Electroosmotic Velocity in an Array of Soft Cylinders

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 j , namely, 

The origin of the cyhndrical coordinate system (r, 0, z) is held fixed on the axis of one cylinder. The polar axis (6 = 0) is set parallel to the applied electric field E so that E is perpendicular to the cylinder axis. Let the electrolyte be composed of M ionic mobile species of valence zt and drag coefficient 2,- (i = 1, 2,. . . , M), and be the concentration (number density) of the /th ionic species in the 

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. 

Consider several limiting cases. In the limit 0, Eq. (22.31) becomes 

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 

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