Hard Rods between Two Walls

Another relatively easy modeling is for hard rods confined between two walls. The mathematics is a little messier and will not be completely given here (see Davis [1] or other statistical mechanic books). The modeling can also include an external field, which is also instructive. Using y and z for the position of the walls, Q, the canonical partition function for this case is 

With a considerable amount of reworking, a reformulation of this is obtained in what is referred to as the p formulation. The p formulation separates the solution into two solutions, one from each wall. The solutions are 

The solution to these equations is rather messy because of the shifts in x that are required. Notice that in Eq. (265) there is a shift from x a/2 to X. Numerical techniques are obviously called for to perform this calculation. Restricting the calculation to one-sized rod is relatively simple for a spreadsheet calculation. Fig. 116 shows a series of calculations for various slit widths (varying distance between the walls) with the chemical potential, temperature held constant and the externally imposed potential, v(x), set to zero. For this calculation, one wall was held as jc=0 and the other wall was allowed to move. Since the center of the rod caimot approach any closer to the fixed wall than the distance a/2, n is zero up to this point. A similar comment is in order for the wall that is allowed to move. 

With the above equations it is simple to add in an external potential to see how the adsorption is affected. In Fig. 117 a catenary potential has been added, that is 

As one would expect, the density is suppressed at the walls and enhanced in the middle with such a field present. 

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