Soft cylinder

Consider the electrostatic interaction between two parallel dissimilar cylindrical soft particles 1 and 2. We denote by di and d2 the thicknesses of the surface charge layers of cylinders 1 and 2, respectively. Let the radius of the core of soft cylinder 1 be fli and that for soft cylinder 2 be U2- We imagine that each surface layer is uniformly charged. Let Z and N, respectively, be the valence and the density of fixed-charge layer of cylinder 1, and Z2 and N2 for cylinder 2. 

Consider first the case of two parallel soft cylinders (Fig. 15.7). With the help of Derjaguin s approximation for two parallel cylinders [8,9] (Eq. (12.38)), namely,... 

FIGURE 15.7 Interaction between two parallel soft cylinders 1 and 2 at separation H. Cylinders 1 and 2 are covered with surface charge layers of thicknesses di and d2, respectively. The core radii of cylinders 1 and 2 are ai and 2> respectively. 

The interaction force Pcy//(H) acting between two soft cylinders per unit length is given by P yz/iH) = —dVcy// H)ldH, which gives [9]... 

Consider next the case of two crossed soft cylinders (Fig. 15.8). Deijaguin s approximation for two crossed cylinders under condition (15.43) is given by Eq. (12.48), namely. 

As in the case of soft spheres, when the thicknesses of the surface charge layers on soft cylinders 1 and 2 are very large compared with the Debye length 1/k, the potential deep inside the surface charge layer is practically equal to the Donnan potential (Eqs. (15.51) and (15.52)), independent of the particle separation H. 

Consider the case where cylinder 1 is a soft cylinder and cylinder 2 is a cylindrical polyelectrolyte. By taking the limit Kd2 1, we obtain from Eqs. (15.66) and (15.71)... 

As in the case of electrophoresis of hard cylindrical particles, the electrophoretic mobility of a soft cylinder depends on the orientation of the particle [44, 47]. 

Consider the electroosmotic hquid velocity 1/ in an array of parallel soft cylinders in a liquid containing a general electrolyte in an applied electric field E[2] (Fig. 22.3). The velocity of the liquid flow, U, is parallel to the applied field E. We assume that the cyhnder core of radius a is coated with an ion-penetrable layer of polyelectrolytes with a thickness d. The polyelectrolyte-coated cylinder has thus an inner radius a and an outer radius b = a + d. We employ a cell model [4] in which each cylinder is surrounded by a fluid envelope of an outer radius c such that the cylinder/ceU volume ratio in the unit cell is equal to the cyhnder volume fraction , namely,... 

FIGURE 22.3 Electroosmosis in an array of parallel soft cylinders (polyelectrolyte-coated cylinders), which consist of the core of radius a covered with a layer of polyelectrolytes of thickness d. Each cylinder is surrounded by a fluid envelope of outer radius c. The volume fraction of the cylinders is given by b/cy, where b = a + d, and the porosity s is given by 8=1 — 0=1— b/cy. The volume fraction of the particle core of radius a is given by 0c = o/cy. The liquid flow U, which is parallel to the applied electric field E, is normal to the axes of the cylinders. 

In the limit a b, the polyelectrolyte layer vanishes and the soft cylinder becomes a rigid cylinder. Indeed, in this limit Eq. (22.31) tends to... 

Ohshima [51] also derived an expression for the electroosmotic velocity in an array of parallel soft cylinders. 

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