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Extended dynamical density functional theory for nonisothermal binary systems including momentum density

Michael te Vrugt, Hartmut Löwen, Helmut R. Brand, Raphael Wittkowski

Abstract

In order to describe the nonisothermal dynamics of two-phase flows or binary mixtures such as colloidal suspensions consisting of colloidal particles and solvent on a microscopic level, we derive a new extended dynamical density functional theory (EDDFT) that includes the total mass density, the local concentration of one species, the total momentum density, and the energy density as variables using the Mori-Zwanzig-Forster projection operator technique. Through the incorporation of the momentum density into EDDFT, not only the diffusive but also the convective dynamics is taken into account. We derive an exact entropy and free-energy functional for the case of hard spheres. The hydrodynamic limit of our new EDDFT and its relation to the mode-coupling theory of the glass transition are discussed. It is shown that EDDFT allows to obtain the correct value for the speed of sound.

Extended dynamical density functional theory for nonisothermal binary systems including momentum density

Abstract

In order to describe the nonisothermal dynamics of two-phase flows or binary mixtures such as colloidal suspensions consisting of colloidal particles and solvent on a microscopic level, we derive a new extended dynamical density functional theory (EDDFT) that includes the total mass density, the local concentration of one species, the total momentum density, and the energy density as variables using the Mori-Zwanzig-Forster projection operator technique. Through the incorporation of the momentum density into EDDFT, not only the diffusive but also the convective dynamics is taken into account. We derive an exact entropy and free-energy functional for the case of hard spheres. The hydrodynamic limit of our new EDDFT and its relation to the mode-coupling theory of the glass transition are discussed. It is shown that EDDFT allows to obtain the correct value for the speed of sound.
Paper Structure (25 sections, 137 equations, 1 figure)

This paper contains 25 sections, 137 equations, 1 figure.

Table of Contents

  1. Introduction
  2. EDDFT including momentum density
  3. Microscopic dynamics
  4. Mori-Zwanzig-Forster projection operator technique
  5. Extended dynamical density functional theory
  6. Momentum density
  7. Entropy, free energy, and adiabatic approximation
  8. Entropy functional
  9. Free-energy functional
  10. Adiabatic approximation for the pressure
  11. Special cases and relation to MCT
  12. Isothermal limit
  13. Hydrodynamic limit
  14. One-component system
  15. Relation to MCT
  16. ...and 10 more sections

Figures (1)

  • Figure 1: Overview over the various levels of approximation in the dynamic equations. The transport equation \ref{['eq:EDDFT_rhom']} for the mass density is exact and left unchanged by any approximation, such that the identical \ref{['eq:EDDFT_rhom_HL1', 'eq:EDDFTm_rhom', 'eq:EDDFTm_rhom_HL1T']} are not listed here. Note that $c$ is not a relevant variable for one-component systems.