Primitive equations description

There have been considerable efforts to move beyond the simplified Gouy-Chapman description of double layers at the electrode-electrolyte interface, which are based on the solution of the Poisson-Boltzmann equation for point charges. So-called modified Poisson-Boltzmann (MPB) models have been developed to incorporate finite ion size effects into double layer theory [61]. An early attempt to apply such restricted primitive models of the double layer to the ITIES was made by Cui et al. [62], who treated the problem via the MPB4 approach and compared their results with experimental data for the more problematic water-DCE interface. This work allowed for the presence of the compact layer, although the potential drop across this layer was imposed, rather than emerging as a self-consistent result of the theory. The expression used to describe the potential distribution across this layer was... 

From the rigorous treatment of the double-layer problem on the molecular level, it becomes clear that the Gouy-Chapman theory of the interface is equivalent to a mean field solution of a simple primitive model (PM) of electrolytes at the interface (6). To consider the correlation between ions, integral equations that describe the PM are devised and solved in different approximations. An exact solution of the PM of the electrolyte can be obtained from the computer simulations. This solution can be compared with the solutions obtained from different integral equations. For detailed discussion of this topic, refer to the review by Camie and Torrie (6). In many cases, the molecular description of the solvent must be introduced into the theory to explain the complexity of the observed phenomena. The analytical treatment in such cases is very involved, but initial success has already been achieved. Some of the theoretical developments along these lines were reviewed by Blum (7). 

The reason why the random flight model has proved so popular theoretically stems from its simplicity, which offers hope for the development of analytic solutions. The problem can usually be cast in the form of a diffusionlike or a Schrodinger-wave-equation-like differential equation, the solutions of which are reasonably well explored. A tendency has developed in recent times to apply extremely sophisticated mathematical procedures to what are really very primitive models for polymer chains (see, e.g. Levine et al., 1978). Whether the ends merit the means in such instances cannot yet be assessed objectively. A strategy that might be more productive in terms of the development of a practical theory for steric stabilization is to aim for a simpler mathematical description of more complex models of polymer chains. It should also be borne in mind in developing ab initio theories that a simple model that may well suffice in polymer solution thermodynamics may be quite inadequate for the simulation of the conformational properties of polymers. Polymer solution thermodynamics seem to be relatively insensitive to molecular architecture per se whereas the conformation of a polymer chain is extremely sensitive to it. 

The evaluation of P requires knowledge about the photoelectron amplitude. It should, of course, be calculated as a continuum amplitude from the Dyson equation, but for a general molecule that is still a tough problem, and one proceeds by making more or less ad hoc choices. The perhaps simplest description of the photoelectron is v kf,r) — (27t) 5 ex.p ikf r). This choice of a plane wave is often referred to as the sudden approximation, or the zeroth-order Born approximation. If a primitive atomic orbital basis aj(r — Pa) is used. 

Primitives and definitions are used to formulate general postulates (e.g., the First and Second Laws, balances of mass, momentum, etc.) valid for all (in fact for a broad class of) material models. Real materials are expressed through special mathematical models in the form of constitutive equations which describe idealized materials expressing features important in assumed applications. Moreover, the same real material may be described by more models with various levels of description. The levels are motivated by the observer s time and space scales— typically the time and space intervals chosen (by the observer) for description of a real material having its own... 

Stochastic equation for reptation dynamics Although the above probabilistic description is quite useful in understanding the essence of reptation dynamics, it becomes progressively more difficult to proceed with the calculation for other types of time correlation function. For example, it is not easy to calculate the mean square displacement of a primitive chain segment (R(s, t)-R(s, 0)) ) by this method. In this section we shall describe a convenient method" for calculating general time correlation functions. 

One of the most fruitful applications of the PB equation is in the description of long cylindrical polyelectrolytes, of which DNA is the prototypical example. The same simplifications used in the planar case with a uniform and constant surface charge density and a restricted primitive model of the electrolyte are assumed here. 

From the preceding sections, it seems evident that a real description of ion specificities in solutions can only be done if the geometry and the properties of water molecules are explicitly taken into account. Such models are called non-primitive or Born-Oppenheimer models. In the 1970s and 1980s, they were developed in two different directions. In particular, integral equation theories, such as the hypernetted chain (HNC) approach, were extended to include angle-dependent interaction potentials. The site-site Ornstein-Zernike equation with a HNC-like closure and the molecular Ornstein-Zernike equation are examples. For more information, see Ref. 17. 

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. 

---