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Igniting homogeneous nucleation

Transient homogeneous nucleation is studied in the limit of large critical sizes. Starting from pure monomers, three eras of transient nucleation are characterized in the classic Becker-D\"oring kinetic equations with two different models of discrete diffusivity: the classic Turnbull-Fisher formula and an expression describing thermally driven growth of the nucleus. The latter diffusivity yields time lags for nucleation which are much closer to values measured in experiments with disilicate glasses. After an initial stage in which the number of monomers decreases, many clusters of small size are produced and a continuous size distribution is created. During the second era, nucleii are increasing steadily in size in such a way that their distribution appears as a wave front advancing towards the critical size for steady nucleation. The nucleation rate at critical size is negligible during this era. After the wave front reaches critical size, it ignites the creation of supercritical clusters at a rate that increases monotonically until its steady value is reached. Analytical formulas for the transient nucleation rate and the time lag are obtained that improve classical ones and compare very well with direct numerical solutions.


I. INTRODUCTION
Homogeneous nucleation occurs in many examples of first order phase transitions [1] such as condensation of liquid droplets from a supersaturated vapor, glass-to-crystal transformations [2], crystal nucleation in undercooled liquids [3], and in polymers [4], colloidal crystallization [5], growth of spherical aggregates beyond the critical micelle concentration (CMC) [6,7], and the segregation by coarsening of binary alloys quenched into the miscibility gap [8,9,10]. In condensed systems, a long time elapses before the nucleation rate (at which stable nucleii larger than the critical size are generated) reaches a steady state, therefore these systems offer excellent opportunities to study time-dependent nucleation [3].
Understanding the kinetics of nucleation and growth beyond the determination of the steady-state nucleation rate is a task of great importance and not yet completely accomplished. For example, it is desirable to obtain a simple asymptotic description of the transient until the steady-state nucleation stage sets in. Moreover, there is no clear distinction between nucleation and growth, and an unified theory of both processes does not exist [3] despite recent attempts at bridging the gap between nucleation and late-stage coarsening theories [11,12,13].
In this paper, we consider the problem of describing the approach to steady-state nucleation within the classical nucleation theory [3]. Thus our starting point is the Becker-Döring (BD) discrete kinetic model of nucleation and indefinite growth of a stable phase from a metastable state [3,14,15]. The BD model contains two kinetic rate constants that are related to each other by assuming detailed balance. To complete the description of the BD equations (BDE), a model for one of the rate constants, usually a discrete diffusivity describing the rate at which a cluster loses one monomer, is needed. In the classical theory, the discrete diffusivity is given by the Turnbull-Fisher (TF) expression which assumes that a monomer has to overcome an activation energy barrier for its transfer across the interface of a cluster. The TF discrete diffusivity is therefore proportional to the surface area of the cluster [16]. Other models are selected so as to yield the known expression for the adiabatic growth of a nucleus of critical size by either diffusion or by heat transfer. The discrete diffusivity of these later models is proportional to the cluster radius.
No matter which discrete diffusivity is used, starting from an initial condition of pure monomers surpassing the CMC, we expect that cluster size increases and stable supercritical nucleii are formed at a nucleation rate that will eventually become stationary at an exponentially small value. After the stationary nucleation has set in, the supercritical clusters continue growing, and the discrete diffusivity of the BDE can be ignored in the description of their growth, which is a pure convection in the space of cluster size. For precipitation processes, this will eventually result in late stage coarsening which we will not study in the present paper.
The small parameter that informs our asymptotic analysis is ǫ = k −1/3 c , where k c is the number of monomers in the critical nucleus, a 'large' quantity that ranges between 20 and 1000 for common materials [3]. Using other small parameters, such as the supersaturation, yields particular cases of our results (cf. Chapter 2 by Neu and Bonilla in Ref. [4]). The analysis of the BDE in the limit as ǫ → 0 distinguishes three well defined stages or eras in the approach to the stationary nucleation rate. Starting from the initial state of pure monomers, a continuous distribution of cluster sizes is established at the monomer's expense during the first era. During the second era, the clusters grow to the critical size in such a way that their size distribution is a traveling wave front in size space. As this wave reaches the critical size, the formation of supercritical nucleii starts, nucleation is ignited, and the nucleation rate increases from zero to its stationary value during the third era. We have obtained two different expressions for the nucleation rate (which is of paramount importance to compare with experiments): (I) a general expression in terms of the instantaneous location of the wave front and its instantaneous width, which solve two given differential equations, and (II) a more explicit description of the nucleation rate in terms of the solution of the linearized wave front position with an origin of time at the time t M needed for the exact wave front to advance from pure monomers to a certain near critical size. Numerical solution of the model confirms all the theoretical predictions.
Most previous studies of transient nucleation considered the Zeldovich-Frenkel equation (ZFE), which is a Fokker-Planck-type equation resulting from taking the continuum limit of the BDE [17]. Zeldovich [17] set the discrete diffusivity equal to its value at the critical cluster size and used a parabolic approximation for the variation of the free energy. The resulting expression for the transient nucleation rate was rather inaccurate [2]. Until the mid 1980s, work on the ZFE was based on similarly uncontrolled approximations [18]. Some of them gave expressions for the nucleation rate and time lag close to the values obtained by numerically solving the BDE for particular parameter values, but were far off for other parameter ranges [19]. Asymptotic theories for the ZFE were elaborated later [20,21,22].
There are two main differences between asymptotic results obtained for the discrete BDE and those obtained for the continuum ZFE: (i) the time lags for transient nucleation are different, as explained by Wu [19], and (ii) the width of the wave front and the time to ignition are different (wider for the ZFE). Nevertheless, other magnitudes such as relaxation times and the stationary nucleation rate are the same for asymptotic approximations of both, the BDE and the ZFE. Thus our simplified theory (II) yields expressions for the nucleation rate that are similar to those found by Shneidman [21] and by Shi et al [22], although their time lags differ from ours, as one would expect from Wu's arguments [19]. For large critical sizes, our approximation (I) is better.
The rest of the paper is as follows. In Section II, we review the Becker-Döring model for nucleation and growth of spherical aggregates with the Turnbull-Fisher (TF) discrete diffusivity [16]. The binding energy of the aggregate with k monomers (k cluster) relative to isolated monomers in solution is (k − 1) times the monomer-monomer bond energy plus a term proportional to the surface area of the aggregate. Beyond a critical density no equilibrium size distribution exists and the aggregates grow indefinitely. The main results of our asymptotic analysis are derived in Section III and compared with the numerical solution of the BDE with the TF discrete diffusion coefficient describing devitrification of lithium disilicate glass. Our results compare favorably with previous theories based on the ZFE, the continuum approximation of the BDE. However, when compared with experimental data for glass disilicate, the theoretical time lag is about 30 times smaller. To improve the agreement with experiments, we propose in Section IV a different discrete diffusion coefficient selected so as to yield the known expression for the adiabatic growth of a nucleus of critical size by heat transfer. The asymptotic theory for the resulting BDE is similar to that explained in Section III, and the resulting time lag is much closer to experimental data. Section V compares our asymptotic results for the transient nucleation rate and for the number of supercritical clusters to previously known analytical formulas (unfortunately all of them dealing with the continuum ZFE, not with the discrete BDE as ours do) [20,21,22,23,24].
Technical matters are relegated to Appendices.

II. KINETIC EQUATIONS AND STATIONARY SOLUTIONS
The model presented here is nucleation in a lattice in which there are many more binding sites, M, than particles, N, [7]. We shall consider the thermodynamic limit, N → ∞ with fixed particle density per site, ρ ≡ N/M. Let p k be the number of clusters with k particles or, in short, k clusters, and let ρ k ≡ p k /M be the density of k clusters. Note that the number densities per site, ρ and ρ k , are both dimensionless. Number densities per unit volume are obtained dividing ρ and ρ k by the molecular volume, v = V /M. Particle conservation implies that the total particle density ρ is constant: In the Becker-Döring kinetic theory of nucleation, a k cluster can grow or decay by capturing or shedding one monomer at a time. Then [7] The monomer density ρ 1 can be obtained from the conservation identity (1) that relates it to the other cluster densities. In these equations,ρ k = dρ k /dt and D ± u k ≡ ±[u k±1 − u k ] are finite differences. t, d k and j k are nondimensional. t and d k are related to the dimensional time t * and decay coefficient d * k as follows Here the factor Ω has units of frequency, it depends on the particular model we choose for d k , and will be determined later. j k is the net rate of creation of a k + 1 cluster from a k cluster (the flux in size space), given by the mass action law. In (3) we have made the detailed balance assumption to relate the kinetic coefficient for monomer aggregation to that of decay of a (k + 1) cluster, d k . This implies that the equilibrium size distribution solving j k = 0 has the formρ In (3) and (5), ε k is the binding energy of a k cluster, required to separate it into its monomer components. Then the total energy measured with respect to a configuration in which all clusters are monomers is − N k=2 p k ε k . For spherical aggregates, This formula holds for k ≫ 1, but we shall use it for all k ≥ 1. αk B T is the monomermonomer bonding energy [6] which, in the case of precipitation of crystals from a solution or segregation by coarsening of binary alloys, may depend on the particle density ρ (volume fraction) through some empirical formulas [14]. In Eq. (6), σ = 2γ s (4πv 2 /3) , where γ s and v = V /M are the interfacial free energy per unit area (surface tension) and the molecular volume, respectively. Note that α and σ are both dimensionless. The correction 3σk B T /2 in (6) ensures that ε 1 = 0, and it improves the agreement between the nucleation rate obtained from the BDE and experiments [19]. More precise atomic models were proposed by Penrose et al [14].
Eqs. (1), (2), (3), (6) and a given discrete diffusivity d k form a closed system of equations that we can solve for an appropriate initial condition. If initially only monomers are present, we have ρ 1 (0) = ρ, and ρ k (0) = 0 for k ≥ 2. Before we obtain formulas for the kinetic coefficient d k , we shall recall the more salient features of the equilibrium size distribution.

A. Equilibrium size distribution
The equilibrium distribution (5) satisfies j k = 0 and it can be written as where g k is the activation energy, equivalently given by ϕ = ln (e α ρ 1 ) .
Rewriting the flux (3) in the BDEs in terms of the activation energy, we obtain Eq. (2) is a spatially discrete Smoluchowski equation with diffusion coefficient d k and drift Notice that v k < 0 for an activation energy that increases with k and v k > 0 for decreasing g k . Hence, g k indicates how the discrete advection v k transports the clusters in size space: subcritical clusters shrink as time elapses while supercritical clusters grow with time.

B. The controlling parameters
The simplest nucleation problem consists of solving the BD equations (1), (2) and (13), with dimensionless activation energy g k = σ k − (k − 1)ϕ, discrete diffusivity d k (to be chosen later) and initial conditions The only parameters left in this initial value problem are ρ and σ. ρ controls the long-time behavior of the BDE: If ρ ≤ ρ c given by (16), ρ k (t) approach their equilibrium values (7), with monomer density ρ 1 that solves Equation (1): The graph of this function is either the solid line or the dashed line in Fig. 1 cluster sizes grow indefinitely whereas their density becomes small. Thus there remains a residual monomer concentration whose density ρ 1 e α → 1 as t → ∞. Summarizing, the union of solid or dashed lines in Fig. 1(b) and the vertical line ρ 1 e α = 1 for ρ > ρ c represents the long-time limit of the monomer concentration as a function of ρ.
Let us identify the controlling parameters ρ and σ in a physical system undergoing homogeneous nucleation. A good experimental example is the transformation of certain silicate glasses to crystals (devitrification) [3]. In particular, abundant data exist for lithium disilicate and we have compiled in Table I appropriate values of parameters characterizing nucleation [2]. In disilicate, the free energy per molecule of the crystal phase in the activation energy (9) is proportional to the undercooling where T m is the melting temperature, ∆S f is the molar entropy of fusion and N A is Avogadro's number. The dimensionless density ρ = e ϕ(0) can be extracted from Eq. (19) as explained in Section III. In energy units, the activation free energy is where a is the radius of a spherical k cluster. From the expression for the volume of this cluster, kv = 4πa 3 /3 (v is the molecular volume), we obtain a = [3v/(4π)] 1/3 k 1/3 , and therefore Comparing (20) with (9) yields σ = (32πv 2 /3) 1/3 γ s /(k B T ), and the critical size .
The other parameters in Table I will be used later to model the discrete diffusivity in the BDE. We observe that the critical size increases with temperature: k c = 18 at 703 K and k c = 34 at 820 K. For other materials, such as undercooled liquid metals, critical sizes can be rather large: liquid iron at maximum undercooling has k c = 494, whereas k c = 2253 for liquid rutenium at maximum undercooling [3].

Parameter Symbol Value
Melting temperature T m 1300 K

hours
Heat capacity per unit volume ρ m c according to (10). Note that s k = 1 in equilibrium. Since g 1 = 0, this equation implies for all t. For the initial condition (17), e ϕ(0)−α = ρ 1 (0) = ρ, and the conservation identity in which we have used (22). In terms of the s k , the flux can be written as and the BDE (2) and (13) becomė for k ≥ 2. Here, The term u k D + s k in Eq. (26) represents discrete advection, with a drift velocity u k = In summary, the transformed nucleation initial-boundary value problem consists of the balance equations (26), the particle conservation equation (24), the boundary condition (23), s 1 = 1, and initial conditions s k (0) = 0 for all k ≥ 2. Its solution gives ϕ(t) and s k (t) for all k ≥ 2 and all t > 0.

D. Stationary solution
The stationary solution of the BDE has a flux independent of cluster size, so that and therefore for k ≥ 2. Since s ∞ = 0, j can be obtained from this expression in terms of an infinite series Substituting back this expression into (28), we obtain Then, ρ k = ρ 1 e −g k s k .

E. Turnbull-Fisher discrete diffusivity
To solve the BDE, we need to establish reasonable models of the kinetic coefficient d k (discrete diffusivity) for the decay of the (k + 1) cluster. A classical formula due to Turnbull and Fisher [16] applies to spherical clusters whose growth is limited by the reaction rate at their boundary: d * k is the product of the number of active sites on the aggregate times the molecular jump rate [2,16] Here D = D 0 e −Q/(RT ) is the diffusion coefficient in the liquid, Q is the activation energy for diffusion (see Table I), R = k B N A is the gas constant and λ = v 1/3 (v is the molecular volume). If we nondimensionalize time as in Eq. (4) with this definition of Ω, we obtain so that the quasistationary valueφ given by Eq. (19) is attained at the end of the initial stage.
In materials such as disilicate glass at the temperatures we consider, the critical size is relatively small. Then ϕ(0) ≈φ, and the initial stage is very short. As the critical size increases (as in the case of undercooled liquid metals), ϕ(0) may differ appreciably fromφ, and the initial stage lasts longer. However, even in such cases, the duration of the initial for disilicate glass are relatively small, between 10 and 50, but they are large for undercooled liquid metals, generally between 100 and 1000. Hence we shall use as a small gauge parameter Our asymptotic analysis will be carried out in the limit ǫ → 0, and therefore k c = ǫ −3 → ∞.

Position of the wave front
In the numerical solutions shown in Fig. 2(a), the graphs of s k vs.
Scaling k f as (same scaling as k c = ǫ −3 ), the right hand side of (37) becomes Eq. (37) can be rewritten as provided we define the slowly varying time scale T = ǫt, and take the limit as ǫ → 0.

Shape of the wave front
The leading edge of the wave front is a layer centered at K(T ) in which s k decreases from 1 to 0 as k increases through it. The continuum representation of s k in this layer is where is S a smooth function of its arguments and X is the scaled displacement from the wave front location at k = K/ǫ 3 , i.e., The scaling exponent p, presumably with 0 < p < 3, is to be determined. The description (42) -(43) should hold as ǫ → 0 with X fixed, so that the layer thickness scales as ǫ −p .

Hence (43) yields
and the limit of (45) as ǫ → 0 is Had we carried out the same analysis for the ZFE, we would have found D(K) ∼ d(ǫ −3 K) ǫ.
This would have resulted in a wider wave front and a longer time to ignition than those described below.

Flux and wave front width
Besides determining the shape of the wave front near its location, Eq. (47) yields the behavior of the flux (creation rate of clusters larger than k) j k near k = k f . If we substitute (32), (42) and (46) into (25): Here is a scaled version of the activation energy (9).
Since j k is proportional to ∂S/∂X, it is convenient to differentiate (47) with respect to X in order to obtain an equation for J ≡ −∂S/∂X, Notice that J is locally conserved, and the following integral conservation identity holds: Eq. (51) has Gaussian solutions satisfying (52), which yields for the wave front profile. Inserting Eq. (53) in Eq. (51), we find the following equation for Since K(T ) is an increasing function, we can express A as a function of K. Inserting (41) in (55), we get

Direct integration of this equation yields
in which q is an arbitrary constant.
After insertion of (53), the flux (49) becomes Here K = K(T ) and A = A(T ) are found by solving the differential equations (41) and (55) with initial conditions K(0) = ǫ 3 and A(0) = 3ǫ 4 /(2φ), respectively. We have to set q = 0 in Eq. (57) for A would become exponentially large as K = ǫ 3 → 0 otherwise. As T → ∞, Notice that the terms proportional to X and X 2 have disappeared from this expression and therefore j ∞ is asymptotically uniform for X = O(1). Eq. (59) is the classical quasi-steady nucleation rate of supercritical clusters due to Zeldovich [17], and it can be directly obtained from the stationary flux (29) in the limit as ǫ → 0.

C. The nucleation rate of supercritical clusters
Let us now study the transient creation rate, in which j ≡ j kc increases from 0 to the steady Zeldovich value (59). As we have just seen, our theory predicts that the wave front  spectively. The flux of clusters with sizes larger than k is then given by Eq. (58). Setting k = k c = ǫ −3 (critical size) and X = (1 − K(T ))/ǫ 3/2 in this equation, we obtain the nucleation rate predicted by our theory, j(t). Its integral over time yields the number of supercritical clusters, N c (t). We shall consider now a different and more explicit approximation of these results.

Linearization of the wave front speed about the critical size
Let us fix k = k c = ǫ −3 (critical size) in the definition (46) of X: We now set X = κ in (58) and perform the limit as ǫ → 0 with κ fixed. The result is as ǫ → 0. The solution of this equation is It is convenient to choose κ M as the value of κ at which the flux j reaches its inflection point. Then we may consider that the wave front has ignited the nucleation of supercritical clusters. Straightforward use of Eqs. (61) and (62) shows that Moreover, T M = ǫt M is the time to ignition, at which the wave front K(T ) reaches the value where t ∞ is the duration of the initial stage. We could have expanded the integral in this expression, but Eq. (66) is better suited for numerical calculation. The nucleation rate is found by inserting Eq. (63) in (61): in which a term of order ǫ 3/2 has been ignored in the exponential.
Integrating j(t) over time, we find the number of supercritical clusters as a function of time. In the limit as t → ∞, this number is N c (t) ∼ j ∞ (t − θ), where the time lag θ is approximately given by θ = t M + τ γ + τ E 1 (e t M /τ ), or  suggested in Ref. [24] and further explained in Appendix C. This latter approximation is the worst one. This is not surprising as such approximation provides the same result for both the discrete BDE and the continuum ZFE.

Comparison between different approximations
For disilicate glass at a lower temperature of 703 K, the critical size is smaller and our approximations deviate more from the numerical solution of the BDE, as shown in Figure   4(a). Integrating j(T ) over time, we find the number of supercritical clusters as a function of time, N c (t), which is depicted in Figs. 3(b) and 4(b). At 703 K, the numerical solution of the BDE with the TF diffusivity yields a time lag θ = 2.6. This value is close to those provided by the linearization approximation, θ = 2.2, and by Eq. (58), θ = 2.3. Thus these analytical approximations to the numerical solution are reasonably good even for a relatively small critical size. However, θ = 2.6 gives 1.6 hours according to Table I, whereas the experimentally measured time lag is about 50 hours, cf. Fig. 5 of Ref. [3]. This discrepancy is due to having used the TF discrete diffusivity, which yields an excessively small time unit, as shown in Table I.

IV. TEMPERATURE DRIVEN GROWTH OF THE NUCLEUS
To improve agreement with experiments, we need a discrete diffusivity different from the TF one. We shall no longer assume that cluster size changes due to the activated transfer of a monomer through the cluster surface as in the TF theory. Instead, we shall assume that the discrete diffusivity in the BDE agrees with an adiabatic temperature driven growth of the nucleus. This yields a different formula for d k which, presumably, is not physically justified for very small cluster sizes. Nevertheless, the numerical solution of the BDE corresponding to thermally driven growth provides a time lag which is much closer to the experimentally measured value for disilicate glass than the TF diffusivity.

A. Discrete diffusivity
Let us assume that there is a nonuniform temperature field about a spherical crystal of radius a (k cluster) that is growing at the expense of the surrounding glass. Eq. (21) shows that a nucleus of critical size grows if (T m −T ) decreases. The same equation yields (T m −T ) at the surface of a critical nucleus with k monomers: At the surface of the crystal, T = T * (k), whereas far from it there is a smaller temperature, Heat transfer from crystal to glass, −ρ m cκ4πa 2 ∂T (a, t)/∂r (κ, c and ρ m are thermal diffusivity, specific heat and mass density, respectively), should equal the increase of energy due to crystal growth, The temperature T (r) is the solution of Laplace's equation with boundary conditions T = T * at r = a, and T = T ∞ infinitely far from the nucleus. The corresponding solution is Using Eq. (69) and the expressions for the radius a and (21) for the critical size k c , we obtain As k → ∞, the flux (13) in the BDE becomes j * k ∼ d * k (e −∂g k /∂k −1) ρ k (written in dimensional units), and therefore Using Eq. (11), this equation can be written as Comparing Eqs. (72) and (73), we obtain As before, we shall absorb the constant Ω in the definition of time according to (4), which yields the following value of the dimensionless discrete diffusivity: Here we shall assume that the thermal diffusivity follows the same Arrhenius law as the diffusion coefficient in the liquid κ = D 0 e Q/(N A k B T ) ; see Kelton et al [2]. With this choice of discrete diffusivity, a numerical solution of the BDE yields a time lag of θ = 46.5 hours at 703 K compared to θ = 1.6 hours previously obtained using the TF diffusivity. The experimentally measured time lag is 50 hours, as shown in Fig. 5, page 94 of Kelton's review [3]. Thus we feel justified in using our formula (74) to solve the BDE for disilicate glass.

B. Asymptotic theory
We have to repeat the arguments given in Section III using the discrete diffusivity (74) instead of the TF expression. One important difference is that time needs to be rescaled as T = ǫ 2 t instead of T = ǫt. Here we shall also use the symbol T for the slow time scale, but remembering that T T DG = ǫ 2 t (for thermally driven growth) instead of T T F = ǫt (TF diffusivity). When necessary, we shall add the labels TF or TDG to the corresponding variables. After the initial discrete stage, our asymptotic theory yields the following results for temperature driven cluster growth, applicable to devitrification of disilicate glass: • The wave front profile S(X, T ), with X = [k − ǫ −3 K(T )]ǫ 3/2 and T = ǫ 2 t, is given by . The front location and its width solve with initial conditions K(0) = ǫ 3 , A(0) = ǫ 3 /2. The latter condition corresponds to q = 0 in Eq. (57). Then A ∼ K/2 as K → 0+, which yields the initial condition for A if K(0) = ǫ 3 . In Eqs. (76) and (77), U(K) and D(K) are defined by Instead of Eq. (49), we get the following approximation for the flux near the wave front: in which G(K) =φ (3K 2/3 /2 − K) and Eq. (53) has been used. Inserting X = [1 − K(T )]/ǫ 3/2 in this equation, we obtain the nucleation rate: • The simplest approximation for the nucleation rate and the time lag yields To obtain these expressions, we have followed the same procedure as in the case of the TF diffusivity. In particular, Eqs.   The stationary nucleation rate is approximated less well by j ∞ in the case of thermally driven growth because of the avoidable singularity of d k at the integer k c , which is slightly different from ǫ −3 .

V. DISCUSION
In this paper, we have studied the case of phase segregation resulting when ρ > ρ c .
Previously, other authors had carried out asymptotic studies of the BDE in the simpler case of subcritical density, ρ < ρ c , in which initial conditions of only monomers, or more general ones, evolve towards the equilibrium distribution. In many cases of polynomial growth for d k , equilibrium is reached via a wave front profile for s k , which is similar to Eq. (54) with A ∝ K δ , and K ∝ T µ , for appropriate positive δ and µ; see Ref. [25] and references cited therein. This advancing and widening wave front leaves in its wake the equilibrium size distribution.
In the more complex case of phase segregation and indefinite aggregate growth considered in this paper, a quasicontinuum wave front of s k emerges after a short transient which is Let us briefly mention several existing asymptotic theories for the ZFE. Shneidman [21] and Shi et al [22] Laplace transformed the continuum ZFE and matched a first stage of pure advection of clusters to a local expansion about the wave front when it is near its final position at the critical size. They obtained our simplest formula for the nucleation rate, as an approximation to the full ZFE. Their results are comparable to those found by means of the Laplace transform and matched asymptotic expansions; see Wu's review [19]. All these authors obtained a transition region for the nucleation rate j(t) that was wider than observed in the numerical solution of the BDE. Several authors also found a nucleation rate for supercritical clusters that did not tend to j ∞ as t → ∞ if k = k c [20,22,24], which is often called the asymptotics catastrophe [26]. Our theory is free from this deficiency: Eq.
(B5) in Appendix B provides the flux at k > k c using the TF diffusivity in which X 0 = ǫ 3/2 (k − k c ). Notice that j k ∼ j ∞ as t → ∞, even after making our simplest approximation: linearization of the wave front about the critical size. To get rid of the asymptotics catastrophe, Maksimov et al [26] assumed that S(X, T ) = (1/2) Given the initial condition S 0 ( which is Eq. (53), up to exponentially small terms.
where t ∞ is the duration of the initial discrete stage in the original time scale. After straightforward calculations, we obtain (66).
Thus we can ignore the exponential integral in (B3), which then yields Eq. (68).
Sometimes it is interesting to calculate the creation rate of clusters of size k > k c . If k is close to critical size, we can write Thus X = X 0 + κ, with X 0 = ǫ 3/2 (k − ǫ −3 ). Inserting X = X 0 + κ and K = 1 − ǫ 3/2 κ in (58), we obtain the creation rate of clusters of size k = ǫ −3 + X 0 ǫ −3/2 : Notice that j k ∼ j ∞ as t → ∞. Thus our asymptotic result for the flux over any cluster size is free from the asymptotics catastrophe [26]: several authors found that their expressions for the flux tend to J = j ∞ as t → ∞ if k = k c . These catastrophes are due to inappropriate assumptions they made in their derivations.

ABOUT THE CRITICAL SIZE
A possible approximation of the wave front equations consists of linearizing the equations for K(T ) and A(T ) about the critical size K = 1: with K(0) = ǫ 3 and A(0) ≈ 0 (initial condition of pure monomers). Both for the TF discrete diffusivity and for temperature driven growth of the nucleus, we have The solutions of Equations (C1) and (C2) can be written in terms of the time scale t as for the TF diffusivity, and the same formulas with τ T DG instead of τ T F for thermally driven growth. Near the critical size, these equations would give an explicit expression of the wave front profile (54), with X = ǫ 3/2 k − ǫ −3/2 K(T ). The nucleation rate of supercritical clusters is then obtained from (58) with X = (1 − K)/ǫ 3/2 , together with (C4) and (C5). For the TF diffusivity, j ≡ j kc is Notice that the same results would have been obtained from the ZFE because the difference between our D(K) and the corresponding one for the ZFE is U(K)/2, which vanishes at K = 1. If we replace 1 instead of [1 − (1 − ǫ 3 )e −t/(2τ ) ] in the previous formulas, we find: