Salt-concentration-dependent nucleation rates in low-metastability colloidal charged sphere melts containing small amounts of doublets

We determined bulk crystal nucleation rates in aqueous suspensions of charged spheres at low metastability. Experiments were performed in dependence on electrolyte concentration and for two different particle number densities. The time-dependent nucleation rate shows a pronounced initial peak, while postsolidification crystal size distributions are skewed towards larger crystallite sizes. At each concentration, the nucleation rate density initially drops exponentially with increasing salt concentration. The full data set, however, shows an unexpected scaling of the nucleation rate densities with metastability times the number density of particles. Parameterization of our results in terms of classical nucleation theory reveals unusually low interfacial free energies of the nucleus surfaces and nucleation barriers well below the thermal energy. We tentatively attribute our observations to the presence of doublets introduced by the employed conditioning technique. We discuss the conditions under which such small seeds may induce nucleation.

Figure 1

AFM images of PS109^* particles deposited on polyvinylpyridine-coated glass slides upon dipping them into the suspension. Scale bar 1 μm. The color-coded $z$-range covers 200 nm from the substrate ($z=0$ nm). (a) Typical scan with no impurities; (b) larger impurity in the lower middle; (c) doublet in the upper middle.

Figure 2

Phase behavior and growth measurements. (a) Phase diagram of PS109^* in dependence on concentration of added electrolyte and number density. Open squares denote samples, for which coexisting phases were observed. Arrows and hatched areas denote the experimental $n$, the observed freezing points, and the range of added electrolyte in the solidification kinetic measurements, respectively. Colors discriminate between the three experiments; red: nucleation experiments at $n=5.4\mu {\mathrm{m}}^{\text{–}3}$; green: nucleation experiments at $n=9.5\mu {\mathrm{m}}^{\text{–}3}$; violet: growth experiments at $n=6.1\mu {\mathrm{m}}^{\text{–}3}$. (b) Growth velocities of wall nucleated crystals as a function of the rescaled energy density Π* as calculated from the experimental parameters and the location of freezing. A WF-growth law fit to the data returns a calibration coefficient of $B=(4.1\pm 0.3)k_{B}T$. Blue and green hatched bars at the top show the matching ranges of metastability covered in the nucleation experiments at $n=5.4\mu {\mathrm{m}}^{\text{–}3}$ and $n=9.5\mu {\mathrm{m}}^{\text{–}3}$, respectively. The red hatched bar indicates the extension of the coexistence region at $n=5.4\mu {\mathrm{m}}^{\text{–}3}$.

Figure 3

Nucleation results. (a) Time-dependent nucleation rates determined from direct video-microscopic observation. The solid line shows the evolution of the fraction of leftover melt volume (right scale). (b) Crystal size distribution for the postsolidification snapshot shown in the inset. Scale bar 1000 μm, image size $8\times 12\mathrm{m}{\mathrm{m}}^2$. Small regions with insufficient color contrast are frequently observed at three-crystallite intersections. We file them under “small uncertain spots.”

Figure 4

Results of crystallization experiments. (a) Crystallite densities versus concentration of added salt. As indicated in the key, densities were derived from sizing at $n=5.4\mu {\mathrm{m}}^{\text{–}3}$ as well as from sizing and counting at $n=9.5\mu {\mathrm{m}}^{\text{–}3}$. Vertical error bars denote the combined statistical and systematic uncertainty of ${\rho }_{\mathrm{c}}$. The error in salt concentration is below symbol size. (b) Nucleation rate densities as a function of concentration of added salt for the two different number densities and evaluation methods as indicated in the key. (c) Nucleation rate densities as a function of metastability for the two different number densities and evaluation methods as indicated in the key. The horizontal error bars denote the systematic uncertainty of Δμ arising from the uncertainty in the determination of the freezing salt concentration combining with the uncertainty in actual salt concentration in the calculation of Π* and the uncertainty of the calibration factor $B$.

Figure 5

CNT parameterization. Symbols as before and indicated in the key. (a) Nucleation rate densities plotted vs $1/{\left(n\mathrm{\Delta }\mu \right)}^2$. Inset: natural logarithm of $J$ plotted vs $1/{\left(n\mathrm{\Delta }\mu \right)}^2$ for the sizing data obtained at $n=5.4\mu {\mathrm{m}}^{\text{–}3}$. Solid lines are least squares fits through three (red) and four (blue) points. (b) Reduced IFE in units of the thermal energy as obtained from the fits versus metastability for the data taken at $n=5.4\mu {\mathrm{m}}^{\text{–}3}$. Solid lines are least-squares linear fits returning the slope $m=C_{\mathrm{T}}$.

Figure 6

Comparison of CNT parameters to those obtained for other charged sphere species. (a) Reduced IFE in dependence on Δμ. Data for PS109^* taken at $n=5.4\mu {\mathrm{m}}^{\text{–}3}$ are compared to two data sets from previous measurements, in which good agreement with the CNT expectations for homogeneous bulk nucleation was observed: PnBAPS68 [29] and PnBAPS70 [75]. Both latter systems show much larger reduced IFE, and these extrapolate to a reduced equilibrium IFE of $1.5k_{B}T$ and $1.7k_{B}T$, respectively. We also display the IFE of HS systems [4, 77], which amounts to $0.51k_{B}T$ (dashed red line). (b) Critical radii in dependence on Δμ. PnBAPS68 [29] and PnBPaPS70 [50, 75] illustrate the behavior expected from CNT for homogeneous bulk nucleation as seen in many previous systems: $r_{\mathrm{crit}}$ decreases with Δμ. For PS109^*, $r_{\mathrm{crit}}$ increases with Δμ. (c) Nucleus free energies ΔG in dependence on nucleus radius as calculated from the obtained CNT parameters. Symbols denote the following: red squares: deionized PnBAPS68 at $n=60.9\mu {\mathrm{m}}^{\text{–}3}$; open green squares: PS109^* at $n=9.5\mu {\mathrm{m}}^{\text{–}3}$; open violet circles: PS109^* at $n=5.4\mu {\mathrm{m}}^{\text{–}3}$ and $c_s=0.65\mu \mathrm{mol}{\mathrm{L}}^{\text{–}1}$. The inset shows a magnification of the free energy curves of PS109^* at $n=5.4\mu {\mathrm{m}}^{\text{–}3}$ with the salt concentration increasing from zero (black solid squares, upper curve at $n=5.4\mu {\mathrm{m}}^{\text{–}3}$) to $c_s=0.65\mu \mathrm{mol}{\mathrm{L}}^{\text{–}1}$ (open violet squares, lowest curve). The arrow denotes increasing salt concentration. Note the low maximum values for PS109^* irrespective of sample parameters and the tendency to decrease further with increasing salt concentration.

Figure 7

Seeded nucleation: (a) Efficiency of seeded nucleation as a function of $n\mathrm{\Delta }\mu$ (lower scale), respectively $n_{\mathrm{seed}}\mathrm{\Delta }\mu$ (upper scale). Data arrange on a straight line (dashed). With increasing $n\mathrm{\Delta }\mu$, efficiencies approach unity from below. (b) Heterogeneous nucleation rate densities as a function of $n\mathrm{\Delta }\mu$. Data arrange on a straight line (solid for comparison) indicating an exponential dependence of $J_{\mathrm{het}}$ on $n\mathrm{\Delta }\mu$.

Figure 8

(a) Conditioning circuit as used in the present crystallization experiments. R: reservoir, P: peristaltic pump, IEX: ion exchange column, C: conductometric cell, ${\mathrm{OC}}_1$, ${\mathrm{OC}}_2$, ${\mathrm{OC}}_3$, … optical flow through cells. All components are connected by gas tight tubings, with flow direction indicated as arrows. After thorough deionization the IEX can be bypassed using the three-way valves (V). Then salt solution can be added to the reservoir under an inert gas atmosphere and its concentration monitored via conductivity. (b) Typical result of prolonged cycling in the peristaltically driven conditioning circuit without integrated filter. The image shows a cropped bright field transmission image (Leitz, N PLAN 10×/0.25; standard video camera). The scale bar is 2.5 μm. The arrow highlights a doublet formed after some cycling in a dilute suspension $(\mathrm{\Phi }={10}^{–4})$ of PnBAPS359 (BASF, Ludwigshafen, Germany).

Figure 9

System characterization. (a) Sample conductivity as a function of number density. The red solid line is a least square fit of Eq. (A1) to the data returning $Z_{\sigma }=459\pm 20e^{-}$. (b) Sample shear modulus as a function of number density. The red solid line is a least-squares fit of Eq. (A5) to the data returning $Z_G=321\pm 18e^{-}$.

Figure 10

Larger objects in colloidal crystals and fluids as observed by different microscopic techniques. (a) Heterogenous doublets; (b) homogeneous doublets from conditioning; (c) larger spheres.

Figure 11

CNT parameterization. (a) Natural logarithm of $J$ plotted vs $1/{\left(n\mathrm{\Delta }\mu \right)}^2$ for the peak data obtained at $n=5.4\mu {\mathrm{m}}^{\text{–}3}$. Solid lines are least squares fits through three (red) and four (blue) points. (b) Natural logarithm of $J$ plotted vs $1/{\left(n\mathrm{\Delta }\mu \right)}^2$ for the sizing (olive) and counting (black) data taken at $n=9.5\mu {\mathrm{m}}^{\text{–}3}$. Solid lines are least-squares fits.

Figure 12

Interfacial free energy as a function of metastability for the sizing and peak data obtained at $n=5.4\mu {\mathrm{m}}^{\text{–}3}$ as denoted by the color code indicated in the key.

Figure 13

Seeding efficiency calculated assuming a constant seed fraction in dependence on metastability. Data shown are for two different number densities and determined by different evaluation approaches, as indicated in the key. Note the pronounced discontinuity at $\mathrm{\Delta }\mu =4k_{B}T$.