Mode coupling approach to the ideal glass transition of molecular liquids: Linear molecules

The mode coupling theory (MCT) for the ideal liquid glass transition, which was worked out for simple liquids mainly by Götze, Sjögren, and their co-workers, is extended to a molecular liquid of linear and rigid molecules. By use of the projection formalism of Zwanzig and Mori an equation of motion is derived for the correlators $S_{{lm,l}^{\prime }{m}^{\prime }}(\mathbf{q},t)$ of the tensorial one-particle density ${\rho }_{\mathrm{lm}}(\mathbf{q},t),$ which contains the orientational degrees of freedom for $l>\mathrm{}0.\mathrm{}$ Application of the mode coupling approximation to the memory kernel results into a closed set of equations for $S_{{lm,l}^{\prime }{m}^{\prime }}(\mathbf{q},t),$ which requires the static correlators $S_{{lm,l}^{\prime }{m}^{\prime }}\left(\mathbf{q}\right)$ as the only input quantities. The corresponding MCT equations for the nonergodicity parameters $f_l^m\mathrm{}\left(q\right)\mathrm{}\equiv f_{lm,lm}\left(q{\mathbf{e}}_3\right)$ are solved for a system of dipolar hard spheres by restricting the values for $l$ to 0 and 1. Depending on the packing fraction φ and on the temperature $T,$ three different phases exist: a liquid phase, where translational (TDOF’s) $\mathrm{}(l=0\mathrm{})\mathrm{}$ and orientational (ODOF’s) $\mathrm{}(l=1\mathrm{})\mathrm{}$ degrees of freedom are ergodic, a phase where the TDOF are frozen into a (nonergodic) glassy state, whereas the ODOF’s remain ergodic, and finally a glassy phase where both, TDOF’s and ODOF’s, are nonergodic. From the nonergodicity parameters $f_0^0\mathrm{}\left(q\right)\mathrm{}$ and $f_1^1\mathrm{}\left(q\right)\mathrm{}$ for $q=0,$ we may conclude that the corresponding relaxation strength of the α peak of the compressibility can be much smaller than the corresponding strength of the dielectric function.