2. Diffusiophoresis of a patch of particles in a dead-end pore geometry

We use dead-end pore geometries with different heights to demonstrate diffusiophoresis of a particle patch and the influence of diffusioosmosis. A finite number of particles are introduced at the inlet of dead-end pores of the same width and length ($w=100\ \mathrm {\mu }$m and $\ell =1$ mm), and different heights ($h=25, 50$ and $100\ \mathrm {\mu }$m; figure 1a,b). Initially, a concentration gradient of NaCl is established by filling the pore with a 10 mM solution and the main channel, in which there is flow, with a 1 mM solution; the injected particles are then advected into the pore by diffusiophoresis. Experimental details for establishing the initial condition are explained in § 5.

Figure 1. Diffusiophoresis and diffusioosmosis-induced dispersion of a patch of particles in a dead-end pore. (a) Schematic of a microfluidic channel with dead-end pores of length $\ell$. Pores with different heights ($h=25,50$ and $100\ \mathrm {\mu }$m) were tested in our experiments. (b) Schematic of an experiment using a particle patch that invades the pore. Details of the experimental set-up are explained in § 5. (c) Fluorescent images obtained from dead-end pore experiments. Particle distributions in pores of three different heights are shown. Scale bar is 100 $\mathrm {\mu }$m. (d) Pore-width-averaged intensity plotted vs distance along the pore ($x$). The profiles are obtained at non-dimensional time $\tau =tD_{s}/\ell ^2=1$, where $D_{s}$ is the solute diffusivity. (e) Peak locations measured in the experiments are plotted vs non-dimensional time. Colours used in all graphs of this article are chosen from the palettes for colour blindness (Krzywinski Reference Krzywinski2020).

Transport of the particle patch in the three pores looks similar at the beginning, but we observe quantitative differences as the particle distribution evolves in pores with different cross-sectional aspect ratios (figure 1c). Because the effect of different $h$ is not captured in the 2-D analyses that neglect the effect of the smallest dimension of the pore, i.e. considering a model pore with $\ell$ and $w$ as its dimensions, the experimental results in figure 1(c) suggest that differences in the particle distribution cannot be simply explained by such 2-D models.

Since the particle distribution varies in the experiments in the three different pores, the peak locations can be compared by analysing the pore-width-averaged intensity profiles (figure 1d). We observe that the peak locations measured in the three pores are similar up to dimensionless time $\tau = t D_{s} / \ell ^2 \approx 0.1$, where $D_{s}$ is the diffusivity of ${\rm Na}^+$ ions, but deviate from each other at later times (figure 1e). The width-averaged intensity plotted vs distance along the pore motivates the question of determining an effective 1-D representation of the particle transport.

As we demonstrate below, diffusiophoresis and diffusioosmosis acting in tandem result in the transport and spreading of the particle patch that is analogous to the flow-driven dispersion that occurs in pressure-driven flow in a rectangular channel. For example, for the flow-driven dispersion in a pipe flow (Taylor Reference Taylor1953; Aminian et al. Reference Aminian, Bernardi, Camassa, Harris and McLaughlin2016), particles move on average with the mean flow speed and dispersion about the mean occurs due to Brownian motion as well as the non-uniform velocity profile throughout the cross-section of the channel. In our experiments, particles translate into the pore with the (transient) diffusiophoretic speed, and the wall-driven diffusioosmotic flow velocity stretches the particle distribution along the pore. Note that mass conservation inside the dead-end pore enforces zero net volumetric flux for the fluid, and the velocity distribution in the channel cross-section drives the axial dispersion of the particle patch. Therefore, for a particle patch that moves along the narrow pore with the diffusiophoretic velocity, we next apply Taylor dispersion analysis, within the lubrication approximation, to account for the flow structure in the rectangular cross-section to obtain a simplified 1-D description of the particle distribution.

3. Derivation of a 1-D cross-sectionally averaged concentration equation for colloids undergoing diffusiophoresis in a diffusioosmotic slip-driven flow

For the channel flow configurations of interest here, the concentration $N$ of colloidal particles undergoing diffusiophoresis can be described by an advective-diffusion equation. The particles diffuse at characteristic rate $D_{p}$ relative to the mean fluid velocity and advect due to both a diffusiophoretic velocity field $\boldsymbol v_{DP}$ and a fluid flow field $\boldsymbol v_{f}$ produced by a diffusioosmotic slip velocity $\boldsymbol v_{s}$ on the channel walls. Both the diffusiophoretic $\boldsymbol v_{DP}$ and diffusioosmotic slip $\boldsymbol v_{s}$ velocities are driven by the diffusion of a background solute with concentration $C$ such that $\boldsymbol v_{DP}\propto \boldsymbol \nabla \log C$ and $\boldsymbol v_{s}\propto \boldsymbol \nabla \log C$. We introduce the particle velocity $\boldsymbol v_{p} = \boldsymbol v_{f} + \boldsymbol v_{DP}$ and write the advective-diffusion equation

(3.1)\begin{equation} \frac{\partial N}{\partial t} + \boldsymbol\nabla\boldsymbol\cdot(\boldsymbol v_{p}N) = D_{p}\nabla^2 N. \end{equation}

Previous studies compare dead-end pore experiments with numerical solutions of the 1-D advective-diffusion equation (Ault et al. Reference Ault, Warren, Shin and Stone2017; Chu et al. Reference Chu, Garoff, Tilton and Khair2020; Gupta et al. Reference Gupta, Shim and Stone2020b; Wilson et al. Reference Wilson, Shim, Yu, Gupta and Stone2020), which exclude $\boldsymbol v_{f}$. While this approach simplifies calculations and provides for a qualitative comparison of the mean colloidal motion, it leaves out the effect of spreading of the particles due to the flow field. By introducing a Taylor dispersion model, we account for flow-driven spreading effects in a 1-D equation for the cross-sectionally averaged colloid distribution. In this section, we derive a general form of this 1-D effective transport equation.

We consider a pore with axial dimension $x=\boldsymbol x\boldsymbol \cdot \boldsymbol e_x$. The typical scale of the fluid velocity ($\boldsymbol {v}_{f}$) is of the order of magnitude of the wall slip velocity ($v_{s} \approx \varGamma _{w}/\ell$, where $\varGamma _{w}$ and $\ell$ are, respectively, the diffusioosmotic mobility and pore length), and an estimate of the Péclet number for the solute, $Pe_{s} = v_{s} \ell /D_{s}$ where $D_{s}$ is the solute diffusivity, yields $Pe_{s} \approx \varGamma _{w}/D_{s} \lesssim 1$ (Gupta et al. Reference Gupta, Shim and Stone2020b). Therefore, inside a narrow dead-end pore, the solute concentration field $C(x,t)$ can be assumed one-dimensional so that 1-D diffusiophoresis of particles is achieved, i.e. ${\boldsymbol v}_{DP}=v_{DP} \boldsymbol e_x$.

Defining the perpendicular components of the position vector as $\boldsymbol x_\perp \equiv \boldsymbol x - x\boldsymbol e_x$, away from the pore entrance the velocity fields have the form of a wall slip velocity $\boldsymbol v_{s}=v_{s}(x,t)\boldsymbol e_x$, fluid velocity $\boldsymbol v_{f}(\boldsymbol x,t)=v_{s}(x,t)f_x(\boldsymbol x_\perp,t)\boldsymbol e_x + \boldsymbol v_{{f}\perp }(\boldsymbol x,t)$, where $\boldsymbol e_x\boldsymbol \cdot \boldsymbol v_{f\perp }=0$, and $\boldsymbol v_{DP}=v_{DP}(x,t)\boldsymbol e_x$, where we assume an incompressible flow such that $\boldsymbol \nabla \boldsymbol \cdot \boldsymbol {v}_{f}=0$. The advective-diffusion equation (3.1) becomes

(3.2)\begin{equation} \frac{\partial N}{\partial t}+\boldsymbol v_{f}\boldsymbol\cdot\boldsymbol\nabla N+\frac{\partial}{\partial x}(v_{DP}N) = D_{p}\nabla^2N. \end{equation}

This equation can be solved numerically for $N$, although to do so would require a 2-D or 3-D computational scheme. To simplify our calculations, we revisit the analysis of Taylor (Reference Taylor1953) to derive a 1-D equation for the cross-sectionally averaged concentration by integrating the effects of the fluid flow, with the important differences that the fluid velocity field in the pore here has zero cross-sectional average and varies in $x$ and $t$ according to $v_{s}$; note that ${\boldsymbol v}_{f}$ varies with the transverse coordinates and time.

We start by defining the perturbation concentration $N'({\boldsymbol x}, t)$

(3.3)\begin{equation} N'({\boldsymbol x}, t) = N({\boldsymbol x}, t) -\langle N \rangle (x,t), \end{equation}

where the cross-sectional average is defined as

(3.4)\begin{equation} {\langle f \rangle } (x,t)=\frac1A\int_Af\, {\rm{d}}A',\end{equation}

for cross-sectional area $A$ with normal vector in the $x$-direction. We expand (3.2) using these definitions to find

(3.5)\begin{equation} \frac{\partial}{\partial t}(\langle N \rangle +N')+v_{{f}x}\frac{\partial\langle N \rangle }{\partial x}+\boldsymbol v_{f}\boldsymbol\cdot\boldsymbol\nabla N'+\frac{\partial}{\partial x}(v_{DP}(\langle N \rangle +N')) = D_{p}\left(\frac{\partial^2\langle N \rangle }{\partial x^2}+\nabla^2N'\right), \end{equation}

and take the cross-sectional average, which yields

(3.6)\begin{equation} \frac{\partial\langle N \rangle }{\partial t}+\langle {\boldsymbol v_{f}\boldsymbol\cdot\boldsymbol\nabla N'}\rangle +\frac{\partial}{\partial x}(v_{DP}\langle N \rangle ) = D_{p}\frac{\partial^2\langle N \rangle }{\partial x^2},\end{equation}

where we have used, by definition, $\langle N' \rangle =0$, $\langle {v_{{f}x}}\rangle =0$ and $\partial \langle N \rangle /\partial \boldsymbol x_\perp = \boldsymbol 0$.

Now we subtract (3.6) from (3.2) to obtain (still an exact expression)

(3.7)\begin{equation} \frac{\partial N'}{\partial t}+v_{{f}x}\frac{\partial\langle N \rangle }{\partial x}+\boldsymbol v_{f}\boldsymbol\cdot\boldsymbol\nabla N'-\langle{\boldsymbol v_{f}\boldsymbol\cdot\boldsymbol\nabla N'}\rangle + \frac{\partial}{\partial x}(v_{DP}N') = D_{p}\nabla^2 N', \end{equation}

at which point we must make approximations to go further. As originally developed by Taylor (Reference Taylor1953), we consider the lubrication approximation

(3.8)\begin{equation} | \nabla_\perp^2N'|\gg \left |\frac{\partial^2N'}{\partial x^2}\right |, \end{equation}

where we define $\nabla _\perp ^2\equiv \nabla ^2-\partial ^2/\partial x^2$. Assuming $O(v_{DP})=O(v_{{f}x})$, $| N' |\ll N$, we apply the approximation

(3.9)\begin{equation} O\left(v_{{f}x}\frac{\partial\langle N \rangle }{\partial x}\right) \gg O\left(\boldsymbol v_{f}\boldsymbol\cdot\boldsymbol\nabla N', \langle{\boldsymbol v_{f}\boldsymbol\cdot\boldsymbol\nabla N'}\rangle, \frac{\partial}{\partial x}(v_{DP}N')\right). \end{equation}

Finally, assuming the perturbation concentration becomes independent of the initial particle distribution, we take the steady-state limit $\partial N'/\partial t\to 0$ to arrive at an equation for the perturbation concentration,

(3.10)\begin{equation} v_{fx}\frac{\partial\langle N \rangle }{\partial x} = D_{p}\nabla_\perp^2N', \end{equation}

which must be solved for $N'(\boldsymbol x)$ by applying no-flux conditions at the pore walls and requiring that $\langle {N'}\rangle =0$. Equation (3.10) is analogous to the equation for the perturbation field of a solute obtained in the original analysis of Taylor (Reference Taylor1953). The steady-state limit neglects the effect on $\partial N'/\partial t$ of the time dependence of $v_{f}$, which arises from the time dependence of $v_{s}$. This is equivalent to assuming that the colloids reach a steady-state distribution throughout the cross-section instantaneously upon a change in $v_{f}$. We justify this assumption by comparing the time scale of particle diffusion through the cross-section, $A/D_{p}$, where $A$ is the cross-sectional area, with the time scale of axial particle transport, $\ell /v_{DP}$. Introducing particle diffusiophoretic mobility $\varGamma _{p}$, we have $v_{DP}\approx \varGamma _{p}/\ell$. Furthermore, as the diffusiophoretic motion cannot outpace its source, the diffusion of the solute, we can write $v_{DP}\lesssim D_{s}/\ell$. Thus, for any system that obeys $A/\ell ^2\ll D_{p}/D_{s}$, the steady-state limit $\partial N'/\partial t\to 0$ is valid.

Motivated by (3.10), it is convenient to define the quantities

(3.11a)\begin{gather} f_x(\boldsymbol x_\perp) = \frac{v_{{f}x}(\boldsymbol x,t)}{v_{s}(x,t)}, \end{gather}

(3.11b)\begin{gather}N'= {-}v_{s}(x,t)\frac{\partial\langle N \rangle }{\partial x}\frac{f_{N'}(\boldsymbol x_\perp)}{D_{p}}, \end{gather}

(3.11c)\begin{gather}\nabla_\perp^2f_{N'} ={-}f_x(\boldsymbol x_\perp), \end{gather}

where (3.11c) is to be solved with no-flux boundary conditions at the pore walls and the requirement that $\langle {f_{N'}}\rangle =0$.

The quantity $\langle {\boldsymbol v_{f}\boldsymbol \cdot \boldsymbol \nabla N'}\rangle$ therefore has the form

(3.12)\begin{equation} \langle{\boldsymbol v_{f}\boldsymbol\cdot\boldsymbol\nabla N'}\rangle = \left\langle v_{s}f_x\frac{\partial N'}{\partial x} + \boldsymbol v_{f\perp}\boldsymbol\cdot\boldsymbol\nabla_\perp N'\right\rangle = \left\langle v_{s}f_x\frac{\partial N'}{\partial x} + \boldsymbol\nabla_\perp\boldsymbol\cdot(\boldsymbol v_{f\perp} N') - N'\boldsymbol\nabla_\perp\boldsymbol\cdot\boldsymbol v_{f\perp} \right\rangle . \end{equation}

Applying the incompressible flow condition

(3.13)\begin{equation} \boldsymbol\nabla\boldsymbol\cdot\boldsymbol v_{f} = f_x\frac{\partial v_{s}}{\partial x} + \boldsymbol\nabla_\perp\boldsymbol\cdot \boldsymbol v_{f\perp} = 0, \end{equation}

we substitute for $\boldsymbol \nabla _\perp \boldsymbol \cdot \boldsymbol v_{f\perp }$ and apply the product rule of differentiation to obtain

(3.14)\begin{equation} \langle{\boldsymbol v_{f}\boldsymbol\cdot\boldsymbol\nabla N'}\rangle = \left\langle f_x\frac{\partial}{\partial x}(v_{s}N') + \boldsymbol\nabla_\perp\boldsymbol\cdot(\boldsymbol v_{f\perp} N') \right\rangle . \end{equation}

With the definitions of the cross-sectional area average in (3.4) and the perpendicular derivative vector $\boldsymbol \nabla _\perp$, denoting $\boldsymbol e_\perp$ as the normal vector to the boundary, using the divergence theorem and noting that there is zero fluid penetration at the pore walls, we finally arrive at

(3.15)\begin{equation} \langle{\boldsymbol\nabla_\perp\boldsymbol\cdot(\boldsymbol v_{f\perp}N')}\rangle = \frac1A\oint \boldsymbol e_\perp\boldsymbol\cdot\boldsymbol v_{f\perp} N'\, {\rm{d}}s = 0, \end{equation}

where the line integral is performed over the boundaries of the pore cross-section. This simplifies (3.14) by eliminating the perpendicular velocity components. Invoking the definition in (3.11c), we have the result

(3.16)\begin{equation} \langle{\boldsymbol v_{f}\boldsymbol\cdot\boldsymbol\nabla N'}\rangle ={-}\frac1{D_{p}} \langle{f_xf_{N'}}\rangle\frac{\partial}{\partial x}\left(v_{s}^2\frac{\partial\langle N \rangle }{\partial x}\right). \end{equation}

Substituting this result into (3.6), we have a 1-D partial differential equation (PDE) for the cross-sectionally averaged colloid concentration

(3.17)\begin{equation} \frac{\partial\langle N \rangle }{\partial t} = \frac{\partial}{\partial x}\left(K\frac{\partial\langle N \rangle }{\partial x}-v_{DP}\langle N \rangle \right), \end{equation}

where the modified diffusion coefficient is defined as

(3.18)\begin{equation} K\equiv D_{p} + \frac1{D_{p}}\langle{f_xf_{N'}}\rangle (v_{s}(x,t))^2. \end{equation}

In Appendix A we calculate $\langle {f_xf_{N'}}\rangle$ by solving (3.11c) for the forms of $v_{{f} x}$ in both the 2-D and 3-D narrow pores depicted in figure 2. We note that the results are equivalent to the well-known 2-D result and additionally the 3-D result introduced by Chatwin & Sullivan (Reference Chatwin and Sullivan1982) upon replacing the diffusioosmotic slip velocity with the mean velocity of a pressure-driven flow. Furthermore, the elements of our Taylor dispersion model have been introduced in part by previous studies. For example, Chu et al. (Reference Chu, Garoff, Tilton and Khair2021) present a Taylor dispersion model of diffusiophoretic motion of particles; however, the flow field is driven by an external pressure gradient and does not include the effect of a diffusioosmotic slip-driven flow field. Furthermore, due to their use of cylindrical geometry, they do not examine the effect of the channel cross-section aspect ratio on the dispersion. It would be straightforward to include features of their model such as a colloidal reaction term, an externally driven flow, and a cylindrical channel geometry into our model. Migacz & Ault (Reference Migacz and Ault2022) also present a Taylor dispersion model of diffusiophoretic colloids with an externally driven flow field; their focus is on a narrow 2-D channel for which they neglect the diffusioosmotic wall slip. It would be straightforward to include in our model the zeta-potential dependence on the solute concentration featured in their model, although to incorporate their characterization of early-time dynamics requires further research. Stone & Brenner (Reference Stone and Brenner1999) present a dispersion model for radial outflow between two disks, which exhibits streamwise variation in the mean velocity. Our model of diffusiophoresis and diffusioosmosis generalizes the above studies by including a multi-directional, longitudinal direction-dependent and time-dependent flow with zero mean in a channel geometry, allowing for accurate direct comparison of the model with dead-end pore diffusiophoresis experiments (see § 5).

Figure 2. Schematic of 2-D and 3-D descriptions of the velocity profile in dead-end pores, depicted in panels (a) and (c), respectively. Coordinate axes are defined with the origin at the upstream corner. Pore length, height and width are defined, respectively, as $\ell$, $h$ and $w$. Velocity profiles are depicted in the $x$-direction. Dispersion of the colloid concentration over time scale $\tau =\tau _1\to \tau _2$ is depicted in the 2-D description (panel (a)) and the cross-sectional average of the colloid concentration is shown in panel (b), where the full width half-maximum $\varDelta$ is visually defined as the distance between the two points in the distribution with values of half the maximum. Panel (d) is a qualitative plot demonstrating the effect of dispersion on a colloid distribution. Introducing a third dimension, as depicted in panel (c), causes an increase in dispersion, as is well known in the Taylor dispersion literature (Doshi, Daiya & Gill Reference Doshi, Daiya and Gill1978; Chatwin & Sullivan Reference Chatwin and Sullivan1982).

4. Analysis of dispersion equations

4.1. Comparison of the dispersion equation with the 2-D advective-diffusion equation

Numerical solutions for the diffusiophoretic motion of colloids in a 2-D dead-end pore (figure 2a) based on (3.1) are compared with numerical simulations of the dispersion model based on (3.17)–(3.18). The 2-D pore is simulated in order to validate our Taylor dispersion model with numerical tests; for comparison with our experiments, a 3-D model must be implemented. We introduce non-dimensional quantities $\boldsymbol X = \boldsymbol x/\ell$, ${\boldsymbol V}\equiv {\boldsymbol v} \ell / D_{s}$ (for all velocities), $\tau \equiv tD_{s}/\ell ^2$ and $n\equiv N/N_0$, where $N_0$ is the initial maximum concentration of the colloid, $\ell$ is the pore length as depicted in figure 2 and $D_{s}$ is the characteristic diffusivity of the diffusing solute whose gradient drives the colloid diffusiophoresis and the diffusioosmotic slip velocity on the pore walls. For simplicity we consider a monovalent, binary salt as the solute, whose concentration can be described by one quantity. We define $C$ as the concentration of the solute, which is made non-dimensional by $c\equiv C/C_0$ for initial maximum concentration $C_0$. Substituting in our non-dimensional quantities, (3.1) becomes

(4.1)\begin{equation} \frac{\partial n}{\partial\tau} + \boldsymbol\nabla\boldsymbol\cdot( \boldsymbol V_{p}n ) = \frac{D_{p}}{D_{s}}\nabla^2n,\end{equation}

where we redefine $\boldsymbol \nabla \equiv [\partial /\partial X,\partial /\partial Y]$ and note that the velocity $\boldsymbol V_{p} = \boldsymbol V_{f} + \boldsymbol V_{DP}$ is the sum of the fluid and diffusiophoretic velocities. We define the non-dimensional modified diffusion coefficient as

(4.2)\begin{equation} \mathcal{K} \equiv \frac{D_{p}}{D_{s}} + \frac{V_{s}^2}{D_{p}/D_{s}}\frac{\langle{f_Xf_{n'}}\rangle}{\ell^2},\end{equation}

where $\langle {f_Xf_{n'}}\rangle /\ell ^2\equiv \langle {f_xf_{N'}}\rangle = (h/\ell )^2/210$, well known for pressure-driven parabolic flow in a 2-D channel, is computed in Appendix A.1, see (A6). We see that introducing the non-dimensional quantities reveals a set of two non-dimensional parameters to describe our system: $D_{p}/D_{s}$ and $h/\ell$. Now, making (3.17) non-dimensional, we have

(4.3)\begin{equation} \frac{\partial\langle n \rangle }{\partial\tau} = \frac\partial{\partial X}\left(\mathcal{K}\frac{\partial\langle n \rangle }{\partial X} - V_{DP}\langle n \rangle \right). \end{equation}

Equation (4.2) can be adapted to describe the solute concentration $c$

(4.4)\begin{equation} \frac{\partial\langle c \rangle }{\partial\tau} = \frac\partial{\partial X}\left(\mathcal{K}_{s}\frac{\partial\langle c \rangle }{\partial X} - V_{f}\langle c \rangle \right), \end{equation}

where we define the solute modified diffusion coefficient as

(4.5)\begin{equation} \mathcal{K}_{s}\equiv 1 + \frac{(h/\ell)^2}{210}V_{s}^2. \end{equation}

As described in § 3, with a solute Péclet number $Pe_{s} \approx \varGamma _{w}/D_{s} \lesssim 1$ the solute concentration is approximately one-dimensional and can be accurately calculated with a 1-D diffusion equation. This ensures furthermore that the solute concentration can be calculated with the dispersion equations (4.4) and (4.5) in the lubrication approximation (Taylor Reference Taylor1953; Aris Reference Aris1956; Chu et al. Reference Chu, Garoff, Tilton and Khair2021), with accuracy greater than or equal to the simple diffusion model. We include the dispersion effect on the solute as a general description of our model. Comparisons between the dispersion model and a 1-D diffusion model of the low Péclet number solute revealed only minor differences.

We assume constant particle and wall zeta potentials and a vanishing ratio of Debye length to particle radius. Therefore, the diffusiophoretic and diffusioosmotic velocities $v_{DP}$ and $v_{s}$ depend on the constant particle and wall zeta potentials $\psi _{p,w}$, the gradient of the logarithm of the solute concentration and the diffusivity difference factor $\beta =(D_+-D_-)/(z_+D_+-z_-D_-)$ of the electrolyte with ionic diffusivities $D_+$ and $D_-$ and valences $z_+$ and $z_-$. The zeta potentials are made non-dimensional by defining $\varPsi = e\psi /(k_{B}T)$, where $e$ is the elementary charge, $k_{B}$ is the Boltzmann constant and $T$ is the absolute temperature of the solution. It is convenient to introduce the velocity prefactor $\alpha = \epsilon k_{B}^2 T^2 / (e^2 \mu D_{s})$, where $\epsilon$ is the electrical permittivity of the solution and $\mu$ is the solution viscosity. We define diffusiophoretic mobility $\varGamma _{p}$ and diffusioosmotic mobility $\varGamma _{w}$, which are made non-dimensional by the solute diffusivity and related to the particle and wall zeta potentials by the approximate form (Prieve et al. Reference Prieve, Anderson, Ebel and Lowell1984; Keh & Ma Reference Keh and Ma2005; Velegol et al. Reference Velegol, Garg, Guha, Kar and Kumar2016)

(4.6)\begin{equation} \frac{\varGamma_{p,w}}{D_{s}} = \alpha\left(\beta\varPsi_{p,w} + \frac{\varPsi_{p,w}^2}8\right). \end{equation}

The diffusiophoretic velocity and the diffusioosmotic slip velocity are then given by

(4.7a)\begin{equation} \boldsymbol V_{DP} = \frac{\varGamma_{p}}{D_{s}}\boldsymbol\nabla \log c, \end{equation}

and

(4.7b)\begin{equation} V_{s} = \frac{\varGamma_{w}}{D_{s}}\left[\frac{\partial \log c}{\partial X}\right]_{Z=0,h/\ell}. \end{equation}

where $Z=0$ and $Z=h/\ell$ correspond to the pore walls.

We can consider the Taylor dispersion limits to simplify our expressions for the velocities. Taylor dispersion most accurately models systems with a low Péclet number, which is the ratio of particle transport time scales

(4.8)\begin{equation} Pe_{TD} \equiv \frac{\text{time scale to diffuse through channel cross section}}{\text{time scale to transport to end of channel}}\ll1 . \end{equation}

In our system, the mechanism of bulk particle transport along the pore length is the diffusiophoretic velocity. If we reintroduce dimensional quantities, we see that $v_{DP}$ scales with $\varGamma _{p}/\ell$. The diffusiophoretic mobility $\varGamma _{p}$ cannot be greater than $D_{s}$, the characteristic diffusivity of the solute. This gives us the condition

(4.9)\begin{equation} \left(\frac h\ell\right)^2\ll \frac{D_{p}}{D_{s}},\end{equation}

for which we expect our Taylor dispersion model to describe 2-D pore diffusiophoresis and diffusioosmosis with reasonable accuracy. The equivalent condition for the dispersion of the solute would be $(h/\ell )^2\ll 1$, which is always true in the lubrication approximation that we have employed for the velocity profiles. Therefore, we can approximate all $c(\boldsymbol X,\tau )$ with $\langle c \rangle (X,\tau )$, giving us the simplified velocities

(4.10a)\begin{equation} V_{DP} = \frac{\varGamma_{p}}{D_{s}}\frac{\partial \log\langle c \rangle }{\partial X}, \end{equation}

and

(4.10b)\begin{equation} V_{s} = \frac{\varGamma_{w}}{D_{s}}\frac{\partial \log\langle c \rangle }{\partial X}. \end{equation}

Simulated colloid concentrations from the 2-D advective diffusion model (see (4.1)) and the 2-D reduced-order dispersion model (see (4.3) and (A6)) are directly compared in figure 3. Particle diffusivity and the ratio $(h/\ell )^2/(D_{p}/D_{s})$ are both varied over three orders of magnitude, demonstrating better agreement of the colloid distributions as the Taylor dispersion limit is better satisfied. Diffusiophoretic focusing is observed in panels (d)–(i), where the colloids increase in concentration as time increases, and is most prominent for the smallest ratios (panels (f) and (i)). This is consistent with the form of (4.2), where an increase in $(h/\ell )^2/(D_{p}/D_{s})$ corresponds to an increase in dispersion. Panels (a)–(c) do not demonstrate diffusiophoretic focusing, however, as the large value for particle diffusivity dominates the dispersion effect.

Figure 3. Comparison of the cross-sectionally averaged diffusiophoretic colloid concentration $\langle n \rangle$ in a dead-end pore with diffusioosmotic slip-driven flow at the walls. Solid lines show numerical solutions to the full 2-D model, (4.1), and dashed lines show numerical solutions to the 1-D model with 2-D dispersion, (4.3) and (A6). The two different numerical solutions have matching initial Gaussian distributions and no-flux conditions at the pore walls and inlet. The diffusiophoretic velocity and the diffusioosmotic slip are driven by a solute diffusing out of the pore into an infinite reservoir. The initial solute condition is $\langle c \rangle (\tau =0)=1$ and the boundary conditions are $\langle c \rangle (X=0)=0.1$ and no-flux conditions at the pore walls. Three diffusivities of the colloidal particles and three pore heights are considered. Excellent agreement is observed between the two models for $(h/\ell )^2=D_{p}/D_{s}/10$ and $(h/\ell )^2=D_{p}/D_{s}/100$, as predicted by the Taylor dispersion limit (4.9), and consistent with the steady-state limit $\partial N'/\partial t\to 0$.

We next report the full width half-maximum (defined in figure 2b), where larger magnitudes indicate greater dispersion. The full width half-maximum as a function of time for the colloid distributions determined from (4.3) and (A6) are displayed in figure 4. The full width half-maximum is compared for varying particle diffusivity across three orders of magnitude and varying pore wall zeta potential: a typical experimental value (Alessio et al. Reference Alessio, Shim, Mintah, Gupta and Stone2021) $\varPsi _{w} = -4$ (panels (a)–(c)), a strongly negative value $\varPsi _{w} = -10$ (panels (d)–(f)) and a positive value $\varPsi _{w} = 4$ (panels (g)–(i)). The strongly negative value corresponds to a large slip velocity and thus to increased dispersion. Equation (4.6), considering that $\beta$ is negative for ${\rm Na}^+$ and ${\rm Cl}^-$, indicates that the positive value corresponds to a small slip velocity and thus to decreased dispersion. Furthermore, each panel shows the full width half-maximum of the colloid distribution for three values of the ratio $(h/\ell )^2/(D_{p}/D_{s})$: 1, 1/10 and 1/100. This effect is weakest for the leftmost column (panels (a), (d) and (g)), as the large value of particle diffusivity dominates the dispersion effect similarly to the simulations of figure 3.

Figure 4. The full width half-maximum $\varDelta (\langle n \rangle )$ (see figure 2b) vs time $\tau$ of colloidal particle distributions $\langle n \rangle$, as obtained by solving (4.3) and (A6). The colloid distributions have Gaussian initial distributions and no-flux conditions at the pore walls and inlet. The diffusiophoretic velocity and the diffusioosmotic wall slip are driven by a solute diffusing out of the pore into an infinite reservoir. The initial solute concentration is $\langle c \rangle (\tau =0)=1$ and the boundary conditions are $\langle c \rangle (X=0)=0.1$ and no-flux conditions at the pore walls. The full width half-maximum is compared for varying particle diffusivity ($D_{p}/D_{s} = 10^{-2}, 10^{-3}, 10^{-4}$) and varying pore wall zeta potential ($\varPsi _{w} = -4, -10, 4$). Furthermore, each panel shows the full width half-maximum for three values of the ratio $(h/\ell )^2/(D_{p}/D_{s})$: 1, 1/10 and 1/100.

4.2. Parameter analysis of the 3-D dispersion equation

To achieve quantitative agreement between the experiments of § 5 and the reduced-order dispersion simulations requires the consideration of the variation of the flow in three spatial dimensions. In Appendix A.2 we introduce the pore width $w$ such that the walls are located at $Y=0$ and $Y=w/\ell$ and $h < w$, and present the corresponding modified diffusion coefficient.

We modify the model in § 4.1 by substituting $\mathcal {K}(X,\tau )$ with the 3-D coefficient, defined in (4.2) and (A14) where $\langle {f_Xf_{n'}}\rangle /\ell ^2\equiv \langle {f_xf_{N'}}\rangle$, into (4.3). Note that we use the same method to make the coefficient non-dimensional, and furthermore that the non-dimensional coefficient for solute dispersion in (4.4) is replaced with the 3-D version in a similar manner. Finally, we include no-flux conditions for each wall of the pore.

The full width half-maximum vs time for colloid distributions determined from (4.3) and (A14) are shown in figure 5(a–c). The results indicate enhanced dispersion as the width of the distribution increases with stronger dispersion. As in § 4.1, we present the effect of pore wall potential $\varPsi _{w}$ on the dispersion in figure 5. Figure 5(d,e) shows the non-dimensional modified coefficient of diffusion vs the non-dimensional particle diffusivity, averaged across the length of the pore and across the timespan $\tau =0\to 0.4$ (panel (d)), and vs time, averaged across the length of the pore (panel (e)). Panel (d) demonstrates a minimum in dispersion for intermediate values of particle diffusivity as predicted by (4.2). Panel (e) highlights how there is a multiple-order-of-magnitude decrease in the modified coefficient of diffusion at early times.

Figure 5. (a)–(c) The full width half-maximum $\varDelta (\langle n \rangle )$ vs time $\tau$ of the colloid distributions $\langle n \rangle$ solved from (4.3) and (A14). The colloid distributions have Gaussian initial distributions and observe no-flux conditions at the pore walls and inlet. The diffusiophoretic velocity and the diffusioosmotic slip are driven by a solute diffusing out of the pore into an infinite reservoir. The initial solute condition is $\langle c \rangle (\tau =0)=1$ and the boundary conditions are $\langle c \rangle (X=0)=0.1$ and no-flux conditions at the pore walls. The full width half-maximum is evaluated for varying pore wall zeta potential ($\varPsi _{w}=-4,-10, 4$). (d) The non-dimensional modified coefficient of diffusion $\mathcal {K}$ vs non-dimensional particle diffusivity $D_{p}/D_{s}$. The double bracket indicates an average over the length of the pore and the timespan $\tau =0\to 0.4$. A minimum is seen in each curve for an intermediate value of particle diffusivity. (e) The non-dimensional modified coefficient of diffusion vs time. The single bracket indicates an average over the length of the pore. Particle diffusivity is chosen to be $D_{p}/D_{s}=10^{-3}$. A sharp decrease is seen for early times. Each panel demonstrates the effect of increasing cross-sectional aspect ratio $h/w$ to decrease dispersion, indicated by decreasing full width half-maximum (a–c) and decreasing modified coefficient of diffusion (d,e).

Each panel of figure 5 demonstrates the trend for dispersion to increase as the aspect ratio $h/w$ decreases. This is a natural consequence of dispersion being enhanced by the influence of the side walls. Furthermore, the inclusion of the 3-D effect of $h/w$ strictly increases the dispersion of the colloid distributions compared with those calculated from the dispersion model of the 2-D channel (Doshi, Daiya & Gill Reference Doshi, Daiya and Gill1978; Chatwin & Sullivan Reference Chatwin and Sullivan1982).

5. Experimental methods and comparison with the 1-D equation

We designed experiments using a dead-end pore with $w=100\ \mathrm {\mu }$m, $h=50\ \mathrm {\mu }$m and $\ell =5$ mm to compare with the model of the 1-D representation of the (3-D) dispersion; $\ell \gg h, w$. Three different particle sizes (diameter $d_{p}=1, 0.5$ and $0.2\ \mathrm {\mu }$m; see Appendix D for particle information) were used to examine the influence of particle diffusivity $D_{p}$, which is related to particle diameter by the Boltzmann constant $k_{B}$, the absolute temperature $T$ of the solution and the viscosity $\mu$ of the solution through the Stokes–Einstein equation

(5.1)\begin{equation} D_{p} = \frac{k_{B} T}{3{\rm \pi}\mu d_{p}}. \end{equation}

In order to establish an initial condition that a finite number of particles are trapped in the inlet region of the pore, we use three successive steps separated by two air bubbles (figure 6a). The polydimethylsiloxane (PDMS) microfluidic channel is made by standard soft lithography and the channel block is bonded to a thin sheet of PDMS to ensure the same surface properties for all walls.

Figure 6. Experiments in a long pore ($w=100\ \mathrm {\mu }$m, $h=50\ \mathrm {\mu }$m and $\ell =5$ mm). (a) Schematic of typical experimental steps. (ai) The pore is initially filled with the 10 mM NaCl solution. The 10 mM NaCl solution with suspended particles, separated from the original solution by a first air bubble, is flowed in the main channel, then comes in contact with the liquid in the pore. (aii) Flow in the main channel introduces penetration of streamlines into the pore at the pore inlet (penetration depth $\approx w$), which allows a patch of particles to form at the inlet region of the pore. Then, separated by the second air bubble, a 1 mM NaCl solution is flowed into the main channel to create a concentration gradient in the pore. (aiii) Finally, we obtain diffusiophoresis of a finite number of particles toward the dead-end. (b) Fluorescent images obtained from the experiments with carboxylate-modified polystyrene (c-PS; diameter $d=0.5\ \mathrm {\mu }$m) particles. Image intensity is enhanced for visualization. Original images are included in Appendix C (figure 10). Scale bar is 100 $\mathrm {\mu }$m.

The pore is initially filled with a 10 mM NaCl solution. Then, the 10 mM NaCl solution with suspended polystyrene (PS) particles, separated by a first air bubble from the original solution, is introduced in the main channel (width, height and length are, respectively, $W=1.2$ mm, $H=200\ \mathrm {\mu }$m and $L =5$ cm) at a mean flow speed $\langle u\rangle \approx 2.5$ mm s$^{-1}$ (figure 6a-i). Once the two solutions come in contact with each other, particles start to accumulate at the pore inlet by the slight penetration of streamlines (penetration depth $\approx w$) (Battat et al. Reference Battat, Ault, Shin, Khodaparast and Stone2019). One minute after the particle suspension flows in the main channel, a second air bubble is introduced, followed by a 1 mM NaCl solution (figure 6a-ii). Once the 1 mM NaCl solution contacts the liquid in the pore, the main channel flow speed is reduced to $\langle u\rangle =25\ \mathrm {\mu }$m s$^{-1}$. Fluorescent images are then recorded every 10 s using an inverted microscope (Leica DMI4000B; figure 6a-iii). Typical experimental images are shown in figure 6(b) as a time sequence.

For the three values of $d_{p}$, we compare the experiments with the 1-D model with 3-D dispersion (figure 7). Details of the model are given in § 4.2. The particle zeta potential has a strong influence on the peak location of the colloid distribution, and the wall zeta potential has a strong influence on the dispersion of the colloid distribution. The dimensionless particle zeta potential $\varPsi _{p}$ and wall zeta potential $\varPsi _{w}$ were calculated as fitting parameters for each experiment. The fitted parameters used in the simulations of each panel are: (a) $\varPsi _{p}=-2.84, \varPsi _{w}=-3.91$, (b) $\varPsi _{p}=-3.31, \varPsi _{w}=-3.26$ and (c) $\varPsi _{p}=-3.04, \varPsi _{w}=-2.06$. The zeta potentials are each calculated within $\varPsi \pm 0.002$ using a least squares method that compares model and experiment.

Figure 7. Comparison of experimental data (solid) with simulated data (dashed) for different colloidal particle diameters. The concentration distributions in a given experiment, in order of peak location from left to right, correspond to times $\tau =0.1, 0.2, 0.3$ and $0.4$. Fitting parameters are the dimensionless particle zeta potential $\varPsi _{p}$ and wall zeta potential $\varPsi _{w}$. Unique values of the fitting parameters were calculated for the simulations of each panel: (a) $\varPsi _{p}=-2.84, \varPsi _{w}=-3.91$, (b) $\varPsi _{p}=-3.31, \varPsi _{w}=-3.26$ and (c) $\varPsi _{p}=-3.04, \varPsi _{w}=-2.06$. Despite a bias due to particle size in the fitted values of the wall zeta potential, there is very good agreement between the experiments and the simulations.

Some variation in particle zeta potentials is expected due to the independent manufacturing of all three sizes of particles used. However, the variation in fitted wall zeta potentials is unexpected as pore walls properties were not changed between experiments. It is possible that deviations from the predicted fluid velocity profile at the pore inlet, not captured by the Taylor dispersion model, enforce a bias of the particle size on the apparent dispersion of the colloid distribution (Battat et al. Reference Battat, Ault, Shin, Khodaparast and Stone2019). The sharp concentration gradient near the pore inlet at $\tau =0$ causes a three to four order-of-magnitude decrease in the modified coefficient of diffusivity over the span of $\tau =0\to 0.05$ (figure 5e), meaning the majority of the dispersion occurs near the inlet during this timespan. A small effect of particle size on the dispersion in this region can strongly impact the apparent dispersion of the colloid distribution throughout the pore. Therefore, it is not feasible with our current set-up to accurately measure the wall zeta potential. In Appendix C, we perform an experiment holding particle size constant and calculate fitted potentials that demonstrate good agreement, supporting our claim that a particle-size dependence of the dispersion in the inlet region may be responsible for variations in the fitted values of the wall zeta potential.

The complex nature of the inlet region in microfluidic dead-end pores may also explain overestimation of the fitted particle zeta potentials. This effect is not due to imprecise measurements. In particular, in Appendix C we demonstrate precise fitting of zeta potentials for particles of the same batch. Full 3-D analysis including complex flow structure near the pore inlet may be helpful to further develop the dispersion system so that a direct mapping of the particle distribution versus the wall or particle potential values is possible for a wide range of surface properties. Additionally, it is possible that differences in manufacturing of each size of particle affected the diffusive behaviour; a wide variation in particle zeta potential within a batch of one size could introduce an apparent spreading that contributes to the dispersion. Despite the imprecise fitted values of wall zeta potential, the very good agreement between the experimental data and the fitted simulation data supports our model.