Heterogeneous Nucleation near Metastable First-Order Bulk and

The effect on this rate of crossing metastable bulk and surface first-order phase transitions is determined. There is partial but not complete wetting...
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Langmuir 2002, 18, 7571-7576

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Heterogeneous Nucleation near Metastable First-Order Bulk and Surface Phase Transitions Richard P. Sear* Department of Physics, University of Surrey, Guildford, Surrey GU2 7XH, United Kingdom Received April 15, 2002. In Final Form: June 21, 2002 Phase transformations such as freezing typically start with heterogeneous nucleation. Here the variation of the rate of heterogeneous nucleation of a new equilibrium phase is studied. The effect on this rate of crossing metastable bulk and surface first-order phase transitions is determined. There is partial but not complete wetting at the metastable bulk transition. The rate of nucleation of the new equilibrium phase changes discontinuously as the metastable transitions are crossed. These discontinuities can be large enough that on crossing a phase transition, the rate of nucleation can jump from a negligible value to an easily observable value; that is, the transformation from one metastable phase to another can trigger nucleation of the equilibrium phase. These findings are relevant to heterogeneous nucleation in liquid alkanes and possibly to that in solutions of some globular proteins.

1. Introduction First-order phase transformations typically start with heterogeneous nucleation. A microscopic nucleus of the new phase forms in the old phase but also in contact with a surface which is itself in contact with the old phase. This is an activated process; the formation of the nucleus costs free energy.1 Thus, the start of the transformation to a new phase is very sensitive to what surfaces are in contact with the old phase and to anything which occurs at these surfaces. Here we examine what happens to the rate of heterogeneous nucleation when the old phase undergoes a first-order bulk phase transition and when at the surface in contact with the old phase there is a first-order surface transition. The metastable bulk or surface phase undergoes a transition to another metastable phase, and this transition effects the nucleation of the equilibrium phase. Phase transitions are associated with singular behavior of the thermodynamic functions, and we find that at both types of metastable phase transition the rate of heterogeneous nucleation of the equilibrium phase is also singular. We find discontinuities in the rate of heterogeneous nucleation, whose size we estimate. A discontinuity is either a jump upward or a jump downward in the rate of nucleation. If it is a jump upward, the nucleation rate may suddenly jump from a negligible value to an observable value as the transition is crossed. In this case, a metastable transition will trigger nucleation of a new phase. This is believed to occur in the crystallization of alkanes with an even number of carbon atoms.2,3 * E-mail: [email protected]. Phone: +44 (0)1483 686793. Fax: +44 (0)1483 686781. (1) In fact, the formation of a new bulk phase at a surface is not always an activated process. If the new phase (here labeled γ) completely wets the interface between the surface and the bulk phase, then as the transition (the R-γ transition) is approached a layer of the γ phase forms at the surface and this can grow into a macroscopic phase. If this occurs, then there is no supercooling at the R-γ transition and so there is no metastability and there is no continuation of the R-β transition past the triple point as the R phase is unstable in this region. Alternatively, if the new phase, the γ phase, does not wet the surfaceR-phase interface but does wet the surface-β-phase interface then the γ phase will always nucleate on approaching the β phase from the R phase. Here we will always assume that the γ phase does not wet any surface and so heterogeneous nucleation is an activated process. (2) Sirota, E. B. Langmuir 1998, 14, 3133.

Motivation for this study is provided by the findings of Sirota and Herhold for alkanes with even numbers of carbon atoms2-4 and by the problem of protein crystallization.5,6 In both cases, the equilibrium transition is fluidto-crystal and we are looking at nucleation of a crystalline phase, and in both cases there is another, metastable, first-order bulk phase transition, a liquid-rotator-phase transition in the case of alkanes and a dilute-solutionconcentrated-solution transition in the case of proteins. By proteins, we mean globular proteins, which are roughly spherical, a few nanometers across, and soluble in salt solution. Also, a surface monolayer of the free surface of liquid alkanes undergoes a first-order transition, sometimes called prefreezing, a few degrees above a bulk transition.7 This motivates our study of the effect of a metastable surface transition on nucleation of the equilibrium phase. Following the pioneering work of ten Wolde and Frenkel,8 there has been considerable theoretical work on the effect of a metastable bulk phase transition on homogeneous nucleation; see refs 9-14. Homogeneous nucleation is nucleation in the bulk of the phase rather than at a surface; see refs 15 and 16 for an introduction to nucleation. In general, phase transitions are associated with singular behavior in the thermodynamic functions, and this work has shown that bulk phase transitions in (3) Sirota, E. B.; Herhold, A. B. Science 1999, 283, 529; Polymer 2000, 41, 8781. (4) Sirota, E. B. J. Chem. Phys. 2000, 112, 492. (5) Rosenberger, F.; Vekilov, P. G.; Muschol, M.; Thomas, B. R. J. Cryst. Growth 1996, 167, 1. (6) Piazza, R. Curr. Opin. Colloid Interface Sci. 2000, 5, 38. (7) Earnshaw, J. C.; Hughes, C. J. Phys. Rev. A 1992, 46, R4494. Wu, X. Z.; Ocko, B. M.; Sirota, E. B.; Sinha, S. K.; Deutsch, M.; Cao, B. H.; Kim, M. W. Science 1993, 261, 1018. Ocko, B. M.; Wu, X. Z.; Sirota, E. B.; Sinha, S. K.; Gang, O.; Deutsch, M. Phys. Rev. E 1997, 55, 3164. (8) ten Wolde, P. R.; Frenkel, D. Science 1997, 277, 1975. (9) Talanquer, V.; Oxtoby, D. W. J. Chem. Phys. 1998, 109, 223. (10) Olmsted, P. D.; Poon, W, C. K.; Mcleish, T. C. B.; Terrill, N. J.; Ryan, A. J. Phys. Rev. Lett. 1998, 81, 373. (11) Sear, R. P. J. Chem. Phys. 2001, 114, 3170. (12) Sear, R. P. J. Chem. Phys. 2002, 116, 2922. (13) Tavassoli, Z.; Sear, R. P. J. Chem. Phys. 2002, 116, 5066. (14) Costa, D.; Ballone, P.; Caccamo, C. J. Chem. Phys. 2002, 116, 3327. (15) Debenedetti, P. G. Metastable Liquids; Princeton University Press: Princeton, 1996. (16) Binder, K. Rep. Prog. Phys. 1987, 50, 783.

10.1021/la0203537 CCC: $22.00 © 2002 American Chemical Society Published on Web 08/08/2002

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Figure 1. A schematic phase diagram in the plane of two field variables, x and the temperature T. It shows the bulk phase diagram consisting of three first-order phase transitions: R-β, R-γ, and β-γ, the solid curves. They meet at the triple point, marked with a black circle. It also shows a surface transition of a surface in contact with the R phase, the dotted curve. The continuation of the R-β transition into the region where the phase γ is the equilibrium phase is indicated by the dashed curve. The question marks indicate that deep in the region where the γ phase is the equilibrium phase the R-β transition and the surface transition may not exist. The arrow indicates a temperature quench of the R phase below the R-γ transition and across both the surface transition in the metastable R phase and the metastable R-β transition.

metastable phases also lead to singular behavior in the homogeneous nucleation rate.8,11-14 Here and in an earlier publication by the author,17 we find that both bulk and surface phase transitions in metastable phases lead to singularities in the variation of the rate of heterogeneous nucleation of the equilibrium phase. Our findings are general but qualitative. We are unable to determine the absolute value of the nucleation rate, which would require detailed calculations on a specific model, but we will determine the generic qualitative variation of this rate across first-order phase transitions. In previous work, we dealt with the variation of the rate of heterogeneous nucleation across a metastable firstorder transition at which there was complete wetting.17 Here we complement that work by looking at a metastable bulk transition at which there is no complete wetting, only partial wetting. See refs 18 and 19 for an introduction to wetting phenomena. Essentially, complete wetting is where as the transition is approached from one side, a layer of the bulk phase about to be formed forms at the surface and grows in thickness until its thickness diverges (in the absence of gravity) at the bulk transition. This smooths the variation in properties of the surface such as the rate of heterogeneous nucleation. Whereas here we find a discontinuity in the rate, in ref 17 we found only discontinuities in its derivatives. Metastable transitions are common, particularly near triple points, where phase transitions cross. Consider the schematic phase diagram in Figure 1. The bulk phase behavior simply consists of three first-order phase transitions between the high-temperature phase, which we call R, and two low-temperature phases, β and γ. For simplicity, we have shown the phase diagram in the plane of of two field variables, the temperature T and another variable, x; for alkanes, x is the number of carbon atoms in the alkane molecule.2,3 The lines of the three equilibrium transitions, R-β, R-γ, and β-γ, are indicated by the solid curves in the phase diagram. They meet at the triple point (17) Sear, R. P. J. Phys.: Condens. Matter 2002, 14, 3693. (18) Schick, M. In Liquids at interfaces, Les Houches XLVIII; Charvolin, J., Joanny, J. F., Zinn-Justin, J., Eds.; Elsevier: Amsterdam, 1990. (19) Bonn, D.; Ross, D. Rep. Prog. Phys. 2001, 64, 1085.

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Figure 2. A schematic of a nucleus of the crystalline phase in contact with the surface (labeled s) below and the bulk (labeled b) above. The nucleus is black, the surface is shaded dark gray, and the bulk is left unshaded. A monolayer at the surface is also shown; it is lightly shaded.

of course. The schematic phase diagram also possesses a surface phase transition, shown by the dotted line. We show the continuation of the R-β transition, and the surface transition in the R phase just above it, into the region where the γ phase is the equilibrium phase.1 The continuation of a transition is, by definition, a transition between two metastable phases and so occurs so long as the metastable system can be supercooled to the temperature at which it occurs. Potentially, there are also similar continuations of the R-γ and β-γ transitions,1 but these are not shown. The question marks in Figure 1 are there to remind us that the continuation of the bulk and surface transitions is only directly observable if the γ phase nucleates very slowly from both the R phase and the β phase at the R-β transition. Finally, although the phase behavior of alkanes is qualitatively like that of Figure 1, the phase diagrams determined for a number of globular proteins20,21 do not have a triple point. This is not important; as we are interested only in a transition which is metastable, it is not necessary for this transition to cross the equilibrium transition (here the R-γ transition) at a triple point and become itself the equilibrium transition. Essentially, although a triple point implies the presence of metastable transitions, it is not a necessary condition for the presence of a metastable transition. Having described the phase behavior, both equilibrium and metastable, that gives rise to the effects we are interested in, in the next section we show that the nucleation rate possesses discontinuities and estimate their size. The final section is a conclusion. 2. Estimation of the Rate of Heterogeneous Nucleation Heterogeneous nucleation is an activated process1,15 and as such occurs at a rate which decreases exponentially with the height of the barrier ∆F* which must be overcome. The barrier height ∆F* is the free energy cost of forming the critical nucleus of the phase γ at the surface and in contact both with the surface and with the bulk phase. Figure 2 is a schematic of the nucleus on the surface. The critical nucleus is the nucleus which has the largest free energy; it is a nucleus at a saddle point. The bulk is in either the R or the β phase. If Nn is the number of nuclei per unit area crossing the barrier per unit time, then Nn is given by an expression of the form15,16

Nn ) στ-1 exp(-∆F*/kT)

(1)

(20) Broide, M. L.; Berland, C. R.; Pande, J.; Ogun, O. O.; Benedek, G. B. Proc. Natl. Acad. Sci. U.S.A. 1991, 88, 5660. (21) Muschol, M.; Rosenberger, F. J. Chem. Phys. 1997, 107, 1953.

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where σ is a surface density, that is, it has dimensions of inverse area, and τ is a characteristic time. We will refer to Nn as the nucleation rate or heterogeneous nucleation rate. The surface is smooth, perfectly planar, and chemically homogeneous. The variation in the rate is generally dominated by the exponential dependence on the barrier height ∆F*, so we focus on this. The free energy barrier comes from the free energy needed to form a microscopic nucleus of the new phase, the phase γ. Within classical nucleation theory for heterogeneous nucleation,15,22 the free energy of formation of a microscopic nucleus is the sum of bulk, surface, and line terms, which scale with the volume, surface area, and edge length, respectively. This division of the free energy is reasonable provided that the nucleus is large in comparison to the characteristic microscopic length scale, the correlation length, or the size of a molecule (which are similar far from a critical point, which we assume we are). Then, for a surface larger than the square of this microscopic length its free energy becomes extensive in the area, with a coefficient which is the interfacial tension, plus a term proportional to its perimeter whose coefficient is the line tension. Also, for a nucleus a few correlation lengths across it makes sense to define an inner core of the nucleus which has a free energy scaling with the volume, with a coefficient equal to the relevant thermodynamic potential of the nucleating phase. We consider a nucleus of volume v, with a total surface area of anb + ans, where ans is the footprint of the nucleus at the surface, that is, the surface area of the nucleus in contact with the surface, and anb is the remaining surface area, the surface area in contact with the bulk which may be in either the R or the β phase. The length of the interface between the nucleus and the surrounding surface phase is l. Note that the interface between the surrounding surface and the nucleus is one-dimensional as it is an interface between the nucleus and a two-dimensional object, the surface. For definiteness, let us consider the system to be a singlecomponent system. Also, let us consider the temperature variation of the nucleation barrier at fixed x (the length of the carbon chain in alkanes). Let TRβ be the temperature of the R-β transition at this value of x. The value of x is below that at the triple point (see Figure 1), so the R-β transition occurs when neither phase is the equilibrium; the equilibrium phase is the γ phase. Then, for the purposes of calculating the free energy cost of a nucleus of the γ phase we work at fixed chemical potential, volume, and temperature so the relevant thermodynamic potential is the grand potential. Thus, what we have rather loosely called the free energy of a nucleus, ∆F, is in fact the excess grand potential when the nucleus is present over that when there is no nucleus. It is given by the expression

∆F ) v(ωγ - ωb) + anbγbγ + ans(γγs - γbs) + lτbγ (2) Navascue´s and Tarazona22 were the first to write down this expression for ∆F in full, including the line tension term, which is essential here to obtain the correct behavior. The first term in the expression for ∆F is the bulk term coming from the difference between the grand potential per unit volume of the nucleating phase, ωγ, and that of the phase it is nucleating in, ωb,

{

ω T>T ωb ) ωR T < TRβ β Rβ

(3)

(22) Navascue´s, G.; Tarazona, P. J. Chem. Phys. 1981, 75, 2441.

with ωR and ωβ being the grand potentials per unit volume in the R and β phases, respectively.23 Similarly, the interfacial tension between the nucleus and the bulk,

{

γ T>T γbγ ) γ Rγ T < TRβ βγ Rβ

(4)

where γRγ and γβγ are the interfacial tensions for the R∠γ and β∠γ interfaces, respectively. The interfacial tensions γRγ and γβγ are for different interfaces and are therefore different, even at TRβ. Also, γRs, γβs, and γγs are the interfacial tensions for the surface-R-phase interface, the surface-β-phase interface, and the surface-γ-phase interface, respectively. The surface-bulk interfacial tension is then

{

γ T>T γbs ) γ Rs T < TRβ βs Rβ

(5)

The third term on the right-hand side of eq 2 is the change in grand potential when the nucleus forms due to the replacement of a surface-bulk interface of area ans with a surface-γ-phase interface. As well as the two surfaces of the nucleus, that in contact with the surface and that in contact with the bulk phase, there is a perimeter of the nucleus in the plane of the surface. Associated with this perimeter is a line tension,τbγ,

{

τ T>T τbγ ) τ Rγ T < TRβ βγ Rβ

(6)

When the bulk is in the R phase, this line tension is τRγ, and when the bulk phase is in the β phase, the line tension is τβγ. Also, when the surface passes through the surface transition, the surface will transform from one surface phase to another. As this happens, τRγ will change discontinuously. 2.1. First-Order Surface Transition. Consider starting off at high temperature, in the phase R. At this value of x, let the surface transition occur at a temperature Ts in the R phase, when the γ phase and not the R phase is the equilibrium phase and so potentially the γ phase can nucleate. Now gradually reduce the temperature. The rate of nucleation Nn is given by eq 1, which depends on the free energy of the critical nucleus, ∆F*. We start our search for singular behavior in our expression for the free energy of a nucleus, ∆F, eq 2. We look for singularities in the volume, area, and perimeter terms that together make up ∆F. As the metastable phase is the R phase, γbγ ) γRγ, γbs ) γRs, and τbγ ) τRγ. The relevant bulk functions, ωR and ωγ, are smooth analytic functions near the surface transition at Ts as they are unaffected by any surface phase transition, as is γRγ, the interfacial tension between the R and γ phases, which is the tension of a different surface. The surface phase transition is in the metastable R phase and is not present in the γ phase; thus the tension for the interface of the γ phase at the surface, γγs, is also a smooth analytic function at Ts. This leaves only γRs and τRγ. At a first-order surface transition, the interfacial tension, here γRs, is continuous but its slope, its temperature derivative, is (23) Throughout, we are neglecting metastability of the R-β transition (for alkanes the liquid-rotator transition); i.e., we are assuming that on cooling/heating through this transition the phase transition happens rapidly and with minimal supercooling/superheating. We do so for simplicity; clearly for some metastable transitions there will also be supercooling and/or superheating and this will alter the effect of this metastable transition on the nucleation of the equilibrium transition. See ref 13 for a discussion of this point with regard to homogeneous nucleation.

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discontinuous. A line tension, here τRγ, is actually discontinuous; it has a nonzero jump at a first-order surface transition. This is just the analogue in two dimensions of a surface tension changing discontinuously when the bulk phase whose surface it is undergoes a first-order transition. Thus, at the surface transition at Ts the free energy ∆F of a nucleus with fixed volume and geometry, eq 2, is singular: it contains singular contributions from γRs and τRγ. The dominant singularity is that in the line tension τRγ as it has a discontinuity, whereas the interfacial tension γRs is continuous; only its slope is discontinuous. If δT is some very small positive increment in temperature, then on going from T ) Ts + δT to T ) Ts - δT the free energy of a nucleus with fixed v, ans, et cetera jumps by an amount given by the length of the perimeter of the nucleus in the plane of the surface times the discontinuity in the line tension τRγ, at the surface transition. Line tensions are of order the thermal energy kT per particle diameter. On crossing a strongly first-order surface transition, the jump in the line tension will be of the same order. For a perimeter of order 10 particle diameters long, we have a discontinuity in ∆F of order 10kT. Now, the nucleation rate is determined by ∆F*, eq 1, the value of ∆F at a saddle point (as a function of v, anb, ans, and l). However, if we assume that the nucleus is a compact object with of order tens of particles in it both above and below Ts, then the change in ∆F* on crossing the transition Ts + δT to Ts - δT will also be of order 10 times the discontinuity in the line tension. Now, if the jump in ∆F* lies within the range 5-30kT, from eq 1 and assuming that on crossing the transition the change in Γ is much less than that in the exponential factor, the change in the nucleation rate lies within the range 102-1013. As ∆F* appears in the exponential, the range is huge. The nucleation rate can either increase or decrease by a factor in this range, depending on whether the line tension is larger above Ts or below it. As an example, an increase by a factor of 105 corresponds to nucleation going from taking a day to occurring in a second. 2.2. First-Order Bulk Transition, with Partial Wetting. Now, again let us start off at high temperature, in the phase R, and work at a fixed value of x where the R-β transition occurs in the phase R when it is metastable. We again search for singular behavior in the free energy of a nucleus, eq 2, as the temperature is reduced. We will assume that there is only partial wetting, no complete wetting, at the R-β transition. Having determined the behavior, we will compare it with that found in the presence of complete wetting in ref 17. The quantities which are properties only of the γ phase are smoothly varying functions; the R-β transition does not affect them. Thus ωγ and γγs are analytic functions. However, the grand potential per unit volume of the metastable phase, ωb, is of course singular at the R-β phase transition. It itself is continuous, but its slope, with respect to temperature, is discontinuous at TRβ. As we have only partial wetting, the R phase does not wet the surface-β-phase interface and the β phase does not wet the surface-R-phase interface; then, all the surface and line tensions, γbγ, γbs, and τbγ, have discontinuities at TRβ. Thus, with partial wetting at the metastable transition at TRβ, the free energy ∆F of a nucleus with fixed volume and geometry, eq 2, is not only singular but also discontinuous; it contains discontinuous contributions from γbγ, γbs, and τbγ. From eqs 4-6, γbγ is γRγ above TRβ and γβγ below, and similarly for γbs and τbγ. There is also a singular contribution from ωb, but the dominant singularity is that from the surface and line tensions as they are themselves discontinuous whereas the bulk grand potential is con-

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tinuous; only its slope is discontinuous. Also, on crossing a first-order surface transition only the line tension jumped; thus, in general, we expect the jump in the nucleation rate on crossing a bulk transition to be much larger than the jump on crossing a surface transition. If δT is some very small positive increment in temperature, then on going from T ) TRβ + δT to T ) TRβ - δT the free energy of a nucleus with fixed v, ans, et cetera jumps by an amount given by the areas of the surfaces times the change in their interfacial tensions plus the length of the perimeter of the nucleus in the plane of the surface times the discontinuity in the line tension. For a strongly first-order transition, the discontinuities in the interfacial and line tensions will be of order the thermal energy kT per molecule at the interface/perimeter. For a total surface with of order tens of molecules plus a perimeter of order 10 particle diameters long, we have a discontinuity in ∆F of order tens of kT. Now, the nucleation rate is determined by ∆F*, eq 1, the value of ∆F at a saddle point. However, if we assume that the nucleus is a compact object with of order tens of particles in it both above and below TRβ, then the change in ∆F* on crossing the transition TRβ + δT to TRβ - δT will also be of order tens of kT. From eq 1, the change in ∆F* of order tens of kT results in a huge change in the nucleation rate. If the jump in ∆F* lies within the range 10-50kT, from eq 1 and assuming that on crossing the transition the change in Γ is much less than that in the exponential factor, the change in the nucleation rate lies within the range 1041021. It will increase or decrease by many orders of magnitude. In principle, the nucleation rate can either increase or decrease, depending on whether the interfacial and line tensions are larger below TRβ or above it. If the phase β is in some sense intermediate between the R and γ phases (as is the case for alkanes), the nucleation rate will increase on cooling below TRβ. The increase in the rate of heterogeneous nucleation may easily be enough to change a rate which is so slow as to be unobservable to a rapid nucleation rate. If this happens, then crossing the R-β transition will act to trigger nucleation of the γ phase. The variation in the rate of homogeneous nucleation is also discontinuous when a first-order phase transition is crossed.13,14 In contrast, when there is complete wetting at the metastable transition at TRβ, the free energy ∆F of a nucleus with fixed volume and geometry, eq 2, although it is still singular, is continuous; only its derivative is discontinuous. This was shown in ref 17 where a vaporliquid transition was studied, in which the liquid phase completely wets the surface-vapor interface. See refs 18 and 19 for an introduction to wetting phenomena. Essentially, if say the β phase completely wets the surfaceR-phase interface then just above TRβ the nucleus forms in a thick layer of the β phase interposed between the surface and the R phase, whereas just below TRβ the nucleus forms in a bulk β phase directly against the surface. As δT f 0, the thickness of the interposed layer of β phase diverges (in the absence of gravity). This diverging layer thickness above TRβ makes the nucleation rate at T ) TRβ + δT approach that at T ) TRβ - δT as δT f 0, and the rate of heterogeneous nucleation is a continuous function of temperature. However, above TRβ there is an R-β interface between the interposed layer of the β phase and the bulk R phase, whereas there is, of course, no such interface below TRβ. The variation of the thickness of this interface as the temperature is varied above TRβ gives a contribution to the temperature derivative of the nucleation barrier above TRβ which is missing below TRβ. This results in a discontinuity in the temper-

Heterogeneous Nucleation near Phase Transitions

ature derivative of the nucleation rate; see ref 17 for details. Note that eq 2 is not appropriate when there is complete wetting. As usual in a classical nucleation theory, it splits the free energy into bulk, surface, and perimeter parts. This is only correct when the nucleus is large with respect to any other length scales. When there is complete wetting near the coexistence there is a thick wetting layer: the thickness of this layer is large, and so the assumption that the nucleus is large with respect to all relevant length scales breaks down. 2.3. Crystallization of Alkanes. Sirota and Herhold have looked at the nucleation from the liquid phase of the triclinic crystalline phase of alkanes with even numbers of carbon atoms;2,3 see also ref 4. These alkanes possess in addition to the liquid (L) and triclinic crystalline (K) phases a rotator crystalline phase (R) which is almost stable at equilibrium. A rotator phase is a type of crystalline phase with rotational disorder of the molecules. Sirota and Herhold find evidence that the K phase nucleates only once a metastable R phase has appeared. The nucleation is heterogeneous. The phases in Figure 1 are labeled R, β, and γ which for alkanes would correspond to the L, R, and K phases, respectively. The x variable would then be the number of carbon atoms in an alkane molecule. Bear in mind that the phase diagram of Figure 1 is just a schematic and not the true phase diagram of alkanes with even numbers of carbon atoms. The experiments find that the measured nucleation rate can be zero until the rotator phase is observed, whereupon the triclinic phase nucleates. This observation of the nucleation rate jumping from a value too small to be detected to observable nucleation, when the rotator phase appears, is just what our theory would lead us to expect. As the rotator phase is a crystalline phase itself, it is reasonable to expect the K-R surface and line tensions to be lower than the K-L tensions. Then, when the metastable liquid transforms to a rotator phase the nucleation barrier drops discontinuously due to the lower surface and line tensions between the triclinic crystalline and the rotator phases in comparison to those between the crystalline and the liquid phase. The surface monolayer of alkane liquids freezes into a rotator-phase-like monolayer at a temperature a little above the temperature of the bulk liquid-rotator transition, sometimes called prefreezing.7 The surface is the liquid-air interface. The rate of heterogeneous nucleation will of course be discontinuous when this freezing occurs. Sirota and Herhold find evidence of a bulk rotator phase when alkanes with 16 and 18 carbon atoms freeze into the triclinic phase. There, the crystallization occurs below the bulk liquid-rotator transition temperature and hence of course also below the prefreezing transition. On cooling to the temperature at which these alkanes crystallize, the nucleation rate has actually gone through two discontinuities. For shorter alkanes, with 14 carbon atoms or fewer, the rotator phase is less stable. It may not be forming at all and hence playing no role in the crystallization. However, the prefreezing is slightly above the bulk transition,7 so even if the bulk rotator phase does not form, a surface monolayer might. It is presumably very difficult experimentally to detect the formation of a shortlived surface monolayer, but such a layer can still significantly alter the barrier to nucleation. Protein solutions also have a metastable transition, a transition between a dilute and a concentrated solution.20,21 A difference between protein solutions and alkanes is that the metastable transition in protein solutions ends at a critical point while the liquid-rotatorphase transition being a liquid-to-crystal transition cannot

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and does not end at a critical point. The importance of a bulk critical point for heterogeneous nucleation is principally that near the critical point, for an attractive surface, there must be complete wetting, as first suggested by Cahn.18,19,24,25 Thus, close to the critical point the results of ref 17 apply, not those obtained here. Far from the critical point, there may be partial wetting, for which the results obtained here apply. 3. Conclusion The rate of heterogeneous nucleation has a discontinuity when a metastable first-order bulk or surface transition is crossed. This is no real surprise as most properties have a discontinuity at a first-order phase transition. For a nucleus of tens of molecules, far from any critical point (bulk or surface) and without complete wetting, the nucleus will be reasonably large with respect to all other relevant length scales. Then classical nucleation theory will be valid and hence so will our rough estimates of the size of the discontinuities. At a surface phase transition, the surface tensions, being the thermodynamic potentials of the surface phases, are themselves continuous and the only reason that there is a discontinuity in the nucleation rate is that the line tension is discontinuous. For this reason, the discontinuity will tend to be much smaller at a metastable surface transition than at a metastable bulk transition. At a bulk transition, the surface tensions are discontinuous as is the line tension. Here we have always assumed that the metastable transitions are strongly first order. A prewetting critical point, which is an example of a surface transition critical point, has been discussed in ref 17. Two sets of experimental systems have been mentioned: alkanes and protein solutions. Both have a firstorder metastable bulk transition: for alkanes (with an even number of carbon atoms), it is a liquid-rotator-phase transition, while for solutions of globular proteins it is a vapor-liquid-like transition between a dilute protein solution and concentrated solution. There is clear evidence of a metastable transition triggering nucleation in the alkanes,3 while proteins are much less well understood.5,6 In protein solutions, crystallization certainly tends to occur in the vicinity of the metastable transition and the nucleation is presumably heterogeneous but precisely how the metastable transition effects the nucleation of the crystalline phase is not known. Indeed, it should be borne in mind that there are many thousands of globular proteins and that they are rather diverse; some may not possess a metastable transition, and some may not even possess a crystalline phase. The results obtained here apply only to proteins with a phase diagram similar to that found for lysozyme and some other proteins,20,21 which may be a considerable fraction of the total but will not be all globular proteins. Here we have seen that if sufficiently far from the critical point there is partial wetting, the nucleation rate will jump upward as the metastable transition is crossed (assuming, very reasonably, that the interfacial tension between the crystal and the concentrated solution is less than that between the crystal and the dilute solution). Throughout this work, we have assumed that heterogeneous nucleation occurs on an infinite, smooth, planar, and chemically homogeneous surface. In an experiment, it is quite possible that none of these things are true; if the nucleus is forming at a solid surface, it may be forming on the surface of a particle of dirt, and so the surface will (24) Cahn, J. W. J. Chem. Phys. 1997, 66, 3667. (25) Indekeu, J. O. Acta Phys. Pol., B 1995, 26, 1065.

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be finite and curved, and it may contain defects such as step defects and impurity atoms. All these deviations from a smooth, infinite, et cetera surface will both shift the nucleation barrier, ∆F*, and effect any surface transition. For example, some patch on the surface of a dirt particle may attract protein molecules more than the rest of the surface and then we would expect the ∆F* to be lower on this patch and that therefore nucleation actually occurs on the patch. Thus, not only would the relevant γγs, τbγ, et cetera be those for the patch but any surface phase transition would be rounded off due to the finite size of the patch, turning the discontinuity in the rate of heterogeneous nucleation into just a very rapid variation

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of the rate with temperature. Of course, a bulk first-order phase transition would be unaffected; the nucleation rate will still jump as the transition is crossed. Thus, future theoretical work on nucleation on less-idealized model surfaces will be required. However, in terms of understanding protein crystallization in particular, the most urgent requirement is for experiments on heterogeneous nucleation on well-characterized surfaces. Acknowledgment. This work was supported by the EPSRC (GR/N36981). LA0203537