Scalar φ⁴ field theory for active-particle phase separation

Abstract

Recent theories predict phase separation among orientationally disordered active particles whose propulsion speed decreases rapidly enough with density. Coarse-grained models of this process show time-reversal symmetry (detailed balance) to be restored for uniform states, but broken by gradient terms; hence, detailed-balance violation is strongly coupled to interfacial phenomena. To explore the subtle generic physics resulting from such coupling, we here introduce ‘Active Model B’. This is a scalar φ⁴ field theory (or phase-field model) that minimally violates detailed balance via a leading-order square-gradient term. We find that this additional term has modest effects on coarsening dynamics, but alters the static phase diagram by creating a jump in (thermodynamic) pressure across flat interfaces. Both results are surprising, since interfacial phenomena are always strongly implicated in coarsening dynamics but are, in detailed-balance systems, irrelevant for phase equilibria.

Introduction

Much recent research has addressed active materials, whose constituent particles violate microscopic time-reversal symmetry (TRS) by continuously converting fuel into motion. Systems with medium- or long-range orientational order, such as swarms of rod-like bacteria¹, can then be viewed as active liquid crystals, which are successfully described by adding minimal active terms to the established continuum equations for liquid crystal hydrodynamics. The resulting field theories involve either vector or tensor order parameters describing the local state of orientational order^2,3.

However, an important alternative paradigm addresses isotropic bulk phases of active (that is, self-propelled) colloidal particles. These can be natural microorganisms such as bacteria or algae^3,4, or synthetic microswimmers^5,6,7,8,9. In many such cases, the dynamics is approximately described by having a fixed relaxation time τ for the direction of self-propelled motion, but a non-trivial dependence of the propulsion speed υ(ρ) on the particle density ρ. If d ln(υ)/d ln(ρ) <−1, steady-state phase separation can be shown to arise through a positive-feedback mechanism in which a slowing of the particles leads to their accumulation, and vice versa^10,11. In biological systems, such slowing could arise through various causes such as a coupling of the quorum-sensing response of bacteria to their motility^4,12. For synthetic swimmers, which are often modelled as ‘active Brownian particles’ (ABPs) whose swimming direction rotates by diffusion, an effective decrease in υ at high density can result instead from collisional interactions^13,14.

This scenario of active-particle phase separation is now fully established in simulation studies^{13,14,15,16,17,18}, and partly confirmed by experiments^19,20. It cannot happen in systems with microscopic TRS for which steady-state accumulation by slowing is forbidden. Indeed, TRS of the steady state, also known as ‘the principle of detailed balance’, ensures that the equilibrium density is controlled solely by conservative inter-particle forces (via the Boltzmann distribution) and not by any density-dependent kinetic coefficients^10,21. For example, in an isothermal suspension of Brownian colloidal particles, the diffusivity decreases strongly at high density, with no effect on the phase diagram. In such ‘passive’ colloids, phase separation requires attractive interactions.

For passive colloidal systems, the resulting phase-separation kinetics is well described by dynamical field theories involving a conserved scalar order-parameter field φ, linearly related to the local density of colloids. Simplifying the underlying free energy to a quartic polynomial in φ with square-gradient terms, and assuming local diffusive dynamics, then gives a theory called ‘Model B’. Such scalar φ⁴ field theories (or phase-field models) have played a pivotal role in understanding phase separation in systems with TRS^22,23. This applies particularly when noise terms are neglected, creating a mean-field model that accurately captures the long-time dynamics of phase separation, which is dominated by the deterministic motion of sharply defined interfaces. In that limit, Model B becomes the simplest form of the Cahn–Hilliard equation²⁴, capturing a celebrated universal result, L~t^1/3, for the dependence of the domain size L on time t²⁵.

Despite the deep distinction between active and passive phase separation, explicit coarse-graining of the active dynamics at large scales establishes a partial mapping between the two cases. This mapping was found first for models of swimming bacteria with discrete reorientations¹⁰, but later extended to ABPs¹¹. At zeroth order in spatial gradients, which is equivalent to considering only systems of uniform density, the mapping allows a bulk free energy to be constructed whose instabilities are those of the active system^10,11. (In what follows, all thermodynamic quantities, such as pressure and bulk chemical potential, refer to those calculated within this mapping.) Detailed balance, while absent microscopically, re-emerges to this order. Recently, however, we studied the leading-order gradient terms and found these to break detailed balance once again¹⁴. This creates a new class of models in which the breaking of TRS is intimately linked to the physics of interfaces.

This feature distinguishes such models from others that address Cahn–Hilliard-like diffusive instabilities in systems without detailed balance (see for example, refs 26, 27, 28, 29, 30). The latter encompass many physical processes but are often too complex for their fundamental physics to be understood. In elucidating the generic physics of active-particle phase separation, it is therefore important to focus on the simplest model of the required structure.

In this article, we present and analyse just such a model. To create our new model, we add the simplest ‘non-integrable’ (as defined below) gradient term to what is otherwise the standard field theory for locally diffusive phase separation, namely Model B^22,23. The chosen gradient term breaks detailed balance in the standard, Passive Model B, which implies that the resulting Active Model B cannot be derived from any free-energy functional.

Below we report simulations of phase kinetics with Active Model B, which echo results found previously using a more elaborate continuum model inspired by ABP simulations¹⁴. We find that the non-integrable term does not greatly alter the dynamical fate of the system. This is remarkable, since coarsening dynamics is controlled by interfacial tension, which vanishes without the gradient terms, and so ought to be sensitive to their form. It also presents a paradox when confronted by another new result: Active Model B admits no static domain-wall solution connecting two bulk regions, whose chemical potential μ₀≡df₀/dφ with the bulk free-energy density f₀ takes the value set by the common tangent construction. (As recalled below, this construction holds at coexistence, independent of gradient terms, for all systems with detailed balance.) We explain the resulting ‘uncommon tangent’ result in terms of an activity-induced analogue of Laplace pressure that arises even across flat interfaces. This insight allows us to explain why active coarsening dynamics, en route to full phase separation, remain similar to that of the traditional Passive Model B.

Results

Active Model B

In line with the principles outlined above, we adopt the following dynamics for a conserved scalar order-parameter field φ(r, t) at position r and time t in d dimensions:

All quantities are made dimensionless by using ‘natural units’; these are υ(0)τ for length and the orientational relaxation time τ for time, where υ(0) is the swim speed of an isolated particle. The composition variable φ is related to the number density ρ(r, t) of active particles by a linear transform φ=(2ρ−ρ_H−ρ_L)/(ρ_H−ρ_L), where ρ_H and ρ_L are the densities of high- and low-density coexisting phases, respectively, as calculated for example from υ(ρ) by the methods of refs 10, 11.

Here equation (1) expresses conservation of φ, while equation (2) states that its mean current J–Λ is proportional to the gradient of a non-equilibrium chemical potential μ obeying equation (3) with a constant λ. Our nomenclature for μ is self-explanatory: even beyond equilibrium, the chemical potential is the quantity whose gradient causes the mean current. The vector Λ is a Gaussian white noise whose variance we take to be constant. This follows standard practice in Passive Model B, although in reality the variance is density dependent, as calculated explicitly for active particles in ref. 10. The noise is often neglected altogether for phase-separation studies³¹ and we generally ignore it below.

In equation (3), μ=μ₀+μ₁ is the sum of bulk and gradient contributions. The bulk part is chosen as the usual Passive Model B form, μ₀=−φ+φ³, so that at zeroth order in a gradient expansion, our Active Model B shares with its passive counterpart the bulk free-energy density of a symmetric φ⁴ field theory, f₀=−φ²/2+φ⁴/4. Note that the phase separation, driven by the negative linear term in μ₀, can arise from activity alone with no need for attractive interactions. The gradient term is the sum of two further terms. The first is an integrable or ‘passive’ piece , which can be written as a functional derivative of some free energy ∫f₁d^dr, while the second is an active part , which cannot.

We now make the standard Passive Model B choice, f₁=(∇φ)²/2, so that . In the passive case, choosing the total free-energy density f=f₀+f₁, in which f₀ is supplemented by the simplest square-gradient term f₁=(∇φ)²/2, captures all universal aspects of the underlying physics, while allowing a vastly simpler analysis of interfacial structure and dynamics than would a more realistic choice of f. The same advantages hold for our Active Model B. For the non-integrable term, we write ; the constant λ is a parameter of the model. This is the simplest addition to μ, at second order in gradients, that cannot be derived from a free-energy or Lyapunov functional. Note that this property is the definition of ‘non-integrable’ for the purposes of the current paper.

Explicit coarse-graining of the dynamics of ABPs^14,32 points to a specific structure of the gradient terms in equation (3) and leads to a gradient term μ₁=−κ(φ)∇²φ with κ(φ)=1+2λφ that combines exactly our λ term with an integrable part that corresponds to f₁=κ(φ)(∇φ)²/2. In the units of Active Model B, the parameter λ, whose sign can be absorbed into that of φ if preferred, is then negative and of order unity for ABPs (see Supplementary Note 1 where an explicit expression for λ is given). The same is true for run-and-tumble bacteria whose dynamics are almost equivalent¹¹. In both cases, λ is set primarily by the shape of the function υ(ρ), parameterizing the decay with density of the mean swimming speed. Meanwhile, dimensioned parameters such as υ(0) and τ serve to set conversion factors between Active Model B units and laboratory ones.

Replacement of the derived coarse-grained form of f₁ with one having constant κ=1 is standard practice for Passive Model B, just as one replaces a complicated coarse-grained f₀ with the standard form f₀=−φ²/2+φ⁴/4. (This is how Passive Model B comes to describe a wide range of microscopic models.) In Active Model B, we make exactly the same simplifications for all the integrable terms, while capturing new physics with the minimal, leading-order TRS-breaking term . Furthermore, we show in Supplementary Note 1 that all possible leading-order current contributions, up to third order in ∇ and second order in φ, are given by combining some choice of free-energy density f with this choice of .

Active Model B’s non-integrable term somewhat resembles the one arising in the celebrated Kardar–Parisi–Zhang equation for nonlinear interfacial diffusion³³, which was constructed on similar minimalist grounds, and likewise supported by direct contact with microscopic arguments for specific examples.

Dynamics of Active Model B

We have explored the dynamics of Active Model B numerically for both symmetric (∫φd^dr=0) and asymmetric (∫φd^dr≠0) quenches in which a uniform initial state, with slight noise, is evolved by equations (1, 2, 3). Our findings are in keeping with a previous study, which showed that the non-integrability leads to at most quantitative but not qualitative changes in coarsening dynamics¹⁴. This applies here also, bearing in mind that the λ term also breaks the φ→−φ symmetry of a symmetric quench. (In two dimensions this results in a droplet rather than a bicontinuous domain structure, resembling a slightly asymmetric quench.) Figure 1 shows snapshots of evolving domain structures for λ=0, −1, −2, with mean initial order-parameter fields corresponding to a symmetric quench (with average density φ₀≡‹φ(r,0)›=0) and a dilute quench (φ₀=−0.4). The case of a dense quench (φ₀=+0.4) is not very different except for a slightly slower coarsening rate. As noted above, results for positive λ can be generated by reversing the sign of φ. Corresponding data for |λ|≤0.1 is not shown, but is very close to the λ=0 case.

Figure 1: Transient domain structures.

Snapshots of evolving phase separation in two dimensions at time t=2,000 and λ=0, −1, −2 for (a) symmetric (φ₀=0) and (b) asymmetric (φ₀=−0.4) quenches. The plots shown have dimensions 256 × 256.

Figure 2 shows the time evolution of the characteristic domain size L(t)~t^α for various λ and φ₀ in two and three dimensions. Here we see some evidence for a downward drift with increasing |λ| of the exponent α from the well-known value α=1/3 of Passive Model B³¹. To distinguish a real asymptotic shift in α from a λ-influenced crossover would require an exhaustive computational study³⁴, which we leave for future work. Note, however, that a similar downward exponent shift was reported in ref. 14, but found there to be reproducible in an integrable Cahn–Hilliard model, albeit with a non-polynomial f₀, for which the asymptotic 1/3 power is assured. This suggests a crossover scenario in which all gradient terms, whether active or passive, gradually merge into an effective interfacial tension that drives t^1/3 coarsening, once L is very much larger than the interfacial width. Further arguments for this outcome are given below and in Supplementary Notes 2 and 3. While the case for a standard asymptotic t^1/3 scaling law is compelling, our next result shows that the λ term in Active Model B is certainly not representable solely by a shift in interfacial tension, since this would have no effect on phase equilibria.

Figure 2: Time evolution of the characteristic domain size.

Numerical results (points with error bars) and least-squares fits (straight lines) for the domain length scale L(t) for different λ and average density φ₀ in both two and three dimensions: (a) d=2, φ₀=0; (b) d=2, φ₀=−0.4; (c) d=3, φ₀=0; (d) d=3, φ₀=−0.4. Notice the apparent downward drift with increasing |λ| of the exponent α from the well-known value α=1/3 of Passive Model B. The error bars denote the s.d. of the numerical data from the corresponding fit curves. The definition of L(t) is given in the Methods section.

Failure of the common tangent construction

In Passive Model B (λ=0), bulk phase separation is characterized by two static coexisting phases of infinite extent that are separated by a planar interface. These bulk phases have densities φ→−1 and φ→1 for |z|→∞, respectively, where z is a coordinate perpendicular to the interface. The densities of the coexisting phases are set by the common tangent construction on f₀(φ), as is fully explained below, and imply μ₀=0 for |z|→∞. Furthermore, the constant densities for |z|→∞ imply μ₁=0 and therefore μ=0 in both bulk phases.

At first sight one expects the common tangent construction still to be possible for finite λ, because the active contribution to the non-equilibrium chemical potential, , vanishes in both bulk phases, and the construction itself cares only about the bulk free-energy density f₀(φ) and not about interfacial tension. This view, however, is mistaken because the construction implicitly assumes integrability, that is, the existence of a free-energy structure everywhere, including any gradient terms. Indeed, we prove here that no similar solution exists at μ=0 for Active Model B, which means that the common tangent construction fails whenever λ≠0.

Instead, for μ₀=0 we find only solutions describing spatially periodic (lamellar) phases. At first sight, these findings suggest that microphase separation, rather than coexisting bulk phases, might be the generic fate of Active Model B. This is, for example, what happens when Passive Model B is coupled to a somewhat different form of activity, namely logistic population growth¹⁵. It is also hinted at by various experiments in which active particles form clusters whose size seemingly remains finite at long times^9,35,36. Such an outcome would, however, be paradoxical given our numerical finding of bulk demixing rather than microphase separation in Active Model B. To resolve this paradox, we show below that a planar interface does exist between fully phase-separated states, but their bulk densities are not given by a common tangent construction on f₀.

To prove this, we look for a fully phase-separated state with a planar interface and take advantage of its two translational invariances to reduce the problem to one spatial dimension. For the required static solution of Active Model B, this means φ=φ(z) where z is a coordinate normal to the interface. To describe static bulk phase separation, the current in equation (2) (with Λ=0), specifically its z component J_z, must vanish. (We assume a uniform current is excluded by boundary conditions at infinity.) This requires μ to be constant. Let us first try to find its value by following the usual equilibrium reasoning. Constant μ requires equality of the bulk chemical potential terms μ₀=df₀/dφ and also of the bulk thermodynamic pressures P=φμ₀–f₀ in the two phases. (Recall that in non-equilibrium systems there is no general relation between the thermodynamic pressure thus defined and the mechanical pressure; we return to this point below.) Together, these conditions imply that a common tangent can be drawn to the curve f₀(φ) at the two coexisting densities. Since f₀ is symmetric, this implies μ→0 for |z|→∞, so that by this argument the non-equilibrium chemical potential μ, which differs from μ₀ only in interfacial regions, is zero everywhere. The static density profile φ(z) is thus given by

By renaming z and φ according to (z→t, φ→x), this equation for the density profile φ(z) can be mapped onto the equation of motion for the trajectory x(t) of a Newtonian particle of unit mass in a symmetric inverted potential U(x)=−f₀(x) under the influence of a velocity-dependent force that is invariant under time reversal. The resulting equation of motion is

This makes calculating the density profile φ(z) the same problem as finding the trajectory x(t) of a Newtonian particle in the potential U(x) with a velocity-dependent force arising from the λ term. In the passive case (λ=0), this Newton mapping is a standard procedure (see ref. 37 and references therein), whose details we recall in the Methods section. In brief, the domain-wall solution is described by a particle that leaves one of the two maxima in U(x) with infinitesimal velocity, travels across the valley in the inverted potential, and then comes to rest at the other maximum.

Given this picture, let us consider the effect of λ≠0, that is, Active Model B. This creates a velocity-dependent force in equation (5), , which has the same sign throughout the trajectory. Because of this, if the Newtonian particle starts very near the top of the first maximum of U(x), that is, if one starts from a large bulk domain close to the bulk density φ₁=±1, the particle either does not make the top of the other maximum or it overshoots. The former gives periodic oscillations between the initial coordinate x₁ and the turning point x₂, which physically describe microphase separation, while the latter gives an unphysical blowup. In the case of microphase separation, the Newtonian particle retraces its steps from the turning point x₂, which is finitely below the second peak, and regains exactly the kinetic energy it lost to the velocity-dependent force during the first half of the cycle. So it arrives exactly at x₁ where it started, comes virtually to rest there, creating another large domain of φ₁, before turning round and starting the cycle again. During this periodic motion, the particle only briefly visits the even peaks, creating a micro-domain of the φ₂ phase whose width is of order of the interfacial width. To make the domain of the second phase any wider, the velocity-dependent term has to be made exponentially small (that is, these domains have a width that varies as −log|λ|). Only this will allow the particle to closely approach the top of the other maximum.

If the Newtonian particle, on the other hand, starts very near the top of the second maximum of U(x), that is, if one starts with a very large domain of the second phase, and λ has the sign that prevents the particle from overshooting when it starts from the first maximum, the λ term gives energy to the particle so that it now overshoots. These arguments confirm that for λ≠0 and μ=0, static bulk phase separation is impossible.

Three resolutions can be envisaged. Either there ceases to be a static solution of Active Model B, or the solutions describing microphase separation become stable or μ is non-zero and the common tangent construction fails. We show next that the last of these resolutions applies.

The uncommon tangent construction

We now generalize the discussion from the previous section to allow a non-zero non-equilibrium chemical potential μ≠0 obeying equation (3). We show that for Active Model B, a solution with a planar interface between two static bulk coexisting phases does after all exist, but is shifted to a non-zero μ=μ_b, so that the coexisting densities are not φ=±1. We also derive the resulting non-equilibrium chemical potential at bulk phase coexistence μ_b as a function of the parameter λ.

For μ≠0 equation (4) for the density profile φ(z) becomes

and the corresponding Newtonian equation is equation (5) with the now asymmetric potential U(x)=μx−f₀(x). We define a critical value of |μ|=μ_c such that, whenever |μ|<μ_c, the potential U(x) has two maxima at positions and , with , and a minimum between them. When μ≠0 the heights of the two maxima of U(x) are different, and the sign of μ decides which maximum is lower. At |μ|=μ_c the minimum and the lower maximum merge into a saddle point, and as |μ| increases beyond the critical value μ_c, there remains only a single maximum. (We find μ_c=3^−1/2–3^−3/2≈0.38.) Notice that μ_c can also be defined as the maximum slope with which two distinct parallel tangents can be drawn on f₀(φ) (see Fig. 3). This maximum slope arises when one such tangent passes through the inflection point of f₀(φ), also known as the spinodal point.

Figure 3: Uncommon tangent construction.

μ is equal in coexisting phases, and the thermodynamic pressures P₁ and P₂ in each phase are the negative intercepts on the vertical axis. Here φ₁ and φ₂ are the densities of the two coexisting phases and ΔP_λ=P₂–P₁ is the pressure difference. This plot shows the construction for λ→∞, where μ₀(φ₁)=μ_c.

Owing to these properties of the potential U(x)=μx−f₀(x), for a Newtonian particle with the equation of motion given in equation (5), there are three different types of motion possible: (i) a periodic oscillatory motion between two turning points x₁ and x₂ with describing microphase separation, (ii) the non-periodic limiting case and that corresponds to bulk phase separation and (iii) divergent solutions corresponding to unphysical density profiles. Since there is only one maximum for |μ|≥μ_c, microphase separation and bulk phase separation are possible only for |μ|<μ_c. For |μ|≥μ_c, on the other hand, no physical solution for the static density profile φ(z) exists.

In the following, we consider the situation |μ|<μ_c and treat the appearance of bulk phase separation as a special limiting case of microphase separation. The density distribution φ(z) for bulk phase separation is a special solution of equation (6) or equivalently equation (5) with U(x)=μx−f₀(x) for an appropriate value μ=μ_b of the non-equilibrium chemical potential, which depends on the parameter λ. Equation (5) with U(x)=μx−f₀(x) is an autonomous nonlinear second-order ordinary differential equation (ODE) that cannot be solved analytically. The function μ_b(λ), however, can be derived from this equation even without solving it. The procedure is detailed in the Methods section and gives an analytic result of 4/15 for the slope of μ_b(λ) at λ=0 as well as an implicit function for μ_b(λ), which is plotted in Fig. 4.

Figure 4: Non-equilibrium chemical potential at bulk phase coexistence.

Analytical result (solid line) and numerical results obtained from simulations using Active Model B (points with error bars that denote the estimated discretization error) for μ_b(λ). For λ→∞, the function μ_b(λ) asymptotically tends to μ_c=3^−1/2–3^−3/2 (dashed line). Only positive λ is shown, since μ_b(λ) has the symmetry property μ_b(−λ)=−μ_b(λ).

With a much higher computational expense, the function μ_b(λ) can also be determined from simulations that solve the dynamical equation (1) of Active Model B by evolving until a steady-state interfacial profile is reached. The results of such simulations are also shown in Fig. 4. The function μ_b(λ) is zero for λ=0, increases monotonically for λ>0, and asymptotically approaches the limiting critical value μ_b(∞)=μ_c for λ→∞. Using the function μ_b(λ), the coexisting densities φ₁=φ(−∞) and φ₂=φ(∞) are given by the smallest and largest solution of U′(φ)=0.

The above arguments demonstrate that solutions describing static bulk phase separation exist for all values of λ, but are shifted to μ=μ_b≠0 and involve coexisting densities φ≠±1. Other static solutions describing, for example, lamellar microphase separation, also exist but are generally not the result of numerically solving equation (1). Such states are generically unstable to the same Ostwald ripening mechanism as in the passive case (see Methods section); this is confirmed in Supplementary Note 3 for the case of droplets in higher dimensions.

Bulk phase coexistence in Active Model B is thus governed by an ‘uncommon tangent’ construction in which tangents to f₀(φ) are parallel, but displaced from each other (see Fig. 3). Defining as usual the bulk thermodynamic pressure by P=φμ₀−f₀, the λ-induced vertical displacement is identified as a pressure difference ΔP_λ between the coexisting bulk phases. The common tangent construction equates both μ₀ and P in these phases; its failure is thus attributable to a λ-induced jump in thermodynamic pressure across the planar interface that separates them. This is a very interesting, and somewhat unexpected, consequence of activity and the resulting breakdown of detailed balance in the interfacial regions.

Active pressure competes with Laplace pressure

The active pressure jump ΔP_λ has no direct counterpart in passive systems. However, for a passive droplet with radius R the interfacial tension γ (where in Passive Model B³⁴) creates a Laplace pressure jump ΔP_L=(d−1)γ/R across the interface. In Supplementary Note 2, we establish that to leading order in large R (that is, small |λ|), a static μ=0 solution exists for a droplet of radius at which the active and Laplace pressures are equal and opposite; common tangency is thereby restored. This can happen only for one of the two possible dispositions of internal and external phases, set by the sign of λ. This result shows that the activity-induced pressure jump ΔP_λ has, within our mapping from active onto passive phase separation, the same ‘thermodynamic’ status as the Laplace pressure.

The existence of these static droplet solutions raises once again the possibility of microphase separation. For instance, one could envisage a state of droplets, each with radius R*, embedded in a continuous phase at zero μ. If stable, such a phase might explain various experiments showing formation of small finite clusters in bacteria and artificial active colloidal particles^9,35,36. (On dimensional grounds, .) However, this would contradict the numerics reported above. We resolve this by noting that even the single-droplet solution is unstable, as shown in Supplementary Note 2. The instability appears to be fundamentally no different from the classical one of a finite fluid droplet in unstable equilibrium with its vapour (see Supplementary Fig. 1). Indeed, in Passive Model B a stationary value of the droplet radius R₀ exists for μ>0, but this is an unstable fixed point separating shrinking from growing droplet states. Moreover, as shown quantitatively in Supplementary Note 3, the active pressure ΔP_λ offsets the relation between μ and the droplet size, but cannot halt the coarsening of an assembly of droplets by Ostwald ripening, in which small ones evaporate and large ones grow (see Supplementary Fig. 2). This reasoning again points firmly towards the usual diffusive growth law L~t^1/3, which holds for Ostwald-like dynamics in both droplet and bicontinuous morphologies²⁵, and explains in large part the remarkable similarity between active and passive phase separation. It also explains why, in the active case, microphase separation is not the generic end point of the dynamics; via Ostwald ripening, coarsening proceeds indefinitely. The deviations from α=1/3 found numerically are, in this view, almost certainly transient or crossover effects.

Mechanical versus thermodynamic pressure

In future work we will generalize this study to allow for coupling of an active scalar field to a momentum-conserving solvent flow. There we will present an active counterpart of ‘Model H’, which describes such coupling for passive systems^22,23. This raises an intriguing issue concerning the nature of the pressure, which we have defined in this paper as P=φμ₀−f to coincide with the standard bulk thermodynamic definition. In passive systems, this is of course equal to the mechanical pressure defined either as a force per unit area on a boundary, or via the diagonal part of the stress tensor. Accordingly for instance, the Laplace pressure jump across a curved interface can be measured directly by a mechanical probe in a passive system.

However, the equivalence of mechanical and thermodynamic pressure in such a system stems ultimately from the fact that the same inter-particle forces determine both mechanics and thermodynamics. In contrast, for active particles, even the integrable part of the free energy has no simple link to inter-particle forces: instead it encodes the effects of density on self-propulsion through the mapping of ref. 10. This means that the quantity P found via that mapping cannot generally be viewed as an actual mechanical pressure. This is unsurprising for a system far from equilibrium; indeed, mechanical and thermodynamic pressures differ even in systems quite close to equilibrium, such as flowing fluids. The active pressure jump across a flat interface could not therefore be measured with a pressure gauge, but is still a pressure jump in the thermodynamic sense that P=φμ−f has different values in the two bulk phases. For a curved interface, exactly the same remarks apply to the Laplace pressure contribution, whenever the phase separation itself, and hence the resulting interfacial tension, is activity-driven. Indeed, the activity-driven interfacial tension is itself not detectable with a tensiometer; its meaning stems from its ability to drive a diffusive flux of active particles via the non-equilibrium chemical potential appearing in equation (2).

When coupling the active system to a momentum-conserving solvent via the Navier–Stokes equation, it is the mechanical pressure that enters, not the thermodynamic one. As we will address in detail elsewhere, in practice this means that an Active Model H must combine the diffusive dynamics of Active Model B developed here with a conceptually separate account of the mechanical forces created by self-propulsion.

That said, if we restrict attention to the class of systems where bulk phase separation is caused solely by inter-particle attractions, so that the only effect of weak activity is to create a non-zero value of λ, even the mechanical pressure in the two bulk phases is then unequal. This is because the gradient terms controlled by λ vanish in bulk: therefore, ordinary equilibrium thermodynamics holds locally, and mechanical and thermodynamic pressure must once again coincide in each bulk phase. The activity represented by λ plays a direct role only near the interface, where it creates the uncommon tangent condition. This obliges the system to develop a real, physically measurable mechanical pressure jump between phases, equal to our active pressure. Note that our active pressure is not just the mechanical pressure that a confined active gas exerts on a container wall³⁸ and that results directly from the increased speed and surface accumulation of active particles compared with passive ones. In contrast to the pressure of an active gas, our pressure jump across the interface between a dilute and a dense phase of active particles is not proportional to the density and its origin is more subtle.

Discussion

We have argued that, when no solvent is present so that dynamics is diffusive, the physics of active-particle phase separation is fully captured by Active Model B. This combines a φ⁴ bulk free energy with passive and active gradient terms in a minimal fashion. It represents an intriguing class of problems involving diffusive phase separation in systems where detailed-balance violations are created primarily by interfaces. The minimalist structure of Active Model B allows not only for efficient simulation, but also for several analytic results to be obtained. These explain why such detailed-balance violations have paradoxically small effects on coarsening dynamics, for which interfacial physics is usually dominant, but large ones on the phase diagram, for which such physics is, at first sight, irrelevant. Somewhat similar equations have been used recently to study crystal growth at finite undercooling^29,30 and state selection in shear-banding rheology³⁹. In these cases, it is known that non-integrable gradient terms can destroy the common tangent construction for phase coexistence^30,39. In Active Model B the physics that replaces common tangency is both simple and remarkable: the equality of chemical potential between phases is maintained, but activity creates a direct analogue of the thermodynamic Laplace pressure operating across the interface between bulk phases which, unlike its equilibrium counterpart, remains finite even for a planar interface.

Our work sheds direct light on how the non-integrable term in Active Model B leads to new and unintuitive physical predictions for active-particle phase separation. The role played by this active term can best be understood in terms of the well-established bulk mapping between active and passive phase separation, as extended here to include an interfacial tension plus an interfacial pressure jump that has no passive counterpart. Our results are all the more powerful and surprising, because we have so far found no convincing route to obtain them by qualitative reasoning applied directly to the motion of active particles. It is very difficult for such reasoning to capture the unusual structure of the problem, which involves restoration of TRS (absent microscopically) at zeroth order in spatial gradients and its loss again at higher order. A fully microscopic interpretation of our results may therefore remain elusive.

Methods

Numerical analysis

To solve Active Model B numerically, a finite-difference scheme with periodic boundary conditions was applied. The initial spatial distribution of the order-parameter field φ was random for the numerical calculations whose results are shown in Figs 1 and 2 and a step function (with φ=−1 for 64≤z≤192 and φ=1 otherwise) for the results shown in Fig. 4. Both for simulations in two and three spatial dimensions, the time step size was Δt=0.001, while the spatial step size was Δz=0.5 for two dimensions and Δz=1 for three dimensions. The lattice size was mainly 256 × 256 for two dimensions with the exception 256 × 50 for the numerical calculations corresponding to Fig. 4 and 128 × 128 × 128 for three dimensions. Finally, the domain length scale L(t) in Fig. 2 was calculated from the inverse of the first moment of the spherically averaged structure factor S(k,t)³⁴:

The spherically averaged structure factor is defined as

where k=||k|| is the modulus of the wave vector k, φ(k, t) is the spatial Fourier transform of the order-parameter field φ(r, t) and denotes an average over a shell in k space at fixed k.

Newton mapping

Here we summarize the Newton mapping for the passive case where λ=0 (see Fig. 5 for a sketch). The inverted potential U(x)=−f₀(x) has two maxima of equal height at x=±1 with a minimum in between. For x <−1 and x>1 it falls off to minus infinity. Notice that the starting position x₁=x(−∞) of the Newtonian particle with the equation of motion (5) corresponds to the density φ₁=φ(−∞) of the first bulk phase. The Newtonian particle can start from rest from one of three qualitatively different domains: (i)−1<x₁<1, (ii) x₁ε{−1,1} and (iii) x₁<−1 or x₁>1. In case (i) the Newtonian particle oscillates indefinitely between its starting position x₁ and a turning point at position −x₁. For the density distribution φ(z) this describes a lamellar state corresponding to microphase separation. (This state could be viewed in one dimension as a series of droplets each of exactly equal size and separation.) Such a state is unstable (in any dimension) to small density variations causing large droplets to grow at the expense of small ones. This is the Ostwald process discussed further in Supplementary Note 3.

Figure 5: Sketch of Newton mapping.

Calculating the density profile φ(z) of a system with free-energy density f₀(φ) is the same problem as finding the trajectory x(t) of a Newtonian particle in the external potential U(x)=−f₀(x). This sketch shows bulk phase separation in Passive Model B for λ=0 and d=1 as an example.

On the other hand, in the more particular case (ii) the Newtonian particle starts from rest infinitesimally below one peak of the potential U(x), where it hovers a long time before starting to move, shoots up the other side and then comes to rest infinitesimally below the other peak. This can be done in either direction and gives the familiar domain-wall solution that describes bulk phase separation. Finally, in case (iii) the Newtonian particle accelerates indefinitely towards x=−∞ or x=∞. This corresponds to a divergent and unphysical solution of equation (4), which is anyway ruled out by the boundary conditions. Imposing φ(z)=±1 at |z|→∞ selects the bulk interfacial profile as the only stable, non-uniform, static solution of Passive Model B.

Determination of the coexistence condition

To find the function μ_b(λ), we first reduce the order of equation (5) with U(x)=μx−f₀(x). The substitution transforms it into a non-autonomous nonlinear first-order ODE. A further non-invertible substitution w(x)=ν²(x) then leads to the linear first-order ODE

This equation describes the squared velocity w(x) of a Newtonian particle, with the equation of motion (5), as a function of the particle’s position x. Equation (9) is much simpler than the ODE (5) with U(x)=μx−f₀(x) and can be solved analytically as

with the initial value w₀=w(x₀) at the initial position x₀=x(0), and the polynomial g_λ(x)=6+12λx+4λ²(3x²−1)+8λ³(−μ−x+x³).

With the help of equation (10) the condition for microphase separation can be written as follows: there are two positions x₁ and x₂ with so that w(x₁)=w(x₂)=0. This condition is equivalent to h_λ(x₁)=h_λ(x₂) with h_λ(x)=g_λ(x)e^−2λx. Bulk phase separation as a limiting case of microphase separation appears for and . The condition for bulk phase separation can thus be written as

Notice that the last term in g_λ(x) and therefore also in h_λ(x) vanishes for . The condition (11) depends only on μ, which we denote as μ_b in the case of bulk phase separation, and λ. This means that bulk phase separation requires a relation between μ_b and λ described by a function μ_b(λ).

For (which also implies ; notice that μ_b=0 for λ=0) the function μ_b(λ) can be found from the implicit equation (11) by expanding the latter in both λ and μ_b. The resulting perturbative solution of equation (11) is given by

Also the property μ_b(−λ)=−μ_b(λ) follows from equation (11), so that there are no terms of even order in equation (12).