The non-equilibrium thermodynamics of active suspensions

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.

💡 Research Summary

This paper develops a comprehensive non‑equilibrium thermodynamic framework for active suspensions composed of self‑propelled colloidal particles, with a particular focus on spherical Janus particles that generate motion through surface catalytic reactions and diffusiophoretic coupling. The authors begin by emphasizing that the internal degrees of freedom of active agents—specifically the three Euler angles that describe particle orientation—must be incorporated explicitly to capture polar, nematic, and higher‑order orientational order. Consequently, they define a six‑dimensional distribution function fC(r,α,t) that depends on the spatial position r and the orientation angles α = (θ, φ, ψ). Integrating over orientation yields the local particle number density nC(r,t); the molecular species in the surrounding fluid are described by their concentrations nκ(r,t), with κ = 0 for solvent and κ = 1…M for solutes.

The macroscopic fields—mass density ρ, momentum density g, velocity v = g/ρ, and total energy density—obey standard conservation laws in the absence of external forces. The authors write a local conservation equation for fC that includes translational fluxes and rotational fluxes (Ω·∇α fC), and they couple it to the advection‑diffusion‑reaction equations for the molecular species. Chemical reactions are written in a generic stoichiometric form, allowing for surface‑catalyzed steps (νC = 0) and bulk steps. From these balance equations they derive a local entropy balance, identifying the entropy production σ as a sum of products of thermodynamic forces (affinities) X and corresponding dissipative currents J.

The next step is to formulate the local Gibbs and Euler relations as functionals of fC and the molecular densities. By taking functional derivatives they obtain explicit expressions for the chemical potentials, temperature, and a torque-like conjugate to the Euler angles. Linear irreversible thermodynamics is then applied: the currents are expressed as linear combinations of the forces, J = L·X, where the phenomenological matrix L is constrained by Curie’s symmetry principle (which forbids couplings between tensors of incompatible rank) and by the Onsager‑Casimir reciprocal relations (Lij = εi εj Lji, with εi the time‑reversal parity). This systematic symmetry analysis reveals that, unlike isotropic passive fluids, active suspensions with anisotropic particles permit mechano‑chemical cross‑couplings because the particle surface breaks the symmetry that would otherwise suppress them.

Specializing to an isothermal, incompressible, dilute suspension, the authors invoke the classic results of Brenner and co‑workers for translational and rotational hydrodynamic friction of arbitrary‑shaped particles. They introduce surface‑specific parameters: catalytic reaction rate constants, diffusiophoretic mobility coefficients, Navier slip length, and particle radius. Using these, they compute the translational propulsion speed v0, the effective viscosity ηeff, and the rotational diffusion‑translation coupling tensors. The full entropy production rate is then written explicitly as a sum of four contributions: (i) chemical reaction dissipation (reaction affinity times reaction rate), (ii) diffusive dissipation (concentration gradients times diffusive fluxes), (iii) viscous dissipation (velocity gradients times stress), and (iv) mechano‑chemical coupling dissipation (terms involving the product of orientation‑dependent fluxes and chemical affinities). Each term is expressed in terms of measurable physical quantities, making the theory directly testable.

The paper proceeds to a detailed case study of spherical Janus particles with a catalytic hemisphere and an inert hemisphere. By solving the chemohydrodynamic problem around a single particle, the authors obtain analytical expressions for the concentration fields, the slip velocity at the surface, and the resulting self‑propulsion velocity. They also calculate the torque generated by asymmetric reaction fluxes, which leads to a steady rotation of the particle’s orientation distribution. The derived constitutive relations predict how the propulsion speed scales with the surface reaction rate, the diffusiophoretic mobility, and the particle size, reproducing experimentally observed speeds of order 10 µm s⁻¹ for Pt‑capped polystyrene spheres in hydrogen‑peroxide solutions.

In the concluding section the authors outline extensions of their framework: inclusion of particle‑particle hydrodynamic interactions at higher concentrations, treatment of compressible or non‑isothermal conditions, and generalization to non‑spherical (e.g., helicoidal) particles where additional coupling tensors appear. They also suggest experimental protocols for measuring the various phenomenological coefficients (e.g., using tracer particles to probe effective viscosity, or optical tweezers to quantify torque) and for validating the predicted entropy production via calorimetric or fluorescence‑based reaction rate measurements.

Overall, the work provides a rigorous, symmetry‑guided, and quantitatively complete thermodynamic description of active suspensions, bridging microscopic chemohydrodynamic mechanisms and macroscopic transport equations, and offering a solid foundation for future theoretical, computational, and experimental studies of active matter.