Method integral equations

We do not go into any detail of the integration methods here, as it seems to us that direct finite difference methods are preferable. 

In descriptions of this problem, the names of Randles [460] and Sevclk [505] are prominent. They both worked on the problem and reported their work in 1948. Randles was in fact the first to do electrochemical simulation, as he solved this system by explicit finite differences (and using a three-point current approximation), referring to Emmons [218]. Sevclk attempted to solve the system analytically, using two different methods. The second of these was by Laplace transformation, which today is the standard method. He arrived at (9.116) and then applied a series approximation for the current. Galus writes [257] that there was an error in a constant. Other analytical solutions were described (see Galus and Bard and Faulkner for references), all in the form of series, which themselves require quite some computation to evaluate. 

---

Rasaiah J C and Friedman H L 1968 Integral equation methods in computations of equilibrium properties of ionic solutions J. Chem. Phys. 48 2742... 

More sophisticated approaches to describe double layer interactions have been developed more recently. Using cell models, the full Poisson-Boltzmann equation can be solved for ordered stmctures. The approach by Alexander et al shows how the effective colloidal particle charge saturates when the bare particle charge is increased [4o]. Using integral equation methods, the behaviour of the primitive model has been studied, in which all the interactions between the colloidal macro-ions and the small ions are addressed (see, for instance, [44, 45]). 

For reasons of space and because of their prime importance, we focus here on free energy calculations based on detailed molecular dynamics (MD) or Monte Carlo (MC) simulations. However, several other computational approaches exist to calculate free energies, including continuum dielectric models and integral equation methods [4,14]. 

The integral equation method is free of the disadvantages of the continuum model and simulation techniques mentioned in the foregoing, and it gives a microscopic picture of the solvent effect within a reasonable computational time. Since details of the RISM-SCF/ MCSCF method are discussed in the following section we here briefly sketch the reference interaction site model (RISM) theory. 

The molecular and liquid properties of water have been subjects of intensive research in the field of molecular science. Most theoretical approaches, including molecular simulation and integral equation methods, have relied on the effective potential, which was determined empirically or semiempirically with the aid of ab initio MO calculations for isolated molecules. The potential parameters so determined from the ab initio MO in vacuum should have been readjusted so as to reproduce experimental observables in solutions. An obvious problem in such a way of determining molecular parameters is that it requires the reevaluation of the parameters whenever the thermodynamic conditions such as temperature and pressure are changed, because the effective potentials are state properties. 

GH Theory was originally developed to describe chemical reactions in solution involving a classical nuclear solute reactive coordinate x. The identity of x will depend of course on the reaction type, i.e., it will be a separation coordinate in an SnI unimolecular ionization and an asymmetric stretch in anSN2 displacement reaction. To begin our considerations, we can picture a reaction free energy profile in the solute reactive coordinate x calculated via the potential of mean force Geq(x) -the system free energy when the system is equilibrated at each fixed value of x, which would be the output of e.g. equilibrium Monte Carlo or Molecular Dynamics calculations [25] or equilibrium integral equation methods [26], Attention then focusses on the barrier top in this profile, located at x. ... 

The solubility of naphthalene in supercritical carbon dioxide fluids has been evaluated by means of the integral equation method (Tanaka and Nakanishi, 1994). 

All the integral equation methods discussed in this chapter are based on an integral representation of the reaction potential. Let us state this point precisely. 

If condition (i) is not satisfied, the integral equation method presented in the previous section needs to be modified. Proceeding as above, it is easy to show that the total electrostatic potential V solution to Equation (1.28) can be decomposed as... 

The range of application of the integral equation method is not limited to the standard dielectric model. It encompasses the cases of anisotropic dielectrics [8] (liquid crystals), weak ionic solutions [8], metallic surfaces (see ref. [28] and references cited therein),. .. However, it is required that the electrostatic equation outside the cavity is linear, with constant coefficients. For instance, liquid crystals and weak ionic solutions can be modelled by the electrostatic equations... 

Lastly, let us mention that the integral equation method applies mutatis mutandis to the case of multiple cavities (i.e. to the case when C has several connected components). This situation is encountered when studying chemical reactions in solution. 

E. Cances and B. Mennucci, New applications of integral equation methods for solvation continuum models ionic solutions and liquid crystals, J. Math. Chem., 23 (1998) 309. 

B. Mennucci, E. Cances and J. Tomasi, Evaluation of solvent effects in isotropic and anisotropic dielectrics, and in ionic solutions with a unified integral equation method theoretical bases, computational implementation and numerical applications, J. Phys. Chem. B, 101 (1997) 10506. 

In a previous contribution in this book, Cancfes has presented the formal background of the integral equation methods for continuum models and has shown how the corresponding equations can be solved using numerical methods. In this chapter the specific aspects of the implementation of such numerical algorithms within the framework of the Polarizable Continuum Model (PCM) [1] family of methods will be considered. 

It should be realized that there is no real purpose in comparing this velocity profile with that given by the exact similarity solution since the integral equation method does not seek to accurately predict the details of the velocity and temperature profiles. The method seeks rather, by satisfying conservation of mean momentum and energy, to predict with reasonable accuracy the overall features of the flow. 

This result can be compared with that deduced from the similarity solution and given in Eq. (3,50). It will be seen that the integral equation method gives the correct form of the result, i.e., Nux Re aPrm but that the coefficient is somewhat in error. This is typical of what can be expected of the integral equation method. 

The problem to which the integral equation method was applied in the above discussion, i.e., flow over an isothermal plate, is, of course, one for which a similarity solution can be found. The usefulness of the integral equation method, however, arises mainly from the fact that it can be applied to problems for which similarity solutions cannot easily be found. In order to illustrate this ability, consider flow over a flat plate which has an unheated section adjacent to the leading edge as shown in Fig. 3.15. 

The integral equation method can also easily be extended to situations that involve a varying freestream velocity. 

Similarity and integral equation methods for solving the boundary layer equations have been discussed in the previous sections. In the similarity method, it will be recalled, the governing partial differential equations are reduced to a set of ordinary differential equations by means of a suitable transformation. Such solutions can only be obtained for a very limited range of problems. The integral equation method can, basically, be applied to any flow situation. However, the approximations inherent in the method give rise to errors of uncertain magnitude. Many attempts have been made to reduce these errors but this can only be done at the expense of a considerable increase in complexity, and, therefore, in the computational effort required to obtain the solution. 

Consider laminar forced convective flow over a flat plate at whose surface the heat transfer rate per unit area, qw is constant. Assuming a Prandtl number of 1, use the integral equation method to derive an expression for the variation of surface temperature. Assume two-dimensional flow. 

The approximate integral equation method that was discussed in Chapters 2 and 3 can also be applied to the boundary layer flows on surfaces in a porous medium. As discussed in Chapters 2 and 3, this integral equation method has largely been superceded by purely numerical methods of the type discussed above. However, integral equation methods are still sometimes used and it therefore appears to be appropriate to briefly discuss the use of the method here. Attention will continue to be restricted to two-dimensional constant fluid property forced flow. 

The results obtained using the integral equation method as given in Eqs. (10.125) and (10.126) agree to within about 29c with the exact result derived earlier. 

As a last example of the use of the integral equation method consider again two-dimensional flow about an isothermal cylinder in a porous medium. The situation considered is shown in Fig. 10.22. 

---