Nanoscale Critical Phenomena in a Complex Fluid Studied by X-Ray Photon Correlation Spectroscopy

The advent of high-speed x-ray photon correlation spectroscopy now allows the study of critical phenomena in fluids to much smaller length scales and over a wider range of temperatures than is possible with dynamic light scattering. We present an x-ray photon correlation spectroscopy study of critical fluctuation dynamics in a complex fluid typical of those used in liquid-liquid extraction (LLE) of ions, dodecane-DMDBTDMA with extracted aqueous $\mathrm{Ce}({\mathrm{NO}}_3{)}_3$. We observe good agreement with both static and dynamic scaling without the need for significant noncritical background corrections. Critical exponents agree with 3D Ising values, and the fluctuation dynamics are described by simple exponential relaxation. The form of the dynamic master curve deviates somewhat from the Kawasaki result, with a more abrupt transition between the critical and noncritical asymptotic behavior. The concepts of critical phenomena thus provide a quantitative framework for understanding the structure and dynamics of LLE systems and a path forward to new LLE processes.

Figure 1

Schematic phase diagram for a liquid-liquid extraction system, expressed as a ternary system between aqueous solution A, organic solvent S, and amphiphilic extractant E. (a),(b) Isothermal cuts above and below, respectively, the temperature at which the 3-phase region and associated critical point appear. Colors denote 1-phase regions (yellow), 2-phase regions (green), and 3-phase regions (orange). (c) Pseudobinary cut at fixed A/E ratio given by dash-dot line in (a),(b), showing the coexistence (solid) and spinodal (dashed) curves.

Figure 2

Time-averaged results. (a) Normalized scattered intensity $I$ as a function of wave number $q$ for different temperatures, plotted on logarithmic scales. Only selected values of the total 20 $T$ and 180 $q$ values measured are shown. Curves are fits with Eq. (1). Inset: Data at all temperatures rescaled using parameters obtained from fits. (b),(c) Plots of fit parameters $I_{q0}$ and $\xi$ vs $T-T_{\mathrm{sp}}$, showing power-law fits. Values with uncertainties are given in the Supplemental Material [35], Table S1.

Figure 3

Dynamics of LLE system. (a) Measured $g^{(2)}(\mathrm{\Delta }t)$ at $T=291.6\mathrm{K}$ and selected $q$, showing fits (curves) using Eq. (4). (b) Extracted fluctuation correlation times $\tau$ as a function of $q$ at selected $T$, showing fits (lines) to obtain power-law slopes $-z_{\mathrm{eff}}$. Inset: Effective dynamic critical exponent $z_{\mathrm{eff}}$ as a function of $T$ for the $q$ range of the measurements.

Figure 4

Reduced diffusivity $D^{*}$ vs reduced wave number $x\equiv q\xi$ for all $q$ and $T$, compared to modified Kawasaki scaling function $\mathrm{\Omega }$ and empirical function ${\mathrm{\Omega }}_{\mathrm{emp}}$. Color of markers indicates $T$ of experimental data points: green (296.2 K) to blue (291.6 K). Residuals of the two fits are compared in [35].