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Active matter, comprising many active agents interacting and moving in fluids or more complex environments, is a commonly occurring state of matter in biological and physical systems. By its very nature, active matter systems exist in nonequilibrium states. In this paper, the active agents are small Janus colloidal particles that use chemical energy provided by chemical reactions occurring on their surfaces for propulsion through a diffusiophoretic mechanism. As a result of interactions among these colloids, either directly or through fluid velocity and concentration fields, they may act collectively to form structures such as dynamic clusters. A general nonequilibrium thermodynamics framework for the description of such systems is presented that accounts for both self-diffusiophoresis and diffusiophoresis due to external concentration gradients, and is consistent with microreversibility. It predicts the existence of a reciprocal effect of diffusiophoresis back onto the reaction rate for the entire collection of colloids in the system, as well as the existence of a clustering instability that leads to nonequilibrium inhomogeneous system states.

Active matter is composed of motile entities or agents interacting with each other either directly or through the velocity and concentration fields of the medium in which they move. Such interactions lead to collective dynamics giving rise to states of matter that may differ from those in equilibrium systems. The study of such collective behavior presents challenges and is currently a topic of considerable scientific interest. Systems with many complex agents can be investigated in different ways. One way is to describe collective dynamics at the macroscale in terms of fields representing the distribution of the agents across the system. These fields are ruled by partial differential equations that are established using general symmetries and experimental observations. Another approach is to model active matter as being composed of active particles moving in space according to specific rules that are postulated on the basis of empirical considerations.

Both of these approaches have been used to explore the origins and types of collective dynamics that can be found in active matter systems, and research on this topic ranges from studies of simple active particle models, often satisfying minimal rules, to suspensions of more complex active synthetic or biological agents [

Systems containing colloidal particles are governed by physicochemical laws, so that their time evolution can be understood from first principles using statistical-mechanical methods. This approach was pioneered by Einstein [

The present paper contributes to the statistical-mechanical and nonequilibrium thermodynamic approaches for active matter systems [

Next, the evolution equation is established for the distribution function of the ensemble of active particles in a dilute colloidal solution. In order to be consistent with microreversibility, the principles of nonequilibrium thermodynamics are used to relate the thermodynamic forces or affinities to the current densities with linear response coefficients satisfying Onsager’s reciprocal relations [

The paper is organized as follows. Section

This section describes the motion of a single spherical Janus colloidal motor of radius

Schematic representation of a Janus particle with its catalytic (C) and noncatalytic (N) hemispheres where the surface reaction (

In order to determine the force and the torque due to diffusiophoresis, as well as the overall reaction rate, the velocity of the fluid and the concentrations of fuel

The coupling between the velocity and concentration fields is established with the boundary conditions [

The velocity field is assumed to vanish at large distances from the particle, so that the fluid is at rest except in the vicinity of the colloid. With the aim of obtaining mean-field equations for a dilute suspension of active particles, we also assume that the concentration fields can have nonvanishing gradients on large spatial scales. Accordingly, the concentration gradients

We suppose that the diffusiophoretic coefficients take the values

Solving equations (

The orientation of the Janus particle is described by the unit vector

Solving the Navier-Stokes equations (

The expressions (

If molecular diffusion is fast enough so that the concentration fields adopt stationary profiles around the catalytic particle in the concentration gradients

The self-diffusiophoretic velocity, expressed in terms of the molecular concentrations

In the absence of a reaction, we recover the diffusiophoretic velocities given in Refs. [

Moreover, if the diffusiophoretic coefficients are the same on both hemispheres

In the presence of a reaction, but without gradients (

The overall reaction rate can be written as follows:

We note that both the self-diffusiophoretic velocity (

We now show that Onsager’s principle of nonequilibrium thermodynamics [

The system we consider is a dilute solution containing the reactive molecular

At the macroscale, the reaction is

The mean concentrations of molecular species are defined by

The Janus particles have a spherical shape so that their random rotational and translational motions are decoupled. In this case, the rotational diffusion operator is given by

Before proceeding with nonequilibrium thermodynamics, we need to identify in equation (

For the rotational degrees of freedom we have

The scalar product between a pair of rotational vectors

In the five-dimensional space, the gradient is given by

Using these notations, equation (

Local thermodynamic equilibrium is assumed on scales larger than the size of the colloidal motors where the description by the mean-field equations (

Here,

Next, we use the principles of nonequilibrium thermodynamics in order to express the current densities in terms of the affinities or thermodynamic forces given in Table

Current densities and corresponding affinities or thermodynamic forces in the active suspension:

Process | Current | Affinity | Dimension |
---|---|---|---|

Reaction | |||

Molecular diffusion of fuel | |||

Molecular diffusion of product | |||

Translational diffusion of colloids | |||

Rotational diffusion of colloids |

According to the Curie principle, there is no coupling between processes with different tensorial characters. However, the Janus particles have a director given by the unit vector

According to Onsager’s reciprocal relations (

We assume that the molecular species

This last assumption consists in neglecting the terms with the coefficient

The scalar coefficient associated with the reaction can be identified as

In equation (

With respect to standard expressions, the terms involving the integral

The conclusion from these considerations is that active matter can be described as generalized diffusion-reaction processes in complete compatibility with microreversibility and Onsager’s reciprocal relations. In this way, the program of nonequilibrium thermodynamics is complete and application of equation (

The second law is satisfied if

The results derived in this section provide the basis for the analysis of collective effects in suspensions of active Janus particles. In Sections

Using a thermodynamic formulation that is consistent with microreversibility, we showed earlier [

We suppose that the colloidal motors are subjected to an external force

Defining the mean value of the

Furthermore, integrating equation (

The equations of motion developed in Section

The equation for the colloidal motors is coupled to the reaction-diffusion equations for the molecular species

The system is driven out of equilibrium if

If the second moment (

If

In the following, we suppose that the diffusion coefficient is the same for both molecular species:

Moreover, consistency with the existence of equilibrium requires that

For this system, there exists a uniform nonequilibrium steady state, where

To analyze the stability of this homogeneous steady state, for simplicity we consider a one-dimensional system where the fields

Nonequilibrium steady state of the one-dimensional system for the parameter values (

The threshold of this clustering instability can be found from a linear stability analysis. Linearizing the equations around the uniform steady state, we find that the perturbations obey

Supposing that the perturbations behave as

These dispersion relations are depicted in Figures

Dispersion relations of linear stability analysis for

Therefore, instability manifests itself if

The dispersion relations can also be obtained from the evolution equation (

The linear stability analysis can be carried out for equation (

The conclusion is that equations (

Autonomous motion is not possible at equilibrium and active matter relies on the presence of nonequilibrium constraints to drive the system out of equilibrium. As a result the theoretical formulations provided by nonequilibrium thermodynamics and statistical mechanics are a natural starting point for the description of such systems.

Many of the active matter systems currently under study involve active agents such as molecular machines or self-propelled colloidal particles with linear dimensions ranging from tens of nanometers to micrometers. The transition from microscopic to macroscopic description for fluids containing active agents of such sizes takes place in the upper range of this scale. Suspensions of active colloidal particles are interesting in this connection since, as described earlier in this paper, the colloidal particles are large compared to the molecules of the medium in which they reside. The dynamics of the suspension can then be described by considering the equations for the positions, velocities, and orientations of the colloidal particles in the medium, or through field equations that describe the densities of these particles.

Nonequilibrium thermodynamics provides a set of principles that these systems must obey. Most important among these is microreversibility that stems from the basic time reversal character of the microscopic dynamics. On the macroscale, this principle manifests itself in Onsager’s reciprocal relations that govern what dynamical processes are coupled and how they are described. For example, for single Janus particles propelled by a self-diffusiophoretic mechanism, microreversibility implies the existence of reciprocal effect where the reaction rate depends on an applied external force [

This paper extended the nonequilibrium thermodynamics formulation to the collective dynamics of ensembles of diffusiophoretic Janus colloids. In particular, we considered Janus colloids driven by both self-diffusiophoresis arising from reactions on the motor catalytic surface as well as motion arising from an external concentration gradient. This latter contribution is essential for the extension of the theory to collective motor dynamics. The resulting formulation is consistent with microreversibility and an expression for the entropy production is provided. From this general formulation of collective dynamics, one can show that if an external force and torque are applied to the system, the overall reaction rate depends on the applied force. In addition, a stability analysis of the equations governing the collective behavior predicts the existence of a clustering instability seen in many experiments of Janus colloids. Such considerations can be extended to ensembles of thermophoretic Janus colloids [

We introduce the fields

The fields

Similar expressions hold for the concentration gradients at large distances:

The fields (

The solution of the equations for

In the equations above, the concentrations

Next, the field

First, we calculate the force (

Then, we calculate the torque (

With the following coefficients associated with diffusiophoresis,

By defining the parameters,

Using the expansion (

Hence, the element of angular integration can be written as

Using the expression (

Consequently, we obtain equations (

Interestingly, the assumption (

In this general case where the matrix of linear response coefficients in equation (

Neglecting the last term in the expression (

Linearizing equation (

Supposing that

In matrix form, we have

The authors declare that there are no conflicts of interest.

Research was supported in part by the Natural Sciences and Engineering Research Council of Canada and Compute Canada. Financial support from the Université libre de Bruxelles (ULB) and the Fonds de la Recherche Scientifique - FNRS under Grant PDR T.0094.16 for the project “SYMSTATPHYS” is also acknowledged.