Capillary charge

A. E. Yaroshchuk, S. S. Durkhin. Phenomenological theory of reverse osmosis in macroscopically homogeneous membranes and its specification for the capillary charged model. J Memb Sci 79 133, 1993. 

Because Barker in this period confined his work to studies at the dme, he focused on and attempted to solve problems peculiar to that electrode. One of those problems is capillary noise or capillary response . These terms refer to the current required to charge the interface between mercury and the film of solution which penetrates some distance into the capillary. Charge must be supplied when the potential is changed, but also the interface itself is unstable, so that the charging current varies from drop to drop. In order to understand the reasoning which led Barker to normal and differential pulse polarography, it is useful to describe in more detail the operation of his square-wave instrument. 

The first term with m = 0 in the right-hand side of Equation 4.167a accounts for the contribution of the capillary charges (or capillary monopoles ). Analytical expressions for the force and energy of interaction between two capillary multipoles of arbitrary order have been derived [324]. 

Kralchevsky et al. [14] also proved from the energy method that Eqn. (3.49) is valid for calculating the horizontal force between the two floating spherical particles. Since r is always positive, the sign of capillary charge Q. depends solely on y/.. If both particles have the same sign of y/., the lateral capillary force between the two floating spheres is attractive otherwise, the force is repulsive. 

Since the contact line moves on the curved surface of the sphere, either r, or y/ is not constant. In order to calculate the capillary charge Q, in Eqn. (3.49), one needs to use the vertical force balance. Because the inclination of the contact line is not significant, the contact line can be assumed horizontal to obtain geometric relations. The volume of the lower part of the sphere immersed in liquid is... 

The last configuration for Cheerios effect in this article is a spherical particle with capillary charge Q floating in vicinity of an infinite vertical wall. In order to find the horizontal capillary force between the wall and particle, one may consider a simple case where the contact angle at the wall is fixed at 0 = 90°, i.e., the boxmdary condition is... 

The capillary charge Q can be calculated by Eqn. (3.57). Since the interface elevation caused by the wall is zero and the elevation caused by the floating particle is negligibly small, the term can be ignored. In addition, Xjf. is small due to the small size and weight of the particle. As a result, Eqn. (3.57) can be reduced to Eqn. (3.60). 

If the elevation of the contact line is zero at the wall, Eqn. (3.60) is valid to calculate the capillary charge Q. For the typical inclined interface near the wall, however, the elevation of the contact line on the particle is no longer small. The term cannot be neglected. Eqn. (3.57) can be used to calculate the capillary charge Q of the particle. The elevation of the contact line on the particle surface h can be given as ... 

For the two particles with capillary charges and Q, Eqn. (3.24) is used to compute the interface deformation close to the particles for small or moderate inter-particle separation quY 1). By setting = in Eqn. (3.24) and using the relations of the bipolar coordinates, one obtains [19]... 

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]). 

The possible relevance of this quadrupole-quadrupole force at colloidal length scales was advanced in references [9,46,47]. As emphasized in reference [9], the natural surface roughness in the nanometer scale of a micrometer-sized spherical particle could conceivably cause a force of this kind. It was also proposed as an explanation of the structures observed experimentally [47] in 2D colloids of nonspherical micrometer particles at a fluid interface. This motivated the theoretical investigation of the anisotropic capillary forces The main difficulty lies in the determination of the capillary charges in terms of given properties of the particles (e.g., wettability and shape). Different simplifications were applied the contact line is approximated by a circle [48-50] or by an expansion in small eccentricity [45] highly elongated shapes are dealt with numerically [51]. 

The capillary multipoles P, of the capillary charge distribution nj oilr) are (see the Appendix) ... 

The capillary force and torque can be derived from an effective interaction potential For two particles, it has the form of a multipole-multipole interaction like Equation 2.14 plus a correction term related to Am in Equation 2.30 and collecting forces and torques contributed by the capillary charge in Il(r), not captured by the multipoles. An immediate conclusion is that, similarly to the deformation by an isolated particle, the capillary force and torque are dominated at large separations by the interaction of the lowest-order nonvanishing multipoles provided their order is < - 2, and in such case pairwise additivity of the capillary force holds. 

Another situation of experimental interest corresponds to nonspherical particles at a curved interface. The lowest-order nonvanishing capillary charge of these particles, in the absence of external... 

Here we recall briefly some results pertaining electrostatics in two dimensions. A distribution Il(r) of capillary charge creates a capillary potential u r) that can be written, provided rt(r) vanishes fast enongh at infinity, as... 

By studying how Q, transforms under a rigid rotation of the capillary charge Il(r), one can show easily that the complex-valued multipole charge can be written as gj = Inserting this into Equation 2.37 one arrives at Equation 2.12. A definition of the multipole charge alternative to Equation 2.38 is provided by application of the residue theorem to Equation 2.37 ... 

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