Surface potential around sphere

Let us turn from planar, parallel surface to spheres. The interaction between spheres can be calculated in different ways, (a) We can start with expressions (4.63-4.66) and apply the Derjaguin approximation [416,427-430). This leads to a good approximation for Ri, R2 3> X.d and short distance, (b) Or, we superimpose the potentials around spheres [430-433]. This is a good approximation for small spheres and large distances, (c) Ohshima solved the linearized Poisson-Boltzmann equation in two dimensions analytically for constant potential conditions [434]. As a leading term, he obtained... 

The first case is relevant in the discussion of colloid stability of section C2.6.5. It uses the potential around a single sphere in the case of a double layer that is thin compared to the particle, Ka 1. Furthennore, it is assumed that the surface separation is fairly large, such that exp(-K/f) 1, so the potential between two spheres can be calculated from the sum of single-sphere potentials. Under these conditions, is approximated by [42] ... 

In view of this equation the effect of the ionic atmosphere on the potential of the central ion is equivalent to the effect of a charge of the same magnitude (that is — zke) distributed over the surface of a sphere with a radius of a + LD around the central ion. In very dilute solutions, LD a in more concentrated solutions, the Debye length LD is comparable to or even smaller than a. The radius of the ionic atmosphere calculated from the centre of the central ion is then LD + a. 

We obtain the potential distribution around a sphere of radius a having a surface potential i/ o immersed in a solution of general electrolytes [9]. The Poisson-Boltzmann equation for the electric potential i//(r) is given by Eq. (1.94), which, in terms of/(r), is rewritten as... 

Figure 3. Cu. Zn superoxide dismutase—electrostatic potential mapped onto the enzyme s molecular surface to show the highly positive potential around the active site channel (S3). The dots are color-coded by electrostatic potential red, <-21kcal/mol yellow, -21 to -7 kcal/ mol green, -7 to +7 kcal/mol cyan, +7 to 21 kcal/mol blue, > 21 kcal/mol. The bound copper ion is shown by the purple sphere.

In order to calculate the maximum rate of the bimolecular reaction, the assumption is made that every time two molecules collide, a chemical reaction occurs. Consider the diffusion of a system of A molecules into stationary B molecules. If every time A collides with B a reaction occurs, the concentration of A at the surface of B must equal zero, while the concentration at a large distance from B is equal to the bulk concentration C o In order to simplify the problem mathematically, the molecules are assumed to be spherical, so that the diffusion process is spherically symmetric, and the potential energy U is assumed to be a function of r only. This model is depicted in Fig. 2-13. A general solution of Eq. (2-85) is still not possible, but if a steady state is assumed (that is, dCJdt = 0), the total flux through the surface of a sphere of radius r around B is constant for all values of r and is... 

These coupled charge and mass transports have given rise to a vast literature that started with Smoluchowski [4], Huckel [5], Debye [6], Henry [7], and Booth [8], who considered limit cases (very thin or very thick double layers) in simple geometries (along a flat wall or around a single sphere) for small surface potentials. The relevant literature has been analyzed in detail by Coelho et al. [9], and it will be simply summarized here by giving some of the major landmarks. 

The potential around a charged sphere. You have an evenly charged spherical surface with radius a and a total charge qm-d medium with dielectric constant D. De-ri e the potential inside and outside the surface by using Coulomb s law and integrating over the charged surface. 

We now consider a particle of mass m free to move around a central point at a constant radius r. That is, it is free to travel anywhere on the surface of a sphere of radius r. To calculate the energy of the particle, we let—as we did for motion on a ring—the potential energy be zero wherever it is free to travel. Furthermore, when we take into account the requirement that the wavefunction should match as a path is traced over the poles as well as around the equator of the sphere surrounding the central point, we define two cyclic boundary conditions (Fig. 9.35). Solution of the Schrodinger equation leads to the following expression for the permitted energies of the particle ... 

Other approximate analytical expressions have been derived [387]. As an example, the calculated electric potential around a sphere of J p = 15 nm in an aqueous medium containing 2 mM monovalent salt is plotted in Figure 4.3 on the right. As for planar surfaces, the linear solution overestimates the potential for surface potentials higher than 50 mV. The decay of the potential is steeper than for planar surface due to the additional factor 1/r. 

Provided that the bulk concentrations remain constant, the flux J of Eq. (9-3) is also constant. Consider a spherical distribution of potential reactants B around a particular molecule of A. The surface area of a sphere at a distance r from A is 4irr2. Thus, the expression for the flow of B toward A is... 

Thus, the potential reaches a maximum at the sphere s center and then decreases as a parabolic function. A completely different behavior is observed outside the sphere. This simple problem allows one to demonstrate again that the potential obeys Poisson s equation. Consider the potential at the point p of an arbitrary body, Fig. 1.12a, assuming that the density may change from point to point. Let us mentally draw a spherical surface around the point p. If its radius is sufficiently small, we can suppose that this sphere is homogeneous. The potential at the point p can be written as... 

Molecular Probe Analysis. In an effort to understand how a molecule is seen by either another molecule or by a surface, molecular probes can be moved around a chemical to map out its surface. These probes include anions and cations (point charges) and hard spheres or can be constructed as a combination of these. The empirical potential energy is computed at a variety of points around the test molecule and an energy surface is thus generated. This can be examined graphically and compared as changes are made to the molecule. 

The potential near an isolated sphere can be found by placing a point charge at the sphere center, where the strength of the point charge is chosen to yield the desired surface properties. Using the cylindrical coordinate system shown in Fig. 1, the potential Fs around the sphere (without accounting for the wall) is then [11,12]... 

Note that the slip velocity at the outer surface of the double layer results in a potential flow around the particle. This flow field, equivalent to that caused by a force quadupole at the origin [10], decays as r-3, which is faster than r ] for sedimentation of a sphere. 

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