Classical Nucleation Theory for Active Fluid Phase Separation M.E. Cates1and C. Nardini2 3 1DAMTP Centre for Mathematical Sciences University of Cambridge Wilberforce Road Cambridge CB3 0WA UK

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Classical Nucleation Theory for Active Fluid Phase Separation

M.E. Cates1and C. Nardini2, 3

1DAMTP, Centre for Mathematical Sciences, University of Cambridge, Wilberforce Road, Cambridge CB3 0WA, UK

2Service de Physique de l’Etat Condensé, CEA, CNRS Université Paris-Saclay, CEA-Saclay, 91191 Gif-sur-Yvette, France

3Sorbonne Université, CNRS, Laboratoire de Physique Théorique de la Matière Condensée, 75005 Paris, France

(Dated: February 14, 2023)

Classical nucleation theory (CNT), linking rare nucleation events to the free energy landscape of

a growing nucleus, is central to understanding phase-change kinetics in passive ﬂuids. Nucleation in

non-equilibrium systems is much harder to describe because there is no free energy, but instead a

dynamics-dependent quasi-potential that typically must be found numerically. Here we extend CNT

to a class of active phase separating systems governed by a minimal ﬁeld-theoretic model (Active

Model B+). In the small noise and supersaturation limits that CNT assumes, we compute analyti-

cally the quasi-potential, and hence nucleation barrier, for liquid-vapor phase separation. Crucially

to our results, detailed balance, although broken microscopically by activity, is restored along the

instanton trajectory, which in CNT involves the nuclear radius as the sole reaction coordinate.

Active ﬂuids dissipate energy at the microscale: each

constituent particle extracts energy from the environ-

ment and uses it to overcome frictional or viscous drag

and create motion [1, 2]. Phase separation is ubiqui-

tous in active systems: as in equilibrium, it can stem

from attractive forces [3, 4], such as adhesion which un-

derlies compartmentalization in biological tissues [5–7].

Phase separation can also emerge for purely repulsive

motile particles [8–10], a situation with no equilibrium

counterpart. Recently it was shown that active phase

separation can displays non-equilibrium features at the

macroscopic scale, such as negative surface tensions [11–

13], mesoscopic currents in the steady state [12, 14–16],

or highly dynamical clustering [17–19]. Below we address

the simplest case where the active system undergoes bulk

ﬂuid-ﬂuid phase separation. Although at ﬁrst sight this

resembles closely the equilibrium case [8, 20–25], detailed

balance remains broken mesoscopically in the presence of

density gradients [25, 26]. In phase-ﬁeld type models, the

resulting interfacial activity alters the binodal densities

at coexistence [27, 28]. It must likewise be accounted for

to properly deﬁne the pressure in particle-based mod-

els [29].

A crucial feature of phase-separating systems is homo-

geneous nucleation, a rare event causing the formation of

a distinct phase by growth of a nucleus within the bulk

of a metastable parent phase. This growth is driven by

noise until a critical radius is reached whereafter it pro-

ceeds spontaneously. In passive ﬂuids, Classical Nucle-

ation Theory (CNT) [30, 31] states that the probability

of nucleating a liquid droplet in a vapor with supersat-

uration is given, within the large deviations limit of

low temperature T, by Pexp (−Ueq(Rc)/kBT). Here,

stands for logarithmic equivalence [32] and kBis the

Boltzmann constant. In three spatial dimensions, the

free energy barrier is given by

Ueq(Rc) = 4π

3σeqR2

c,eq +O(Rc, T )d= 3 (1)

in terms of the critical radius is Rc,eq = 2σeq/(f0(φs)∆φ−

∆f)and σeq is the surface tension of the interface. Here,

∆φ=φ2−φ1and ∆f=f(φ2)−f(φ1)where φis the

order parameter (e.g., particle density); f(φ)is the corre-

sponding free-energy density; φ1,2represent respectively

the vapor and liquid binodals, and φs=φ1+. CNT

holds for small supersaturation ( |φ1|) such that the

critical nucleation radius Rcis large compared to the

interfacial width. It assumes that the nucleus remains

almost spherical, which is true for ﬂuid-ﬂuid phase sep-

aration in the regime just delineated. CNT equally de-

scribes nucleation of vapor from liquid by interchanging

1↔2. The vast literature on CNT has inter alia aimed

at testing it experimentally and numerically [31, 33]; at

improving its predictions beyond the limit of small su-

persaturation [34]; at describing systems where multiple

pathways to nucleation are present [35], and at assessing

the relative importance of homogeneous and heteroge-

neous nucleation [36].

It has been suggested that CNT might be extended

to address nucleation in phase-separating active sys-

tems [37–39], but there has been limited progress along

these lines so far. We are aware of one study, restricted to

hard-core non-Brownian particles, which assumes a nu-

cleation pathway via single-monomer attachments, and

requires ﬁtting parameters to get quantitative agreement

with simulations [38]. Below we address instead CNT via

statistical ﬁeld theory. Here we will ﬁnd that the stan-

dard analysis for passive systems can be extended with

surprising completeness to the active case.

Classical nucleation theory is one prominent instance

of large deviation theory (LDT) [32, 40], which ad-

dresses rare events in settings ranging from solid state

physics [41] and physical chemistry [42] to ﬁnance [43],

turbulence [44, 45], and geophysical ﬂows [46, 47]. In

thermal equilibrium systems, event rates can be found

from the free energy barrier, e.g. via (1) above (although

dynamical methods can also be used [48]). By working

with the free energy, one also accesses the typical dy-

namics of the rare event: time-reversal symmetry ensures

arXiv:2210.05263v2 [cond-mat.soft] 13 Feb 2023

that the most probable route up the barrier (the so-called

instanton path) is the time-reversal of the noiseless (re-

laxational) downward path [49, 50].

The situation is very diﬀerent in non-equilibrium sys-

tems such as active matter. Within LDT [32, 40] the free

energy is replaced by the quasi-potential [49, 50], but this

is unknown a prori, and the instanton is not in general

the time-reversal of the relaxational path. Computing

the quasi-potential and/or instanton represents an intrin-

sically dynamical problem which, even in the small noise

limit of LDT, is rarely achievable analytically. (Only for a

few minimal models was the quasi-potential found either

exactly [51–53], or by perturbation theory [54–56].) Even

from a numerical perspective, studying rare events with-

out detailed balance is much more complex than at equi-

librium; dedicated algorithms developed for this task [57]

include cloning [58, 59], instanton-based codes [60–62],

and other approaches [63–67]. Accordingly, while intense

research into rare events in active systems was recently

initiated [66–72], this has been mainly numerical.

In this Letter we extend CNT to active ﬂuid phase

separation, using statistical ﬁeld theory. We can thereby

access analytically nucleation rates and quasi-potentials

for a generic class of non-equilibrium, many-body sys-

tems. This is possible because, although activity breaks

detailed balance, this is restored along the instanton tra-

jectory, which in CNT involves a single reaction coordi-

nate (the droplet radius) with noise that we infer from

the inﬁnite-dimensional Langevin equation for the order

parameter ﬁeld. Our results are given for Active Model

B+ (AMB+) [12, 25], a canonical ﬁeld theory for active

phase separation. However, the analysis route just out-

lined should be open whenever CNT’s precept of a single

reaction coordinate is applicable.

In their simplest form [12, 27, 73], statistical ﬁeld theo-

ries of active phase separation only retain the evolution of

a composition or density ﬁeld, φ. (Hydrodynamic [16, 74]

or polar [75] ﬁelds can be added if required.) Their con-

struction proceeds via conservation laws, symmetry ar-

guments, and an expansion in φand its gradients, along

lines long established for Model B, which describes pas-

sive phase separation [76–78]. In the active case, locally

broken time-reversal symmetry implies that new non-

linear terms are allowed. The ensuing minimal theory,

AMB+, includes all terms that break detailed balance

up to order O(∇4, φ2)[12, 25]:

∂tφ=−∇ · J+√2DΛ(2)

J/M =−∇µλ+ζ(∇2φ)∇φ(3)

µλ[φ] = δF

δφ +λ|∇φ|2.(4)

Here F=Rdrhf(φ) + K(φ)

2|∇φ|2i, with f(φ)a double-

well local free energy density, and Λis a vector of zero-

mean, unit-variance, Gaussian white noises. Below we

choose unit mobility (M= 1); set Kconstant (though

our results can be extended to any K(φ)>0); as-

sume constant noise D; and choose f(φ)as a quartic

polynomial. These are standard simpliﬁcations for pas-

sive Model B, which is recovered, setting D=M kBT,

at vanishing activity (λ=ζ= 0) [76], and leads to

(1). As shown in [12, 28] the explicit coarse-graining of

quorum-sensing particle models leads ζ= 0, while non-

vanishing ζand λare obtained when two-body forces are

included [12, 79]. Note that the ζterm in (3) can be

written, via Helmholtz decomposition, as −∇µζ+∇∧A,

whose second, divergenceless part does not aﬀect the φ

dynamics in (2). Thus we deﬁne a total chemical poten-

tial µ=µλ+µζ, with µζnonlocal in φ[80].

Let us denote by φ1,2the binodal densities at which

bulk vapor and liquid phases coexist. Within LDT, these

can be calculated at mean ﬁeld (D→0) level; with-

out activity this amounts to global minimization of F.

For AMB+ they are instead found by changing variables

from φand fto ψand g: these solve K∂2ψ/∂φ2=

(ζ−2λ)∂ψ/∂φ and ∂g/∂ψ =∂f/∂φ, where in uniform

bulk phases ∂f/∂φ =µas deﬁned previously. It follows

that ψ=K(exp[(ζ−2λ)φ/K]−1) /(ζ−2λ)[12, 28]. In

transformed variables, the binodal densities φ1,2obey the

usual equilibrium conditions: µ1=µ2and (µψ −g)1=

(µψ −g)2[12, 28]. This change of variables vastly sim-

pliﬁes the mathematical construction of phase equilibria

but we show in [80] how our main results can be found

without them.

Figure 1. A nucleating liquid (orange) droplet in a vapor

(blue) environment, showing the notation used in the text.

We now consider, as in Figure 1, the nucleation of a

liquid droplet of mean radius Rtin a supersaturated va-

por with density at inﬁnity φs=φ1+(vapor-in-liquid

nucleation can of course be addressed likewise). We de-

tail our analysis in d= 3 but our main results are valid

in dimensions d≥2; the case d= 2 involves bound-

ary terms and we treat it separately below. Following

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ClassicalNucleationTheoryforActiveFluidPhaseSeparationM.E.Cates1andC.Nardini2,31DAMTP,CentreforMathematicalSciences,UniversityofCambridge,WilberforceRoad,CambridgeCB30WA,UK2ServicedePhysiquedel'EtatCondensé,CEA,CNRSUniversitéParis-Saclay,CEA-Saclay,91191Gif-sur-Yvette,France3SorbonneUniversité,CNRS,...

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Classical Nucleation Theory for Active Fluid Phase Separation M.E. Cates1and C. Nardini2 3 1DAMTP Centre for Mathematical Sciences University of Cambridge Wilberforce Road Cambridge CB3 0WA UK

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