Electrostatic analogy

Figure 17. Stereoview of a cast of the HIV-1 protease binding site in the X-ray crystal structure [72] generated using CAVITY [77]. The CAVITY display is color-coded by electrostatics. Analog 29 is shown in yellow.

The electrostatic analogy is clear and the molecular and heat fluxes to the growing crystal can be written... 

FIGURE 23 Analyzers (a-c) Castaing-Henry analyzer (d) Q filter and (e) electrostatic analog of the Q filter. 

A synthetic adduct of thiamin diphosphate-methyl acetylphosphonate had been prepared by Kluger and Pike [58]. They confirmed this proposed mechanism of inhibition and provided detail information about the enzyme s active site. Sodium (9-methyl acetylphosphonate 1-1 added to thiamin to form methyl 2-hydroxy-2-(2-thiamin)ethylphosphonate 14 (Scheme 1.15), which in turn served as a structural and electrostatic analog of reactive intermediate a-lactyl-TPP in which the planar carboxylate was replaced by a tetrahedral phosphonate. 

We have just described the linearized theory of capillarity. In the electrostatic analogy the field M(r ) is identified with a 2D electrostatic potential ( capillary potential ) and Il(r ) with a charge density ( capillary charge ) Equation 2.8 reduces to the Poisson equation of electrostatics and Equation 2.9 relates the tensor Tn, which has the form of Maxwell s stress tensor, with the electric force exerted on the capillary charge n(rn) (also the usual boundary conditions imposed on the interface have a close electrostatic analogy [34,35]). 

Consider now a collection of particles. The electrostatic analogy establishes that the interfacial deformation can be written as a superposition of expansions of the form of Equation 2.12, one centered at each particle, but with certain multipole charges Qf + 5Qf (i =1,2, N), which do not have to be identical with the charges as if each particle were isolated ... 

The charges Qf which the fth particle has when isolated at the interface can be termed permanent capillary multipoles, as opposed to the capillary multipoles dQf induced by the presence of the other particles. The induced multipoles arise in general because it is obvious that a superposition of profiles as if the particles were isolated cannot satisfy in general the boundary conditions at all the contact lines (see the sketch in Figure 2.5). In general, the induced multipoles will depend on the three-dimensional (3D) positions and orientations of all the particles through the boundary conditions at each particle. Thus, there is not a simple constitutive relation between 8<2f and the deformation field, and the electrostatic analogy seems of limited use in this respect instead, would have to be determined by a detailed solution of the problem near the particles. 

Note that, because of the implicit dependence of 5Qf on the interfadal deformation brought about by the other particles, Equation 2.13 is not a linear superposition. However, by definition, the corrections 5Qf vanish in the limit of well-separated particles, Ir - r,l —> >. As a consequence, asymptotically for large separations the deformation is dominated by the linear superposition of the deformation due to the lowest-order nonvanishing permanent capillary chaige of each particle. This is the so-called superposition approximation introduced in reference [5] and frequently used in the literature (either in the force or in the energy approaches). As we see, it is a straightforward consequence of the electrostatic analogy in the absence of induced capillary multipoles violations of this approximation are possible in the context of the linearized theory of capillary due to capillary polarization effects. 

When the interface is already curved by itself, there can arise a capillary force on isolated particles. In the limit of small interfacial curvature the electrostatic analogy holds again, with the curved interface playing the role of an external electric field for example, a neutral, spherical particle at a minimal surface is pulled to regions of larger curvature. Thus, a novel phenomenology is predicted adding to the effects of particle-particle interaction just mentioned. 

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