Crystal nucleation in the presence of a metastable critical point

JOURNAL OF CHEMICAL PHYSICS

VOLUME 109, NUMBER 1

1 JULY 1998

Crystal nucleation in the presence of a metastable critical point V. Talanquer Facultad de Quı´mica, UNAM. Me´xico, 04510, D. F. Me´xico

David W. Oxtoby The James Franck Institute, The University of Chicago, Chicago, Illinois 60637

~Received 14 January 1998; accepted 27 March 1998! Density functional theory is applied to the study of crystal nucleation in the presence of a metastable critical point. A phenomenological model for fluids with short range interactions is developed to study the influence of critical density fluctuations on the structure of the critical nucleus and the height of the barrier to nucleation. Our results show dramatic changes in the nature of crystal nucleation near the metastable critical point, with nucleation rates increasing several orders of magnitude; this behavior has important consequences for nucleation of colloids and proteins from solution. A nonmonotonic dependence of the critical cluster size on supersaturation is observed under these conditions. © 1998 American Institute of Physics. @S0021-9606~98!51325-6#

I. INTRODUCTION

for the transition means that the two need not go together; one of the two can dominate the critical nucleation and serve as the prerequisite for the other one. In our earlier work,7 such cases with a nonlinear dependence of one order parameter on the other through the nucleus serve as indicators for the breakdown of simple classical theories of nucleation; the critical nucleus for modest undercoolings can have a structure that differs sharply from the eventual stable phase. In particular, for small-molecule liquids ~which tend to be quite incompressible! changes in density can only be achieved after crystalline structure appears, so the critical nucleus is dominated by changes in the structural order parameter.8 For colloids and proteins, on the other hand, the presence of a metastable critical point under the liquid-crystal phase coexistence line means that fluctuations in density ~concentration! can become very large. This suggests the reverse possibility: that the formation of the critical nucleus may be an aggregation process ~involving disordered clusters of particles! with periodic crystalline order appearing only later and playing a much smaller role in nucleation. The goal of this paper is to investigate this possibility. Such a mechanism for protein crystallization from solution is suggested by recent computer simulations by ten Wolde and Frenkel.9 These authors found that far from the metastable critical point in a colloidal system the critical nucleus was ‘‘classical’’ in that its degree of crystalline order was comparable to that in a stable bulk crystal. In other words, the number of ‘‘crystalline’’ particles was comparable to the total number of particles in the critical nucleus. Close to the metastable critical point, in the other hand, the critical nucleus was a highly disordered aggregate of particles. Moreover, these authors suggested that the conditions of optimal crystal growth ~minimum crystal nucleation barrier! corresponded to exactly this ‘‘nonclassical’’ critical nucleus. In this paper we apply the methods of density functional theory to the same problem. Density functional theory7 has advantages of speed relative to computer simulation, and it also helps to identify the key physical features involved in

The fact that liquids and solutions can be undercooled below the thermodynamic freezing transition ~in some cases by hundreds of degrees! suggests a kinetic barrier to crystallization. In most cases this nonequilibrium response results from the slowness of crystal nucleation. In the absence of heterogeneous impurities that can catalyze nucleation, the critical fluctuation needed to access the crystalline phase can be many times larger than the free energies readily obtained thermally. Only through deep undercooling ~or high supersaturation in the case of solutions! can this barrier be reduced sufficiently for crystals to form on a laboratory time scale. However, in this case kinetic factors related to a slowing down of diffusive processes can then prevent either nucleation or crystal growth in spite of the large thermodynamic driving force. Some melts or solutions thus are extremely difficult to crystallize. This problem is particularly acute for protein crystallization, one of the most important aspects technologically, where achieving conditions for growth of high-quality crystals is a necessary prerequisite to obtaining atomic-level structures from x-ray diffraction. The large, globular molecules that proteins in solution represent have different interactions than the small molecules whose crystal growth has been more extensively studied to date, and this may have a significant effect on their crystal nucleation. This conclusion is also consistent with experiments and computer simulations of colloidal particles,1–6 for which the range of interaction is short compared with the particle size ~in contrast to small molecules, where the two lengths are comparable!. The effect is to depress the putative gas–liquid critical point below the liquid–solid coexistence curve, entirely changing both the equilibrium phase diagram and the crystal nucleation behavior. Crystal nucleation and growth involve changes in both the periodic structure of the molecules making up the new phase and in the average particle density ~or concentration, for a solution!. The fact that there are two order parameters 0021-9606/98/109(1)/223/5/$15.00

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J. Chem. Phys., Vol. 109, No. 1, 1 July 1998

V. Talanquer and D. W. Oxtoby

the problem. It provides both a physically realistic and computationally efficient way of studying phase transitions. Our goal in this paper is not to start with a first-principles calculation ~building a free-energy functional based on a realistic potential for colloids or proteins!. Rather, we take a phenomenological approach, constructing a free-energy functional that is physically reasonable but in which the parameters are adjusted to give equilibrium phase diagrams comparable to those found in real colloidal systems. Our focus is therefore on the nucleation behavior from such a functional, not on the microscopic origins of the functional itself. We describe the model in Sec. II and present the results in Sec. III. The conclusion from our work is that the nature of crystal nucleation changes qualitatively near an underlying metastable critical point, and that the barrier to nucleation is lowered substantially there, increasing nucleation rates by tens of orders of magnitude. The consequences for nucleation of colloids or proteins from solution are substantial.

II. MODEL

The proposed free-energy functional for the inhomogeneous solid–fluid system has the basic form V @ r ~ r! ,m ~ r!# 5

E

dr@ f ~ r ~ r! ,m ~ r!! 2 m r ~ r!#

1 1 2

E

2 @ a1a m m 2s ~ r !# r 2 ,

~1!

where f is a local function of the average number density of dissolved molecules r~r! and the structural order parameter m(r), and m is the chemical potential. The two squaregradient terms in this relationship account for nonlocal effects in the system, with the constants K r and K m related to correlation lengths for the order parameters r and m. The local variables r~r! and m(r) take the values r 5 r s , m 5m s , and r 5 r f , m f 50, at the bulk solid and bulk fluid phases, respectively. We follow van der Waals in writing the free energy per unit volume for the fluid, and then add to it a quadratic dependence on the structural order parameter m. The nature of this contribution is consistent with similar expressions obtained by more sophisticated models where the average density is expressed in terms of very flat overlapping Gaussians,8 f f ~ r ,m ! 5kT r @ ln r 212ln~ 12b r !# 2a r 2 1kT a 1 m 2 . ~2! Here, k is Boltzmann’s constant, T is the absolute temperature, and a 1 , a, and b are phenomenological parameters. The latter parameters, a and b, are chosen in order to include, respectively, the effect of long range attractive forces between dissolved molecules and the molecular size. The free-energy density for the solid is built in a phenomenological way and is proposed to have the form,

~3!

where ¯r is a function of r and m, and plays the role of a weighted or coarse grained averaged density.10,11 Intuitively, at high densities, crystallization should decrease the ‘‘effective’’ number of neighboring molecules for each molecule in the system. This can be attained through the coarse grained density ¯r by assigning ¯r 5 r @ 12 a 3 m ~ 22m !# ,

~4!

where a 3 is an additional phenomenological parameter. This expression assures that the local value of m s ( r ), given by

U

]fs ]m

50,

~5!

m s~ r !

is equal to 1 in the close-packing limit (b¯r 51). Finally, the phenomenological parameter a m in Eq. ~3! has been introduced to mimic the effect on the solid free energy of changing the range of the attractive interactions between dissolved molecules in the system. For a homogeneous system, the structure of Eqs. ~2! and ~3! determines the nature of the coexisting phases at each temperature. The chemical potential along the solid and fluid branches is given by

m5

dr@ K r ~ ¹ r ~ r!! 2

1K m r 2s ~ ¹m ~ r!! 2 # ,

f s ~ r ,m ! 5kT r @ ln r 212ln~ 12b¯r !# 1kT @ a 1 m 2 1 a 2 #

S

] f ~ r ,m ! ]r

D

,

~6!

T

and any pair of coexisting phases with densities r a and r b should satisfy the coupled equations

m a ~ r a ! 5 m b ~ r b ! and v a 5 v b , where v i 5 f i 2 m i r i is the thermodynamic grand potential of phase i. The minimum condition in Eq. ~5! determines the equilibrium value of the structural parameter m s , for the solid phase at coexistence. The properties of inhomogeneous systems, like the surface tension of a planar interface, or the work of formation of a critical nucleus in a given metastable state, can be derived from the solutions to the Euler–Lagrange relations associated with the conditions

dV dV 50 and 50, d r ~ r! d m ~ r! when applied to Eq. ~1!, with f ( r ,m) defined as the minimum of f f and f s . The resulting ordinary differential equations are coupled through the local free energy f , and can be solved by a conventional shooting method under appropriate boundary conditions. For solid–fluid interfaces, however, special care should be taken in following the proper solution at the intersection of the corresponding free-energy branches.

III. RESULTS A. Phase diagrams

The well-known phase diagram for the van der Waals model has a critical point at




225

FIG. 2. Number of solidlike particles N c in critical clusters as a function of the excess number of particles N in a system with a m 54.0. The supersaturation Dm decreases continuously as we move along these curves to higher values of N c . The number of solidlike particles N c starts growing rapidly with N at D m /kT c '0.75 for T r 51.025.

B. Critical nuclei

FIG. 1. Phase diagram for the proposed model in reduced temperature T r 5T/T c and density b r units. ~a! a m 50. ~b! a m .0. The lower dome in part ~b! locates the metastable fluid–fluid coexistence curve.

b r c5

Recently, ten Wolde and Frenkel linked the presence of a metastable fluid–fluid critical point to the observed enhancement of crystal nucleation in solutions of globular proteins under certain conditions.9 Their computer simulations showed that this phenomenon can induce a drastic change in the pathway for the formation of a critical nucleus. Hence, in this work we focus on understanding the nucleation behavior under such conditions. Following ten Wolde and Frenkel we analyzed the structure of critical nuclei at different undercoolings DT r or different supersaturations Dm, by determining the number of solidlike particles belonging to a given crystal nucleus, defined by

1 8a and kT c 5 , 3 27b

and these features characterize the fluid–fluid coexistence curve in the phase diagrams of our model as depicted in Fig. 1. Here, we have chosen a 1 51/4, a 2 52, and a 3 53/10 for our calculations, and studied the effect of an increasing value of a m ~mimicking a decreasing range of the effective attractive interactions between dissolved molecules!. In this and the following figures, densities are expressed in dimensionless units of b 21 and all temperatures are referred to the van der Waals critical value (T r 5T/T c ). In Fig. 1~a! (a m 50) the horizontal solid line locates a triple point where the solid phase coexists with the dilute and dense fluid phases. An increasing value of a m drives the triple point towards the fluid–fluid critical point until the only stable coexisting phases are fluid and solid. Beyond this point, however, the fluid–fluid coexistence curve survives in the metastable regime below the fluid–solid coexistence curve @Fig. 1~b!#. This behavior is in qualitative agreement with that observed in computer simulations of colloidal systems with varying ranges of attractive interactions between particles.3

FIG. 3. Parametric variation of the structural order parameter m with the average density r through the critical nucleus for liquidlike ~T r 51.0, D m /kT c 50.9074! and crystal-like ~T r 51.2, D m /kT c 51.3675! clusters. The nuclei are similar at the center but differ sharply as they contact the surrounding fluid.


226



FIG. 4. Number of solidlike particles N c in critical clusters as a function of the excess number of particles N. The reduced temperature T r 51.025 is kept constant in these calculations performed at different values of a m .

N c5

E

drm ~ r!@ r ~ r! 2 r f # ,

~7!

as a function of the excess number of particles N5

E

dr@ r ~ r! 2 r f # .

~8!

For temperatures well above the metastable critical point, our model predicts that the number of solidlike particles in the critical nucleus, N c , increases in proportion to the excess number N, as the supersaturation Dm decreases towards the binodal ~see results for T r 51.2 in Fig. 2; K r 5a/2 and K m 5a/8 in all our calculations!. These results also provide useful information about the basic features of growing clusters along the lowest free-energy path to the critical nucleus at the particular working temperature. If we assume that the internal structure of a cluster largely depends on the interactions among its particles, but not on the properties of the surrounding fluid, it is reasonable to expect that the properties of all density fluctuations in a given state of supersaturation can be derived from those of clusters in equilibrium with their fluid at other Dm.12 From this point of view, results in Fig. 2 should be similar to those obtained by following the lowest free-energy path to the critical nucleus. As the temperature decreases towards the critical value T c , the structure of the clusters starts changing. Critical nuclei of intermediate size develop an interfacial fluid layer ~note the long tail in liquidlike clusters in Fig. 3!. At higher temperatures ~crystal-like clusters in Fig. 3!, the structural order parameter increases first with almost no change in the average density ~note the nearly vertical slope in Fig. 3!. The number of solid particles in critical nuclei in this supersaturation range increases slowly with N, up to a critical size where crystallization is strongly enhanced ~see results for T 51.025 in Fig. 2!. In accordance with the computer simulations of ten Wolde and Frenkel,9 our results suggest that, close to the critical temperature T c , the first step towards the critical nucleus is the formation of a liquidlike cluster. Then, at a certain critical size, the crystallite grows at a fast pace

FIG. 5. Number of solidlike particles N c in critical clusters as a function of the excess number of particles N in a system with a m 54.0. The supersaturation Dm decreases continuously as we move along these curves to higher values of N c . D m /kT c 50.9045 at the metastable critical point at T r 51.0.

inside the liquidlike droplet. Beyond this point, the increase in N c is proportional to the increase in the excess number of particles N. This particular behavior, characteristic of temperatures above but close to the fluid–fluid critical point, is very sensitive to the structure of the phase diagram. Figure 4 shows that there is a set of intermediate values of a m , for which the formation of a fluid ‘‘wetting’’ layer affects the internal structure of the critical nuclei across a wider range of sizes. At higher values of this parameter ~a m 55.0 in Fig. 4!, where the equilibrium crystallization curve further separates from the metastable critical point, small clusters become essentially liquidlike, up to a region where different supersaturations lead to critical clusters of similar size but different internal structure ~liquidlike or solidlike!. Beyond this point, nuclei tend to be more crystal-like than equivalent clusters of particles with longer-range attractive interactions. However, the proximity of the metastable critical point and the equilibrium crystallization curve for low values of a m can hinder the appearance of liquidlike clusters at all supersaturations, as suggested by the case a m 53.0 in Fig. 4. The description of the structure of critical nuclei at the critical temperature T c deserves particular attention and is summarized in Fig. 5. At this temperature, small clusters also exhibit the formation of a fluid wetting layer as the supersaturation is reduced. The thickness of the interfacial layer increases as Dm approaches the location of the metastable critical point, where it diverges; the height of the nucleation barrier at this particular point remains finite, but exhibits an infinite slope as a function of Dm. Although the excess number of particles N grows without limit at the divergence, the number of solidlike particles remains essentially constant under such conditions ~see Fig. 5!. For lower supersaturations, the wetting layer starts shrinking; the critical nucleus decreases in size and extension up to a point where crystallization starts being favored. From this point on, the increase in N c is proportional to N as one moves towards the binodal. At temperatures lower than T c , the thickness of the wet-




FIG. 6. The barrier to crystal nucleation DV/kT at different reduced temperatures around the metastable critical point. Results obtained by keeping the classical work of formation DV cl /kT5150 in Eq. ~9! for a system with a m 54.0.

ting layer in crystal-like clusters diverges as the dilute branch of the spinodal is approached from the fluid–solid binodal. However, smaller crystal-like nuclei are allowed to form under these conditions. Beyond the fluid spinodal there is a sudden change in the nature of the metastable phase ~dense fluid!, and small liquidlike clusters start developing ~see Fig. 5!. At these lower temperatures, the homogeneous nucleation of dense fluid droplets can compete with crystal nucleation. In fact, this process is always energetically favored at some point before reaching the fluid spinodal ~where the work of formation DV of the fluid nuclei is equal to zero!. C. The nucleation barrier

Ten Wolde and Frenkel showed that the height of the barrier to crystal nucleation, close to the critical temperature T c , is expected to be much lower than at either higher or lower temperatures.9 This fact can be verified with our model by evaluating the work of formation of a critical nucleus, DV, in systems with a comparable degree of undercooling. In particular, we chose states such that the classical barrier to nucleation DV cl5

16p g 3 3 r 2s D m 2

~9!

227

is equal to 150kT at every temperature T. The surface tension g in this relation can be evaluated in a consistent way through Eq. ~1!, and the associated Euler–Lagrange equations for a planar solid–fluid interface. The behavior of the nucleation barrier as predicted by our model for temperatures around T c is shown in Fig. 6. The minimum in this curve occurs at temperatures very slightly above the critical value. It should be pointed out that the internal structures of those nuclei on the branches on either side of the minimum are remarkably different. At higher temperatures, nucleation proceeds through the formation of crystal-like clusters, while at lower temperatures liquidlike clusters with an extended wetting layer are favored. This is an important result because if the presence of a metastable critical point is to be used to selectively speed up the rate of crystal nucleation, but not the rate of crystal growth, experiments should be carried out at temperatures close to, but just above the metastable critical point. Deeper quenches will increase crystal growth more than nucleation rates.

ACKNOWLEDGMENTS

We thank Dr. Yu Chen Shen for many helpful discussions. This work was supported by the National Science Foundation through Grant No. CHE94-22999. Support to V. Talanquer from the Facultad de Quı´mica at UNAM is also gratefully acknowledged.

1

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