Equilibrium fluid-crystal interfacial free energy of bcc-crystallizing aqueous suspensions of polydisperse charged spheres

The interfacial free energy is a central quantity in crystallization from the metastable melt. In suspensions of charged colloidal spheres, nucleation and growth kinetics can be accurately measured from optical experiments. In previous work, from these data effective nonequilibrium values for the interfacial free energy between the emerging bcc nuclei and the adjacent melt in dependence on the chemical potential difference between melt phase and crystal phase were derived using classical nucleation theory (CNT). A strictly linear increase of the interfacial free energy was observed as a function of increased metastability. Here, we further analyze these data for five aqueous suspensions of charged spheres and one binary mixture. We utilize a simple extrapolation scheme and interpret our findings in view of Turnbull's empirical rule. This enables us to present the first systematic experimental estimates for a reduced interfacial free energy, ${\sigma }_{0,bcc}$, between the bcc-crystal phase and the coexisting equilibrium fluid. Values obtained for ${\sigma }_{0,bcc}$ are on the order of a few $k_{B}T$. Their values are not correlated to any of the electrostatic interaction parameters but rather show a systematic decrease with increasing size polydispersity and a lower value for the mixture as compared to the pure components. At the same time, ${\sigma }_0$ also shows an approximately linear correlation to the entropy of freezing. The equilibrium interfacial free energy of strictly monodisperse charged spheres may therefore be still greater.

Figure 1

Dependence of the reduced interfacial free energies on metastability $\mathrm{\Delta }\mu$ for the indicated species. Interfacial free energies $\gamma$ (in $J/m^2$) were normalized by $n^{-2/3}$ and plotted in units of $k_{B}T$ as quoted from the original literature or re-evaluated from the original data using a corrected value for $\mathrm{\Delta }\mu$. Solid lines are least-square fits of $\sigma ={\sigma }_0+m\mathrm{\Delta }\mu$. Note that here and throughout, we adopt the convention to consider the crystal as educt and the melt as product, as to obtain positive values for $\mathrm{\Delta }\mu ,\mathrm{\Delta }{H}_f$, and $\mathrm{\Delta }{S}_f$.

Figure 2

Extrapolation scheme based on Turnbull's rule: ${\sigma }_0=C_T\mathrm{\Delta }{H}_f/N_A=C_T{T}_M\mathrm{\Delta }{S}_f/N_A$.

Figure 3

Correlations between the extrapolated reduced equilibrium IFEs and various particle characteristics. (a) Effective charges $Z_{\mathrm{eff},G}$; (b) the number averaged mean particle diameter $2a$; (c) effective surface potential ${\mathrm{\Psi }}_{\mathrm{eff}}$; (d) effective temperature, $T_{\mathrm{eff}}=k_{B}T/V\left(d_{NN}\right)$, at melting; (e) coupling strength at melting. No clear correlation of ${\sigma }_0$ to any of these interaction parameters is obtained.

Figure 4

Correlation between the extrapolated reduced equilibrium IFEs and the polydispersity index PI. A clear anticorrelation is observed: ${\sigma }_0$ decreases with increasing PI for all single component samples. The case of the mixture is denoted by the horizontal bar. Its length and position denote the range of PIs covered by the involved single component samples. Note that its position does not correspond to the effective PI of the bimodal size distribution of the mixture. This quantity would be much larger, but cannot be obtained from the equation for monomodal PIs: $\text{PI}=s_a/\overline{a}$. Note further that its value lies significantly below those of the corresponding single component species. For comparison, we also show values for the equilibrium IFE as obtained from simulations of macroscopic flat fluid-crystal interfaces of monodisperse HSs (solid horizontal line with thickness corresponding to the spread of published data) and a point Yukawa system (dashed line). (For further details, see text.)

Figure 5

Correlation between the extrapolated equilibrium reduced IFE and the enthalpy of fusion equaling the entropy of fusion times the melting temperature. A good linear correlation (correlation coefficient $r=0.988$) is observed as expected from Turnbull's rule and its interpretation by Laird [41, 53].

Figure 6

Turnbull plot of the reduced equilibrium interfacial free energy vs the equilibrium enthalpy of fusion. Shown are our data for bcc-crystallizing colloids (circles), simulation data for bcc-crystallizing metals (up triangles) [85], bcc-crystallizing point Yukawa systems (down triangles) [84], and fcc crystallizing metals (diamonds) [85] as well as experimental data for fcc crystallizing metals (squares) [46]. Lines correspond to the indicated average values of Turnbull coefficients as quoted from [84, 85], and [46] for simulation and experimental data, respectively. For our data we find an average value of $C_{T,bcc}=0.31\pm 0.03$ (thick solid line) which appears to be remarkably close to that expected for bcc metals.

Figure 7

(a) Equilibrium reduced IFEs ${\sigma }_0$ vs $m$. No correlation between these two quantities is observed. (b) Plot of $T_M\mathrm{\Delta }{S}_f=\mathrm{\Delta }{H}_f$ vs the polydispersity index PI. A clear decrease is observed. In addition, note the strong decrease also for the value of the mixture (horizontal bar), as compared to those of the two pure components (leftmost circles).

Figure 8

Correlations between the fitted slope $m$ and various particle characteristics. (a) Effective charges ${}_{\mathrm{eff}.G}$; (b) the number averaged mean particle diameter $2a$; (c) effective surface potential ${\mathrm{\Psi }}_{\mathrm{eff}}$; (d) effective temperature, $T_{\mathrm{eff}}=k_{B}T/V\left(d_{NN}\right)$, at melting; (e) coupling strength at melting, $\kappa d_{NN}$; and (f) polydispersity index PI. Only in (e), a weak correlation is observed with a slightly negative slope and a correlation coefficient of $r=0.53$. Note that in (f), $m$ does not vary with PI.

Figure 9

Typical wall crystal growth curves obtained for different particle volume fractions as indicated.

Figure 10

Growth velocities of PS90 in dependence on the reduced density. The solid line is a fit using Eq. (C1) with Eq. (C2). The fit parameters obtained are $B_{PS90}=4\pm 0.6$ for the proportionality constant and $v_{\infty }=8.4\pm 0.3\mu \text{m/s}$ for the limiting velocity.

Figure 11

Density dependence of the interfacial free energies $\gamma$ of PnBAPS68 obtained from different evaluation schemes. Solid symbols denote use of CNT with $J_0$ calculated using Eq. (C6) in Eq. (C5) with $A=1$ and $D_{\mathrm{eff}}=0.1D_0$ and different input data: average nucleation rate density $J_{AVR}$ (squares), maximum nucleation rate density $J_{\mathrm{MAX}}$ (diamonds), or steady state nucleation rate density $J_{SS}$ (stars); hatched area denotes the maximum systematic change by setting $D_{\mathrm{eff}}=D_0$ (upper bound) or $D_{\mathrm{eff}}={10}^{-3}{D}_0$ (lower bound). Open symbols denote data from two different experimental runs using a corrected version graphical evaluation [146]. The solid line gives a least-square linear fit to these data. The dashed red line gives the $\gamma \left(n\right)$ values obtained from the fit of Eq. (C8) to the data as shown in Fig. 12. All curves increase linearly with $n$ for $n>n_F$. Inset: time dependent nucleation rate densities for three densities $n$ as indicated. With increased $n$, transient effects become more pronounced. Solid lines are the fits of Kashchiev's theory yielding $J_{SS}$ [60].

Figure 12

Nucleation rate densities of PnBAPS68 as measured by video microscopy $\left(n=18\text{–}19.9\mu {\text{m}}^{-3}\right)$, postsolidification crystal size analysis $\left(n=18\text{–}33\mu {\text{m}}^{-3}\right)$, and static light scattering $\left(n=25\text{–}67\mu {\text{m}}^{-3}\right)$. Note the excellent agreement between data derived from different methods. The solid line is a least-square fit of Eq. (C8) to the data using $A,D_L^S\left(n\right)$, and $\gamma \left(n\right)$ as free parameters. An excellent description of the experimental data can be observed. The obtained $\gamma \left(n\right)$ are shown in Fig. 11 as a dashed red line.

Figure 13

Nucleation rate densities of different colloidal species vs volume fraction. Open symbols denote charged sphere systems with their diameters indicated. Closed symbols refer to data obtained for two HS systems (PMMA890: [34]; PMMA402: [29]. Note the low volume fractions of CS and the steepness of the increase in $J$