Solute dispersion boosts the phoretic removal of colloids from dead-end pores

TL;DR

This work addresses whether solute-front dispersion in porous media suppresses diffusiophoretic migration of colloids from dead-end pockets. By integrating microfluidic experiments, 2D simulations, and a 1D analytical model for slowly varying solute profiles, it shows that front dispersion does not weaken, and can even enhance, phoretic extraction because persistent gradients extend the forcing $U_{DP}=\Gamma_p\nabla\ln c$ over time. The analysis remains robust to mobility models, including concentration-dependent $\Gamma_p(c)$ and various electrolytes, providing a unified description of sharp and error-function–type solute histories. The findings suggest diffusiophoresis can play a significant role at larger scales in filtration, remediation, and targeted delivery within porous media, where tuning solute dispersion could optimize particle removal. The framework combines experiments, simulations, and theory across linear and diffuse solute front regimes.

Abstract

Predicting and controlling the transport of colloids in porous media is essential for a broad range of applications, from drug delivery to contaminant remediation. Chemical gradients are ubiquitous in these environments, arising from reactions, precipitation/dissolution, or salinity contrasts, and can drive particle motion via diffusiophoresis. Yet our current understanding mostly comes from idealized settings with sharply imposed solute gradients, whereas in porous media, flow disorder enhances solute dispersion, and leads to diffuse solute fronts. This raises a central question: does front dispersion suppress diffusiophoretic migration of colloids in dead-end pores, rendering the effect negligible at larger scales? We address this question using an idealized one-dimensional dead-end geometry. We derive an analytical model for the spatiotemporal evolution of colloids subjected to slowly varying solute fronts and validate it with numerical simulations and microfluidic experiments. Counterintuitively, we find that diffuseness of solute front enhances removal from dead-end pores: although smoothing reduces instantaneous gradient magnitude, it extends the temporal extent of phoretic forcing, yielding a larger cumulative drift and higher clearance efficiency than sharp fronts. Our results highlight that solute dispersion does not weaken the phoretic migration of colloids from dead-end pores, pointing to the potential relevance of diffusiophoresis at larger scales, with implications for filtration, remediation, and targeted delivery in porous media.

Figures (11)

• Figure 1: The dispersion of solute front does not weaken the diffusiophoretic removal of colloids. (a–c) Numerical simulations; (d) Simulations and experiments. (a) We displace an aqueous solution of colloids with salt concentration $c_0$ with a colloid-free solution with salt concentration $c_1$ at constant flow rate. Medium disorder leads to the heterogeneous distribution of flow velocity magnitude, solute concentration, diffusiophoretic velocity magnitude and particle density. Three fields of view (FOVs) are highlighted by white boxes. (b) Cross-sectionally averaged profiles of the normalized solute concentration (red dashed), particle density (black solid), and diffusiophoretic velocity magnitude (purple dash-dotted), shown at the same time as panel (a). (c) Temporal evolution of the average solute concentration and diffusiophoretic velocity magnitude within the three FOVs. The solute profiles are fitted using the solution to the 1D advection–diffusion equation (Eq. \ref{['eq:1Dsoln']}). The inset presents the fitted characteristic solute transition time $\tau_\text{erf}$ for each FOV. (d) Temporal evolution of the average particle density across the three FOVs, in the presence (single lines), and absence of solute gradients (double solid lines). Symbols represent the corresponding experimental observations.
• Figure 2: The evolution of colloid density in dead-end pores exposed to (a) sharp, and (b) diffuse solute fronts. (c) The dispersion of solute front is represented by the evolution of the fluorescein light intensity at the inlet of the dead-end pores. (d) The evolution of colloid density exposed to sharp versus diffuse solute fronts. For both cases; snapshots at $t/\tau_\text{s}=0,\,2,\,4$. (e) The evolution of fraction of residual particles over time showing the crossover at "late times". Symbols: experiments; solid lines: 2D simulations. The blue dash–dot line indicates the analytical prediction of the final residual (equation \ref{['eq:General_N_Ave_ND']}), consistent with both. In the presence of sharp solute front, colloids migrate toward the inlet of the dead-end pore, leading to a non-uniform density profile (f). However, in the diffuse front case, colloidal density in the pore remains nearly uniform (g). For the sharp front, data points correspond to $t/\tau_\text{s}=0,\,0.1,\,0.2,\,0.5,\,1,\,2$, and for the diffuse front, we have $t/\tau_\text{s}=0,\,0.25,\,0.5,\,1,\,2,\,4$. Solid lines represent smoothed density profiles obtained via a five-point adjacent-averaging method. (h) Two-dimensional simulation domain. Colour map: particle density after the flushing stage; streamlines: background flow showing the primary recirculation in the pore. (i,j) Simulated particle density profiles for the sharp (i) and diffuse (j) fronts at the same sampling times as in (f,g).
• Figure 3: Transport of solute and particles in a one-dimensional dead-end pore. (a) Schematic of the setup: The initial solute concentration and particle density inside the pore are $c_0 = 1$ and $n_0 = 1$, respectively. The inlet boundary conditions are given by $c_\text{in}(t)$ and $n_1 = 0$. As shown on the right, $c_\text{in}(t)$ increases linearly from 1 to 100 over the transition time $T$ and remains at 100 thereafter. (b) Maximum diffusiophoretic velocity over time for different normalized solute transition times $T/\tau_\text{s} \in \{0,\,1,\,10\}$ (red: 0, yellow: 1, blue: 10). (c, d) Evolution of the solute field (c) and diffusiophoretic velocity field (d) for $T/\tau_\text{s} = 0,\,1,\,10$. Sampling times are $t/\tau_\text{s} = \{0.001,\,0.02,\,0.05,\,0.1,\,1\}$ for $T/\tau_\text{s} = 0$, $\{0.01,\,0.1,\,0.5,\,1,\,2\}$ for $T/\tau_\text{s} = 1$, and $\{0.1,\,0.5,\,1,\,2,\,10\}$ for $T/\tau_\text{s} = 10$. (e) Particle trajectories $x(t,\xi_0)$ as functions of both time $t$ and initial position $\xi_0$, using the same sampling times. (f) Particle density fields: non-diffusive particles from particle tracking using Eq. \ref{['eq:Udp_Expression1']} (solid lines) and diffusive particles from the continuum model in Eq. \ref{['eq:ADE_Particle']} (symbols), evaluated at the same sampling times.
• Figure 4: Longer solute transition times slow the extraction of particles from dead-end pores but ultimately enhance their overall removal. (a) Temporal evolution of the fraction of residual particles in dead-end pores for different normalized solute transition times ($T/\tau_\text{s} = 0,\,1,\,10,\,100$), derived from the non-diffusive particle trajectories (Eq. \ref{['eq:Udp_Expression1']}), is shown by black lines. The cyan solid line is the envelope of these curves, indicating the minimal achievable residual particle fraction at any given time. (b) Final residual particle fraction (evaluated once the solute gradient has vanished) as a function of the solute transition time. Results are obtained by solving continuous equations (Eq. \ref{['eq:ADE_Particle']}) for diffusive particles (diamond symbols) and via particle tracking (Eq. \ref{['eq:Udp_Expression1']}) for non-diffusive particles (black solid lines). The magenta dash-dot line indicates the residual particle fraction at the moment when the inlet solute concentration reaches its final value, assuming particles are non-diffusive. The yellow solid line represents the prediction from our analytical model (Eq. \ref{['eq:final_residual']}) that accounts for both diffusiophoresis and particle diffusion.
• Figure 5: Analytical and numerical results for particle transport in a dead-end pore with large normalized solute transition times, $T/\tau_\text{s}$. (a, b) Residual particle fraction (a) and breakthrough curves (b) for $T/\tau_\text{s} = 10^1, 10^2$, and $10^3$, compared to a control case without solute gradients. Symbols indicate numerical results, and solid lines show analytical predictions, demonstrating good agreement across a range of solute transition times.