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 rela- tive 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 suffi- cient even close to the critical point and that the classical "capillarity approxi- mation" 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 mixtu- res (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 in- terpretation 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 nuc- leation 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 concen- tration 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 dif- ferent 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 ap- preciable 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 con- densation. 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 be- havior near the critical point, like criti- cal 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 sys- tems. 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 oc- curs 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.