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 [1–11]. The collective behavior in systems where the active agents are chemically propelled colloids, the subject of this paper, has also been the topic of experimental and theoretical research [12–25].
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  and Smoluchowski [27–29] at the beginning of the twentieth century and systematically developed since then for passive colloidal particles [30–33]. In active matter, the colloidal particles are propelled with energy supplied by the surrounding solution, so that the description should be extended to include the molecular concentrations of fuel and product powering their motion, in addition to the velocity field of the fluid. Through such an approach, active matter can be described from the scale of a single colloidal motor moving in the surrounding fluid, up to the macroscale where many colloidal motors generate collective motion by interaction. At the macroscale, collective dynamics is described in terms of the distribution function giving the orientation as well as the position of the colloidal motors. This statistical-mechanical approach has the advantage that the parameters characterizing active matter at the macroscale can be deduced from the microscopic level of description. The knowledge of these parameters in terms of the properties of materials composing the colloidal motors and the surrounding solution is fundamental for engineering active systems.
The present paper contributes to the statistical-mechanical and nonequilibrium thermodynamic approaches for active matter systems [34–42], and considers systems whose active agents are Janus colloids with catalytic and noncatalytic faces moving by diffusiophoresis generated by chemical reactions taking place on their catalytic faces or caps [43, 44, 40]. Because of diffusiophoresis, the velocity and concentration fields are coupled together in the fluid around the Janus particle . We start from the calculation of the diffusiophoretic force and torque on a single Janus particle moving in a fluid in the presence of molecular species corresponding to the fuel and the product of the reaction taking place on its catalytic surface. The concentrations of these molecular species may develop gradients on large scales under nonequilibrium conditions, and these gradients should be included in the calculation of the force and torque. The resulting diffusiophoretic force and torque enter the coupled Langevin equations ruling the displacement, rotation, and overall reaction of a single active particle.
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 [46–53]. This method allows us to obtain all the possible couplings compatible with microreversibility, including a priori unexpected reciprocal effects. Moreover, this method provides an expression for the entropy production rate density for active matter in agreement with the second law of thermodynamics and including the contribution of the reaction powering activity. Through this procedure, macroscopic evolution equations are obtained that govern the collective dynamics of colloidal motors coupled to the molecular concentrations of fuel and product. These equations can be shown to generate the reciprocal effect of diffusiophoresis back onto the reaction rate that has been obtained previously for a single particle [39, 40], but now at the macroscale. Furthermore, pattern formation due to a clustering instability manifests itself under nonequilibrium conditions induced by a bulk reaction replenishing the solution with fuel.
The paper is organized as follows. Section 2 is devoted to the dynamics of a single colloidal motor. The force and torque due to diffusiophoresis are deduced by solving the diffusion equations for the molecular concentrations coupled to the Navier-Stokes equations for the fluid velocity, including the contributions of concentration gradients at large distances from the particle. These contributions were neglected previously [39, 40] and are calculated in detail here. In Section 3, the diffusiophoretic force and torque obtained in Section 2 are incorporated into the evolution equation for the distribution function describing the ensemble of colloidal motors, and the entropy production rate density is explicitly obtained. Two implications of these results are presented in Sections 4 and 5. First, the reciprocal effect due to the diffusiophoretic coupling of an external force and torque back onto the reaction rate is recovered, now at the level of the collective dynamics. Second, a clustering instability leading to pattern formation is shown to manifest itself. The conclusions of the research are given in Section 6. The appendices provide additional details of the calculations.
2. Diffusiophoresis and Colloidal Motors
This section describes the motion of a single spherical Janus colloidal motor of radius that is propelled by self-diffusiophoresis generated by a reversible reaction, with rate constants taking place on its catalytic surface, as depicted in Figure 1. In this reaction, is the fuel and the product, which are present in the solution surrounding the particle. Moreover, the concentrations of the and molecular species are assumed to have gradients with at large distances from the particle that also contribute to motion by diffusiophoresis; thus, the motion of the particle is determined by processes in the fluid surrounding the particle.
2.1. Chemohydrodynamics around a Colloidal Motor
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 and product should be obtained by solving the Navier-Stokes equations for the fluid velocity coupled to the advection-diffusion equations for the molecular concentrations with : where is the constant mass density (the fluid being assumed to be incompressible), the hydrostatic pressure, the shear viscosity, and the molecular diffusivity of species .
The coupling between the velocity and concentration fields is established with the boundary conditions [40, 54] where is the unit vector normal to the solid surface, , is the slip length, , denotes the transpose, is the diffusiophoretic coefficient of species coupling the velocity field to the corresponding concentration field because of different interactions between the solid surface with the molecules of different species. The velocity field inside the solid particle is given by in terms of the translational and angular velocities of the particle, respectively, denoted by and , and position of the center of mass of the particle. Equations (5) and (6) are the boundary conditions on the components of the velocity field that are, respectively, normal and tangential to the interface , which is located on the sphere . The last equation, i.e., equation (7) is the boundary condition for the two reacting species , where is the stoichiometric coefficient of species in the reaction ( and ), and are the forward and reverse surface rate constants per unit area.
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 are taken to exist at large distances from the colloidal particle.
We suppose that the diffusiophoretic coefficients take the values and on the catalytic and noncatalytic hemispheres, respectively, while the surface rate constants per unit area take positive values on the catalytic hemisphere and vanish on the noncatalytic hemisphere, . Using spherical coordinates with polar angle defined with respect to the axis of the cylindrical symmetry of the Janus particle, we have where denotes the Heaviside function such that on hemisphere and is zero otherwise. The catalytic hemisphere is taken as , and the noncatalytic hemisphere as .
Solving equations (2)–(4) with the boundary conditions (5)–(7), the velocity and concentration fields can be obtained in the vicinity of the colloidal motor [45, 40, 41]. Accordingly, the force and the torque exerted by the fluid on the motor, as well as the overall reaction rate at its catalytic surface, are given by the following surface integrals at the fluid-colloid interface , where is the pressure tensor of the fluid. The fluctuating contributions from thermal noise can also be included [40, 41].
2.2. Coupled Langevin Equations for the Motor
The orientation of the Janus particle is described by the unit vector attached to the axis of the cylindrical symmetry of the Janus particle and pointing towards the catalytic hemisphere. Accordingly, the displacement and rotation of the particle are ruled by in terms of the translational and rotational velocities. These velocities, as well as the number of reactive events taking place on the particle, are governed by the following coupled Langevin equations [39, 40, 41]: where and denote the mass and inertia tensor of the motor, is the translational friction coefficient, the rotational friction coefficient, and the diffusiophoretic force and torque, and the external force and torque exerted on the particle, while and are the contributions to the force and torque due to thermal fluctuations. The overall net reaction rate is , is the reciprocal contribution of diffusiophoresis back onto the reaction rate, and is the fluctuating reaction rate. If the Janus particle has a magnetic dipole and is subjected to an external magnetic field , then the external torque would be given by . In the overdamped regime, the coupled Langevin equations are obtained by neglecting the inertial terms in equations (11) and (12).
Solving the Navier-Stokes equations (2) and (3) coupled to equation (4) for molecular concentrations with the boundary conditions (5)–(7), the force and the torque exerted on a spherical particle of radius in a fluid with shear viscosity and the overall net reaction rate are given by  expressed in terms of the surface average
The expressions (14) and (15) find their origin in the generalization of Faxén’s theorem to a sphere moving in a time-dependent velocity field [55, 56]. When writing these equations, we have taken into account the possibility that the diffusiophoretic coefficients and the surface rate constants may be nonuniform on the particle surface.
2.3. Motion in Molecular Concentration Gradients
If molecular diffusion is fast enough so that the concentration fields adopt stationary profiles around the catalytic particle in the concentration gradients , the diffusiophoretic translational and rotational velocities can be written as follows (see Appendix A): where the parameters , , and are given in equations (A.35)–(A.40) in terms of the diffusiophoretic coefficients , the rate constants per unit area , the slip length , the molecular diffusivities , and the geometry of the Janus particle. The identity matrix is , while
The self-diffusiophoretic velocity, expressed in terms of the molecular concentrations extrapolated to the center of the particle, is since the parameters may be written in the forms and (see Appendix A, equations (A.33)-(A.34).
Moreover, if the diffusiophoretic coefficients are the same on both hemispheres , the angular velocity is equal to zero, .
In the presence of a reaction, but without gradients (), we have and the linear velocity reduces to the contribution of self-diffusiophoresis, , characterizing the activity of the Janus particle.
The overall reaction rate can be written as follows: in terms of rate constants and a parameter given in equation (A.45). In the absence of the concentration gradients, we recover the expression obtained in Ref. . In the presence of the concentration gradients , there is an extra contribution depending on the direction of the Janus particle. However, this last term is normally negligible because we typically have for micrometric particles and macroscopic gradients of molecular concentrations.
We note that both the self-diffusiophoretic velocity (21) and the leading term of the reaction rate (23) are proportional to each other. Their ratio defines the self-diffusiophoretic parameter which was introduced in Refs. [39, 40], where the last equality was obtained using .
3. Active Suspension of Colloidal Motors
3.1. Onsager’s Reciprocal Relations
We now show that Onsager’s principle of nonequilibrium thermodynamics [46–53] can be used to establish coupled diffusion-reaction equations of motion for active matter that are consistent with microreversibility. According to Onsager’s principle, currents are related to thermodynamic forces (or affinities) by where the linear response coefficients satisfy the Onsager reciprocal relations, if the affinities are even under time reversal. The thermodynamic entropy production rate density is given by where is Boltzmann’s constant.
3.2. Mean-Field Equations for the Active Suspension
The system we consider is a dilute solution containing the reactive molecular and species together with colloidal motors in an inert solvent . The motors are spherical Janus particles and, as described in Section 2, have hemispherical catalytic surfaces where the reaction takes place. Moreover, we suppose that the solution is globally at rest, so that the velocity field is equal to zero on scales larger than the size of colloids. The solution is described at the macroscale in terms of the molecular densities and , as well as the distribution function of the colloidal motors, , where is the position and is the unit vector giving the orientation of the Janus particles (expressed in spherical coordinates in the laboratory frame). The distribution function is defined as where are the positions and orientational unit vectors of the colloidal motors. For a dilute suspension, the evolution equation of this distribution function can be deduced from the Fokker-Planck equation for the probability that a single colloidal motor is located at the position with the orientation [39, 40, 41]. Once, this distribution function is known, we can obtain the successive moments of : where , is the density or concentration of colloidal motors, is the polar order parameter or polarization of the colloidal motors, and is the traceless order parameter analogous to that for apolar nematic liquid crystals expressed in terms of the tensor (20) and, thus, satisfies .
At the macroscale, the reaction is with the rate constants . For the colloidal suspension treated here, the reaction should be described by a reaction rate density that is proportional to the distribution function of colloidal motors and determined by the surface reaction taken into account with the boundary conditions (7) in Section 2.
The mean concentrations of molecular species are defined by , where is the volume fraction of the suspension. Their corresponding gradients are related to those considered in Section 2 by for a dilute enough suspension. The coupled diffusion-reaction equations for the different species take the following forms: where are the molecular current densities, is the total drift velocity obtained by adding the drift velocity due to the external force to the diffusiophoretic velocity (18) giving with the self-diffusiophoretic velocity (equation (21)) now expressed in terms of the mean concentrations , and the inverse temperature . In equation (34), is an effective translational diffusion coefficient related to the effective translational friction coefficient by Einstein’s formula and is an effective rotational diffusion coefficient related to the effective rotational friction coefficient by . Since the shear viscosity increases as with the volume fraction of the suspension [26, 31], both friction coefficients and also increase, and the diffusion coefficients decrease. In particular, it is known that . A similar dependence on the volume fraction is expected for the parameters , , , and given in Appendix A, since these parameters are proportional to the diffusiophoretic coefficients that are known to be inversely proportional to shear viscosity, [54, 57, 58]. The effects of this dependence would manifest themselves if the colloidal suspension became dense enough. Here, such effects are assumed to play a negligible role.
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 expressed in terms of the rotational energy associated with the torque exerted by an external magnetic field on some magnetic dipole of the particle  and that due to the diffusiophoretic effect, we have the following:
3.3. Translational and Rotational Current Densities
Before proceeding with nonequilibrium thermodynamics, we need to identify in equation (34) the current densities associated with the translational and rotational movements of the colloidal motors. The distribution function for colloidal Janus particles is defined in the five-dimensional space , where are the Cartesian coordinates for the position and the spherical coordinates for the orientation . Vector calculus is used in these coordinates to obtain the corresponding gradients and divergences .
For the rotational degrees of freedom we have
The scalar product between a pair of rotational vectors is given by , and the scalar product of such a vector with itself is denoted . In spherical coordinates, the rotational gradient and divergence are given, respectively, by 
In the five-dimensional space, the gradient is given by and the divergence of a five-dimensional vector, , is
Using these notations, equation (34) can be written in the form of a local conservation law involving the five-dimensional current density, , as with translational current density where is the translational potential energy due to the external force , rotational current density and their translational and rotational divergences, and , where is the operator (37).
3.4. Nonequilibrium Thermodynamics of the Active Suspension
Local thermodynamic equilibrium is assumed on scales larger than the size of the colloidal motors where the description by the mean-field equations (33) and (34) is valid and the fluid is at rest. According to this assumption, thermodynamic quantities can be locally expressed in the active suspension in terms of the molecular densities and the distribution function . Furthermore, we suppose that the system is isothermal and isobaric and the solution is dilute in the species , , and . The appropriate thermodynamic potential is thus Gibbs’ free energy given by the following volume integral of the corresponding density: where the first term is the contribution from the solvent of density , the next terms from fuel and product dilutely dispersed in the solvent , and the last terms from all the orientations of the colloidal motors moving in the mechanical potential energies due to the external force and the external torque exerted by the magnetic field on the magnetic dipoles of the colloids. We can thus deduce the following chemical potentials:
Here, , where is the standard chemical potential of species and is the standard concentration. Since the solution is dilute, we have taken the solvent density to be essentially uniform in space and constant in time.
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 1. For the reaction (32), the affinity is given by and the corresponding current density is the rate density introduced in equation (33). At chemical equilibrium, we have , , and . In the linear regime close to equilibrium where , the chemical affinity (51) can be approximated as where is the diffusivity of the reaction taking place on the colloidal motors [39, 40]. For the diffusion processes of species , the affinity associated with the current density is given by in terms of the chemical potential . For molecular species, the gradient is tridimensional in Euclidian space, so that . For the colloid with chemical potential (50), the affinity is given by the five-dimensional gradient (42) as if the magnetic field is uniform. In this five-dimensional space, the associated current density given by equations (45) and (46) can thus be written in the following form: where is the identity matrix. In this form, we see that the first term is related to the reaction affinity since the self-diffusiophoretic velocity can be written as . The next two terms can be related to the affinities of molecular species, and the last term to the affinity of the colloidal species.
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 and we have adopted a description in terms of the distribution function for the Janus particles. Consequently, it is possible that a vectorial process such as diffusion may be coupled to a scalar process such as reaction if it is polarized by the unit vector . If we introduce the densities for Janus particles having their orientation in cells with a size of , along with the associated current densities, we may write a general coupling (25) of the following form: up to possible nonlinear contributions that may be required in order for the reaction rate to obey the mass-action law. In equation (57), we have that is , , , , , , , , and (for ).
According to Onsager’s reciprocal relations (26), the linear response coefficients should obey for and where again denotes the transpose.
We assume that the molecular species and undergo Fickian diffusion without cross-diffusion, so that and that the reaction rate does not depend on the gradients or , whereupon
The scalar coefficient associated with the reaction can be identified as and the linear response coefficients , , and in equation (57) can be determined using the current density (55), as described in Appendix B. As a consequence of Onsager’s reciprocal relations, we can conclude that the reaction rate and the current densities should be given by
In equation (62), the second term describes the reciprocal effects of diffusiophoresis back onto reaction. The second term in equation (63) is due to cross-diffusion between the molecular and colloidal species due to diffusiophoresis. We see that the linear response coefficients depend on the unit vector in a manner similar to that already shown in Refs. [39, 40].
With respect to standard expressions, the terms involving the integral in equation (63) are required in order to satisfy Onsager’s reciprocal relations and for these quantities to be compatible with microreversibility. However, these extra terms can be shown to be negligible, although the reciprocal terms are not negligible in equations (45) and (46). In order to show that the extra terms are negligible, we suppose that the self-diffusiophoretic and diffusiophoretic velocities take the typical value . According to Ref. , the molecular gradients used in experiments of diffusiophoresis are of the order of , so that diffusiophoretic parameters have the value . Moreover, we have , but since , the effect of the coefficients is again of the same order of magnitude as and . Molecular diffusivities typically have the value , while the translational diffusion coefficient of a micrometric colloidal particle is of the order of . The molecular concentrations used in experiments on self-diffusiophoresis are about , while the density of micrometric colloidal particles is approximately , or lower. If we assume that the molecular and colloidal gradients take comparable values , the ratio between the extra term and the standard molecular diffusion term in equation (63) is given by which shows that the second term in equation (63) is negligible. Accordingly, the standard Fickian expressions are very well justified for the molecular current densities. In the presence of colloidal motors, the expressions compatible with microreversibility are nevertheless given by equations (62) and (63). In contrast, the terms associated with the diffusiophoretic parameters in the colloidal current density (55) have effects that are not negligible.
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 (27) gives the following expression for the thermodynamic entropy production rate density: