Soft spheres interactions

Aggregates are viewed as soft spheres in such a manner that, over a range of distance r - r, the interparticle force increases with the displacement from the equilibrium distance r  

Dynamic modulus varies with strain because of a deagglomeration-reagglomeration process of soft spheres. 

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Early molecular dynamics simulations focused on spherically shaped particles in zeolites. These particles were either noble gases, such as argon, krypton, and xenon, or small molecules like methane. For these simulations, the sorbates were treated as soft spheres interacting with the zeolite lattice via a Lennard-Jones potential. Usually the aluminum and silicon atoms in the framework were considered to be shielded by the surrounding oxygen atoms, and no aluminum and silicon interactions with the sorbates were included. The majority of those studies have concentrated on commercially important zeolites such as zeolites A and Y and silicalite (all-silica ZSM-5), for which there is a wealth of experimental information for comparison with computed properties. 

In the previous section, non-equilibrium behaviour was discussed, which is observed for particles with a deep minimum in the particle interactions at contact. In this final section, some examples of equilibrium phase behaviour in concentrated colloidal suspensions will be presented. Here we are concerned with purely repulsive particles (hard or soft spheres), or with particles with attractions of moderate strength and range (colloid-polymer and colloid-colloid mixtures). Although we shall focus mainly on equilibrium aspects, a few comments will be made about the associated kinetics as well [69, 70]. 

Charged particles in polar solvents have soft-repulsive interactions (see section C2.6.4). Just as hard spheres, such particles also undergo an ordering transition. Important differences, however, are that tire transition takes place at (much) lower particle volume fractions, and at low ionic strengtli (low k) tire solid phase may be body centred cubic (bee), ratlier tlian tire more compact fee stmcture (see [69, 73, 84]). For tire interactions, a Yukawa potential (equation (C2.6.11)1 is often used. The phase diagram for the Yukawa potential was calculated using computer simulations by Robbins et al [851. 

The hard sphere (HS) interaction is an excellent approximation for sterically stabilized colloids. However, there are other interactions present in colloidal systems that may replace or extend the pure HS interaction. As an example let us consider soft spheres given by an inverse power law (0 = The energy scale Vq and the length scale cr can be com-... 

DPMs offer a viable tool to study the macroscopic behavior of assemblies of particles and originate from MD methods. Initiated in the 1950s by Alder and Wainwright (1957), MD is by now a well-developed method with thousands of papers published in the open literature on just the technical and numerical aspects. A thorough discussion of MD techniques can be found in the book by Allen and Tildesley (1990), where the details of both numerical algorithms and computational tricks are presented. Also, Frenkel and Smit (1996) provide a comprehensive introduction to the recipes of classical MD with emphasis on the physics underlying these methods. Nearly all techniques developed for MD can be directly applied to discrete particles models, except the formulation of particle-particle interactions. Based on the mechanism of particle-particle interaction, a granular system may be modeled either as hard-spheres or as soft-spheres. ... 

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. 

In the following, we focus on the soft-sphere method since this really is the workhorse of the DPMs. The reason is that it can in principle handle any situation (dense regimes, multiple contacts), and also additional interaction forces—such as van der Waals or electrostatic forces—are easily incorporated. The main drawback is that it can be less efficient than the hard-sphere model. 

Fig. P4.9 The soft-sphere DEM method for a two-dimenstional assembly, demonstrated by the interaction of two disks, x and y. Positive Fn and Fs are as shown.

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 Chapter 11, we derived the double-layer interaction energy between two parallel plates with arbitrary surface potentials at large separations compared with the Debye length 1/k with the help of the linear superposition approximation. These results, which do not depend on the type of the double-layer interaction, can be applied both to the constant surface potential and to the constant surface charge density cases as well as their mixed case. In addition, the results obtained on the basis of the linear superposition approximation can be applied not only to hard particles but also to soft particles. We now apply Derjaguin s approximation to these results to obtain the sphere-sphere interaction energy, as shown below. 

If sphere i were not hard but ion-penetrable (a soft sphere), with sphere j i,j = 1, 2 i j) being a hard sphere with constant surface charge density, then the interaction energy would be equal to the sum of only V ° R) and l ), namely,... 

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... 

As in the case of two interacting soft plates, when the thicknesses of the surface charge layers on soft spheres 1 and 2 are very large compared with the Debye length 1/k, the potential deep inside the surface charge layer is practically equal to the Donnan potential (Eqs. (15.51) and (15.52)), independent of the particle separation H. In contrast to the usual electrostatic interaction models assuming constant surface potential or constant surface... 

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