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Density corrections to transport coefficients from time correlation functions Alavi, Saman
Abstract
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 threebody 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 threeparticle 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 iVparticle resolvent to a tractable form.
Item Metadata
Title 
Density corrections to transport coefficients from time correlation functions

Creator  
Publisher 
University of British Columbia

Date Issued 
1999

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 threebody 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 threeparticle 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 iVparticle resolvent to a tractable form.

Extent 
9156034 bytes

Genre  
Type  
File Format 
application/pdf

Language 
eng

Date Available 
20090702

Provider 
Vancouver : University of British Columbia Library

Rights 
For noncommercial purposes only, such as research, private study and education. Additional conditions apply, see Terms of Use https://open.library.ubc.ca/terms_of_use.

DOI 
10.14288/1.0061538

URI  
Degree  
Program  
Affiliation  
Degree Grantor 
University of British Columbia

Graduation Date 
199911

Campus  
Scholarly Level 
Graduate

Aggregated Source Repository 
DSpace

Item Media
Item Citations and Data
Rights
For noncommercial purposes only, such as research, private study and education. Additional conditions apply, see Terms of Use https://open.library.ubc.ca/terms_of_use.