Assignment
(a) A 3-D liquid crystalline fluid consists of N rod-like molecules of length l. Each rod can occupy one of V available lattice sites, i.e., the center of gravity of each rod can be at one of the V lattice sites indicated by the dots in the first figure. The rods can orient themselves arbitrarily. The number of directions each rod can point in per unit solid angle is λ. Consider a dilute system (N<
(b) The system described in part (a) is brought into contact with a zeolite (a porous material). The particular zeolite consists of many cylindrical tubes and each tube contains a 1-D array of L lattice sites lying on its center axis (as shown in the second figure).
The total number of lattice sites in all the tubes in the zeolite is Vz. The rods are free to move between the two systems so one rod can now occupy any of the lattice sites both inside and outside the zeolite. The interesting problem arises when the diameter of the tubes, d, is smaller than l, so the rods in the tubes have fewer ways to orient themselves because of the constraint of the walls of the tubes. Calculate the number of rods, M, absorbed by the zeolite in equilibrium. Hence determine a relation between the densities in the zeolite, Φz , and in the solution outside the zeolite, Φs , where the density in each region is defined as the number of rods per site in that region.
Hint for the number of directions in a 3-D space: A rod with indistinguishable ends which can orient freely within an angle Φ of a certain direction (indicated by the cones around the z axis in the figure below) has Ω possible orientational configurations, where Ω= 2πλ(1 - cos Φ).
Hint for (b): Find the number of directions a rod can point in when it is absorbed by the zeolite. Express the entropy of the whole system as a function of the number of rods absorbed by the zeolite and then find M by maximizing this entropy.