Electrostatic Interaction Between Soft Particles

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. 

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The potential distribution outside the surface charge layer of a soft particle with surface potential j/g is the same as the potential distribution around a hard particle with a surface potential xj/g. The asymptotic behavior of the potential distribution around a soft particle and that for a hard particle are the same provided they have the same surface potential xj/o- The effective surface potential is an important quantity that determines the asymptotic behaviors of the electrostatic interaction between soft particles (see Chapter 15). 

In Chapter 1, we have discussed the potential and charge of hard particles, which colloidal particles play a fundamental role in their interfacial electric phenomena such as electrostatic interaction between them and their motion in an electric field [1 ]. In this chapter, we focus on the case where the particle core is covered by an ion-penetrable surface layer of polyelectrolytes, which we term a surface charge layer (or, simply, a surface layer). Polyelectrolyte-coated particles are often called soft particles [3-16]. It is shown that the Donnan potential plays an important role in determining the potential distribution across a surface charge layer. Soft particles serve as a model for biocolloids such as cells. In such cases, the electrical double layer is formed not only outside but also inside the surface charge layer Figure 4.1 shows schematic representation of ion and potential distributions around a hard surface (Fig. 4.1a) and a soft surface (Fig. 4.1b). 

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. 

The elastic stress may be external or internal. External stresses are exerted on the chromatin during the cell cycle when the mitotic spindle separates chromosome pairs. The 30-nm fiber should be both highly flexible and extensible to survive these stresses. The in vitro experiments by Cui and Bustamante demonstrated that the 30-nm fiber is indeed very soft [66]. The 30-nm fiber is also exposed to internal stresses. Attractive or repulsive forces between the nucleosomes will deform the linkers connecting the nucleosomes. For instance, electrostatic interactions, either repulsive (due to the net charge of the nucleosome core particles) or attractive (bridging via the lysine-rich core histone tails [49]) could lead to considerable structural rearrangements. 

Though the electrostatic interactions certainly predominate the bonds between charged particles in the gaseous state, the covalent contribution may nevertheless not be negligible, especially not in bonds formed between acceptors and donors of otherwise markedly soft behaviour, i. e. of especially high covalent bonding capacity. 

The stability of colloidal systems consisting of charged particles can be essentially explained by the Derjaguin-Landau-Verwey-Overbeek (DLVO) theory [1-7]. According to this theory, the stability of a suspension of colloidal particles is determined by the balance between the electrostatic interaction and the van der Waals interaction between particles. A number of studies on colloid stability are based on the DLVO theory. In this chapter, as an example, we consider the interaction between lipid bilayers, which serves as a model for cell-cell interactions [8, 9]. Then, we consider some aspects of the interaction between two soft spheres, by taking into account both the electrostatic and van der Waals interactions acting between them. 

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