Experiments

  

Figure 2: $52 \times 52~\mu{\mathrm m}^2$ fields of view of the (111) surfaces of the colloidal crystals at (a) $\phi = 0.011$ and (c) $\phi = 0.026$. The spheres' trajectories over 10 seconds appear in (b) and (d) for the same fields of view.

The colloidal suspensions used in this study consist of polystyrene sulfate spheres 0.654 $\pm$ 0.009 $\mu$m in diameter (Catalog # 5065, Duke Scientific, Palo Alto, CA), dispersed in deionized water. These spheres develop strongly negative surface charges due to dissociation of their ionic surface groups. Pair interaction measurements indicate an effective surface potential of $\zeta_0 = -140$ meV [12]. An aqueous suspension of these spheres was purified by 3 months of dialysis against deionized water followed by tumbling with mixed-bed ion exchange resin. Samples of the cleaned suspension were filled into sample chambers at volume fractions $\phi = 0.011 \pm 0.003$ and $0.026 \pm 0.004$. Both suspensions crystallized into face-centered cubic (FCC) structures. The remainder of this Article concerns the properties of these two crystals.

The sample container depicted schematically in Fig. 1 is created by hermetically sealing the edges of a #1 glass coverslip to a glass microscope slide with a high-purity UV curing adhesive (Norland Optical Adhesive Type 88, Norland Products, New Brunswick, NJ). The enclosed volume has a visible area of $1 \times 1.5~{\mathrm cm}^2$and is roughly $40~\mu$m thick. Access to the sample volume is provided by two glass tubes extending from holes drilled through the slide. All parts were thoroughly cleaned before assembly [13] and the finished sample cells were flushed with deionized water in a nitrogen-purged glove box before filling. After filling, the tubes serve as reservoirs of colloid and contain mixed bed ion exchange resin to maintain the chemical purity of the interior. The ends of the tubes are continually flushed with water-saturated Ar to prevent the suspension from becoming acidified by absorbing CO₂ from the air. Applying an alternating pressure difference across the tubes causes the colloid to flow back and forth through the sample cell and past the ion exchange resin, and further lowers the ionic strength of the suspension. Filled sample cells were allowed to equilibrate for several weeks before the measurements we describe below were performed.

Observations were made with an Olympus IMT-2 inverted optical microscope. A combination of a 100$\times$ N.A. 1.4 oil immersion objective and a 10$\times$ video eyepiece provides a total magnification of 172 nm/pixel on the attached charge-coupled device camera, so that a single sphere's image subtends an area of roughly $4 \times 4$ square pixels. At this magnification, the 615$\times$470 pixel field of view corresponds to a visible area of 106$\times$82 $\mu$m². The $\pm$100 nm depth of focus of the optical system is comparable to the diameter of a sphere so that only a single crystal layer is resolved at a given focal depth. We record the spheres' motions at 1/30 second intervals using a SONY EVO-9650 Hi-8 computer-controlled video deck. The video tapes are then digitized with a Data Translation DT-3851A frame grabber before being analyzed. Figures 2(a) and (c) show typical digitized images of the two crystals' (111) planes in the layer closest to the glass wall.

Each sphere in Fig. 2 appears as a bright blur on a dark background. We use precision image analysis techniques [4] to locate the spheres' centroid to within 20 nm in the plane. Their locations in a sequence of video frames are then linked into trajectories using a maximum likelihood algorithm [4]. The data for this study consists of the time-resolved sphere trajectories

$$\rho({\mathbf r},t) = \sum_{i=1}^{N} \delta({\mathbf r} - {\mathbf r}_i(t)),$$ (1)

for $N \approx 1000$ spheres in the field of view, where ${\bf r}_i(t)$ is the position of sphere i at time t. Since we image only one layer of spheres, the experimentally obtained trajectories constitute a two-dimensional projection of the three-dimensional crystal's surface layer. Similar data sets were obtained for the second and third layers by adjusting the microscope's focus. Spheres' out-of-plane motions are small enough that they never move completely out of focus unless they hop to another plane, a comparatively rare event. Because imaging contrast degrades rapidly with depth into the crystal, the data presented below were obtained from the surface layer. Consistent results, albeit with considerably worse statistics, were obtained also from the second layer.

Small variations in atmospheric pressure cause the crystals to drift slightly relative to the imaging volume during data acquisition. To avoid artifacts introduced by this secular drift, we subtract off the mean displacement over the field of view at each time step. Typical trajectory sequences appear in Figs. 2(b) and (d).

Once the spheres' centroids have been located in the field of view, we apply computational geometry techniques [14] to analyze their distribution. In particular, we use the Delaunay triangulation to uniquely identify the set of nearest neighbors for each sphere in a snapshot. Given the nearest-neighbor connectivity, we then calculate lattice descriptors such as the mean nearest-neighbor separation and the principal reciprocal lattice vectors. For the relatively dilute crystal shown in Fig. 2(a), the mean separation is $a = 3.52 \pm 0.12~\mu$m. Focusing up through the crystal reveals a $2.5~\mu$m interlayer spacing. This is consistent with the FCC structure and is inconsistent with a body-centered cubic (BCC) lattice whose equivalent interlayer spacing would be only half as big in this projection. The corresponding lattice constant in the FCC crystal's (111) plane is $d = 3.05 \pm 0.10~\mu$m. By an equivalent analysis, the dense crystal also adopts the FCC structure, with $a = 2.50 \pm 0.10~\mu$m and $d = 2.17 \pm 0.09~\mu$m.

  

Figure 3: Pair correlation functions for the two crystals in this study. (a) $\phi = 0.011$. (b) $\phi = 0.026$. Data points calculated according to Eq. (2) are overlaid with non-linear least-squares fits to Eq. (3).

The extent of the crystals' ordering can be measured with the time-averaged pair correlation function

$$g({\mathbf r}) = \frac{1}{\tau N \rho} \, \int_0^\tau dt \,... ...- {\mathbf r}, t ) \, \rho( {\mathbf x}, t ) \, d{\mathbf x},$$ (2)

where $\rho$ is the mean number density of spheres. The pair correlation functions for the crystals in this study, averaged over angles, appear in Fig. 3. We estimate the crystals' correlation lengths by comparison with ideal triangular lattices. An ideal triangular lattice's pair correlation function, g_a(r), has delta function peaks at spacings parameterized by the nearest-neighbor spacing a. These peaks are broadened by noise and measurement error in real crystals. Their magnitude falls off with separation owing to an accumulation of lattice defects and distortions. Thus, the experimental pair correlation function is quite well fit by the form [15]

$$g(r) = \left[ \int_0^\infty g_a(r-x) \, e^{ -c x^2 } \, dx - 1 \right] \: e^{ - \frac{r}{\xi}} + 1.$$ (3)

The simplistic exponential model for the decay of correlations tends to underestimate the height of the first peak in g(r)but is still useful for quantifying the extent of ordering. The correlation lengths, $\xi = 8 a$ and 10 a, for the dilute and dense crystals respectively are consistent with the observed separation between lattice defects such as those visible in Fig. 2.

The angle-averaged structure factor,

$$S(q) = 1 + 2 \pi \, \int_0^\infty \left[ g(r) - 1 \right] \, J_0(qr) \, r \, dr,$$ (4)

offers additional insights into the suspensions' thermodynamic state. This form is appropriate for our two-dimensional data sets. Rather than providing a view of the crystals' three-dimensional structure, Eq. (4) yields a particular projection along the [111] direction. Structure factors for the crystals in this study appear in Fig. 4.

  

Figure 4: Structure factors for the two crystals averaged over 100 video frames. The dashed line at S(q) = 2.85 represents the Hansen-Verlet criterion. (a) $\phi = 0.011$. (b) $\phi = 0.026$.

Hansen and Verlet [16] observed that the structure factor's first peak at q = q₀ reaches the universal value S(q₀) = 2.85 when fluids freeze to FCC crystals. Figure 4 reveals that both suspensions are crystallized according to the Hansen-Verlet criterion.

The spheres' dynamics further allow us to assess the crystal's thermodynamic state. Building upon a suggestion by Lindemann [17], Gilvarry [18] postulated that crystals melt when the ensemble averaged r.m.s. lattice displacement

$$\Delta = {\langle \left\vert {\mathbf r}_i(t) - \bar {\mathbf r}_i \right\vert^2 \rangle_{t,N}}^{\frac{1}{2}},$$ (5)

of atoms labeled i about their mean positions, $\bar {\mathbf r}_i$, reaches about 10 percent of the crystals' nearest neighbor spacing, a. The average in Eq. (5) is taken over time and the ensemble of particles. Cho refined the estimated threshold for FCC crystals to $\Delta / a = 0.0967$ by compiling experimental data for a variety of metals [19]. The higher-density crystal in this study has $\Delta / a = 0.072 \pm 0.006$ and so is crystalline according to the Lindemann criterion. The lower-density crystal has $\Delta / a = 0.110 \pm 0.010$. Even though the entire field of view is well ordered and frozen according the the Hansen-Verlet criterion, this suspension is likely to be in the crystal-fluid coexistence regime. Proximity to the melting point is desirable for our purposes, not only because we are interested in measuring colloidal crystals' properties near melting, but also because softer crystals' dynamics are more easily measured by real space imaging.

Having established the crystals' states, we are in a position to probe the interactions responsible for their structure and dynamics. This is a useful goal in itself since measuring particle interactions is an important component in process control for a wide range of industrial applications [20]. Direct imaging is almost entirely noninvasive and nonperturbative and so offers possible benefits over electrokinetic and mechanical methods commonly used.