Integral equations hypernetted-chain approximation

Another possible approach solving the equilibrium distribution for an electric double layer is offered by integral equation theories [22]. They are based on approximate relationships between different distribution functions. The two most common theories are Percus-Yevick [23] and Hypernetted Chain approximation (HNQ [24], where the former is a good method for short range interactions and the latter is best for long-range interactions. They were both developed around 1960, but are still used. The correlation between two particles can be divided into two parts, one is the direct influence of particle j on particle i and the other originates from the fact that all other particles correlate with particle j and then influence particle i in precisely... 

Theory nowadays overcomes the limitation of concentration range by integral equation and simulation methods. Mean spherical approximation (MSA) and hypernetted chain approximation (HNC) are the most important features yielding modern analytical transport equations over extended concentration ranges. Nevertheless, the IcCM expressions maintain their importance. They are reliable expressions for the determination of limiting values of the transport properties at infinite dilution of the electrolyte as a convenient basis for the provision of... 

A number of approximate integral equations for the radial distribution function g(r) of fluids have been proposed in recent years. Two particularly useful approximations are the Percus-Yevick (PY)1,2 and the Convolution Hypernetted Chain (CHNC)3-4 equations. In this paper an efficient numerical method of solving these equations is described and the results obtained bv applying the method to the PY equation are discussed. A later paper will describe the behavior of the... 

Much more flexible is to describe the solute-solvent system within the supercell technique employing a set of 3D plane waves. The convolution in the OZ equation can be then approximated by means of the 3D fast Fourier transform (3D-FFT) procedure. For the first time, this approach was elaborated by Beglov and Roux [16] for numerical solution of the 3D-OZ integral equation with the hypernetted chain (HNC) closure for the 3D distribution of a simple LJ solvent around non-polar solutes... 

During the last 20 years, a quantitative description of conductance and self-diffusion up to 1-M solutions has been achieved by the use of modern gij functions coming from integral equation techniques such as the hypernetted chain (HNC) equation or mean spherical approximation... 

To complete our theoretical treatment of liquids, we need a procedure to calculate g(r). It so happens that there is no exact equation for g(r), but there are several accurate approximate equations. Four equations that have had some success for fluids are the Kirkwood equation, the Born-Green-Yvon equation, the Percus-Yevick equation and the Hypernetted chain equation (McQuarrie, 1976). All four of these equations are integral equations for g(r) in terms of u(r). 

Baxter (1968b) showed that the Ornstein-Zernike equation could, for some simple potentials, be written as two one-dimensional integral equations coupled by a function q(r). In the PY approximation for hard spheres, for instance, the q(r) functions are easily solved, and the direct-correlation function c(r) and the other thermodynamic properties can be obtained analytically. The pair-correlation function g(r) is derived from q(r) through numerical solution of the integral equation which governs g(r) for which a method proposed by Perram (1975) is especially useful. Baxter s method can also be used in the numerical solution of more complicated integral equations such as the hypernetted-chain (HNC) approximation in real space, avoiding the need to take Fourier transforms. An equivalent set of relations to Baxter s equations was derived earlier by Wertheim (1964). 

Indeed, nature is much more comphcated. Even systems that show the same Hofmeister series may have different interactions that nnderpin the behaviour. As ion behaviour depends on the environment, it is also difficult to interpret widely different experiments by the same type of interactions. That the same series is found in two experiments is not a proof that the same interactions govern both experiments. This is a frequent mistake. How subtle these interactions can be is illustrated in Chap. 11, where the potentials of mean force from molecular dynamics simulations are taken and inserted into Poisson-Boltzmann calculations. Figure 4 of this chapter demonstrates that the double layer pressure increases in the non-Hofrneister sequence NaCl < Nal < NaBr at large separations (> 1.0 nm). Frequent changes in this sequence occur as the surfaces approach closer to each other, including NaCl < NaBr < Nal and NaCl > NaBr > Nal. Even if the underlying approximations do not fully reflect the real world, this result is far too complicated to be interpreted by simple rules. Note that integral equations such as the hypernetted chain theory are a more powerful alternative to the Poisson-Boltzmann equation, see Chap. 10. 

---