A theory for the phase behavior of mixtures of active particles

Abstract

Systems at equilibrium like molecular or colloidal suspensions have a well-defined thermal energy k_BT that quantifies the particles' kinetic energy and gauges how “hot” or “cold” the system is. For systems far from equilibrium, such as active matter, it is unclear whether the concept of a “temperature” exists and whether self-propelled entities are capable of thermally equilibrating like passive Brownian suspensions. Here we develop a simple mechanical theory to study the phase behavior and “temperature” of a mixture of self-propelled particles. A mixture of active swimmers and passive Brownian particles is an ideal system for discovery of the temperature of active matter and the quantities that get shared upon particle collisions. We derive an explicit equation of state for the active/passive mixture to compute a phase diagram and to generalize thermodynamic concepts like the chemical potential and free energy for a mixture of nonequilibrium species. We find that different stability criteria predict in general different phase boundaries, facilitating considerations in simulations and experiments about which ensemble of variables are held fixed and varied.

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1 Introduction

Active matter systems like colonies of bacteria and self-propelled synthetic microswimmers are a rich area of study for soft matter. The fundamental and seemingly elementary ability of self-propulsion allows active systems to free themselves from classical thermodynamic constraints and to control their own motion and the surrounding environment. Their inherently nonequilibrium properties engender intriguing behavior such as spontaneous self-assembly and pattern formation,^1,2 making active matter a fascinating but challenging system to study.

Recently a new “swim pressure” concept was introduced—namely, all active entities exert a unique mechanical pressure owing to their self-motion.^3,4 This perspective was applied⁵ to predict the self-assembly of a suspension of active particles into regions of dense and dilute phases observed in both experiments and simulations.^6–10 The usefulness of the mechanical pressure to illuminate active matter's physical principles begs the question: what is the temperature of active matter? Do active swimmers “thermally equilibrate” with their surroundings? Although it is clear that the mechanical pressure can be quantified and is valid out of equilibrium, it is uncertain whether the notion of a temperature exists and can be explained in basic physical quantities.

To understand the temperature of active matter, we shall first discuss a simple experiment involving passive Brownian suspensions (i.e., no self-propulsion) which can be rigorously related to conventional thermodynamic quantities like the temperature and free energy. Suppose we have a purely Brownian suspension with thermal energy (k_BT)_H that is separated by a thermally-insulated partition from another Brownian system with a different temperature (k_BT)_C, as shown in Fig. 1. The partition is suddenly removed and the particles at different temperatures are allowed to mix. The “hot” and “cold” particles undergo many collisions, share their kinetic energy with each other, and eventually equilibrate to a common temperature (k_BT)_eq.

Fig. 1 Schematic of the mixing process of purely Brownian suspensions (top) and active systems (bottom) that are initially at two different “temperatures.” The Brownian particles thermally equilibrate their thermal energy k_BT whereas the active swimmers do not share their characteristic “energy scale” k_sT_s ≡ ζU₀²τ_R/2.

Now suppose we do the same mixing experiment with self-propelled swimmers at two different activity levels. For simplicity we consider self-propelled spheres of radii a that translate with an intrinsic swim speed U₀, reorient with a reorientation time τ_R, and experience a hydrodynamic drag factor ζ from the surrounding continuous Newtonian fluid. Their motion can be described as a random-walk process for times t > τ_R with a diffusivity D^swim = U₀²τ_R/2 in two dimensions (2D). Their characteristic “energy scale” is not the thermal energy k_BT = ζD₀ where D₀ is the Stokes–Einstein–Sutherland translational diffusivity, but comes from their self-propulsive activity, defined as k_sT_s ≡ ζD^swim = ζU₀²τ_R/2 (see later section for a more detailed treatment).

A system of “hot” active swimmers with (k_sT_s)_H is initially separated from “cold” swimmers with (k_sT_s)_C as shown in Fig. 1. When the partition is removed, the swimmers with different activity levels spontaneously mix and undergo collisions with each other. When a swimmer collides into another swimmer, it displaces the body by its size a until they move completely clear of each others' trajectories. After the collision, each swimmer then continues its motion with the same activity it had initially—there is no sharing of kinetic activity (k_sT_s) upon collisions. This implies that the swimmers' activity scale k_sT_s ≡ ζU₀²τ_R/2 does not get shared via collisions and thus does not “equilibrate” like the temperature of a classical fluid k_BT. This simple experiment already reveals the richness and challenge to understand the “temperature” of nonequilibrium active systems.

A simple multicomponent mixture of self-propelled particles with two different activities is an ideal system to discover and study this problem in greater detail. Previous studies have provided various interpretations of the temperature in a nonequilibrium active matter system.^11–13 We discuss a new perspective by developing a mechanical pressure theory for predicting the phase behavior of a mixture of active swimmers over the entire phase space of the system. Our theory applies in general to a multicomponent suspension with swimmers of different activities, but perhaps the most straightforward mixture is that of active self-propelled particles and passive Brownian particles in a single solvent. In this mixture we must treat active swimmers and passive particles as independent species, because their compositions vary in space due to the phase-separating behavior of active suspensions. This is true in general for multicomponent systems—in a simple polymeric solution of polyethylene in benzene, the polyethylene molecules do not all have the same number of segments or molecular weight, and thus generally need to be treated as different components. Experiments also often use mixed solvents in which the solvent composition inside a polymer coil (or gel) is in general different from the outer regions, as certain solvent species preferentially remain inside (or outside) the polymer coil.¹⁴

We consider a simple mixture of spherical active and passive Brownian particles with equal size a; the passive particles translate by Brownian motion but are otherwise inactive (see Appendix A for the equations of motion). We do not include the effects of hydrodynamic interactions, and there is no polar order of the swimmers or any large-scale collective motion (e.g., bioconvection). We find that many new insights about the temperature of active matter can be obtained from this simple system.

In the next section we further extend the mixing example discussed above (Fig. 1) by analyzing the effects of adding a small concentration of passive Brownian particles into an active system. We analyze the quantities that “equilibrate” in an active system by studying the collisions between a swimmer and a passive particle. In Section 3 we develop a simple mechanical theory by identifying the different contributions that make up the total active pressure of the mixture. Since active matter is an inherently nonequilibrium system, we do not rely upon the thermodynamic free energy or chemical potential to predict the phase behavior of the system. Unlike these thermodynamic quantities, the mechanical stress (or pressure) is defined out of equilibrium and can be used to analyze mechanical instability of active matter. We then take our equation of state to compute what would be the nonequilibrium analogs of the free energy and chemical potential. Lastly, we analyze different stability criteria, facilitating discussion about the variables that may be held fixed and varied in experiments and computer simulations.

2 Do active particles “thermally” equilibrate?

From the mixing process in Fig. 1 we learned that the characteristic activity scale of the swimmers do not equilibrate (i.e., (k_sT_s)_H ≠ (k_sT_s)_C) unlike the thermal energy k_BT of passive Brownian particles. To gain further insight into the quantities that get shared in an active system, suppose now that we have a dilute concentration of passive bath particles in a sea of active swimmers. The motion and behavior of passive bath particles are influenced markedly by the swimmers' reorientation Péclet number Pe_R ≡ a/(U₀τ_R), a ratio of the swimmer size a to its run length U₀τ_R.

Swimmers with run lengths small compared to their size (Pe_R ≫ 1) reorient rapidly and take small swim steps behaving as Brownian walkers. When a swimmer takes a step and collides into a bath particle, the passive particle gets a displacement of order the swimmer's step size . After many such collisions, the change in the translational diffusivity of the passive bath particle is (D^bath − D₀) ∼ U₀(U₀τ_R)ϕ_a, where D₀ = k_BT/ζ is the Stokes–Einstein–Sutherland diffusivity of an isolated bath particle and ϕ_a is the area (or volume in 3D) fraction of the swimmers. In this limit active swimmers repeatedly displace the bath particle by their run length U₀τ_R, which allows the bath particle to sense the activity or ‘temperature’ of the swimmers via collisions. In other words, the bath particle behaves as a ‘thermometer’ of the active suspension,¹² where the collisional displacements it receives from the swimmer can be used to infer the swimmers' characteristic ‘energy scale’ k_sT_s = ζU₀²τ_R/2. This activity scale is analogous to the thermal energy k_BT, the kinetic activity of passive Brownian particles, which can also be probed by analyzing the collisions between two passive particles. In this sense a suspension of swimmers with small run lengths U₀τ_R < a behaves similarly to a purely Brownian suspension with an effective ‘temperature’ k_sT_s. For active Brownian particles, this contribution is in addition to the thermal k_BT that gets shared as usual as a result of translational Brownian motion. However, one would not be able to distinguish between the two contributions because the dynamics of swimmers with Pe_R ≫ 1 is equivalent to that of passive Brownian particles. If we placed active swimmers that behave identically to passive Brownian particles behind an osmotic barrier, we would not be able to distinguish one from the other. In this sense a Brownian particle can be interpreted as a “swimmer” having an extreme value of the reorientation Péclet number Pe^B_R → ∞.

The swim activity k_sT_s can also be understood by comparing the statistical correlation of the self-propulsive swim force, F^swim ≡ ζU₀ = ζU₀q where q is the unit orientation vector specifying the swimmer's direction of self-propulsion, to that of the Brownian force, F^B. The swim force correlation 〈F^swim(t)F^swim(t′)〉 = (ζU₀)²〈q(t)q(t′)〉 = (ζU₀)² exp(−(t − t′)/τ_R) turns into a delta-function correlation 〈F^swim(t)F^swim(t′)〉 ∼ (ζU₀)²τ_Rδ(t − t′) as τ_R → 0.¹⁵ Recall that as τ_R → 0 the active swimmers behave as random Brownian walkers, which have the white noise statistics where the overline indicates an average over the solvent fluctuations. A comparison of these two correlations again suggests that the swimmers' kinetic activity can be interpreted by k_sT_s ≡ ζU₀²τ_R/2.

For swimmers with run lengths large compared to their size, (Pe_R ≪ 1), we observe a different behavior. Colliding into a bath particle, the swimmer continues to push the bath particle until it moves completely clear of the swimmer's trajectory. The bath particles receive a displacement of upon colliding with a swimmer, not the run length U₀τ_R. Therefore the length scale associated with collisions is the swimmer size a, and the change in the long-time diffusivity of the bath particles (D^bath − D₀) ∼ U₀aϕ_a. Unlike the limit of Pe_R ≫ 1 discussed above, here the bath particles cannot probe the activity or ‘temperature’ of the swimmers because it only receives a displacement of its size a, even though the swimmers actually diffuse with their swim diffusivity D^swim ∼ U₀²τ_R. The ratio of the two diffusivities (D^bath − D₀)/D^swim ∼ U₀aϕ_a/(U₀²τ_R) = ϕ_aPe_R, suggesting that the reorientation Péclet number Pe_R ≡ a/(U₀τ_R) is the quantity that gets shared between the swimmers via collisions for small Pe_R.⁵ This implies that the swimmers' energy scale k_sT_s = ζU₀²τ_R/2 does not get shared in the collisions and thus does not represent the ‘temperature’ in the classical sense.

The bath particles' entirely different behavior for large and small Pe_R reveals the richness and challenge to understanding the ‘temperature’ of nonequilibrium active systems. This marked change in the quantity that gets shared in active systems is due to the capability of swimmers to have run lengths U₀τ_R that can be small or large compared to their size a. This is a key fundamental difference between the swimmers' activity k_sT_s and the thermal energy k_BT. In a classical molecular fluid, k_BT is always the quantity that equilibrates because the displacements of a passive Brownian particle are small compared to its size a (or any other length scale), i.e. Pe^B_R ≡ a/(U_Bτ_B) → ∞ where U_B = D₀/a is the characteristic speed of a Brownian step and τ_B is its momentum relaxation timescale.

Moreover, the swimmers must continuously collide with the passive particle to impart information about their kinetic activity, k_sT_s—even after many collisions, the passive particle only possesses k_BT units of thermal energy once all collisions stop. This is in stark contrast with a molecular or kinetic fluid particle that is able to completely transmit its kinetic activity to another particle upon collisions. If a molecular fluid particle with initially zero activity is placed inside a container full of fluid particles with energy k_BT, the inactive particle would collide repeatedly and eventually attain the thermal energy k_BT. Furthermore, it will keep its k_BT activity even when the other particles are removed. In contrast, a passive particle would cease to move (aside from its translational Brownian motion) if active swimmers are removed because of the damping due to the solvent. In this sense the temperature of an active nonequilibrium fluid is not well defined, as each swimmer has its own unique intrinsic kinetic activity that does not get shared and equilibrated.⁵

In pursuant of the discussion above we conducted Brownian dynamics (BD) simulations (see Appendix) and computed the long-time self diffusivity , where x_d is the position of the passive bath particles. As shown in Fig. 2, for small Pe_R we indeed find that D^bath = D₀ + U₀aϕ_a/2 fits the data for all ϕ ≲ 0.4. At higher ϕ the passive particles are trapped into clusters by the swimmers and D^bath decays to 0. Fig. 2 suggests that the parameter Pe_R gets shared upon swimmer collisions and not the scale k_sT_s.

Fig. 2 Long-time self diffusivity of a passive particle as a function of the total area fraction for different values of the active swimmer fraction x_a. The known Brownian diffusivity D₀ was subtracted from the results. The solid line is the analytical theory and symbols are Brownian dynamics (BD) simulations. All data collapse onto a single curve when the diffusivity is scaled with U₀ax_a/2.

Finally, an important concept here is that the departure induced by a swimmer is the same whether it collides into a passive particle or into another active swimmer. In both cases the displacement due to the collision is the swimmer size a (for small Pe_R), and this is the key idea underlying the mechanical theory which we explain below.

3 Mechanical theory

Our theory applies in general to a mixture of active swimmers with different activity levels, but here we focus on a mixture of active swimmers and passive particles. Since a passive particle behaves equivalently to an “active” particle with a very small reorientation time and step size, this system corresponds to the limiting case of a mixture of one group of swimmers with a finite, nonzero Pe_R and another group of “swimmers” with Pe_R → ∞. The general case is a mixture of active swimmers with two different, finite Pe^a_R and Pe^d_R. However, the active and passive limit is interesting from an experimental perspective because a mixture of passive and active particles is easy to make. Mixtures of swimmers with different, finite Pe_R are difficult to analyze because of the inherent variations in activity in living organisms and in synthetic self-propelled particles due to fabrication defects.

We are now in a position to derive a simple mechanical pressure theory to predict the phase behavior of a mixture of active and passive particles. The total active pressure of the mixture is given by

Π^act = Π^swim + Π^P_a + Π^P_d + nk_BT, (1)

where Π^swim is the ‘swim pressure’, Π^P_a and Π^P_d are the interparticle pressure contributions of the active swimmer and passive particle, respectively, and nk_BT is the Brownian osmotic pressure. It is permissible to add the separate contributions of the pressure in what appears to be a superposition; this is true in general for molecular, Brownian and active systems. Eqn (1) is the additional pressure contribution due to the particles (both active and passive); the solvent pressure p_f is arbitrary and constant in our analyses.

In general Π^act is a function of (ϕ, x_a, Pe_R, k_sT_s, k_BT), where ϕ is the total area fraction (ϕ = ϕ_a + ϕ_d), ϕ_a and ϕ_d are the area fractions of the active and passive particles, respectively, x_a = ϕ_a/ϕ is the active swimmer composition, the reorientation Péclet number Pe_R ≡ a/(U₀τ_R) is the ratio of swimmer size a to its run length U₀τ_R, k_sT_s ≡ ζU₀²τ_R/2 is the swimmers' characteristic ‘energy scale’ as discussed earlier, and k_BT is the thermal energy. We can also express the active pressure using the area fractions of the active and passive particles, Π^act(ϕ_a, ϕ_d, Pe_R, k_sT_s, k_BT). To reduce the number of parameters, we take equal size active and passive particles a_a = a_d = a and assume that swimmer reorientation is thermally induced so that the translational and reorientational diffusivities are related via the Stokes–Einstein–Sutherland expressions: (D₀/a²)/τ_R = 4/3. Thus the ratio of the thermal energy to the swim activity is k_BT/(k_sT_s) = 8Pe_R²/3. This is not a requirement; one can also vary a swim Péclet number, Pe_s ≡ U₀a/D₀ in addition to the reorientation Péclet number Pe_R ≡ a/(U₀τ_R).

We now explain the independent pressure contributions in detail below. The theory is presented for 2D, but it is straightforward to generalize to 3D.

3.1 Swim pressure of active swimmers, Π^swim

The swim pressure is defined as the first moment of the swim force Π^swim = −n_a〈x·F^swim〉/2 (in 2D), where n_a is the number density of swimmers and the angle brackets denote an average.³ It is permissible for computing the stress to interpret the self-propulsion of an active swimmer as arising from a swim force, F^swim ≡ ζU₀,‡ where U₀ = U₀q; U₀ is the swimming speed and q is the unit orientation vector defining the swimmer's direction of self-propulsion. Physically, F^swim represents the force required to prevent an active swimmer from moving, for example by optical tweezers. The origin of the swim pressure stems from the notion that confined self-propelled bodies exert a pressure on the container boundaries as they collide into the surrounding walls. The same notion applies to molecular gases that collide into the container walls to exert a pressure or to colloidal solutes that collide into a semipermeable membrane to exert an osmotic pressure. The swim pressure is the “osmotic” pressure of active particles.

A dilute system of purely active swimmers exerts an ‘ideal-gas’ swim pressure given by Π^swim = n_aζU₀²τ_R/2 = n_ak_sT_s in 2D.³ The swim pressure is a single-particle self contribution in which the relevant length scale (i.e. moment arm) is the swimmers' run length U₀τ_R. As discussed earlier the ratio of the swimmer size a to the run length U₀τ_R is the reorientation Péclet number Pe_R ≡ a/(U₀τ_R), and this parameter impacts the phase behavior of active systems.⁵ For large Pe_R the swimmers take small swim steps and behave as Brownian walkers, exerting the swim pressure Π^swim = n_aζU₀²τ_R/2 = n_ak_sT_s for all concentrations.

For small Pe_R the swimmers have large run lengths and undergo many collisions with passive particles and other swimmers in a time τ_R. The average distance traveled by a swimmer between reorientation events is reduced and the same is true for the swim pressure. Extending the results for a purely active system,⁵ we take (for small Pe_R)

Π^swim = n_ak_sT_s(1 − ϕ − 0.2ϕ²), (2)

where n_a is the number density of active swimmers, k_sT_s ≡ ζU₀²τ_R/2 is the characteristic ‘energy scale’ of a swimmer. Inside the parenthesis of eqn (2) is the total area fraction because both active and passive particles hinder the run length of an active swimmer. Recall our discussion from Section 2 that the displacement induced by a swimmer is the same whether it collides into another swimmer or a passive particle. For a dilute system ϕ → 0 we recover the ‘ideal-gas’ swim pressure Π^swim = n_ak_sT_s. As the area fraction increases, both passive and active particles collide and obstruct the motion of swimmers, decreasing the run length and therefore the swim pressure. The decrease in Π^swim is the principle destabilizing term that facilitates a phase transition in active systems. This is fundamentally different than a purely Brownian system where repulsive interactions (e.g., excluded volume) necessarily increase the pressure and have a stabilizing effect. Recall the concept that a passive Brownian particle with the thermal energy k_BT is equivalent to a “swimmer” with Pe^B_R → ∞. In this work we focus on small Pe_R since this is the limit that engenders interesting phase behavior in active matter.

Fig. 3 confirms that all data from BD simulations collapse onto eqn (2). To better understand eqn (2), we can analyze the limits for large and small concentrations of active swimmers relative to passive particles. Expanding the swim pressure for small ϕ_d/ϕ_a = (1 − x_a)/x_a, we find

The first term on the right is the swim pressure for a purely active system, and the second term is the leading-order correction of the hindrance provided by passive particles. As expected, it is a 2-body correction of an active swimmer colliding into a passive particle, ∼n_aϕ_d.

Fig. 3 Swim pressure exerted by active swimmers in a mixture as a function of the total area fraction ϕ = ϕ_a + ϕ_d for different values of active composition x_a = ϕ_a/ϕ and fixed Pe_R ≡ a/(U₀τ_R) = 0.1. Subscripts “a” and “d” refer to active and passive particles, respectively. The solid curve is the mechanical theory eqn (2) and the symbols are BD simulations. The swimmer activity k_sT_s ≡ ζU₀²τ_R/2.

In the other limit of small concentration of active swimmers relative to passive particles (i.e., small ϕ_a/ϕ_d = x_a/(1 − x_a)), we find

Unlike the large active concentration limit, the reduction in the swim pressure is caused entirely by the sea of passive particles. Due to the small concentration of swimmers, a swimmer exerts the self-term ‘ideal-gas’ swim pressure n_ak_sT_s but does not hinder the motion of other active swimmers.

3.2 Interparticle (collisional) pressure

In addition to the swim pressure, which is a single-particle contribution to the mechanical pressure, there is also an interparticle (or collisional) pressure arising from interactions between the particles. Since two bodies are required for an interaction (or collision for a hard-sphere potential) and the relevant length scale is the particle size a, the interparticle pressure scales as Π^P ∼ n²ζU₀a³ ∼ nk_sT_sPe_Rϕ, fundamentally different from the swim pressure. Furthermore, the interparticle pressure monotonically increases with concentration for a repulsive potential and helps stabilize a system. The competition between the destabilizing effect of the swim pressure and the stabilizing effect of the interparticle (or collisional) pressure controls the phase behavior of active systems. For clarity we split the interparticle pressure into two contributions—collisions induced by active swimmers and passive particles separately.

3.2.1 Active swimmer, Π^P_a. Extending the nonlinear microrheology analysis,³ the collisional pressure contribution for active swimmers (for small Pe_R) iswhere k_BT is the thermal energy and g(ϕ) is the pair-distribution function at contact. The first and second terms in the bracket are the collisional pressures due to self-propulsion and Brownian fluctuation, respectively. The former scales as ∼n_anζU₀a³ whereas the latter scales as ∼n_anζ(D₀/a)a³; the characteristic Brownian speed D₀/a replaces the swim speed U₀ in the collisional pressure arising from thermal noise. We again use the total area fraction in eqn (5) since the active swimmers impart the same departure whether they collide with a passive or an active particle. Rigorously, the pair-distribution function is different for each pair, i.e., g_aa(ϕ), g_ad(ϕ), etc., but we assume that they are all the same and equal to g(ϕ) since we have taken a_a = a_d. We adopt g(ϕ) = (1 − ϕ/ϕ₀)⁻¹ where ϕ₀ is the area fraction at close packing (ϕ₀ = 0.9 in this study).^5,18

3.2.2 Passive particle, Π^P_d. The collisional pressure contribution of a passive particle is given byThe first term in the brackets is the interparticle pressure due to collisions with active swimmers, which scale as n_dn_aζU₀a³ because these collisions are induced only by the active swimmers. The second term is the usual Brownian collisional pressure. Unlike eqn (5) we see that the collisional pressure of passive particles has an additional dependence on the active-swimmer fraction x_a. If there are no active swimmers (i.e., x_a = 0) then eqn (6) reduces to the usual collisional pressure of Brownian hard-spheres.¹⁹

Fig. 4 graphs the sum of the collisional pressures of the contributions from both active and passive particles as a function of the total area fraction. We see a dependence on the composition of active swimmers x_a especially at high area coverage. We assume that swimmer reorientation is thermally induced so that the translational and reorientational diffusivities are related via the Stokes–Einstein–Sutherland expressions, (D₀/a²)/τ_R = 4/3, and the ratio of the thermal energy to the swim activity is k_BT/(k_sT_s) = 8Pe_R²/3.

Fig. 4 Collisional pressure exerted by active and passive particles Π^P = Π^P_a + Π^P_d for fixed Pe_R ≡ a/(U₀τ_R) = 0.1 as a function of the total area fraction ϕ = ϕ_a + ϕ_d and different values of active composition x_a = ϕ_a/ϕ. The solid curve is the mechanical theory eqn (5) and (6) for x_a = 0.3, and the symbols are BD simulations. We take the swimmer reorientation to be thermally induced so that k_BT/(k_sT_s) = 8Pe_R²/3.

4 Phase behavior

Experiments and computer simulations have shown that a suspension of purely active particles may self-assemble into regions of dense and dilute phases, resembling an equilibrium liquid–gas coexistence.^6–10 The source of this phase separation is that swimmers collide and obstruct each others' movement, causing large clusters to form at sufficiently high concentrations.¹³ Now, if this active system also contained passive Brownian particles, recent computer simulations²⁰ and experiments²¹ have shown that the composition of passive particles inside the dense cluster phase is generally larger than that in the dilute phase, as they tend to stay inside the cluster once they are pushed into one by an active swimmer. In contrast, the active swimmers prefer to swim freely in the dilute phase because their activity allows them to escape the dense clusters.

Theory and simulations have produced phase diagrams for a suspension of purely active swimmers,^{5,8,15,22–24} but a mixture of active and passive particles is yet to be thoroughly analyzed. Recently Stenhammar et al.²⁰ conducted Brownian dynamics simulations of a mixture of active and passive Brownian particles and used a kinetic model to locate the phase boundaries. The kinetic model based upon Redner et al.⁸ accurately predicts many regions of phase space, but due to the theory's inherent assumptions the lower spinodal boundary is not well characterized.

Our theory is based upon the new ‘swim pressure’ perspective which accurately predicts the phase behavior of a system of active swimmers.^3–5 Others have subsequently used the swim pressure to study phase-separating active systems.^25,26 We now have eqn (1), an equation of state that allows us to predict the phase behavior of the active/passive mixture.

Interpreting the total density derivative of the active pressure as a global mechanical instability, (∂Π^act/∂ϕ)_xa,Ts,PeR = 0, we can identify the regions of stability in the phase diagram. This is a purely mechanical definition of the spinodal and does not rely upon thermodynamic arguments. As shown by the red curve in Fig. 5, our prediction agrees well with Stenhammar et al.'s²⁰ simulation data. Here the spinodal and the simulation data correspond to a global dense/dilute-phase separation based upon fluctuations in the total particle—active plus passive—density. This is different from the phase separation that may occur locally within each phase, as commonly seen in immiscible polymer mixtures. There are no adjustable parameters in the comparison.

Fig. 5 Phase diagram in the Pe_R − ϕ plane in 2D for a fixed active swimmer composition x_a = 0.5. The colorbar shows the active pressure scaled with the swim activity k_sT_s = ζU₀²τ_R/2. The open and filled symbols are simulation data of Stenhammar et al.²⁰ with a homogeneous and phased-separated state, respectively. The solid and dashed red curves are the spinodals delineating the regions of stability based upon fluctuations in the total particle density and the thermodynamic definition, respectively.

Compared to a purely active swimmer system, onset of phase transition occurs at lower Pe_R when passive particles are present. For x_a = ϕ_a/ϕ = 0.5 shown in Fig. 5, phase transition is possible for Pe_R ≲ 0.025, compared to Pe_R ≲ 0.04 for a purely active system x_a = 1. Therefore, given a fixed total area fraction the presence of passive particles makes it more difficult for phase separation to occur, which may be an important consideration in the design of experiments of active systems.

In Section 2 we discussed that the reorientation Péclet number Pe_R is the quantity that gets shared upon collisions between swimmers for Pe_R ≪ 1. Using the swimmer activity k_sT_s ≡ ζU₀²τ_R/2, we can rewrite Pe_R ≡ a/(U₀τ_R) = ζU₀a/(2k_sT_s), which is interpreted as the interactive energy of the swimmer (ζU₀)a to its swim activity scale k_sT_s. In Fig. 5 phase separation becomes possible for small Pe_R, or large k_sT_s. In contrast, phase transition in a classical thermodynamic system is usually driven by attractive enthalpic interactions and becomes possible for small k_BT (low temperatures). If k_sT_s is interpreted as the “temperature” of active matter, Fig. 5 suggests that mixtures of active and passive particles may exhibit a lower critical solution temperature (LCST) transition,²⁷ commonly seen in thermosensitive polymer systems.²⁸ The LCST phase transition is dominated by entropy—as Pe_R decreases (k_sT_s increases), the run length of the swimmer increases, and the particle becomes effectively larger in size and has less space available for entropic mixing.⁵ However, because Pe_R is the quantity that gets shared upon collisions for Pe_R ≪ 1 (and not the activity k_sT_s), the activity k_sT_s does not play the same role as the thermal energy scale k_BT in LCST phase transitions of polymer mixtures. This further verifies that the “temperature” of active matter is an elusive quantity that does not have a direct mapping to the temperature of an equilibrium system.

5 Limits of active pressure

Recent experiments by Kümmel et al.²¹ analyzed the phase behavior of a mixture of passive particles with a small concentration of active swimmers (ϕ_a ≈ 0.01). They observed swimmers gathering and compressing the passive particles into clusters. By varying the concentration of passive particles, they observed a phase separation of the mixture even at very small active swimmer concentrations.

Our BD simulations agree qualitatively with the experiments.²¹ The active swimmers create tunnels in the sea of passive particles, which open a path for other trailing swimmers to move through. This leads to the formation of large clusters composed of purely passive particles and individual swimmers moving in the dilute phase, as shown in simulation images in Fig. 6. Based upon our mechanical theory, there is an equality between the Brownian collisional pressure of the dense passive clusters and the swim pressure of the dilute active swimmers compressing the crystals. A video of the BD simulation is available in the ESI.†

Fig. 6 Phase diagram in the Pe_R − ϕ plane in 2D for different active swimmer compositions x_a = ϕ_a/ϕ. The solid curves are the spinodals delineating the regions of stability based upon fluctuations in the total particle density. The two-phase region diminishes as x_a decreases. Steady-state images from BD simulations are shown for Pe_R = 0.01, x_a = 0.05 at ϕ = 0.35 (left) and ϕ = 0.6 (right), corresponding to a homogeneous and phased-separated state, respectively. The red and white circles are the active and passive particles, respectively.

To model these observations, it is instructive to analyze the limits of the active pressure for large and small concentrations of active swimmers relative to passive particles. Since the mechanical pressure exerted by a system of purely active swimmers and purely Brownian particles are known, we can interrogate the effect of adding a small amount of passive or active particles into the suspension. This may be particularly useful for further experimental pursuits of active/passive mixtures.

In the limit of small active swimmer concentration relative to passive particles, the active pressure is

where the first term on the right is the osmotic pressure of a purely Brownian suspension:

Π^osm = n_dk_BT(1 + 2ϕ_dg(ϕ_d)), (8)

and the second term in eqn (7) isIn this limit, the swim pressure and swimmers' interparticle collisions appear in the leading-order correction. Taking the global density fluctuation ∂Π^act/∂ϕ = 0, we find that the spinodal qualitatively agrees with the experiments of Kümmel et al.²¹—a lower spinodal boundary of ϕ ∼ 0.45 and the divergence of the interparticle pressure near close packing. A phase diagram in the Pe_R − ϕ plane for different active swimmer compositions is shown in Fig. 6. As x_a decreases the spinodal curve lowers to smaller Pe_R because phase separation becomes more difficult to observe with a smaller fraction of swimmers. For smaller x_a, the Brownian crystals have more time to melt and dissolve into a homogeneous system, and hence the swimmer must have a small Pe_R that is in commensurate with the small x_a. Kümmel et al.²¹ report phase separation in swimmers with Pe_R ≈ 0.04, but our theory suggests that Pe_R must be smaller (Pe_R ≲ 0.01) for phase separation to be possible at the small concentration of active swimmers used in their study.

In the other limit of large active swimmer concentration relative to passive particles, we expect phase behavior similar to those observed in purely active suspensions.⁵ The passive particles can act as nucleation sites for cluster formation, which may spark an earlier onset of phase separation. The active pressure has the form

where the first term on the right is the active pressure for a purely active swimmer suspension⁵ (i.e., eqn (1) with ϕ_d = 0):and the second term in eqn (10) isAs expected the leading-order correction to the swim and interparticle pressures scales as ∼n_aϕ_d. As shown in Fig. 6, the spinodal curve for x_a ≈ 1 remains high because phase separation is dominated by the hindered motion of the active swimmers.

6 ‘Thermodynamic’ quantities

Thermodynamic quantities like the chemical potential and free energy are defined only for equilibrium systems. However, standard macroscopic mechanical balances can be applied to define quantities that are nonequilibrium analogs for active systems.^3,5 Here we extend the derivation of the nonequilibrium free energy and chemical potential to mixtures of active and passive particles, and interpret these quantities as a natural extension for nonequilibrium systems.

The virtual work done by an external mechanical force (i.e., stress) due to an infinitesimal change in the system volume δV is given by δW = −ΠδV where Π is the applied mechanical pressure. One can interpret this virtual work as the change in Helmholtz free energy of the system due to an applied mechanical stress, as is commonly done in elasticity theory.²⁹ Upon carefully imposing incompressibility of the solvent, one can relate the nonequilibrium free energy to the mechanical pressure of a multicomponent mixture as¹⁴

where N_c is the number of species in the mixture and f(0) is the free energy density of the pure solvent (which is arbitrary and constant in our analysis). We interpret eqn (13) as the definition of the free energy for nonequilibrium active systems with Π^act in place of Π. For our two-component (plus the solvent) system, we have N_c = 2 and the nonequilibrium free energy f^act(ϕ_a, ϕ_d, T_s, Pe_R) can be defined asThe general solution iswhere ν ≡ πa² is the projected area of a particle. This definition for the nonequilibrium free energy agrees with the true thermodynamic free energy for molecular or colloidal solutes in solution (i.e., f^act(ϕ_a = 0, ϕ_d, T_s, Pe_R) = f^osm).¹⁴ To gain further insight into the free energy, in the Appendix we analyze the limits of f^act for our mixture for large and small concentrations of active swimmers relative to passive particles.

As done previously for a purely active system,³ we can derive the nonequilibrium chemical potential for multicomponent mixtures using purely mechanical arguments (see Appendix C). For a mixture of active and passive particles, it is given by

Again this expression agrees with the rigorous thermodynamic definition of the chemical potential for mixtures of molecular solutes in solution.¹⁴ The chemical potential for each species i in a multicomponent system can thus be obtained fromwhere the reference states were absorbed into the free energy. We can invoke eqn (15) and (1) to obtain the chemical potential for the active (μ^act_a) and passive (μ^act_d) species.

From the thermodynamics of mixtures, the stability criterion using the free energy is given by det(∂²f/∂ϕ_i∂ϕ_j) = 0.¹⁴ For our system this reduces to

This gives us the reorientation Péclet number as a function of the active and passive concentrations, Pe_R = Pe_R(ϕ_a, ϕ_d).

The dashed curve in Fig. 5 is the spinodal curve using eqn (18) for a fixed active swimmer fraction x_a = 0.5. This spinodal boundary does not agree with the simulation data of Stenhammar et al.,²⁰ as eqn (18) predicts a different phase boundary than those observed in a simulation. The simulations reflect a global dilute/dense phase separation based upon fluctuations in the total particle (both active and passive) density. In contrast, eqn (18) interrogates the stability of the free energy due to fluctuations in the active particle concentration while keeping the passive particle concentration fixed, and vice versa.

This facilitates an important consideration in both experiments and simulations about which variables are held fixed and varied. Depending on the ensemble of variables that are held fixed (active swimmer density, composition, etc.), the theory predicts in general different phase boundaries. To produce a phase diagram in a simulation, one typically fixes the overall swimmer composition x_a and swimmer Pe_R, and varies the total area fraction ϕ or vice versa. This corresponds to a global dense/dilute-phase separation based upon fluctuations in the total particle density, which is well described by the mechanical instability criterion (∂Π^act/∂ϕ)_xa,Ts,PeR = 0, as shown by the red solid curve in Fig. 5.

In the experiments of Kümmel et al.,²¹ the active swimmer area fraction (ϕ_a = 0.01) and Péclet number were held fixed, and the passive particle area fraction (ϕ_d) was varied. The ensemble of variables that we fix and vary must therefore be considered when we predict of the phase behavior of active mixtures. It is likely that one can conduct an experiment or simulation where the phase behavior agrees with the thermodynamic spinodal det(∂²f/∂ϕ_i∂ϕ_j) = 0 (red dashed curve in Fig. 5). There remains much more to the phase portrait than the existing studies and our mechanical theory have revealed.

7 Conclusions

We developed a simple mechanical theory to address an important question in active matter: do active particles thermally equilibrate, and if so, what is the quantity that gets shared upon collisions? We found that the swimmers' activity k_sT_s ≡ ζU₀²τ_R/2 does not have the same properties of the thermal energy k_BT. The swimmers' capability to have run lengths U₀τ_R small or large compared to their size a (and other length scales in the problem) distinguishes them from passive Brownian particles whose step size is smaller than any other length scale in the system.

We discovered that for Pe_R ≡ a/(U₀τ_R)≪ 1 the quantity that gets shared upon collisions is Pe_R, not the scale k_sT_s. This was seen in the simple mixing experiment in Fig. 1 and from analyzing the motion of a passive particle as a probe to measure the kinetic activity of the swimmers (k_sT_s). The notion of the swimmers' energy k_sT_s and/or Pe_R being shared via collisions is an interesting concept that may facilitate further theoretical and experimental studies.

Another fundamental difference between an active system and a classical fluid was found by observing the motion of a passive particle in a sea of active swimmers. Even after undergoing many collisions with swimmers, the passive bath particle ceases to move (aside from its translational Brownian motion) if the swimmers are removed because of the damping by the solvent. In contrast, a passive bath particle placed inside a classical molecular or colloidal solution keeps its k_BT activity even when the other particles are removed. Because the swimmers must continuously collide into the passive bath particle to impart information about their kinetic activity, there is no “thermal equilibration” that takes place in an active suspension.

To understand the temperature and phase behavior of active matter, we studied a mixture of active and passive Brownian particles. Our theory applies more generally to a mixture of active systems with different activities. In fact, we showed that a passive Brownian particle behaves equivalently to a “swimmer” with Pe_R → ∞, so the active/passive mixture corresponds to a limiting case of a mixture of active systems with different activities. A swimmer that takes small steps and reorients rapidly is indistinguishable from a purely Brownian particle if it is placed behind an osmotic barrier. For a mixture of active particles with different, finite Pe_R, we would simply write the swim and collisional pressures for each individual species Pe⁽¹⁾_R, Pe⁽²⁾_R, etc. The total active pressure of the system is a sum of the contributions from all species, as in eqn (1).

By understanding the dependence of the active swimmer composition x_a and the total area fraction ϕ in each of the active pressure contributions, we obtained an explicit equation of state for the active/passive mixture. The key principle in deriving the equation of state was that a swimmer imparts the same displacement whether it collides into another swimmer or a passive particle. We found that the swim pressure decreases with increasing area fraction and is the destabilizing term that leads to a phase separation in active systems. In contrast, the interparticle (collisional) pressure increases monotonically with the area fraction and helps to stabilize the suspension from phase separation. The competition between these two effects is determined by the reorientation Péclet number, Pe_R ≡ a/(U₀τ_R). The spinodal specifies the regions in the phase diagram where these two opposing effects cancel precisely, and these regions were identified in the Pe_R − ϕ space for our mixture.

We corroborated our theory with recent simulations²⁰ and experiments²¹ of active/passive mixtures. Our simple model may be a useful tool for predicting phase behavior in both experiments and simulations, as many regions of phase space are difficult to explore because of experimental and computational challenges of covering the parameter space.

We found that different stability conditions give rise to different phase boundaries, facilitating considerations in simulations about which variables are held fixed and varied. The derivative of our active pressure with respect to the total area fraction predicts accurately the global dense/dilute phase transitions observed in simulations. To predict the local phase separation within the dense or dilute phase (as in immiscible polymer mixtures), a different stability criterion is required. Finally, we extended the mechanical theory to determine the nonequilibrium chemical potential and free energy for a mixture of active and passive species.

Extension of our theory to 3D and for different particle size ratios is straightforward. In 3D the characteristic activity scale becomes k_sT_s ≡ ζU₀²τ_R/6 instead of ζU₀²τ_R/2 due to the extra degree of freedom. For a mixture of particles with different sizes a and b, the pair-distribution function adjusts to different collision pairs g_aa(ϕ), g_ad(ϕ), etc because now the particle–particle separation at contact is different. For a polydispersed active system, the large clusters are no longer crystalline and are less stable than those in a monodisperse system. Therefore the two-phase region in Fig. 5 shrinks and shifts to smaller Pe_R.

In our model we neglected hydrodynamic interactions between the particles, which may contribute additional terms such as the “hydrodynamic stresslet”³⁰ to the active pressure. We also did not consider the effects of polar order and alignment of the swimmers, which are not necessary for phase-separating systems.

Appendix

A. Micromechanical equations of motion

The active particle dynamics are governed by the N-particle Langevin equationwhere U is the translational velocity, ζ is the hydrodynamic drag factor, F^swim ≡ ζU₀ = ζU₀q is the self-propulsive swim force, U₀ is the swim speed, θ specifies the swimmers' direction of motion q = (cos θ, sin θ), F^P is the interparticle force between the particles to enforce no overlap, Λ_T and Λ_R are unit random normal deviates, τ_R is the orientation time of the swimmer, and D₀ is the Stokes–Einstein–Sutherland translational diffusivity. The passive Brownian particles are governed by the same equation but without the self-propulsive force:where the subscript “d” indicates a passive particle. For simplicity in this work we considered spherical particles with the same size for active and passive particles so that ζ = ζ_d. The left-hand side of eqn (A1) and (A3) is zero since inertia is negligible for colloidal suspensions. A more detailed discussion concerning the origin of the swim force and the role of hydrodynamic interactions is available elsewhere.¹⁷

B. Limits of active free energy

To gain further insight into the free energy, we analyze the limits of f^act for our mixture system for large and small concentrations of active swimmers relative to passive particles. Expanding the active free energy for small ε = ϕ_d/ϕ_a = (1 − x_a)/x_a, we find in the limit of large active concentrationwhere ν ≡ πa² is the projected area of a particle and the first term on the right is the active free energy for a purely active system:⁵and the second term in eqn (B1) is

Expanding the swim pressure for small ε′ = ϕ_a/ϕ_d = x_a/(1 − x_a), we find in the limit of small active concentration

where the first term on the right is the osmotic pressure of a purely Brownian suspension:and the second term in eqn (B4) isThe influence of the swim pressure and swimmers' interparticle collisions are present in the correction term.

C. Mechanical derivation of the chemical potential for multicomponent systems

The number density of an N_c-component system§ satisfies the conservation equationwhere j_i = n_iu_i = n_i〈u〉 + j^rel_i is the particle flux of species i, j^rel_i = n_i(u_i − 〈u〉) is the flux of species i relative to the suspension average velocity 〈u〉, which is defined as , and u_i and u_f are the number averaged velocity of swimmer species i and fluid at a continuum point, respectively. The total volume (or area) fraction of the particles is . Incompressibility requires the suspension-average velocity (particles plus the fluid) to satisfy ∇·〈u〉 = 0.

We apply an averaged macroscopic mechanical momentum balance to obtain an expression for j^rel_i. Following the standard Irving–Kirkwood approach, we obtain

where σ^act = σ^swim + σ^P is the active stress and the left-hand side is zero since inertia is negligible for colloidal systems. Using the relative flux j^rel_i = n_i(u_i − 〈u〉) we arrive at a relationship between the active particle flux and gradients in the active stress:We did not rely upon the notion of a thermodynamic chemical potential or the free energy to arrive at this expression.

We can use our mechanical derivation to define a nonequilibrium chemical potential by analogy to the quantity whose gradient would drive a flux:

where again . This definition is analogous to that of a thermodynamic system where the relative flux is driven by gradients in the thermodynamic chemical potential. Substituting eqn (C4) into eqn (C3) and using the definition Π^act ≡ −tr σ^act/2, we arrive atFor a two-component (active and passive) system, we have n_a(∂μ^act_a/∂ϕ) + n_d(∂μ^act_d/∂ϕ) = (1 − ϕ_a −ϕ_d)∂Π^act/∂ϕ, as given in the main text.

This relationship between the chemical potential and pressure is equivalent for a system of passive Brownian particles and active swimmers with small τ_R. We thus interpret μ^act as a natural definition and extension of the chemical potential for nonequilibrium systems.

Comparison to thermodynamics. From equilibrium thermodynamics,¹⁴ the chemical potential of species i for a multicomponent system is given bywhere ν is the volume (or area) of a particle. The free energy is related to the osmotic pressure by Taking the density derivative of both eqn (C6) and (C7) and combining the results, we obtainwhich is identical to eqn (16) of the main text, a result obtained using a mechanical derivation.

Therefore the mechanical derivations of the stress, momentum balance, and flux are in full agreement with thermodynamics.

Acknowledgements

SCT is supported by the Gates Millennium Scholars fellowship and the National Science Foundation Graduate Research Fellowship under Grant No. DGE-1144469. This work is also supported by NSF Grant CBET 1437570.

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Footnotes

† Electronic supplementary information (ESI) available. See DOI: 10.1039/c5sm01792k

‡ This however does not imply that the intrinsic swimming mechanism generates a long-range (1/r) Stokes velocity field as does an external force.^16,17

§ There are N_c + 1 total components, including the solvent.

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