R EPORTS

of correlation functions yield coefficient of determi- where ⌫ is the gamma function, defined as ⌫(n) ϭ References and Notes 2 ͐ ϱ Ϫx nϪ1 1. C. A. Angell, K. L. Ngai, G. B. McKenna, P. F. McMillan, nation R values greater than 0.99 in most cases. 0 e x dx. S. W. Martin, J. Appl. Phys. 88, 3113 (2000). Standard errors for the fitting parameters tend to be 36. The orientational transient can be calculated from 5 2. M. D. Ediger, Annu. Rev. Phys. Chem. 51, 99 (2000). extremely small (1 part in 10 ) but provide little the polarized fluorescence data with 3. H. Sillescu, J. Non-Cryst. Solids 243, 81 (1999). meaningful basis for comparison because of the large ͑ ͒ ␤ ␶ Ϫ // t covariance of and . A useful measure of this error ␪͑t͒ ϭ tan 1ͫͱ ͬ 4. M. D. Ediger, C. A. Angell, S. R. Nagel, J. Phys. Chem. ␤ Ќ͑t͒ 100, 13200 (1996). determines that a 1% variation in or a 5% variation in ␶ is sufficient to reduce the R2 value from 0.99 to Molecules were determined to have exchanged environ- 5. J. T. Fourkas, D. Kivelson, U. Mohanty, K. A. Nelson, ␶ Eds., Supercooled Liquids: Advances and Novel Appli- 0.98. The rotational correlation time C is defined as ments when the average angle jump changes by more cations, vol. 676 of ACS Symposium Series (American the integral from zero to infinity of the correlation than 2 SD from the previous average angle jump. Chemical Society, Washington, DC, 1997). function, but for a stretched exponential function, 37. C. Bennemann, C. Donati, J. Baschnagel, S. C. Glotzer, this integral can be conveniently written in terms of Nature 399, 246 (1999). 6. G. Harrison, The Dynamic Properties of Supercooled ␶ ␤ Liquids (Academic Press, New York, 1976). the and parameters as 38. D.A.V.B. is a Research Corporation Cottrell Scholar and a Camille and Henry Dreyfus Foundation New 7. T. Gleim, W. Kob, K. Binder, Phys. Rev. Lett. 81, 4404 ϱ Faculty Awardee. This work is supported by grants (1998). ␶ 1 8. D. B. Hall, A. Dhinojwala, J. M. Torkelson, Phys. Rev. ␶ ϭ ͵ ͑ ͒ ϭ KWW ⌫ ͩ ͪ from the Research Corporation. C C t dt ␤ ␤ Lett. 79, 103 (1997). KWW KWW 9. G. Adam, J. H. Gibbs, J. Chem. Phys 43, 139 (1965). 0 10 October 2000; accepted 28 February 2001 10. H. Wendt, R. Richert, Phys. Rev. E. 61, 1722 (2000). 11. E. R. Weeks, J. C. Crocker, A. C. Levitt, A. Schofield, D. A. Weitz, Science 287, 627 (2000). 12. C.-Y. Wang, M. D. Ediger, J. Phys. Chem. B 104, 1724 (2000). 13. I. Chang, H. Sillescu, J. Phys. Chem. B 101, 8794 Real-Space Imaging of (1997). 14. K. L. Ngai, J. Phys. Chem. B 103, 10684 (1999). 15. R. Richert, J. Phys. Chem. B 101, 6323 (1997). Nucleation and Growth in 16. E. V. Russell, N. E. Israeloff, Nature 408, 695 (2000). 17. W. E. Moerner, Science 265, 46 (1994). 18. P. F. Barbara, W. E. Moerner, Acc. Chem. Res. 29, 561 Colloidal Crystallization (1996). 1 1 2 2 19. H. P. Lu, X. S. Xie, Nature 385, 143 (1997). U. Gasser, * Eric R. Weeks, † Andrew Schofield, P. N. Pusey, 20. S. Nie, D. T. Chiu, R. N. Zare, Science 266, 1018 D. A. Weitz1 (1994). 21. D. A. Vanden Bout et al., Science 277, 1074 (1997). 22. R. M. Dickson, A. B. Cubitt, R. Y. Tsien, W. E. Moerner, Crystallization of concentrated colloidal suspensions was studied in real space Nature 388, 355 (1997). with laser scanning confocal microscopy. Direct imaging in three dimensions 23. T. Basche, S. Kummer, C. Braeuchle, Nature 373, 132 (1995). allowed identification and observation of both nucleation and growth of crys- 24. The single-molecule environment is probe depen- talline regions, providing an experimental measure of properties of the nucle- dent, with its size determined by the dye used. ating crystallites. By following their evolution, we identified critical nuclei, 25. J. J. Macklin, J. K. Trautman, T. D. Harris, L. E. Brus, determined nucleation rates, and measured the average surface tension of the Science 272, 255 (1996). 26. The sample is held atop a closed loop piezo scanning crystal-liquid interface. The structure of the nuclei was the same as the bulk stage and illuminated with the second harmonic of a solid phase, random hexagonal close-packed, and their average shape was rather Nd:yttrium-aluminum-garnet laser through a 1.25 nu- nonspherical, with rough rather than faceted surfaces. merical aperture (NA) oil immersion objective in the epi configuration. By scanning the sample in the focus of a laser beam, we can observe the fluorescence from in- The study of the structure, growth, and prop- the surface energy and bulk energy is reflect- dividual molecules. The fluorescence signal from the erties of crystals is one of the most important ed in the free energy for a spherical crystallite molecules is passed through a dichroic beam splitter to areas of solid state physics (1). Although a remove excitation light and split into two orthogonal 4␲ 2 3 polarizations, which are imaged onto the front of two great deal is known about the behavior of ⌬G ϭ 4␲r ␥ Ϫ r ⌬␮n (1) single-photon-counting avalanche photodiodes. For bulk crystals, considerably less is known 3 temperature-dependent data, the sample was held at about their earliest stage, when they nucleate where r is the radius, ␥ is the free energy of constant temperature in an Oxford Microstat-N cryo- and grow from a liquid. Homogeneous nucle- the crystal-liquid interface per unit area, ⌬␮ stat, and the fluorescence was excited and collected through a 0.6-NA long working distance air objective. ation occurs when small crystalline regions is the difference between the liquid and solid All other aspects of the experiment remain unchanged. form from structural fluctuations in a liquid chemical potentials, and n is the number den- 27. The Tg of the polymer was measured by differential cooled below its freezing point. The growth sity of particles in the crystallite (2). The size scanning calorimetry. ϭ ␥ ⌬␮ 28. T. Ha, T. A. Laurence, D. S. Chemla, S. Weiss, J. Phys. of these regions depends on a competition of the critical nucleus is rc 2 /( n), Chem. B. 103, 6839 (1999). between a decrease in bulk energy, which corresponding to the maximum of ⌬G (Eq. 29. R. M. Dickson, A. P. Bartko, J. Phys. Chem. B. 103, favors growth, and an increase in surface 1). Despite its crucial role, very little is 3053 (1999). 30. R. M. Dickson, D. J. Norris, Y.-L. Tzeng, W. E. Moerner, energy, which favors shrinkage. The smallest known about this nucleation process, primar- Science 274, 966 (1996). crystals continually form by fluctuations but ily because of the difficulty of directly ob- 31. Measurement of the full three-dimensional orientation then typically shrink away because of the serving the nuclei in real space. (27, 28) is not required because the dynamics are high surface energy. Growth becomes ener- No experiment has ever directly measured isotropic in three dimensions and thus revealed fully in the fluctuations of the dichroism. Additionally, the use getically favorable only when the crystallites the size of the critical nucleus. It cannot be of a high NA objective (NA ϭ 1.25) leads to a slight reach a critical size. The competition between determined theoretically—even the surface contamination of the dichroism signal due to the out- tension is unknown. Furthermore, an addi- of-plane component of the fluorescence that manifests itself as an additional background but does not adverse- 1Department of Physics and Division of Engineering tional important yet unresolved question in- ly affect the measured dynamics (28). and Applied Sciences, Harvard University, Cambridge, volves the internal structure of critical nuclei, 32. J. A. Veerman, M. F. Garcia-Parajo, L. Kuipers, N. F. MA 02138, USA. 2Department of Physics and Astron- because nucleation need not occur via the van Hulst, Phys. Rev. Lett. 83, 2155 (1999). omy, University of Edinburgh, Edinburgh, Scotland stable bulk phase (3). General arguments 33. T. Ha, J. Glass, T. Enderle, D. S. Chemla, S. Weiss, Phys. EH9 3JZ, UK. Rev. Lett. 80, 2093 (1998). based on symmetry suggest that nucleation 34. G. E. P. Box, G. M. Jenkins, Time Series Analysis Forecast- *To whom correspondence should be addressed. E- may proceed by a body-centered cubic (bcc) ing and Control (Holden-Day, San Francisco, 1976). mail: [email protected] 35. For all correlation functions, the first point containing †Present address: Physics Department, Emory Univer- structure as an intermediate phase, with the the noise is not shown and not included in the fit. Fits sity, Atlanta, GA 30322, USA. final equilibrium solid phase having a differ-

258 13 APRIL 2001 VOL 292 SCIENCE www.sciencemag.org R EPORTS ent symmetry (4). Computer simulations of a manner, but the boundaries of the phase dia- the sample before observation. Heterogeneous Lennard-Jones system found critical nuclei gram are shifted to slightly lower ␾ values (11). nucleation at the walls of the sample chamber with a face-centered cubic (fcc) core and a Colloidal (6–8, 13–16) and protein (17) sys- was suppressed by fusing colloidal particles of predominantly bcc surface layer (5). Howev- tems have been valuable for studying crystal a smaller size onto the walls before adding the er, light-scattering measurements from hard nucleation, but these studies did not directly colloidal samples. Further, we focused at least 8 spheres (6–8), which provide an average visualize the critical nuclei and thus were un- and typically 15 particle diameters from the over crystalline regions of all sizes, suggest able to directly observe their properties. walls to ensure that we studied bulk homoge- that nucleation occurs through the random We report on three-dimensional real- neous nucleation. hexagonal close-packed (rhcp) phase, a ran- space imaging of the nucleation and growth We observed the growth of crystallites dom mixture of hexagonal close-packed of nearly hard-sphere colloidal crystals. We by extracting the particle centers from the (hcp) and fcc-like stackings of planes with observed the formation of the smallest nuclei, raw images with an accuracy of ϳ0.04 ␮m hexagonal order, and this is supported by which shrink far more frequently than they (23). From these particle positions, crystal- recent computer simulations (9). The impor- grow; by measuring their size distribution, line regions were identified with an algo- tant questions about the structure and size of we were able to directly determine the sur- rithm that finds ordered regions, indepen- the nucleating crystallites can best be re- face tension of a colloidal crystal. We were dent of any particular crystal structure (5, solved by direct experimental observation in also able to determine the size of the critical 24). It relies on the assumptions that the real space. nuclei by observing when the probability to neighbors of a particle in a crystal lattice The small length and time scales that char- grow exceeds the probability to shrink. We are arranged in a particular orientation acterize homogeneous crystal nucleation have found that the structure of critical nuclei is about the particle and that this orientation to date precluded this direct real-space observa- rhcp, the same as that of the bulk. The critical of neighbors is the same for nearby parti- tion of critical nuclei in atomic or molecular nuclei have rough surfaces, and their average cles. Quantitatively, this is achieved by systems. By contrast, however, colloidal sys- shape is ellipsoidal. calculating local bond-order parameters tems can be studied directly, because their larg- We used poly(methyl methacrylate) spheres (25) that are based on spherical harmonics er size and concomitant slower time scale sterically stabilized with poly-12-hydroxy- Ylm and are a measure of the orientation of makes them much more experimentally acces- stearic acid (18, 19), with a radius a ϭ 1.26 ␮m the neighbors around a particle. Two adja- sible. Colloidal particles can serve as good and a polydispersity of Ͻ5%. They were sus- cent particles with similar orientation of models for atomic or molecular materials; they pended in a mixture of decahydronaphthalene show an analogous phase diagram (10), and the and cyclohexylbromide, which closely matches volume exclusion that results in crystallization both the refractive index and the density of the of colloidal particles also plays a dominant role particles. To make the particles visible with in traditional crystallization (2). Concentrated fluorescence microscopy, we dyed them with suspensions of hard-sphere colloids crystallize rhodamine, which gives the particles a small ␾ ϭ because of entropy (11): The free volume for charge, resulting in freezing at f 0.38 and ␾ ϭ ordered spheres (maximum packing volume melting at m 0.42 (20). We used a fast laser fraction ␾ϭ0.74) is greater than that of disor- scanning confocal microscope, which allowed dered spheres (maximum packing ␾Ϸ0.64), us to measure the three-dimensional arrange- and thus, the entropy is higher in the crystalline ment of colloidal particles in suspension (21, state. The phase behavior is determined solely 22). We observed a 58 ␮mby55␮mby20␮m by the volume fraction ␾, with freezing occur- volume, containing ϳ4000 particles. The total ␾HS ϭ ϳ ␮ ring at f 0.494 and melting occurring at volume of our samples was 50 l, and they ␾HS ϭ 0.545 (HS, hard sphere) (12). Weakly contained a short piece of paramagnetic wire, m Fig. 2. The difference pg – ps of the probabilities charged colloids behave in essentially the same which we agitated with a magnet to shear melt with which a crystallite is growing or shrinking plotted against the normalized average crystal- lite radius ͗r͘/a (bottom) and the number of Fig. 1. Four snapshots ␾ϭ during crystallization particles M in the crystallite (top) for 0.47. of a sample with ␾ϭ The size of the critical nucleus lies in the range 0.45. The red spheres in which pg – ps goes from negative to positive are drawn to scale and values. represent particles that were identified as crystal-like. The parti- Table 1. The experimentally measured bond-order cles in the metastable histograms for the crystal regions were least liquid state are shown squares fit with the histograms for crystals and by the blue spheres, liquid. For three samples, the fraction of each reduced in size for structure type found by the fit is given, with the clarity. (A) Time t ϭ sum of all the fractions constrained to be 1. The 20 min after shear values in parentheses are the uncertainty of the ϭ last digit. The sum of the squared differences melting, (B) t 43 ⌬2 min, (C) t ϭ 66 min, between the measurement and the fit, ,isa and (D) t ϭ 89 min. measure for the quality of the fit, with smaller values indicating a better fit. The fit for the ␾ϭ 0.45 sample is shown in Fig. 4.

␾⌬2 bcc fcc hcp Liquid

0.49 0.021 0.00(3) 0.10(3) 0.32(6) 0.58(14) 0.45 0.006 0.00(1) 0.58(7) 0.20(3) 0.22(6) 0.43 0.006 0.00(3) 0.34(5) 0.25(4) 0.41(10)

www.sciencemag.org SCIENCE VOL 292 13 APRIL 2001 259 R EPORTS ϭ their neighbors have similar bond-order pa- ticles were present. This is shown in two or shrink (27). Because pg ps at the rameters and are therefore said to be joined early-time snapshots (Fig. 1, A and B), critical size, we plot the difference pg – ps by a crystal-like “bond” (26). Even in the where we represent crystal-like particles as as a function of crystallite radius and par- disordered liquid, crystal-like bonds are not red spheres and liquid-like particles as blue ticle number M in Fig. 2 for a sample with uncommon; thus, only particles with eight spheres, shown with a reduced diameter to ␾ϭ0.47. We found an abrupt change from

or more crystal-like bonds are defined as improve visibility. Typically, these sub- negative to positive values of pg – ps (28), being crystal-like. All particles in a crys- critical nuclei contained no more than 20 allowing us to identify the critical size, tallite with perfect fcc, hcp, rhcp, or bcc particles and shrank to reduce their surface which is 60 Ͻ M Ͻ 160, in good agreement structure are correctly recognized, whereas energy. After a strongly ␾-dependent peri- with recent computer simulations (9). This Ϸ only an insignificant number of particles in od of time, critical nuclei formed and rap- corresponds to rc 6.2a, assuming a spher- the liquid are found to be crystal-like. idly grew into large postcritical crystallites ical nucleus. The volume fraction of the At the beginning of an experiment, sam- (Fig. 1, C and D). By following the time nuclei is larger than the ␾ value of the ples started in the metastable liquid state, evolution of many crystallites, we deter- fluid; above coexistence, the difference is but because of random structural fluctua- mined the size dependence of the probabil- ⌬␾ ϭ 0.012 Ϯ 0.003, independent of ␾, ⌬␾ Ͼ tions, subcritical nuclei of crystal-like par- ities pg and ps with which crystallites grow where increases slightly for M 100. We can understand this ⌬␾ value as result- ing from the higher osmotic pressure exert- Fig. 3. A snapshot of a ed by the fluid on the nuclei (16), whereas crystallite of postcritical ␾ϭ in the coexistence regime, ⌬␾ must reflect size in a sample with ␾ 0.47 is shown from three the evolution of to the higher value, different directions (A ultimately attained by the crystallites, ⌬␾ϭ␾ Ϫ␾ through C). The 206 red where m f. The nucleation rate spheres represent crystal- densities were slower than 5 mmϪ3 sϪ1 for like particles and are ␾Ͻ0.45, as well as for ␾Ͼ0.53. Values drawn to scale; the 243 of the order of 10 mmϪ3 sϪ1 were found for extra blue particles share Ͻ␾Ͻ Ͻ␾Ͻ at least one crystal-like 0.45 0.53. However, for 0.47 “bond” to a red particle 0.53, the average size of the nuclei began to but are not identified as grow immediately after shear melting; thus, crystal-like and are re- there was little time for the sample to equil- duced in size for clarity. ibrate after shear melting, and we were not (D) A cut with a thickness able to observe the formation process of of three particle layers through the crystallite, il- critical nuclei entirely. Our nucleation rate lustrating the hexagonal densities are of the same magnitude as structure of the layers. values obtained by small-angle light scat- Blue, red, and green tering from hard spheres (29, 30) but are spheres represent parti- two orders of magnitude larger than those cles in the different layers obtained from Bragg scattering (31). (front to rear). This cut was taken from the re- The direct imaging afforded by confocal gion that is indicated by microscopy enabled us to determine the the green boxes in (A) and structure and shape of individual nuclei. In (B). The particle diameter in (D) is reduced in order to improve the visibility of the second and third Fig. 3, A through C, we show a crystallite layers. that is slightly larger than the critical size. Again, the red spheres represent crystal- like particles, and the blue ones depict par- Fig. 4. (A) q , q , and 4 6 ticles in the liquid state that are on the w6 bond-order parame- ter histograms for fcc (blue curves), hcp (red curves), bcc (black curves), and liquid (purple curves). (B) The measured bond-order histograms (black plots) of a sample with ␾ϭ 0.45 are shown togeth- er with a least squares fit (blue curve) using the bond-order histo- grams from (A). The re- sults of this and other structure fits are sum- marized in Table 1. Fig. 5. The number of nuclei N(A) (circles) is shown as a function of the nucleus surface area A approximated by an ellipsoid (␾ϭ0.445). From the fit (line) with the function N(A) ϭ ⅐ ␥ constant exp[–A /(kBT)], the surface tension ␥ϭ 2 ␥ 0.026kBT/a is determined. The values of for all samples are given in the inset as a function of ␾.

260 13 APRIL 2001 VOL 292 SCIENCE www.sciencemag.org R EPORTS surface of the crystallite and have at least ellipsoid axes was never higher than 0.65 Ϯ 4. S. Alexander, J. P. McTague, Phys. Rev. Lett. 41, 702 one crystal-like bond with a red particle. 0.15 for nuclei of all sizes. This suggests that (1978). 5. P. R. ten Wolde, M. J. Ruiz-Montero, D. Frenkel, The cores of the crystalline nuclei are the effects of the nonspherical shape could be J. Chem. Phys. 104, 9932 (1996). formed from hexagonal layers, as shown by included in any future modification of the 6. P. N. Pusey, W. van Megen, Nature 320, 340 (1986). the cut through the center of this crystallite theory. Furthermore, we found that the radius 7. P. N. Pusey et al., Phys. Rev. Lett. 63, 2753 (1989). 8. J. Zhu et al., Nature 387, 883 (1997). (Fig. 3D), and their surfaces are more dis- of gyration rg of crystallites was related to the 9. S. Auer, D. Frenkel, Nature 409, 1020 (2001). ordered. The interface between the highly number of particles M within each crystallite 10. P. N. Pusey, in Liquids, Freezing and the Glass Transi- ϰ d tion, D. Leveesque, J. P. Hansen, J. Zinn-Justin, Eds. ordered cores and the amorphous liquid as M(rg) rg f with the fractal dimension ϭ Ϯ ␾ (Elsevier, Amsterdam, 1991), pp. 763–942. exterior is not sharp; this decrease in order df 2.35 0.15 for all values of ; the 11. A. K. Sood, in Solid State Physics, H. Ehrenbach, D. from the interior to the exterior is in agree- fractal behavior presumably reflects the Turnbull, Eds. (Academic Press, Boston, 1991), vol. 45, ment with computer simulations of a Len- roughness of their surfaces. pp. 1–73. nard-Jones system (5) and density function- The interfacial tension between the 12. B. J. Alder, W. G. Hoover, D. A. Young, J. Chem. Phys. 49, 3688 (1968). al calculations for hard spheres (32). crystal and fluid phases is a key parameter 13. K. Scha¨tzel, B. J. Ackerson, Phys. Rev. Lett. 68, 337 To determine the crystal structure of the that controls the nucleation process, yet ␥ is (1992). ordered cores, we used the rotational in- difficult to calculate (1, 2) and to measure 14. P. Bartlett, R. H. Ottewill, P. N. Pusey, Phys. Rev. Lett. 68, 3801 (1992). variants q4(i), q6(i), and w6(i)(33), which experimentally, but with our data, we can 15. Y. He, B. J. Ackerson, W. van Megen, S. M. Underwood, are quantitative measures for the local or- directly measure ␥ by examining the statis- K. Scha¨tzel, Phys. Rev. E 54, 5286 (1996). der around particle i, and distinguished be- tics of the smallest nuclei. For r ϽϽ r , the 16. B. J. Ackerson, K. Scha¨tzel, Phys. Rev. E 52, 6448 c (1995). tween fcc, hcp, bcc, and liquid (5, 9, 24). surface term in Eq. 1 dominates the free 17. S.-T. Yau, P. G. Vekilov, Nature 406, 494 (2000). Because of thermal fluctuations, even ho- energy of the crystallites, and consequent- 18. L. Antl et al., Colloids Surf. 17, 67 (1986). mogeneous crystals have a distribution of ly, we expect the number of crystallites to 19. S.-E. Phan et al., Phys. Rev. E 54, 6633 (1996). ϰ ␥ 20. We determined the particle radius by measuring the these parameters. Nevertheless, these dis- be N(A) exp[–A /(kBT )], where A is the volume per particle at random close packing (␾ ϭ ␾ ϭ rcp tributions are distinct for every crystal surface area, which we approximate by an 0.64) and rhcp ( rhcp 0.74) with the microscope. structure, as shown by the calculations (34) ellipsoid. In Fig. 5, we show that this pro- We obtained the same particle radius from these two points in phase space; therefore, we used this radius in Fig. 4A. These are compared to measure- portionality does indeed hold, allowing us to calculate ␾ of all other samples from the mea- ments of experimental crystalline nuclei, to obtain values for the surface tension. As sured volume per particle. ␥Ϸ 2 21. A. van Blaaderen, P. Wiltzius, Science 270, 1177 shown by a black curve in Fig. 4B. The shown in the inset, 0.027kBT/a and blue curve is a least squares fit with the may decrease slightly with increasing ␾ (1995). 22. E. R. Weeks, J. C. Crocker, A. C. Levitt, A. Schofield, distributions from Fig. 4A, and the fit pa- values. This value of ␥ is in reasonable D. A. Weitz, Science 287, 627 (2000). rameters are summarized in Table 1. Dur- agreement with density functional calcula- 23. J. C. Crocker, D. G. Grier, J. Colloid Interface Sci. 179, ing the nucleation process, bcc order is tions for hard spheres and Lennard-Jones 298 (1996). 24. P. J. Steinhardt, D. R. Nelson, M. Ronchetti, Phys. Rev. completely absent; in particular, as shown systems (32). By contrast, an older density B 28, 784 (1983). in Fig. 4, the q4 distribution for bcc is functional study (35) and computer simula- 25. The normalized order parameter qˆl (i) of particle i has markedly different from the measured dis- tions of hard spheres (9, 36) gave results (2l ϩ 1) complex components B͑i͒ tribution. Instead, the main contributions to approximately four times as large. Our 1 qˆ ͑i͒ ϭ ͸Y ͑ˆr ͒ the structure of the experimental crystalline measurement of a low value of ␥ is consis- lm NB͑i͒ lm ij jϭ1 nuclei are due to fcc, hcp, and liquid. The tent with the observed rough surfaces of the where N is a normalization factor such that ⌺ * ϭ mixture of fcc and hcp is the signature of crystallites; this may reflect the effects of mqˆlm(i)qˆlm(i) 1, B(i) is the number of neighbors rhcp structure and is consistent with the the softer potential due to the weak charges of particle i, ˆrij is the unit vector pointing from particle i to its jth neighbor, and Y is a spherical stacked hexagonal planes in Fig. 3D. The of our particles. Approximating the critical lm Ϸ harmonic function. Two particles are defined as liquid contribution in the fit is mainly due nucleus as an ellipsoid, with Mc 110, we neighbors if their separation is smaller than a cutoff ϭ ␮ 2 ⌬␮ Ϸ value of 2.83a, chosen to be close to the minimum to the surface of the crystallites. These obtain Ac 880 m , 0.13kBT, and results are in agreement with those expect- ⌬G(A ) Ϸ 7.4k T. between the first and second peaks of the pair c B correlation function g(r) for our samples. The “bond” ed for hard spheres, which also crystallize The direct observation of the structure, between two neighboring particles i and j is recog- in the rhcp structure (7), confirming that dynamics, and evolution of small crystalline nized as crystal-like if the complex inner product ⌺ * the critical nuclei are formed only from the nuclei allowed us to characterize the key mqˆlm(i)qˆlm( j) exceeds a threshold value of 0.5. As in other studies (5, 9), we used the l ϭ 6 order bulk solid phase, rather than forming from processes of nucleation and growth of colloi- parameter. a different metastable solid phase. dal crystals and to quantitatively determine 26. In this context, “bond” denotes the direction that In classical nucleation theory (2), it is the important parameters that control this connects a particle with one of its neighbors and not a bond in the chemical sense. assumed that the nuclei have a spherical process. These results also provide crucial 27. Whether a nucleus was fully visible or not was de- shape due to surface tension. To the best of experimental input to help guide any future cided by looking for particles in the nucleus that were our knowledge, this assumption has never refinements of theories for nucleation and closer than 0.8a to an edge of the observed volume. been experimentally verified. As shown by growth of colloidal crystals. Moreover, the Nuclei that did not meet this criterion were not used to determine the probabilities pg and ps. A typical the example in Fig. 3, our data indicate that ability to obtain three-dimensional movies of measurement contained between 1500 and 2500 the crystallite surfaces were quite rough, sug- the dynamics of colloidal solids will allow subcritical and critical nuclei that were fully visible gesting that the surface tension is low. Con- the direct observation and study of other im- and could be tracked for some time. 28. Larger nuclei were not observed as often as smaller sistent with this, the crystallite shapes ap- portant phenomena, such as defect motion, ones, because they have a higher free energy and, peared rather nonspherical. To confirm this, effects of impurities, and crystallization of consequently, do not form as often. Thus, the bin size ͗ ͘ we approximated each nucleus with an ellip- colloidal alloys. on the average crystallite radius r axis of Fig. 2 is increased logarithmically, which also reflects the ac- soid and found that the average nucleus is curacy with which the critical size can be identified. best described by an ellipsoid whose major References and Notes 29. Nucleation density rates obtained with particles of axes have the ratio 1:1.8:3.2. Nuclei contain- 1. T. Palberg, J. Phys. Condens. Matter 11, R323 (1999). different sizes can be compared when expressed in Ͼ 2. K. F. Kelton, in Solid State Physics, H. Ehrenbach, D. dimensionless form using the intrinsic length and ing 50 particles were found to be somewhat Ϫ5 Turnbull, Eds. (Academic Press, Boston, 1991), vol. 45, time scales. For this comparison, D0/(2a) is used as less anisotropic than smaller ones, but on Ϸ pp. 75–177. the unit of the nucleation rate density, where D0 average, the ratio of the shortest and longest 3. W. Ostwald, Z. Phys. Chem. 22, 289 (1897). 0.085 ␮m2 sϪ1 is the free-particle diffusion constant.

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30. K. Scha¨tzel, B. J. Ackerson, Phys. Rev. E 48, 3766 wl(i) is a third-order rotational invariant defined by calculated in the same way as for particle positions (1993). from a measurement. The liquid histogram was taken ͑ ͒ ϭ ͸ ͩ lllͪ 31. J. L. Harland, W. van Megen, Phys. Rev. E 55 3054 wl i from observations shortly after shear melting, when m1 m2 m3 (1997). m1,m2,m3 the sample was in the liquid state. m ϩ m ϩ m ϭ 0 32. R. Ohnesorge, H. Lo¨wen, H. Wagner, Phys. Rev. E 50, 1 2 3 35. D. W. Marr, A. P. Gast, Phys. Rev. E 47, 1212 (1993). ϫ ͑ ͒ ͑ ͒ ͑ ͒ 4801 (1994). qlm1 i qlm2 i qlm3 i 36. R. L. Davidchack, B. B. Laird, Phys. Rev. Lett. 85, 4751 33. ql(i) is a second-order rotational invariant of the where (: : :) stands for a Wigner 3j symbol. (2000). ϩ (2l 1)–component bond-order parameter ql(i) for 34. The histograms for fcc, bcc, and hcp order were 37. This work was supported by NSF (grant particle i and is defined by obtained from computer-generated crystal data. Ran- DMR-9971432) and NASA (grant NAG3-2284). U.G. 1 dom fluctuations were added to the particle positions also acknowledges support by the Swiss National l 4␲ 2 in such a way that the width of the first peak of g(r) Science Foundation. ͑ ͒ ϭ ͫ ͸  ͑ ͒2ͬ ql i 2l ϩ 1 qlm i for the computer-generated data matched the exper- m ϭϪl imental value. The bond-order histograms were then 20 December 2000; accepted 28 February 2001

catalysis, and the photoinduced redox reactions. Capturing a Photoexcited According to previous studies (20–22)ina weakly coordinating or noncoordinating sol- Molecular Structure Through vent, the 3d8 electrons of the Ni(II) ion in the square planar geometry of NiTPP adopt a sin- glet spin state 1(3d )2, S , whereas these elec- Time-Domain X-ray Absorption z2 0 trons in the octahedral geometry of NiTPP-L2 3 prefer the triplet spin state (3dx2-y2, 3dz2), T0. Fine Structure This difference reflects the response of the 3d and 3d orbitals in the square planar and Lin X. Chen,1* Wighard J. H. Ja¨ger,1 Guy Jennings,2 x2-y2 z2 octahedral ligand fields. Starting with NiTPP- David J. Gosztola,1 Anneli Munkholm,1† Jan P. Hessler1 L2, a laser pump pulse induces an electronic transition from the triplet ground state, T0,toan The determination of the structure of transient molecules, such as photoexcited excited triplet state, T * (Fig. 1B). This state states, in disordered media (such as in solution) usually requires methods with high decays and undergoes intersystem crossing to an temporal resolution. The transient molecular structure of a reaction intermediate excited singlet state, S *. This excited state ejects two ligands L to produce the square planar produced by photoexcitation of NiTPP-L2 (NiTPP, nickeltetraphenylporphyrin; L, piperidine) in solution was determined by x-ray absorption fine structure (XAFS) geometry (20–22). If the solvent were noncoor- data obtained on a 14-nanosecond time scale from a third-generation synchrotron dinating or weakly coordinating, this reaction source. The XAFS measurements confirm that photoexcitation leads to the rapid sequence would be completed. However, in a removal of both axial ligands to produce a transient square-planar intermediate, strongly coordinating solvent, S0 is unstable and NiTPP, with a lifetime of 28 nanoseconds. The transient structure of the photo- the system will return to T0, by recombining dissociated intermediate is nearly identical to that of the ground state NiTPP, with two piperidine ligands to form an octahe- suggesting that the intermediate adopts the same structure as the ground state in drally coordinated NiTPP-L2 (Fig. 1). The tran- a noncoordinating solvent before it recombines with two ligands to form the more sient structures involved in this recombination are not known. A penta-coordinated NiTPP-L stable octahedrally coordinated NiTPP-L2. molecule could be involved in the recombina-

Photoexcited states of molecules are often stud- Here we present time-domain laser pump/ tion, the square planar NiTPP could return to T0 ied with “pump-probe” methods, in which a x-ray probe XAFS studies using x-ray pulses via a concerted axial chelation with two piperi- short optical pulse excites the ground state, and from a third-generation synchrotron source. dine molecules, or an intermediate square pyra- a second probe pulse interrogates the excited (The experiments were conducted at a wig- mid NiTPP-L structure could be important. Be- state. For structural studies, substantial progress gler beamline at the Basic Energy Sciences cause it is difficult to identify a unique optical has been made in generating ultrashort x-ray Synchrotron Research Center at Argonne Na- absorption spectrum for each possible interme- pulses (1–4) that have been used to obtain tional Laboratory’s Advanced Photon diate involved in photodissociation, transient time-resolved x-ray diffraction information Source.) Although the duration of the x-ray optical absorption spectra of NiTPP in coordi- with picosecond resolution (5–7). Because pulses (100 ps at full width at half-maximum) nating solvent alone are not sufficient to identify many light-driven processes occur in disordered is too long to follow atomic displacements the intermediates.

media, techniques that do not rely on the long- due to the photoexcitation, it is suitable for The excited-state dynamics of NiTPP-L2 range order of a system must be developed. The capturing intermediate structures with sub- have been studied by ultrafast optical transient approach of laser pump/x-ray probe x-ray ab- nanosecond resolution. We are able to take absorption and Raman spectroscopy (20–25). sorption fine structure (XAFS) was proposed “snapshots” of molecules when the popula- The intersystem crossing from T *toS * and the

earlier (8–10), and transient molecular struc- tion of the laser-generated transient species subsequent dissociation to S0 occur in less than tures, generated in light-induced processes, reaches a maximum; in this case, the species a few hundred picoseconds (22). In a strongly were captured without using the temporal res- is a reaction intermediate with a 28-ns life- coordinating solvent at room temperature, this olution of the source (11–19). time that results from the photodissociation state of NiTPP has a lifetime of 28 ns, because

of NiTPP-L2 (NiTPP, nickeltetraphenylpor- it must react with two piperidine molecules to phyrin; L, piperidine) in solution (Fig. 1A). reform NiTPP-L2. We can therefore use x-ray 1Chemistry Division and 2Materials Science Division, The photoinduced processes of hemelike pulses from a synchrotron source to determine Argonne National Laboratory, Argonne, IL 60439, porphyrin derivatives are examples of how, the local structure of these intermediates and USA. through electron transfer, the coordination ge- compare them with the structure of ground state *To whom correspondence should be addressed. E- ometry of a metal ion is altered by the electronic NiTPP. In addition, it is possible to determine mail: [email protected] †Present address: LumiLeds Lighting, 370 West structure of its macrocycle ligands. These mol- whether an intermediate species with only one Trimble Road, San Jose, CA 95131, USA. ecules can serve as models for oxygen transport, axial ligand is present (22). The experiment was

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