopic structure and thermodynamics of a core-softened model fluid from the second-order integral equations theory

We have studied the structure and thermodynamic properties of isotropic three-dimensional core-softened fluid by using the second-order Ornstein-Zernike integral equations completed by the hypernetted chain and Percus-Yevick closures. The radial distribution functions are compared with those from singlet integral equations and with computer simulation data. The limits of the region of density anomaly resulting from different approximate theories are established. The obtained results show that the second-order hypernetted chain approximation can be used to describe both the structure and the density anomaly of this model fluid. Moreover, we present the results of calculations of the bridge functions.

Evaluation of bridge function for hard sphere fluid confined in a narrow slit-pore via TMMC Mayer-sampling

The bridge function required to yield a singlet integral equation (IE) up to the second order in density expansion for the hard sphere fluid confined in a slit-pore is evaluated. The slit-fluid bridge function can be divided into wall-particle bridge diagrams with h b-bond, which were evaluated by recently proposed Transition Matrix Monte Carlo (TMMC) Mayer-sampling method. The bulk-fluid total correlation function h b(r) used in cluster integrals is determined by solution of the bulk-fluid Ornstein–Zernike (OZ) equation with a hypernetted chain closure (HNC). The calculation is performed for the reduced density of bulk fluid in equilibrium with the fluid in slit-pores from 0.3 to 0.7 with narrow slit width of 3.0σ and 4.0σ. The quantity of the slit-fluid bridge function is assessed by comparison of the density profile obtained from the singlet IE theory and the grand canonical Monte Carlo (GCMC) simulation. Good agreement between the proposed approach and the GCMC data is observed. The reduced normal pressure is also calculated, and agrees well with the simulation data at low to medium densities but becomes a little larger at high density. It is expected that the result can be improved by adding higher order bridge coefficients. The direct evaluation of the slit-fluid bridge function seems to be practical since a great improvement of the quality of the singlet IE theory has been achieved for predicting the structural and thermodynamic properties of fluids confined in narrow slit pores.

Protein solvation from theory and simulation: Exact treatment of Coulomb interactions in three-dimensional theories

Solvation forces dominate protein structure and dynamics. Integral equation theories allow a rapid and accurate evaluation of the effect of solvent around a complex solute, without the sampling issues associated with simulations of explicit solvent molecules. Advances in integral equation theories make it possible to calculate the angle dependent average solvent structure around an irregular molecular solution. We consider two methodological problems here: the treatment of long-ranged forces without the use of artificial periodicity or truncations and the effect of closures. We derive a method for calculating the long-ranged Coulomb interaction contributions to three-dimensional distribution functions involving only a rewriting of the system of integral equations and introducing no new formal approximations. We show the comparison of the exact forms with those implied by the supercell method. The supercell method is shown to be a good approximation for neutral solutes whereas the new method does not exhibit the known problems of the supercell method for charged solutes. Our method appears more numerically stable with respect to thermodynamic starting state. We also compare closures including the Kovalenko-Hirata closure, the hypernetted-chain (HNC) and an approximate three-dimensional bridge function combined with the HNC closure. Comparisons to molecular dynamics results are made for water as well as for the protein solute bovine pancreatic trypsin inhibitor. The proposed equations have less severe approximations and often provide results which compare favorably to molecular dynamics simulation where other methods fail.

The special features of the thermodynamic characteristics of hydration of univalent ions according to the reference interaction site model

The dependence of the Gibbs energy of hydration of univalent ions was calculated on the basis of integral equations for a molecular liquid. Calculations were performed in the approximation of the reference interaction site model (RISM). The relation between the energy of hydration of ions and local electrostatic field microinhomogeneities in solution was studied. It was shown that the best variant was the RISM with the hypernetted chain closure, partial inclusion of orientation effects, and semiempirical corrections for the excluded volume and the presence of H-bonds with solutes. Predictions based on the selected variant of the model were in close agreement with the experimental data. The analysis performed indicated directions for the improvement of the RISM method.

Nanoparticles in aqueous media: crystallization and solvation charge asymmetry

We examine the issue of whether dispersion forces can lead to crystallization in a system of charged nanoparticles in aqueous solution with NaCl salt. To this end, we determine the effective pair potential (EPP) among the nanoparticles starting from a model system that explicitly includes the salt ions and the water molecules, using the well-tested simple point charge extended (SPC/E) model for the latter. The two-particle correlations among the components of this model system are determined using the reference interaction site model (RISM) equation complemented with the hypernetted-chain (HNC) closure. The EPP at infinite nanoparticle dilution is obtained from these correlations after contracting the salt ions and water molecules following the method presented in P. Gonzalez-Mozuelos, J. Phys. Chem. B, 2006, 110, 22702. The dressed-interaction-site theory (DIST) discussed in that work shows that the corresponding EPP has a short-range contribution plus a screened electrostatic (Yukawa) potential with renormalized charges and dielectric constant. A polynomial-fitting scheme is devised to quantify the dependence of the effective electrostatic paramenters on the underlying salt concentration. As such, we derive the phase diagram for our system, using a mean field approach based upon the computed EPP, for a range of (finite) nanoparticle densities and salt concentrations and demonstrate crystallization. Findings from our model also suggest the possibility of crystallization occurring preferentially among nanoparticles with negative charges than those with positive charges of the same magnitude and thus exhibiting charge asymmetry due to solvation effects.

Statistical entropy and clustering in absence of attractive terms in the interparticle potential

Recently a new intriguing class of systems has been introduced, the so-called generalized exponential models, which exhibit clustering phenomena even if the attractive term is missing in their interaction potential. This model is characterized by a index n which tunes the repulsive penetrability of the potential. This family of potentials can represent the effective interactions for a large number of soft matter systems. In this paper we study the structural and thermodynamic properties in the fluid regime of the generalized exponential model with a value of index n suggested by Mladek et al. [B. M. Mladek, G. Kahl, and C. N. Likos, Phys. Rev. Lett. (2008)] to fit the effective potential of a typical amphiphilic dendrimers. We use the conventional approach of the liquid state theory based on the hypernetted chain closure of the Ornstein-Zernike equation together with some Monte Carlo numerical simulations. Moreover, we try to detect qualitatively the freezing line exploiting the predictive properties of a one-phase rule based on the expansion of the statistical entropy.

Structure of penetrable sphere fluids and mixtures near a slit hard wall: A modified bridge density functional approximation

The modified density functional theory, which is based both on the bridge density functional and the contact value theorem, has been proposed for the structural properties of penetrable sphere fluids and their mixtures near a slit hard wall. The Verlet-modified bridge function proposed by Choudhury and Ghosh [J. Chem. Phys. 119, 4827 (2003)] for one-component system has been extended for fluid mixtures. The radial distribution functions obtained from the Verlet-modified bridge function are in excellent agreement with computer simulations over a wide range of density and temperature and better than those obtained from the standard integral theories including the Percus-Yevick and hypernetted-chain closures. The calculated particle density distributions confined in a slit pore are also found to be reasonably good compared to the simulation data. Even for high density systems the accuracy of the hypernetted-chain and the mean-field approximation functionals increase with increasing temperature. However, the agreement between theory and simulation slightly deteriorates in the systems of low temperature.

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.

Determination of interaction potentials of colloidal monolayers from the inversion of pair correlation functions: A two-dimensional predictor-corrector method

The structure and stability of colloidal monolayers depend crucially on the effective pair potential u(r) between colloidal particles. In this paper, we develop a two-dimensional (2D) predictor-corrector method for extracting u(r) from the pair correlation function g(r) of dense colloidal monolayers. The method is based on an extension of the three-dimensional scheme of Rajagopalan and Rao [Phys. Rev. E 55, 4423 (1997)] to 2D by replacing the unknown bridge function B(r) with the hard-disk bridge function B(d)(r); the unknown hard-disk diameter d is then determined using an iterative scheme. We compare the accuracy of our predictor-corrector method to the conventional one-step inversion schemes of hypernetted chain closure (HNC) and Percus-Yevick (PY) closure. Specifically we benchmark all three schemes against g(r) data generated from Monte Carlo simulation for a range of 2D potentials: exponential decay, Stillinger-Hurd, Lennard-Jones, and Derjaguin-Landau-Verwey-Overbeek. We find that for all these potentials, the predictor-corrector method is at least as good as the most accurate one-step method for any given potential, and in most cases it is significantly better. In contrast the accuracy of the HNC and PY methods relative to each other depends on the potential studied. The proposed predictor-corrector scheme is therefore a robust and more accurate alternative to these conventional one-step inversion schemes.

Microscopic structure and thermodynamics of a core-softened model fluid: Insights from grand canonical Monte Carlo simulations and integral equations theory

We have studied the microscopic structure and thermodynamic properties of isotropic three-dimensional core-softened model fluid by using extensive grand canonical Monte Carlo computer simulations and Ornstein-Zernike integral equations with hypernetted chain and Rogers-Young closures. Applied simulation tools permit to obtain insights into the properties of the model in addition to available molecular dynamics data of other authors. We discuss equation of state in the density-chemical potential projection and explore the temperature dependence of the chemical potential along different isochores. The limits of the region of anomalous behavior of the structural and thermodynamic properties are established by investigating derivatives resulting from the equation of state, pair contribution to excess entropy, and translational order parameter. Besides, we evaluate the dependence of the heat capacity on temperature and density. The microscopic structure is discussed in terms of the pair distribution functions and corresponding structure factors. We have established that the hypernetted chain approximation is not successful to capture the region of anomalies in contrast to Rogers-Young approximation, but is very accurate for high fluid densities. Thus we have studied the onset for crystallization transition within this theoretical framework. Moreover, using the replicated Ornstein-Zernike integral equations with hypernetted chain closure, we explore the possibility of glass transition and described it in terms of transition density and chemical potential.

Reference interaction site model investigation of homonuclear hard dumbbells under simple fluid theory closures: Comparison with Monte Carlo simulations

We revisit the thermodynamic and structural properties of fluids of homonuclear hard dumbbells in the framework provided by the reference interaction site model (RISM) theory of molecular fluids. Besides the previously investigated Percus-Yevick (PY) approximation, we test the accuracy of other closures to the RISM equations, imported from the theory of simple fluids; specifically, we study the hypernetted chain (HNC), the modified HNC (MHNC) and, less extensively, the Verlet approximations. We implement our approach for models characterized by several different elongations, up to the case of tangent diatomics, and investigate the whole fluid density range. The theoretical predictions are assessed against Monte Carlo simulations, either available from literature or newly generated by us. The HNC and PY equations of state, calculated via different routes, share on the whole the same level of accuracy. The MHNC is applied by enforcing an internal thermodynamic consistency constraint, leading to good predictions for the equation of state as the elongation of the dumbbell increases. As for the radial distribution function, the MHNC appears superior to other theories, especially for tangent diatomics in the high density limit; the PY approximation is better than the HNC and Verlet closures in the high density or elongation regime. Our structural analysis is supplemented by an accurate inversion procedure to reconstruct from Monte Carlo data and RISM the "exact" direct correlation function. In agreement with such calculations and consistent with the forecast of rigorous diagrammatic analysis, all theories predict the occurrence in the direct correlation function of a first cusp inside the dumbbell core and (with the obvious exception of the PY) of a second cusp outside; the cusps' heights are also qualitatively well reproduced by the theories, except at high densities.

A core-softened fluid model in disordered porous media. Grand canonical Monte Carlo simulation and integral equations

We have studied the microscopic structure and thermodynamic properties of a core-softened fluid model in disordered matrices of Lennard-Jones particles by using grand canonical Monte Carlo simulation. The dependence of density on the applied chemical potential (adsorption isotherms), pair distribution functions, as well as the heat capacity in different matrices are discussed. The microscopic structure of the model in matrices changes with density similar to the bulk model. Thus one should expect that the structural anomaly persists at least in dilute matrices. The region of densities for the heat capacity anomaly shrinks with increasing matrix density. This behavior is also observed for the diffusion coefficient on density from independent molecular dynamics simulation. Theoretical results for the model have been obtained by using replica Ornstein–Zernike integral equations with hypernetted chain closure. Predictions of the theory generally are in good agreement with simulation data, except for the heat capacity on fluid density. However, possible anomalies of thermodynamic properties for the model in disordered matrices are not captured adequately by the present theory. It seems necessary to develop and apply more elaborated, thermodynamically self-consistent closures to capture these features.