Nucleation Theory and Dymamics of First-Order Phase Transitions Near a Critical Point K

NUCLEATION THEORY AND DYMAMICS OF FIRST-ORDER PHASE TRANSITIONS NEAR A CRITICAL POINT K. Binder

To cite this version:

K. Binder. NUCLEATION THEORY AND DYMAMICS OF FIRST-ORDER PHASE TRANSI- TIONS NEAR A CRITICAL POINT. Journal de Physique Colloques, 1980, 41 (C4), pp.C4-51-C4-62. 10.1051/jphyscol:1980409. jpa-00219924

HAL Id: jpa-00219924 https://hal.archives-ouvertes.fr/jpa-00219924 Submitted on 1 Jan 1980

HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. JOURNAL DE PHYSIQUE Colloque C4, suppUment au n " 5, Tome 41, mai 1980, page C4-51

NUCLEATION THEORY AND DYNAMICS OF FIRST-ORDER PHASE TRANSITIONS NEAR A CRITICAL POINT

K. Binder

IFF, KFA Jiilich, 0-5170 Jiilich, Postfach 1a3, PJ-Germany

Abstract.- A brief review of nucleation theory and its problems is given. !-lain em- phasis is on nucleation close to the critical point of liquid-gas systems or liquid liquid mixtures, and the interpretation of recent experiments in these systems. ~t is shown that critical slowing down leads to considerable enhancement of the relative undercooling at the limit of metastability close to the critical point. The role of pre-exponential factors on the nucleation rate is discussed, and scaling laws for nucleation near the critical point are derived. Computer simulations of the lattice gas model are used to show that the one-cluster-coordinate theory is sufficient even close to the critical point and that the classical "capillarity approximation" underestimates the energy barriers even at fairly low temperatures.

1. Introduction.- Nucleation theory consi- high barriers the conventional Becker- ders the rate of phase transformation of me- Daring nucleation theory /8/ is a reasona- tastable states. The mechanism one has in ble approximation, and typically the limi- mind is the formation of small nuclei of ted accuracy of experimental data does not the new (stable) phase by thermal fluctua- warrant to consider a more sophisticated tions. These nuclei then grow to form large theory. In solid systems, on the other hand, macroscopic domains of this new phase. the atomic mobility is much smaller and In a liquid-gas system, such initially hence the energy barriers relevant for the metastable states can in principle be pro- observation of homogeneous nucleation would duced by quenching the system from a tempe- be much smaller. But there in most cases rature in the one-phase region to a point T within the co-existence curve, keeping the density constant (Fig-lA). Similar quenches can be performed in liquid or solid mixtures (Fig. lB) . Very beautiful experiments of this type have in fact been performed by Goldburg and coworkers /1,2/ and by Knobler and others /3?6/, and will be described in the next section. The present paper will be largely concerned with the theoretical interpretation of these experiments, emphasi- zing the regime close to the critical point Tc. The reason is that varying the tempera-

ture T and hence the parameter E = AT&= (Tc-T)/Tc changes the relevant time scales (because of "critical slowing down" /7/) , and hence a more stringent test of the theory is possible than with ordinary nucleation experiments (done far away from Tc). I In fluid systems away from Tc the attempt coex ccclt I'21 -C +-A~!oex frequency is so high that nucleation beco- Fig.1.- Phase diagram of a gas-fluid system (A, mes appreciable whenever the energy barrier temperature T plotted versus density p) and a bina- becomes less than about 50 kgT. For such ry mixture (B, temperature T plotted versus concentration C). Quenches from one-phase equilibrium states to metastable states in the two-phase region are indicated.

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1980409 04-52 JOURNAL DE PHYSIQUE heterogeneous nucleation" (at lattice de- 2. pxerimental situation. - The experiments f ects, surf aces etc. ) is more important. usually are started in a thermal equilibrium The situation is also more complicated sin- one-phase state outside the coexistence cur- ce anisotropy energy, elastic energy etc. ve. In a sudden quench the systkm is brought have to be included when discussing the for- to a point inside the coexistence curve, mation energy of a nucleus. Therefore, only where thermal equilibrium would require a fluid systems will be considered explicite- first-order phase transition to a mixture ly in the present paper, but the general of macroscopic domains of the two stable concepts are clearly more general. phases. (In practice this is done e.g . for As far as static properties near the cri- mixtures with an inverted miscibility gap, tical point Tc are concerned, all liquids where the "quench" is a sudden rise of tem- as well as liquid mixtures belong to the perature produced by microwave heating) . same "universality classK /9/. Even none- One then asks whether for a considered un- quilibrium systems as the electron-hole li- dercooling 6T an appreciable fraction of quid, which occurs in semiconductors at the metastable phase transforms within a very low temperatures /lo/ (Fig. 2) , should characteristic time rcond [e .g ., whetb.er en belong to the same class ."u niversclit.ytlmeans aazeciable fraction of supersaturated gas has condensed]. In these thermal equilibrium nucleation experiments the time rcond is typically of the order of 1 s. A very different regime of times would be accessible electron- hole- gas I in laser-induced nucleation in solids a coex~stencecurve KT 1 0-9 s] , but, in that case, one no longer deals with thermal equilibrium. In the ecpi- electron- hole-l ~qu~d libriumcase, the undercooling 6T where appreciable phase transformation sets in can be measured by light scattering, direct mi- croscopic observation or measuring the la- tent heat which is associated with tne condensation. Varying the density of the fluid, Fig. 2. - "Phase diagram" of the (nonequilibrium)ele- or the concentration of the mixture, one ctron-hole-gas-liquid system in Ge (Temperature plotted versus carrier density, according reference can measure this relative undercooling /lo/. Open and full circles denote various experi- 6T/AT, where the phase transformation starts mental data (see /lo/ while full curve is a theore- T~~~~=1 E tical interpretation derived from equation (17)/lo/. to occur within s, for various (figure 3). that parameters describing the singular behavior near the critical point, like critical exponents, scaling Bunctions etc.are t'ne same for all systems belonging to the same class, which in our case is the class of the Ising or lattice gas model. Hence one can also expect that the same description of large droplets (or "clusters") applies near the critical point in all these systems. Thus we will use "computer experid ments" /11/ on lattice gases, which are so 'L much easier to perform than computer expe- Fig -3.- Relative undercooling 6T/AT [A% = AT-6T, riments on more realistic systems, in order* cf . !?ig.l] , at which the phase transformation occurs during r =I sec, plotted versus distance cond to check the assumptions made in the theory from the critical point. Circles denote data for which is outlined below. lutidine-water mixture /I/, squares C~H~I,-C~F~L, mixtures /1/, crosses 'He/!e/s/ and triangles Con/6/. Dashed.curve is the result of 'fclassical" nucleation theory while dash-dotted curve is the prediction of /13/. From reference /I/. C4-53

Various systems like lutidine-water mixtur- activate process where clusters exceeding a res (circles) /l/, C-JHI 1,-0-7^1^ mixtures /4/ certain critical size have to be formed in 3 (squares), He /5/ (crosses) and C02 /6/ order to enable the phase transformation (triangles)fall within error bars on the to occur.Let us describe all clusters by a same universal curve, which is a constant certain set of coordinates : their "size" far away from T , but rises steeply close I [i.e., the amount of order parameter as to T . All these authors /1-6/ have taken sociated with a cluster], their surface all precautions against heterogeneous nucle- area s, etc. Then the formation free energy ation ; thus one can be relatively sure F (l, s,...) of a cluster in the metastable that this behaviour is characteristic of state will exhibit a saddle point geometry homogeneous nucleation. The theoretical (Fig.4) : for small clusters the surface analysis which will be given below will area is relatively large, and although show that what is measured here is some their bulk energy is negative, F is still sort of universal dynamic scaling function positive due to unfavorable surface energy. for this highly nonlinear dynamic response For large clusters, the negative bulk ener of the system. gy wins, and hence F is steadily decreasing The curves included in the figure are there. theoretical predictions. The dashed hori F(l.s)n zontal curve is the "classical" Becker- DSring nucleation theory, as adapted to the critical region by Langer and Turski /12/. Note that corrections discussed in the nu / $\\ \ \ \ cleation literature over decades such as Lothe-Pound translation-rotation correc tions /8/, replacement partition function corrections /8/ etc. would not change much- the agreement with experiment would become even worse. The dash-dotted curve is the prediction which follows from the theory developed by Stauffer and the present au Fig.4.-Schematic plot of the variation of the clus thor /13/. Although the agreement is not ter formation free energy F (£, s) as a function of perfect, the theory does give the right two "cluster coordinates" (e.g., size Si and excess surface area s-s . ). The coordinates of the saddle trend. Note that in neither calculation • mm +4 point are denoted by & . are there adjustable parameters. Next, the basic ingredients of this the Steady-state nucleation is then descri ory /13/ will be outlined, and the open bed /13/ by equations equivalent to Maxwell's questions will be pointed out. It will be equations for steady-state currents in the - -»- shown that lack of knowledge on the free "cluster-size space" {H}= {H,s,...} : -»• -*- energy barriers of the "critical clusters" VxE U) = 0, (1) still hamper the calculation of this dyna and the continuity equation : mic scaling function of nucleation rates. v'j(T) = 0. (2) It is this point where computer simulations Here the "field"is : on lattice gases are helpful. In the last part of this paper, hence, the direct simu EU) = -V, (3) lation of the equilibrium between a cluster with a potential : and surrounding supersaturated gas will be = -n(l)/n(I), (4) _ -»- briefly discussed. where n («.) is the cluster concentration 3. Nucleation theory and its check up by which is established in the' steadv state computer experiments.- As is well known, nucleation orpcess, and n (X.) i's defined by homogeneous nucleation is a thermally the Boltzmann factor : c4-54 JOURNAL DE PHYSIQUE

3 n (R) exp [- F /kB~]. formula /13/ : 3 3 This is a hypothetical cluster concentra- J a exp{-F(~*)/kB~)d~(Ak~(R*)A*)/l~*l. (11) tion which would be in equilibrium in the In the conventional Becker-Dijring nuclea- metastable state. tion theory, one uses a single cluster coor- 3 3 dinate R only, i.e. all properties of the For the current density j (R) we then have the relation involving a "conductivity clusters are uniquely fixed bg7 their size, 3 fluctuations in the cluster pro~ertiesto tensor" 9 (R) : be described by other coordinates being neglected. IrTith one coordinate only, the steady state solution for both n(R)/n (2) (6 -+ and J is easily found. It reads : Here g (R) is a tensor of cluster reaction rates describincr ~urelvkinetic factors. This set of equations is completed by spe- cifying boundary conditions for the goten- and : tial. Since small clusters are in thermal r 1. equilibrium, -we have : $ (0) =-n (0)/n (0) = -1. (7 The saddle point approximation to this for- Further,since there are not yet any infini- mula (quadratic expansion at the "critical tely large domains present, we have : cluster size" Rf) :

The nucleation rate is then the total cur- yields a similar expression as above, but rent in large distances from the " source" the ree exponential factor is different : at the origin : with one coordinate one does not take gro-

J = d;. (9) perly the available phase space into ac- count. Ve conclude that fluctuations will Just as a battery supplies electrons to affect the preexponential factor of the maintain an electric current, thermal f luc- nucleation rate. tuations supply small clusters (in this -+ The question is, how serious is this continuum description at R = 0) which are neglect in practice, and how accur,ate is fed into the nucleation process and main- the one-cluster-coordinate nucleation theo- tain the steady-state nucleation current. ry? This is a question which can be answe- This description is the more accurate red from Monte Carlo simulations of the two the more cluster coordinates are used. dimensional lattice qas model. These atoms However, the ??robla with this approach is 5. 3 are restricted to occupy sites of a given twofold : (i)the functions n ($1 , g(R) are lattice (e.g. the square lattice) . Atas not known explicitly very well ; (ii)even which are nearest neighbors attract each if they were known the problem isnon tri- other and, in addition, there may be a bin- vial, since there is no general solution ding energy to the lattice or substrate, if for conduction in an inhomogeneous and ani- one considers the lattice to be constituted sotropic medium. What one can do is a sad- by the "absorption sites" produced by the dle point approximation appropriate for F surface layer of an underlying substrate (%*) >> kgT : one expands I?(%)at the saddle lattice. This is also a crude model of crys- point I* up to quadratic terms : tal growth from the gas phase in the limit 3 3 -+ 3 F(R) = F(R*) + (l-%*)g(~-~*)+... (10) where the crystal growsby layer by layer The matrix 5 has one negative eigen-value addition. First of alllone has to obtain (-g) and otherwise positive ones, since $*- the functions R(R) and n (R) which occur is a saddle point. Introducing an effective in the integrals in equations (12), (131, cross section area 2 of the saddle point figure 5. It turns out that the data are region, one finds the £allowing Arrhenius quite well approximated by a power law for R(R) /13/ : Note that in figure 5 also the "classi- r cal" Becker-D6ring "capillarity approxima- R(R) = RR- , r=2-vz/(B6) and Ram. (I5) tion" is included. There, one assumes that

12 I& 260 5&l the surface free energy of the droplets is grecisely obtained by using the surface tension fs of an infinite flat surface, and hence :

n (a)=n (1) exp (-fsS,-Apa) , (18) where SR is the surface area of a droplet (assumed spherical in the classical theory) . In the two dimensional lattice gas, fs is known rigorously from Onsager's exact solution /15/, and hence equation (18) can be evz.lua.ted unambiguously. Obviously, it does not fit the computer experiment "data" in

Fig.5.- Log-log plot of cluster concentration n(R) the considered range of cluster sizes. One and cluster reaction rate R(R) [insert] plotted ver- does expect, however, that this classical sus R, as obtained from Monte Carlo simulation of the two dimensional lattice gas model at Tw0.96 T expression becomes valid asymptotically for /13/. Predictions of the Fisher model, equation very large clusters. In the direct simula- (171, and the classical capillarity approximation, equation (181, are included. Note that straight tion, where one just counts the actual ave- lines indicate power-law behavior, and W(R, l)/nR is rage numbers of clusters of all sizes, one the part of the cluster reaction rate where only mnomer condensation or evaporation is included, and cannot go to such large cluster sizes, however, since these very large clusters are too seldom. Hence, later on, another method Here., z is the exponent describing critical of simulation will be reported where such slowing down /7/, v is the exponent of the large critical clusters are studied direct- correlation length of order parameter fluc- -v ly. The present cluster sizes in figure 5 tuations (5 E /9/) , and 61 6 are the are relevant for energy barriers FRX/kgTX critical exponents describing the singular 10 or less : there, distinct deviations from behavior of the order parameter p-pCrit : classical the'ory do occur. The "data" of figure 5 now provide the 1/6 (P-P~~~~)~~=~a E~~ (p-pCrit) E=O =(AP) , input necessary to evaluate unambiguously (16) the nucleation rate as well a steady-state where Ap is the reduced chemical potential cluster distribution [~qs.(121, (1311. On difference (p-uc) /kgT, uc being the chemi- the other hand, one may record the actual cal potential at the coexistence curve. ii(R) which occurs in the simulation : since Similarly, the cluster concentration is then there are no adjustable parameters wha- fitted by an expression dominated by a po- tsoever, one obtains a meaningful compari- wer law, namely the Fisher droplet model son between computer experiment and the one- 1141 : cluster-coordinate nucleation theory. Figure 6 shows the 'outcome of the compu- n(~)=~R-(~+~'~)ex~(-~ER'/~~-A~.IR) and Ram. ter experiment. (17) However, the prefactors R, q and b cannot Cluster concentrations Ea(t) are plotted be determined in general from any ,first versus time t, the parameter of the curves R. ~rinciples; but it is amusing to note that is the cluster size Three different equation (17) not only fits the lattice gas quenches are shown. After some time lag, data of figure 5 but provides a satisfac- the cluster concentrations always reach so- tory fit to the equation of state of the me nearly stationary state. Ln these flat parts, the cluster concentrations do not electron-hole droglet system (Fig .2) . C4-56 JOURNAL DE PHYSIQUE

~ig.6.- Average cluster concentration ii (t) of the R two dimensional lattice gas model atTx0.96 T plot- Fig.7.- Steady-state cluster concentration fi: (R)/ ted versus time after quenches from Av= 0 to A$=O .03 n(R) obtained from figure 5 and equation (13){das- (a) , 0.024 (b) and 0.016 (c) . Parameter of the cur- hed curves) and from direct simulation { points, ves is R. Relaxation time in equilibrium (r ) as obtained from "flat parts" of figure 6) for the well as time-scale for the first-order phasgptransi- three quenches shown in figure 6. Both the size 9. tion (rAH)are indicated. From /13/. of clusters with volume VRz~dand the critical 5 ?J cluster size R% are indicated. From /13/. become truly constant, but decrease slowly : this happens partly because of the neglec- for liquids near their critical point, since ted cluster coagulation events, and partly they belong to the same static universality because the"supersaturation" of the lattice class. gas decreases by cluster formation, and 4.Scaling laws for nucleation near a criti- thus also concentrati,ons of small clusters cal point.- [That is known about the general change with time. Anyhow, we may take the behavior of the cluster concentration n(R) flat parts at least as rough estimates of and reaction rate R(R)near Tc? As for other quantities near Tc/9/, one can write down what the steady state values should be. "scaling expressions" : these functions do Figure 7 now compares the " observed" values (points) to the calculations using not depend on the three variables R, E, and Ap separately, but apart from a "scale equations (12) , (13) [broken curves] . Since - factorn- only on two scaled variables A~R~, there are no adjustable parameters, not EQ~, y even in the coordinate scales, one does in where the scaling powers x, in our case are /13/ x=l,y=l/BG : fact get significant agreement, fitting not only one curve but a whole family of curves Hence, the result is that the one-cluster- coordinate approximation is accurate, even for such low barrier heights of order FRg/ k Tz10. But it is important to use accura- B Both the, exponent (2+1/6) of the "scale te expressions for the functions R (R) and n Eactor" and the exponents x, y are obtained (R). One expects that the same result holds by considering the contribution of the clus- scaling" /7/ to obtain the scaling behavior ters to the free energy : nucleation theory of R(L) as given in equation (20) . Clearly, considers clusters much larger than the co- the power-law prefactors of equations (19) rrelation length 6 (Pig .a), which are basi- and (20) have dominated the behavior of R cally non-interacting apart from random co- (9,) and n (9,) in the simulations (Fig. 5) . llisions where the considered cluster reac- Let us examine the explicit exgressions tions take place. Hence the contribution of of the previous section in the light of this general scaling analysis. The reaction rate, equation (15), is indeed a special case of the scaling assumution, equation 'L (20) [ with ~:l]; similarly, the Fisher dro~letmodel [~q.(1711 is a s>ecial case 'I, of equation (19) with N= exp(-bE9,1/BB-hv9,). The classical expression, equation(la), on the other hand, is not fully consistent with scaling. To examine this point further we have to express the surface area SR in terms of R , the amount of order garameter Fig.8.- Density profile of a "classical" droplet, associated with a cluster of volume V R' with linear dimensions much larger than the corre- For compact clusters we have, in d dimen- lation length (or domain wall thickness) 5 [ left part1 and of a "nonclassical" droplet of smaller sion, St = v~'-'/~, where ;is a constant size, where interior and surface region are no lon- of order unity. From figure 8 we note : ger clearly distinguishable Cright /13/.

Ehese large clusters to the pressure, PI follows from the ideal gas law for a mix- where B is a constant, and hence S R = ture of clusters : (;/Bl-l/d) &-8 (l-l/d) &l-l/d

where po is a "background" term (which also contains the contribution of the more strongly interacting clusters of the scale of the correlation length 5 or smaller). Ye require now that both po and the cluster contribution to the pressure have the same scaling behaviour as p itself, i.e. its singular ??art psins must scale as : /9/

Combining equations (21) and (22) sets the exponents to be used in equation (191, but

does not determine the explicit structure Fig.9.- Scaled relaxation time of the first-order of the "scaling function" 8. But since two phase transition in the two dimensional lattice gas model plotted versus scaled supersaturation. scale factors q and b have been taken out, Points are simulation results (for E k 0.1, 0.04 one can invoke the universality principle and 0.02, respectively), while curve is a theoreti- % cal formula /18/ analogous to equation (32). From /9/ to conclude that N should be universal /lo/ - (i.e., the,same for all fluids including ?raking use of the critical behaviour of the the three dimensional lattice gas). By a surface tension fs= ;c (d-l) where similar argument /13,16/, one can relate ', is a constant, we hence find : R(L) to the dynamic response of the system in equilibrium states, and invoke "dynamic C4-58 JOURNAL DE PHYSIQUE n(e) =n(l) exp[fid-l)v~R-~~a~=n(l)exp scaling function 3 is universal, i.e. the [-(gsjB1-l/d) (l-l/d) (dv-B) E1-l/d,Au same for each of the dynamic universality classes which were just mentioned. =n ( 1) exp [-f S/B '-lid) (E~%)'-'/~-AUR], (24) One can use the equation of state equa- where in the last step the scaling law dv= tion (22), to relate the "scaled sugersatu- 86 + B /9/ was used. Thus we note that the ration" Au/(be) B6 to the relative supercoo- exponential in equation (24) has the cor- ling 6T/AT. One finds that /13/ : rect scaling form - but the preexponential scaling power R -(2+1/6) is missing! Hence, % classical nucleation theory, in its stan- where Au is again a universal function. As dard form, cannot provide us with the cor- a result : rect critical behavior of the nucleation rates. The general scaling exgressions for n X where J is again (another) universal func- (2) and R (a) can be used to derive general tion of its argument. scaling laws for the nucleation rate, howe- A simple explicit estimate of 7 is obtai- ver. From equations (13) , (19) , and (20) we ned by amending the classical expression find for the droplet concentration, Eq. (24) with the scaling power R -(2+1/6) which should occur in any scaled expression for the droplet concentration. One then finds /13/ : dJ= ( ,,/AT) (j-ll2)(k6-l) exp [-xclass ( ST/AT) -Ir

(29) Hence, it turns out that the nucl-eation where the universal constant xclass =%4 0.7 rate depends on the chemical potential diE- [as estimated /13/ from surface tension ference and hence on the supersa-turation in measurements]. Due to the use of the saddle the same scaled form [ A~E-~'] as the equa- point aggroximation [ Eq. (14)I~q.(29) is tion of state, equation (22): accurate for small 6T/AT only, but the . -'L above approximation for n(&) cln::hou: is not J = E' JJ(A~/(~E)"1 where j=BG+B+vz. (26) expected to yield accurate results for The exponent j of the attempt frequency 6T/AT + 1. It was equation (291, however, can be interpreted as follows : J is the which was used in the comparison to the da- number of supercritical clusters per unit ta (Fig .3) and hence it is not too surpri- time and an3. Since nucleation starts with sing that systematic deviations between d spontaneously formed clusters of size 5 , theory and ex~erimentoccur for larger whose number scales as E B6+8{cf.Eqs . (21) , 6T/AT. (22)}, we get the critical behavior of J by Before discussing on how one can make dividing this number by their relaxation progress to better determine the scaling I time (which diverges like a E-"1, in agre- function N, let us note that it is actually ement with equation (26) . Vhile Bd+B x 2 -not J which is measured : one rather mea- and v = 2/3 for all systems of interest to sures the time rcond after which an appre- us here /9/, the dynamic exponent z depends ciable fraction of the metastable sugersatu- on the type of system considered /7/ : for rated phase is "condensed". This fraction Eluids z x3, while for solid mixtures (or- is, for a liquid-gas system : der parameter conserved) z x 4, and for second-order structural transitions in solids (order parameter not conserved) zz 2. Plhile the constant J in equation (26) depends on where B(t-t') is the radius of a droplet at the material under consideration, the time t which was nucleated at time t1after the physical origin of the correction fac- the quench. Neglecting the decrease of su- tors which modify the classical expression persaturation during the phase transforma- for the droplet concentration, equation (1 8) , 2 2 tion, one has J(t)W J and R (t-tl)=R* +B near a critical point. Consider critical (Gl/AT)D (t-t'), where D is the diffusion clusters which are not much larger than the coefficient [DCDE"] and R* the size of the correlation length 5 of order parameter critical nucleus. However, neglecting R', fluctuations (Fig.8) . Since the thickness one finds : of the droplet-gas interface is also given by a distinction between bulk and surface of a cluster then becomes questionable. Hence, there is no reason to assume that the classical expression for the droplet free The condensation time T~~~~ is defined when energy is still correct if we deal with the reaction reaches completion, i.e. when such strongly fluctuating (partially "rami- the left hand side of equation (31) is of fied") clusters. Of course, one can still order unity. Thus, it follows that /13/: define an effective surface free energy of the cluster by writing :

ef f but Fsurf (R) is no longer proportional to It 1s this formula which [fo; icond=1 s ad using equation (29) for J] was compared vt-'Id as the cluster size decreases : ra- to experiment in figure 3. Langer and ther,one expects a "crossover" as shown Schwartz/17/ recently pointed out that the schematically in figure 10. agreement between theory and experiment becomes better, when one does neglect neither R* nor the decrease of supersaturation "' "'t during the droplet formation. However, the uncertainty about preexponentional factors iU in the scaling function J(GT/AT) is not removed by this analysis. On the other hand, a treatment similar to the above equations (30) to (32) has been applied to the relaxation of the two dimensional lattice gas /18/, figure 9. In that case, n(R)was well- represented by equation(l7), and hence there was no ambiguity in calculating 5 (Ap/(bc) from equation (13), (17). It turns out that the computer experiment data have the expected dynamic scaling pro- perty, and moreover they agree nicely with Fig.10.- Schematic plot of effective cluster surface ePf the predicted scaling function for the re- free energy F (R) plotted versus V crosso- surf i-l'd. lexation time. Hence, basically the same ver from the classical regime, where mean fluctua- approach works for both two dimensional tion of cluster surface area is small as compared to the mean cluster surface area, to the nonclassi- lattice gases (Fig. 9) and for three dimen- cal strongly fluctuating regime is.indicated. sional real fluids and fluid mixtures d eH For VR% 5 , we must have FsUrf (2) of the (fig.3). order of the thermal energy kgT, because 5 .Corrections to the Classical "~a~illari- such clusters are typical for that tempera- t17 Approximaltion" for the Droplet Forma- ture (i.e., forming spontaneously in fairly tion Free Energy.- Let us now come back to high concentration). Asymptotically for C4-60 JOURNAL DE PHYSIQUE

V" -+ - the classical expression is reached, Feff i.e.' l-l/d. A crossover surf (a) ( V,/S away from the classical expression sets al- ef f ready in at V = V where Fsurf(R) dif- R cross fers from the classical value by, say, 10% (note that the famous rotation-translation 17 correction factor 10 to the nucleation rate /8/ is removed if one changes the surface tension by about 15%! In looking for preexponential correction factors to the classical expression, deviations of a few percent do matter) effectively in this crossover regime, fluctuations 8.5% in the surface area start to become comparable to the surface area S R itself. The description given qualitatively in figure 10, also applies far away from the critical point, Fig.11.- Schematic sketch of finite box (of volume 5 V) containing a liquid cluster ( of size II) in equi- where is of the order of the distance librium with surrounding supersaturated gas. between neighboring atoms. But, while near ef f as a function of cluster size R and chemical Tc the function Fsurf (R) is universal (apart potential difference. Since the total parti- from scale factors), far away from Tc the deviations from the classical expression cle number in the box is held constant, the for small cluster sizes reflect details of system is constrained to move on the dash- the potential between neighboring atoms, dotted curve. This is the reason why the resulting equilibrium stable rather than and hence depend on the material conside- is unstable : if the cluster is small, most of red : no "universal" description is then possible. the particles are in the gas phase : although the barrier is low, the total free Recently a simulation method has been energy is high. Conversely, if the super- developed /19/ by which one can study Feff saturation 5s very small and the cluster surf (R) more directly in this crossover regime. This is done by putting a liquid is large, the free energy of the gas is cluster into a finite box and by studying small but the barrier is high. Hence in between there is a relatively more favora- the resulting equilibrium between the clus- ble situation, where the cluster is in ter and the surrounding supersaturated gas stable equilibrium with the surrounding gas in the box (Fig -11) . This approachfaiffers £ram some previous attempts along similar lines in the box, since the total free energy has a minimum there. The fact that a cluster of /20/ where the atoms in the box were critical size is in stable equilibrium with considered as part of the cluster : sucla a the surrounding supersaturated gas is pos- treatment is reasonable only if /19/ the sible, since we only consider aabox of gas density is le,ss than about of the finite volume V with a total particle number Liquid densi-k.7, because then the sunersatu- N strictly constant. The equilibrium of the rated gas does not contribute much to the cluster with the gas at constant chemical 'mtal :Free energy of the box, and no serious C .C potential 11 would be an unstable equili- error in F::;~ (R) is introduced. For higher brium, of course. A detailed analysis of gas densities, however, the error does 12 become serious and hence the new method of the situation shown in figure yields the result /19/ that the equilibrium is obtai- Reference /19/ should be used. ned for : The equilibrium is described qualitatively in figure 12. Here the free energy of the total system (cluster + gas) is shown result of the simulation at T/TCw 0.59,the broken curve is the prediction of classical theory. Crosses are more accurate estimates from low temperature expansions /19/, which are available for very small cluster sizes only. The full curve was obtained by integrating the cluster internal energy frm zero temperature to the considered temperature. This indirect method agrees well with the direct results based on equation (34) , and proves the consistency of the various numerical procedures. Although the data are no longer in the region near the critical point here,classical theory still predicts values of Fsurfeff which are too small and, hence, too small energy barriers for most of the cluster sizes studied. This discre- pancy goes in the same direction as the Fig.12.- Schematic plot of the free energy of the box (Fig.11) plotted versus size R of the cluster power law correction applicable near T C . and chemical potential p. Free energy of the gas is It : indicated by dashed straight line. Due to constraint remains an open question, however to of constant total particle number the system is cors- what extent similar deviations occur-in re- /19/. trained to move on dash-dotted curve alistic three dimensional systems far away Hence, one jus/t observes in the simulation from Tc? which cluster size R* is in equilibrium 6. Conclusions.-In this paper it was asked: with which chemical potential difference does homogeneous nucleation theory provide 81.1 : from equation (34) one then reads off us with an accurate description of first aFeffsurf/a%l,: Varying N/V one varies the order transition kinetics? First, it was size R* at which equilibrium is obtained. shown that the classical one-cluster-coordinate nucleation theory, where fluctuations Figure 13 shows some results which were in cluster properties are neglected, may obtained with this technique for the two dimensional lattice gas. Circles are the yield inaccurate preexponential factors. Computer simulations of the lattice gas model show that, even near the critical point, the one-cluster- coordinate nucleation theory is quite sufficient, provided that one uses the appropriate expressions for cluster concentration n (R) and reaction rate R(R). The scaling behavior of these functions near the critical point of a second-order transition was discussed, and a dynamic scaling formula for the nucleation rate derived. It is crucial to consider the combined effects 05 nucleation and growth for an in- eff ~ig.13 .-Log-log plot of aF ($)/a% versus R for terpretation of the experimental enhancement surf the two dimensional lattice gas model at T/T ,+,'0.4. of the accessible undercooling before the Points are computer simulation results [obtgined transition occurs near the critical point from ~q.(34j3, crosses are low temperature expansion results, the dashed straight line is the clas- (Pig .3) . For nucleation in fluids far away sical capillarity approximation, and the full strai- from Tc, the energy barriers at this metas- ght line was obtained from the same simulation via f integrating the cluster internal energy over inter- tability limit are of the order of F(R )/ mediate temperatures. From /19/. kgT&50, and corrections to the classical C4-62 JOURNAL DE PHYSIQUE

theory are hardly important. Due to criti- /9/ For a recent review on static critical phenomena, see Fisher, M.E., Rev. cal slowing down near Tc the relevant bar- Mod .Phys. 46, (1974) 597 ; Levelt- rier heights are lower, and corrections Sengers, J.M.H., Hocken, R. and ~engers,J.V. Phys. Today become the more important the closer one , 30, (1977) 42. comes to Tc. The accuracy of the present /lo/ Reinecke, T.L. and Ping, S.C., Phys. data does not yet allow an unambiguous as- Rev. Lett. 35, (1975) 311. sessment of this crossover away from the /11/ Binder, K. (ed.) Monte Carlo Methods in Statistical Physics (Springer, classical behaviour (Fig but it is .lo) , Berlin-Heidelberg-New York) 1979. hoped that such data should be available /12/ Langer, J.S. and Turski, L.A., Phys. in the near future. Note also that a study Rev. @, (1973) 3230. of this crossover to low barrier heights is /13/ Binder, K. and Stauffer, D. Adv. Phys. 25, (1976) 343. interesting because, then, the density of - /14/ Fisher, M.E., Physics (1967) 255. the critical nuclei strongly increases. 3 Ultimately, one enters a regime of general /15/ Onsager, L. , Phys. Rev. 65, (1944) 117. /16/ Binder, K., Phys. Rev. (1977) instability of the system and a broad class m, 4425 and references therein. of long-wavelength fluctuations (spinodal /17/ Langer, J.S. and Schwartz , A. J. ,unpu- decomposition) /16/. published. The regime of lower barrier heights is /IS/ Binder, K. and Pliiller-IQumbhaar , H., Phys. Rev. (1974) 2328. relevant, away from Tc, if one considers %, /19/ Binder, K. and Kalos, M.H., J. Statist. times for the completion of the phase tran- Phys. (in press) . sitions which are considerably smaller than /20/ Abraham, F.F., Homogeneous Nucleation 1 s, as e.g. in laser-induced nucleation Theory (Academic Press, New York) 1974. where the time is 10-~s.It must be noted, however, that present nucleation theories work onlyfor situations very close to thermal equilibrium, where the system reaches its final tem?erature very quickly in comparison with the time needed to complete the transition. An extension of nucleation theory to systems far away from equilibrium does not seem to be straightforward. Pluch remains to be done before a satisfactory theory for the kinetics of all kinds of first-order phase transformations is at hand.

References

/1/ Schwartz, A.J., Krisl~namurthy, S. and Goldburg, VJ. I., pre~rint. /2/ Goldburg, :I. I., priv. comm. /3/ !