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The non-equilibrium thermodynamics of active suspensions

Pierre Gaspard

TL;DR

The study extends non-equilibrium thermodynamics to active suspensions by incorporating the orientation degrees of freedom of colloidal particles, formulating a six-dimensional distribution function $f_{\rm C}({\bf r}, {\pmb{\alpha}}, t)$ and deriving the local Gibbs–Euler relations to obtain the entropy production. It derives complete non-equilibrium constitutive relations, constrained by Curie symmetry and Onsager–Casimir reciprocity, and provides explicit expressions for the entropy production rate, including couplings between chemical reactions, diffusiophoresis, diffusion, and momentum transport. The authors apply the framework to isothermal, incompressible, dilute suspensions of spherical Janus particles, calculating diffusion coefficients, self-diffusiophoretic propulsion, interfacial transport coefficients, and the full entropy production density, as well as the dynamics of polar and nematic order via moments of the colloidal distribution. These results offer a predictive thermodynamic lens to quantify energy transduction, efficiency, and inter-process couplings in active matter, and set the stage for extensions to other particle shapes, external fields, and denser regimes with potential instabilities and phase behavior.

Abstract

Active suspensions composed of self-propelled colloidal particles are considered. Their propulsion of is generated by chemical reactions occurring by heterogeneous catalysis and diffusiophoresis coupling the concentration gradients of reacting molecular species to the fluid velocity. By this mechanism, chemical free energy is transduced into mechanical motion. The non-equilibrium thermodynamics of such active suspensions is developed by explicitly taking into account the internal degrees of freedom of active particles, which are the Eulerian angles specifying their orientation. Accordingly, the distribution function of colloidal particles is defined in the six-dimensional configuration space of their position and their orientation, which fully characterises polar, nematic, and higher orientational orders in the active system. The local Gibbs and Euler thermodynamic relations are expressed in terms of the colloidal distribution function, the dynamics of which is ruled by a six-dimensional local conservation equation. All the processes contributing to the entropy production rate are derived from the local conservation and kinetic equations for colloids, molecular species, mass, linear momentum, and energy, identifying their thermodynamic forces, also called affinities, and their dissipative current densities. The non-equilibrium constitutive relations are obtained using the Curie symmetry principle and the Onsager-Casimir reciprocal relations based on microreversibility. In this way, all the mechanochemical coupling coefficients are completely determined for isothermal, incompressible, dilute suspensions composed of spherical Janus particles on the basis of the interfacial properties between the fluid solution and the solid particles and chemohydrodynamics. The complete expression of the entropy production rate is established for such active systems.

The non-equilibrium thermodynamics of active suspensions

TL;DR

The study extends non-equilibrium thermodynamics to active suspensions by incorporating the orientation degrees of freedom of colloidal particles, formulating a six-dimensional distribution function and deriving the local Gibbs–Euler relations to obtain the entropy production. It derives complete non-equilibrium constitutive relations, constrained by Curie symmetry and Onsager–Casimir reciprocity, and provides explicit expressions for the entropy production rate, including couplings between chemical reactions, diffusiophoresis, diffusion, and momentum transport. The authors apply the framework to isothermal, incompressible, dilute suspensions of spherical Janus particles, calculating diffusion coefficients, self-diffusiophoretic propulsion, interfacial transport coefficients, and the full entropy production density, as well as the dynamics of polar and nematic order via moments of the colloidal distribution. These results offer a predictive thermodynamic lens to quantify energy transduction, efficiency, and inter-process couplings in active matter, and set the stage for extensions to other particle shapes, external fields, and denser regimes with potential instabilities and phase behavior.

Abstract

Active suspensions composed of self-propelled colloidal particles are considered. Their propulsion of is generated by chemical reactions occurring by heterogeneous catalysis and diffusiophoresis coupling the concentration gradients of reacting molecular species to the fluid velocity. By this mechanism, chemical free energy is transduced into mechanical motion. The non-equilibrium thermodynamics of such active suspensions is developed by explicitly taking into account the internal degrees of freedom of active particles, which are the Eulerian angles specifying their orientation. Accordingly, the distribution function of colloidal particles is defined in the six-dimensional configuration space of their position and their orientation, which fully characterises polar, nematic, and higher orientational orders in the active system. The local Gibbs and Euler thermodynamic relations are expressed in terms of the colloidal distribution function, the dynamics of which is ruled by a six-dimensional local conservation equation. All the processes contributing to the entropy production rate are derived from the local conservation and kinetic equations for colloids, molecular species, mass, linear momentum, and energy, identifying their thermodynamic forces, also called affinities, and their dissipative current densities. The non-equilibrium constitutive relations are obtained using the Curie symmetry principle and the Onsager-Casimir reciprocal relations based on microreversibility. In this way, all the mechanochemical coupling coefficients are completely determined for isothermal, incompressible, dilute suspensions composed of spherical Janus particles on the basis of the interfacial properties between the fluid solution and the solid particles and chemohydrodynamics. The complete expression of the entropy production rate is established for such active systems.

Paper Structure

This paper contains 48 sections, 155 equations, 2 figures, 1 table.

Table of Contents

  1. Introduction
  2. Macroscopic description of active suspensions
  3. Mesostructure and mesodynamics of active suspensions
  4. The macrofields of the suspension
  5. The distribution function of the colloidal particles
  6. The densities of molecular species
  7. The mass density
  8. The velocity field
  9. The total and internal energy densities
  10. The local conservation and balance equations
  11. The local conservation equation for the colloidal particles
  12. The local balance equations for the molecular species
  13. The local conservation equation for mass
  14. The local conservation equation for linear momentum
  15. The local conservation equation for the total energy
  16. ...and 33 more sections

Figures (2)

  • Figure 1: Illustrative example of an active suspension, which is here composed of Janus particles. These axisymmetric particles are self-propelled through the surrounding fluid by chemical reactions occurring at their catalytic hemisphere and by diffusiophoresis coupling the concentration gradients of the reactants to the fluid velocity. The case of Janus particles is considered in section \ref{['sec:Janus']}.
  • Figure 2: Schematic representation of a Janus particle of radius $R$ with catalytic (c) and non-catalytic (n) hemispheres. The chemical reaction ${\rm A} \, \rightleftharpoons \, {\rm B}$ is catalysed by the c-hemisphere. $\bf u$ denotes the unit vector along the axis of axisymmetry and pointing in the direction of the catalytic hemipshere.