Correlations and scaling properties of nonequilibrium fluctuations in liquid mixtures

Abstract

Diffusion in liquids is accompanied by nonequilibrium concentration fluctuations spanning all the length scales comprised between the microscopic scale a and the macroscopic size of the system, L. Up to now, theoretical and experimental investigations of nonequilibrium fluctuations have focused mostly on determining their mean-square amplitude as a function of the wave vector. In this work, we investigate the local properties of nonequilibrium fluctuations arising during a stationary diffusion process occurring in a binary liquid mixture in the presence of a uniform concentration gradient, ∇c 0 . We characterize the fluctuations by evaluating statistical features of the system, including the mean-square amplitude of fluctuations and the corrugation of the isoconcentration surfaces; we show that they depend on a single mesoscopic length scale l = √ aL representing the geometric average between the microscopic and macroscopic length scales. We find that the amplitude of the fluctuations is very small in practical cases and vanishes when the macroscopic length scale increases. The isoconcentration surfaces, or fronts of diffusion, have a self-affine structure with corrugation exponent H = 1/2. Ideally, the local fractal dimension of the fronts of diffusion would be D l = d − H , where d is the dimensionality of the space, while the global fractal dimension would be D g = d − 1. The transition between the local and global regimes occurs at a crossover length scale of the order of the microscopic length scale a. Therefore, notwithstanding the fact that the fronts of diffusion are corrugated, they appear flat at all the length scales probed by experiments, and they do not exhibit a fractal structure.

Nonequilibrium fluctuations during diffusion in liquid layers

“…As in any linearized model, the key assumption is that fluctuations represent small perturbations of the macroscopic variables. This assumption is confirmed by experiments and simulations, even under microgravity conditions where the amplitude of the fluctuations is not damped by external forces [12,13,[18][19][20]. Notwithstanding the effectiveness of linearized hydrodynamics, recent theoretical work has found that the cumulative contribution of secondorder terms to the equations is far from being negligible.…”

“…The procedure is similar to that followed in the three-dimensional case [19]. The calculation requires the introduction of a small-wavelength cutoff Q.…”

“…(23); we consider a system with a constant concentration difference c at the boundaries, with a concentration gradient ∇c = c/L, where L is the size of the region over which the concentration gradient is present. For three-dimensional fluids, the root-mean-square amplitude of the fluctuations is [19] …”

“…Further theoretical work showed that the fronts of diffusion exhibit a self-affine structure rather than a self-similar one, as it is often the case of interfaces in anisotropic systems [44]. The scaling law is compatible with a fractal structure only at length scales that are not actually covered by the fluctuations (smaller than the molecular size) [19]. In the case of diffusion in 2D, we find that the scaling laws of the isoconcentration curves are not compatible with a fractal structure at any length scale.…”

The NEUF-DIX space project - Non-EquilibriUm Fluctuations during DIffusion in compleX liquids

“…However, theory and experiments refer only to ideal conditions as for example stationary states, small gradients, and diluted systems. In recent years a number of theories [23][24][25][26][27][28] and some preliminary experiments [29][30][31] have indicated the importance of a deeper understanding of NEFs in more general cases.…”