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Apr 26, 2013 - ... to a Metastable Condensation Transition in Polymer–CO2 Mixtures .... C. Forest , P. Chaumont , P. Cassagnau , B. Swoboda , P. Son...
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Discontinuous Bubble Nucleation Due to a Metastable Condensation Transition in Polymer−CO2 Mixtures Xiaofei Xu,† Diego E. Cristancho,‡ Stéphane Costeux,‡ and Zhen-Gang Wang*,† †

Division of Chemistry and Chemical Engineering, California Institute of Technology, Pasadena, California 91125, United States The Dow Chemical Company, Midland, Michigan 48674, United States

‡

S Supporting Information *

ABSTRACT: We combine a newly developed density-functional theory with the string method to calculate the minimum free energy path of bubble nucleation in compressible polymer−CO2 mixtures. Nucleation is initiated by saturating the polymer liquid with high pressure CO2 and subsequently reducing the pressure to ambient condition. Below a critical temperature, we find that there is a discontinuous drop in the nucleation barrier with increased initial CO2 pressure, as a result of an underlying metastable transition from a CO2-rich-vapor phase to a CO2-rich-liquid phase. This phenomenon is different from previously proposed nucleation mechanisms involving metastable transitions.

SECTION: Glasses, Colloids, Polymers, and Soft Matter

T

Bubble nucleation in polymer−CO2 mixtures is a problem of great practical interest to manufacturing of polymer foams since it plays a crucial role in determining the cell size and pore density of the foam material.11 A quantitative theory that predicts the bubble nucleation behavior in polymer−CO2 mixtures as a function of the pressure, temperature, and composition can be a valuable tool for rationally controlling the foaming conditions. As a fundamental phenomenon, the study of bubble nucleation in a binary fluid mixture is complicated by the finite compressibility of the mixture and possible interplay between liquid−vapor transition and liquid−liquid phase separation.10,12,13 We have recently developed a density functional theory (DFT) that can quantitatively describe the thermodynamic bulk and interfacial properties of polymer-CO2 mixtures in a wide temperature and pressure range.14 In this Letter, we combine this DFT with the string method15 to make quantitative predictions for bubble nucleation in polymer−CO2 mixtures, using poly(methyl methacrylate) (PMMA) as an example for the polymer. We focus on the experimentally relevant temperature and pressure conditions, i.e., temperatures near the critical temperature of CO2, and pressures in the range from ambient pressure to tens of mega Pascals (MPa). A full account of the theory as well as more extensive discussions of the results, including applications to other polymer−CO2 mixtures, will be presented elsewhere. For interested readers, some details are also provided in the Supporting Information.

he role of metastable, intermediate phases in the nucleation of stable phases has been of abiding interest in the crystal growth and nucleation community1,2 since the early work of Oswald.3 In more recent years, considerable attention has been focused on how nucleation in a main phase transition is affected by a second, metastable phase transition.1,4−7 Computer simulation by ten Wolde and Frenkel4 and subsequent density functional calculation by Talanquer and Oxtoby8 showed that crystal nucleation in colloidal systems with short-range attractions can be significantly enhanced by density fluctuation near a metastable liquid−liquid critical point. Experimentally, Galkin and Vekilov5 demonstrated that in some protein solutions the crystal nucleation rate passes through a maximum in the vicinity of the metastable liquid−liquid coexistence. Nucleation below the metastable critical point was examined in more detail by Tavassoli and Sear9 using an Ising-type phenomenological model, who found a discontinuous jump in the nucleation barrier upon crossing the metastable transition line. In all these studies, the metastable phase transition is between the initial metastable parent phase and another intermediate phase. Here, we report a different type of phenomenon where the metastable phase transition is between dif ferent states of the nucleus. We use the example of a specific molecular system, polymer−CO2 mixtures, for which we predict that discontinuous nucleation can occur as a result of a metastable condensation transition in the CO2-rich nucleus10 and that this behavior should be observable under realistic experimental conditions. We show that the discontinuity is connected to a buried spinodal in the metastable state. © 2013 American Chemical Society

Received: March 11, 2013 Accepted: April 26, 2013 Published: April 26, 2013 1639

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{ρj(r, 1)}(a well-developed bubble) via a path that passes through the transition state (saddle point on the free energy surface). It is defined such that the tangent along the path is parallel to the free energy gradient,

The DFT we use for the polymer−CO2 is built on a coarsegrained molecular model in which the CO2 molecule is modeled as a sphere, and the polymer is modeled as a freely jointed chain of tangentially connected spheres. The excluded volume of the species is represented by hard-core interactions. Energetic interactions are described by the attractive part of the Lennard-Jones potential. In addition, a weak association interaction is included between the CO2 molecules. The DFT is constructed from the perturbed-chain-statistical-associatingfluid-theory equation of state,16 with a Helmholtz free energy functional expressed as a sum of an ideal-gas term and excess terms accounting for contributions from excluded volume effects, correlations due to chain connectivity, and association and dispersion interactions. The details for the molecular model and free energy functional are given in our recent publication.14 In the present work, we take our polymer to be PMMA of molecular weight 89230 g/mol, which is modeled with 2855 identical units. For sufficiently long chains, the thermodynamic behavior becomes insensitive to chain lengths, so we take this single value of the chain length as representative of long-chain PMMA. Inasmuch as bubble nucleation can be considered an activated rare event, we adopt a thermodynamic approach in which nucleation is treated as a localized fluctuation in an open system with a fixed chemical potential for each species set by the metastable bulk phase. We consider the nucleation of a single spherical CO2 bubble from the bulk PMMA−CO2 mixture, with the origin of the coordinate taken to be at the center of bubble (see Figure 1). Because of spherical symmetry, the density profiles are only functions of the radial coordinate.

⎧ ⎛ ⎞ ⎪ δW − ⎜⎜τ1∗ δW ⎟⎟τ1 = 0 ⎪ ⎝ δρ1 ⎠ ⎪ δρ1 ⎨ ⎛ δW ⎞ ⎪ δW ⎟⎟τ2 = 0 ⎪ − ⎜⎜τ2∗ ⎪ δρ ⎝ δρ2 ⎠ ⎩ 2

(1)

where τj(r , s) =

∂ρj (r , s) ∂s

⎛ ∂ρ (r , s) ∂ρ (r , s) ⎞1/2 j ⎜⎜ j ⎟ * ∂s ⎟⎠ ⎝ ∂s

is the normalized tangent along the path and ∗ denotes the inner product defined as f∗g = ∫ f(r)g(r) dr. Equation 1 is then solved by using the string method,15 which is a modified steepest descent algorithm, ⎧ ⎡ ⎛ ⎞ ⎤ ⎪ ∂ρ1(r , s ; t ) = −⎢ δW − ⎜τ ∗ δW ⎟τ ⎥ + k τ ⎜1 ⎟1 11 ⎪ ∂t ⎢⎣ δρ1 ⎝ δρ1 ⎠ ⎥⎦ ⎪ ⎨ ⎡ ⎪ ∂ρ (r , s ; t ) ⎛ δW ⎞ ⎤ δW ⎪ 2 ⎟⎟τ2 ⎥ + k 2τ2 = −⎢ − ⎜⎜τ2∗ ⎪ ∂t ⎢⎣ δρ2 ⎝ δρ2 ⎠ ⎥⎦ ⎩

(2)

where k1 and k2 are Lagrange multipliers introduced to enforce the particular parametrization of the string one chooses (for example, by normalizing its arc length),17 and t is a fictitious time for evolving the equations on the free energy landscape. We adopt an explicit forward time splitting to solve eq 2. The density profile of the terminal state is determined by a constrained method18 of minimizing the grand potential by specifying the bubble radius for a given value of the density of one of the component, e.g., the PMMA (see section II of the Supporting Information). The iteration starts with a set of initial density profiles (t = 0) between the initial state (s = 0, the uniform metastable bulk) and the terminal state (s = 1, a well developed bubble). States between s = 0 and s = 1 are obtained by linear interpolation. After each iteration, we reparametrize the states of density profile equidistantly along the path. The process ends when the maximum difference of free energy along the path between two consecutive iterations is less than 10−5. Nucleation is initiated by a pressure drop from a high initial pressure P0 at the coexistence (i.e., a CO2 saturated polymer liquid) to the ambient pressure 0.1 MPa. The density is allowed to relax, while maintaining the same composition (weight fraction of CO2) at the new pressure value, resulting in a metastable bulk state highly supersaturated with CO2. The initial pressure P0 thus controls the composition of the metastable state and its degree of supersaturation. The terminal state is a well developed bubble whose center densities are determined by equality of chemical potential with the bulk metastable phase

Figure 1. Illustration of a nucleated CO2 bubble in the metastable bulk PMMA−CO2 mixture.

The key properties of interest are the free energy barrier (the free energy at the transition state) and the nucleation path. In a mean-field framework, nucleation proceeds along the minimum free energy path (MFEP) on the functional surface of grand potential W[ρ1(r), ρ2(r)], where ρ1 and ρ2 are the number densities of CO2 and PMMA segments, respectively. The MFEP {ρj(r, s)|j = 1, 2; 0 ≤ s ≤ 1}(where s is the normalized reaction coordinate on the path) connects the initial state {ρj(r, 0)}(the metastable bulk state of the mixture) and the final state

μj (ρ1center , ρ2center ) = μj (ρ1bulk , ρ2bulk ),

j = 1, 2

(3)

In the region near the critical temperature of CO2, a CO2rich liquid (CRL) and a CO2-rich vapor (CRV) can coexist with the polymer-rich phase, giving rise to an equilibrium triple 1640

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line.19 The effects of this equilibrium triple line on nucleation in generic binary mixtures 20 and in model polymer−CO 2 mixtures10 have been previously elucidated. However, in the CO2 supersaturated metastable polymer-rich phase at ambient pressure, at the same respective chemical potential for CO2 and PMMA, there can also exist metastable CRV and CRL phases whose densities are determined by the same equation as eq 3. These metastable phases are lower in their grand potential density than the parent metastable phase (which is just the negative of the ambient pressure, i.e., −0.1 MPa). Since the initial stage of nucleation occurs at constant chemical potential, it is these metastable phases, rather than the equilibrium CRV at the ambient pressure, that form the nuclei of the initial bubble nucleation. Figure 2a shows the grand potential density

phase; the metastable CRV and CRL do not exist by themselves. It is also important to note that the solubility of CO2 in the metastable polymer-rich phase remains a continuous function of pressure at these temperatures (see Figure S2 of the Supporting Information), i.e., there is no phase transition in the bulk parent phase. This feature is essentially different from previously proposed nucleation mechanisms2,4,5,8,9 involving a metastable transition that occurs in the initial bulk parent phase. The “phase diagram” shown in Figure 2b provides a global map for the nucleation behavior. For T ≥ Tc, the mixture nucleates to a supercritical CO2-rich bubble. For T < Tc, below the metastable spinodal of the CRL, CRL does not exist, and so nucleation can only be to the CRV state. In the region above the metastable spinodal of the CRV, the mixture can only nucleate a CRL bubble. In the region between these two spinodals, both CRL and CRV phases can exist as metastable states having lower free energy than the parent metastable phase. The nucleating bubble can therefore correspond to either the CRV or the CRL phase; the actual nucleation path is the one of the lowest free energy barrier. We have calculated the nucleation properties to both CRV and CRL. The results are shown in Figure 3a,b. Clearly, for all P0, nucleation to the

Figure 2. (a) Grand potential density of the metastable CO2-rich phases determined from the equal chemical potential condition with the metastable parent PMMA-rich liquid at the ambient pressure 0.1 MPa for T = 301 K (blue lines), 304 K (green lines), 307.6 K (red lines), and 310 K (dark green lines). The solid lines at T = 301K and 304 K are CRV phases and the dashed lines are CRL phases, respectively. The solid circles denote the metastable condensation points, and the solid square is the metastable critical point. (b) Phase diagram for the metastable CO2-rich phases in the T − P0 plane at the nucleation pressure of 0.1 MPa.

Figure 3. (a,b) Free energy barrier and radius of the critical nucleus as a function of the initial pressure. The colors of the curves correspond to the same temperatures as in Figure 2a. The solid circles and solid square also denote the same corresponding points as in Figure 2a. (c) The free energy barrier as a function of temperature for P0 = 39.45 MPa.

(i.e., negative of the pressure) of these two metastable phases as a function of the initial pressure P0. At T = 301 K, the CRV crosses the CRL phase at P0 = 38.5 MPa, corresponding to a metastable condensation transition. The CRV phase persists until P0 ≤ 45.1 MPa, and the CRL phase starts to appear at P0 ≥ 33.2 MPa. These pressures define the respective spinodals for the CRV and CRL phases. Between these spinodals, both CRV and CRL phases can exist. As temperature increases, the spinodal pressure of the CRV decreases, while that of the CRL increases, until they converge to a critical point at Tc =307.6 K. For T > Tc, the CO2-rich metastable phase becomes supercritical. Figure 2b shows the metastable spinodal lines as well as condensation transition line in the T-P0 plane. We emphasize that this metastable “phase diagram” is only meaningful in the context of the metastable PMMA-rich parent

CRL phase has a lower barrier. While this behavior is consistent with Ostwald’s step rule3 before the condensation point, since the free energy of the CRL is closer to the metastable parent phase than CRV, this rule does not apply after the condensation point. Exceptions to Ostwald’s rule have also been reported in the crystallization of minerals.21,22 To understand the difference between nucleation to the CRV branch and nucleation to the CRL branch, we examine the evolution of the dimensionless density (packing fraction) profiles along the MFEP. The packing fraction is defined as ηi = 1641

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depression of the PMMA, even though the final, well-developed bubble has a lower CO2 density than in the PMMA-rich parent phase. Thus, the first stage in the nucleation is an overshoot of the density of CO2. The center CO2 density then begins to decrease beginning at panel c. Looking at the free energy values with these profiles, it is clear that the steepest increase in the free energy on its way to the critical nucleus (panel f) is the formation of the low density CO2 hole in an overly dense CO2 droplet. The density of the PMMA in the nucleus, on the other hand, decreases monotonically. The nucleation to the CRV branch thus seems to involve two stages: first, the formation of a small liquid-like nucleus, then the formation of a vapor-like core inside the liquid-like droplet when the droplet becomes sufficiently large. Note also that there is a wetting layer of the higher-density CRL at the interface of the nucleus, whose role is similar to that for the crystallization in the presence of a buried liquid−liquid transition.4,8 In contrast to nucleation to the CRV branch, the MFEP for the formation of the CRL bubble is simpler, as shown in Figure 5. Here, the density of CO2 in the nucleus increases monotonically to a value that is higher than the density in the bulk phase. Initially, the CO2 density peaks at the center. However, as the critical nucleus is approached, there is a redistribution of the CO2, giving a slight enrichment at the interface of the nucleus. As shown in Figure 3a, for temperatures below the metastable critical point, the nucleation barrier to the CRL branch is always lower than to the CRV branch, and so the preferred nucleation path is the one to the CRL branch. This suggests the following scenario for the nucleation behavior in the mixture. For initial pressure P0 below the spinodal value of the CRL phase, only a CRV nucleus is possible, which, however, requires a rather high nucleation barrier, making bubble nucleation an unlikely event. Upon increasing P0 to the spinodal of the CRL, there is a precipitous drop in the nucleation barrier, as the nucleating phase now becomes CRL. In other words, we expect a sudden increase in the nucleation rate as a function of the initial pressure P0. We note that the location of this discontinuity for the PMMA-CO2 system is within the range of experimental pressures for polymer foaming.23 Therefore, the phenomenon predicted here should be readily observable. Because the equilibrium coexisting CO2-rich phase at the nucleation pressure (i.e., 0.1 MPa) is a vapor, the CRL nucleus will eventually turn into CRV. This process may take the form of another nucleation of the CRV from a well developed CRL bubble, which, however, will require another barrier. The more likely scenario is that as the CRL bubble grows, depletion of CO2 from the polymer-rich parent phase becomes significant, and with the decrease in the chemical potential of CO2, the CRL bubble evaporates to become CRV, with no barrier or a much reduced barrier. Since the process involves the growth phase and a changing chemical potential, it can no longer be treated within the existing thermodynamic framework in which nucleation takes place at constant reservoir conditions supplied by the bulk metastable parent phase. Regardless of how the CRV appears in the CRL bubble, bubble formation for initial pressures exceeding the spinodal value of the CRL will be a two-step process, first the formation of a CRL nucleus and then the transformation of CRL into a CRV. Two-step nucleation processes have also been suggested for a number of systems.2 While going through the CRL branch leads to a faster nucleation in the formation of the bubbles, the resulting gas

(π/6)ρiσ3i , where σi is the diameter of CO2 molecules (i = 1) or PMMA segments (i = 2). Figures 4 and 5 show, respectively,

Figure 4. (a−g) Evolution of the dimensionless density profiles for a vapor-like bubble along the MFEP for P0 = 38.5 MPa and T = 301 K. The weight fraction of CO2 in the parent bulk phase is 28.20%. The excess free energy (relative to the metastable bulk) is shown above each panel. Panel f is the critical nucleus. (h) Excess free energy of bubble formation (ΔW = W − Wbulk) as a function of the PMMA bulk 2 deficiency defined as V2 = −4πσ32∫ ∞ 0 drr [ρ2(r) − ρ2 ]. The solid circles correspond to the state of the nucleus shown in panels a−g.

Figure 5. (a−g) Evolution of the dimensionless density profiles for a liquid-like bubble along MFEP at P0 = 38.5 MPa and T = 301 K. The weight fraction of CO2 in the parent bulk phase is 28.20%. Panel f is the critical nucleus. (h) Excess free energy of bubble formation as a function of the PMMA deficiency.

the evolution of a vapor-like and liquid-like bubble along MFEP at the metastable condensation point (P0 = 38.5 MPa) at T = 301 K, where the CRV and CRL have the same free energy (grand potential density). The number above each panel is the free energy at that state. For nucleation to the CRV branch, the bubble starts as a density enhancement of CO2 and a density 1642

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(5) Galkin, O.; Vekilov, P. G. Control of Protein Crystal Nucleation Around the Metastable Liquid−Liquid Phase Boundary. Proc. Natl. Acad. Sci. U. S. A. 2000, 97, 6277−6281. (6) Chung, S. Y.; Kim, Y. M.; Kim, J. G.; Kim, Y. J. Multiphase Transformation and Ostwald’s Rule of Stages during Crystallization of a Metal Phosphate. Nat. Phys. 2009, 5, 68−73. (7) Olmsted, P. D.; Poon, W. C. K.; McLeish, T. C. B.; Terrill, N. J.; Ryan, A. J. Spinodal-Assisted Crystallization in Polymer Melts. Phys. Rev. Lett. 1998, 81, 373−376. (8) Talanquer, V.; Oxtoby, D. W. Crystal Nucleation in the Presence of a Metastable Critical Point. J. Chem. Phys. 1998, 109, 223−227. (9) Tavassoli, Z.; Sear, R. P. Homogeneous Nucleation Near a Second Phase Transition and Ostwald’s Step Rule. J. Chem. Phys. 2002, 116, 5066−5072. (10) Müller, M.; MacDowell, L. G.; Virnau, P.; Binder, K. Interface Properties and Bubble Nucleation in Compressible Mixtures Containing Polymers. J. Chem. Phys. 2002, 117, 5480−5496. (11) Nawaby, A. V.; Handa, Y. P.; Liao, X.; Yoshitaka, Y.; Tomohiro, M. Polymer−CO2 Systems Exhibiting Retrograde Behavior and Formation of Nanofoams. Polym. Int. 2007, 56, 67−73. (12) Binder, K.; Müller, M.; Virnau, P.; MacDowell, L. G. Polymer Plus Solvent Systems: Phase Diagrams, Interface Free Energies, and Nucleation. Adv. Polym. Sci. 2005, 173, 1−104. (13) Zeng, X. C.; Oxtoby, D. W. Binary Homogeneous Nucleation Theory for the Gas−Liquid Transition - A Nonclassical Approach. J. Chem. Phys. 1991, 95, 5940−5947. (14) Xu, X. F.; Cristancho, D. E.; Costeux, S.; Wang, Z.-G. DensityFunctional Theory for Polymer−Carbon Dioxide Mixtures: A Perturbed-Chain SAFT Approach. J. Chem. Phys. 2012, 137, 054902. (15) E, W. N.; Vanden-Eijnden, E. Transition-Path Theory and PathFinding Algorithms for the Study of Rare Events. Annu. Rev. Phys. Chem. 2010, 61, 391−420. (16) Gross, J.; Sadowski, G. Perturbed-Chain SAFT: An Equation of State Based on a Perturbation Theory for Chain Molecules. Ind. Eng. Chem. Res. 2001, 40, 1244−1260. (17) E, W. N.; Ren, W.; Vanden-Eijnden, E. Simplified and Improved String Method for Computing the Minimum Energy Paths in BarrierCrossing Events. J. Chem. Phys. 2007, 126, 164103. (18) Wood, S. M.; Wang, Z.-G. Nucleation in Binary Polymer Blends: A Self-Consistent Field Study. J. Chem. Phys. 2002, 116, 2289−2300. (19) Xu, X. F.; Cristancho, D. E.; Costeux, S.; Wang, Z.-G. DensityFunctional Theory for Polymer−Carbon Dioxide Mixtures. Ind. Eng. Chem. Res. 2012, 51, 3832−3840. (20) Granasy, L.; Oxtoby, D. W. Cahn−Hilliard Theory with TripleParabolic Free Energy. I. Nucleation and Growth of a Stable Crystalline Phase. J. Chem. Phys. 2000, 112, 2399−2409. (21) Deeiman, J. C. Breaking Ostwald’s Rule. Chem. Erde:Geochem. 2001, 61, 224−235. (22) Hedges, L. O.; Whitelam, S. Limit of Validity of Ostwald’s Rule of Stages in a Statistical Mechanical Model of Crystallization. J. Chem. Phys. 2011, 135, 164902. (23) Ruiz, J. A. R.; Pedros, M.; Tallon, J. M.; Dumon, M. Micro and Nano Cellular Amorphous Polymers (PMMA, PS) in Supercritical CO2 Assisted by Nanostructured CO2-philic Block Copolymers - One Step Foaming Process. J. Supercrit. Fluids 2011, 58, 168−176. (24) van Konynenburg, P. H.; Scott, R. L. Critical Lines and PhaseEquilibria in Binary van der Waals Mixtures. Phil. Trans. R. Soc. A 1980, 298, 495−540.

bubbles can be quite large since the CRV is formed at a later stage. In order to produce small bubbles with high density, Figure 3c suggests that it is better to induce nucleation in the supercritical states, where the nucleation barrier decreases smoothly with increasing initial pressure and the critical nucleus sizes are relatively small. In conclusion, we have combined a newly developed DFT with the string method for calculating the MFEP to investigate bubble nucleation in PMMA−CO2 mixture induced by saturating the PMMA with CO2 and subsequently decreasing the pressure to ambient conditions. In the vicinity of a metastable condensation of the CO2-rich phase, the nucleated state jumps discontinuously from the CRV branch to the CRL branch with a precipitous reduction in the free energy barrier and the radius of critical nucleus, as a function of increased initial pressure. The location of this discontinuity coincides with the spinodal of the metastable CRL phase, and is within the experimental range of pressures used for foaming. Because of the intervention of the CRL, the formation of CO2-rich gas bubbles will be a two-step process. The quantitative accuracy of our DFT enables specific numerical predictions that can be directly tested by experiments. Our work also suggests that for the purpose of producing nanocelluar polymer foams, the nucleation temperature should be above the critical temperature of the metastable condensation. Although our results are obtained for a specific system, the phenomenon of a metastable transition in the state of the nucleus should be a general one within the same universality class24 to which the CO2−polymer mixtures belong. We therefore expect similar nucleation mechanisms to be manifested in many different systems.

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ASSOCIATED CONTENT

S Supporting Information *

A brief description of the molecular model, key ingredients of the DFT, the numerical procedures, and solubility data are provided in the Supporting Information. This material is available free of charge via the Internet at http://pubs.acs.org.

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AUTHOR INFORMATION

Corresponding Author

*Electronic mail: [email protected]. Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS The Dow Chemical Company is acknowledged for funding and for permission to publish the results. The computing facility on which the calculations were performed is supported by an NSFMRI grant, Award No. CHE-1040558.

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REFERENCES

(1) Sear, R. P. Nucleation: Theory and Applications to Protein Solutions and Colloidal Suspensions. J. Phys.: Condens. Matter 2007, 19, 033101. (2) Vekilov, P. G. Two-Step Mechanism for the Nucleation of Crystals from Solution. J. Cryst. Growth 2005, 275, 65−76. (3) Ostwald, W. Studies of the Formation and Conversion of Solid Body. Z. Phys. Chem. 1897, 22, 289−330. (4) ten Wolde, P. R.; Frenkel, D. Enhancement of Protein Crystal Nucleation by Critical Density Fluctuations. Science 1997, 277, 1975− 1978. 1643

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