Solute-mediated colloidal vortex in a microfluidic T-junction

Haoyu Liu Amir A. Pahlavan Corresponding author [email protected] Department of Mechanical Engineering and Materials Science, Yale University, New Haven, Connecticut 06511, USA

Abstract

Solute gradients next to an interface drive a diffusioosmotic flow, the origin of which lies in the intermolecular interactions between the solute and the interface. These flows on the surface of colloids introduce an effective slip velocity, driving their diffusiophoretic migration. In confined environments, we expect the interplay between diffusiophoretic migration and diffusioosmotic flows near the walls to govern the motion of colloids. These near-wall osmotic flows are, however, often considered weak and neglected. Here, using microfluidic experiments in a T-junction, numerical simulations, and theoretical modeling, we show that the interplay between osmotic and phoretic effects leads to unexpected outcomes: forming a colloidal vortex in the absence of inertial effects, and demixing and focusing of the colloids in the direction opposite to what is commonly expected from diffusiophoresis alone. We show these colloidal vortices to be persistent for a range of salt types, salt gradients, and flow rates, and establish a criterion for their emergence. Our work sheds light on how boundaries modulate the solute-mediated transport of colloids in confined environments.

^†^†preprint: APS/123-QED

The discovery of colloidal migration in response to solute gradients dates back to the works of Derjaguin more than half a century ago [1] and the later developments by Prieve, Anderson and others [2, 3, 4, 5, 6]. While sophisticated technologies for manipulation of colloids and biomolecules in microfluidics using external electric/magnetic fields have been developed over the past two decades [7, 8, 9, 10], using a “pinch of salt” to guide the motion of colloids offers a tempting alternative [11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 29, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70] that biology seems to have benefited from as evidenced by the role of diffusiophoresis in cargo transport and phase separation within the living cells [71, 72, 73, 74, 75, 76, 77].

Solute gradients in confined and crowded environments also drive diffusioosmotic flows next to the confining surfaces. These flows have been utilized for directional pumping and trapping of DNA and exosomes [78, 79, 80]. However, they are often considered weak, leading to small changes to the diffusiophoretic migration of colloids, and therefore neglected. A few recent studies have reported on the emergence of solute-driven vortices and convection rolls near surfaces, leading to the trapping and entrainment of colloids [81, 82, 83]. Yet, how and under what circumstances diffusiophoresis and diffusioosmosis conspire to lead to these reported behaviors has remained elusive.

Here, we use microfluidic experiments and 3D numerical simulations to demonstrate that solute-driven colloidal vortices emerge due to a subtle interplay between solutal gradients perpendicular to the walls, driving the phoretic migration of colloids toward the walls, and solutal gradients parallel to the walls, sweeping the colloids by the diffusioosmotic flows. We show the universality of these vortices for a range of salt types, salt gradients, and flow rates, and demonstrate that the vortex location, which coincides with the colloidal focusing line $x_{f}$ scales as $\textrm{Pe}^{-1/3}$ , where $\textrm{Pe}=UH/D_{s}$ with $U$ as the flow velocity, $H$ as the channel height, and $D_{s}$ as the solute diffusivity. We finally demonstrate that the competition between diffusioosmotic mobility $\Gamma_{w}$ and diffusiophoretic mobility $\Gamma_{p}$ governs the formation of these colloidal vortices.

Refer to caption — Figure 1: Solute gradients lead to the emergence of a colloidal vortex and its unexpected focusing in a microfluidic T-junction. (a) Setup schematic: we inject solute concentrations $c_{L}$ and $c_{R}$ from the left and right branches, respectively. The colloidal particles (green dots) are injected from the right side. The red box demonstrates the field of view in our experiments. (b) In the absence of solute gradients, colloids simply follow the flow streamlines. Introducing a solute gradient (1mM vs 0.01mM NaCl here), however, leads to the spiral-like motion of colloids (insets), and the emergence of a focusing band in the lower solute concentration region, in contrast with the previous observations [11].

We use a microfluidic T-junction setup, which is schematically shown in Fig. 1 (a), with height $H=50\leavevmode\nobreak\ \mu\textrm{m}$ , width $W=500\leavevmode\nobreak\ \mu\textrm{m}$ , and length $L=4000\leavevmode\nobreak\ \mu\textrm{m}$ . The mid-channel width is set as $2W$ to keep the mean flow velocity constant. We inject solutions with salt concentrations $c_{L}$ and $c_{R}$ from the left and right sides, respectively; colloids are introduced from the right branch [84]. In the absence of solute gradients, i.e., the control case, colloids occupy half the channel and move downstream without any noticeable diffusion (Fig. 1 (b)). We use particles with 1 micron diameter, with diffusivity $D_{p}\approx 10^{-13}\textrm{m}^{2}/\textrm{s}$ , which should be contrasted with the solute diffusivity of $D_{s}\approx 10^{-9}\textrm{m}^{2}/\textrm{s}$ . For the experiments shown in Fig. 1 (b), the particle Peclet number is therefore $\textrm{Pe}_{p}=UH/D_{p}\approx 10^{5}$ , indicating the dominance of advection over diffusion for the particles.

Adding a “pinch of salt” to either channel, however, changes the picture completely. We expect the negatively-charged colloids to migrate toward the higher salt concentration for the salts we use (LiCl, NaCl, KCl) [11]. Surprisingly, however, we observe the focusing of colloids in the lower salt concentration region, i.e., when $c_{L}>c_{R}$ , colloids move to the right side, and when $c_{L}<c_{R}$ , a focusing line emerges on the left side (Fig. 1 (b)), in contrast to what is expected [11].

To gain insight into this unexpected focusing, we conducted 3D numerical simulations of the flow, solute, and particle transport. The flow field is governed by the Stokes equations:

	$\displaystyle 0$	$\displaystyle=-\nabla p+\mu\nabla^{2}\mathbf{u},$		(1)
	$\displaystyle 0$	$\displaystyle=\nabla\cdot\mathbf{u},$		(2)

where $\mathbf{u}$ is the flow velocity, $p$ represents the fluid pressure, and $\mu$ is the fluid viscosity. The solute $c$ and colloid $n$ fields are governed by advection-diffusion equations:

	$\displaystyle\frac{\partial c}{\partial t}$	$\displaystyle=-\nabla\cdot\left(\mathbf{u}c\right)+D_{s}\nabla^{2}c,$		(3)
	$\displaystyle\frac{\partial n}{\partial t}$	$\displaystyle=-\nabla\cdot\left(\left(\mathbf{u}+\mathbf{u}_{dp}\right)n\right)+D_{p}\nabla^{2}n,$		(4)

where the colloid diffusiophoretic velocity is $\mathbf{u}_{dp}=\Gamma_{p}\nabla\ln c$ with $\Gamma_{p}$ as the diffusiophoretic mobility [3, 6, 35, 84]. The flow velocity on the walls of the channel is determined by the diffusioosmotic slip velocity $\mathbf{u}_{do}=-\Gamma_{w}\left(\nabla\ln c-\left(\mathbf{e}_{n}\cdot\nabla\ln c\right)\mathbf{e}_{n}\right)$ , where $\mathbf{e}_{n}$ is the unit normal vector on the boundary. We solve these equations using the open-source software OpenFoam [85, 84].

In a Lagrangian frame, the colloidal transport can be described as $d\mathbf{x}_{p}/dt=\mathbf{u}+\mathbf{u}_{dp}$ , where $\mathbf{x}_{p}(t)$ represents the particle location at time $t$ . Following the trajectories of particles obtained from the simulations, we gain insight into the focusing mechanism (Fig. 2). In the control case, where solute gradients are absent, colloids simply follow the flow streamlines as evident from the absence of mixing of the colored particles introduced upstream in the four quadrants (Fig. 2 (a, b)), which remain in their corresponding quadrants as they travel along the channel (Fig. 2 (c)). In this case, there is no colloidal motion in the z direction, perpendicular to the fluid flow.

In the presence of salt gradients, both diffusiophoretic migration of colloids and diffusioosmotic flows emerge. To disentangle these two effects, we first turn-off the osmotic flows near the walls, setting $\Gamma_{w}=0$ . In this case, colloids deviate from the flow streamlines, and move toward the higher salt concentration side due to diffusiophoresis (Fig. 2 (d)). This behavior is consistent with the observations of Abécassis et al. [11]; however, opposite of what we observed in our experiments (Fig. 1). We therefore conclude that diffusiophoresis alone cannot explain our observations.

Accounting for the diffusioosmotic flows near the channel walls changes the particle trajectories drastically (Fig. 2 (e)). The colloids move out-of-plane in the $z$ direction, and get dragged along the walls due to the diffusioosmotic flows, leading to their spiraling motion, and focusing on the right side of the channel, opposite of what diffusiophoresis alone does (Fig. 2 (f)). The presence of diffusioosmosis, therefore, does not simply lead to a weak quantitative drift in the colloidal trajectories, but leads to significant qualitative changes, and an effective demixing of the colloidal field that certainly cannot be described using a diffusive framework.

The out-of-plane motion of colloids is driven by the solute gradients in the $z$ direction. The presence of shear near the wall and its absence in the mid-plane leads to a differential solute diffusion and is the reason for the emergence of solute gradients in the $z$ direction, driving the motion of colloids both within and perpendicular to the x-y plane (Fig. 3 (a-c)). In the mid-plane of the channel, solute diffusion competes with advection along the channel ( $U\partial c/\partial y\approx D_{s}\partial^{2}c/\partial x^{2}$ ), leading to $\delta_{s}/H\sim(y/H)^{1/2}\textrm{Pe}^{-1/2}$ scaling for the solute diffusion width $\delta_{s}$ . Near the walls, however, the presence of shear fow $\dot{\gamma}z$ changes this scaling ( $\dot{\gamma}z\partial c/\partial y\approx D_{s}(\partial^{2}c/\partial x^{2}+\partial^{2}c/\partial z^{2})$ ), leading to $\delta_{s}/H\sim(y/H)^{1/3}\textrm{Pe}^{-1/3}$ scaling for the diffusion width [86, 87, 84].

Once near the walls, the colloids experience the competing effects of diffusiophoresis and diffusioosmosis. These osmotic flows create a fluid vortex, which is responsible for the spiral-like motion of the colloids [84]. To identify these osmotic near-wall vortices, we rely on the second invariant of the velocity gradient tensor, $\mathbf{A}\equiv\nabla\mathbf{u}$ , known as the Q criterion [88]. The eigenvalues of the velocity gradient tensor satisfy the characteristic equation $\lambda^{3}+P\lambda^{2}+Q\lambda+R=0$ , where $P$ , $Q$ , and $R$ are the invariants [89]. These invariants can be written in terms of the symmetric rate-of-strain tensor $\mathbf{S}=(\nabla\mathbf{u}+\nabla\mathbf{u}^{T})/2$ and antisymmetric vorticity tensor $\bm{\Omega}=(\nabla\mathbf{u}-\nabla\mathbf{u}^{T})/2$ , where $P=-\textrm{tr}(\mathbf{A})=0$ for incompressible flows, $R=-\textrm{det}(\mathbf{A})$ , and

Q=(\|\bm{\Omega}\|^{2}-\|\mathbf{S}\|^{2})/2,

(5)

with $Q>0$ identifying a vortex as the region, where vorticity dominates over the rate of strain [84] (Fig. 3 (d)). The colloid focusing can therefore be understood as the interplay between the diffusiophoretic migration of colloids away from the mid-plane toward the walls and entrainment by the diffusioosmotic vortex near the walls.

The focusing of colloids near the walls creates a local maximum in the colloid density field, where $\nabla n\approx 0$ . The colloid evolution equation near this local maximum can therefore be simplified to

\frac{\partial n}{\partial t}\approx\underbrace{-n\nabla\cdot\mathbf{u}_{dp}}_{\text{compressibility}}+\underbrace{D_{p}\nabla^{2}n}_{\text{diffusion}},

(6)

which balances the colloid accumulation due to the effective compressibility $\nabla\cdot\mathbf{u}_{dp}<0$ with the colloid diffusion away from the peak, where $\nabla^{2}n<0$ (Fig. 3 (e,f)). This effective compressibility of the colloid density field creates an effective particle source near the mid-plane and an effective particle sink near the walls [46].

The competition between diffusioosmosis and diffusiophoresis modulates the focusing location of colloids near the wall. At steady state and neglecting particle diffusion, this balance can be written as $\nabla\cdot\left((\mathbf{u}+\mathbf{u}_{dp})n\right)=0$ , which further simplifies to

\underbrace{\left(\Gamma_{w}-\Gamma_{p}\right)\frac{\partial\ln{c}}{\partial x}}_{\text{osmosis vs phoresis}}\left(\frac{\partial\ln{n}}{\partial x}\right)\approx\underbrace{\Gamma_{p}\frac{\partial^{2}\ln{c}}{\partial x^{2}}}_{\text{compressibility}},

(7)

leading to the following scaling for the particle focusing location

x_{f}/H\sim\left(\frac{\Gamma_{w}-\Gamma_{p}}{\Gamma_{p}}\right)\underbrace{\left(\frac{y}{H}\right)^{1/3}\textrm{Pe}^{-1/3}}_{\delta_{s}/H},

(8)

which suggests that the solute diffusion width near the walls sets the length scale for the colloidal focusing (Fig. 4 (a)). The dependence of the focusing location on the strength of solute gradients can be inferred from the evolution of colloids in the Lagrangian frame near the wall, which can be written as $dx_{p}/dt=(\Gamma_{w}-\Gamma_{p})\partial\ln c/\partial x$ , suggesting $x_{f}\sim\ln(c_{L}/c_{R})$ . This scaling argument for the colloid focusing location is supported by our experimental observations and numerical simulations (Fig. 4 (b-c)), where $x_{f}$ is defined as the distance between the central line $x=0$ and the focusing band at location $y=2W$ . In the experiments, $x_{f}$ is obtained by averaging a series of steady-state snapshots of the colloid field, and the error bars represent the standard deviation of measurements [84]. We further demonstrate in the Supplementary Materials that this scaling is consistent with self-similarity arguments for the solute and colloid fields [84].

Our scaling argument suggests that the focusing side is determined by the ratio $\Xi\equiv\Gamma_{w}/\Gamma_{p}$ : when osmotic mobility is stronger than the phoretic mobility ( $\Xi>1$ ), focusing occurs on the lower concentration side as we have shown. However, when phoretic mobility dominates ( $\Xi<1$ ), we expect the focusing on the higher concentration side. For a colloid with a given $\Gamma_{p}$ value, we can modulate the diffusioosmotic mobility of the surfaces by using different materials (glass versus PDMS) or by adding a buffer solution [84]. We observe that adding the buffer solution completely eliminates the diffusioosmotic flows, explaining why the early work of Abécassis et al. [11] had not observed these flows (Fig. 5 (a,i)). On a PDMS-coated channel, we do observe focusing to the right, but weaker than that of the glass (Fig. 5 (a,ii-iii)). While we cannot directly measure the diffusioosmotic mobility of these surfaces, we can infer them from the comparison of the colloid focusing location in the experiments and simulations (Fig. 5 (b)). Our simulations show that for $\Xi<1$ , the focusing location becomes insensitive to the value of diffusioosmotic mobility. However, for $\Xi>1$ , the focusing location keeps moving further to the right; this increase is initially almost linear, before plateauing to an upper limit, which is determined by the solute diffusion width [84]. We note here that we have used a constant zeta potential in all our simulations so far; considering variable zeta potential models leads to quantitative changes in our simulation results as discussed in the Supplementary Materials [84].

Our observations suggest that the subtle interplay between diffusiophoresis and diffusioosmosis can be utilized to infer the diffusioosmotic mobility of the surfaces, or separate particles based on their zeta potential or size. Beyond microfluidic applications for particle sorting and separation, our findings point to the importance of accounting for the often-ignored diffusioosmotic flows in the solute-mediated transport of colloids in complex environments, from contaminant removal in porous materials to cargo delivery in living cells [90, 91, 72, 73, 75]. We further expect these osmotic flows to play role in modulating reactions in subsurface flows [92, 93], phase separation in living cells [76, 77], and ionic transport in porous electrodes [94, 95, 96, 97, 98, 99, 100, 101].

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