Large Spherical Soft Particles

We derive approximate mobility formulas for the simple but important case where the potential is arbitrary but the double-layer potential still remains spherically symmetrical in the presence of the applied electric field (the relaxation effect is neglected). Further we treat the case where the following conditions hold 

Equation (21.63) covers a plate-like soft particle. Indeed, in the limiting case of oo, the general mobility expression (Eq. (21.41)) reduces to 

In Fig. 21.2 we plot the function/(d/a), which varies from 2/3 to 1, as dia increases. For d a (J dla) 1), the polyelectrolyte layer can be regarded as planar, while for d a (f dla) 2/3), the soft particle behaves like a spherical polyelectrolyte with no particle core. 

Equation (21.51) consists of two terms the first term is a weighted average of the Dorman potential i/ don and the surface potential ij/o- It should be stressed that only the first term is subject to the shielding effects of electrolytes, tending zero as the electrolyte concentration n increases, while the second term does not depend on the electrolyte concentration. In the limit of high electrolyte concentrations, all the potentials vanish and only the second term of the mobility expression remains, namely. 

Equation (21.62) shows that as k co, p tends to a nonzero limiting value p°°. This is a characteristic of the electrokinetic behavior of soft particles, in contrast to the case of the electrophoretic mobility of hard particles, which should reduces to zero due to the shielding effects, since the mobility expressions for rigid particles (Chapter 3) do not have p°°. The term p°° can be interpreted as resulting from the balance between the electric force acting on the fixed charges ZeN)E and the frictional force yu, namely. 

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A question often asked is whether the parabolic energy wells as predicted by Pieranski have an activation barrier that prevents the particle from falling in spontaneously. One can argue that, especially for a large spherical particle, upon its approach to the soft interface, the interface needs to deform and liquid has to drain. This event adds an activation barrier that needs to be overcome for the particle not to bounce off the interface, and clearly the interfacial tension between the two soft bulk phases (liquid-liquid and liquid-air) and the viscosity of both phases play key roles. Note that a potential hydrophobic effect [28] can counterbalance such a barrier because the dewetting of the liquid between a hydrophobic particle and the hydrophobic liquid phase, or air, stimulates long-range attraction and eases the adhesion process. 

We argue that the above features of star dynamics are generic for soft systems of the core-shell type for which stars serve as prototype. Support for this comes from the dynamic light scattering (DLS) investigation of large block copolymer micelles, where all three relaxation modes, i.e., cooperative, structural and selfdiffusion are observed [188]. In particular, the star model discussed above applies to core-shell particles with a small spherical core relative to the chain (shell) dimensions. For a surface number density a = f / (47i r ) the polymer layer thickness under good solvent conditions is L ... 

Multiarm star polymers have recently emerged as ideal model polymer-colloids, with properties interpolating between those of polymers and hard spheres [62-64]. They are representatives of a large class of soft colloids encompassing grafted particles and block copolymer micelles. Star polymers consist of f polymer chains attached to a solid core, which plays the role of a topological constraint (Fig. Ic). When fire functionality f is large, stars are virtually spherical objects, and for f = oo the hard sphere limit is recovered. A considerable literature describes the synthesis, structure, and dynamics of star polymers both in melt and in solution (for a review see [2]). 

In Vol. 1, Sect. 3.3. we discussed how elastic and inelastic collisions contribute to the broadening and shifts of spectral lines. In a semiclassical model of a collision between partners A and B, the particle B travels along a definite path r(t) in a coordinate system with its origin at the location of A. The path r t) is completely determined by the initial conditions r(0) and (dr/df)o and by the interaction potential V(r, Ex, E-b), which may depend on the internal energies Ex and b of the collision partners. In most models a spherically symmetric potential V r) is assumed, which may have a minimum at r = ro (Fig. 8.1). If the impact parameter b is large compared to tq the collision is classified as a soft collision, while for b hard collisions occur. 

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