Nonequilibrium fluctuations in free diffusion processes.

Diffusion is the fundamental mass transfer mechanism in many natural and technological processes. The diffusive transport can be interpreted by the simple molecular random walk model. A more refined description requires the understanding of direct interaction between the diffusing particles and possibly hydrodynamic interactions. Both types of interactions may produce appreciable changes in the magnitude of the effective diffusion coefficient $D$ but, at any extent, diffusion is believed to give rise to an intimate and homogeneous remixing on matter. The general belief is that while the process occurs over quite microscopic distances, nothing peculiar should occur at any other lengthscale, except the molecular one where the random molecular diffusion takes place.

It has been recently shown that, quite unexpectedly, gigant fluctuations are present during the diffusive remixing of two miscible phases of a binary mixture not too far from its critical point [13]. A fluctuating hydrodynamic description has been developed [20], which indicates that gigant nonequilibrium fluctuations should be present during the diffusive remixing of fluids in general; moreover, it has been shown that the fluctuations can be considered the origin of the whole Fick flow [21].

The presence of the fluctuations has been detected experimentally during the free diffusive remixing occurring in ordinary liquid mixtures and in macromolecular solutions [22,23]. The measurements concerned an ordinary, low molecular weight liquid mixture, an aqueous solution of a low molecular weight solid, a polymer solution and a protein solution, thus giving evidence that these anomalous fluctuations are a universal feature associated with spontaneous diffusion across a macroscopic gradient.

A free diffusion experiment begins filling a cell with the two liquids, with the denser solution in the lower part to avoid convective instability. The two horizontal layers are initially separated by a fairly sharp meniscus. As soon as the two liquids came into contact, the diffusive remixing begins, and the meniscus repidly becomes smeared. The concentration profile inside the sample, initially a step function as a function of the height $z$, gradually evolves into an s-shaped function [24], until eventually, after a few days, the concentration becomes uniform throughout the sample.

During the free diffusion process described above, intense fluctuations arise. Their power spectrum $S\left(\vec{q}\right)$, with $\vec{q}$ in the horizontal direction, is given by:

$$S\left(q\right)= S_0\frac{1}{1+\left[\frac{q}{q_{RO}}\right]^4}.$$ (9.1)

The roll-off wave vector $q_{RO}$ is given by:

$$q_{RO}=\sqrt[4]{\frac{\beta g \nabla c}{\nu D}}$$ (9.2)

where $g$ is the gravity acceleration, $\nu$ is the kinematic viscosity, $D$ is the diffusion coefficient, and $\beta=\frac{1}{\rho}{ \frac{\partial \rho}{\partial c}}_p,T$ is the solutal expansion coefficient, a quantity that increases as increases the mismatching of the two liquids. The gradient $\nabla c$ can be assumed roughly connstant in the region between the fluids, where diffusion takes place, and vanishes outside. The sample-dependent prefactor in Eq. (9.1) is given by:

$$S_0=K_B T\left(\frac{\partial n}{\partial c}\right)^2 \frac{\Delta c}{\rho \beta g}$$ (9.3)

where $\Delta c$ is the total concentration difference across the sample.

The power spectrum $S\left(q\right)$ dislpays a $q^{-4}$ power low divergence at large wave vectors, $q\gg q_{RO}$, and a saturation at a constant value at small wave vectors, $q\ll q_{RO}$. The $q^{-4}$ power low is interpreted as the result of of the coupling of velocity fluctuations with concentration fluctuations, while the saturation is due to a stabilizing effect of gravity on long wavelength fluctuations.

Moreover, the roll-off wave vector where the transition between the two regimes occurs gets smaller as $\beta g \nabla c$, and the the low wave vector value of the power spectrum $S_0$ is roughly constant as free diffusion takes place, since the concentration near the upper and lower windows of the cell are initially constant.

The nonequilibrium concentration fluctuations are originated from the coupling of velocity fluctuations with concentration fluctuations, due to the presence of a macroscopic concentration gradient. This can be understood by simple naive arguments, discussed in detail in [20] and [25]. Suppose that a small parcel of fluid of linear size $a$ undergoes a velocity fluctuation. This fluctuation will displace the parcel until the viscous drag will stop it in a time given approximately by $\tau_{visc}=a^2/\nu$, $\nu$ being the kinematic viscosity. If the displacement of the parcel occurs in a direction parallel to the macroscopic concentration gradient, the parcel will be surrounded by fluid with different concentration. The life time of this concentration fluctuation is $\tau_{diff}=a^2/D$, and is much larger than the viscous time $\tau_{visc}$, as $D\ll\nu$. Thus, in the presence of a macroscopic gradient, the effect of a short living velocity fluctuation is to induce a long lasting concentration fluctuation. Once a concentration fluctuation has been created, two mechanisms may contribute to its relaxation: diffusion and buoyancy. If the spatial extent of the fluctuation is small, then the fluctuation will soon disappear due to diffusion. This mechanism gives rise to the $q^{-4}$ divergence of the static power spectrum at high wavevectors. As the wavevector increases, the velocity fluctuation lives for a shorter time, and can displace the parcel of a smaller amount, and this gives a factor $q^{-2}$; moreover, the displaced parcel will be dissipated as $q^{-2}$. However, if the fluctuation is large enough, the buoyancy force acting on it will be able to restore the fluctuation in the layer of fluid having the same density in a time shorter than the diffusive one. This gives rise to the frustration of the $q^{-4}$ divergence at smaller wavevectors.

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