Partition function electrostatic

Here SU = AU — AC/is the electrostatic contribution to the solute-solvent interactions. The final expression is another partition function formula,... 

The cell model is a commonly used way of reducing the complicated many-body problem of a polyelectrolyte solution to an effective one-particle theory [24-30]. The idea depicted in Fig. 1 is to partition the solution into subvolumes, each containing only a single macroion together with its counterions. Since each sub-volume is electrically neutral, the electric field will on average vanish on the cell surface. By virtue of this construction different sub-volumes are electrostatically decoupled to a first approximation. Hence, the partition function is factorized and the problem is reduced to a singleparticle problem, namely the treatment of one sub-volume, called cell . Its shape should reflect the symmetry of the polyelectrolyte. Reviews of the basic concepts can be found in [24-26]. 

In Section III we consider the problem of an electron gas in a lattice of positive point charges from a general point of view. The difficulties related to the long-range character of the electrostatic forces are discussed, and a certain class of divergences are eliminated from the grand partition function through the application of the condition of electroneutrality of the whole system. 

Most cell model calculations to-date have been performed on electrostatically stabilized dispersions. The canonical partition function for such a system is given by... 

In order to use this transformation for the Hamiltonian as represented by Eq. (6.68), microscopic density terms that are quadratic in nature need to be written in the form given on the left-hand side in Eqs. (6.77) and (6.78). Electrostatic terms in He are already in the appropriate form. It is only the terms in Hw that needs to be rewritten. This can be achieved by rewriting in terms of order parameters and total density. For an n component system, all microscopic densities can be described by n—1 independent order parameters (due to the incompressibility constraint serving as the nth relation among the densities). There are many different ways of defining these order parameters. One convenient definition, which makes mathematics simple, is the deviation of densities of solutes from the solvent density, that is, defining c )j(r) = Qj(r)—Qj(r) forj = 1,2,... (n—1), wherej is the index for different solutes (monomers, counterions, and the salt ions). Using the transformation for each quadratic term in the Hamiltonian (cf. Eq. (6.68)), the partition function becomes... 

Now, using the methods of collective variables (cf. Section 6.4.2.1) for decoupling all the interactions except the electrostatics and the Hubbard-Stratonovich transformation [14, 55] (cf. Section 6.4.2.2) for the electrostatic part in Eq. (6.124), the partition function can be written as integrals over the collective densities and corresponding fields so that Eq. (6.124) becomes... 

Here, Wp, Wg are the collective fields experienced by the monomers and solvent, respectively, and Qp,Qg represent their respective collective densities. All charged species (excluding the ion-pairs formed due to adsorption of counterions) experience a field t ) (which is equivalent to the electrostatic potential), t) and u are Lagrange s multipliers corresponding to, respectively, the incompressibility and net charge constraints in the partition function. 

In the SCLF model, adsorption is depicted as the result of nonCoulombic interactions between the solute and solvent molecules and the electrostatic surface potential. The solution is divided into lattice sites which are arrayed in layers parallel to the surface. The equilibrium distribution of solute molecules between the solution and surface is determined from the partition function, Q, for a solution mixture [50, 52, 55] constituted of an entropic, and an energetic part, U, of general form [e.g., see Eq. (8) in Ref 53] ... 

Limitations of the SCLF method include (1) electrostatic Coulombic interactions between the solute and surface moleucles are ignored, (2) the dependence of calculated adsorption results on the model parameters (such as the solute-solute, solute-surface and solute-solvent Flory-Huggins interaction parameters, the lattice site size, etc.) is difficult to determine, (3) adsorption must be determined by simultaneous solution of probability density equations derived from partition functions so that no single analytical adsorption equation is possible, (4) effects of pH on surface sites can only be considered implicitly through the Flory-Huggins interaction parameters, and (5) the Flory-Huggins interaction parameters do not allow explicit consideration of the molecular or chemical characteristics of the surface site molecules. 

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