Density corrections to transport coefficients from time correlation functions
Creators: name: Alavi, Saman
Publisher: name: University of British Columbia
Year: 2009

description: A new method for deriving first order density corrections to transport coefficients using projection operators in the time correlation function formalism is developed. Low and moderately dense gas transport coefficients are standardly calculated from a form of the generalized Boltzmann equation. This equation being solved to first order density corrections for repulsive potentials at the binary collision level by Snider and Curtiss and later extended to include the effects associated with the static presence of a third particle on a binary collision by Hoffman and Curtiss. Rainwater and Friend added extra contributions for the presence of bound pairs when the molecules have an attractive potential. They utilized the Stogryn - Hirschfelder theory for the bound pairs and performed detailed numerical calculations of the resultant formulas. While the numerical calculations give good agreement with experiment, questions remain as to the nature and rigor of the assumptions made in obtaining the final formulas, especially the ad hoc addition of bound pair contributions to the density corrections of systems with repulsive potentials, and the extent that these approximations affect the final numerical results. To study these questions, the time correlation function formulas for the transport coefficients were chosen as an alternative route to determine first order density corrections. The time correlation formulas are formally exact and so the density corrections can be usefully compared to those of the generalized Boltzmann equation. Kawasaki and Oppenheim had previously derived formal expressions for first order density corrections to the shear viscosity for a gas of molecules with a repulsive potential, but their results had not been reduced to a form that could be directly compared to those of Snider and Curtiss. As a first step in the study of the time correlation function formalism, the density corrections of Kawasaki and Oppenheim are shown to be equivalent to those of Snider and Curtiss along with an additional correction for three-body collisions. The projection operator method developed in this thesis does not have the infinite series resummation procedure used by Kawasaki and Oppenheim and is an alternative route to obtaining density corrections from the time correlation functions. At low pressures, projection operators are defined which only consider kinetic contributions to the flux function and expressions for the lowest order transport coefficients along with their higher moment corrections are derived. These expressions are consistent with the solution of the Boltzmann equation. The first order density correction from bound pairs on the transport coefficients are approximated by treating the system as a binary gas mixture consisting of free molecules and bound pairs. The results of viewing the system from the molecular picture and the atomic picture with appropriate projection operators are shown to be consistent with one another and also with the Boltzmann equation for binary mixtures. Density corrections in moderately dense gases also arise from potential contributions to the flux. Projection operators which account for both the kinetic and potential flux contributions are required in order to derive explicit expressions for the first order density corrections to the viscosity and thermal conductivity. It is observed that these corrections are consistent with those of Snider and Curtiss with the added Hoffman and Curtiss correction and a term which takes explicit account of three-particle collisions. In the treatment of mixtures and potential interaction effects, the calculation of a transport coefficient is reduced to an equivalent matrix inversion problem. The binary collision expansion of the respective resolvent in the matrix elements in these formulas allows the transport coefficient to be expressed in terms of integrals over functions of the intermolecular potential. The projection operator for each system is determined in a straightforward manner with reference to the particular flux tensor in the time correlation formula. Reduction of the general formula to relations suitable for numerical calculation involves the resolvent expansion onto the appropriate projected subspace, and the subsequent binary collision expansion to reduce the iV-particle resolvent to a tractable form.