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Numerical Solutions of the Integral Equation

Also shown in Fig.3,4 are the numerical results for the full self-consistent integral equation, (14). This we have solved iteratively after the fashion of Broyles [22]: an initial guess input function tex2html_wrap_inline980 is used in computing the right hand side of Eq.(14) giving an output function tex2html_wrap_inline982 , and the next input function is chosen to be:


where tex2html_wrap_inline984 is a constant chosen between 0 and 1 for optimal convergence of the process. For computational purposes, the function tex2html_wrap_inline884 has to be specified on a two dimensional grid tex2html_wrap_inline992 (assuming unbroken symmetry about the axis normal to the plates), with h being a fixed parameter during the iterations. The function is obtained by fitting bi-cubic splines between grid points. A fine grid means more grid points where evaluations of the function n has to be made, whereas a coarse one would not supply a satisfactory accuracy (which we define to be convergence of the force to within tex2html_wrap_inline998 of the maximum value of tex2html_wrap_inline728 for any given force curve at fixed tex2html_wrap_inline714 ).

The bottleneck of the iterative process is the integral contained in tex2html_wrap_inline886 , which is of high dimension, and makes precise evaluations very difficult. This problem is alleviated by splitting the integral tex2html_wrap_inline886 into two more straightforward ones:


where tex2html_wrap_inline1008 , tex2html_wrap_inline1010 , tex2html_wrap_inline1012 and f and g are two dimensionless functions:



Substituting f and g into Eq.(34), together with the relations between tex2html_wrap_inline1022 's, gives the original Eq.(15). Note that f(t,t') can be calculated and tabulated beforehand to speed up the iteration procedure, and tex2html_wrap_inline1026 is a simple integral with its integrand specifying a straight line segment in the tex2html_wrap_inline1028 plane.

It turns out the number of iterations needed to converge Eq.(33) is of the order 5-15 with tex2html_wrap_inline984 between 0.1 and 0.5, and rather insensitive to the initial guess function tex2html_wrap_inline980 , which is therefore always set to the constant bulk value tex2html_wrap_inline700 . Not surprisingly, the process is a lot faster for smaller concentrations ( tex2html_wrap_inline1042 ), and it gets increasingly difficult when tex2html_wrap_inline714 approaches 2, which we found to be the maximum value for which converged results could be obtained in reasonable time (5-10 hrs CPU per tex2html_wrap_inline728 value on a DEC Alpha 600/5/266). In order to have the fastest convergence, a two dimensional adaptive grid was adopted which becomes finer wherever the function n varies most rapidly.

After numerically solving Eq.(14) for tex2html_wrap_inline884 , and following Eq.(4), we can find:


We found that this leads, with a high degree of numerical accuracy and irrespective of the value of tex2html_wrap_inline714 to:


so that the resulting pressure tex2html_wrap_inline964 is once more identical to the second virial result [21, 18], which is known to be exact in the limit of tex2html_wrap_inline814 which we have adopted.

The depletion force for general separation h is then given by equations (3, 37) as:


As in the perturbative treatments of Section 3, the force between spheres of radius R is then found by the Derjaguin integral, Eq.(1); the interaction energy between two such spheres can then be found by a further integration.

next up previous
Next: Discussion Up: Theory of the Depletion Previous: Perturbation Treatment

Yong Mao
Tue Sep 17 16:40:38 BST 1996