HNC integral equation

Vlachy, V., Marshall, C.H., and Haymet, A.D.J. Highly asymmetric electrolytes - a comparison of Monte-Carlo simulations and the HNC integral-equation. Journal of the... 

The remainder of this chapter centers upon the calculation of the equilibrium properties of MM-level models. Such models with ion concentrations of up to 10 molecules/cm correspond to ionic solutions with total ionic concentrations up to about 1M. This concentration is roughly a tenth of the ionic concentration in a molten salt it is low enough so that many approximation methods that are quite satisfactory for BO-level models at densities up to a tenth that of the liquid may be used to calculate the measurable properties of MM-level models for the solutions. A typical approximation method of this kind is the HNC integral equation (Section 7). 

The integral equations themselves are obtained by substituting the various approximations to Cab into the Omstein-Zernike equation [Eq. (142)]. Substitution of Eq. (156) gives the HNC integral equation... 

The ionic HNC integral equation has been used extensively to calculate the thermodynamic properties of appropriately chosen MM-level models for... 

Once potential parameters have been determined, we can start calculation downward following arrow in the figure. The first key quantity is radial distribution function g(r) which can be calculated by the use of theoretical relation such as Percus-Yevick (PY) or Hypemetted chain (HNC) integral equation. However, these equations are an approximations. Exact values can be obtained by molecular simulation. Ifg(r) is obtained accurately as functions of temperature and pressure, then all the equilibrium properties of fluids and fluid mixtures can be calculated. Moreover, information on fluid structure is contained in g(r) itself. 

Figure 24 A simple model (the left panel) for the lock-key system represented by a substrate with a hemispherical pocket and big spherical particle, respectively, in HS solvent. The right panel plots the corresponding PMF under different conditions from direct sampling, HNC integral equation, MC-PDT, and MC-DFT. After Jin and Wu (2011).

EoS based on perturbation theory and computer simulation results validity of the Barker-Henderson theory Derivation of the BGY, PY, and HNC integral equations in 2D Fourth virial coefficient Fifth virial coefficient Quantum corrections to the third and fourth virial coefficients quantum corrections to the Helmholtz free energy Results at very high densities EoS based on computer simulation results and on the five first virial coefficients No influence of number of particles for states far from the phase transitions values of RDF... 

The results were analysed using HNC calculations described in another chapter of this book. The ion-ion correlations in the electrolyte and the ionic profiles in the vicinity of the water-air interface were calculated within the HNC integral equation approximation at the Primitive Model level of description (ionic spheres immersed in a continuous dielectric solvent). The (solvent-averaged) ion-ion interaction potential y(r) is the sum of a hard-sphere contribution (radii ), a generic Coulombic Contribution ZiZje / 47T oer) (valency Z, dielectric constant e = 78) and a specific dispersion contribution. ... 

The theoretical approach based on the HNC integral equation is described in the context of ionic specificity. Two levels of description of the water medium are considered. Within the Primitive Model (continuous solvent), ionic specificity is introduced via effective, solvent-averaged, dispersion forces. The agreement with experimental data in bulk or at air-water interfaces is only partial and illustrates the limits of that approach. Within the Born-Oppenheimer model, the molecular HNC equation is solved with an explicit description of the solvent molecules (SPC water). Ionic and solvent profiles in bulk and at interfaces are enriched by short-range osdUated structures. The ionic polaris-ability is introduced via the self-consistent mean-field theory, the polarisable ions carrying an effective, fixed dipole moment. The study of the air-water interface reveals the limits of the conventional HNC approach and the needs for improved integral equations. 

Fig. 1. Concentration profiles near a macroion (gm = 30 A and —20 o) in a monovalent electrolyte solution of ionic strength 1.0 M for (a) NaCl and (b) Nal. Open circles represent counterion concentrations and dark circles the co lons. Solid lines are numerical solutions of the non-linear Poisson-Boltzmann equation and dashed lines are for the OZ-HNC integral equation. ...

The integral equation approach has also been explored in detail for electrolyte solutions, with the PY equation proving less usefiil than the HNC equation. This is partly because the latter model reduces cleanly to the MSA model for small h 2) since... 

The MS closure results from s = 2. The HNC closure results from s = 1. In the latter two expressions, additional adjustable parameters occur, namely ( for the RY closure and for the BPGG version of the MS approximation. However, even when adjustable, these parameters cannot be chosen at will, as they should be chosen such that they eliminate the so-called thermodynamic inconsistency that plagues many approximate integral equations. We recall that a manifestation of this inconsistency is that there is a difference between the pressure as computed from the virial equation (10) and as computed from the compressibility equation (20). Note that these equations have been applied to a very asymmetric mixture of hard spheres [53,54]. Some results of the MS closure are plotted in Fig. 4. The MS result for y d) = g d) is about the same as the MV result. However, the MS result for y(0) is rather poor. Using a value between 1 and 2 improves y(0) but makes y d) worse. Overall, we believe the MS/BPGG is less satisfactory than the MV closure. 

In the theoretical approaches of Poisson-Boltzmann, modified Gouy-Chapman (MGC), and integral equation theories such as HNC/MSA, concentration or density profiles of counterions and coions are calculated with consideration of the ion-waU and ion-ion in-... 

The name, DLYO, originates from the first letter in the surname of the four authors (Derjaguin, Landau, Verwey and Overbeek) from two different groups, which originally published these ideas. The theory is based on the competition between two contributions, a repulsive electric double layer and an attractive van der Waals force [4,5]. The interaction in the electric double layer was originally obtained from mean field calculations via the Poisson-Boltzmann equation [Eq. (4)]. However, the interaction can also be determined by MC simulations (Sec. II. B) and by approximate integral equations like HNC (Sec. II. C). This chapter will focus on the first two possibilities. 

One other second-order integral equation calculation has been made. Henderson and Sokolowski [50] and Henderson et al. [51] have calculated the correlation functions of a hard sphere mixture, when X2 — 0, using the PY2 theory. They find excellent agreement with the HC formulae, Eqs. (42) and (43) with Eq. (41) for g (d ). In particular, the limit 2 = 0, the HNC/PY/PY approximation for 22( 22) is rather good. 

For central force models of water, the bridge function is essential for accurate solutions of the integral equations. Thuraisingham and Friedman found that at room temperature the HNC [Bab T) = 0] approximation for the CF model was in very poor agreement with... 

In what follows we discuss the phase behavior of the Stockmayer fluid in the presence of disordered matrices of increasing complexity. All results are based on a variant of the HNC equation [see Eq. (7.49)], which yields very good results for bulk dipolar fluids [268, 322]. Moreover, subsequent studies of dipolar hard-sphere (DHS) fluids [defined by Eq. (7.59) with ulj = 0] in disordered matrices by Fernaud et al. [323, 324] have revealed a very good performance of the HNC closure compared with parallel computer simular tion results. The integral equations are solved numerically with an iteration procedure. To handle the multiple angle-dependence of the correlations... 

Figure G.l Flow scheme of the numerical solution of the integral equations consisting of the OZ equations [see Eqs. (G.18) and (G.22)) together with the HNC closure [see Eqs. (G.19) and (G.20)]. The acronyms FT and FT denote Fourier transformation and Fourier inversion, respectively Ac is defined in Eq. (G.27) and S is a threshold value set according to the desired accuracy.

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