Capillary Waves at the Transition from Propagating to Overdamped Behavior

We measure the dispersion relation of capillary waves on a liquid surface by heterodyne x-ray photon correlation spectroscopy near the transition from propagating to overdamped dynamic behavior. A strong deviation of the propagation frequency from the small-damping result ${\omega }_p\propto k^{3/2}$ is observed long before the actual transition where $(\partial {\omega }_p/\partial k)<0$ and ${\omega }_p$ tends to zero. This behavior is successfully described by expressions derived within linear response theory.

Figure 1

Time-correlation functions of the scattered intensity at $q=5.6\times {10}^{-6}{\AA }^{-1}$ taken at different temperatures. The data have been shifted along the ordinate.

Figure 2

Experimentally determined dispersion relations of the propagation frequency ${\omega }_p$ (circles) and the damping constant $\Gamma$ (squares) for the $\mathrm{\text{water}}/\mathrm{\text{glycerol}}$ mixture at (a) $30°\mathrm{C}$ and (b) $12°\mathrm{C}$. The data deviate significantly from the small-damping expressions ${\omega }_0=q^{3/2}\sqrt{\sigma /\rho }$ and ${\Gamma }_0=2\eta q^2/\rho$ [$\eta =9.1\mathrm{c}\mathrm{P}$, $\sigma =0.047\mathrm{N}/\mathrm{m}$, $\rho =1156\mathrm{k}\mathrm{g}/{\mathrm{m}}^3$ ($30°\mathrm{C}$) and $\eta =23.7\mathrm{c}\mathrm{P}$, $\sigma =0.050\mathrm{N}/\mathrm{m}$, $\rho =1156\mathrm{k}\mathrm{g}/{\mathrm{m}}^3$ ($12°\mathrm{C}$)] at both temperatures. At $T=12°\mathrm{C}$ a crossover (${\omega }_p=0$) is evident at $q\approx 8\times {10}^{-6}{\AA }^{-1}$. In the overdamped region ($q>8\times {10}^{-6}{\AA }^{-1}$) a clear offset of the data (squares) from the strong-damping result $\Gamma =\sigma q/(2\eta )$ (dotted line) is observed in (b).

Figure 3

The dynamic structure factor $S(q,\omega )$ calculated at $k=1,2,\dots \times {10}^{-6}{\AA }^{-1}$ (top to bottom). The inset shows the crossover behavior of the center frequency ${\omega }_p$ (circles) and the HWHM $\Gamma$ (squares) where ${\omega }_p$ is well described by Eq. (5) (solid line). The dashed lines in the inset show ${\omega }_0=k^{3/2}\sqrt{\sigma /\rho }$ and ${\Gamma }_0=2\eta k^2/\rho$. The dotted line illustrates the strong-damping result $\Gamma =\sigma q/(2\eta )$.

Figure 4

Experimental data (from Fig. 2) that are well described by linear response theory (solid lines). The insets show inverse Fourier transformed time-correlation functions (circles) measured at $30°\mathrm{C}$ and $12°\mathrm{C}$ with $q=3.5\times {10}^{-6}{\AA }^{-1}$, and the corresponding calculation of $S(\omega )$ (solid line) following Eq. (3).