Non-Gaussianity-induced enhanced target-finding dynamics of confined colloids

TL;DR

The study addresses how confinement alters first-passage times for diffusive particles, revealing that non-Gaussian displacements near boundaries can either slow or speed target finding. Using Lorenz-Mie holography, they track 3D colloid trajectories and develop bulk and confinement FPT theories with a potential $U_{eq}(z)$ and position-dependent mobility; the bulk FPTD is $f(t)= L / sqrt(4 π D_0 t^3) exp(-L^2/(4 D_0 t))$. In wall-parallel geometry the FPTD remains bulk-like with an effective diffusivity $D_x$ and the most-likely time $T^{max}$ increases with confinement parameter $Λ= a_p/l_B$, whereas in wall-normal geometry non-Gaussian, hydrodynamics-induced tails increase large displacements and reduce the MFPT relative to a Gaussian reference, with the effect growing with $Λ$. The results quantify how near-wall physics and rare-event statistics govern target-search efficiency, with implications for confined chemistry and biology near boundaries.

Abstract

The encounter of diffusing entities underlies a wide range of natural phenomena. The dynamics of these first-passage processes are strongly influenced by the geometry of the system, for example through confining boundaries. Confinement, which alters the diffusion of microscopic particles through both conservative and hydrodynamic interactions, emerges as a key ingredient for modeling realistic environments. In this Letter, we investigate the impact of confinement on the first-passage statistics of a diffusing particle. This diffusive motion is probed, with nanometric precision, by combining \textit{state-of-the-art} holographic microscopy with advanced statistical inference methods. Our experimental and numerical results provide a comprehensive understanding of this process, which is governed by the coupling between gravitational, screened electrostatic and hydrodynamic forces, as well as thermal fluctuations. We further show that confinement can either slow down or enhance the typical first-passage kinetics, depending on the experimental parameters and considered direction of space. In particular, the observed boost in wall-normal target-finding efficiency appears to be a direct consequence of the non-Gaussian displacement distribution induced by the near-surface effects, and the associated increased probability of large displacements. As the latter are rare events, our findings may be of relevance to rationalize confined chemical reactions, or biological \textit{winners-take-all} stochastic processes near boundaries.

Figures (4)

• Figure 1: Monitoring confined diffusion via Mie holography lavaud2021stochastic. (a) Three-dimensional trajectory of a spherical polystyrene colloid of radius $a_\mathrm{p}=1.48 \,\mathrm{\mu m}$ diffusing in a viscous liquid near a rigid glass wall. (b) Raw experimental (Exp) hologram, and corresponding theoretical hologram, with their respective radial intensity profiles. (c) Experimental (exp) equilibrium probability distribution function, $P_\mathrm{eq}$, for the altitude $z$ of the colloid (gray circles), with its corresponding fit to Eq. \ref{['eq:Peq']} (black line), leading to $B = 6.96$, $l_\mathrm{D} = 33 \, \mathrm{nm}$, and $l_\mathrm{B} = 604 \, \mathrm{nm}$. (d) Mean Square Displacements $\langle \Delta_\tau q^2\rangle$ as functions of the time lag $\tau$, in the $x$-direction (red) and the $z$-direction (blue). The corresponding colored solid lines depict the best fits to Eq. \ref{['eq:MSD']} in both directions, leading to $D_x = 0.542\, D_0$ and $D_z = 0.268\, D_0$. The equilibrium time is indicated with the dot-dashed vertical line. (e) Probability Distribution Function, $P(\Delta_\tau x)$, of the colloid's in-plane displacements $\Delta_\tau x$ over a time $\tau$, for $\tau = 50\,$ms (red dots) and $1\,$s (orange dots). The corresponding colored solid lines depicts the best fits to the theoretical prediction detailed in Refs. lavaud2021stochasticalexandre2023non, with the same parameter values as in panel (c). The black dashed line corresponds to the Gaussian prediction of Eq. \ref{['eq:gauss']}, with the diffusion coefficient $D_x$ from panel (d). (f) Same as panel (e) but in the $z$-direction.
• Figure 2: First-passage statistics of a spherical polystyrene colloid ($a_\mathrm{p} = 1.46 \, \mathrm{\mu m}$) diffusing in the bulk of an ethylene-glycol-water mixture (weight ratio of 42% in ethylene glycol). (a) Trajectory along $x$. The colloid initially starts at $x=x_0=0$ and reaches the target located at $x=x_0+L=L$, at time $t$. (b) Distribution $f(t)$ of the first-passage times $t$, for a target located at a distance $L = 500\, \mathrm{nm}$ from the initial position. Experimental data (gray circles) and theoretical prediction (gray line) of Eq. \ref{['eq:FPTD']}, with the diffusion coefficient $D_0 = 4.53.10^{-2} \, \mathrm{\mu m ^2/s}$ (see Eq. \ref{['eq:D0']}).
• Figure 3: First-passage-time statistics in confinement for an wall-parallel search with a target located at a distance $L = 500 \, \mathrm{nm}$ from the initial position. (a) Dimensionless distributions $F(T)$ of the dimensionless first-passage times $T$ of a colloid diffusing, either in the bulk (gray) or near a wall (red). Circles correspond to experimental (exp) data, upside-down triangles to Langevin simulation (LS) data (see Eq. \ref{['eq:Langevin_confined']}a), and the solid lines to the theoretical predictions of Eq. \ref{['eq:FPTD']} (bulk, $D_0 = 4.53.10^{-2} \, \mathrm{\mu m^2 / s}$) and Eq. \ref{['eq:ansatz']} (confined, $D_0 = 15.1.10^{-2}\, \mathrm{\mu m^2 / s}$, $D_x = 8.18.10^{-2} \, \mathrm{\mu m^2 / s}$). (b) Relative delay $\Delta T^\mathrm{max} / T_\mathrm{bulk}^\mathrm{max}$ (see Eq. \ref{['rtd']}) of the most-likely dimensionless first-passage time as a function of the confinement parameter $\Lambda = a_\mathrm{p} / l_\mathrm{B}$. Orange hexagons correspond to experiments, gray hexagons to simulations, and the gray line to the associated theoretical prediction (see SI), with no adjustable parameter.
• Figure 4: First-passage-time statistics in confinement for a target located along $z$ at a distance $L = 300 \, \mathrm{nm}$ from the initial position. Note that the latter is randomly selected from $P_{\textrm{eq}}(z)$ for pre-thermalized samples. (a) Dimensionless distribution $F(T)$ of the dimensionless first-passage times $T$ for a colloid diffusing either in the bulk (gray) with $D_0 = 4.53.10^{-2} \, \mathrm{\mu m^2 / s}$, or near a wall (blue) with $D_0 = 7.14.10^{-2}\mathrm{\mu m^2 / s}$ and $D_z = 4.3.10^{-3} \mathrm{\mu m^2 / s}$. Circles correspond to experimental data, upside-down triangles to simulation data (see Eq. \ref{['eq:Langevin_confined']}b), the gray solid line to the bulk theoretical prediction of Eq. \ref{['eq:FPTD']}, and the blue dashed line to simulation data when replacing the $z$-dependent diffusivity $d_z(z)=k_{\textrm{B}}\Theta/\gamma_z(z)$ by $D_z$ within Eq. \ref{['eq:Langevin_confined']}b. (b) Relative reduction $\Delta \overline{T} / \overline{T^\mathrm{G}}$ (see Eq. \ref{['eq:MFPT']}) of the dimensionless mean first-passage time as a function of the confinement parameter $\Lambda = a_\mathrm{p} / l_\mathrm{B}$. Langevin simulations (LS) are indicated with gray hexagons, while experiments (exp) are indicated with orange hexagons.