Nonequilibrium thermodynamics of nucleation.

Nonequilibrium thermodynamics of nucleation M. Schweizer and L. M. C. Sagis Citation: The Journal of Chemical Physics 141, 224102 (2014); doi: 10.1063/1.4902885 View online: http://dx.doi.org/10.1063/1.4902885 View Table of Contents: http://scitation.aip.org/content/aip/journal/jcp/141/22?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Potential and flux field landscape theory. II. Non-equilibrium thermodynamics of spatially inhomogeneous stochastic dynamical systems J. Chem. Phys. 141, 105104 (2014); 10.1063/1.4894389 Mesoscopic nonequilibrium thermodynamics approach to non-Debye dielectric relaxation J. Chem. Phys. 132, 084502 (2010); 10.1063/1.3314728 Multicomponent nucleation: Thermodynamically consistent description of the nucleation work J. Chem. Phys. 120, 3749 (2004); 10.1063/1.1643711 Homogeneous nucleation in inhomogeneous media. II. Nucleation in a shear flow J. Chem. Phys. 119, 9888 (2003); 10.1063/1.1614777 Thermodynamic properties of molecular clusters in nucleation AIP Conf. Proc. 534, 233 (2000); 10.1063/1.1361854

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THE JOURNAL OF CHEMICAL PHYSICS 141, 224102 (2014)

Nonequilibrium thermodynamics of nucleation M. Schweizer1,a) and L. M. C. Sagis1,2,b) 1 2

ETH Zurich, Department of Materials, Polymer Physics, Vladimir-Prelog-Weg 2, 8093 Zurich, Switzerland Food Physics Group, Wageningen University, Bornse Weilanden, 6708 WG Wageningen, The Netherlands

(Received 11 September 2014; accepted 17 November 2014; published online 8 December 2014) We present a novel approach to nucleation processes based on the GENERIC framework (general equation for the nonequilibrium reversible-irreversible coupling). Solely based on the GENERIC structure of time-evolution equations and thermodynamic consistency arguments of exchange processes between a metastable phase and a nucleating phase, we derive the fundamental dynamics for this phenomenon, based on continuous Fokker-Planck equations. We are readily able to treat nonisothermal nucleation even when the nucleating cores cannot be attributed intensive thermodynamic properties. In addition, we capture the dynamics of the time-dependent metastable phase being continuously expelled from the nucleating phase, and keep rigorous track of the volume corrections to the dynamics. Within our framework the definition of a thermodynamic nuclei temperature is manifest. For the special case of nucleation of a gas phase towards its vapor-liquid coexistence, we illustrate that our approach is capable of reproducing recent literature results obtained by more microscopic considerations for the suppression of the nucleation rate due to nonisothermal effects. © 2014 AIP Publishing LLC. [http://dx.doi.org/10.1063/1.4902885] I. INTRODUCTION

Nucleation is the onset of a new phase, and can be observed in a variety of phenomena such as condensation of gases, solidification of melts, or crystallization. The initial step of many phase transformations is the formation of small nuclei of the new phase within the bulk of a metastable phase. The nuclei, initially in an entropically unfavorable state, can grow solely due to density fluctuations in the ambient phase. When however a critical size is reached, they grow deterministically. Nucleation theories aim to predict the rate at which clusters of critical size are formed. The predictions of classical nucleation theory often deviate from experimental and simulation results by orders of magnitude.1 Many studies try to overcome this by incorporating, apart from the nuclei cluster size, additional dynamic variables in their models,2–4 such as the nuclei energy, volume, and velocity. Others have introduced effects of external shear5 and temperature gradients3 on the nucleus. Effects of temperature gradients have not received much attention since the external gradients to which the microscopic nuclei are exposed must be extremely large to be relevant. The other effects have been analyzed on an individual basis, without allowing for general coupling mechanisms between them, and without a rigorous incorporation of the environment dynamics. Several studies6–9 indicate that when the energy of nuclei is chosen as an independent fluctuating state variable, on top of their size fluctuations, this represents a relevant extension of classical nucleation theory, resulting, for instance, for the nucleation of liquid drops towards their liquid-gas coexistence, in a correction of the nucleation rate by 2–3 orders of magnitude. This goes along with exa) Electronic mail: [email protected] b) Electronic mail: [email protected]

0021-9606/2014/141(22)/224102/13/$30.00

tensive studies10–12 on two-phase systems where the coupling of energy and mass exchange have been demonstrated convincingly to be significant. In particular, in evaporation and condensation processes the exchange of particles between the phases is always accompanied by an exchange of heat, so that mass and energy transfer appear highly coupled. Even for single phase systems the Dufour effect (an energy flux driven by concentration gradients) and Soret effect (mass fluxes driven by temperature gradients) illustrate that coupling of energy and mass-fluxes are relevant. Up to now, no completely predictive theory of nucleation exists. Literature is mainly focused on formulating extended versions of the Becker-Doering dynamics based on discrete birth and death equations.6, 9, 13 For the extended models there is no generally accepted procedure to yield the corresponding continuum limit.14 Constructing the proper continuum limit from discrete rate equations can be circumvented by working directly on the continuum level. Notably, with respect to nonisothermal nucleation, Fokker-Planck equations have been obtained from nonequilibrium statistical mechanics based on projection operator techniques.15, 16 The statistical treatment however invokes rather uncontrolled approximations and is valid only provided the nucleating cores can be attributed thermodynamic bulk and interfacial properties, which in view of their size is a rather questionable assumption. A variety of kinetic models17, 18 develop continuum equations by studying the escape dynamics of monomers near the surface of a nucleus. These models are strictly valid only for nucleation of a gas phase towards its liquid-vapor coexistence and also require thermodynamic concepts such as the surface tension of a nucleating core. As an alternative to previous approaches, we formulate here a continuum nucleation theory solely based on thermodynamic consistency arguments within the GENERIC19

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framework. We do not focus on improving the lack in coincidence of theoretical, experimental, and molecular dynamics predictions, but formulate general equations for the nucleation process based on the assumption that the relevant nuclei variable are their size, momentum, and energy. Our work goes beyond the existing literature in that we rigorously treat the mass, energy and momentum exchange between the nucleating phase and the metastable surrounding, and keep track of the tiny space fractions that are occupied by nuclei. This essentially extends our approach to even incorporate nuclei volume fluctuations. During the nucleation process the properties of the background medium can be altered significantly, so that the exchange of mass, energy and momentum of the nucleation centers and their surrounding must be taken into account. Even minor modifications of nucleation energy landscapes lead to orders of magnitude different nucleation rates, and consequently it is of utmost importance to consistently keep track of the exchange dynamics of clusters and the metastable phase, and in particular to deal with time-dependent environments. Our formulated dynamics in its most generality does not require concepts such as surface tension or other intensive thermodynamic quantities of microscopic nuclei nor the existence of a well-defined nuclei bulk phase. This is especially important for many substances for which the size of critical nuclei is only composed of a few tens of atoms so that the assignment of a nuclei bulk phase, or the frequently imposed capillarity approximation is doubtful.20 Our approach allows the choice of a completely general nuclei entropy depending on the aforementioned state variables and is hence potentially applicable to study important modifications21 of classical nucleation theory under incorporation of non-isothermal effects. We successfully illustrate that for the prototypical example of liquid drop nucleation our thermodynamic inspired approach can indeed reproduce the fundamental results of a recent more microscopic approach by Barrett6 when we impose continuum assumptions on the nuclei structure. Barrett studied the formation of nuclei in a supersaturated gas composed of monomers of mass m within a discrete rate theory approach and improved the original theory by Feder et al.22 by incorporating the temperature dependence of the surface tension of nuclei. He fixed the rate at which nuclei exchange mass and energy with their surrounding by resampling to microscopic detailed balance arguments. A cluster partition function was used to obtain averaged thermodynamic properties of the nuclei. In the vincinity of the saddle point of the free energy landscape that the nuclei have to surpass, where the nuclei attain their critical size, an approximate solution was given to Barrett’s rate equation whose parameters only contain the averaged thermodynamic properties of the nuclei and surrounding; it was shown that when the nuclei energy is allowed to fluctuate, the nucleation rate Jnon-iso as compared to the rate Jiso from the isothermal theory at temperature T deviates as 1 Jnon-iso = . (1) 2 HBarret Jiso Barrett 1 + c+ 1 ( ˆ 2 )(1+φ) dγ

Here, HBarret = kmLT − 12 − 3(n∗2)1/3 T A∗ (γs − T dTs ) is related to B the change in the mean nuclei energy with respect to its mass

at the critical nucleus size n∗ and cˆ = 32 is the heat capacity of the gas phase reduced by kB . We introduced the latent heat of vaporization L, the surface tension γ s and the parameter φ ≥ 0 regulating the ability of the nuclei to accumulate energy without mass transfer. Barrett showed that depending on the thermo-physical properties the incorporation of nonisothermal effects can suppress nucleation significantly. A bottle neck of Barrett’s work is the absence of a thermodynamic definition of the cluster temperature. Assigning a temperature to a nucleus which is inherently a small thermodynamic system is a controversial topic.23 Within our approach the definition of a cluster temperature is manifest and in particular we will highlight that subcritical nuclei exhibit non-Gaussian temperature fluctuations around the thermodynamic “local equilibrium temperature” as recently discussed by Wedekind.23 In particular, we show that for the particular case of nucleation in argon the “local equilibrium temperature” of clusters is smaller than the ambient vapor temperature for subcritically sized clusters and larger for critically and super-critically sized clusters. This insight traces back to the brute-force molecular dynamics results by Wedekind23 that have later been reconsidered by Schmelzer et al.24 Other theoretical studies25, 26 come to different conclusions and within the nucleus temperature definition of Wyslouzil and Seinfeld,25 or Barrett,26 the cluster temperature is always higher than the ambient temperature. As the simulation work of Wedekind et al.23 suggests, the conclusion drawn on the cluster temperature dependence of the nuclei size is strongly dependent on the inherent definition of the cluster temperature. This paper is organized as follows: first we setup the GENERIC for nucleation processes. Subsequently, we show that we recover the key results of classical nucleation theory in the isothermal case. We then discuss the non-isothermal nucleation dynamics, including the feedback effects on the environment. Finally, we will illustrate our approach on the nucleation of a gas phase towards its vapor liquid equilibrium where we assume that the nucleus can be assigned thermodynamic properties. We will show that in this limit we recover the recent result, Eq. (1), of Barrett and go beyond it by using our thermodynamic inspired approach to discuss the cluster temperature of subcritical, critical, and supercritical clusters. II. OVERVIEW OF GENERIC FRAMEWORK

In the GENERIC formalism19 the evolution equations for a set of relevant state variables x s describing a nonequilibrium system is a simple combination of mechanics- —expressed in a term of the Hamiltonian form - and a relaxation - expressed in the form of a gradient Ginzburg-Landau type dynamics that guarantees conservation of the energy, momentum, and mass, and the growth of the entropy.27 The dynamic behavior of a system can then be captured by a single compact equation involving two brackets, dA(x s ) = {A(x s ), E(x s )} + [A(x s ), S(x s )], (2) dt where A is an arbitrary observable, E is the total energy of the system, and S is the total entropy. The dynamics is split into a reversible Poisson bracket {A, B} and the dissipative


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part [A, B]. The choice of the brackets depends on the chosen system variables x s and is restricted by thermodynamic consistency requirements. In particular, the Poisson-Bracket must be anti-symmetric, fulfill the Jacobi identity and the entropy must be in its null-space, i.e., {A, S} = 0, ensuring that reversible dynamics does not lead to entropy production. Correspondingly, the dissipative bracket must be positive definite, to guarantee a positive entropy production, symmetric, and the energy must be in its null-space, i.e., [A, E] = 0. The latter criterion, known as energy-degeneracy of the dissipative bracket, ensures that irreversible dynamics does not lead to energy production. The GENERIC framework will allow us to incorporate the highly important coupling mechanisms between a nucleating core and its metastable surrounding. In particular, through the degeneracy requirements novel terms in the nucleation barrier emerge—these terms must clearly be present to guarantee the thermodynamic consistency of the process. GENERIC is particularly well suited for nonisothermal nucleation, when the nucleus temperature differs from the surrounding temperature. In that case the entropy is the fundamental thermodynamic potential rather than the free energy. Since GENERIC seeks dynamics directly from the entropy, treating nonisothermal nucleation will be relatively straightforward.

J. Chem. Phys. 141, 224102 (2014)

We denote by v = ρu the velocity field of the metastable phase. The system is hence fully characterized by the state variables x s = (ρ, u, e, f ). IV. ISOTHERMAL NUCLEATION THEORY

In this section we will seek the GENERIC building blocks for the isothermal nucleation theory, where the internal state of the nucleus is characterized by x c = (vc , n). We start by postulating the expressions for the Hamiltonian E(x s ) and total entropy S(x s ), and subsequently discuss the specific forms of the Poisson and dissipative brackets. A. Energy and entropy

For convenience we first introduce the notation (...) = d x c (...)f (r, x c ) for the ensemble average at a particular location r and double brackets (...) = d 3 rd x c (...)f (r, x c ) for spatially and ensemble averaged quantities. Then we can decompose the contributions of the total energy E(x s ) of the system into parts corresponding to the metastable environment and nuclei parts as follows:

E(x s ) =

3

d r

III. FORMULATION OF THE NUCLEATION PROBLEM

We are investigating the nucleation of a one component system in a metastable phase. For illustration purpose we will consider the metastable phase composed of gas particles at a supersaturated state that can assemble and form nuclei. The formalism is however more general and can equivalently describe much richer systems—the only requirement is that their dynamics can be captured by the subsequent set of variables and fields. Every nucleus is characterized by its internal state x c . In the case of isothermal nucleation theory this state is given by x c = (vc , n), where vc is the nucleus velocity and n its cluster size. For non-isothermal nucleation processes we will extend the state space by incorporating the nucleus internal energy, so that x c = (vc , n, c ). To describe the dynamics of the process, we use the density ft (r, x c ) of nuclei with degree of freedom x c at location r at time t; we will subsequently suppress the time argument in the notation. The mass of one unit or atom is m, so that the total nucleus mass is M(n) = nm. Every nucleus possesses a volume V (n) that solely depends on the cluster size n. We do not consider the case of shape fluctuations of the nucleus. From the nuclei distribution func tion the total number of nuclei Nc = d 3 rd x c ft (r, x c ) can be obtained. The total number of nuclei in a nucleation process are not necessarily constant since nuclei can be spontaneously created from density fluctuations in the metastable environment. The amount of nuclei in the system is hence a non-conserved quantity, however, the total particle number of the system composed of particles in the ambient phase and Nc is indeed a conserved quantity (see also the remark about boundary conditions on f in Sec. IV D. The dynamics of the metastable media is governed by its density ρ(r), the momentum density u(r) and the internal energy density (r) at the position r. Here, ρ, u, and exclude the nuclei contributions.

u2 + 2ρ

1 M(n)vc 2 + c (n) 2 2 u + V (n) . − 2ρ +

(3)

The energy of the nuclei is further decomposed into a kinetic part and an internal part. In case the nucleus is sufficiently large to possess a bulk phase, the internal energy c (n) can be split into bulk and surface terms. The last term in Eq. (3) takes into account the excluded volume that is occupied by the nucleus and where hence the metastable phase is absent. This subtle term arises since a fragment of a new phase is created at the expense of the metastable phase. The entropy of the system is given by

S(x s ) = d 3 r s(ρ, ) + kB V S (x c ) − log (f ) −s(ρ, )V (n)

(4)

and consists of the entropy of the metastable phase and a contribution involving the nucleus parts. The latter consists of the usual logarithmic contribution known for distribution functions, and an additional entropic potential kB V S (r, n). The presence of the entropic potential is a result of coarse-graining from the atomistic level to our current mesoscopic level.19 In particular, since the presence of a nucleus requires the absence of bulk phase at the corresponding nucleus location, the entropy (or work) of formation of a single nucleus of size n is naturally given by the difference kB V S − V (n)s(ρ, u, e), so that kB V S represents the effective entropy associated with the nucleus. The nucleus entropy kB V S must be independent of position. We here carefully separated entropic parts and energetic parts, so that the nucleus entropy kB V S does not depend on the nucleus velocity. The nucleus entropy is connected to


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the number of microscopic states the nuclei can occupy and hence must be independent of the motion of the nucleus. B. Reversible structure of evolution equation

We decompose the Poisson structure governing the reversible dynamics into parts involving solely the nucleus contributions, {A, B}f , parts containing only the hydrodynamic fields of the metastable phase, {A, B}H , and an exchange bracket {A, B}ex , so that the total Poisson bracket is given by {A, B} = {A, B}H + {A, B}f + {A, B}ex .

(5)

This Poisson-Bracket must be constructed such that all symmetries are obeyed for all observables A. In the Appendix we show how the Poisson structure can be obtained by studying the related brackets of the effective bulk densities ρ¯ = ρ(1 − V ), u¯ = u(1 − V ), and s¯ = s(1 − V ), and assuming that the distribution f is naturally convected according to the usual geometric structure familiar from Boltzmann equations. We obtain 1 ∂B ∂A ∂ · {A, B}H = − d 3 r ρ ∂ u ∂ r 1 − V ∂ρ 1 ∂A ∂B ∂ · − ∂ u ∂ r 1 − V ∂ρ 1 ∂B ∂A ∂ 3 · − d ru ∂ u ∂ r 1 − V ∂ u 1 ∂A ∂B ∂ · − ∂ u ∂ r 1 − V ∂ u 1 ∂B ∂A ∂ · − d3r ∂ u ∂ r 1 − V ∂ 1 ∂A ∂B ∂ · − ∂ u ∂ r 1 − V ∂ p ∂B ∂A ∂ 3 · − d r ∂ u ∂ r 1 − V ∂ ∂B ∂ p ∂A − · , (6) ∂ u ∂ r 1 − V ∂ Here, p denotes the thermodynamic pressure of the metastable phase. The nucleus distribution function is convected as19 ∂ δA ∂ δB f · {A, B}f = d 3 rd x c M ∂ r δf ∂vc δf ∂ δA ∂ δB · . (7) − ∂ r δf ∂vc δf The expression for the Poisson-bracket (6) for the hydrodynamic fields of the metastable phase is standard and its first instances in connection with the GENERIC framework can be traced back to the work of Grmela and Öttinger28 where it is shown that this Poisson bracket indeed reproduces the reversible parts of the evolution equations of hydrodynamics. Alternatively, the brackets are constructed28 based on the representation of the continuous group of space transformations on the space of hydrodynamic fields. The basic idea

of such a construction is that the Poisson-bracket only involves kinematic effects and these are closely related to spacetransformations.28 In similar lines the bracket (7) for the convection of distribution functions was understood.27, 28 In order to highlight the structure of the exchange bracket {A, B}ex we introduce the notation δA δA δA +u + ( + p) , (8) A˜ = ρ δρ δu δ so that the bracket takes the form ∂ δA ∂ V f 3 ˜ · B {A, B}ex = − d rd x c M ∂vc δf ∂ r 1 − V ∂ V ∂ δB ˜ · A . (9) − ∂vc δf ∂ r 1 − V It can be verified that the resulting total Poisson bracket {A, B} is such that the total mass of the system (10) M = d 3 r ρ + M(n) − ρV (n) , the total momentum M = d 3 r u + M(n)vc − uV (n),

(11)

and the entropy S fall in its null-space, i.e., {A, M} = 0, {A, M} = 0 and {A, S} = 0, so that not only energy, but also mass, momentum, and entropy are conserved under reversible dynamics. Similar bracket structures for exchange mechanisms of two phases have been obtained by Ref. 29, where boundary conditions at the interface between two phases have been taken into account explicitly. An analogous term to Eq. (9) was there added to ensure consistency of the GENERIC with the chain rule of calculus in the presence of moving and deforming interfaces. The term was noticed29 to be crucial in obtaining mass and momentum conservation across phase boundaries. And also here we remark that without the exchange term {A, B}ex these conservation laws cannot be guaranteed. In our theory, the presence of a nucleating phase only modifies the metastable phase slightly, so that the resulting theory is more an effective medium theory than a theory that can cleanly distinguish two distinct bulk phases through the presence of a localized interface. The Poisson structure {A, B}ex implicitly assumes that there is no reversible exchange of mass between the nucleus and the metastable phase on our coarse-grained description of nucleation. Note, in equilibrium there is no entropy production, but on the microscopic level nuclei can nevertheless grow and shrink by the exchange of particles with the ambient phase. This effect is however not detectable on the coarse-grained level through the use of a macroscopic distribution function f. In equilibrium, f remains stationary and formulated differently, our proposed Poisson-structure assumes that any time rate of change in f caused by exchange of mass between the nucleus and its surrounding is of entropic origin. The recently developed “Entropic Nucleation Theory”30 supports the fact that mass transfer of the nucleus and the surrounding is a purely dissipative effect. Since reversible dynamics cannot create entropy, mass-exchange must hence be incorporated in the subsequent dissipative bracket structure.


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C. Irreversible structure of evolution equations

We split the total dissipative bracket [A, B] into a part [A, B]H attributed solely to the hydrodynamic fields (ρ, u, ), with the familiar irreversible dynamics of hydrodynamics, and a bracket [A, B]ex governing the irreversible exchange dynamics, [A, B] = [A, B]H + [A, B]ex .

(12)

The hydrodynamic part [A, B]H is, as usual,19 given by 1 ∂A s ∂ 3 [A, B]H = d r 2ηT ∂ r 1 − V ∂ u ∂A ∂ 1 ∂B s 1 s − κ : 1 − V ∂ ∂ r 1 − V ∂ u ∂B 1 − κs 1 − V ∂ 1 ∂A ∂ · + d 3 r κT ∂ r 1 − V ∂ u ∂A ∂ 1 ∂B 1 · − trκ 1 − V ∂ ∂ r 1 − V ∂ u ∂B 1 − trκ 1 − V ∂ 1 ∂A ∂ + d 3 r λT ∂ r 1 − V ∂ 1 ∂B ∂ , (13) × ∂ r 1 − V ∂ where η is the viscosity, κ is the bulk viscosity, λ is the thermal conductivity, and κ is the velocity gradient tensor in the metastable phase. A superscript “s” on top of a tensor denotes the symmetrized traceless part of that tensor. For the exchange dynamics [A, B]ex the dissipative structure for the nuclei is obtained as

∂ δA (M − ρV ) δA − [A, B]ex = d 3 rd x c d x c ∂ x c δf 1 − V δρ

Mvc − uV δA − · 1 − V δu 1 2 M(v − vc ) + c − V δA 2 − 1 − V δ ·T D(r, n)f δ(x c − x c )

M − ρV δB ∂ δB · − ∂ x c δf 1 − V δρ

M v c − uV δB − · 1 − V δu 1 2 M (v − v c ) + c − V δB 2 . − 1 − V δ

(14)

Here all primed variables depend on x c rather than x c , so that M = M(n ), V = V (n ) and f = f (r, x c ). This bracket results solely from the requirement of having a double derivative term in n in the final nucleus distribution evolution equa-

tion, together with the thermodynamic consistency requirements that the total mass, momentum, and energy must clearly be in the null-space, i.e., [A, M]ex = 0, [A, M]ex = 0, and [A, E]ex = 0, so that mass, momentum, and energy are conserved quantities during the exchange process form the metastable bulk phase into the nuclei. Feedback effects on the environment are completely incorporated: we see that any growth of the nucleus size is always accompanied by momentum and energy exchange between the two phases. The phenomenological diffusion tensor D(r, n) can be split into parts corresponding to Brownian motion of the nucleus in the metastable medium associated with the friction coefficient ξ , and parts regulating the nucleus growth, usually denoted by Dnn , ξ (n)I3×3 0 D= , (15) 0 Dnn (n) where I3×3 is the 3 × 3 unit matrix and we assumed that transfer mechanisms are decoupled. Both diffusion coefficients depend on the nucleus size, but are assumed to be independent of its velocity.

D. Nucleus evolution equation

Through the GENERIC equation (2) the dynamics of the host medium and the nuclei are now completely specified. We will restrict this discussion to the nucleus evolution equation and postpone the dynamics of the host medium to the discussion of nonisothermal nucleation. The nuclei evolution equation is given by 1 ∂ ∂ ∂f ∂ · f Vp =− · vc f + ∂t ∂r M ∂vc ∂r ∂ ∂ ∂ (v − v)f + kB T ξ · f ∂vc c ∂vc ∂vc

∂ ∂ ∂ f −f VS − c + kB T Dn,n ∂n ∂n ∂n T

2 p v − vc μ 1 − M − V −M . (16) T T 2 T + ξM

Here, T and μ are the temperature and chemical potential of the metastable phase. The last two terms in the second line describe the Brownian motion of the nuclei in the metastable medium. The term involving the thermodynamic pressure implies tilted diffusion in case the metastable medium has a pressure gradient. In a pressure gradient particles from the metastable medium tend to hit the nucleus stronger from one side than from the other. This term was already recovered by Bashkirov in his statistical mechanics of Brownian motion.31 Unlike other attempts to describe Brownian motion in a gas,32 we recover this term since we keep rigorous track of the tiny volume terms in our derivation. The remaining term describes the mass-accumulation and loss of the nucleus. We realize that the mass, momentum, and energy degeneracy requirements of the dissipative bracket modify the entropic landscape of the nucleation. In particular, nucleation takes place in the


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effective entropic potential p μ 1 (v − vc )2 S − M = kB V S − c − V − M kB Veff . T T T 2 T (17) The modifications involving velocities are not of an entropic origin. They appear solely as a result of the mass, momentum and energy degeneracy of the GENERIC friction matrix. In principle, Eq. (16) must be supplemented by boundary conditions on f (x c ) for small nuclei sizes nc = 1. From a rigorous point of view this would require us to introduce in addition to the Poisson- and dissipative brackets also the corresponding boundary brackets for the distribution function f (x c ). Guidelines for such a procedure have recently been developed.29, 33 Here we will motivate the boundary conditions on f (x c ) from a physical perspective when we apply our framework to a specific example (see also Sec. VI). To investigate this expression for the effective entropic potential in more detail, we assume now that the isothermal nucleation takes place in a host medium that acts as a heat bath, that is, the environment pressure p and chemical potential μ are fixed control variables. Then there are no spatial gradients in the state variables of the metastable gas phase. In addition we assume that the nucleus is sufficiently large so that it can be attributed thermodynamic bulk and surface properties. Since we consider the process to be isothermal, the nucleus temperature coincides with the heat-bath temperature T. The internal energy of the nucleus is composed of bulk cb and surface contributions cs , i.e., c = cb + cs ,

(18)

where the surface contribution is taken in the equimolar location of the dividing surface between the nucleating and metastable phase. The entropy of the nucleus can also be decomposed into surface Scs and bulk contributions, cb + pc V − μc M + Scs , (19) T where pc is the bulk pressure and μc is the bulk chemical potential of the nucleus. Since −γs A = T Scs − cs , where γ s is the surface tension of the nucleus and A its area, the effective entropic potential is hence given by kB V S =

S =M kB Veff

(μ − μc ) (p − pc ) 1 (v − vc )2 γ −V − M − s A. T T 2 T T (20)

The Brownian motion of the nucleus relative to the velocity of the host medium always increases the nucleation barrier. Without the Brownian motion we recover the nucleation barrier of classical nucleation theory. But here, we solely derived this result by exploiting the symmetries of the dissipative bracket structure. So that mass exchange between two media seems to be governed quite universally by chemical potential and pressure differences. V. NONISOTHERMAL NUCLEATION

Subsequently, we will seek the GENERIC building blocks for non-isothermal nucleation, where the internal state

of the nucleus is characterized by x c = (vc , n, c ), that is, the previously non-fluctuating internal energy c (n) of the nucleus now becomes an independent dynamic variable c . We can directly adopt the previously introduced Poisson structure. In particular, only a slight modification of the dissipative bracket (14) allows us to incorporate the nonisothermal dynamics. Since contrary to the isothermal case, the state space x c of the nucleus is now extended, the only modification to obtain the friction matrix is to extend the tensor D of isothermal nucleation theory by one additional row and column to account for energy fluctuations, ⎛ ⎞ 0 0 ξ (n)I3×3 ⎜ 0 Dnn (n) Dn (n) ⎟ D=⎝ (21) ⎠, c 0 Dn (n) D (n) c

c

c

where we assumed that energy fluctuations are decorrelated from Brownian motion of the nucleus. As is known from the study of interfacial systems, the off-diagonal elements Dn, c describing the interplay between mass and energy exchange of the nucleus with the metastable surrounding are highly significant.10–12 In single phase systems the coupling of mass and energy fluxes result in the Soret and Dufour effects. If we consider for illustration purposes a metastable gas phase where liquid drops are nucleating, then the coupling is related to the heat exchange in evaporation and condensation of particles. When a particle evaporates from an nucleus, then the potential energy of this particle must be converted to kinetic energy in the gas phase and vice versa for the reverse process of particle capturing. In that sense the loss and accumulation of particles in a nuclei leads to energy exchange with their surroundings. We further assumed that the coefficients do not depend on the nucleus internal energy c . Such a dependency is completely natural and can readily be implemented. In particular, in nuclei with high internal energy, nuclei constituents are more likely to escape, so that the coefficients would indeed depend on c . Notice that we could further extend the state space of the nuclei x c and treat the volume term V as an additional independent variable. The only modification of the GENERIC necessary to incorporate this variable would be to extend the diffusion tensor by one additional row and column, to incorporate the fluctuations of the nuclei volume and their coupling to the remaining exchange mechanisms. A. Nucleus evolution equation

In the non-isothermal case the dynamics of the nucleus are given by ∂f 1 ∂ ∂ ∂ · f Vp =− · vc f + ∂t ∂r M ∂vc ∂r ∂ ∂ ∂ (v − v)f + kB T ξ · f ∂vc c ∂vc ∂vc

∂ ∂ ∂ + kB T Dnn VS f −f ∂n ∂n ∂n p μ 1 (v − vc )2 − M − V −M T T 2 T + ξM


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S ∂ ∂V 1 f −f − ∂c ∂c T

S ∂ ∂V 1 ∂ kB T D f −f − + c c ∂c ∂c ∂c T

∂ S ∂ ∂ f −f V kB T Dn + c ∂c ∂n ∂n

2 p v − vc μ 1 − M − V −M . (22) T T 2 T +

∂ k T Dn c ∂n B

Naturally, the driving force for energy fluctuations is the difS and ference of the associated inverse nucleus temperature ∂V ∂ c

inverse bulk temperature T1 . This equation however is also valid for small nuclei that cannot be attributed bulk and interfacial thermodynamic properties. Assuming now that the nuclei can be attributed thermodynamic properties, we can proceed as before in Sec. IV D to get further insight into the driving force for mass-transfer associated with the effective entropic potential p μ 1 (v − vc )2 S − M . = kB V S − V − M kB Veff T T 2 T

(23)

It is useful to switch to the nucleus temperature Tc for further analysis, instead of the nucleus internal energy c . The temperature Tc ( c , n) can be obtained by inverting the relationship c = c (Tc , n). We can split the nucleus energy into a bulk part cb and surface contribution cs . The surface energy is evaluated in the equimolar location of the dividing surface separating the phases. Then we can rewrite the nuclei entropy as VS =

cb + pc V − μc M + Scs . Tc

(24)

With the surface tension γs A = cs − Tc Scs and the surface dγ entropy Scs = −A dTs , we can write the effective potential in c the form p 1 p μ μc 1 S −V − − c + cb =M − kB Veff T Tc T Tc Tc T 1 dγs γs 1 γ s − Tc − − A− Tc T Tc dTc 1 (v − vc )2 . − M 2 T

(25)

Setting formally T = Tc recovers the landscape of the isothermal case. Apart from the kinetic energy contribution in S coincides with Eq. (25), the effective entropic potential kB Veff the equilibrium work of formation of nuclei, as obtained in recent work by Alekseechkin.13 We recover this term in the limit where the nuclei have thermodynamic properties within a dynamical theory for the first time. However, we clearly realize that solely the work of formation cannot be used to estimate nucleation barriers since cross correlations with energy fluctuations are present. These cross correlations generally modify the nucleation barrier significantly.

B. Medium evolution equations and entropy production

In the following we show the hydrodynamic evolution equations that govern the dynamics of the metastable host medium. We will assume that the nucleus is incompressible so that the volume of the nuclei is given by V (n) = nm/ρc , where ρ c is the nucleus density. It is expected that all transport equations obey to leading order O(V 0 ) in the volume term the usual hydrodynamic equations. The volume term then modifies the reversible and irreversible dynamics. This originates from the fact that through the growing cluster phase the metastable phase is continuously extruded. Terms involving the dissipative transport coefficients related to the tensor (21) are responsible for the irreversible exchange of quantities between the nuclei phase and the metastable surrounding. For the mass density we obtain from the GENERIC framework, ∂ ρ ∂V ∂ ∂ρ =− · (ρv) + v· − · V v c ∂t ∂r 1 − V ∂r ∂r ∂V S ∂Dnn ρ mkB T 1− + Dnn eff − 1 − V ρc ∂n ∂n S ∂V 1 . (26) − + Dn c ∂c T While the first term on the right-hand side describes the reversible convection of mass, the second describes reversible changes due to spatial changes of the occupied volume of the nucleus. The remaining terms define source or sink terms due to the mass exchange with the nucleus. We obtain for the momentum density ∂ ∂ V ∂ ∂u =− · (uv) − ·π − ·τ ∂t ∂r ∂r 1 − V ∂ r u ∂V ∂ + v· − · V vc 1 − V ∂r ∂r ∂V S ∂Dnn u mkB T vc − + Dnn eff − 1 − V ρc ∂n ∂n S ∂V 1 − + Dn c ∂c T −

2 1 M kB ξ (v − v c ) , 1 − V

(27)

where π = 2ηκ s + κtrκI3×3 is the extra stress tensor of the metastable phase, τ = π − pI3×3 is the stress tensor. We further obtain for the energy balance ∂ ∂ ∂ ∂ =− · (v) − π : v− · jq ∂t ∂r ∂r ∂r ∂ V ∂ V τ: v− · jq − 1 − V ∂r 1 − V ∂ r ( + p) ∂V ∂ v· − · V v c + 1 − V ∂r ∂r


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∂Dnn 1 2 (v − v c ) − 2 ρc ∂n S S ∂V ∂V 1 − + Dnn eff + Dn c ∂n ∂c T 2 (v − v c ) kB T Mξ M −1 + 1 − V T ∂D S ∂V 1 kB T c n + − − D c c 1 − V ∂c T ∂n ∂V S + D n eff , c ∂n mkB T − 1 − V

system is dS = dt

∂ S Veff − log (f ) , ∂n

c

(28)

(29)

roughly interpreted as a generalized chemical potential difference between the metastable phase and nuclei, the thermodynamic force S ∂V 1 ∂ log (f ) , (30) X = − − c ∂c T ∂c associated with temperature differences of the metastable phase and nuclei, and finally ∂ log (f ) 1 1 v − vc X Mv = , (31) M − c M 2 T ∂vc related to differences in velocities between the two phases. We also introduce the associated thermodynamic fluxes from the metastable phase into the nucleating phase, corresponding to mass-flux, Jn = T Dnn Xn + T Dn X , c

c

(32)

and analogously the internal energy flux, J = T D n Xn + T D X , c

c

c

c

c

(33)

and momentum flux, J Mv = MT ξ X Mv . c

c

(34)

Notice, that with these fluxes the Fokker-Planck equation for nonisothermal nucleation (22) can be cast into the compact form ∂ ∂f 1 ∂ ∂ =− · vc f + · f Vp ∂t ∂r M ∂vc ∂r −

J Mv ∂ ∂ ∂ c f− Jn f − · J f. ∂vc M ∂n ∂c c

η κ λ ∂ 1 d 3 r 2 κ s : κ s + (tr κ)2 − jq · T T T ∂r T

+ Xn Jn + X J + X Mv · J Mv ,

where j q = −λ ∂∂r T is the diffusive heat flux. In order to study the entropy production in the nucleation process, it is useful to look at thermodynamic forces and fluxes in the mass, momentum, and internal energy exchange mechanisms. To this extent we define the thermodynamic force Xn =

(35)

The fluxes, Eqs. (32)–(34), are of irreversible origin, and in Eq. (35) we realize that there is an additional influx into the state-space variable vc , which however as discussed is of reversible origin. The total entropy production in the

c

c

c

(36)

and separates into entropy production due to irreversible processes in the environment and due to exchange processes with the nuclei. By virtue of the thermodynamic forces we realize that the fundamental reason for the latter entropy production is due to chemical potential, temperature, and velocity differences between the metastable bulk and nuclei. The entropy production is hence characterized by the different intensive states of the nuclei and their surrounding.

VI. NUCLEATION OF A GAS PHASE TOWARDS LIQUID-VAPOR COEXISTENCE

Subsequently we will study the nuclei evolution equation for the isothermal and nonisothermal nucleation processes when the metastable phase is a heat bath with fixed temperature T and density ρ. In this case where we study the initial stage of nucleation we can neglect the dynamics of the environment. We assume that the nucleus can be attributed bulk and interfacial thermodynamic properties. As a specific model system we choose a metastable gas phase in a supersaturated state in which high density incompressible nuclei are formed and grow. We will treat the gas phase under the assumptions of ideality and investigate the instances that lead to suppression of the nucleation rate as compared to the isothermal case.

A. Thermodynamic model

The hypothesis of local equilibrium which is inherent in the definition of a nuclei entropic potential implies that the nuclei thermodynamic equations of state are the same whether they are in global equilibrium with the surrounding gas phase, or they are in nonequilibrium with their environment. We S in terms of exuse this notion to rewrite Eq. (25) for Veff perimentally accessible quantities. We introduce the equilibeq,n eq,n rium pressure pc (Tc ) and chemical potential μc (Tc ) corresponding to an equilibrium state of a nucleus of size n in coexistence with its vapor at temperature Tc , or respectively its inverse 1/(kB Tc ) = ∂V S (n, c )/∂c . Introducing similarly the corresponding gas pressure peq, n (Tc ) and chemical eq,n potential μeq, n (Tc ), the equilibrium conditions are pc (Tc ) eq,n eq,n eq,n − p (Tc ) = 2γs (Tc )/R(n) and μc (Tc ) = μ (Tc ), where R(n) is the nucleus radius. We also introduce the nuclei eq,∞ eq,∞ quantities pc (Tc ) and μc (Tc ), and respective gas varieq, ∞ eq, ∞ (Tc ) and μ (Tc ) corresponding to the equiables p librium coexistence values of a nucleus with size n → ∞ for which the interface separating the nuclei from the ameq,∞ bient phase can be considered flat and therefore pc (Tc ) eq,∞ eq,∞ eq,∞ (Tc ) and μc (Tc ) = μ (Tc ). The fact that the sur=p face tension γ s (Tc ), considered as a property of the nucleus, is solely a function of the nucleus temperature together with the


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incompressibility assumption leads to34 eq,n

μc (Tc ) − μeq,∞ (Tc ) =

1 eq,n pc (Tc ) − peq,∞ (Tc ) . ρc (37)

In the following we assume, as is usual for nucleation of a gas phase towards its liquid-vapor coexistence, that ρ eq, n (Tc ), ρ eq, ∞ (Tc ) ρ c and that the latent heat of evaporation L is a function of temperature and obeys mL(Tc ) = mL0 p p + (cp − cc )Tc , where cp and cc are the heat capacities at constant pressure of the vapor and cluster. This ensures the known thermodynamic result that the derivative of the latent heat with respect to the temperature equals the difference in heat capacity of the nuclei and vapor phase. From the Clapeyron relation we then conclude eq,∞ 1 (T ) 1 p = mL(Tc ) − log peq,∞ (Tc ) Tc T T Tc p

+ 1− c . − cp − cc log T T (38) We also fix the equation of state for the nucleus bulk energy. For this purpose we use the Clausius jump relationship35, 36

global equilibrium. The so-called local equilibrium temperature Tloc introduced by Wedekind et al.23 was then shown within molecular dynamics simulations of nucleation of liquid argon drops to be smaller than the ambient temperature T for subcritical clusters and always larger for super critical clusters. Around the critical size, the clusters were observed to be hotter than the bath temperature T as later also argued by Schmelzer et al.,24 where the molecular dynamics simulations of Wedekind et al.23 were reconsidered. Within our thermodynamic analysis we however find corrections to the temperature fluctuations as proposed by Wedekind. Up to our best knowledge we are the first one obtaining the corrections involving the surface tension in Eq. (40). Although in many studies the change of surface tension with temperature is ignored, for a variety of substances dγ s /dTc is comparable to γ s /Tc and therefore non-negligible.38 The surface tension correction to the nucleation potential vanishes in two prominent instances often encountered in literature; when γ s is a constant independent on temperature and when γ s is a linear function of temperature. Finally, we specify further the functional form of the dissipative diffusion coefficients. We assume that the velocity friction is given by the Einstein-Smoluchovski scaling, −1/3

ξ = ξ0 ρc

eq,n

1 cb (n, Tc ) − V eq,n (Tc ) dμc (Tc ) eq,n = μc (Tc ) − Tc eq,n V ρc − ρ (Tc ) dTc cs (n, Tc ) 2 − , R(n) A(n)(ρc − ρ eq,n (Tc )) (39) where the second term on the right-hand side involves the curvature correction. Using the Clapeyron equation again the simple form cb (n, Tc ) = 3/2nkB Tc − n mL(Tc ) − kB Tc is implied. Using Eqs. (37)–(39) we find for the effective nucleation potential 1 v2 γ (T ) S A − nm c = nkB log(S) − s kB Veff T 2 T T T p c + 1− c + ncc log T T T dγs (γ (T ) − γs (Tc )) − 1− c A+ s A. T dTc T

M −5/3 ,

(41)

where ξ 0 is a positive constant. The accumulation of mass and energy must scale with the surface area of the nucleus. This is related to the fact that microscopically mass and energy exchange goes along with colliding particles. We hence set D(nc , c ) = D0 ρ

M ρc

23 .

(42)

The mass density of the gas phase ρ is incorporated since the exchange of mass and energy is related to the hitting frequency of gas molecules onto the nuclei and this scales linearly with ρ.

B. Stochastic process behind nucleus evolution equation

(40) Here, we introduced the saturation ratio S = ρ/ρ eq, ∞ (T). The first line involves the contributions to the effective potential from classical nucleation theory. The second line involves non-isothermal corrections and is known from an analysis of temperature fluctuations of small systems37 and then later used to define a thermodynamic local equilibrium tempera23 ture of clusters Tloc . S Resampling to the equilibrium distribution feq ∝ exp Veff of Eq. (22), the nucleus temperature Tc fluctuates around the ambient temperature T for any fixed nucleus size n in a non-Gaussian way. Wedekind et al.23 argued that in nonequilibrium the nucleus temperature Tc fluctuates around a temperature Tloc in the same way as the nucleus temperature fluctuates around the bath temperature T in

We seek a numerical solution of the Fokker-Planck equation describing the nucleus evolution equation. This is done by mapping the equation to the appropriate Brownian dynam v ics. Given the Wiener Increments dWt c , dWtn , and dWt c , the appropriate stochastic processes for the position of the nuclei r tc , the velocity v tc , size n ct , and energy tc in the Ito sense are given by39 d r tc = v tc dt,

v dv tc = −ξ M v tc − v dt + 2kB T ξ dWt c , (43) S S ∂V ∂V ∂Dnn 1 dn ct = kB T Dnn eff + + Dn dt − c ∂n ∂n ∂c T

D D + σnn dWtn + σn dWt c , c


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S ∂Dn ∂Veff c + d tc = kB T Dn c ∂n ∂n + D c

c

∂V S 1 − ∂c T

density of the nuclei ρ c = 2, for the friction coefficient ξ 0 = 1 and finally the surface tension was chosen to be of the linear form γ s (Tc ) = 3.5 × (1 − Tc ). We compared our numerical results to an approximate solution of the stationary Fokker-Planck equation obtained via the well-known saddle-point approximation described in Refs. 22 and 40. We obtain

dt

D + σD n dWtn + σ dWt c . c

In order to define the matrix σ we define the 2 × 2 ma˜ 12 = D ˜ 21 = Dn , and D ˜ 22 ˜ with entries D ˜ 11 = Dnn , D trix D c ! ˜ is given by = D . Then σ = 2kB T D c c √ √ (44) σ D = v − λ− v − + v + λ+ v + , where 1 ˜ (tr D ± 2

λ± =

! ˜ 2 − 4det D) ˜ (tr D)

1

2 2 Dn, + λ ± − D c

(45)

c

λ ± − D c c . (46) Dn, c

c

In the stochastic representation (43) the different driving forces are readily identified. C. Detailed results

We simulated the homogeneous nucleation process with Brownian dynamics. The nuclei were initially set to size n = 1 and their velocity was set to a Maxwellian corresponding to the temperature T. When nuclei shrink to sizes smaller than n = 1, we reset solely their size to n = 1. Since nucleation is a rare event, initial conditions do not influence the nucleation rates. We then evaluated the nucleation rate Jiso and Jnon-iso for the isothermal and nonisothermal case by counting the number of nuclei per unit time that pass the size nmax much larger than the critical nuclei size n∗ of the isothermal nucleation theory 2A(1)γs (T ) 3 . (47) n∗ = 3T log(S) For the nonisothermal nucleation process the critical cluster size is not well-defined and would in principle depend on the temperature Tc of the specific nucleus. After a nucleus reached the size nmax we reset its size, velocity, and energy to the previously discussed initial conditions. This procedure solves the isothermal Fokker-Planck equation (16), or respectively the nonisothermal analogue (22), in the stationary limit. As a specific model system we investigated the nucleation of argon droplets in its vapor. In the text, we set the temperature at the critical point kB Tkrit = 1, the vapor mass density at the critical point ρ krit = 1, the mass of monomers m = 1 and the entry D0, nn of the diffusion tensor D0 in Eq. (42) to D0, nn = 1 to define the dimensionless units. In the following we use this convention to present the simulation results. As model parameters we used for the heat capacp ity cc = cV , for the heat of evaporation L0 = 4.7, for the mass

(48)

where λmin is the negative eigenvalue of the 2 × 2 ma˜ ∗ G ∗ . Here the superfix * indicates that the functions trix D are evaluated at the saddle point (vc = 0, n = n∗ , Tc = T ) of S . We defined kB Veff ⎛ ⎞ ω2 1 ζ 2 ⎝ 1 − ζ2 ζ ⎠ . (49) G= 2 1 1 σ 2 c

and v± = !

|λ | 1 Jnon-iso = √ min , Jiso Dnn ω |detG|

c

ζ

ζ

The coefficients are given by d 2 γs p , σ2 = ncc T − T 2 A c dTc2 T =T c 1 d 2 A = −γs (T ) , dn2 n∗ σn2 ω2 =

σ2

c

σn2

(50)

,

5 dA dγs ζ = mL(T ) − kB T − γ (T ) − T , 2 dn n∗ s dTc T =T c

p where cc

is the heat capacity at constant pressure of the liquid. The reduction in nucleation rate as given by Eq. (48) is independent of the Brownian motion of the nuclei characterized by the friction coefficient ξ . We can interpret T G −1 as the variance of the fluctuating quantities n and c . Then the magnitude of fluctuations in energy roughly scale with σ2 while c nuclei size fluctuations scale with σn2 . In equilibrium this becomes evident by approximating the distribution function of nuclei near the saddle point S

feq ∝ exp Veff S

mnvc 2 ∗ ≈ exp Veff (vc = 0, n , T c = T ) exp − 2kB T 1 × exp − (51) (n, c )G ∗ (n, c )tr , 2kB T and in the stationary nonequilibrium case a more involved solution exists22, 40 around the saddle point that however offers the same interpretation of σ2 and σn2 . In the limit of small enc ergy (or identically nuclei temperature) fluctuations as compared to nuclei size fluctuations, identified with ωs /ζ s 1, we obtain Jnon−iso 1 = . (52) 2 s ∗ ζ +D ∗ Jiso ωs 1 Dnn n ζ c 2 1+ ∗ D∗ Dnn

− c c

∗ Dn

c


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Tloc

iso

Jiso 0.10

T

0.04 T

0.56

0.08

0.02 T

0.06

0.4

T

0.32 5

5

n

10

0.04 0.02 0.02 0

1

2

3

4

5

6

log S

FIG. 1. Reduction in nucleation rate due to nonisothermal effects for argon at various vapor temperatures T and supersaturation ratios S. Solid lines are the predictions of Feder22 where the temperature dependence of the surface tension is neglected. Dashed lines are the predictions of Barret’s rate theory approach6 and coincides with the saddle point approximation of our theory. The symbols are the result of the Brownian dynamics simulations of our theory.

If we now take the statistical expressions obtained from rate theory for the coefficients of the diffusion matrix,38 Dn /Dnn c = 2kB T , D /Dnn = 6(kB T )2 (1 + φ) and we neglect the c c heat capacity of the interface related to the second derivative of temperature of the surface tension, we recover Eq. (1) of Barrett’s work. This is remarkable since Barrett’s work was based on statistical arguments within a rate theory approach while we entirely relied on thermodynamic arguments. Here, φ ≥ 0 defines the ability of a cluster to exchange energy without mass transfer. For φ = 0 energy transfer is exclusively caused by mass transfer. We compare the saddle point solution (52) in different temperature regimes with the solution of the Brownian dynamics simulation as visualized in Fig. 1 for the coefficients of the diffusion matrix as proposed by Barrett with φ = 0. We also show the original theoretical results by Feder22 that neglect the temperature dependence of the surface tension. We observe that the saddle-point approximation underestimates the solution obtained within Brownian dynamics. This highlights that nucleation of droplets does not happen entirely through the region captured by the saddle point approximation. This goes hand in hand with our analysis of the local equilibrium temperature Tloc of clusters in Fig. 2 for a supersaturation ratio of log (S) = 3.5 and ambient temperature T = 0.4. The local equilibrium temperature Tloc of nuclei is obtained, see also Fig. 3, by fitting the temperature distribution PT in the Brownian dynamics simulations with PT c c ∝ exp{nccV [log(Tc /Tloc ) + (1 − Tc /Tloc )]}. We generally observe that the local equilibrium temperature of clusters is smaller than the ambient temperature for subcritically sized clusters, and larger for critically and supercritically sized clusters which has been confirmed by molecular dynamics simulations by Wedekind et al.23 and later more extensively studied by Schmelzer et al.24 In general, the conclusion drawn on the cluster temperature and its cluster size dependence crucially depends on the chosen definition of the cluster temperature as already indicated by the work of Wedekind et al.23 This is also illustrated in the extensive study25 of the tempera-

0.04 FIG. 2. Local equilibrium temperature of clusters as defined in Ref. 23 in dependence of the deviation n of the cluster size from its critical size n∗ ≈ 15 for argon at T = 0.4. Within our approach (dashed line: fit to the Brownian dynamics simulations) we observe that critically sized clusters are generally warmer than the bath temperature as confirmed by the recent molecular dynamics simulations by Wedekind et al.23 and was later argued by Schmelzer et al.24 Also shown is the original prediction by Feder22 (solid line).

ture in nonisothermal nucleation, and the more recent work by Barrett,26 where with their definition of the cluster temperature, the cluster temperature is always higher than the ambient temperature. We have also studied, see Fig. 4, the reduction in nucleation rate when we vary the coupling coefficient Dn c 2 from the case of completely coupled p = Dn /(Dnn D ) c c c to completely decoupled p = 0 heat and mass exchange between nuclei and surrounding. We observe that when the exchange processes are highly coupled, the nonisothermal rate is drastically suppressed. But even when the processes are decoupled, nucleation is suppressed. Hence, temperature fluctuations push the nuclei to an unfavorable entropic state so that the energetic barrier they need to surmount to reach the critical size is generally larger. In a recent study9 by Horst et al. it was argued within mesoscopic nonequilibrium thermodynamics that the nonisothermal nucleation rate is suppressed by Jnon-iso /Jiso = (1 − p)/(1 + L(T)Dne /Dee ). In their study, the cluster temperature Tc was not allowed to fluctuate, but PTc 5 4 3 2 1 0 0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

Tc

FIG. 3. Temperature distribution PT for argon for different sizes of the nuc

clei (black curve: subcritically sized cluster with size n∗ − 5, red curve: supercritically sized cluster with size n∗ + 5, dots are the corresponding Brownian dynamics simulation data). Also shown is the bath temperature T = 0.4 (black vertical line) and the local equilibrium temperatures as defined in Ref. 23 (dashed vertical line).


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iso

Jiso 1.0 0.8 0.6 384

0.4 0.2 0.0 0.0

0 D2 n 0.2

0.4

0.6

0.8

1.0 Dnn D

FIG. 4. Reduction in nucleation rate due to nonisothermal effects in dependence of the coupling strength between mass and energy exchange. For rapid equilibration of the nuclei temperature φ → ∞ the predictions of our theory (dashed lines: saddle point approximation, symbols: Brownian dynamics simulation) coincides with the predictions (solid lines) of Horst et al.9 where temperature fluctuations are neglected.

rather set to a size dependent value for all nuclei. That energy fluctuations lead to significant corrections for small values of φ is illustrated in Fig. 4 where we compare the result of Horst et al.9 with our framework. It is the limit φ → ∞ identified with rapid thermal equilibration in which our results coincide with the study by Horst et al.9 VII. CONCLUSION AND OUTLOOK

We have formulated the dynamics of nucleation within the GENERIC framework and realized that its structural restrictions are such that the final nucleation dynamics that emerges is of a very specific form. Our equations are constructed by exploiting structural and thermodynamic arguments19 and therefore apply to a wider spectrum of systems that are composed of a metastable phase treatable with hydrodynamic equations, and nuclei described by internal state variables. The structure of our transport equations is complete in the sense that all structural requirements of the GENERIC formalism are satisfied. In that sense our framework offers a starting point for further studies. In particular, the dissipative bracket for the exchange dynamics between the nucleating phase and the ambient phase can be extended to incorporate the effect of temperature gradients or shear-flows on nucleation. We studied the nucleation of liquid drops in a metastable gas phase and recovered a recent literature result obtained within a more microscopic rate theory approach6 for the suppression of the nucleation rate due to nonisothermal effects. More quantitatively, we chose the thermo-physical properties of argon and were able to reproduce the insight by Wedekind et al.23, 24 obtained with brute force molecular dynamics simulations of the nucleation process. In particular, Wedekind et al. found that the local equilibrium temperature of subcritically sized nuclei is always smaller than the ambient temperature in contrast to critically- and supercritically sized clusters which are warmer than their surrounding. Specifically, we showed within our simulations that the nuclei temperature fluctuates in a non-Gaussian way around the local equilibrium temperature as introduced by Wedekind et al.23, 24 We

have seen that the isothermal and nonisothermal nucleation rates differ most significantly if the energy fluctuations of the nuclei are suppressed. That is, when nuclei cores are hardly thermalized through the heat bath. Our work can form the basis of further investigations that seek more quantitative predictions. For quantitative predictions the development of a proper statistical mechanics model connecting atomistic degrees of freedom with our coarse grained model would be desirable. The aforementioned universality implies that such a time and length scale bridging scheme should end up in the thermodynamic model that we constructed, but however with atomistic expressions for the irreversible transport coefficients and the entropic potential of the nuclei. While nucleation within classical physics is more and more understood, nucleation in the quantum regime lacks still largely a conceptual basis. When the temperature of the metastable phase becomes sufficiently low, nuclei no longer reach the critical size by thermal fluctuations, since these are exponentially suppressed, but by tunneling. Recently, a beautiful generalization of the classical GENERIC to quantum systems has been developed.41 This theory might shed light on the understanding of the nucleation processes in the quantum regime. This is, for instance, relevant in the study of liquid Helium at low temperature,42 or the onset of superconductivity where magnetic field lines slowly start penetrating the superconductor.43 Such a quantum nucleation theory is particularly challenging since the mass and momentum of the nuclei cannot easily be chosen as independent variables; in the conversion of the classical Hamiltonian to its quantum counterpart a spurious ordering problem would arise,44 known as the position dependent mass problem in super-symmetric quantum field theories. ACKNOWLEDGMENTS

We thank Hans Christian Öttinger for various hints and ideas to improve this work significantly. APPENDIX: PROOF OF JACOBI IDENTITY

In the following we prove that the proposed Poissonbracket (5) indeed satisfies the Jacobi identity. We introduce the effective bulk densities ρ¯ = ρ (1 − V ), u¯ = u (1 − V ) and s¯ = s (1 − V ), where s is the entropy of the metastable phase. These densities are convected by means of the standard Poisson structure for hydrodynamics,19 ∂B ∂ ∂A ∂A ∂ ∂B H 3 · − · {A, B} = − d r ρ¯ ∂ u¯ ∂ r ∂ ρ¯ ∂ u¯ ∂ r ∂ ρ¯ ∂B ∂ ∂A ∂A ∂ ∂B 3 · − · − d r u¯ ∂ u¯ ∂ r ∂ u¯ ∂ u¯ ∂ r ∂ u¯ ∂B ∂ ∂A ∂A ∂ ∂B . · − · − d 3 r s¯ ∂ u¯ ∂ r ∂ s¯ ∂ u¯ ∂ r ∂ s¯ (A1) The distribution f is convected by {A, B}f in Eq. (7). Obviously, the bracket {A, B} = {A, B}H + {A, B}f


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satisfies the Jacobi identify. By means of the successive variable transformation ⎛ ⎞ ⎛ ⎞ ⎛ ⎞ ρ ρ ρ¯ ⎜u⎟ ⎜u⎟ ⎜ u¯ ⎟ ⎜ ⎟ → ⎜ ⎟ → ⎜ ⎟, (A2) ⎝⎠ ⎝s⎠ ⎝ s¯ ⎠ f f f the brackets transform as {A, B}H →{A, B}H and {A, B}f → {A, B}f + {A, B}ex . So that the bracket {A, B} and {A, B} in Eq. (5) are equivalent and hence {A, B} automatically inherits all symmetries of {A, B} , including the Jacobi-identity. 1 S.

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