Diffusiophoresis and diffusioosmosis in tandem: Two-dimensional particle motion in the presence of multiple electrolytes

Diffusiophoresis is the movement of colloidal particles due to a gradient in the concentration of a solute. Previous studies on diffusiophoresis have focused largely on one-dimensional colloid transport due to the gradient of a single electrolyte. Recent studies have considered two-dimensional geometries including dead-end pores, multiple electrolytes, and a background flow field due to diffusioosmosis. In this work, we develop a model of the time-dependent diffusiophoretic compaction of colloids in a two-dimensional pore due to the gradient of multiple electrolytes in tandem with a diffusioosmotic slip-driven background flow field, which builds upon these recent studies by combining each of these effects. We simulate this model for varying properties of the pore walls and colloidal particles. Furthermore, we conduct experiments varying the initial ion combinations and total solute concentration, which show good qualitative agreement with the simulations. Our results indicate that diffusiophoretic compaction can be increased or decreased by manipulating electrolyte combinations, total solute concentration, wall charge, and particle diffusivity; each effect can modify the particle velocity, with varying strength, in unison or in opposition to the other effects. By offering a larger toolbox to manipulate colloidal particles, our results on diffusiophoretic and diffusioosmotic motion in tandem and in the presence of multiple electrolytes can be exploited for lab-on-a-chip and biophysics applications.

Figure 1

Schematic of the transport of colloidal particles inside a pore in the presence of a three-ion solute concentration gradient. An example of a microscopic view of a single colloidal particle is shown to the right. The ion concentration gradients induce a diffusioosmotic slip velocity, which generates a flow field ${\mathbf{v}}_{\text{f}}$ in the solution, as well as a diffusiophoretic velocity ${\mathbf{v}}_{\text{DP}}$ on the particles. The net velocity of the particles ${\mathbf{v}}_{\text{p}}={\mathbf{v}}_{\text{DP}}+{\mathbf{v}}_{\text{f}}$ is a superposition of diffusiophoretic and fluid velocities. The pore at $\tau \approx 0$ shows the initial state of the colloidal particles in a nonzero solute gradient, and the pore at $\tau \approx 1$ shows the colloidal particles in a compacted state after the ions have diffused throughout the length of the pore.

Figure 2

Simulated codependent diffusion of two solutes, NaCl and KCl, which share a common anion. We consider the three ions ${\mathrm{Na}}^{+} \left(c_1\right), {\mathrm{Cl}}^{-} \left(c_2\right)$, and ${\mathrm{K}}^{+} \left(c_3\right)$ diffusing in a finite pore geometry with diffusioosmotic wall slip. Initial and boundary conditions are $c_1(X,Y,\tau =0)=1,c_2(X,Y,\tau =0)=1,c_3(X,Y,\tau =0)=0,c_1(X=0,Y,\tau )=0,c_2(X=0,Y,\tau )=1,c_3(X=0,Y,\tau )=1,\frac{\partial c_i}{\partial X}(X=1,Y,\tau )=0,\frac{\partial c_i}{\partial Y}(X,Y=\pm h/\ell ,\tau )=0$. (a)–(c) Show the concentration of each respective ion as a color map of the pore at three different times. Each color map also includes a vector plot of the fluid velocity ${\mathbf{V}}_{\text{f}}$ induced by the diffusioosmotic wall slip. (d)–(f) Show the $Y$-averaged concentration of each respective ion at the same three times. Note that the vertical axis of (e) spans a much smaller range than the vertical axes of (d) and (f). We use the parameters $\ell /h=10, {\mathrm{\Psi }}_{\text{w}}=-4$, and those identified in Table 1.

Figure 3

Simulated colloidal particle velocity ${\mathbf{V}}_{\text{p}}$, which is the sum of the diffusiophoretic velocity ${\mathbf{V}}_{\text{DP}}$ and the fluid velocity ${\mathbf{V}}_{\text{f}}$. A finite pore geometry is considered with initial condition $n(X,Y,\tau =0)=1$ and boundary conditions $n(X=0,Y,\tau )=0,\frac{\partial n}{\partial X}(X=1,Y,\tau )=0,\frac{\partial n}{\partial Y}(X,Y=\pm h/\ell ,\tau )=0$. Each wall reference potential given in Table 1 was simulated. (a)–(e) Show the velocity vector fields. (f)–(j) Show the $Y$-averaged colloidal particle velocity magnitude $〈\left|\right|V_{\text{p}}\left|\right|〉$. Three values of time are used to show the system immediately following the initial conditions, at an intermediate stage, and approaching steady state. We use the parameters $\ell /h=10, {\mathcal{D}}_{\text{p}}={10}^{-4}$, with other values given in Table 1.

Figure 4

Simulated colloidal particle concentration $n$ for different particle diffusivities ${\mathcal{D}}_{\text{p}}$. A finite pore geometry is considered with initial condition $n(X,Y,\tau =0)=1$ and boundary conditions $n(X=0,Y,\tau )=0,\frac{\partial n}{\partial X}(X=1,Y,\tau )=0,\frac{\partial n}{\partial Y}(X,Y=\pm h/\ell ,\tau )=0$. The wall reference potential is ${\mathrm{\Psi }}_{\text{w}}^{*}=-4$. (a)–(c) Show the concentrations for each particle diffusivity in the pore using a color map. (d)–(f) Show the $Y$-averaged concentrations for each particle diffusivity. Three values of time are used to show the system immediately following the initial conditions, at an intermediate stage, and approaching steady state. We use the parameters $\ell /h=10, {\mathrm{\Psi }}_{\text{w}}=-4$, with other values given in Table 1.

Figure 5

Comparison between the experimental (solid) and simulated (dashed) time evolution of the colloid concentration for an initial condition of 10 mM KCl in the main channel and 10 mM NaCl in the pore, which is condition iv from Table 2. (a) Shows the experimental images of the pore and (b) shows the simulated data at various values of $\tau$. The experimental data qualitatively match the trends in the simulated data well throughout time, including the parabolic contours due to the diffusioosmotic slip-induced flow field [(a) and (b)] but shows some quantitative differences [(c) and (d)]. (c) Shows the $Y$-averaged colloid concentrations vs $X$ for each value of $\tau$. (d) Shows the center-line $(Y=0)$ colloid concentrations vs $X$ for each value of $\tau$.

Figure 6

Comparison between the experiments (solid) and simulations (dashed) for initial conditions of 10 mM KCl in the main channel varying pore concentrations: 2.5 mM NaCl and 7.5 mM KCl, 5 mM NaCl and 5 mM KCl, 7.5 mM NaCl and 2.5 mM KCl, and 10 mM NaCl, which are conditions i–iv from Table 2. We note that, for each condition, the total concentration of ${\mathrm{Cl}}^{-}$ is conserved and the concentrations of ${\mathrm{Na}}^{+}$ and ${\mathrm{K}}^{+}$ are varied. (a)–(d) Compare the simulated data (top) with experimental images at $\tau =0.5$. These four experiments demonstrate that creating a larger concentration gradient of $\mathrm{KCl}$ increases colloidal compaction. The experimental data qualitatively match the trends in the simulated data well throughout time, including the parabolic contours due to the diffusioosmotic slip-induced flow field [(a)–(d)] but show some quantitative differences [(e) and (f)]. (e) Shows the $Y$-averaged colloid concentrations versus $X$ for each condition. (f) Shows the center-line $(Y=0)$ colloid concentrations versus $X$ for each condition.

Figure 7

Comparison between the experiments (solid) and simulations (dashed) for initial conditions of 10 mM KCl in the main channel and 10 mM NaCl in the pore, and 1 mM KCl in the main channel and 1 mM NaCl in the pore, which are conditions iv and v from Table 2. The comparison demonstrates the effect of absolute concentration. (a), (b) Compare the simulated pores (top) to the experimental images at $\tau =0.5$ for conditions iv and v, respectively. We observe deeper colloidal compaction with lower ion concentrations because of a change in the zeta potentials. The experimental data qualitatively match the trends in the simulated data well throughout time, including the parabolic contours due to the diffusioosmotic slip-induced flow field [(a) and (b)] but show some quantitative differences [(c) and (d)]. (c) Shows the $Y$-averaged colloid concentrations versus $X$ for each condition. (d) Shows the center-line $(Y=0)$ colloid concentrations versus $X$ for each condition.

Figure 8

Comparison between the experiments (solid) and simulations (dashed) for initial conditions of 10 mM KCl in the main channel and 5 mM NaCl and 5 mM KCl in the pore, and 5 mM KCl in the main channel and 5 mM NaCl in the pore, which are conditions ii and vi from Table 2. The comparison demonstrates the effect of the background solute. (a), (b) Compare the simulated pores (top) to experimental images at $\tau =0.5$ for conditions ii and vi, respectively. We see deeper colloid compaction in the absence of background $\mathrm{KCl}$ because lower ion concentrations result in larger zeta potentials. The experimental data qualitatively match the trends in the simulated data well throughout time, including the parabolic contours due to the diffusioosmotic slip-induced flow field [(a) and (b)] but show some quantitative differences [(c) and (d)]. (c) Shows the $Y$-averaged colloid concentrations versus $X$ for each condition. (d) Shows the center-line $(Y=0)$ colloid concentrations versus $X$ for each condition.

Figure 9

Mean intensity from the overlaid pore images is plotted versus time. The particle mass conservation in the compaction experiments is maintained after overlaying results from five pores.