On cooperative modes of reorientation in liquid crystals

The role of cooperative modes of reorientation in liquid crystals for the spectral densities of importance in magnetic resonance relaxation is examined from the viewpoint of microscopic models that can be solved analytically. These include linear arrays of rods in an applied external field whose orientations are strongly coupled and whose motion is described by overdamped and coupled diffusion equations. Also considered are Cayley-tree models, which, at least for nearest neighbors, can mimic higher dimensions than one. All these models lead to the result that the qth normal mode of relaxation has a decay rate that depends upon q as sin2(qa/2), where a is the lattice spacing. Using simple scaling, it is argued that if a is replaced by a′≡La, so our model consists of blocks of L rodlike molecules with renormalized parameters, then molecular detail becomes less important and the model arrays become more physically representative. These results are then utilized to extend the standard hydrodynamic theory for long wave vectors q to short wave vectors approaching molecular dimension. The spectral densities obey scaling (i.e., are nearly independent of a′) and are nearly identical to the hydrodynamic result for lower frequencies. However, the new results do show small departures at the higher frequencies usually studied by NMR.