Spherical polyelectrolyte sphere

Figure 6.8 shows as a function of the ratio dia of the polyelectrolyte layer thickness d to the core radius a for two values of Q (5 and 50) at = 10 . Note that as dIa tends to zero, the polyelectrolyte-coated particle becomes a hard sphere with no polyelectrolyte layer, while as dia tends to inhnity, the particle becomes a spherical polyelectrolyte with no particle core. Approximate results calculated with Eq. (6.155) for Q = 5 (low charge case) and Eq. (6.168) for Q = 50 (high charge case) are also shown in Fig. 6.8. Agreement between exact and approximate results is good. For the low charge case, the surface potential is essentially independent of d and is determined only by the charge amount Q. In the example given in Fig. 6.8, for the high charge case, the particle behaves like a hard particle with no polyelectrolyte layer for dia 10 and the particle behaves like a spherical polyelectrolyte for dia 1.

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. 

Note that the following exact expression for the electrostatic interaction between two porous spheres (spherical polyelectrolytes) for the low charge density case has been derived [5,6] (Eq. (13.46)) ... 

If we further take the limit Kd 1 in Eq. (15.61), then we obtain the electrostatic interaction energy for the case where sphere 1 is a spherical polyelectrolyte and sphere 2 is a hard sphere, namely. 

Electrokinetic equations describing the electrical conductivity of a suspension of colloidal particles are the same as those for the electrophoretic mobility of colloidal particles and thus conductivity measurements can provide us with essentially the same information as that from electrophoretic mobihty measurements. Several theoretical studies have been made on dilute suspensions of hard particles [1-3], mercury drops [4], and spherical polyelectrolytes (charged porous spheres) [5], and on concentrated suspensions of hard spherical particles [6] and mercury drops [7] on the basis of Kuwabara s cell model [8], which was originally applied to electrophoresis problem [9,10]. In this chapter, we develop a theory of conductivity of a concentrated suspension of soft particles [11]. The results cover those for the dilute case in the limit of very low particle volume fractions. We confine ourselves to the case where the overlapping of the electrical double layers of adjacent particles is negligible. 

Water-soluble poly(acrylic acid)-based nanocapsules with reversible pH- and ionic strength-dependent swelling transition were prepared by Meier et al. [224], During this transition gated pores in the spherical polymer shells are opened (closed), which enables free molecular exchange between the interior of the hollow sphere and the bulk medium. This pH-switchable control of permeability can trigger release of encapsulated cargo from the polyelectrolyte spheres. 

The exponent b depends also on the polyelectrolyte linear charge density p and on the polyelectrolyte stiffness [130], which make it a nonuniversal characteristic for the used polyelectrolytes. Typically, polyelectrolytes with smaller p reveal a stronger dependence of <7 on The polyelectrolyte-sphere binding affinity was shown to decrease with the polymer persistence length and to increase with the polyelectrolyte linear charge density. Note that for complexation of polyelectrolytes of different p with the spherical dimethyl dodecylamine oxide (DMDAO) micelles, a modified dependence on the critical micelle charge density has been suggested, namely cT(. [128]. 

Mei Y, Wittemann A, Shartna G, et al Engineering the interaction of latex spheres with charged surfaces AFM investigation of spherical polyelectrolyte brushes on mica. Macromolecules 36 3452-3456, 2003. 

Fig. 13.3 Schematic representation of capsules prepared by LbL adsorption of oppositely charged polyelectrolytes onto spherical charged spheres, followed by dissolution of sacrificial templates. During LbL coating, various charged elements (such as metal, magnetic or fluorescent nanoparticles) can be incorporated within the multilayers to add functionahty to the capsules. Reproduced from [40] with permission from Elsevier...

MUle, M., Vandeikooi, G. Electrochemical properties of spherical polyelectrolytes. 1. Impermeable sphere model. J. Colloid Interface Sci. 1977, 59(2), 211-224. 

Several models of polyelectrolyte thermodynamics have been proposed and have been recently reviewed [84], Historically, two general approaches [84,85] have been used to model polyelectrolyte thermodynamics, spherical and chain models. In the former approach the coiled polyions are treated as spherical domains with charge density distributed continuously within the sphere... 

Hard-sphere or cylinder models (Avena et al., 1999 Benedetti et al., 1996 Carballeira et al., 1999 De Wit et al., 1993), permeable Donnan gel phases (Ephraim et al., 1986 Marinsky and Ephraim, 1986), and branched (Klein Wolterink et al., 1999) or linear (Gosh and Schnitzer, 1980) polyelectrolyte models were proposed for NOM. Here the various models must be differentiated in detail—that is, impermeable hard spheres, semipermeable spherical colloids (Marinsky and Ephraim, 1986 Kinniburgh et al., 1996), or fully permeable electrolytes. The latest new model applied to NOM (Duval et al., 2005) incorporates an electrokinetic component that allows a soft particle to include a hard (impermeable) core and a permeable diffuse polyelectrolyte layer. This model is the most appropriate for humic substances. 

FIGURE 15.1 A soft sphere becomes a hard sphere in the absence of the surface layer of polyelectrolyte while it tends to a spherical polyelectrol) te (i.e., porous sphere) when the... 

Consider the limiting case of xdi 1 and xd2 1. In this case, soft plates and soft spheres become planar polyelectrolytes and spherical... 

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,... 

Blum, L., Kalyuzhnyi, Yu.V., Bernard, O., and Herrera-Pacheco, J.N. Sticky charged spheres in the mean-spherical approximation A model for colloids and polyelectrolytes. Journal of Physics - Condensed Matter, 1996, 8, No. 25A, p. A143-A167. 

The adsorption of a polyelectrolyte onto a spherical particle has been studied using Monte Carlo simulations [35]. In particular, the critical sphere radius has been determined as a function of k. Aside from the interactions considered in our model. 

Fig. 10 Critical surface charge densities obtained by the WKB approach for polyelectrolyte adsorption onto planar, cylindrical, and spherical surfaces. The asymptotic scaling relations for a cylinder (rod) (45) and a sphere (53) are indicated by dotted lines [48]...

Comparing the results obtained by the WKB method with the exact solutions for the planar and spherical surface, we find, within 2% error, quantitative agreement in the planar case. For a sphere, we find the same asymptotic dependence of critical adsorption behavior for a wide range of geometries. The main advantage of the WKB method is a unified approach for the various geometries based on the same level of approximations. It can be applied at the same level of complexity to virtually any shape of the polylectrolyte-surface adsorption potential. Recent advances in polyelectrolyte adsorption under confinement [49,167] and adsorption onto low-dielectric interfaces [50] have been presented. 

---