The equilibrium intrinsic crystal–liquid interface of colloids
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Edited by David Chandler, University of California, Berkeley, CA, and approved July 2, 2009 (received for review May 1, 2009)
Abstract
We use confocal microscopy to study an equilibrated crystal–liquid interface in a colloidal suspension. Capillary waves roughen the surface, but locally the intrinsic interface is sharply defined. We use local measurements of the structure and dynamics to characterize the intrinsic interface, and different measurements find slightly different widths of this interface. In terms of the particle diameter d, this width is either 1.5d (based on structural information) or 2.4d (based on dynamics), both not much larger than the particle size. This work is the first direct experimental visualization of an equilibrated crystal–liquid interface.
The interface between crystal and liquid phases of a material governs phenomena such as wetting, lubrication, and crystal nucleation (1, 2). Interfaces are poorly defined at the atomic level: Capillary waves cause fluctuations in the interface position (3–6), and locally the structure varies in a smooth way from ordered to disordered (2). Existing literature makes a distinction between the intrinsic interface (presumed to be sharp) (3), and the observed surface blurred by capillary waves (4, 7). The standard equilibrium interface profile is well defined and contains fluctuations at all length and time scales, due to these capillary waves. Because of the rapid time scales of capillary-wave fluctuations, along with the small length scales at the interface, it is difficult to study these interfaces directly (1). Thus, computer simulations provide useful information about model crystal–liquid interfaces, such as hard-sphere systems (2, 8, 9) and Lennard–Jones systems (10, 11).
Recently, crystal–liquid interfaces were directly studied in colloidal suspensions by using confocal microscopy (6, 12). Colloids are systems of solid particles in a liquid and are a good model system for phase transitions (5, 6, 13, 14). Microscopy allows direct observation of structure and dynamics of the colloidal particles (15). However, the previous experiments focused on nonequilibrium cases where samples were crystallizing and did not provide data on equilibrium interfaces, such as those studied by simulation (2, 8–10, 16). Furthermore, these experiments did not examine the intrinsic interface, perhaps because they were nonequilibrium studies, and thus crystalline particles were present in the “liquid” side and vice versa, which confuse the structure near the interface.
In this work, we present confocal microscope observations of an equilibrated colloidal crystal–liquid interface. By following the positions of several thousand colloidal particles on both sides of the interface, we directly visualize the interface. An example of our data is shown in Fig. 1A, where blue particles are crystalline and yellow/red particles are liquid-like. This interface has a low surface tension, and we see capillary waves. We are able to remove the influence of these capillary waves from the data and measure the intrinsic surface profile. In particular, we find that capillary waves cause an apparent broadening of the surface, but the structure of the intrinsic surface is characterized by a width of only 1.5d (in terms of the particle diameter d). This is the first direct experimental visualization of an equilibrated interface. A precise definition of the intrinsic interface has not been made before, and we show that several plausible definitions give slightly different results.
Pictures showing the crystal–liquid interface. (A) Picture of the liquid–crystal interface showing a slice that is two crystalline layers thick (4.8 μm). Particles are colored according to the number of ordered neighbors they have, No. Blue particles have No ≥ 8, and darker blue indicates higher No. Yellow, orange, and red particles have No < 8, with red particles having No = 0, and the brightest yellow particles having No = 7. The image has been slightly rotated (by 3° around an axis parallel to gravity) to view the crystalline particles in-plane. (B) Here the particles are colored according to how much time they spend as “crystalline” particles (with No ≥ 8). White particles spend all of their time as crystalline or liquid-like, and the colored particles fluctuate over the ≈1-hr experiment. Blue particles spend nearly all their time as crystalline, light green corresponds to those which spend half their time in each state, and red particles spend nearly all their time as liquid-like. Although the coloring is based on the time average of the data, the positions are considered at the same time as those shown in A.
Our colloidal sample is described in Materials and Methods. Here, we briefly note that in our solvent the colloidal particles have a slight charge. Experimentally, we observe that the freezing transition volume fraction is at ϕfreeze = 0.43, and the melting transition is at ϕmelt = 0.49, to be compared with the hard-sphere values of ϕfreeze = 0.494 and ϕmelt = 0.545. Our measured values are similar to those seen for other experiments with similar colloidal samples (12).
Results
At each time step, we determine the crystalline region by using the method of bond-order parameters (6, 12, 17, 18). To do this, each particle i at each time is characterized by a normalized order parameter q̂l(i) with (2l + 1) complex components
where 𝒩 is a normalization factor, such that Σmq̂lm(i)q̂lm*(i) = 1, B(i) is the number of neighbors of particle i, r̂ij is the unit vector pointing from particle i to its jth neighbor, and Ylm is a spherical harmonic function. Following prior work, we use l = 6 (12, 18). The neighbors of a particle are defined as those with centers separated by <1.41d, which corresponds to the first minimum of the pair correlation function g(r) for the liquid region. Two neighboring particles are termed “ordered neighbors” if the complex inner product Σmq̂lm(i) q̂lm*(j) exceeds a threshold value of 0.5. For each particle, the number of ordered neighbors No is determined. Following the usual convention, particles with No ≥ 8 are classed as crystalline particles, and the other particles are liquid-like particles (18). The advantages of using bond-order parameters are that they are local measures of order, are somewhat insensitive to variations
in the number of neighbors each particle has, and do not depend on the specific type of crystal (17, 18).
Fig. 1A shows a snapshot of the sample, colored according to the bond-order parameter method. The crystalline side of the interface (blue particles) is composed of hexagonal layers in random stacking [a mixture of face-centered-cubic stacking (abcabc..) and hexagonal-close-packed stacking (ababab..)]. This stacking is similar to that seen in growing colloidal crystallites (12). In particular, the data shown in this article are for a sample where nine hexagonal layers are imaged, with stacking abacbcbac. Two other regions of this sample were imaged and the results presented below do not vary in any significant way.
Typically, between two different phases, there will always exist a surface energy defining the energetic cost of maintaining the interface. For fluid–fluid interfaces, this is also known as the surface tension, although the concept is relevant for solid–liquid interfaces as well [for example, in nucleation of crystals (12)]. For a hard-sphere system, this surface energy is entropic in origin, reflecting the difficulty of packing particles at the interface. Right at the interface, particles can pack neither optimally for the crystal state nor optimally for the liquid state, thus resulting in an entropic penalty that gives rise to a surface tension. In our experiment, the interface shown in Fig. 1 is rough and varies in time due to surface capillary waves. Because the surface energy is not extremely large compared with kBT (the thermal energy, based on the temperature T and Boltzmann's constant kB), thermal fluctuations allow these capillary waves to be observed in our experiment. The temporal variability of the sample is indicated in Fig. 1B, which colors only the particles that we observe to spend some time in both crystalline and liquid structures for the duration of our experiment (3,750 s). As can be seen, the exact position of the interface thus fluctuates by as much as ≈4 particle diameters (in a “peak-to-peak” sense).
To determine the surface tension, we follow the procedure of ref. 5. Due to a slight purposeful density mismatch (described in Methods and Materials), the crystal–liquid interface is nearly parallel to the xy plane; we rotate the data by 15° to align the interface with the xy plane. Based on the new z axis, at each time we locate the maximum z position of crystalline particles as a function of x and y (coarse-grained in x and y over a distance 1.035d, the spacing between crystalline layers, where d is the diameter of an individual particle), giving us the interface height h(x, y, t). Although our data are only over a short spatial extent in x and y (9d × 24d), we have good temporal data (0 ≤ t ≤ 3,750 s), and thus we calculate the temporal dynamical correlation function
where h′(x, y, t) = h(x, y, t) − 
h(x, y, t)
t represents the fluctuations of the interface about its time-averaged position, and the angle brackets 
… 
x,y,t indicate an average over space and time. We plot this dynamic correlation function in Fig. 2(circles). The intercept is at gh(Δt = 0) ≈1.1d2; thus, the root-mean-square width of the capillary wave fluctuations is approximately d, as compared with the slightly larger amplitude of the peak-to-peak fluctuations seen in Fig. 1B.
Interfacial height correlation function. The circles show the temporal correlation function of interface position, averaged over x and y (Eq. 2). The line is a fit to Eq. 5, with interfacial stiffness γ̃ = 0.93 nN/m = 1.2 kBT/d2 and a capillary time τ = 750 s. The small k modes give rise to the very steep drop of the correlation at short time scales, and the exact shape of the theoretical curve for Δt < 25 s depends on how the numeric integration of Eq. 5 is done.
The capillary waves are limited by the interfacial stiffness γ̃, rather than the surface tension γ (16). For a crystal–liquid interface, γ is usually anisotropic and depends on the crystal orientation. Fluctuations in the surface
thus depend both on γ (related to interfacial stretching) as well as second derivatives of γ with respect to angles away from
the interface normal (related to interfacial bending). The orientational average of γ̃ is the bulk surface tension γ, but
measuring this requires vastly more data than we have (8, 16, 19). To extract the interfacial stiffness γ̃, we fit gh(Δt) by using the results of capillary wave theory (5, 20). Overdamped capillary waves with wave number k should decay as
with gravitational acceleration g, density difference Δρi (across the interface), and viscosity η (equal to the sum of the viscosities of the two phases) (20). By equipartition, the amplitude of the Fourier component hk of the interface displacement contributes as
with L the lateral system size, and using the capillary length ξ =
(21). As in ref. 5, we define the nondimensional k̄ = ξk, and then combine these two results to calculate the theoretical dynamic correlation function as an integral
with τ = ξη/ γ̃ as the capillary time, the characteristic time scale for decay of interfacial fluctuations (21). In particular, τ corresponds to the time scale for the slowest decaying wavelength, which is at the capillary length scale
ξ.
We vary τ and γ̃ to find the best fit to our data, and plot this as the solid line in Fig. 2, finding excellent agreement using γ̃ = 1.2 kBT/d2. Our value of γ̃ = 1.20 ± 0.05 (in units of kBT/d2) is a similar order of magnitude to previously found values of γ for hard spheres, which range from 0.11 to 0.78 (8, 9, 12, 22–24). Our measured γ̃ is larger than these values of γ, perhaps due to the above-noted anisotropy of γ̃, for which we may have a stiff direction (8).
We note several limitations with our measurement of the interfacial stiffness. First, our result is calculated for one particular interface orientation, rather than averaging over many different crystal orientations. Second, the theory of refs. 5 and 21 is derived for liquid–gas interfaces, where the surface tension is isotropic. As noted above, the local interfacial stiffness is an adequate replacement for the isotropic surface tension (8). Third, the theory also assumes that both phases have a well-defined viscosity, whereas one of our phases is a colloidal crystal. The crystalline phase could be considered as a liquid with a very large viscosity, but this would then predict a very large capillary time, in contrast with our observed τ = 750 s. This value is consistent with a viscosity η = 50 mPa·s ≈ 20 η0 (compared with the solvent viscosity η0). This value is approximately half of what could be expected for high-volume fraction colloidal samples at volume fractions ϕ = 0.43 − 0.49 (25), given that η here represents the sum of the viscosities of the two phases. To summarize, our crystal-liquid interface has capillary waves which fluctuate in a way consistent with predictions for liquid–gas capillary waves, but interpretations of this observation should be done cautiously.
Additionally we note that the space-averaged height 
h(x,y,t)
x,y fluctuates but does not monotonically increase or decrease, confirming that we observe an equilibrium interface rather than
a system which is crystallizing or melting. However, as can be seen in Fig. 1B, the exact interface position fluctuates only within a certain range over the duration of our experiment, precluding us from
a true ensemble-averaged measurement. For a longer duration experiment, the time-averaged shaded region in Fig. 1B would presumably meander less and be more uniform across the y direction.
We now turn to the structural details of the crystal–liquid interface. As mentioned above, we rotate the data around the x axis by 15° clockwise (as seen from the view in Fig. 1), so that the crystal structure is aligned with the new z axis; see Fig. 3A Inset. We then plot the number density d3n(z) as a function of z/d in Fig. 3A and see oscillations similar to what has been seen in experiments (6) and simulations (2, 10, 11). The smooth line in Fig. 3A is a hyperbolic tangent fit to the data. Following previous work, we define the interfacial width as the 10–90 width, where
the hyperbolic tangent function goes from 10% of its value to 90% of its value. We find W10−90 = 9.6d, quite similar to what has been seen previously (2, 6). Some of this width is due to capillary waves, and some may be due to particles in the crystalline lattice (but near the
interface) having larger fluctuations around their mean positions (2). We checked for this latter influence by low-pass filtering (2) the density profile of Fig. 3A and found a similar W10−90 width, although this width depends on the details of the filter—this filtering also shifts the halfway point of the hyperbolic
tangent fit toward the liquid side by 4d. We also calculate the width based on the structural ordering, by plotting the average number of ordered neighbors 
No
as a function of z′/d in Fig. 3B. Based on this measure, the profile is slightly sharper, with W10−90 = 6.1d; this sharpness agrees with previous results (2, 6).
Two views of the interfacial profile in the “laboratory” reference frame. (A) Interfacial profile showing the number density profile d3n(z) as a function of z/d. The smooth curve is a hyperbolic tangent fit to the envelope, with 10–90 width of 9.6d. Inset shows the crystal orientation used to determine this curve, after the sample has been rotated slightly. (B) Bond-order profile showing the average number of ordered neighbors particles have, as a function of z/d. The dotted curve is again a hyperbolic tangent fit, with a 10–90 width of 6.1d. By definition, 
N0
= 8 at z = 0. For both A and B, the data are time averaged over 2,000 s (80 3D images).
However, Fig. 1 implies that measurements that average over x and y will artificially broaden the interface, given that the interface is not flat. Due to the fractal structure of the capillary waves, the interfacial width depends on the total system size. Our measured widths depend on the size of the observation region and would increase if the observed region was larger and encompassed more of the sample (7). Our data allow us a closer examination of the detailed behavior near the interface. For each particle, we calculate the distance s to the instantaneous interface position by using the procedure illustrated in Fig. 4.
Sketch defining the distance from the interface s. For crystalline particles, the distance s is found to the closest noncrystalline particle (any particle with No < 8). This distance is defined to be negative. For liquid particles, the distance s is the closest distance to the nearest crystalline particle (No ≥ 8). Based on this definition, the region −d < s < d is excluded, as no two particles are closer than their diameters d. Thus, we shift the distance s by +d/2 for crystalline particles, and by −d/2 for liquid particles. This shift leaves a gap of size d, reflecting that the interface truly lies between the crystalline and liquid particles.
To examine the transition from order to disorder, we calculate the average number of ordered neighbors 
No
now as a function of s, and plot the result in Fig. 5A. By definition, No ≥ 8 for crystalline particles and No ≤ 7 for liquid-like particles, thus producing a gap in 
No
at the interface. Fig. 5A shows that based on the ordering, the interface is quite sharp, with a width W10−90 = 1.5 d, in contrast to the interpretation of Fig. 3. By this measure, the interface is locally a sharp, well-defined interface, as has been conjectured (3, 4).
Three views of the intrinsic interface profile. (A) Bond-order profile (solid line), showing the average number of ordered neighbors of particles as a function of s/d. s is the distance from the interface, as described in Fig. 4. The dotted line is a hyperbolic tangent fit with W10−90 = 1.5d. (B) Average distance Δr moved in a given Δt as indicated, as a function of s/d. The dotted lines are hyperbolic tangent fits, each with W10−90 = 2.4d. Here s is the distance measured at the particle's initial position at the start of the displacement Δr. (C) Nondimensional number density d3n as a function of s/d.
Fig. 5B shows the mobility Δr/d, measured across the interface (defined for two different time scales Δt). Here we see the interfacial width is W10−90 = 2.4d, significantly broader than that determined by the structure but still fairly narrow, supporting the interpretation of a
meandering but sharp interface. On the liquid side, the mobility is still increasing for s>d where 
No
has reached the steady-state liquid value; the halfway point of the hyperbolic tangent fit is at s = 0.2d for the Δt = 75 s data and at s = 0.8d for the Δt = 750 s data. This result is understandable as the crystalline particles form a relatively immobile region: Thus, the more
mobile liquid-like particles are slowed by proximity to the crystal, similar to a hydrodynamic “no-slip” condition. Examining
the individual components of motion (Δx, Δy, Δz) we find no significant differences, indicating that the motion is isotropic, as seen in simulations (2). Although our data for Δt > 750 s are noisy, we can estimate that the diffusivity on the liquid side is ≈50× smaller than the diffusivity in a dilute
suspension, in reasonable agreement with hard-sphere simulations (2).
To examine the intrinsic density profile, we plot the number density d3n(s) in Fig. 5C. The crystal structure has a fixed orientation and, thus, relative to the curving interface, the layers seen in Fig. 3A diminish away from the interface (the region s < 0). On the liquid side (s > 0), layering relative to the intrinsic interface is seen with two clear layers—fewer layers than the crystal side. This finding reinforces that the layering shown in Fig. 3A for the region z > 0 is mostly due to the uneven interface.
Discussion
Note that in this work we have studied slightly charged particles rather than ideal hard spheres. Perhaps surprisingly, our results compare reasonably well with hard-sphere simulations such as those in refs. 2, 8, 9, and 24. Thus, the influence of the charges shift the phase boundaries as noted above (a freezing volume fraction ϕmelt = 0.49 for our experiments as compared with 0.545 for hard spheres), but the charges do not seem to strongly modify the structure or dynamics of the interface. It is likely that particle motion is somewhat slowed due to smaller effective cage sizes (26) and thus modifies the plateau levels of Fig. 5B. However, this slowness does not seem to modify the extent of the transition region of the interface, and indeed one might expect that a long-range interaction would only broaden the interface, thus making it more striking that we observe such narrow interface widths.
We have studied the equilibrium crystal–liquid interface of a dense colloidal suspension. We observe a rough interface with width 6–10d, depending on the property examined. Capillary waves account for some of this broadening, although overall the mean-square fluctuations from capillary waves have magnitude gh(Δt = 0) ≈1.1d2. Thus, there may be a base structure comprised of crystalline facets, as suggested by the darkest particles shown in Fig. 1B. These facets may be slower to rearrange, and their potential existence likely broadens the observed interface width.
We also studied the profile in the direction locally perpendicular to the rough interface, and find that this intrinsic interface is much sharper, 1.5d and 2.4d based on structure and dynamics, respectively. These results emphasize that the observed broadness of the interface is only apparent, and depends on the measurement length scales (7) and time scales (2). The underlying spatial transition from one phase to the other is quite sharp, confirming the classic picture (3, 4). Our data provide a useful test for future models of the intrinsic profile.
Materials and Methods
Our samples are composed of colloidal polymethyl(methacrylate) (PMMA) particles, sterically stabilized to prevent aggregation
(13). The particles have mean diameter d = 2.30 ± 0.02 μm and a polydispersity of ≈5%. The solvent is a mixture of cyclohexylbromide and decalin, chosen to closely
match the density and index of refraction of the particles, with viscosity η0 = 2.25 mPa·s (27). However, we intentionally slightly density-mismatch the fluid by adding an excess of decalin, which is more dense than
the particles. This mismatch lets us gravitationally induce a crystal–liquid transition within our sample chamber. Our particle–fluid
density mismatch is less than Δρ ≈5 × 10−4 g/mL, giving us a gravitational scale height kBT/ ΔρV g > 130 μm, using the temperature T (295 K for these experiments), the particle volume V =
d3, and the gravitational acceleration g. In our solvent, the particles have a slight charge, shifting the freezing transition volume fractions to ϕfreeze = 0.43 and the melting transition to ϕmelt = 0.49, similar to values for other weakly charged colloidal PMMA experiments (12). The existence of the charge is also apparent from comparisons of the pair correlation function g(r) taken at different volume fractions. For hard spheres, the first peak of g(r) should always be at r = d, whereas for our samples we find the peak at r > d, and the peak position varies slightly with ϕ. However, this peak position is still fairly close to d; for example, in the crystalline region we find the peak at r = 1.04d, and in the liquid region we find the peak at r = 1.15d, suggesting that the charging is not too extreme (28, 29).
Our microscope sample chambers have dimensions 0.4 mm × 3 mm × 30 mm and are stored with the long dimension oriented vertically. Because of the slight density mismatch between the solvent and the particles, gravity sets up a slight concentration gradient. We let the samples equilibrate in the sample chamber for more than one month before taking data. We make one important modification to our microscope: The microscope base is tilted 90° so that the objective lens points horizontally (30). The sample is placed on the microscope stage with the orientation kept the same as the storage conditions. Thus, we study a completely equilibrated sample at a stable crystal–liquid interface, and the direction of gravity (−ẑ) points perpendicular to the optical axis.
The particles are dyed with rhodamine 6G so that they can be viewed with a laser-scanning confocal microscope (15, 27). We acquire images of size 20 × 55 × 60 μm3 once every 25 s, where the long direction (z) is parallel to gravity. The images were taken from 20 to 40 μm away from the coverslip to avoid direct influence of the walls. As noted above, the full sample size is 0.4 × 3 × 30 mm3, where the smallest dimension is x, and the largest direction is z; thus, the imaged region is only a small volume within the sample. The 3D images are analyzed to determine particle positions with a resolution of 0.1 μm parallel to the optical axis (x) (27, 31), and a resolution of 0.05 μm perpendicular to the optical axis (y and z). Because this is a dense sample, particles do not move far between images, and thus we follow their motion by using standard particle-tracking techniques (31).
Acknowledgments
We thank D. G. A. L. Aarts, M. Asta, G. C. Cianci, B. B. Laird, W. K. Kegel, and T. Witten for helpful discussions. We thank A. B. Schofield and W. C. K. Poon (University of Edinburgh, Edinburgh, United Kingdom) for our colloidal samples. This material is based upon work supported by National Science Foundation Grant 0239109.
Footnotes
- 1To whom correspondence should be addressed. E-mail: weeks{at}physics.emory.edu
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Author contributions: J.H.-G. performed research; J.H.-G. and E.R.W. analyzed data; E.R.W. designed research; and E.R.W. wrote the paper.
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The authors declare no conflict of interest.
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This article is a PNAS Direct Submission.
