Nonequilibrium fluctuations during diffusion in liquid layers

Theoretical analysis and experiments have provided compelling evidence of the presence of long-range nonequilibrium concentration fluctuations during diffusion processes in fluids. In this paper, we investigate the dependence of the features of the fluctuations from the dimensionality of the system. In three-dimensional fluids the amplitude of nonequilibrium fluctuations can become several orders of magnitude larger than that of equilibrium fluctuations. Notwithstanding that, the amplitude of nonequilibrium fluctuations remains small with respect to the concentration difference driving the diffusion process. By extending the theory to two-dimensional systems, such as liquid monolayers and bilayers, we show that the amplitude of the fluctuations becomes much stronger than in three-dimensional systems. We investigate the properties of the fronts of diffusion and show that they have a self-affine structure characterized by a Hurst exponent $H=1$. We discuss the implications of these results for diffusion in liquid crystals and in cellular membranes of living organisms.

Figure 1

Graphical representation of the diffusion coefficient as the integral of the velocity power spectrum for $\omega =0$. The top and bottom panels correspond to the 3D and 2D cases, respectively. The diffusion coefficient is the area below the curve, between the low- and high-wave-vector cutoffs $Q$ and $\mathrm{\Theta }$. The parameters are $Q=2\pi /1\mathrm{cm}, a=0.4\mathrm{nm}, \mathrm{\Theta }=\pi /\left(2a\right), q_C=3.8\times {10}^5{\mathrm{m}}^{-1}$.

Figure 2

Root-mean-square amplitude of the nonequilibrium fluctuations of concentration in three-dimensional and two-dimensional fluids and in liquid layers surrounded by less viscous fluids. The amplitude is represented as a fraction of the macroscopic concentration difference $\mathrm{\Delta }c$ across the layer of thickness $L$ over which diffusion takes place. The scale on the upper axis assumes $a=0.4\mathrm{nm}$. The value of $q_C$ is $3.8\times {10}^5{\mathrm{m}}^{-1}$ and $2\times {10}^6{\mathrm{m}}^{-1}$, respectively, for the liquid crystal layer suspended in air and for the lipid bilayer suspended in water.

Figure 3

Sections of the fluid slab where diffusion takes place. For each row, the image on the left is the concentration profile, represented in gray scale; the right image represents the displacement of the concentration with respect to the constant-gradient macroscopic concentration profile. Isoconcentration curves, i.e., the fronts of diffusion, are also shown. The radius of the molecule is $a=0.4\mathrm{nm}$ for all the sections. Top row: Three-dimensional fluid. Center row: Two-dimensional fluid; the simulation should be representative of a liquid crystal film suspended in medium vacuum. Bottom row: Liquid layer with viscous drag of the surrounding fluid; the parameters used are those for a lipid bilayer in water.