Crossover model for the work of critical cluster formation in nucleation theory

Crossover model for the work of critical cluster formation

in nucleation theory

V. I. Kalikmanova)

University of Delft, Department of Geotechnology, Laboratory of Applied Geophysics & Petrophysics, P.O. Box 5028, 2600 GA, Delft, The Netherlands

共Received 6 July 2004; accepted 20 August 2004兲

We propose a relation for the work of critical cluster formation in nucleation theory W for the systems with long-range interparticle interactions. The method of bridge functions is used to combine the system behavior at sufficiently small quenches, adequately predicted by the classical nucleation theory, with nonclassical effects at deep quenches in the vicinity of the thermodynamic spinodal, described within the framework of the field theoretical approach with an appropriate Ginzburg-Landau functional. The crossover between the two types of nucleation behavior takes place in the vicinity of the kinetic spinodal where the lifetime of a metastable state is of the order of the relaxation time to local equilibrium. We argue that the kinetic spinodal corresponds to the minimum of the excess number of molecules in the critical cluster. This conjecture leads to the form of W containing no adjustable parameters. The barrier scaling function⌫⫽W/Wcl, where Wclis the classical nucleation barrier, depends parametrically on temperature through the dimensionless combination of material properties. The results for argon nucleation are presented. © 2004

American Institute of Physics. 关DOI: 10.1063/1.1806404兴

I. INTRODUCTION

Scaling relations play an important role in nucleation theory increasing our ability to describe nucleation phenom-ena in complex substances for which microscopic description becomes intractable in view of the lack of knowledge of intermolecular interactions. A number of such approaches appeared in recent years.1 The main quantity of interest in these models is a barrier scaling function—the ratio of the nucleation barrier W to its value Wclpredicted by the classi-cal nucleation theory共CNT兲

⌫⫽ W

Wcl. 共1兲

The most recent model proposed by Kashchiev2predicts⌫ to be a universal function of the quench depth, independent of either the material properties or temperature. This attractive feature of the theory of Kashchiev2 is, however, overshad-owed by a number of controversial statements used in its formulation as discussed recently in Ref. 3.

The CNT assumes that the system is in a metastable state 共e.g., supersaturated vapor for the case of gas-liquid transi-tion兲 and its thermodynamics can be described by quasi-equilibrium theory. This assumption leads to the capillarity approximation which presents the free energy barrier for a cluster formation as a sum of bulk and surface terms. The latter is a cost to form a droplet surface characterized by a plain layer surface tension ␥_⬁ which is independent of the quench depth 共supersaturation兲. A cluster 共droplet兲 is as-sumed to be a compact spherical object. It is established that the nucleation behavior at low supersaturations is well de-scribed by the CNT.

At deep quenches the system becomes unstable; in the theory of phase transitions the boundary between metastable and unstable region is given by a thermodynamic spinodal being a locus of points corresponding to a divergent com-pressibility. Rigorously speaking the transition from meta-stable to unmeta-stable states does not reduce to a sharp line but rather represents a region of a certain width which depends on the range of interparticle interactions.4Nucleation in this region cannot be described by the CNT and a more general formalism is needed. A field theoretical approach to nucle-ation pioneered by Cahn and Hilliard5 and developed by Langer6 is based on the Ginzburg-Landau theory of phase transitions. Langer showed that if interparticle interactions are short-range the nucleating droplet has still a compact form and can be characterized by a sharp interface in agree-ment with the CNT predictions.

Langer’s model, however, cannot be applied to systems with long-range interactions. In this case one can describe the system behavior in the framework of a mean-field ap-proach leading to an existence of a well-defined共mean-field兲 spinodal. Mean-field theory becomes asymptotically accurate in the limit of infinite-range interactions and hence a spin-odal exists in the same limit. Therefore, for systems with a sufficiently long-range interaction potential the concept of a spinodal can be applicable. Note in this respect that a spin-odal is unambiguously determined for van der Waals fluids. At the spinodal the surface tension vanishes which means that capillary forces can no longer sustain the compact form of a droplet. Indeed, Klein and co-workers7,8 showed that clusters in the vicinity of a spinodal are ramified fractal ob-jects.

When fluctuations are taken into account the kinetic con-siderations show that the lifetime of a metastable state

de-a兲_{Electronic mail: V.Kalikmanov@tnw.tudelft.nl}

8916

creases becoming of the order of the relaxation time of the system to local equilibrium before the thermodynamic spin-odal is reached. This situation corresponds to a kinetic

spinodal9which is a physical crossover between metastable and unstable states. Beyond the kinetic spinodal the concept of quasiequilibrium, lying in the heart of the CNT, becomes inapplicable.

In the present paper we discuss nucleation in the systems with long-range interaction potentials focusing on the gas-liquid transition. We use the method of bridge functions to-gether with plausible physical arguments to combine the CNT behavior at low supersaturations with nonclassical spinodal effects at high supersaturations. We argue that the kinetic spinodal corresponds to the minimum of the excess number of molecules in the critical cluster over that present in the mother phase. This conjecture leads to the nucleation barrier W which is the simplest smooth interpolation be-tween the two regimes. At the same time W and⌫ parametri-cally depend on temperature through the material param-eters, such as saturation pressure, plain interface surface tension, equilibrium liquid density, and the vapor pressure at the mean-field spinodal.

The paper is organized as follows. In Sec. II we briefly discuss the CNT result for the nucleation barrier using the Gibbs approach to thermodynamics of curved surfaces. In Sec. III A we summarize the field-theoretical description of nucleation near the mean-field thermodynamic spinodal us-ing the Ginzburg-Landau free energy functional and deter-mine the scaling laws for the size and amount of particles in the critical cluster in this domain. Section III B is devoted to the role of fluctuations in the nucleation phenomenon which lead to a concept of a kinetic spinodal. Using the method of bridge functions, the condition of kinetic spinodal and for-mulating an ansatz for the crossover between the CNT-dominated regime and the spinodal-CNT-dominated regime, we construct in Sec. IV the nucleation barrier and the barrier scaling function for the entire range of supersaturations. In Sec. V we apply our theory to argon for which the thermo-dynamic spinodal is obtained from the van der Waals equa-tion of state. Appendix presents calculaequa-tions of the supersatu-rated vapor density and the thermodynamic spinodal for van der Waals fluids.

II. GIBBS THERMODYNAMICS OF CURVED

SURFACES AND CLASSICAL NUCLEATION THEORY In this section we briefly discuss the CNT prediction for the free energy barrier paying attention to the underlying assumptions. Consider a system ‘‘droplet⫹vapor’’ put in contact with a reservoir at a temperature T⬍Tc (Tc is the

critical temperature兲 and pressure pv which exceeds the co-existence共saturation兲 pressure psat(T). The system thus finds itself in a metastable state; the degree of metastability is conventionally described by the supersaturation ratio:

S⫽ p

v

psat共T兲 .

If S is not high the lifetime of the metastable state is much larger than the relaxation time of the system to its local

equi-librium and one can use quasiequiequi-librium theory considering a droplet to be in equilibrium with the surrounding super-saturated vapor and applying the Gibbs equilibrium thermo-dynamics of curved surfaces10,11to this metastable state. As-suming that an arbitrary chosen dividing surface is spherical with a radius R, one obtains the reversible work of a droplet formation in the capillarity approximation10

W⫽⫺⌬p4␲

3 R

3_⫹4␲R2_{␥共R兲, 共2兲}

where the surface tension␥(R) refers to the chosen dividing surface;⌬p⫽pl_⫺pv_{, p}l_{being the pressure of the bulk liquid}

held at the same temperature T and the same chemical po-tential ␮ as the supersaturated vapor: ␮⫽␮v( pv)⫽␮l( pl). For the surface of tension with R⫽Rt,␥(Rt)⫽␥ttaking into

account the Laplace equation ⌬p⫽2␥t/Rt, we find from

共2兲: W⫽(16␲/3)␥_t3/⌬p2. Since ␥_t is not known, the CNT approximates it by the plain interface value␥_⬁(T), resulting in the classical nucleation barrier

Wcl⫽16␲

3 ␥⬁3

共⌬p兲2. 共3兲

The nucleation barrier is the leading order contribution to the nucleation rate J which is the number of critical clusters formed per unit volume per unit time:

J⫽Ke⫺W/kBT_. 共4兲

Here kBis the Boltzmann constant and K is the kinetic pref-actor. In the CNT kinetics of cluster formation is described by a set of elementary processes which change the size of a cluster by attachment to its surface or loss from its surface of one molecule. The kinetic prefactor can be well approxi-mated by12 K⬇共␳ v_兲2 ␳l

冑

2␥_⬁ ␲m1 , 共5兲

where m1is the mass of a molecule. Assuming that the liquid is incompressible and the vapor is ideal 共which is a reason-able assumption not too close to Tc) we write down the

Kelvin equation13relating⌬p to S, ⌬p⫽␳eq

l

kBT ln S,

where␳_eql (T) is the equilibrium liquid number density. At the spinodal the supersaturation reaches its maximum Ssp corre-sponding to the maximum of⌬p:

⌬psp⫽␳eq

l

kBT ln Ssp.

Introducing another useful measure of metastability ␩⫽⌬p/⌬psp⫽ln S/ln Ssp, 0⭐␩⭐1 we present共3兲 as Wcl⫽cb 1 ␩2, cb共T兲⫽ 16␲ 3 ␥_⬁3 共⌬psp兲2 . 共6兲

A general result in nucleation theory which holds for all cluster sizes down to atomic scale共where the capillarity ap-proximation fails兲 and which is independent of a particular model is known as the nucleation theorem.14,15It states that

the excess number of molecules in the critical cluster over that present in the mother phase⌬n

* is related to the

nucle-ation barrier via, ⳵W

⳵⌬␮⫽⫺⌬n*,

where⌬␮⫽␮v( pv)⫺␮l( pv). Using the standard thermody-namic relationships this expression can be rewritten as

冉

␳eq

l

⌬psp

冊

⳵W

⳵␩ ⫽⫺⌬n*. 共7兲

Applying the nucleation theorem to the CNT we have, ⌬n * cl_⫽

冉

2cb␳eq l ⌬psp

冊

␩ ⫺3_,

which in combination with the Laplace equation yields the scaling relation for the critical cluster,

⌬n_*cl_⬃R3_{, 共8兲}

showing共in agreement with the assumptions made兲 that the critical cluster in the CNT is a compact spherical object.

It is tempting to apply the CNT to the entire domain 0 ⭐␩⭐1. By doing so one would find that as spinodal is ap-proached, lim ␩→1 Wcl⫽cb⫽const, lim ␩→1⌬n* cl_⫽2cb␳eq l ⌬psp ⫽const. On the other hand the field theoretical arguments, presented in the following section, supported by the density functional calculations,17show that as the system approaches the mean-field spinodal the nucleation barrier vanishes, while⌬n

*

di-verges. Moreover, the nucleating cluster at the same limit is not a compact object but a fractal, correspondingly a divid-ing surface cannot be characterized by a sdivid-ingle parameter (R). These results, however, are not in contradiction with the classical theory since the latter is simply not designed to be applicable in the spinodal region where its basic assumption—the concept of quasiequilibrium—becomes meaningless. Note in this respect that the statement used by some authors 共see Ref. 2兲, that the CNT is thermodynami-cally inconsistent because it predicts a finite value of W at the spinodal, is irrelevant.

III. NONCLASSICAL NUCLEATION A. Nucleation in the vicinity of the thermodynamic spinodal

To analyze the nucleation behavior in the vicinity of the thermodynamic spinodal we apply field theoretical consider-ations following Refs. 5, 6 and 8. We start with Landau expansion18 for the free energy density in powers of the or-der parameter m. g⫽g0⫹ a 2m 2_⫹b 4m 4_⫺mh, where a⫽a0t, t⬅共T⫺Tc兲/Tc, a0⬎0,b⬎0,

h is the external field conjugate to m. For the gas-liquid

transition m⫽␳⫺␳c共where␳cis the critical density兲 and h

⫽⌬␮; at the spinodal h⫽hsp, while at the binodal h⫽0. Below Tc, a⬍0 and the free energy density has a

double-well structure with the local minimum at m⫽m

*

corre-sponding to the given metastable state and the global mini-mum at m⫽m_glob corresponding to the thermodynamically stable liquid state to which the system evolves 共see Fig. 1兲. The two minima are separated by the energy barrier. Stan-dard analysis19,20yields

m *共h兲⫽2

冑

s 3bcos

冉

␣⫹ 2␲ 3

冊

, 共9兲

where s⫽⫺a⬎0 and cos 3␣⫽3) 2 b1/2 s3/2h, 0⭐␣⭐ ␲ 6. 共10兲

The maximum supersaturation corresponds to hsp

2 ⫽(4/27)s3_{/b; at this field the local minimum m}

sp becomes an inflection point—this is the case of spinodal decomposi-tion. Thus, nucleation takes place for the field interval 0 ⬍h2_⬍h

sp 2

. For the states close to m

*the free energy density

can be expanded in powers of␾⫽(m⫺m

*)/m*: g⫽g *⫹ b2 2 ␾ 2_⫺b3 3 ␾ 3_{⫹O共␾兲}4_{, 共11兲} where g

* is the free energy density of the metastable state

m * and b2共h兲⫽m_* 2 ⳵ 2_g ⳵m2

冏

m⫽m* , b3共h兲⫽⫺m_* 3 1 2 ⳵3_g ⳵m3

冏

m⫽m* . 共12兲 The term linear in ␾vanishes since m⫽m

* is a local

mini-mum of g. At the spinodal

b2共h⫽hsp兲⫽0, b3共h⫽hsp兲⬎0.

Let us study the system behavior near the mean-field spinodal 共i.e. when h is close to hsp) using an appropriate Ginzburg-Landau free energy functional which describes the

FIG. 1. Schematic plot of Landau free energy density g for T⬍Tc. At h

⫽0 共long dashed line兲 there are two equal minima corresponding to the

coexisting states. At 0⬍h⬍hsp 共solid line兲 the left, local, minimum m_*

corresponds to a metastable state 共supersaturated vapor兲, while the right, global, minimum mglobrefers to a stable state共liquid兲; the two states are

separated by the energy barrier. At h⫽hsp 共short dashed line兲 the local

state of the system undergoing a first-order phase transition characterized by a conserved scalar order parameter ␾(r) ⫽关m(r)⫺m_*兴/m_*.21Using Eq. 共11兲 this functional reads,

F关␾共r兲兴⫽F_*⫹

冕

dr

冋

c0 2 兩ⵜ␾兩 2_⫹b2 2 ␾ 2_⫺b3 3 ␾ 3

册

_{, 共13兲} where F

* is the free energy of the metastable state m

⫽m_*, out of which nucleation starts. The square-gradient term is an energy cost to create an interface between the phases; c0⬎0 is related to the correlation length in the system.22 Following Ref. 8 we associate the critical cluster with the saddle point of the free energy functional共13兲. If the saddle point ␾共r兲 is found, its substitution into F yields the nucleation barrier W⫽F⫺F *⫽

冕

dr

冋

c0 2 兩ⵜ␾兩 2_⫹b2 2 ␾ 2_⫺b3 3 ␾ 3

册

_{. 共14兲} To analyze this expression we proceed by performing a set of scaling transformation of the variables. Rescaling the order parameter␾₁⫽(b₃/c₀)1/3␾ and denoting

⑀⫽b2共b3 2 c0兲⫺1/3 共15兲 we rewrite共14兲 as W⫽c0

冕

dr

冋

1 2

冉

c0 b3

冊

2/3 兩ⵜ␾1兩2⫹ ⑀ 2␾1 2_⫺1 3␾1 3

册

.

The next transformation rescales the spatial coordinates r1 ⫽(b3/c0)⫺1/3r yielding W⫽b3

冕

dr1

冋

1 2兩ⵜ1␾1兩 2_⫹⑀ 2␾1 2_⫺1 3␾1 3

册

,

whereⵜ1⫽⳵/⳵r1. And finally, further rescaling is useful,

␾1⫽⑀␾˜ , r1⫽⑀⫺1/2˜,r 共16兲

with the help of which W takes the form,

W⫽⑀3/2b3

冕

dr˜

冋

1 2兩ⵜ˜␾˜兩 2_⫹1 2˜␾ 2_⫺1 3˜␾ 3

册

_{, 共17兲}

where ⵜ˜⫽ (⳵/⳵˜). The saddle point of the functional isr given by the Euler-Lagrange equation:

ⵜ˜2_{␾˜⫽␾˜⫺␾˜}2_{. 共18兲}

The critical cluster is the nontrivial solution of Eq.共18兲 van-ishing at infinity. The existence of such solutions was proved for sufficiently large bounded domains.23Without presenting the full form of the solution it is instructive to study its behavior at large r, i.e., far from the center of mass of the cluster. In this domain the amplitude of the droplet is small and we can neglect the second term in Eq. 共18兲 which leads to the equation

ⵜ˜2_{␾˜⫽␾˜ , large r˜,}

whose spherically symmetric solution vanishing at infinity is the screened Coulomb function:

␾ ˜_⫽C ⬎ e⫺r˜ r ˜ , large r˜,

where C_⬎is a constant. Returning to the units of Eq.共16兲 it reads

␾1共r1兲⫽C⬎

冑

⑀

e⫺冑⑀r1 r1

, large r₁.

Since ␾1(r1) is the density fluctuation associated with a nucleus, its decay length

R

*⬇⑀⫺1/2, ⑀→0 共19兲

characterizes the size of the critical nucleus. The latter di-verges as the spinodal is approached. In the same limit the nucleation barrier 共17兲 decreases as

W⬃⑀3/2, ⑀→0. 共20兲

Finally we must relate ⑀ to the physical parameters of the system, i.e. to determine the scaling of⑀near spinodal to the leading order in (h⫺hsp); obviously ⑀(h⫽hsp)⫽0. From Eq. 共15兲 and the definition of b2 it follows that⑀is propor-tional to the curvature of the Landau free energy at the meta-stable state, ⑀⬃b2⬃g2共h兲⬅ ⳵2_g ⳵m2

冏

_m⫽m *(h) ⫽⫺s⫹3b m * 2_.

Substituting h⫽h_sp⫺u, where u is a 共small兲 deviation of the external field from its value at the spinodal, into Eqs.共9兲 and 共10兲, we obtain to the leading order in u,

g2⫽2共3bs兲1/4

冑

hsp⫺h.

Since hsp⫺h⬃⌬psp⫺⌬p⫽⌬psp(1⫺␩), we have

⑀⬃共1⫺␩兲1/2_{. 共21兲}

Substituting Eq.共21兲 into Eqs. 共20兲 and 共19兲 we find that in the vicinity of the thermodynamic spinodal the nucleation barrier vanishes as

W_sp⫽c_sp共T兲 共1⫺␩兲3/4_{, c}

sp共T兲⬎0, ␩→1⫺, 共22兲 while the radial extent of the critical cluster diverges as

R

*⫽R0共1⫺␩兲⫺1/4, R0共T兲⬎0, ␩→1⫺. 共23兲 The excess number of molecules in the critical cluster in the spinodal region is found from the nucleation theorem共7兲:

⌬n_*⫽3₄

冉

csp␳eq

l

⌬psp

冊

共1⫺␩兲

⫺1/4_{, ␩→1}⫺_{. 共24兲}

Thus, near the spinodal ⌬n

* diverges. Comparison of Eqs.

共23兲 and 共24兲 predicts the scaling, ⌬n_*⬃R_*.

This result supports the conjecture of Klein7that the critical cluster near the spinodal is a ramified fractal 共chain-like兲 object. This is distinctly different from the scaling ⌬n

*

⬃R3 _{near the binodal predicted by the CNT关cf. Eq. 共8兲兴.} Note, that the nucleation barrier proposed in Ref. 2,

WK⫽cb

冉

1 ␩⫹ 1 2

冊

2 共1⫺␩兲, 0⭐␩⭐1

scales near the thermodynamic spinodal as

WK⫽ 9

which implies using the nucleation theorem that⌬n *,Kat the spinodal is finite ⌬n_*,K⫽ 9 4

冉

cb␳eq l ⌬psp

冊

⫹O共1⫺␩ 兲, ␩→1⫺

in contradiction with the nonclassical results given by Eqs. 共22兲 and 共24兲.

B. Role of fluctuations: Kinetic spinodal

The condition b2⫽0 describes the mean-field thermody-namic spinodal in the absence of fluctuations. However, their role in nucleation phenomena is important. One is facing the question: how deep the quench can be so that the concept of quasiequilibrium invoked in CNT can be considered still valid? To find an answer it is necessary to compare two characteristic times:共i兲 the time tMnecessary to form a

criti-cal cluster which is a lifetime of the metastable state and共ii兲 the relaxation time tR during which the system settles in this

state. The first quantity can be related to the nucleation rate by using its definition: tM⫽1/(JV). To find tR one must

study the dynamics of the metastable state. Since the order parameter ␾共r兲 in the Ginzburg-Landau functional 共13兲 is a conserved variable, its evolution is governed by the Cahn-Hilliard dissipative dynamics:24

⳵␾ ⳵t ⫽⌫0ⵜ

2␦F

␦␾⫹␨, 共25兲

where ⌫0 is a transport coefficient and ␨(r,t) is a noise source共which models thermal fluctuations兲 satisfying

具

␨共r,t兲␨共r

⬘

,t

⬘

兲

典

⫽⫺2T⌫₀ⵜ2␦_共r⫺r

_⬘

兲␦_共t⫺r

_⬘

_兲

to ensure that the equilibrium distribution associated with Eq.共25兲 is given by the Boltzmann statistics. From the solu-tion of Eqs. 共25兲 and 共13兲, obtained by Patashinskii and Shumilo9共see also Ref. 25兲, it follows that

tR⫽

16c0 ⌫0b2 2,

implying that when the system approaches the thermody-namic spinodal (b2→0) its relaxation time diverges. The relation between tM and tR established in Ref. 9 is

tM⫽tR

冉

4␲␹ ␭0

冊

exp

冉

␹W kBT

冊

, where ␹⫽共b2c0兲3/2 kBTb3 2 , 共26兲

and ␭0⬇8.25. Clearly, the concept of quasi-equilibrium is meaningful for the metastable states satisfying the require-ment: tMⰇtR. In the opposite case this concept becomes

irrelevant. The boundary between these two domains is called a kinetic spinodal9 and is defined by the condition

tM⬵tR yielding

␹W kBT⫽1.

Using Eq.共26兲 it can be written as

W⫽共kBT兲

2_b 3 2

c₀3/2b₂3/2 . 共27兲

The quantities b2 and b3 are expressed in terms of the de-rivatives of the free energy as given by Eq. 共12兲 and hence can be calculated given the equation of state:

b2⫽␳2 ⳵␮ ⳵␳

冏

_␳⫽␳v ⫽␳⳵_⳵␳p

冏

␳⫽␳v , 共28兲 b3⫽⫺ 1 2␳ 3⳵ 2_␮ ⳵␳2

冏

␳⫽␳v ⫽b2⫺ 1 2␳ ⳵b2 ⳵␳

冏

_␳⫽␳v . 共29兲

The parameter c0 can be well approximated by 25

c0⬵kBT␳c

1/3

. 共30兲

Hence, all parameters in Eq.共27兲 are temperature dependent material properties which can be determined from the equa-tion of state p⫽p(␳,T). In particular from Eq. 共28兲 it fol-lows that b2 is the inverse isothermal compressibility of the vapor at the metastable state.

IV. NUCLEATION BARRIER: A CROSSOVER MODEL At low supersaturations, where the capillarity approxi-mation holds, the CNT expression 共6兲 accurately describes the nucleation barrier; at high supersaturations in the spin-odal region the capillarity approximation fails and W is given by the field theoretical model共22兲. To obtain the nucleation barrier for the entire range of 0⭐␩⭐1 we propose a smooth interpolation between the two regions introducing a bridge function B(␩):

W共␩兲⫽cb␩⫺2⫹csp共1⫺␩兲3/4⫹B共␩兲, 共31兲 which should be smooth and satisfy the boundary conditions,

␩2_{B共␩兲→0 for ␩→0}⫹_, 共1⫺␩兲⫺3/4_关c

b⫹B共␩兲兴→0 for ␩→1⫺.

Obviously, these constraints do not define B(␩) in a unique way. We choose the simplest form B(␩)⫽⫺cb satisfying

both boundary conditions. Then from Eq. 共31兲,

W⫽cb关␩⫺2⫹␬共1⫺␩兲3/4⫺1兴, 0⭐␩⭐1, 共32兲

with the yet unknown dimensionless parameter ␬

⬅csp(T)/cb(T)⬎0. Let us consider the excess number of

molecules in the critical cluster applying the nucleation theo-rem 共7兲 to 共32兲, ⌬n_*⫽

冉

cb␳eq l ⌬psp

冊

冋

2␩⫺3⫹3 4␬共1⫺␩兲 ⫺1/4

册

_{, 0⭐␩⭐1.} 共33兲 The qualitative behavior of⌬n

*(␩) is shown in Fig. 2. We

distinguish two domains of the metastable states: the ‘‘CNT domain’’ and the ‘‘spinodal domain.’’ In the CNT domain ⌬n_* is decreasing with the quench depth ␩ while in the spinodal domain⌬n

*(␩) is increasing tending to infinity as

the thermodynamic spinodal is approached. At some ␩¯ it

reaches the minimum⌬n

*. It is plausible to assume that the

CNT-dominated regime covers the range of quench depths up to␩¯ and at␩¯ the spinodal-dominated behavior takes over.

We associate the crossover between these two domains with the kinetic spinodal. This ansatz stems from the argu-ment that the kinetic spinodal is the physical limit of appli-cability of quasiequilibrium approach on which the CNT is based. These considerations lead to the set of equations:

d⌬n * d␩

冏

_␩¯⫽0, 共34兲 W共␩¯兲⫽共kBT兲 2_关b 3共␩¯兲兴2 c₀3/2关b₂共␩¯兲兴3/2 . 共35兲 Equation共34兲 yields ␬⫽32共1⫺␩¯兲 5/4 ␩ ¯4 . 共36兲

Substituting it into Eq.共35兲 we obtain the crossover equation determining ␩¯ : 1 ␩ ¯2⫹ 32共1⫺␩¯兲2 ␩ ¯4 ⫺1⫽ 共kBT兲2关b3共␩¯兲兴2 cbc0 3/2 关b2共␩¯兲兴3/2 . 共37兲

The crossover is expected to be close to the thermodynamic spinodal关later we verify this assumption analyzing the solu-tion of Eq.共37兲兴. It is then plausible to expand b₂ and b₃ in density in the vicinity of the spinodal keeping the linear or-der terms. For b2 we have

b2⫽␰共␳v⫺␳sp

v_{兲, ␰⫽}⳵b2

⳵␳

冏

_sp 共38兲

关the constant term vanishes since b2(␳sp

v_{)⫽0]. From Eqs.} 共29兲 and 共38兲 b3⫽␰共 1 2␳v⫺␳sp v_{兲. 共39兲}

Both b2 and b3 have the dimensionality of pressure, there-fore it is convenient to rewrite them using the reduced vari-ables ␳*⫽␳v/␳c, T*⫽T/Tc, p*⫽pv/ pc 共the subscript c

refers to the critical point兲:

b2⫽pc⌽2共␳*,T*兲, b3⫽pc⌽3共␳*,T*兲, 共40兲 where ⌽2共␳*,T*兲⫽␻共␳*sp⫺␳*兲, ⌽3共␳*,T*兲⫽ ␻ 2共2␳sp*⫺␳*兲, 共41兲 and ␻共T*兲⫽⫺

冉

␳c pc

冊

␰⫽⫺

冉

␳c pc

冊冉

␳ ⳵2_p ⳵␳2

冊

␳⫽␳_spv⬎0. 共42兲

Substitution of Eqs. 共40兲 and 共41兲 into Eq. 共37兲 yields the crossover equation in the form,

1 ␩ ¯2⫹ 32共1⫺␩¯兲2 ␩ ¯4 ⫺1⫽

冑

␻Zc 4⌳ 关共2␳sp*⫺␳*共␩¯兲兴 2 关␳sp*⫺␳*共␩¯兲兴3/2 , 共43兲

where Zc⫽pc/(␳ckBTc) is the critical compressibility factor,

and

⌳共T兲⫽16₃␲ ␥⬁ 3

kBT⌬psp

2 . 共44兲

Having solved it for␩¯ one can estimate the excess number of

molecules in the critical cluster at the crossover. From Eqs. 共33兲 and 共36兲 we find ⌬n_*⫽⌬n_*共␩¯兲⫽

冉

2⌳ ln S_sp

冊冉

12⫺11␩¯ ␩ ¯4

冊

. 共45兲

Finally, the barrier scaling function共1兲 reads

⌫共␩兲⫽1⫺␩2_⫹␬␩2_{共1⫺␩兲}3/4_{, 共46兲}

where␬is given by Eq.共36兲 with ␩¯ being a solution of the

crossover equation 共43兲. We stress that ⌫ parametrically de-pends on temperature through␬.

The behavior of⌫共␩兲 for various values of␬is shown in Fig. 3. If ␬⬍1: ⌫⬍1 for all values of supersaturation im-plying that the nucleation rate J exceeds the rate Jclpredicted by the CNT. In the opposite case of ␬⬎1, the function ⌫ exceeds unity at some range of supersaturations correspond-ing to 0⭐␩⭐␩

*, beyond which it becomes less than unity.

In Fig. 3 we also show Kashchiev’s universal function2 ⌫K共␩兲⫽共1⫹

1 2␩兲

2_{共1⫺␩兲, 共47兲}

FIG. 2. Excess number of molecules in the critical cluster⌬n_*as a function

of the quench depth ␩ given by Eq.共33兲. In the CNT domain ⌬n_* is

decreasing while in the spinodal domain it is increasing tending to infinity; at␩¯ it reaches the minimum value⌬n

*.

FIG. 3. Barrier scaling function共46兲 共solid lines兲; labels are the correspond-ing values of␬. Also shown is Kashchiev’s function⌫K2given by Eq.共47兲

which is close 共though not identical兲 to our result when ␬ ⬇0.2. Note that according to Eq. 共47兲 the nucleation rate predicted by the model2 exceeds the classical one for all temperatures and all supersaturations.

V. RESULTS AND DISCUSSION

We apply the present theory to argon. Empirical correla-tions for its bulk properties are given in Ref. 27 For the plain interface surface tension we use the general correlation for nonpolar substances.27 To a good approximation argon can be considered a van der Waals fluid. From the van der Waals equation of state we derive the vapor density at the super-saturated state␳v( pv,T) and the vapor part of the thermody-namic spinodal 共see the Appendix兲. Figure 4 shows the model parameters Ssp, ⌬n_*, ␬ for the temperature range 0.3⬍T/Tc⬍0.8. The solution ␩¯ of the crossover equation,

shown on the right y axis, is close to unity an agreement with the assumptions resulted in Eqs.共38兲 and 共39兲. The minimum excess number of molecules in the critical cluster ⌬n

*

共at-tained at the crossover兲 increases with temperature. The pa-rameter␬also increases with T; the value␬⫽1, at which the qualitative change in the behavior of the barrier scaling func-tion occurs, corresponds for argon to T/Tc⬇0.39. A general

conclusion is that the temperature rise shifts the boundary between the CNT domain and the spinodal domain in the direction of the former.

In experiment one measures directly the nucleation rate

J as a function of temperature and supersaturation共see Ref.

28兲, hence it is convenient to relate ⌫ to J. Assuming following12that the kinetic prefactor K is well approximated by the CNT we find from Eq.共4兲

J共T;␩兲

Jcl共T;␩_{兲 ⫽}exp

再

⌳

冋

1⫺⌫共␩;T兲

␩2

册冎

. 共48兲

It is instructive to study this ratio at a fixed value of Jcl_{. With} this in mind we choose the quench depth␩⫽␩₁ correspond-ing to a typical value of Jcl₍␩

1)⫽1 cm⫺3s⫺1. This implies using Eqs.共4兲 and 共5兲 that␩₁ satisfies,

⌳ ␩1 2⫽⌿⫹2␩1ln Ssp, where ⌿共T兲⫽ln

冋

psat 2 ␳eq l _共k BT兲2

冑

2␥_⬁ ␲m1

册

. 共49兲

Figure 5 shows the ratio J(T;␩1)/Jcl(T;␩1) for the same temperature interval as in Fig. 4共solid line兲; the value of␩1 is shown on the right y axis共dashed line兲.

Thus, for argon the CNT underestimates the nucleation rate at lower temperatures for all supersaturations. At higher temperatures the CNT overestimates J for 共relatively兲 small

S and underestimates J for共relatively兲 high S. This result is

an agreement with the density functional calculations of nucleation in Lennard-Jones fluids.26 It is not clear whether this tendency remains true for more complex substances. This is a problem to be studied in future work.

At this stage comparison with experiment can be only qualitative since for the spinodal properties we used the sim-plest van der Waals equation of state. It would be desirable to verify the predictions of the theory for the substances for which the thermodynamic spinodal can be accurately esti-mated.

ACKNOWLEDGMENTS

It is a pleasure to thank M. E. H. van Dongen, V. Holten, and D. Labetskii for helpful discussions. The author is indebted to H. Bruining, L. Dyadkin, E. Slob, and D. Smeulders for valuable comments.

APPENDIX: SUPERSATURATED VAPOR DENSITY AND THERMODYNAMIC SPINODAL

FOR VAN DER WAALS FLUIDS

In reduced units ␳*⫽␳/␳c, T*⫽T/Tc, p*⫽p/pc the

van der Waals equation of state reads13

p*⫽⫺3␳*2_⫹8␳*T*

3⫺␳*. 共A1兲

Solving this cubic equation for the vapor density ␳*v( p*) we obtain using the standard methods:19

␳*v_{共p*,T*兲⫽1⫹2}

冑

_1⫺

冉

8T*⫹p*

9

冊

cos

冉

␣⫹ 2␲

3

冊

,

FIG. 4. Model parameters for argon. Solid lines共left y axis兲: Ssp,⌬n_*,␬.

Dashed line共right y-axis兲 is the locus of crossover points␩¯ .

FIG. 5. log10(J/J cl

) for argon as given by Eq.共48兲 共solid line兲; the quench depth␩1corresponding to J

cl_{⫽1 cm}⫺3_s⫺1_{关Eq. 共49兲兴 is shown on the right}

where␣is given by cos共3␣兲⫽ 1⫺

冉

4T*⫺p* 3

冊

冋

1⫺

冉

8T*⫹p* 9

冊册

.

The spinodal equation⳵p*/⳵␳*⫽0 is

T*⫽␳*

4 共3⫺␳*兲

2_.

Solving for the spinodal vapor density ␳_sp*vwe obtain ␳sp*

v

共T*兲⫽2⫺2 cos共1

3␤兲, ␤⫽arccos共1⫺2T*兲. 共A2兲

The parameter␻defined in Eq.共42兲 is

␻⫽6␳sp* v ⫺ 48T*␳sp* v 共3⫺␳sp* v 兲3. 共A3兲

Substitution of Eq.共A2兲 into Eq. 共A1兲 yields the vapor pres-sure at the spinodal,

p_sp*v⫽8关4T*⫺3 cos共 1 3␤兲⫹3 cos共 2 3␤兲兴sin 2_共1 6␤兲 1⫹2 cos共1 3␤兲 , 共A4兲

from which the maximum supersaturation is

Ssp共T*兲⫽ p_sp*v_共T*兲 p_sat*_共T*_兲 ⫽

冋

1 p_sat*_共T*兲

册

⫻

再

8关4T*⫺3 cos共 1 3␤兲⫹3 cos共 2 3␤兲兴sin 2_共1 6␤兲 1⫹2 cos共13␤兲

冎

. 共A5兲

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