Anisotropic colloidal particles in critical fluids
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18.In particular, we consider a particle size much larger than the extrapolation length l of the particle surface (Refs. 1, 2, 4, and 5). We do not discuss crossover phenomena which may occur for particle surfaces with a weak preference for one of the two solvent phases and where l is of mesoscopic size. To estimate the effect of the extrapolation length on density profiles or correlation functions, consider them for fixed point conditions but for a boundary surface which is shifted by l. For a symmetric dumbbell of two touching spheres with positive l, thus take the surfaces of two slightly overlapping spheres with a diameter L+2l and center-to-center distance L, intersecting in a circle with diameter and at an angle see Eqs. (5.11) and (5.12). Concentrating, for simplicity, on the symmetry plane of the dumbbell, the normalized profile is then given by the product of and see Eq. (5.14). This confirms the expectation that l drops out and the fixed point profile (2.28) of two touching spheres is reproduced provided (i) the particle size is much larger than l so that the first factor tends to see Eq. (5.11), (5.3), or (5.8) and (ii) r is much farther from the sphere surfaces than l so that and the second factor tends to
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30.This relation between the free energy and the stress tensor average (2.16) of the film specifies our normalization of the stress tensor. It coincides with the normalization in Ref. 28 and implies that the integral of the average in the half space over a plane with distance from the surface which is smaller than the distance equals the derivative
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32.For a sphere the small particle expansion has been checked for temperatures above see, e.g., Refs. 9, 16, and 17.
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