Non-Gaussian Diffusion Near Surfaces

We study the diffusion of particles confined close to a single wall and in double-wall planar channel geometries where the local diffusivities depend on the distance to the boundaries. Displacement parallel to the walls is Brownian as characterized by its variance, but it is non-Gaussian having a nonzero fourth cumulant. Establishing a link with Taylor dispersion, we calculate the fourth cumulant and the tails of the displacement distribution for general diffusivity tensors along with potentials generated by either the walls or externally, for instance, gravity. Experimental and numerical studies of the motion of a colloid in the direction parallel to the wall give measured fourth cumulants which are correctly predicted by our theory. Interestingly, contrary to models of Brownian-yet-non-Gaussian diffusion, the tails of the displacement distribution are shown to be Gaussian rather than exponential. All together, our results provide additional tests and constraints for the inference of force maps and local transport properties near surfaces.

Figure 1

Schematic. A particle of radius $a$ diffuses in two dimensions, between two walls separated by a distance $2H_p$.

Figure 2

(a) Experimental longtime PDF $p_0$ for position $z$, as a function of the distance $H+z$ to the bottom wall. Solid lines show the best fit to the Gibbs-Boltzmann distribution of Eq. (2), using $V(z)$ in Eq. (16), with $B=5.0$, $l_D=88\mathrm{nm}$, and $l_B=526\mathrm{nm}$ at room temperature. Here, $H_p=40\mathrm{\mu }\mathrm{m}$. Error bars give 95% confidence intervals. (b) Experimentally measured horizontal ($i=\parallel$) and vertical ($i=\perp$) local diffusion coefficients $D_i(z)$, normalized by the bulk value $D_0$. Solid lines are theoretical predictions $D_i(z)$ using Eqs. (17) and (18). (c) Experimentally measured second cumulant $⟨X_t^2{⟩}_c$. The solid line corresponds to Eq. (6), with $⟨D_{\parallel }{⟩}_0=0.58D_0$. The slope triangle indicates an exponent 1. (d) Experimentally measured (green stars) and numerically simulated (blue dots) fourth cumulant $⟨X_t^4{⟩}_c$. The red solid line represents the prediction of Eq. (9), while the black dashed and dotted lines are its short-time and longtime asymptotics of Eqs. (10) and (11), with no adjustable parameter. The slope triangles indicate exponents 2 and 1. (e) Non-Gaussianity parameter $\alpha \equiv ⟨X_t^4{⟩}_c/⟨X_t^2{⟩}_c^2$ as a function of time $t$, computed from the data and theoretical predictions in (c) and (d). The red dashed line shows the predicted $\sim 1/t$ decay (see SM [35]). (f) PDF in horizontal displacement, at time increment $t=0.01\mathrm{s}$, from experiments (green stars) and numerical simulations (blue dots). The Gaussian tail predicted in Eq. (S45) (see SM [35]) is also shown (solid line), as well as a simple Gaussian distribution (dashed line) with the proper mean and variance of the data.