Linear superposition approximation

The superposition approximation (SA) suggested in Refs. Ill and 259 is essentially a nonlinear theory that cannot be represented in the form of Eqs. (3.707). The same is true for the extended version of SA [260]. For this reason, we focus on two derivatives of these theories linearized near the equilibrium the linearized superposition approximation (LSA) and the linearized extended superposition approximation (LESA). It was found that LSA developed in a number of works [139,175,255,260] isinfact identical to IET (see Table VIII). They both have the same concentration-independent kernel S(j ). As for LESA, it was, strictly speaking, created for the reactions in the ground state [241,242], but can be easily extended to the case of equal lifetimes, uA = uc-... 

Linear Superposition Approximation for the Double-Layer Interaction of Particles at Large Separations... 

Arbitrary Potentials Derjaguin s Approximation Combined with tbe Linear Superposition Approximation... 

In Chapter 11, we derived the double-layer interaction energy between two parallel plates with arbitrary surface potentials at large separations compared with the Debye length 1/k with the help of the linear superposition approximation. These results, which do not depend on the type of the double-layer interaction, can be applied both to the constant surface potential and to the constant surface charge density cases as well as their mixed case. In addition, the results obtained on the basis of the linear superposition approximation can be applied not only to hard particles but also to soft particles. We now apply Derjaguin s approximation to these results to obtain the sphere-sphere interaction energy, as shown below. 

Note that the boundary condition (13.6) states that membranes 1 and 2 are electrically transparent to each other. As a result, the solution to Eqs. (13.2)-(13.4) is obtained by a linear superposition approximation (LSA), which, for the case of rigid membranes, holds only at large membrane separations. 

Comparison is made with the results for the two conventional models for hard plates given by Honig and Mul [11]. We see that the values of the interaction energy calculated on the basis of the Donnan potential regulation model lie between those calculated from the conventional interaction models (i.e., the constant surface potential model and the constant surface charge density model) and are close to the results obtained the linear superposition approximation. 

The first-order correction to the linear superposition approximation Vlsa is given by the sum of the second and third terms on the right-hand side of Eqs. (14.23), (14.29) and (14.34), each corresponding to the image interaction of one sphere with respect to the other sphere. We denote this image interaction by Vimage nd expressed it as... 

FIGURE 14.4 Comparison of the linear superposition approximation V lsa = 1 ° the image interaction correction Fimage = their sum ULSA + Fmage, and the full... 

Fig. 14.5), then the interaction energy shows a minimum. That is, the interaction force, which is attractive at large separations, may become repulsive at small separations. As Fig. 14.5 shows, the change in sign of the interaction force or the appearance of the extremum in the interaction energy occurs when the contribution of the image interaction correction exceeds that of the leading term (or the linear superposition approximation term). 

The leading term of V R) and V iR) agrees with Eq. (11.82) obtained by the linear superposition approximation. 

A simple approximate analytic expression for P ih) can be obtained using the linear superposition approximation (LSA) (Chapter 11). In this approximation, y h/2) in Eq. (15.34) is approximated by the sum of the asymptotic values of the two scaled unperturbed potentials ys(T) that is produced by the respective plates in the absence of interaction. For two similar plates. 

Now that we know the individual dynamics of boimd, free and bulk water, our aim is to find the d)mamics of a water molecule that was initially bound and later is still bound, free, or escaped to the bulk. Towards this goal we use a linear superposition approximation to express the mean square displacement of such a water molecule as ... 

Finally, the linear superposition approximation (LSA) is a simple way to try to take into account that real interfaces most often behave neither as constant potential surfaces nor as constant charge ones LSA is a simple average between those two cases for the linear PB, it results in... 

This non-linear superposition approximation clearly requires that the plate separation, h, be large compared with the Debye length. 

By solving the linearized Poisson-Boltzmann equation through a multipole expansion, Russel ef al. mapped out regions where the Derjaguin and linear superposition approximations are valid with small potentials. 

Colloidal particles snspended in a solvent can acquire charge by two ways. The surface groups can dissociate or the ions from solution can bind to the particle surface (Israelachvili, 1992 Russel, 1989 Hunter, 1987). The charge boimd to the particle surface is balanced by a diffuse region of ions in solution, called the diffuse double layer. When two like particles (radius a) approach one another, the double layers overlap and the particles feel a repulsion. Exact analytic expressions for the electrostatic potential energy ( Ve) for all values of the particle surface potential are difficult to compute and therefore analytical approximations or numerical solutions are used. Eor interacting particles with low surface potential e fa/kT < 1), can accurately be calculated using the linear superposition approximation (Israelachvili, 1992 Russel, 1989 Hunter, 1987)... 

The electrostatic free energy of a thin liquid film can be approximated with different expressions, depending on the specific conditions. " For arbitrarily charged droplets and high electrolyte concentrations i.e. weakly overlapped double layers) one may use the non-linear superposition approximation ... 

If the conditions for applying the non-linear superposition approximation are violated, but the particles are weakly charged (or with low surface potential), other... 

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