Brownian motion in a critical mixture:<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:mi>K</mml:mi></mml:math>-dependent diffusion

Abstract

A light-scattering investigation of Brownian motion in the critical mixture nitroethane-isooctane is reported, along with a theory to explain the results. The diffusion constant of $0.3\ensuremath{-}\ensuremath{\mu}$ Teflon particles was measured by autocorrelation spectroscopy, at various values of scattering wave vector $\stackrel{\ensuremath{\rightarrow}}{\mathrm{K}}$. The experimental results exhibit a $K$ dependence stronger than can be accounted for by a simple dynamic scaling approach to the shear viscosity. The theory presented takes into account the nonlocality of the shear viscosity and its effect on the diffusion of small material particles. In addition, we develop a theory to explain the effect of a specific interaction between the particles and the two fluid components. Defining an effective viscosity as ${\ensuremath{\eta}}_{\mathrm{eff}}\ensuremath{\equiv}\frac{{k}_{B}T}{6\ensuremath{\pi}\mathrm{RD}}$, where $D$ is the diffusion coefficient, the theoretical results may be expressed as ${\ensuremath{\eta}}_{\mathrm{eff}}=\ensuremath{\eta}({\ensuremath{\xi}}_{1})+G{\ensuremath{\alpha}}^{2}{\ensuremath{\xi}}_{2}^{2}[\frac{1}{2}\ensuremath{-}(\frac{R}{{\ensuremath{\xi}}_{2}}){K}_{1}(\frac{2R}{{\ensuremath{\xi}}_{2}})]$, ${\ensuremath{\xi}}_{1}^{\ensuremath{-}2}\ensuremath{\equiv}{(0.94R)}^{\ensuremath{-}2}+{\ensuremath{\xi}}^{\ensuremath{-}2}+{a}^{2}{K}^{2}$, ${\ensuremath{\xi}}_{2}^{\ensuremath{-}2}\ensuremath{\equiv}B\ensuremath{\alpha}+{\ensuremath{\xi}}^{\ensuremath{-}2}+{a}^{2}{K}^{2}$, where $\ensuremath{\eta}(\ensuremath{\xi})$ is the viscosity expressed as a function of the correlation length $\ensuremath{\xi}$, $a$ is the dynamic scaling parameter ($a=1.12\ifmmode\pm\else\textpm\fi{}0.1$), and ${K}_{1}(x)$ is the Bessel function of imaginary argument. The parameters $G$ and $B$ are nonadjustable and depend on the particle size $R$, the intermolecular spacing, and the Ornstein-Zernike parameter. The parameter $\ensuremath{\alpha}$ is a measure of the surface interaction between the particles and the fluids. The resulting expression fits the data within the experimental error of 1%.