Hard sphere limit

Home, D.S. (2003). Casein micelles as hard spheres limitations of the model in acidified gel formation. Colloids and Surfaces A Physicochemical and Engineering Aspects, 213,255-263. 

In this section, we develop the formulation of the nonbonded interactions in the hard sphere limit. Although the simulations are based on the repulsive part of unb r) as defined in Eq. (5), the hard sphere formulation permits a simpler physical interpretation that is valuable. 

By the same mathematical technique used in the theory of simple fluids1 the hard-sphere limit for Eq. (24)... 

Stillinger, F. H., and Weber, T. A., Hidden structure in liquids. Phys. Rev. A 25, 978 (1982). Stillinger, F. H., and Weber, T. A., Inherent structure-theory of liquids in the hard-sphere limit. 

Multiarm star polymers have recently emerged as ideal model polymer-colloids, with properties interpolating between those of polymers and hard spheres [62-64]. They are representatives of a large class of soft colloids encompassing grafted particles and block copolymer micelles. Star polymers consist of f polymer chains attached to a solid core, which plays the role of a topological constraint (Fig. Ic). When fire functionality f is large, stars are virtually spherical objects, and for f = oo the hard sphere limit is recovered. A considerable literature describes the synthesis, structure, and dynamics of star polymers both in melt and in solution (for a review see [2]). 

Other residual properties can be obtained via Legendre transforms. Note that in the zero-density limit, these residual properties all go to zero, as they should. Further, in the hard-sphere limit a = 0) these expressions revert to the Carnahan-Starling expressions (4.5.5)-(4.5.7), as they should. 

Another parameter that is sensitive to the macromolecular architecture is the Huggins constant, / h, calculated by dilute solution viscometry measurements. For linear polymers, kn varies between 0.5 and 0.6 in 0-solvents and is equal to 0.3 in good solvents. For star polymers, in both good and 0-solvents, the values of fen increase with inaeasing / from the value obtained for linear polymers and reach values slightly lower than unity, which is the hard sphere limit (Tables 3 and 4). 

Figure 37 Relative zero-shear viscosity (normalized to the solvent tis) as a function of the effective volume fraction

An example of this scaling is shown in Figure 2. The reason for this can be shown to be rigorous in the hard-sphere limit by formal expansion of the time correlation function about / = 0. For a time correlation function, C(t) (normalised so that C(0) = 1) we find... 

In most practical cases this fixed point is reached quickly. For example, widi g = 3 and m = 4 we are already dealing with subunits of 3 100 monomers, for which the hard sphere limit usually holds well. 

An interesting point concerning the encounter theory is that the results remain well defined even in the "sticky" hard sphere limit, in which... 

Figure 2.1 Trajectories for two hypothetical spherical molecules. Two trajectories are shown by arrows. Trajectory I (dashed line) is the trajectory that would be taken in the absence of any attractive intermolecular forces, which would see the two molecules miss each other. This trajectory is modified by long range attractive forces (trajectory 2, solid line) which create an effective collision cross section that is larger than the hard-sphere limit. In this particular case, the modification leads to a successful collision. The effective increase in collision cross section over the hard sphere limit (nfrj + t2) where r, and t2 are the radii of the spherical particles) is very significant if one of the molecules is charged, since this can give rise to substantial charge-induced dipole attractive forces.

From the shapes of the curves in Figures 8.14-8.16, we can conclude that the JKR model is already a good approximation for values of P, > 3. To justify the use of the DMT model, the value of p, should be at least lower than 0.1 but better even lower than 0.01. Especially in Figure 8.16, it becomes obvious that prediction of the DMT model that the rupture of the contact occurs at zero contact radius will be fulfilled only in the limit of pn —> 0, which corresponds to the hard sphere limit of Bradlqr. 

Figure 11 clearly shows that Ry/Rg increases from less than unity to the hard sphere limit as the number of arms in the star increases. The values of Rj/Rq for the 0 case are always slightly higher than for the good solvent. Note also that the ratio Ry/Rc for 64- and 128-arm stars is larger than the hard sphere limit. Plots of RfjlRc as a fimction of the star functionality are essentially identical to the Ry/Rg plot in Figure 11. For linear polymers Ry/Rffhas been found to vary between 1.13 [40,43] and 1.03 [35] but for stars with many arms Rjj = within the limits of experimental error [21,33,40].

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