Charged particles boundary conditions

To be sure, any particle represented by this archetype, will, in general, not be in a specific state. Therefore, the actual charge-wave will be an infinite sum over the complete set of ft s, but with the same spin-magnetic moment. This sum will superpose these ft s concentrically, as they are representing only one charge. The boundary conditions determine the weights of the various components. 

We have used the procedure of Marcelja et al (22) to compute r for spherical molecules with the same charge and volume as 200 bp rodlike DNA. Each molecule is at the center of a Wigner-Seitz cell of volume (4/3)with bulk salt concentrations spanning the experimental range (2). The nonlinear Poisson-Boltzmann e( uation is solved numerically with appropriate boundary conditions at the particle surface and the cell boundary. The results are that F = 155 at about 3 mg/mL DNA (twice the experimental concentration) with no added salt, but F is always < 155 for added salt in the experimental range. For NaPSS, with dp 3800 at 1-4 X 10 mg/mL, F > 300, consistent with the observation of a structure factor maximum. 

The constant A must equal zero for the potential / to fall to zero at a large distance away from the charge and the constant B can be obtained using the second boundary condition, that / = /o at r = where a is the radius of the charged particle and /o the electrostatic potential on the particle surface. Thus we obtain the result that... 

In order to calculate the matrix elements with the Coulomb operator Vc, one again uses Slater determinantal wavefunctions, for the intermediate state xp(Mp, t) as well as for the complete final state which contains the doubly charged ion, f, and the two ejected electrons, x<, (Ka, Kb). Assuming that there is no correlation between the two escaping electrons and that their common boundary condition applies separately to each single-particle function, the directional emission property is included in the factors f( ka) and f( kb), and one gets for this Coulomb matrix element C... 

Even with these useful results from statistical mechanics, it is difficult to specify straightforward criteria delineating when the Poisson-Boltzmann or linear Poisson-Boltzmann equations can be expected to yield quantitatively accurate results for particle-wall interactions. As we have seen, such criteria vary greatly with different types of boundary conditions, what type of electrolyte is present, the electrolyte concentration and the surface-to-surface gap and double layer dimensions. However, most of the evidence supports the notion that the nonlinear Poisson-Boltzmann equation is accurate for surface potentials less than 100 mV and salt concentrations less than 0.1 M, as stated in the Introduction. Of course, such a statement might not hold when, for example, the surface-to-surface separation is only a few ion diameters. We have also seen that the linear Poisson-Boltzmann equation can yield results virtually identical with the nonlinear equation, particularly for constant potential boundary conditions and with surface potentials less than about 50 mV. Even for constant surface charge density conditions the linear equation can be useful, particularly when Ka < 1 or Kh > 1, or when the particle and wall surfaces have comparable charge densities with opposite signs. 

We consider spherical particles, and, without loss of generality, the fixed charge in the membrane is assumed to be negative. For simplicity, we assume that the distribution of fixed charge is uniform. The variation of the electrical potential is governed by Eq. (1) with m = 2, and the boundary conditions described by Eqs. (2)-(5). We use the approximate perturbation solution expressed by Eqs. (37) and (38)-(40). Suppose that the membrane is thick, and i Don and [/d are related by Eq. (48). 

To solve for the electrical potential, ion concentrations and fluid flow around a particle, one needs to know the boundary conditions in such a system. Far from the particle, the electrical field due to the interaction between the charges in the double layer and on the particle surface vanishes. Thus the... 

Figure 10.7 shows the potential distribution surrounding two spheres of the same size with the same surface potentials, one with constant potential and the other with constant chaise boundary conditions as they are moved together. At small separations, the potential distribution between the particles is again greatly effected by the other particles. The surface of the sphere with the constant charge boundary... 

The surface charge density cr of the particle is related to the particle surface potential i/ o obtained from the boundary condition at the sphere surface,... 

Other kinematic regions require a complete description of the collision, which may be facilitated by including the boundary condition for the three charged particles in the final state. This is nontrivial because there is no separation distance at which the Coulomb forces in the three-body system are strictly negligible. The pioneering experiments of Ehrhardt et al. (1969) are of this type. 

This potential is valid for arbitrarily widely separated spheres, but it breaks down when the Debye double layers begin to overlap significantly (Russel et al. 1989). However, for small separations, kD < 2, the Derjaguin approximation may still be used, as long as D < a-, that is, Ka 2. An analytic expression can then be obtained if the potential is small enough that the Poisson-Boltzmann equation can be linearized (Russel et al. 1989, p. 117). For small separations between particles, a choice must be made between a constant-potential or a constant-charge boundary condition. For a constant-potential boundary condition, one can write the approximate expression... 

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