Hard Sphere Repulsion

In this formula, r is the interatomic separation, and e is a constant that represents the potential at the energy minimum defined by 

Because this potential includes positive and negative terms (from the attractive and repulsive forces), the curve will have a minimum point. 

FIGURE 1.5 The Lennard-Jones potential plotted as the interaction energy between two particles as a function of separation distance r. This potential represents both the van der Waals attraction and the hard sphere repulsion. 

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Gilman [124] and Westwood and Hitch [135] have applied the cleavage technique to a variety of crystals. The salts studied (with cleavage plane and best surface tension value in parentheses) were LiF (100, 340), MgO (100, 1200), CaFa (111, 450), BaFj (111, 280), CaCOa (001, 230), Si (111, 1240), Zn (0001, 105), Fe (3% Si) (100, about 1360), and NaCl (100, 110). Both authors note that their values are in much better agreement with a very simple estimate of surface energy by Bom and Stem in 1919, which used only Coulomb terms and a hard-sphere repulsion. In more recent work, however, Becher and Freiman [126] have reported distinctly higher values of y, the critical fracture energy. ... 

The interaction between ions of the same sign is assumed to be a pure hard sphere repulsion for r < a. It follows from simple steric considerations that an exact solution will predict dimerization only if i < a/2, but polymerization may occur for o/2 < L = o. However, an approximate solution may not reveal the fiill extent of polymerization that occurs in a more accurate or exact theory. Cummings and Stell [ ] used the model to study chemical association of uncharged atoms. It is closely related to the model for adliesive hard spheres studied by Baxter [70]. 

As the tip is brought towards the surface, there are several forces acting on it. Firstly, there is the spring force due to die cantilever, F, which is given by = -Icz. Secondly, there are the sample forces, which, in the case of AFM, may comprise any number of interactions including (generally attractive) van der Waals forces, chemical bonding interactions, meniscus forces or Bom ( hard-sphere ) repulsion forces. The total force... 

Figure C2.6.5. Examples of tire AO potential, equation (C2.6.12). The values of are indicated next to tire curves. The hard-sphere repulsion at r = 7 has not been drawn.

In the classical contact mode (Fig. 6a) AFM measures the hard-sphere repulsion forces between the tip and the sample. As a raster-scan drags the tip over the sample surface, the detector measures the vertical deflection of the cantilever, which indicates the local sample height. A feedback loop adjusts the position of the cantilever above the surface as it is scanned and monitors the changes in the surface height, generating a 3D image—a decisive advantage of AFM over TEM [3]. 

Before moving on to further discussions, we shall say a few words on dispersive or vdWs interactions. These are composed of an attractive force that arises from a sophisticated quantum-chemical short-range interaction of electrons, and an even shorter range, i.e., almost hard sphere repulsion of the cores of the atomic... 

The first term can be identified with a "hard-sphere" repulsion term, while the second accounts for attractive forces. The second term can be rewritten as,... 

The behaviour of the repulsive term of the lattice EOS is more complicated and will not be discussed in detail. At liquid-like densities this repulsion term is a better approximation to the hard spheres repulsion than the van der Waals repulsion term. At gas-like densities, the repulsion term of the lattice model and the van der Waals EOS have the same functional form. 

In eq 3.1, the activity coefficients appear as a result of the hard-sphere repulsions among the droplets. Since the calculations focus on the most populous aggregates, the hard-sphere repulsions will be expressed in terms of a single droplet size corresponding to the most populous aggregates. One can derive expressions for the activity coefficients y ko of a component k in the continuous phase O starting from an equation for the osmotic pressure of a hard-sphere fluid,3-4 such as that based on the Carnahan—Starling equation of state (see Appendix B for the derivation) ... 

The dispersion forces play an important role only for separations less than about 5 A. The problem is that in this region, they seem to play an enormous role (curve 1 in Figure 1). A hard-sphere repulsion (Bom interaction) is typically... 

For the values of the parameters employed (a relatively large Hamaker constant), the potential barrier is only a few kT or less hence, the apoferritin should coagulate at almost all the concentrations studied. Since experiment shows that the proteins did not coagulate, another repulsion should be present, at least al low separation distances. This repulsion, while essential for the stability of the system, did not affect much, because of its short range, the behavior of the second virial coefficient. In the calculation of the second virial coefficient, it was assumed that the distance of closest approach between apoferritin proteins cannot be less than 8 A. This value leads to a dimensionless second virial coefficient for the hard spheres repulsion of 4.8 instead of 4. 

In recent years, a number of investigators have studied the phase equilibria of simple fluids in pores by the application of density functional theory. Semina] studies were carried out by Evans and his co-workers (1985,1986). Their approach was considered to be the simplest realistic model for an inhomogeneous three-dimensional fluid . The starting point was a model intrinsic Helmholtz free energy functional F(p), with a mean-field approximation for the attractive forces and hard-sphere repulsion. As explained in Section 7.6, the equilibrium density profile of the fluid in a pore was obtained by minimizing the grand potential functional. 

The F[p(r)] functional can be separated into an ideal gas term and contributions from the repulsive and attractive forces between the adsorbed molecules (i.e. the fluid-fluid interactions). Hard-sphere repulsion and pairwise Lennard-Jones 12-6 potential are usually assumed and a mean field treatment is generally applied to the long-range attraction. However, the evaluation of the density profile of an inhomogeneous hard-sphere fluid presents a special problem since its free energy is... 

Many approaches already do this, for example, by incorporating known chemical constraints, densities or hard-sphere repulsions. Many of the emerging methods described below have this flavor, and as time goes on our ability to complex our data and our modeling approaches will only increase. 

This involves a modified core term where the hard core or hard sphere repulsion is replaced by a non-hard sphere short range repulsion which is proportional to This replaces the Debye-Hiickel core term. The Gurney term remains proportional to the volume of overlap of the co-spheres. 

The inclusion of many types of interactions is one of the respects in which this treatment is superior to the Debye-Hiickel treatment. In the Debye-Hiickel treatment the only interactions considered are the hard sphere repulsion and the long range coulombic interactions between the ions which are taken to be the only factor giving rise to non-ideality. 

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