Spherical soft particle

Consider a dilute suspension of polyelectrolyte-coated spherical colloidal particles (soft particles) in a salt-free medium containing counterions only. We assume that the particle core of radius a (which is uncharged) is coated with an ion-penetrable layer of polyelectrolytes of thickness d. The polyelectrolyte-coated particle has thus an inner radius a and an outer radius b = a + d (Fig. 6.4). We also assume that ionized groups of valence Z are distributed at a uniform density N in the poly electrolyte 

FIGURE 6.4 A polyelectrolyte-coated spherical particle (a spherical soft particle) in a free volume of radius R containing counterions only, a is the radius of the particle core. b — a + d. dis the thickness of the polyelectrolyte layer covering the particle core. (blR) equals the particle volume fraction (j . From Ref. [9]. 

We assume that each sphere is surrounded by a spherical free volume of radius R (Fig. 6.4), within which counterions are distributed so that electroneutrality as a whole is satisfied. The particle volume fraction (f is given by 

In the following we treat the case of dilute suspensions, namely, 

We denote the electric potential at a distance r from the center O of one particle by i/ (r). Let the average number density and the valence of counterions be n and —z, respectively. Then from the condition of electroneutrality in the free volume, we have 

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SPHERICAL SOFT PARTICLE 4.3.1 Low Charge Density Case... 

Transcendental equations for determining the relationship between don and jo can be obtained by using methods similar to those used for spherical soft particles, as shown below [15]. 

In this chapter, we give approximate analytic expressions for the force and potential energy of the electrical double-layer interaction two soft particles. As shown in Fig. 15.1, a spherical soft particle becomes a hard sphere without surface structures, while a soft particle tends to a spherical polyelectrolyte when the particle core is absent. Expressions for the interaction force and energy between two soft particles thus cover various limiting cases that include hard particle/hard particle interaction, soft particle/hard particle interaction, soft particle/porous particle interaction, and porous particle/porous particle interaction. 

FIGURE 20.3 Interaction between two similar spherical soft particles. 

ANALYTIC APPROXIMATIONS FOR THE ELECTROPHORETIC MOBILITY OF SPHERICAL SOFT PARTICLES... 

The electrophoretic mobility of spherical soft particles in a concentrated suspension is defined by = UIE. It must be mentioned here that the electrophoretic mobility fx in this chapter is defined with respect to the externally applied electric field E so that the boundary condition (22.8) has been employed following Levine and Neale [5]. There is another way of defining the electrophoretic mobility in the concentrated case, where the mobility /i is defined as /i = U/ E), (E) being the magnitude of the average electric field (E) within the suspension [8, 19-21]. It follows from the continuity condition of electric current that K E) = K°°E, where K and K°° are, respectively, the electric conductivity of the suspension and that of the electrolyte solution in the absence of the particles. Thus, jx and ix are related to each other by = K /K°°. For the dilute case, there is no difference between jx and jx. ... 

Consider spherical soft particles moving with a velocity U (electrophoretic velocity) in a liquid containing a general electrolyte in an applied electric field E. Each soft particle consists of the particle core of radius a covered with a polyelectrolyte layer of thickness d (Fig. 22.1). The radius of the soft particle as a whole is thus b = a + d. We employ a cell model [8] in which each sphere is surrounded by a concentric spherical shell of an electrolyte solution, having an outer radius of c such that the particle/cell volume ratio in the unit cell is equal to the particle volume fraction (j) throughout the entire suspension (Fig. 22.1), namely,... 

Consider a suspension of Vp identical spherical soft particles in a general electrolyte solution of volume V. We define the macroscopic electric field in the suspension (E), which differs from the applied electric field E. The field (E) may be regarded as the average of the gradient of the electric potential (= in the... 

Consider a spherical soft particle moving with a velocity Uexp(—icot) in a liquid containing a general electrolyte in an applied oscillating electric field E exp(—icot), where co is the angular frequency (2n times frequency) and t is time (Fig. 25.1). The dynamic electrophoretic mobility /i(co), which is a function of co, of the particle is defined by... 

For the limiting case of a 0, the particle core vanishes, so a spherical soft particle becomes a spherical polyelectrolyte. In this case, Eq. (25.27) tends to... 

Consider a dilute suspension of Np spherical soft particles moving with a velocity U exp(—/fflf) in a symmetrical electrolyte solution of viscosity r] and relative permittivity r in an applied oscillating pressure gradient field Vp exp(—imt) due to a sound wave propagating in the suspension, where m is the angular frequency 2n times frequency) and t is time. We treat the case in which m is low such that the dispersion of r can be neglected. We assume that the particle core of radius a is coated... 

FIGURE 26.3 A spherical soft particle in an applied pressure gradient field, a = radius of the particle core. = thickness of the polyelectrolyte layer covering the particle core. 

For a spherical soft particle, an approximate expression for p(co) for the dynamic electrophoretic mobility is given by Eq. (25.45), which is a good approximation when the following conditions are satisfied ... 

In this chapter, we first present a theory of the primary electroviscous effect in a dilute suspension of soft particles, that is, particles covered with an ion-penetrable surface layer of charged or uncharged polymers. We derive expressions for the effective viscosity and the primary electroviscous coefficient of a dilute suspension of soft particles [26]. We then derive an expression for the effective viscosity of uncharged porous spheres (i.e., spherical soft particles with no particle core) [27]. 

FIGURE 27.1 Function Q Xa, alb) for a suspension of uncharged spherical soft particles as a function of Xa for several values of alb. The line for a b= 1 corresponds to a suspension of spherical hard particles. From Ref. 26. 

For a suspension of charged spherical soft particles carrying low ZeN, the electroviscous coefficient p can be calculated via Eq. (27.53) as combined with Eq. (27.54) for L ku, la, alb). Figure 27.2 show some examples of the calculation of L Ka, la, alb) as a function of Ka obtained via Eq. (27.54) at la = 50. This figure shows that as Ka increases, L(Ka, la, alb) first decreases, reaching a minimum around ca. no =10, and then increases. Note that the primary electroviscous coefficient p for a suspension of hard particles (shown as a curve with alb = 1 in Eig. 27.2) does not exhibit a minimum. 

Huang PY, Keh HJ (2012) Diffusiophoresis of a spherical soft particle in electrolyte gradients. J Phys Chem B 116 7575-7589... 

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