Nondiffusive decay of gradient-driven fluctuations in a free-diffusion process

We report the results of an experimental study of the static and dynamic properties of long wavelength concentration fluctuations in a mixture of glycerol and water undergoing free diffusion. The shadowgraph method was used to measure both the mean-squared amplitude and the temporal correlation function of the fluctuations for wave vectors so small as to be inaccessible to dynamic light scattering. For a fluid with a stabilizing vertical concentration gradient, the fluctuations are predicted to have a decay rate that increases with decreasing wave vector $q$, for wave vectors below a cutoff wave vector $q_C$, determined by gravity and the fluid properties. This behavior is caused by buoyant forces acting on the fluctuations. We find that for wave vectors above $\sim q_C$, the decay rate does vary in the normal diffusive manner as $D{q}^2$, where $D$ is the mass diffusion coefficient. Furthermore, for $q\lesssim q_C$ we find that longer wavelength fluctuations decay more rapidly than do shorter wavelength fluctuations, i.e., the behavior is nondiffusive, as predicted.

Figure 1

Schematic diagram of the apparatus showing the sample cell (SC) and the shadowgraph arrangement. A clean diverging beam with a nearly Gaussian profile exited from the monomode fiber at F and was collimated by lens L1, with focal length $20\mathrm{cm}$. Mirror M1 redirected the beam vertically downward, where it passed through the sample cell parallel to the concentration gradient. A second mirror, M2, together with lens L2, of focal length $15\mathrm{cm}$, imaged the intensity distribution in the desired visualization plane onto the CCD sensor S.

Figure 2

Scaled difference of two shadowgraph images taken at $t\cong 600\mathrm{s}$, using a nominal visualization distance of $-48\mathrm{cm}$. The fluctuations are visible as the lighter and darker patches. The actual amplitude of the intensity variations was about $\pm 4$ gray levels, and the image has been gray scaled so that $\pm 5$ gray levels cover the range from black to white.

Figure 3

Log-log graph of data for $S\left(q\right)T\left(q\right)$ taken at 600 (squares), 20 760 (circles), 72 000 (triangles), and 156 600 (diamonds) seconds after establishing the gradient. The solid curves are the result of fitting a simple function to the data (see text) to extract values for the crossover wave vector ${\overline{q}}_C$. The effect of the shadowgraph transfer function is evident for both the data and the fits. As can be seen, there is relatively little change in the data between $t=600\mathrm{s}$ and $t=20760\mathrm{s}$, i.e., before the diffusing zone reaches the boundaries. However, once the concentration at the boundaries begins to change, at about $25000\mathrm{s}$, the signal amplitude falls dramatically.

Figure 4

The crossover wave vector ${\overline{q}}_C$ vs time, deduced by fitting static data of the type shown in Fig. 3. The solid curve is the result of vertically averaging the theoretical result for $S\left(q\right)$ to obtain theoretical values for ${\overline{q}}_C$. As can be seen, the data are not consistent with the theory. Because ${\overline{q}}_C$ varies only weakly with the gradient and fluid properties, we believe this discrepancy is significant.

Figure 5

Predicted correlation functions for four wave vectors as calculated for $t=600\mathrm{s}$. The correlation functions were calculated by vertically averaging the theoretical result for $S(q,\omega )$ and Fourier transforming with respect to $\omega$ to obtain $G\left(\tau \right)$. One can see immediately that the theory predicts a smaller decay rate for $q=200{\mathrm{cm}}^{-1}$ than it does for any of the other wave vectors, i.e., the predicted behavior is strongly nondiffusive. One should also note that the theory predicts nonexponential decay at the lower wave vectors.

Figure 6

Measured correlation functions for the same wave vectors as Fig. 5, obtained for $t=600\mathrm{s}$. We observe the predicted nondiffusive behavior, and we observe curvature for the lowest wave vector, much like that shown in Fig. 5. The slight curvature predicted for $q=100{\mathrm{cm}}^{-1}$ was not within our ability to detect.

Figure 7

The decay rate $\Gamma \left(q\right)$ vs $q$ for data taken at $t=14880\mathrm{s}$ [$z_{\mathrm{o}}=-48\mathrm{cm}$ (squares)] and $t=15420\mathrm{s}$ [$z_{\mathrm{o}}=-36\mathrm{cm}$ (circles)]. As can be seen, the decay rate reaches a minimum near $q_C$ (${\overline{q}}_C=126.2$ and $124.6{\mathrm{cm}}^{-1}$ for $t=14880$ and $15420\mathrm{s}$, respectively). The solid line is the theoretical result, obtained as discussed in the text, including the concentration dependence of both $\beta$ and $\nu$.

Figure 8

Experimental (symbols) and theoretical (solid curves) results for $\Gamma \left(q\right)$ vs $q$ for data taken at $t=600$ (squares), 3200 (circles), 19 000 (triangles), and 72 000 (diamonds) seconds. The theoretical results were obtained by vertical averaging, using concentration-dependent fluid properties, as described in the text.

Figure 9

An effective dynamic cutoff wave vector equal to the wave vector of minimal decay rate vs time for both data and theory.