Additivity of the electrostatic interaction

Suppose we have three molecules A, B, C, intermolecular distances are long and therefore it is possible to use the polarization perturbation theory, in a veiy similar way to that presented in the case of two molecules (p. 692). In this approach, the unperturbed Hamiltonian represents the sum of the Hamiltonians for the isolated molecules A, B, C. Let us change the abbreviations a little bit to be more concise for the case of three molecules. A product fimction j/A,nA 4 B B C,nc will be denoted by In n nc) = where ha, b, c (= 2.) stand 

Let us recall that the electrostatic interaction energy Ee]st(.ABC) of the ground-state ha =0,riB = 0, nc = 0) molecules is defined as the first-order correction to the energy in the polarization approximation perturbation theoiy  

The integration in the last formula goes over the coordinates of all electrons. In the polarization approximation, the electrons can be unambiguously divided into three groups those belonging to A, B and C. Because the zero-order wave function lO OfiOc) represents a product 0 lO 0c), the integration over the electron coordinates of one molecule can be easily performed and yields 

let us look at the last formula. We easily see that the individual terms simply represent the electrostatic interaction energies of pairs of molecules AB, BC and AC, that we would obtain in the perturbational theory (within the polarization approximation) for the interaction of AB, BC and AC, respectively. Conclusion  

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Additivity of the Electrostatic Interaction Exchange Non-additivity Induction Non-additivity... 

Polyelectrolytes provide excellent stabilisation of colloidal dispersions when attached to particle surfaces as there is both a steric and electrostatic contribution, i.e. the particles are electrosterically stabilised. In addition the origin of the electrostatic interactions is displaced away from the particle surface and the origin of the van der Waals attraction, reinforcing the stability. Kaolinite stabilised by poly(acrylic acid) is a combination that would be typical of a paper-coating clay system. Acrylic acid or methacrylic acid is often copolymerised into the latex particles used in cement sytems giving particles which swell considerably in water. Figure 3.23 illustrates a viscosity curve for a copoly(styrene-... 

As a next step, the resulting surface must be divided into segments. For the subsequent calculation of the electrostatic interactions, it is important to know the area, shape, and position of each segment. In addition, it is desirable to achieve reasonably homogeneous segment areas in order to reduce the total number, M, of segments on the molecule, because the time requirements of the algorithms for the solution of the CSM boundary equations typically scale with the third power of M and the memory-demand scales with M2. 

Calculations of the deposition rate show a dramatic dependence on the mechanism by which the surface charge is generated, in addition to the dependence on the charge itself. This reflects the importance of surface chemistry in particle deposition. Characterization of the electrostatic interactions, involving a given surface, require electrokinetic measurements on that surface under a broad spectrum of electrolyte conditions so that the number density and dissociation constant of ionizable surface groups can be deduced a single electrokinetic measurement is not sufficient. 

Two infinite-size plates are immersed in a semidilute solution of polyelectrolyte in a good solvent which also contains the small ions of a salt. One of the plates is located at x=0 and the other one at x=D. The system is considered to be in contact with a large reservoir, which contains a polyelectrolyte/salt solution. In addition to the electrostatic interactions, the segments of the polymer have a van der Waals interaction —UkT with the plates. In the mean field approach, the intra- and interchain interactions together with the electric field induced by the surface charges of the plates and polyelectrolyte molecules are expressed as an external potential. Within the mean-field approximation, the free energy of the system with respect to that in the reservoir can be expressed as the sum of three contributions, the polymer contribution Fpol, the salt ions contribution Fion and the electrostatic field contribution Fels,... 

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