Equilibrium between a Droplet and Surrounding Vapor: A Discussion

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Equilibrium Between a Droplet and Surrounding Vapor: A Discussion of Finite Size Effects Andreas Troester, Fabian Schmitz, Peter Virnau, and Kurt Binder J. Phys. Chem. B, Just Accepted Manuscript • DOI: 10.1021/acs.jpcb.7b10392 • Publication Date (Web): 08 Dec 2017 Downloaded from http://pubs.acs.org on December 12, 2017

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The Journal of Physical Chemistry

Equilibrium Between a Droplet and Surrounding Vapor: A Discussion of Finite Size Effects Andreas Tröster,∗,† Fabian Schmitz,‡ Peter Virnau,‡ and Kurt Binder∗,‡ Vienna University of Technology, Institute of Material Chemistry, Getreidemarkt 9, A-1060 Wien, Austria, and Institut f¨ ur Physik, Johannes Gutenberg-Universität Mainz, Staudinger Weg 9, D-55099 Mainz, Germany E-mail: [email protected]; [email protected]

Abstract In a theoretical description of homogeneous nucleation one frequently assumes an “equilibrium” coexistence of a liquid droplet with surrounding vapor of a density exceeding that of a saturated vapor at bulk vapor-liquid two-phase coexistence. Thereby one ignores the caveat that in the thermodynamic limit, for which the vapor would be called supersaturated, such states will at best be metastable with finite lifetime, and thus not be well-defined within equilibrium statistical mechanics. In contrast, in a system of finite volume stable equilibrium coexistence of droplet and supersaturated vapor at constant total density is perfectly possible, and numerical analysis of equilibrium free energies of finite systems allows to obtain physically relevant results. In particular, such an analysis can be used to derive the dependence of the droplet surface tension γ(R) on the droplet radius R by computer simulations. Unfortunately, however, the precision of the results produced by this approach turns out to be seriously affected ∗ †

To whom correspondence should be addressed Vienna University of Technology, Institute of Material Chemistry, Getreidemarkt 9, A-1060 Wien, Aus-

tria ‡

Institut f¨ ur Physik, Johannes Gutenberg-Universität Mainz, Staudinger Weg 9, D-55099 Mainz, Germany

1

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by a hitherto unexplained spurious dependence of γ(R) on the total volume V of the simulation box. These finite size effects are studied here for the standard Ising/lattice gas model in d = 2 dimensions and an Ising model on the face-centered cubic lattice with 3-spin interaction, lacking symmetry between vapor and liquid phases. There also the analogous case of bubbles surrounded by undersaturated liquid is treated. It is argued that (at least a large part of) the finite size effects result from the translation entropy of the droplet or bubble in the system. This effect has been shown earlier to occur also for planar interfaces for simulations in the slab geometry. Consequences for the estimation of the Tolman length are briefly discussed. In particular, we find clear evidence that in d = 2 the leading correction of the curvature-dependent interface tension is a logarithmic term, compatible with theoretical expectations, and we show that then the standard Tolman-style analysis is inapplicable.

Introduction Understanding the properties of liquid droplets in supersaturated vapor, or the related problem of vapor bubbles in undersaturated liquid, is of central importance for the description of condensation and evaporation processes in fluids; related problems can also be addressed for fluid-solid transitions, and solid-solid phase transformations. 1–5 The key concept is based on estimating the free energy barrier ∆F ∗ that needs to be overcome in a homogeneous nucleation process, and the subsequent growth of nuclei of the new (stable) phase; see e.g. 1,2,6–41 for some pertinent references. The classical theory of nucleation describes the formation free energy ∆F (Vn ) of a nucleus as function of its volume Vn in terms of a competition of a volume term and a surface term. In d-dimensional systems (d = 2, 3), this can be written as

∆F (Vn ) = −Vn ∆p + Ad Vn1−1/d γ∞ = −Vn ∆µ(ρ` − ρv ) + Ad Vn1−1/d γ∞

2


.

(1)

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Here ∆p is the pressure difference between the nucleus and its environment, Ad is the surface area of a d-dimensional unit volume (A3 = (36π)1/3 ), and γ∞ is the surface tension of an (infinitely large) planar liquid-vapor interface in thermal equilibrium. The second expression on the right hand side of Eq. (1) holds to leading order in an expansion of ∆p in powers of the chemical potential difference ∆µ at the coexistence curve (where ∆µ = 0). Here ρ` and ρv are the densities of coexisting bulk liquid and vapor phases. Eq. (1) is doubtful since it uses a thermal equilibrium description in an out of equilibrium situation, so ∆p is not really well defined, and moreover the nuclei of physical interest are of nanoscopically small size only. Hence the so-called “capillarity approximation”, i.e. the use of γ∞ rather than γ(R), for a spherical nucleus with Vn = 4πR3 /3, is problematic. Analytical theories often are based on the concept of computing the density profile of a (spherically symmetric) droplet coexisting with the surrounding metastable phase. 25–30,42–45 However, these treatments also require the (doubtful 10–12,23 ) assumption that a uniquely defined free energy density f (ρ) exists for homogeneous states at densities throughout the two-phase coexistence region, in between the vapor density ρv , and the liquid density ρ` . An alternative description deals with nucleation as a kinetic phenomenon, studying the time-dependence of a distribution of nuclei n(V, t) as a function of time t elapsed after the system was suddenly brought out of equilibrium. 7–9,14,15,17,21–24,31,32,34–37,40,41 However, in practice such descriptions need as an input the (hypothetical) nucleus size distribution associated with a metastable state in equilibrium,

nms V ∝ exp[−∆F (Vn )/kB T ] ,

(2)

for which the knowledge of ∆F (Vn ) is again needed. If one tries to avoid this problem by using computer simulations that study n(Vn , t) directly, one needs a criterion to decide which particles belong to the nucleus, or do not. Even for the simplistic lattice gas/Ising model this is a complicated problem, 40,41 and many treatments along such lines (e.g. 14,15 ) hence are of doubtful validity. The merit of these kinetic approaches is that they predict the 3



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nucleation rate J(t), which also can be extracted from experiments, at least in principle. In the steady state case, the nucleation rate Jss , is related to the barrier ∆F ∗ (which follows from Eq. (1) via ∂(∆F (Vn )/∂Vn )Vn∗ = 0) as

ß

Jss = ωZ exp(−∆F ∗ /kB T ) = ωZ exp −

ãd ãòd−1 ™ 1Å 1 ãd−1 1− (Ad γ∞ /[∆µ(ρ` − ρv d d

(3)

where ω is a kinetic prefactor and Z ∝ (∆µ)(d+1)/2 the Zeldovich 8 factor. However, for many cases of practical interest the volume Vn∗ of the critical droplet is not large enough to assume that the quasi-macroscopic description in terms of Eq. (1) would be justified. A more rigorous approach would be required, which is hampered by the fact that the assumed metastable reference state is ill-defined. The ill-definedness of the thermodynamic properties of metastable states is related to the fact that first order phase transitions are associated with an essential singularity 10–12,23 at the coexistence curve. However, Langer 10–12 and G¨ unther et al. 19 proposed an analytic continuation of the thermodynamic potential in the complex plane, suggesting that the imaginary part of the free energy essentially describes the lifetime of the metastable state (apart from a kinetic prefactor). Their theory reproduces Eq. (3), but with a different prefactor Z, namely 19     (∆µ)−1

Z = ZG ∝    (∆µ)−7/3

(d = 2)

.

(4)

(d = 3)

The difference between ZG and the original Zeldovich factor has been attributed to the effect of droplet shape fluctuations, 10,19 leading to an additive logarithmic correction to ∆F (Vn ), of the form −τ kB T ln Vn , with τ = 5/4 (d = 2) or τ = 13/9 (d = 3). Indeed simulations (e.g. 34 ) have revealed strong fluctuations in the shape of critical droplets in d = 2, and evidence for Eq. (4) was obtained for the d = 2 Ising model, 36,37 but more work clearly would be needed to test Eq. (4) for d = 3 by simulations. We note that a different

4


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value of the exponent τ has been used in nucleation studies (e.g. 14,15,17,24,46 ) on the basis of Fisher’s 47 droplet model; however, it has been recognized 48–50 that this model can at best describe nuclei of size comparable to a correlation volume ξ d , where ξ is the correlation length of density fluctuations in the fluid at the coexistence curve, and hence describe only barriers ∆F ∗ of a few kB T . 17 The result of Refs. 10–12,19 that the leading correction to Eq. (1) is of order ln Vn is at variance with the phenomenological theory for the surface tension of curved interfaces. 51–57 The standard result is a prediction that there occurs a correction involving the Tolman length 53 δ

γ(R∗ ) = γ∞ /(1 + (d − 1)δ/R∗ ) ,

(5)

R∗ being the radius of the critical nucleus. If Eq. (5) holds the leading correction to Eq. (1) would be of order Vn(d−2)/d = Vn1/3 in d = 3 rather than only ln Vn . However, both magnitude and sign of δ have been the subject of a longstanding controversy, 58–68 and even its very existence has been doubted 58 (see also 3–5 for a discussion and further references). Analytical work on droplet shape fluctuations in terms of an expansion around a Cahn-Hilliard-type 42 description of a droplet coexisting with supersaturated vapor 43–45 in d = 3 has revealed both the predicted logarithmic correction and (as leading term) a Tolman correction (∝ V 1/3 ), 51–53 if the system lacks symmetry between vapor and liquid. One might think that this issue could have been easily settled by computer simulations, but this is not the case: simulations that directly study the decay of metastable states are very expensive, when a wide range of nucleation barriers need to be covered. In addition, a precise definition of what particles belong to a nucleus (needed to identify its volume Vn and radius R) is difficult even for the simple lattice gas model, 40,41 and not yet well understood for off-lattice models of fluids. This problem also hampers direct studies of nucleation barriers by biased sampling. 58 As an alternative, a thermodynamic method for the estimation of γ(R) was recently 5



proposed. 3–5,69–71 It is based on the analysis of the equilibrium between a nucleus of finite volume Vn and surrounding phase in a finite simulation box of volume Vbox at total density ρ in between the bulk coexistence densities. While for a given choice of supersaturation S = ρ/ρυ −1 for nucleation of liquid droplets from the vapor, or undersaturation S = ρ/ρ` for nucleation of vapor bubbles from the liquid, this equilibrium always is unstable for Vbox → ∞, for large but finite Vbox (by “large” we mean Vbox ξ d ) a range of S exists where this equilibrium is stable. 72 It is both possible to “measure” the free energy excess of such an inhomogeneous system relative to a corresponding homogeneous system, that has the same chemical potential, and to identify the nucleus volume from an extension of the lever rule for two-phase coexistence, thus avoiding the need of identifying the vapor-liquid interface on the single particle level in the system configurations that are analyzed. While first results on ∆F ∗ and γ(R) were rather encouraging, 3–5 they were clearly hampered by strong finite size effects (i.e., dependence of the results on Vbox ) which hitherto remained unexplored. However, similar finite size effects also occur in studies of two-phase coexistence involving a liquid slab separated from the vapor by two on average planar flat interfaces. 73–79 For the latter problem, one may choose Vbox = Ld−1 Lz with Lz > L and periodic boundary conditions throughout. Recently a more detailed understanding of the finite size effects (of order L−(d−1) ln Lz and L−(d−1) ln L and const./Ld−1 ) has been established, 80,81 in terms of the translational entropy of the slab, and fluctuations termed as “domain breathing”. The aspects of these entropic corrections relevant in the present context are summarized in the Supporting Information accompanying the present paper. Thus in the present work we reconsider this method to study the surface free energy of droplets and bubbles, examining these finite size effects in more detail. In Sec II, we briefly summarize the pertinent theoretical background, and in Sec. III we discuss numerical results for two models: the two-dimensional Ising model on the square lattice, and the Ising model with three-spin interaction on the face-centered cubic (fcc) lattice. Preliminary results (without analysis of finite size effects), on the fcc Ising model were described in Ref. 69 while

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the results for the d = 2 Ising model are new. Sec. IV summarizes our conclusions.

THEORETICAL BACKGROUND Phenomenological introduction of the Tolman length Recognizing the difficulties related to the somewhat diffuse microscopic profile of real interfaces and the resulting inherent arbitrariness accompanying any precise definition of a “cluster criterion”, the standard approach resists the temptation to preassign all particles as belonging to one of two coexisting phases from the outset. Rather than that, one instead follows an idea of Gibbs, who introduced the concept of a so-called “dividing surface” to describe the thermodynamics of a coexisting liquid (`) and vapor (υ) phase. In this approach, the existence of a diffuse interface is accounted for in a somewhat indirect fashion. First it is replaced by a mathematically sharp interface perpendicular to the gradient of the total density ρ(~x) but at a position which is arbitrary for the moment. The total volume V is thus perfectly separated into two subvolumes V = V` + Vv , with no residual inhomogeneous interfacial volume separating them. All other extensive thermodynamic variables O (like particle number N , Helmholtz free energy F , grand potential Ω, . . . ) are boldly replaced by the product of their bulk densities o` and ov of the corresponding coexisting phases and multiplied by the subvolumes V` , Vv . The true spatial dependence of the corresponding volume densities of these extensive quantities is therefore cartooned by two constant densities with a discontinuous jump at the dividing surface. To account for the apparent error resulting from such a crude description, Gibbs introduced so-called excess quantities as correctors to the corresponding extensive observables that need to be subtracted to restore their true physical values. By construction, these excess quantities scale with the total area A of the dividing surface. In particular, the proportionality factor Γ ≡ N x /A between dividing surface area A and excess particle number N x = N − N` − Nv is defined as the adsorption. In general excess quantities generally depend on the particular choice of dividing surface. Interestingly, 7



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for the particular case of a planar interface the excess grand potential Ωx turns out to be independent of the choice of dividing surface, which motivates the definition of the surface tension 55

γ ≡ Ωx /A = (F x − µN x )/A

(6)

also for nonplanar dividing surfaces, where µ is the chemical potential and N the particle number in the system. For any arbitrarily chosen dividing surface, the resulting thermodynamic description of the system is perfectly legal, but the underlying formalism turns out to be quite cumbersome. However, two particular choices, which we illustrate here for the simple example of a one component system, are immediately recognized to lead to drastic simplifications. 1. The so-called equimolar surface corresponds to choosing the position of the dividing surface such that the adsorption Γ ≡ 0 vanishes, and yields an intuitively appealing division of the total volume shared by the two coexisting phases. In fact, asking for vanishing adsorption is tantamount to imposing the simple lever rule

N = ρ` V` + ρv Vυ = (ρ` − ρυ )V` + ρv V

(7)

i.e.

V` ≡ xV,

x≡

ρ − ρv V ρ` − ρv

(8)

where x denotes the volume fraction of the liquid. 2. Consider the equilibrium between a spherical droplet and surrounding vapor in a finite cube of total volume V = Ld in d dimensions. Let p` > pv denote the bulk pressures of these coexisting phases. Choosing an arbitrary radius R, we define a spherical Gibbs dividing surface of area A(R), which separates a droplet volume Vl (R) from the 8


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surrounding vapor of volume Vv (R) = V − V` (R). In particular, let Re denote the radius of the equimolar surface defined above. In d = 3 dimensions, definition (6) can then be rephrased in the form

F = −p` V` − pv Vv + µN + Aγ

.

(9)

The total free energy F on the left hand side of this equation can, of course, not depend on the arbitrary choice of R. On the right hand side, however, the subvolumes V` and Vv and the area A do depend on R, which must be compensated by an R-dependence of the surface tension γ = γ(R). One can thus take advantage of the vanishing

0 ≡

∂F ò = −A(R)(p` − pv ) ∂R T,µ ñ ô dγ A(R) +(d − 1) γ + A(R) R dR ï

(10)

of the formal variation of F with R, the so-called notional derivative, 55 which must not be confused with the variational effect of actually changing the physical size of the droplet. The right hand side is reminiscent of the classical Laplace equation, provided we can get rid of the spurious unknown derivative

î

dγ dR

ó

. In fact, for the particular

choice R = Rs of the so-called surface of tension 51–53,55,56 implicitly defined by the requirement ñ

dγ dR

ô

≡0 ,

(11)

R=Rs

the Laplace equation p` − pυ = (d − 1)γ(Rs )/Rs

(12)

holds exactly. Eq. (11) indicates that Rs is an extremal point of the function γ(R). In

9



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fact, one can show that knowledge of Rs and γ(Rs ) completely determines γ for all R according to the universal formulas 55  Å ã2    1 + 1 R−Rs

R

γ(R) 2 R Rs = Å ã2 γ(Rs )   1 + 1 R−Rs Rs +2R 3 R Rs

(d = 2)

.

(13)

(d = 3)

from which we learn that γ(R) takes on its single global minimum at R = Rs . Fixing the value of the arbitrary parameter R to Re or Rs by condition (7) or (11), respectively, means that e.g. in the grand-canonical description Re = Re (T, µ) and Rs = Rs (T, µ). In particular, this implies that while a variation of the arbitrary parameter R merely constitutes an infinitesimal reparametrization of our description, variations of Rs or Re correspond to real physical changes of the physical state of the system. Whereas both parameters are equally capable of encoding this information, in view of the considerable technical simplifications offered by relation (11) it is standard practice to reparametrize the theory in terms of the variables T and Rs , i.e. to regard µ = µ(T, Rs ) as a function of T and Rs . Consequently, the equimolar radius Re = Re (T, µ) is also reparametrized as Re = Re (T, Rs ). The difference

δ(T, Rs ) = Re (T, Rs ) − Rs

(14)

which generally must be expected to be a nontrivial function of Rs , and its limiting value, the notorious “Tolman length”

δ∞ (T ) := lim δ(T, Rs )

(15)

Rs →∞

played a key role in the further development of the theory. In fact, provided δ(T, Rs ) ≈ δ∞ (T ) is assumed to be independent of Rs and of microscopic size, it is possible to show that 53 Å γ∞ (T ) 2δ∞ (T ) ` ã2 =1+ +2 + ..., γ(T, Rs ) Rs Rs

10


Rs → ∞ .

(16)

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Here l is another microscopic length. Unfortunately, phenomenological thermodynamics can not yield any more explicit results on γ(T, Rs ), and even the existence of the limit in Eq. (15) has been questioned in the literature (see 56 for a discussion, where Eqs. (15), (16) were confirmed by a Landau theory treatment). As emphasized in the introduction, the situation described by Eq. (9) can be a stable equilibrium for constant µ, T and V → ∞ only when also V` → ∞. Thus the theory summarized in Eqs. (9)- (15) is to be understood as a description of how the thermodynamic limit is approached. For any finite value of the simulation volume Vbox , however, fluctuations that were not addressed in Eqs. (9)- (15) need to be considered. We have summarized here these classical arguments 55 in order to emphasize that no essential difference between d = 2 and d = 3 arises with respect to Eqs. (14)-(16). However, as we shall see, in d = 2 the leading correction to γ∞ (T ) due to finiteness of Rs are due to statistical fluctuations, and then Eqs. (14)-(16) in fact are meaningless.

Phenomenological considerations on phase coexistence in finite volumes for arbiWhile in the thermodynamic limit phase coexistence exists for ρ exceeding ρcoex v trarily small x, this is not true for finite Vbox , due to the occurrence of the droplet evaporation/condensation transition, 3,72,82–84 see Fig. 1. This droplet evaporation/condensation transition occurs since for small droplets the free energy cost of forming the interface is too high, in comparison with the state where the excess density contained in the small droplet is dissolved in the homogeneous vapor, which hence in the finite system is the stable phase up to the density ρ1 . In the regime from about ρ1 to about ρ2 (see Fig. 1) the equilibrium in the finite volume is described by coexistence of a (spherical) liquid droplet of volume V` (i.e., the nucleus volume Vn , cf. Eq. (1)) surrounded by vapor that in comparison to bulk coexistence conditions is supersaturated. Since the supersaturation decreases with increasing droplet volume, the variation of the droplet volume as a function of the total density in the box for finite box volumes is slightly nonlinear (Fig. 1). When one studies this situation by a mean 11



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field theory (describing the cost of forming a droplet by Eq. (1), and disregarding hence is of order L−d/(d+1) . the problem that γ(R) may differ from γ∞ ) one finds 83 that ρ1 − ρcoex v ) is of order unity, so near ρ2 we expect hVn i ∝ Ld already, − ρcoex )/(ρcoex However, (ρ2 − ρcoex v ` v as predicted by the lever rule. Since Vn is not a conserved quantity, we expect that then the fluctuation hVn2 i − hVn i2 ∝ Ld as well, while near ρ1 despite the smallness of hVn i also the fluctuation will be larger, since the system jumps between the two states shown by the dots in Fig. 1. The relative mean square fluctuations of the droplet volume are of order unity near ρ = ρ1 and of order L−d for ρ1 ρ < ρ2 . The chemical potential in finite size systems in the canonical ensemble (Vbox , N and T are fixed) is also a fluctuating quantity, and we expect fluctuations of order hµ2 i−hµi2 ∝ L−d as well, 85 apart from the vicinity of the droplet condensation/evaporation transition where (in d = 3) a larger fluctuation (corresponding to jumps between the two states denoted by dots in Fig. 1b) occurs, since both of these states have an excess chemical potential of order L−d/(d+1) . 3,83,84 From Fig. 1b) it is seen that states with densities ρ, in the region ρ1 < ρ < ρ2 (choosing ρ such that the regions of strong fluctuations near ρ1 and ρ2 , that we have discussed above qualitatively, are avoided) have a chemical potential hµi for which also a homogeneous vapor < ρv < ρ1 ) exists. Likewise there exists for this state (with a density ρv in the region ρcoex υ (Fig. 2). chemical potential also a state of homogeneous liquid with density ρ` , for ρ` > ρcoex ` To obtain hµL (T, ρ)i = (∂h∆fL (T, ρ)i/∂ρ)T,L

(17)

from the free energy density h∆fL (T, ρ)i, which can be sampled very efficiently by the successive umbrella sampling method, 86 we carry out the derivative in Eq. (17) numerically. Of course, applying a straightforward finite difference type of scheme to accomplish this task inevitably amplifies any residual numerical noise present in the underlying sampled free energy data. For moderate to large system sizes such a brute force approach frequently results in an unacceptably high numerical noise contamination of the resulting derivative, irrespective of the fact that any statistical noise of ∆fL (T, ρ) may be invisibly small at least on the scale 12


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of Fig. 2. To overcome this nuisance we recommend smoothening the free energy data by so-called Beziér interpolation. The nuts and bolts of this procedure, which nowadays may be regarded as standard in the field, are explained in great detail e.g. in Ref. 87 Indeed, Fig. 2 illustrates that the statistical noise in the resulting chemical potential hµi is hardly visible. In previous work, 3–5 it was already shown that for ρ distinctly smaller than ρ1 (L) systematic finite size effects on both h∆fL (T, ρ)i and hµL (T, ρ)i) were completely negligible. The result is not surprising, since finite size effects in the bulk homogeneous phases in systems with periodic boundary conditions are expected to be 88,89 of relative order exp(−L/ξ), where ξ is a bulk correlation length. In the case of Fig. 2, liquid-vapor criticality occurs for 90 kB T /J ≈ 2.27, and since the order parameter ρ` − ρv for kB T /J = 2.0 is close to saturation already, and it is known that there ξ is of the order of the lattice spacing only, and terms of order exp(−L/ξ) are hence negligible (lengths are measured in units of the lattice spacing throughout). For ρ exceeding ρ1 (L) in Fig. 1, however, finite size effects are very strong, of course, but entirely due to phase coexistence, including also the fluctuation effects that are additionally possible in this situation. From the triple of values ρυ , ρ, ρ` and Eq. (7) it is straightforward to obtain the equimolar droplet volume, hVè i. Since we know that in thermal equilibrium hµi is spatially homogeneous also in a system exhibiting two-phase coexistence where both density and pressure are not homogeneous, we can conclude that the free energy density of the vapor surrounding the droplet is the same as the free energy density h∆fL (ρv )i of the vapor having the density ρv (that we can read off from Fig. 2). Given then the free energy density at the density ρ, which we also observe in the simulation, we can conclude that this free energy density is enhanced relative to h∆fL (ρv )i due to terms related to the droplet, and due to additional fluctuation contributions to the free energy, that are specific for phase coexistence in a finite volume Ld (and possibly depend on the magnitude of this volume). One such fluctuation term obviously results from fluctuations in the position of the center of mass of the droplet in the simulation box, i.e. giving rise to the translational entropy of

13



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the droplet,

Ftransl = −kB T ln(Vbox /`d0 )

(18)

where `0 is a reference length that will be discussed in the Supporting Information. Of course, for macroscopic two-phase coexistence (Vbox → ∞, V` /Vbox = x > 0) this term is negligible, but it is important for finite Vbox . The situation where a liquid droplet coexists with surrounding vapor (for ρ1 < ρ < ρ2 ) or the inverse situation where a bubble coexists with surrounding liquid is qualitatively analogous to the slab configuration that exists for ρ )/2. There one finds a liquid domain separated from the vapor by + ρcoex near ρd = (ρcoex ` v two planar interfaces connected into themselves via the periodic boundary conditions. This situation is conveniently studied in a Ld−1 Lz geometry with Lz > L, and varying Lz at fixed L the presence of the translational entropy has been clearly established. 80,81 Since we rely here on several aspects of this study, the essential features of this situation, in particular the arguments that clarify the length `0 , will be summarized in the Supporting Information. In these studies of planar interfaces, an additional contribution giving rise to a sizedependent correction to the interface tension has been discovered, the so-called “domain breathing effect”. 80,81 It could be expected, that an analogous effect exists in the present problem, too. This problem is rather subtle and will be discussed in the Supporting Information. Arguments discussed in the Supporting Information show that this fluctuation in the size of a droplet, of average radius hRe i extracted via Eq. (17) from the observation of the densities ρ` and ρv as indicated in Fig. 2, using 4πhRe i3 /3 = hV` i, does occur and depends on the size of the domain hRe i itself, but does not depend on the linear dimension of the box as long as hV` i/V is very small. Of course, for the case of planar interfaces, studied in a V = Ld−1 Lz geometry, the domain volume is of the same order as V itself, and thus there is no contradiction between the arguments described in the Supporting Information and the previous work. 80,81 The fluctuation in the volume of the droplet can be alternatively 14


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interpreted as the “uniform mode” of the interfacial capillary wave spectrum. Both this volume fluctuation and the nonuniform fluctuations of the droplet may show up in corrections to γ(Rs ), or equivalently γ(Re ), and actually such corrections have been identified in various treatments, as discussed already in the introduction. Of course, the condition that hV` i/V 1 breaks down when the density ρ in Fig. 2 approaches the transition density ρ2 . Then the droplet radius is of the same order as the box linear dimension L, and a nontrivial size dependence of γ(Rs ), γ(Re ) on L is expected. However, this regime is not useful for considering nucleation phenomena, since large fluctuations in droplet shape controlled by the boundary conditions applied to the simulation box will gradually come into play. In d = 2, we will encounter fluctuations from circular to slab-like domains or vice versa, 91 and in d = 3, from spherical to cylinder-like shape of the domain. Only when these fluctuations are completely negligible can we learn anything on the droplets that appear as transient objects in nucleation phenomena. We now wish to make contact with the quasi-macroscopic theory of Sec. 2.1, as summarized in Eq. (5)-(16). Note that this approach implicitly assumes that the center of mass of the droplet remains fixed at the origin of the coordinate system, and the boundaries of the considered system are very far away and need not be considered. Obviously, these assumptions are not appropriate for a discussion of the simulation. Since in the simulation no “demon” is applied that fixes the center of mass of the droplet in a prescribed position, we need to replace Eq. (9) by

F − Ftransl (V ) = −p` V` (R) − pv Vv (R) + µN + A(R)γ(R),

(19)

where R defining the radius of the dividing surface is then fixed at R = Rs by the condition Eq. (11) as previously. In Eq. (19) now F is the exact free energy of the unconstrained finite system in which a stable equilibrium with a finite value of Rs is established. Of course, the exact free energy F is not really available, it is estimated only within some statistical error from the simulations, such as shown in Fig. 2, but in principle such errors can be made 15



Page 16 of 39

arbitrarily small, if only enough computational resources are invested. Since no physical quantities can depend on the arbitrary radius R that only is needed to define Rs , from Eq. (9) or Eq. (19), respectively, we simply can repeat the constructions of Ref., 5 where Ftransl (V ) was neglected, now avoiding this neglect. Fig. 3 shows that for radii Rs of interest, indeed a significant change of both Rs and γ(Rs )/kB T occurs.

NUMERICAL RESULTS FOR SELECTED MODELS The Two-Dimensional Ising Model As described in more detail by Schmitz et al., 81 the free energy density ∆fL (T, ρ) of L × L lattice gases and its derivative µL (T, ρ), cf. Eq. (17), has been obtained as a function of density ρ, for a broad range of linear dimensions L, namely L =50,60,70,80,90,100,110 and 120 lattice units (see Fig. 2 for a typical example). Using these data one can follow the procedure explained in Ref. 5 and outlined in Fig. 3 to obtain for each choice of L both Rs and γ(Rs ) for a suitable range of ρ. I.e., taking advantage of the fact that at kB T /J = 2.0 the lattice anisotropy of the Ising model interfacial tension can be safely neglected,

γ(R) = [L2 ∆fL (T, ρ) − R2 π∆fL (T, ρ` ) − (L2 − R2 π)∆fL (T, ρv ) + 2kB T ln(L/`0 )]/(2πR) ,(20)

ρv , ρ, ρ` being the three densities defined in Fig. 2 that characterize the considered equilibrium, ∆FL (T, ρv ), ∆FL (T, ρ) and ∆FL (T, ρ` ) the corresponding free energies, and R is the radius of the dividing surface; as noted in Eqs. (11), (13) and illustrated in Fig. 3, the minimum of γ(R) for each choice of T, L, and ρ yields then both Rs and γ(Rs ). Fig. 4 gives an example for this construction, using `0 equal to the lattice spacing. We emphasize that the inclusion of the translational entropy of the droplet (i.e. the last term on the right hand side of Eq. (20) is crucial for a reasonable data collapse; part C of the SI gives details on the

16


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failure of this construction when the translational entropy is omitted. Of course, by using the notation γ(R) in Eq. (20) we have already expressed our hope that a meaningful result for γ(Rs ) which does not depend significantly on the arbitrary choice of the simulation box linear dimension L will be obtained. However, Fig. SI1 show that this expectation dramatically fails if we would ignore the translational entropy correction in Eq. (20), there would be no tendency whatsoever for the relevant part of the curves (highlighted by colors in Fig. 4) to superimpose on a L-independent master curve. The decreasing part of the curves on the left side of the graph correspond to the region in Fig. 2 where actually the slab configuration occurs, and the right most part of the data are affected by the droplet evaporation/condensation transition and hence also are useless in the context of the present analysis. It is the remaining part of the data where one stays away from these limits (highlighted by the colors) that should converge to a L-independent result for γ(Rs ), and this is indeed the case, but only provided that the translational entropy is included (cf. part C of the SI). The same behavior is observed when one analyzes the adsorption Γ(Rs ), Fig. 4b. However, including the translational entropy correction as written in Eq. (20) yields a spectacular improvement (Fig. 4a): in the relevant regime the data for the larger choices of L do converge to a master curve that is well fitted by 5 ln Rs c γ(Rs ) =1+ + γ∞ 4πγ∞ Rs Rs

.

(21)

with c = 1.2. Note that the coefficient 5/(4πγ∞ ) of the logarithmic correction is exactly what has been predicted earlier, by field-theoretic methods 19 describing capillary-wave type shape fluctuations of the circular droplet, and this result has been independently confirmed by an umbrella sampling Monte Carlo study of geometric Ising clusters (defined at low temperatures in terms of contours enclosing reversed spins). 36,37 The present method for estimating γ(Rs ) is completely different from the previous methods, 19,36,37 and unlike 36,37 it is applicable at all temperatures T less than the bulk critical temperature Tc , while the use 17



Page 18 of 39

of geometric Ising clusters is legitimate only for 40,41 T much less than Tc . Thus we consider the strong evidence for the predicted logarithmic correction as rather satisfactory. As expected, also the corresponding data for the adsorption Γ(Rs ) converge for large L to a nontrivial master curve (Fig. 4b). However, if one naively applies the phenomenological theory 51–53,55 to extract a Tolman length é

Ñs

δ(Rs ) = Rs

2Γ(Rs ) 1+ −1 Rs ∆ρ

≈

Γ(Rs ) , ∆ρ

(22)

one obtains the rather unexpected result that both Γ(Rs ) as well as δ(Rs ) seems to approach negative limiting values for Rs → ∞, possibly even logarithmically diverging towards minus infinity (Fig. 4b,c)! Of course, this result is at variance with the expectation from Landau theory 43,44,55,56 that for systems with perfect symmetry between liquid and gas (which holds for the Ising/lattice gas model) δ(Rs ) → 0 for Rs → ∞. We suggest that the reason for this problem is that in d = 2 a Tolman type correction, Eq. (5), is not the leading correction describing the curvature dependence of γ(Rs ). Eqs. (9)-(16) ignore fluctuation effects from the outset; in d = 3, however, a combination of the Tolman-type theory with capillary-wave type droplet shape fluctuations, has been worked out 43–45 and yields B 1 2δ∞ A ln Rs γ(Rs ) + =1− + 2 γ∞ Rs γ∞ Rs γ∞ Rs2

,

d = 3,

(23)

where the universal constant A = −7/(12π) agrees with the prediction derived from field theoretic methods, 19 while the constant B is non-universal. Thus, in d = 3 dimensions for systems lacking a precise liquid-gas symmetry the Tolman correction for Rs → ∞ is the leading correction term while in d = 2 it is not. If one applies the Tolman-type analysis nevertheless in d = 2, one picks up the leading correction in Eq. (21), i.e. a logarithm, and hence the interpretation in terms of the Tolman “length” δ∞ (T ) in Eq. (16) becomes meaningless. The numerical values of Rs – and consequently those of γN (Rs ) as shown in Fig. 4 –

18


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are obtained from minimizing γ(R), implementing the prescription Eq. (11), (13), which in turn relies heavily on “classical” thermodynamical reasoning. In comparison, the definition of the equimolar radius Re merely requires knowledge of the densities entering the lever rule Eq. (8). Nevertheless, as Fig. 5 illustrates, the equimolar interface tension γ(Re ) exhibits a behavior similar to that of γ(Rs ). In fact, failing to include translational entropy corrections to γ(Re ) is found to produce a spread similar to that for γ(Rs ) in the numerically obtained values for the equimolar interface tension in different box sizes (Fig. SI2a). Once again, including a corresponding correction with normalization length of `0 = 1.0 tremendously reduces this spread (Fig. SI2b), allowing to anticipate a common master curve that exhibits a logarithmic correction to the equimolar version of the Tolman formula (cf. Eq. (12.11) of Ref. 54 ) which is well described by Eq. (21) by simply changing the value of c to 0.8 (Fig. 5c).

The face-centered cubic Ising model with three-spin interaction Landau theory implies δ∞ (T ) ≡ 0 for systems possessing strict symmetry between liquid and vapor, such as the standard Ising model with pairwise interactions. Then Eq. (23) would imply that even in d = 3 the logarithmic term (ln Rs )/Rs2 in fact is the leading curvature correction. However, it is clear that this term will be very small numerically for large enough values for Rs , where the methods described here are applicable. In addition, there do hardly exist systems in nature for which this Ising-symmetry is strictly realized. In 5 it was attempted to carry out the analysis based on Eqs. (9)-(17) for the Lennard-Jones (LJ) fluid, which indeed lacks this symmetry. The analysis in 5 missed to include in the analysis translational entropy term, Eq. (18), of droplets and bubbles. The translational entropy term is only fully sampled when the center of mass of the droplet (or bubble) has diffused over the whole available box volume in the course of the simulation. However, since only local moves of the particles were used in these simulations 5 it is not clear whether the configuration space of the model system has been well enough explored. Indeed, in the study of planar interfaces for the LJ model 81 clear evidence was obtained that the translation entropy of the interfaces 19



Page 20 of 39

is only fully “measured” in the simulations when non-local moves of the particles, leading to a much more rapid “diffusion” of the domain in the simulation box, are included. Also the statistical accuracy of the data of Ref. 5 is somewhat unsatisfactory, and hence a study of the problem for off-lattice fluid models must be left for future work. However, already in 69 the idea was exploited that the Ising model on the face centered cubic (fcc) lattice with triplet interactions

H = −J

X

Si Sj Sk − H

X

Si ,

Si = ±1,

(24)

i

hijki

is a model with large asymmetry between its vapor-like and fluid-like phases, and hence a Tolman length δ∞ (T ) appreciably distinct from zero can be expected. 63 Since Eq. (24) is a lattice model, and using a single spin-flip algorithm in conjunction with Wang-Landau sampling the full equilibrium free energy density fL (T, ρ) at a temperature T ≈ 0.7T0 (kB T0 /J = 11.39 ± 0.01 is the transition temperature of this model) has been obtained, for L = 18, 20, 22, 24, 26, 28 and 30. 69 But in the analysis of these data 69 the inclusion of the translational entropy was missed, and in the following we hence shall explore the consequences when this entropy {Eq. (18)} is included, applying the d = 3 analog of Eq. (20) in order to carry out a re-analysis of the data of Ref. 69 Again (due to the high coordination number 12 of the fcc lattice), the anisotropy of the interfacial tension at the chosen temperature can be neglected, both droplets and bubbles can be assumed to be practically spherical, in spite of the underlying lattice structure. Numerical data for γ(Rs )/γ∞ computed from our present method, in which translational entropy effects are taken into account, are shown in Fig. 6. The reader is invited to compare these results to those obtained by a previous analysis 69 (reproduced in Fig. SI3) in which translational entropy contributions of droplets and bubbles had been ignored. For the convenience of the reader, part C of the SI includes plots analogous to Fig. 6 of a corresponding evaluation of the same data set in which translational entropy effects are ignored.

20


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Indeed, as Fig. SI3 illustrates, even when ignoring effects of translational entropy the tendency of the data to collapse on master curves was much better than in d = 2 (cf. Fig. SI1), albeit still far from perfect. Note that both master curves must end in the point γ(Rs → ∞)/γ∞ = 1, so the curve referring to bubbles must come from above to this limit, and hence clearly even for the largest accessible values of Rs , Rs ≈ 7, the data for γ(Rs )/γ∞ for bubbles would be far from their asymptotic behavior for large Rs when translational entropy corrections are ignored. However, when the translational entropy term is included, the collapse on master curves for γ(Rs )/γ∞ indeed is now significantly improved, and the data for 1/Rs ≤ 0.15 for both bubbles and droplets are compatible with the two leading terms in Eq. (23), i.e. a linear variation with 1/Rs , with δ∞ ≈ −0.36 (bubbles) while δ∞ ≈ +0.36 (droplets) when we always take Rs positive. Fortunately, this result is close to the previous estimate (δ∞ ≈ 0.33 69 ), but clearly the present analysis is much more convincing. We also emphasize that the present result does not depend in a sensitive way on the choice `0 = 1 for the normalization length of the translational entropy in Eq. (18), see the discussion of this problem in the Supporting Information. Of course, it is a bit disappointing nevertheless that the data clearly do not yet warrant a search for the higher order terms in Eq. (23): the logarithmic correction of the radiusdependent surface tension that was so overwhelming in d = 2, has no significant traces in our d = 3 data yet; the residual finite box-size effects do not warrant a search for these higher order terms here; of course, in d = 2 the logarithmic term was the dominant correction {Eq. (21)} and hence it is clear that the situation had to be different.

CONCLUSIONS In this paper, we have presented a re-analysis of the method to extract the curvaturedependent surface tension γ(Rs ) of droplets from the analysis of the equilibrium conditions of the droplet or bubble coexisting in a finite volume with surrounding fluid, that for an

21



infinite volume would be supersaturated or undersaturated, respectively. Unlike the original proposal of Block et al., 4,5 the translational entropy of the droplet or bubble in the considered volumes is taken into account, since in a Monte Carlo simulation context it is natural that the center of mass of the droplet or bubble is not fixed at a particular position in space, unlike methods based on the density functional approach 42–45 where it is natural to fix the center of the droplet or bubble at the origin of a (spherical) volume. For the first time, also an extension of the methodology to d = 2 dimensions is presented. We show that in this case the original methodology of Block et al. 4,5 fails completely, since the simulation results for σ(Rs ) are strongly affected by the finiteness of the linear dimension L of the box, irrespective of how large L is chosen (Fig. SI1). We demonstrate that the situation is significantly improved when the translational entropy of the droplet (Eq. (18), leading to the last term on the right hand side of Eq. (20)) is taken into account (Fig. 4). It should be noted that for practically accessible values of L and Rs residual finite size effects still are present for several reasons: the influence of the droplet evaporation/condensation transition at the density ρ1 , and the droplet slab transition at the density ρ2 (Fig. 1) disturbs the analysis; also the chemical potential in a finite system is not strictly constant (as presumed in the analysis) but fluctuating. But the data clearly indicate (Fig. 4) that the collapse on a L-independent nontrivial master curve for σ(Rs ) improves as L increases. This master curve, for the Ising model at kB T /J = 2.0 in d = 2 (i.e., T /Tc ≈ 0.881, where the droplet shape is hardly affected by the lattice anisotropy and hence circular) is not consistent with the standard description 51–53,55,56 in terms of the Tolman length, since the leading correction in the expansion of γ(Rs ) is logarithmic, γ(Rs ) = γ∞ + (5/4π) ln Rs /Rs + higher order terms, rather than of the Tolman form, γ(Rs )/γ∞ = 1 − δ∞ /Rs , Eqs. (5), (16). We note that actually the existence of the leading logarithmic correction (and its universal coefficient (5/4π) in d = 2 dimensions) can be inferred from previous work using field theoretic methods for nucleation theory 19 and umbrella sampling Monte Carlo simulations for Ising clusters. 36,37 It is nevertheless very gratifying that the present approach confirms these findings com-

22


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pletely independently. We speculate that the conclusion of the standard approach 44,45,56 that δ(Rs ) → 0 for Rs → ∞ when liquid-vapor symmetry occurs, as is true for the Ising lattice gas model, is inapplicable when the Tolman correction is not the leading term. In the case of d = 2 dimensions, capillary-wave type droplet shape fluctuations provide the strongest correction, rendering the standard approach 51–53,55,56 less useful. However, in d = 3 dimensions for systems lacking vapor-liquid symmetry, such as the Ising model on the fcc lattice with triplet interaction, the Tolman length does yield the leading correction, cf. Eq. (23), and actually the numerical results (Fig. 6, SI4) comply nicely with the standard theory. Also in this case the translational entropy of droplets and bubbles matters (Fig. SI3), including this correction greatly improves the collapse on the expected master curves for γ(Rs ), δ(Rs ) and Γ(Rs ); in the regime where we observe a reasonable collapse the latter two quantities simply are constants, and agree with the theoretical relation δ(Rs ) = Γ(Rs )/∆ρ {Eq. (22)}, ∆ρ being the density difference between coexisting bulk vapor and liquid. Our data, however, do not allow to conclude anything about the next order logarithmic term in d = 3 (Eq. (23)). As a final caveat, we recall that in simulation approaches for droplets or bubbles in off-lattice models one has to clarify whether the applied methodology does allow for a translational motion of the droplet in the simulation box, or not; we speculate that translational droplet motions are included for nonlocal Monte Carlo algorithms (as used in 81 ) but are not included in strictly microcanonical Molecular Dynamics runs.

Acknowledgement The authors declare no competing financial interest. A.T. acknowledges support by the Austrian Science Fund (FWF) Project P27738-N28. Supporting Information: The Supporting Information summarizes in Part A what is known about the translational entropy of planar interfaces in the Ising model. Part B summarizes comments on the ”domain breathing” effect, and Part C provides further details 23



on the numerical results and their analysis.

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(58) ten Wolde, P.R.; Frenkel, D. Computer Simulation Study of Gas-Liquid Nucleation in a Lennard-Jones System. J. Chem. Phys. 1998, 109, 9901-9918. (59) Van Giessen, A.E.; Blokhuis, E.M.; Bukman, D.J. Mean Field Curvature Corrections to the Surface Tension. J. Chem. Phys. 1998, 108, 1148-1156. (60) Moody, M.P.; Attard, P. Curvature Dependent Surface Tension from a Simulation of a Cavity in a Lennard-Jones Liquid Close to Coexistence. J. Chem. Phys. 2001, 115, 8967-8977. (61) Blokhuis, E.M.; Kuipers, J. Thermodynamic Expressions for the Tolman Length. J. Chem. Phys. 2006, 124, 074701. (62) Barrett, J.C. Some Estimates of the Surface Tension of Curved Interfaces Using Density Functional Theory. J. Chem. Phys. 2006, 124, 144705. (63) Anisimov, M.A. Divergence of Tolman’s Length for a Droplet near the Critical Point. Phys. Rev. Lett. 2007, 98, 035702. (64) Van Giessen, A.E.; Blokhuis, E.M. Direct Determination of the Tolman Length from the Bulk Pressures of Liquid Drops via Molecular Dynamics Simulations. J. Chem. Phys. 2009, 131, 164705. (65) Xuo, Y.-Q.; Yang, X.-C.; Cui, Z.-X.; Lai, W.P. The Effect of Microdroplet Size on the Surface Tension and Tolman Length. J. Phys. Chem. B 2011, 115, 109-112. (66) Horsch, M.; Hasse, H.; Shchekin, A.K; Agarwal, A.; Eckelsbach, S.; Vrabec, J.; Mueller, E.A.; Jackson, G. Excess Equimolar Radius of Liquid Drops. Phys. Rev. E 2012, 85, 031605. (67) Hommann, A.-A.; Bourasseau, E.; Stoltz, G.; Malfreyt, P.; Strafella, L.; Ghoufi, A. Surface Tension of Spherical Drops from the Surface of Tension. J. Chem. Phys. 2014, 140, 034110. 29



(68) Bruot, N.; Caupin, F. Curvature Dependence of the Liquid-Vapor Surface Tension Beyond the Tolman Approximation. Phys. Rev. Lett. 2016, 116, 056102. (69) Tröster, A.; Binder, K. Positive Tolman Length in a Lattice Gas with Three-Body Interactions. Phys. Rev. Lett. 2011, 107, 265701. (70) Das, S.K.; Binder, K. Universal Critical Behavior of Curvature-Dependent Interfacial Tension. Phys. Rev. Lett. 2011, 107, 235702. (71) Tröster, A.; Binder, K. Microcanonical Determination of the Interface Tension of Flat and Curved Interfaces from Monte Carlo Simulations, J. Phys.: Condens. Matter 2012, 24, 2844107. (72) Binder, K.; Kalos, M.H. “Critical Clusters” in a Supersaturated Vapor: Theory and Monte Carlo Simulation. J. Stat. Phys. 1980, 22, 363-396. (73) Binder, K. Monte Carlo Calculation of the Surface Tension for Two- and ThreeDimensional Lattice-Gas Models. Phys. Rev. A 1982, 25, 1699-1709. (74) Berg, B.A.; Hansmann, U.; Neuhaus, T. Properties of Interfaces in the Two and Three Dimensional Ising model. Z. Phys. B 1993, 90, 229-239. (75) Hunter, J.E.; Reinhardt, W.P. Finite-size Scaling Behavior of the Free Energy Barrier between Coexisting Phases: Determination of the Critical Temperature and Interfacial Tension of the Lennard-Jones Fluid. J. Chem. Phys. 1995, 103, 8627-8637. (76) Chen, L.J. Area Dependence of the Surface Tension of a Lennard-Jones Fluid from Molecular Dynamics Simulations. J. Chem. Phys. 1985, 103, 10214-10216. (77) Potoff, J.J.; Panagiotopoulos, A.Z. Surface Tension of the Three-Dimensional LennardJones Fluid from Histogram-Reweighting Monte Carlo Simulations. J. Chem. Phys. 2000, 112 , 6411-6415.

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(78) Errington, J.K. Evaluating Surface Tension Using Grand-Canonical Transition-Matrix Monte Carlo Simulation and Finite-Size Scaling. Phys. Rev. E 2003, 67, 012102. (79) Santra, M.; Chakrabati, S.; Bagchi, B. Gas-Liquid Nucleation in a Two Dimensional System. J. Chem. Phys. 2008, 129, 234704. (80) Schmitz, F.; Virnau, P.; Binder, K. Determination of the Origin and Magnitude of Logarithmic Finite-Size Effects on Interfacial Tension: Role of Interfacial Fluctuations and Domain Breathing. Phys. Rev. Lett. 2014, 112, 125701. (81) Schmitz, F.; Virnau, P.; Binder, K. Logarithmic Finite-Size Effects on Interfacial Free Energies: Phenomenological Theory and Monte Carlo Studies. Phys. Rev. E 2014, 90, 012128. (82) Biskup, M.; Chayes, L.; Kotecky, R. On the Formation/Dissolution of Equilibrium Droplets. Europhys. Lett. 2002, 60, 21-27. (83) Binder, K. Theory of the Evaporation/Condensation Transition of Equilibrium Droplets in Finite Volumes. Physica A 2003, 319, 99-114. (84) MacDowell, L.G.; Virnau, P.; M¨ uller, M.; Binder, K. The Evaporation/Condensation Transition of Liquid Droplets. J. Chem. Phys. 2004, 120, 5293-5308. (85) Landau, L.D.; Lifshitz, E.M. Statistical Physics; Pergamon Press: Oxford, 1958. (86) Virnau, P.; M¨ uller, M. Calculation of Free Energy through Successive Umbrella Sampling. J. Chem. Phys. 2004, 120, 10925-10930. (87) Bachmann, M. Thermodynamics and Statistical Mechanics of Macromolecular Systems; Cambridge Univ. Press: Cambridge, UK., 2014. (88) Barber, M.N. in Phase Transitions and Critical Phenomena, Vol. 8, Chapter 2 (C. Domb amd J.L. Lebowitz, eds.) Academic Press: London, 1981. 31



(89) Privman, V.P. (ed.) Finite Size Scaling and the Numerical Simulation of Statistical Systems; World Scientific: Singapore, 1990. (90) Onsager, L. Crystal Statistics. I. A Two-Dimensional Model with an Order-Disorder Transition. Phys. Rev. 1944, 65, 117-149. (91) Moritz, C.; Dellago, C.; Tröster, A. Interplay of Fast and Slow Dynamics in Rare Transition Pathways: The Disk-to-Slab Transition in the 2d Ising Model. J. Chem. Phys. 2017, 147, 152714.

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Page 32 of 39

Page 33 of 39

vapor+ spherical droplet

vapor+ liquid cylinder

hV i

homogeneous vapor

lever rule

(a) ρcoex v

ρ1

ρ2

droplet evaporation/ condensation transition

(b)

droplet shape transition sphere * ) cylinder

µ

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


ρcoex v

ρ1

ρ2

ρ Figure 1: a) Qualitative sketch of the variation of the average liquid droplet volume hV` i in a simulation box of volume Vbox = Ld plotted versus the density ρ. In the thermodynamic , according to the lever rule (broken straight limit, hV` i is simply proportional to ρ − ρcoex v line). In the finite system, the regime of stability of the homogeneous vapor extends roughly up to ρ1 , the density at which the droplet evaporation/condensation transition occurs. Note that ρ1 ∝ 1/Ld/(d+1) , and this transition in mean field theory is a sharp first order transition, associated with jumps of both hV i (a) and the chemical potential (b), as indicated by the broken straight lines ending at coexisting states (denoted with dots). In reality this transition is rounded by finite size (full curve). For densities ρ in between about ρ1 and about ρ2 the equilibrium state consists of a droplet coexisting with surrounding vapor (which is supersaturated in comparison with a vapor at µcoex , since hµi exceeds µcoex ). At around ρ2 the droplet undergoes a shape transition from circular shape to a slab-like domain (in d = 2) or from spherical to cylindrical shape (in d = 3). For more explanations see the text.

33



0.06

βµ(T, ρ)

0.04 0.02 0.00 0.0

0.2

−0.02

0.4

0.6

0.8

1.0

0.6

0.8

1.0

ρ

βµ ρl

−0.04

ρ ρ`

−0.06 0.009 0.008 0.007

βf (T, ρ)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Page 34 of 39

0.006 0.005 0.004 0.003 0.002 0.001 0.000 0.0

0.2

0.4

ρ Figure 2: Illustration of the numerical procedure to determine the coexistence density triplets (ρυ , ρ, ρ` ) for a 2d Ising lattice gas for a temperature kB T /J = 2.0 and a twodimensional square box with linear dimensions L = 60 and periodic boundary conditions throughout. Taking the total density ρ = 0.109 as an example, the corresponding liquid and vapor densities at a common chemical potential of hβµL i = 0.0369 are ρv = 0.05 and ρ` = 0.962, respectively. The lower part of the figure shows the corresponding excess free energy density h∆fL i/kB T as a function of density.

34


Page 35 of 39

γ (noF S) (R) 0.40

γ (F S) (R) (Re , γ (noF S) (Re )) (Re , γ (F S) (Re )) (F S)

(Rs

γ (F S)(R), γ (noF S)(R)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


(F S)

)), γ (F S) (Rs

))

(noF S) (noF S) (Rs )), γ (noF S) (Rs ))

0.35

Re

(F S)

Rs

0.30

(noF S)

Rs

0.25

0.20 10

15

20

25

30

35

R Figure 3: Plot of γ(R) vs. R, showing both the case with finite size corrections (labelled by (FS)) and without them (labelled by (noFS)) for the Ising model in d = 2 with L = 90 at kB T /J = 2.0 and total density ρ = 0.1834. While Re (black vertical line) does not show any significant finite size corrections, γe (Re )/kB T is raised when finite size corrections are included. As a consequence, a shift of both Rs (blue vertical line) and γ(Rs )/kB T (blue curve) results.

35



1.45 (a)

1.40

γ(Rs)/γ∞

1.35 1.30

1 + 4πγ5 ∞ lnRRs s + 1.2 Rs

1.25 1.20

1 + 4πγ5 ∞ lnRRs s

1.15 1.10 1.05 1.00 0.00

0.01

0.02

0.03

(γ∞ ≈ 0.228) 0.04

0.05

0.06

0.07

0.08

0.05

0.06

0.07

0.08

0.05

0.06

0.07

0.08

1/Rs

0 (b)

Γ(Rs)

−1

L=50 L=60 L=70 L=80 L=90 L=100 L=110 L=120

−2 −3 −4 −5 −6 0.00

0.01

0.02

0.03

0.04

1/Rs

0 (c) −1

δ(Rs)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Page 36 of 39

−2 −3 −4 −5 0.00

0.01

0.02

0.03

0.04

1/Rs Figure 4: (a) Normalized radius-dependent surface tension γ(Rs )/γ∞ for the twodimensional Ising model at temperature kB T /J = 2.0. The dotted curve shows the result due to the predicted leading logarithmic correction, 19,36,37 1 + (5/4πγ∞ )Rs−1 ln Rs , while the broken curve shows a fit in which a subleading correction c/Rs with c = 1.2 is also included. The eight choices of linear dimension L of the L × L simulation box are indicated in the legend of part (b). Note that γ∞ ≈ 0.228 is known from Onsager’s exact solution. 90 (b), (c) Adsorption Γ(Rs ) and Tolman length δ(Rs ) plotted vs. 1/Rs , obtained from the same analysis as part (a). Only the parts of the curves highlighted by color refer to regions ρ1 < ρ < ρ2 of density in which droplets are stable. 36


Page 37 of 39

1.6 (c) 1.5

1.4

1 + 4πγ5 ∞ lnRRs s + 0.8 Rs

γ(Re)/γ∞

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


1.3

1 + 4πγ5 ∞ lnRRs s

1.2

1.1

1.0 0.00

0.02

0.04

(γ∞ ≈ 0.228) 0.06

0.08

0.10

1/Re Figure 5: Equimolar interface tension γ(Re ) obtained by including translational entropy corrections. Different colors refer to the same choices of L as in Fig. 4. Dotted curve shows the leading term of Eq. (21), while broken curve gives a full fit of Eq. (21) using a value of c = 0.8.

37



1.20

(a)

1.15 bubbles

γ(Rs)/γ∞

1.10 1.05 1.00 0.95 0.90

droplets

0.85 0.80 0.00

0.05

0.10

0.15

0.20

0.25

0.30

1/Rs 0.25

(b) L=18 L=20 L=22 L=24 L=26 L=28 L=30

Γ(Rs)

0.20 0.15

droplets bubbles

0.10 0.05 0.00 0.00

0.05

0.10

0.15

0.20

0.25

0.30

0.20

0.25

0.30

1/Rs

1.0 (c) 0.8

±δ(Rs)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Page 38 of 39

0.6 0.4

droplets: +δ(Rs )

0.2

bubbles: −δ(Rs )

0.0 0.00

0.05

0.10

0.15

1/Rs Figure 6: Normalized radius-dependent surface tension γ(Rs )/γ∞ for the three-dimensional fcc Ising model with triplet interactions, plotted vs. 1/Rs when the contribution of translational entropy of droplets and bubbles, respectively, is included in the analysis. A normalization length of `0 = 1.0 in Eq. (20) is assumed. Note (i) the pronounced improvement of the mutual consistency of the data obtained for different linear box size L (ii) the reduction of the 1/Rs -dependence of both the adsorption and the Tolman length in comparison to Ref., 69 where translational entropy corrections had been neglected (see also Fig. SI3). Our improved estimate δ∞ ≈ 0.36 is again represented by the black horizontal dashed line. 38


Page 39 of 39

droplet evaporation/ condensation transition

droplet shape transition sphere * ) cylinder

µ

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


ρcoex v

ρ1

ρ2 ρ

Figure 7: TOC figure

39


Equilibrium between a Droplet and Surrounding Vapor: A Discussion

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