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Morphology of Crystals Made of Hard Spheres Yueming He, B. Olivier, and Bruce J. Ackerson* Department of Physics and Center for Laser Research, Oklahoma State University, Stillwater, Oklahoma 74078-0444 Received September 30, 1996. In Final Form: December 2, 1996X The growth and morphology of crystals of hard colloidal spheres is monitored by a special photographic technique. Crystals are observed to grow in size and increase in number density, initially. This is followed by a decrease in size and an increase to saturation in the number density. During the growth stage the crystals evidence irregular rectangular shapes, and an internal substructure is evident within the larger crystals. The observed size decrease suggests a breakup of crystals to a length scale of the internal structure. A simple model for surface roughening, which has been used to classify crystal morphology, is examined for hard sphere crystals. Hard sphere crystals should be classified as “rough” and subject to growth instabilities. The length scale for simple planar and spherical instabilities is examined for hard sphere crystals and is found to be smaller than the observed length scales.
Introduction A thermally activated collection of identical hard spheres is one of the simplest classical many-body systems. Because the hard sphere interaction is infinitely repulsive at contact and zero otherwise, the potential energy of the system is the same for all particle configurations. As a result, temperature is not an important thermodynamic parameter but particle density is. Because there are no attractive forces to hold hard sphere crystals together, the existence of an order/disorder phase transition was debated.1 For an isolated system the equilibrium phase must be determined by the maximization of entropy, and a (disordered) fluid appears to have a larger entropy than a (ordered) crystal phase at the same density. However, computer simulations2,3 established the existence of a transition, with the volume fraction of spheres being 0.494 at the freezing point and 0.545 at the melting point. The system is only half full of spheres when freezing begins, and the equilibrium phase maximizes the system entropy. More recent simulations4 have confirmed the existence of a complex entropy driven transition suggested by experiments:5,6 a mixture of two different radii hard spheres will form a complex AB13 structure having 112 particles in a unit cell. Density functional theories7 suggest that these order/disorder transitions are driven by a competition between the large global entropy of the liquid state and the large local entropy of the crystal phase. While computer simulations have been important in studying hard sphere systems, experiments are also possible. Hard sphere interactions have been approximated at the colloidal particle level using steric stabilization to minimize the van der Waals attraction.8,9 The particles are very uniform in size with diameters in the colloidal range between 0.1 and 1 µm. Suspensions of * Corresponding author. X Abstract published in Advance ACS Abstracts, February 15, 1997. (1) Uhlenbeck, G. E. In The Many Body Problem; Percus, J. K., Ed.; Wiley Interscience: New York, 1963; p 498. (2) Alder, B. J.; Hoover, W. G.; Young, D. A. J. Chem. Phys. 1968, 49, 3688. (3) Hoover, W. G.; Ree, F. H. J. Chem. Phys. 1968, 49, 3609. (4) Eldridge, M. D.; Madden, P. A.; Frenkel, D. Nature 1993, 365, 35. (5) Bartlett, P.; Ottewill, R. H.; Pusey, P. N. J. Chem. Phys. 1990, 93, 1299. (6) Bartlett, P.; Ottewill, R. H.; Pusey. P. N. Phys. Rev. Lett. 1992, 68, 3801. (7) Baus, M. Mol. Phys. 1983, 50, 543. (8) de Hek, H.; Vrij, A. J. Colloid Interface Sci. 1981, 84, 409. (9) Antl, L.; Goodwin, J. W.; Hill, R. D.; Ottewill, R. H.; Owens, S. M.; Papworth, S.; Waters. J. A. Colloids Surf. 1986, 17, 67.
S0743-7463(96)00943-2 CCC: $14.00
these particles in organic solvents evidence properties consistent with expected hard sphere behavior. Sedimentation velocities as a function of volume fraction10,11 show neither a decrease due to longer ranged particle repulsions nor an increase due to particle attractions as compared to predicted theoretical results for hard spheres.12 These suspensions evidence a fluid to crystal phase transformation with an increasing volume fraction of particles,10,13,14 which has a coexistence region width in volume fraction equal to one tenth the freezing value ((φmelting - φfreezing)/φfreezing ∼ 0.1). This is expected for hard spheres but not for softer repulsive potentials.15 Shear induced microstructures have been understood in terms of the steric interaction of hard spheres.16 The crystal structure in these suspensions has been determined using light diffraction, the analogue of X-ray powder diffraction for atomic systems.17 The crystal structure data are described well as a registered random stacking of close packed planes of spheres. This can be considered as a highly faulted face-centered cubic or threedimensional hexagonal close-packed structure. Facecentered cubic structures are favored energetically when there are long range repulsions between particles extending to second nearest neighbors. Hexagonal close-packed structures are favored energetically for attractive interactions of second nearest neighbors. For hard spheres there is no energy of interaction between second nearest neighbors, and a random close-packed structure results. Recently small angle light scattering has been used to monitor the nucleation and growth process in suspensions of hard spheres.18-20 The scattered intensity distribution scales during nucleation and growth, while the characteristic length increases approximately as a power law in time with exponents distributed between half and unity, the expected limiting values of diffusion- and interface(10) Paulin, S. E.; Ackerson, B. J. Phys. Rev. Lett. 1990, 64, 2663. (11) Buscall, R.; Goodwin, J. W.; Ottewill, R. H.; Tadros, Th. F. J. Colloid Interface Sci. 1982, 85, 78. (12) Beenakker, C. W. J.; Mazur, P. Physica (Amsterdam) 1984, 126A, 349. (13) Pusey, P. N.; van Megen, W. Nature 1986, 320, 340. (14) Underwood, S. M.; Taylor, J. R.; van Megen, W. Langmuir 1994, 10, 3550. (15) Hansen, J. P.; Schiff, D. Physics 1973, 25, 1281. (16) Ackerson, B. J. J. Rheol. 1990, 34, 553. (17) Pusey, P. N.; van Megen, W.; Bartlett, P.; Ackerson, B. J.; Rarity, J. G.; Underwood, S. M. Phys. Rev. Lett. 1989, 63, 2153. (18) Scha¨tzel, K.; Ackerson, B. J. Phys. Rev. Lett. 1992, 68, 337. (19) Scha¨tzel, K.; Ackerson, B. J. Phys. Rev. E 1993, 48, 3766. (20) He, Y.; Ackerson, B. J.; Underwood, S. M.; van Megen, W.; Scha¨tzel, K. Phys. Rev. E, accepted.
© 1997 American Chemical Society
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limited growth, respectively. This growth is followed by a crossover region where the characteristic length scale is observed to decrease in time before increasing again during ripening at larger times. This reduction in crystal size was not expected or understood. While microphotography has been used to study the growth and morphology of colloidal crystals in chargestabilized particle suspensions,21 it has not been applied to suspensions of hard spheres. For charge-stabilized particle suspensions, crystals appear spherical, not faceted, indicating growth of crystals with (microscopically) rough surfaces.22,23 The crystal size grows linearly with elapsed time, indicating interface-limited growth. On the other hand, unstable Mullins-Sekerka24,25 growth also has been observed.26 It is not clear what the morphology of hard sphere crystals should be given the nature of the interparticle interaction and given that the crystal microstructure is highly faulted or random. Computer simulations are not yet powerful enough to answer questions involving such long times and large numbers of particles. Are the crystals faceted, spherical, or dendritic? What can be learned about the mysterious crystal size reduction in the crossover region between growth and ripening? In this article we examine the growth and morphology of hard sphere crystals using a special microphotographic technique. We evaluate theories for roughening and growth instabilities in the hard sphere limit. The crystal length scale reduction in time, seen in small angle light scattering experiments, is examined photographically. Experimental Procedure The sample used in these experiments is comprized of 0.22 µm radius poly(methyl methacrylate) (PMMA) spheres with a polydispersity of 0.07 (standard deviation to mean radius) suspended in a mixture of tetralin and decalin. The solvent ratio is chosen to match the refractive index of the particles and render the suspensions transparent. The particles are sterically stabilized against aggregation with a ∼10 nm thick surface layer of hydroxystearic acid. The volume fraction of the sample studied drifted slightly during the observation time but was determined by weighing to be in the range 0.522 < φ< 0.524. These volume fractions are referenced to the freezing volume fraction as described elsewhere.10,13 This sample at equilibrium is in the liquid/crystal coexistence region. Before measurements the sample is shear melted27 to an amorphous metastable fluid state and the elapsed time, t, is measured from the cessation of shear melting. Since crystals appear to the eye approximately 1 h after shear melting, there is sufficient time for convection to cease without disturbing the nucleation and crystal growth. The sample is contained in a cuvette (1.0 cm × 1.0 cm × 5.0 cm) with one of the vertical rectangular faces perpendicular to the optical axis of the camera to minimize image distortions. A Nikon series N2000 camera is used with Kodachrome DX 35 mm slide film. The camera optics affords a magnification of approximately 2X in the slide image with further magnification (approximately 20X) in making Figure 1. The sample is illuminated using a 500 W tungsten light source which is collimated and weakly focused with a cylindrical lens (f ) 50 cm) into a thin vertical sheet having an minimum half width of approximately 100 µm. The variation of the sheet width over the size of the cell is 4%. The angle the optical axis of the camera makes with the plane of the sheet of light corresponds to the first-order Bragg scattering angle for one wavelength in the visible spectrum. For this radius particle and the wavelength λ ) 514.5 (21) Aastuen, D. J. W.; Clark, N. A.; Cotter, L. K.; Ackerson. B. J. Phys. Rev. Lett. 1986, 57, 1733. Errata: Phys. Rev. Lett. 1986, 57, 2772. (22) Jackson, K. A. In Liquid Metals and Solidification; ASM: Cleveland, 1958; p 174. (23) Jackson, K. A.; In Progress in Solid State Chemistry; Reiss, H., Ed.; Pergamon Press: New York, 1967; Vol. 4, p 53. (24) Mullins, W. W.; Sekerka. R. F. J. Appl. Phys. 1963, 34, 323. (25) Mullins, W. W.; Sekerka, R. F. J. Appl. Phys. 1964, 35, 444. (26) Gast, A. P.; Monovoukas, Y. Nature 1991, 351, 553. (27) Ackerson, B. J.; Clark, N. A. Phys. Rev. Lett. 1981, 46, 123.
Langmuir, Vol. 13, No. 6, 1997 1409 nm, the Bragg angle is approximately 55°. Exposure times ranged from 30 to 90 s depending on the width of the incident sheet and the actual elapsed time during the crystallization process. The exposure times were, however, much smaller than times corresponding to observable changes in the crystal growth. The growth of an individual crystal, as shown in Figure 2, is captured from original photographic slides by the following imageprocessing procedure. The slides are illuminated with diffuse light and digitized using a CCD camera and high-resolution frame grabber. The resolution, however, was limited by the camera to approximately 640 × 480 pixels on a 3/4 in. active surface. The magnification was increased by a factor of 9X using a continuously focusing microscope attached to the camera. Care was taken to insure a uniformly illuminated photograph in order to minimize artifacts which could arise from nonuniform background fields. The image-processing consisted of contrast enhancement of regions of interest. The crystal size data shown in Figure 3 and the crystal density data shown in Figure 4 were determined from measurement in areas of approximately half a square centimeter at the earliest time, where there are order ten crystals, to areas one quarter this size at large times, where there are well over a hundred crystals in the target area. The crystal size is determined by measuring the largest dimension and the dimension orthogonal to it for each crystal. The product of these two dimensions for each crystal is averaged over the sample distribution at a given time, and the root of this average is the reported size. The crystal density is determined by counting the crystals in the target area and dividing by the area and the 100 µm illumination thickness. Figure 5 reports density data for all crystals at the earliest times and Bragg crystals for a full range of times.
Results and Discussion Figure 1 shows images from photograph slides taken at the first-order Bragg angle at t ) 135 min (a) and t ) 242 min (b) after shear melting. The lighter crystals have a set of crystallographic planes “properly” oriented to Bragg scatter the incident beam. The darker structures are crystals having non-Bragg scattering orientations relative to the incident sheet of light. These “dark” crystals scatter less light than the surrounding metastable fluid, which is near its maximum scattered intensity at this angle. Thus, all crystals within the illuminated region are visible when the crystal size is comparable to the sheet width. Bragg scattering crystals are observed first in time, presumably due to the large scattered intensity and smallness of the crystal size compared to the sheet width in present studies. Other photographic observations21 have not used a thin sheet illumination and only observe the Bragg scattering crystals. In these experiments light from the foreground and background obscures the nonBragg scattering crystals. Our sheet illumination produces greater clarity of the crystal image, at least in the center of the field of view, where the camera object plane and the sheet illumination overlap. The crystals shown in Figure 1 are irregular, largely with rectangular or butterfly shapes. The outer edges are reasonably well defined, having sharp corners in some cases. This is suggested by recent surface tension calculations of Marr and Gast28 which indicate a fairly sharp transition between crystal and liquid, but the surface tension of different crystallographic planes is nearly the same, suggesting spherical equilibrium shapes. At the larger time shown in Figure 1b, the crystals display extensive internal structure. These characteristic morphologies were found throughout the sample for all data runs. At even larger times the dark crystals appear to fill the volume of the sample. Here it becomes difficult to resolve individual crystals, and the internal structure length scale becomes the dominant visible feature. The internal structure suggests several mechanisms for the formation of these crystals. The crystals may be (28) Marr, D. J. W.; Gast, A. P. Phys. Rev. E 1995, 52, 4058.
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Figure 1. Photographs of dark and Bragg scattering hard sphere crystals: (a, left) 135 min; (b, right) 242 min after shear melting.
formed from the aggregation of independently grown crystals, from autonucleation, or from unstable dendritic growth. Autonucleation is the nucleation of crystals having a different crystallographic orientation at the surface of an existing crystal. This can be ruled out as a mechanism by noting that the Bragg scattering crystals generally have the same dimensions as the neighboring dark crystals. If autonucleation were responsible for the substructure, the Bragg scattering should have this smaller length scale. For the same reason the aggregation model may be ruled out, because the aggregate structure is unlikely to share a common crystal plane across all aggregates. Unstable dendritic-like growth remains a possible explanation. Jackson22,23 introduced a parameter which proved extremely useful in correlating crystal growth morphologies. This parameter, R ) βLξ ) 2β, is the product of the inverse thermal energy, β, with the heat of fusion, L, and the ratio of the number of bonds per atom on the surface to the number per atom in the bulk, ξ. Also it may be expressed in terms of the bond energy, . For systems where R is small, crystals exhibit nonfaceting morphologies but show faceting when R > 2. Low values of R correspond to microscopically “rough” surfaces which may advance isotropically or exhibit growth instabilities and dendritic structures. Large values of R correspond to microscopically “smooth” surfaces which grow anisotropically with facets observed in at least some growth directions. For large undercoolings and large R, autonucleation results. The transition between rough and smooth surfaces is captured in a simple, single-layer, mean field model for
the surface Helmholtz free energy, F,22,23 where
∆F/NkT ) RF(1 - F) + F ln(F) + (1 - F) ln(1 - F) The first term represents the combined bond energy and entropy contribution to the free energy in moving an atom from the fluid to the crystal interface. This term includes the effect of interaction between surface atoms through the quadratic F dependence. The entropic contribution from counting the possible combinations of surface occupancy states is represented by the logrithmic terms. Here N is the number of surface sites, F is the fraction of sites occuppied, and β ) 1/kT. For R < 2, the free energy has a minimum at F ) 0.5 and the surface is considered rough. For R > 2, the free energy has minima approaching F ) 0 or 1 and the surface is considered smooth (i.e. nearly empty or full). For hard sphere systems there is no attractive interaction between particles and therefore no bond energy. If R is zero, then clearly the surface is rough. There is, however, a contribution to R from the change in entropy in moving a particle from the liquid to the crystal. The entropy change between the liquid at the freezing volume fraction and the crystal at the melting volume fraction has been determined from computer simulations3 to be Sc - Sf ) -1.06NAk, where Sc is the bulk crystal entropy, Sf is the bulk liquid entropy, NA is the number of liquid particles transferred to the solid surface, and k is the Boltzmann constant. This change is negative, indicating that the entropy of the crystal is lower than that of the liquid, and unfavorable, if entropy is to be maximized. (The reader should not be surprised by this
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Figure 2. Time sequence of the growth of a single crystal showing the development of protrusions at the corners of a generally rectangular shape. The times in seconds are 3120 (a), 3360 (b), 3660 (c), 4020 (d), 4320 (e), 4740 (f), 5160 (g), 5580 (h), and 6000 (i).
Figure 3. Average crystal size versus elapsed time since shear melting. An initial power-law growth is indicated by the solid line before the size reduction is observed. The open circles represent the average radius of the dark crystals. The closed squares represent the average radius of the Bragg scattering crystals.
result because the coexisting liquid is at one volume fraction and the crystal at another. For a metastable fluid at equilibrium densities corresponding to a fully crystalline solid, the entropy of the metastable fluid is lower than that of the corresponding equilibrium crystal, in agreement with our understanding that entropy is maximized in equilibrium.3,29) Thus the parameter R ) 1.06 for hard spheres corresponds to rough surfaces. These systems are not expected to be faceted or evidence autonucleation. Rather they correspond more closely to growth of metallic crystals from the melt and are subject to growth instabilities. Figure 2 follows the growth of a crystal which has been isolated from the original photographic slides. The crystal is initially roughly rectangular but later develops protrusions at its corners. The protrusions have a width ∼100 µm at later times. If unstable growth produces an internal structure in the crystals, it must be reconciled with the (29) Ackerson, B. J. Nature 1993, 365, 11.
Figure 4. Bragg scattering crystal density as a function of elapsed time since shear melting.
Figure 5. Initial linear dependence of the Bragg scattering crystal density (circles) and total crystal density (squares) as a function of elapsed time since shear melting. The slope of the initial linear increase gives an estimate of the nucleation rate density.
Bragg scattering of crystals in Figure 1 which often have a length scale on the order of the crystal size and not the observed internal length scale. Evidently the different
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substructures within a single dark crystal do share a common crystallographic plane. Eventually this crystal grows into a neighboring crystal which will influence its growth morphology. The Mullins-Sekerka instability length scale24,25,30 for a planar interface has been estimated using thermodynamic and kinetic results determined for hard spheres.31 The instability length scale is given as the sqare root of a capillary length and diffusion length product. The ratio of the capillary length to the particle radius is found to be 1.9 at volume fraction 0.52, when using the surface tension for hard spheres quoted by Marr and Gast,28 the miscibility gap given by the difference in the coexisting equilibrium fluid volume fraction and crystal volume fraction, and the equation of state for the metastable hard sphere fluid.31 The diffusion length is calculated using the dilute solution diffusion constant as an approximation for the collective diffusion constant in the diffusion equation. Estimates of the growth velocity of the crystal/ fluid interface will depend on the choice of self-diffusion constant.31 Using the short time self-diffusion constant gives 108 for the ratio of the diffusion length to the particle radius at volume fraction 0.52, while the long time selfdiffusion constant gives 806. These values produce instability length scales of 20 µm (short time self-diffusion constant) and 55 µm (long time self-diffusion constant). The observed crystal growth is not linear in time as assumed above, but approximating a growth velocity from the change in radius divided by elapsed time for the period that crystals are increasing in size produces instability length scale estimates between 20 and 55 µm. These calculated length scales are two to three times smaller than the observed substructure length scale. The spherical harmonic instabilities of a growing sphere30 may be more appropriate for our system, since the crystals are not large compared to the internal structure length scale. The lowest order harmonics become unstable first, and the radius at the instability depends on the critical nucleation radius. For hard spheres at volume fraction 0.52 the critical nucleus radius is 1.54 µm, so the lowest (second harmonic) instability occurs at 11 µm and the fourth order harmonic is unstable at 25 µm. Again these sizes are much smaller than the observed values. This suggests that the observed length scales result from higher order nonlinearities. Figure 3 shows the average crystal size as a function of elapsed time. The initial growth is fit by a power law R ∼ tδ with exponent δ ) 0.65. This power law exponent is consistent with that observed in small angle scattering measurments20 and is intermediate between diffusionlimited growth (δ ) 0.5) and interface-limited growth (δ ) 1.0). Theoretical calculations for hard sphere crystal growth31 show that such “pseudo” power law behavior with exponents between interface- and diffusion-limited growth can occur as transients. In the coexistence region the asymptotic behavior is expected to be diffusion-limited growth. After the initial growth, the average crystal size is observed to decrease. The decrease in a characteristic length was first observed in small angle light scattering18-20 and not understood. The direct observations, presented here, indicate that the crystals grow independently from the melt and exhibit an internal structure with a characteristic length scale before overlapping or touching at the completion of the growth phase. At this time the sample appears full of crystals and the only
obvious length scale is the crystal internal structure scale. The Bragg scattering length scale also evidences a decrease, indicating that the large crystals have broken up and undergone some reorientation. Theoretical calculations31 and Bragg angle scattering measurements32 have indicated that the growing crystals are compressed to higher than equilibrium densities during growth. When a full complement of crystals is created at the completion of growth and the metastable fluid density is reduced to nearly the equilibrium value, the compressed crystals must then be in an unstable situation in which expansion fractures them and/or dissolution occurs along the internal structure boundaries. It is also possible that gravitationally induced stresses assist in the breakup of crystals. In any case the observed decrease in the characteristic length scale, observed in small angle scattering, is real. Figure 4 shows the crystal density in the sheet volume as a function of elapsed time. The total crystal density is indicated for early times, and the Bragg crystal density is shown for the full time range. The ratio of the total number of crystals to the number of Bragg crystals is ∼5. To obtain better statistics at early times, when there are few crystals in the scattering volume, this ratio could be used to estimate the total number of crystals from the Bragg crystal count in a much larger scattering volume. The initial growth in crystal density is linear in elapsed time, and the slope is a measure of the crystal nucleation rate density. For this volume fraction the nucleation rate density is 3.4 × 106 crystals/(m3 s). Nucleation rate densities are also estimated from small angle light scattering20 by assuming the equilibrium complement of the crystal is present when the growth size reaches its maximum value and using this value to determine the total number of crystals present in the sample. The resulting nucleation rate density at volume fraction 0.52 is 1.3 × 106 crystals/(m3 s). This compares well with the result above, given that the nucleation rate is a strong function of particle volume fraction and that there is a small amount of solvent loss due to evaporation between the two measurements. The crystal density continues to increase after the average crystal size begins its decrease, crystal breakup accounting for the increased number of crystals. Eventually the crystal size becomes constant and the crystal density saturates. In conclusion, we find that hard sphere crystals have a complex structure, which is neither faceted nor spherical. The transition zone from fluid to crystal is reasonably sharp due to the surface energy. There is evidence for extensive internal structure, which we associate with growth instabilities and which is expected from a model of surface roughening by Jackson. The crystals evidence a catastophic decrease in size, which is due to a breakup, induced either by gravitaional stresses or by overcompression during the growth process. The reduced size is similar to the internal structure observed during growth. This is very complex behavior for a collection of particles which interacts via hard core repulsions.
(30) Langer, J. S. Rev. Mod. Phys. 1980, 52, 1. (31) Ackerson, B. J.; Scha¨tzel, K. Phys. Rev. E 1995, 52, 6448.
(32) Harland, J. L.; Henderson, S. I.; Underwood, S. M.; van Megen, W. Phys. Rev. Lett. 1995, 75, 3572.
Acknowledgment. The authors gratefully acknowledge support of this work by NASA through Grant NAG31624. W. van Megen and S. M. Underwood are thanked for supplying the samples used in these experiments. Robert Ascio is thanked for crystal size and density measurements. LA9609433
