Interaction between soft spheres

FIGURE 20.3 Interaction between two similar spherical soft particles. 

We treat the case in which the number density of molecules within the surface layer is negligibly small compared to that in the particle core. The van der Walls interaction energy is dominated by the interaction between the particle cores, while the contribution from the surface charge layer can be neglected. Thus, we have (from Eq. (19.31)) 

In this situation, F(kH) shows a maximum (F(kH) = 0.6476) in the region kH 0. In case (i), we have further three possible cases. 

In this case, Eq. (20.25) has no root, so V(kH) decreases monotonically as kFI decreases. 

In this situation, F(kH) shows no maximum, decreasing monotonically as kH 

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In a hard-sphere system, the trajectories of particles are determined by momentum conserving binary collisions. The interactions between particles are assumed to be pair-wise additive and instantaneous. In the simulation, the collisions are processed one by one according to the order in which the events occur. For not too dense systems, the hard-sphere models are considerably faster than the soft-sphere models. Note that the occurrence of multiple collisions at the same instant cannot be taken into account. 

Regarding the primary coordination sphere of iron, the identity of the axial ligand is critical to determining the chemical reactivity of heme. In general, the interaction between the axial ligand and heme iron follow the hard-soft acid-base principle... 

One model which has been extensively used to model polymers in the continuum is the bead-spring model. In this model a polymer chain consists of Nbeads (mers) connected by a spring. The easiest way to include excluded volume interactions is to represent the beads as spheres centered at each connection point on the chain. The spheres can either be hard or soft. For soft spheres, a Lennard-Jones interaction is often used, where the interaction between monomers is... 

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. 

Consider the electrostatic interaction between two dissimilar spherical soft spheres 1 and 2 (Fig. 15.3). We denote by and the thicknesses of the surface charge layers of spheres 1 and 2, respectively. Let the radius of the core of soft sphere 1 be a and that for sphere 2 be a. We imagine that each surface layer is uniformly charged. Let Zi and N, respectively, be the valence and the density of fixed-charge layer of sphere 1 and Z2 and N2 for sphere 2. 

FIGURE 15.3 Interaction between two soft spheres 1 and 2 at separation H. Spheres 1 and 2 are covered with surface charge layers of thicknesses di and CI2, respectively. The core radii of spheres 1 and 2 are and a2, respectively. 

FIGURE 15.4 Scaled electrostatic interaction energy V p(/T)= (oi+ 2)/ Usp(W) between two dissimilar soft spheres with fixed charges of like sign as a function of scaled sphere separation calculated with Eq. (15.44) at various values of Kd = Kd2 — Kd, where and C2 (defined hy Eqs. (15.46) and (15.47)) are kept constant at a2ld = 0.5. From Ref. [3]. 

Scaled electrostatic interaction energy V p(/7) = fir oK (fli+ 2)/, (H) between two dissimilar soft spheres with fixed charges of unlike sign as a... 

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. 

Several models have been developed to describe these phenomena quantitatively, the main difference being the interaction potential between the particles. There are two major approaches the hard sphere and the soft sphere. The hard sphere assumes that the only interaction between particles is a strong repulsion at the point of contact. The soft sphere is more realistic and assumes a potential with a barrier and a primary minimum like in DLVO theory (Figure 11.8). 

Star polymers are known to interact through an ultrasoft pair potential that is very different from that of the other soft spheres described above [123]. The energy of interaction between two identical stars with effective diameter <7 is of the form ... 

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