Particle dispersion in porous media: Differentiating effects of geometry and fluid rheology

We investigate the effects of geometric order and fluid rheology on the dispersion of micron-sized particles in two-dimensional microfluidic porous media. Particles suspended in a mixture of glycerol and water or in solutions of partially hydrolyzed polyacrylamide (HPAM) polymers were imaged as they flowed through arrays of microscale posts. From the trajectories of the particles, we calculated the velocity distributions and thereafter obtained the longitudinal and transverse dispersion coefficients. Particles flowed in the shear-thinning HPAM solution through periodic arrays of microposts were more likely to switch between streamlines, due to elastic instabilities. As a result, the distributions of particle velocity were broader in HPAM solutions than in glycerol-water mixtures for ordered geometries. In a disordered array of microposts, however, there was little difference between the velocity distributions obtained in glycerol-water and in HPAM solutions. Correspondingly, particles flowed through ordered post arrays in HPAM solutions exhibited enhanced transverse dispersion. This result suggests that periodic geometric order amplifies the effects of the elasticity-induced velocity fluctuations, whereas geometric disorder of barriers effectively averages out the fluctuations.

Figure 1

(a) Steady-shear viscosity as a function of shear rate for the 0.1% HPAM (closed diamonds) and 90% glycerol (open circles) solutions. The red line represents a fit to the Carreau model, Eq. (1). (b) Linear dynamic oscillatory shear-based elastic ($G^{\prime }$, closed) and viscous ($G^{\prime \prime }$, open) moduli of the 0.1% HPAM solution as a function of angular frequency $\omega$. Inset: the complex viscosity ${\eta }^{*}$ and the viscosity $\eta$ obey an empirical Cox-Merz rule [51].

Figure 2

(a–d) Brightfield micrographs of the post arrays used in the experiments. Angles indicate the orientation of the director relative to the incident flow (along $x$). The red dot in (a) indicates the size of the particles relative to the post array. (a) Square $0^{\circ }$ post array. (b) Square ${45}^{\circ }$ post array. (c) Hexagonal post array. (d) Randomly distributed post array. (e–h, m–p) Representative trajectories of 2 $\mu \mathrm{m}$ PS particles and (i–l, q–t) COMSOL finite-element simulations of flow in the post arrays. The average velocity $\sim 250\mu \mathrm{m}{\mathrm{s}}^{-1}$ is constant in all images. In (e–l), the fluid was 90% glycerol-water: (e, i) square $0^{\circ }$ array (experimental flow rate: 0.4 $\mu \mathrm{l}{\min }^{-1}$); (f, j) square ${45}^{\circ }$ array (0.6 $\mu \mathrm{l}{\min }^{-1}$); (g, k) hexagonal array (0.6 $\mu \mathrm{l}{\min }^{-1}$); (h, l) randomly distributed array (0.3 $\mu \mathrm{l}{\min }^{-1}$). In (m–t), the fluid was 0.1% HPAM: (m, q) square $0^{\circ }$ array (0.3 $\mu \mathrm{l}{\min }^{-1}$); (n, r) square ${45}^{\circ }$ array (0.3 $\mu \mathrm{l}{\min }^{-1}$); (o, s) hexagonal array (0.2 $\mu \mathrm{l}{\min }^{-1}$); (p, t) randomly distributed array (0.1 $\mu \mathrm{l}{\min }^{-1}$). The colorbars in (e–t) indicate the magnitude of the longitudinal velocity $v_x$. The scale bar for all images [shown in (d)] is 50 $\mu \mathrm{m}$. The black arrows in (m) indicate recirculating trajectories.

Figure 3

Example trajectories of 2-$\mu \mathrm{m}$ particles in 90% glycerol-water solution that were flowed through square arrays oriented $0^{\circ }$ [(a) and (b), 0.4 $\mu \mathrm{l}{\min }^{-1}]$ or ${45}^{\circ }$ [(c) and (d), 0.6 $\mu \mathrm{l}{\min }^{-1}]$ with respect to the flow. From top to bottom in each panel: position, velocity angle, longitudinal velocity $v_x$, transverse velocity $v_y$, longitudinal velocity fluctuations $v_x^2/〈v_x^2〉$, and transverse velocity fluctuations $v_y^2/〈v_y^2〉$. The selected particles had similar average velocities ($\sim 250\mu \mathrm{m}{\mathrm{s}}^{-1}$). The vertical dashed lines in (b) and (d) indicate the horizontal position at which the particle switched from one streamline to another. The value $x=0$ is not physically meaningful.

Figure 4

Example trajectories of 2-$\mu \mathrm{m}$ particles in 0.1% HPAM solution that were flowed through square arrays oriented $0^{\circ }$ [(a)–(c), 0.6 $\mu \mathrm{l}{\min }^{-1}]$ or ${45}^{\circ }$ [(d)–(f), 0.6 $\mu \mathrm{l}{\min }^{-1}]$ with respect to the flow. From top to bottom in each panel: position, velocity angle, longitudinal velocity $v_x$, transverse velocity $v_y$, longitudinal velocity fluctuations $v_x^2/〈v_x^2〉$, and transverse velocity fluctuations $v_y^2/〈v_y^2〉$. The selected particles had similar average velocities ($\sim 250\mu \mathrm{m}{\mathrm{s}}^{-1}$). The vertical dashed lines in (b) and (e) indicate the horizontal position at which the particle switched from one streamline to another; the vertical dotted lines in (c) and (f) indicate the horizontal position at which the particle interacted with the stagnant recirculating regions downstream of the posts. The insets in (c) and (f) highlight segments of each particle trajectory that exhibit recirculation. The value $x=0$ is not physically meaningful.

Figure 5

Distributions of normalized longitudinal ($x$, top row) and transverse ($y$, bottom row) velocities of 2 $\mu \mathrm{m}$ particles in (a, c, e, g, i, k) 90% glycerol-water or (b, d, f, h, j, l) 0.1% HPAM solutions in post arrays of various configurations. Post configuration: (a, b, g, h) random array; (c, d, i, j) square $0^{\circ }$ array; (e, f, k, l) square ${45}^{\circ }$ array. Symbols indicate flow rates: 0.1 $\mu \mathrm{l}{\min }^{-1}$ (squares), 0.3 $\mu \mathrm{l}{\min }^{-1}$ (triangles), 0.5 $\mu \mathrm{l}{\min }^{-1}$ (pentagons).

Figure 6

Normalized longitudinal $[D_L\left(\tau \right)/D_0$ (a–d)] and transverse $[D_T\left(\tau \right)/D_0$ (e–h)] dispersion coefficients as a function of the normalized lag time $\tau =t〈v_x〉/d_c$ for 2 $\mu \mathrm{m}$ particles in 90% glycerol-water (a, c, e, g) or 0.1% HPAM (b, d, f, h) solutions. The posts were randomly oriented (a, b, e, f) or arranged in a square lattice oriented ${45}^{\circ }$ with respect to the incident flow (c, d, g, h). The insets in (a) and (e) illustrate how error bounds on the asymptotic dispersion value were obtained for one set of data (particles in glycerol-water flowed at 0.5 $\mu \mathrm{l}{\min }^{-1}$ through a random array). Symbols indicate flow rates: 0.1 $\mu \mathrm{l}{\min }^{-1}$ (squares), 0.3 $\mu \mathrm{l}{\min }^{-1}$ (triangles), 0.5 $\mu \mathrm{l}{\min }^{-1}$ (pentagons).

Figure 7

Normalized asymptotic (a) longitudinal ($D_L/D_0$) and (b) transverse ($D_T/D_0$) dispersion coefficients as a function of Péclet number $\mathrm{Pe}$. The data for the 0.4 $\mu \mathrm{m}$ PS particles in glass bead packed beds is taken from Ref. [4]. Error bars indicate the bounds on the asymptotic coefficients estimated from the time-dependent dispersion functions as illustrated in Fig. 6.

Figure 8

Normalized asymptotic (a) longitudinal ($D_L/D_0$) and (b) transverse ($D_T/D_0$) dispersion coefficients as a function of Weissenberg number Wi. In both panels, ${\mathrm{Wi}}_c$ is set to 1. The data for the 0.4 $\mu \mathrm{m}$ PS particles in glass bead packed beds is taken from Ref. [4]. Error bars indicate the bounds on the asymptotic coefficients estimated from the time-dependent dispersion functions as illustrated in Fig. 6.

Figure 9

Comparison of transverse dispersion coefficients obtained for 2-$\mu \mathrm{m}$ particles in HPAM solution in a square ${45}^{\circ }$ array at a flow rate of 0.3 $\mu \mathrm{l}{\min }^{-1}$. (a) Extracted from the asymptotic value of the time-dependent dispersion coefficient $D_T\left(t\right)$ (shown here as a function of the distance traveled, $〈v_x〉t$). (b) Extracted from the long-time slope of the mean-square displacement as a function of time.

Figure 10

Longitudinal (a–d) and transverse (e–h) velocity autocorrelation functions as a function of the normalized lag time $\tau =t〈v_x〉/d_c$ of 2-$\mu \mathrm{m}$ particles in 90% glycerol-water (a, c, e, g) or 0.1% HPAM (b, d, f, h) solutions. The posts were randomly oriented (a, b, e, f) or arranged in a square lattice oriented ${45}^{\circ }$ with respect to the incident flow (c, d, g, h). The red line indicates a VACF of zero. Symbols indicate flow rates: 0.1 $\mu \mathrm{l}{\min }^{-1}$ (squares), 0.3 $\mu \mathrm{l}{\min }^{-1}$ (triangles), 0.5 $\mu \mathrm{l}{\min }^{-1}$ (pentagons).

Figure 11

Longitudinal (a–d) and transverse (e–h) velocity autocorrelation functions (VACFs) as a function of distance traveled, $〈v_x〉t$, for 2-$\mu \mathrm{m}$ particles in 90% glycerol-water (a, c, e, g) or 0.1% HPAM (b, d, f, h) solutions. The posts were randomly oriented (a, b, e, f) or arranged in a square lattice oriented ${45}^{\circ }$ with respect to the incident flow (c, d, g, h). The red line indicates a VACF of zero. Symbols indicate flow rates: 0.1 $\mu \mathrm{l}{\min }^{-1}$ (squares), 0.3 $\mu \mathrm{l}{\min }^{-1}$ (triangles), 0.5 $\mu \mathrm{l}{\min }^{-1}$ (pentagons).

Figure 12

Distributions of normalized longitudinal ($x$, top row) and transverse ($y$, bottom row) velocities of 2-$\mu \mathrm{m}$ particles in (a, c) 90% glycerol-water or (b, d) 0.1% HPAM solutions in a hexagonal post array. Symbols indicate flow rates: 0.1 $\mu \mathrm{l}{\min }^{-1}$ (squares), 0.3 $\mu \mathrm{l}{\min }^{-1}$ (triangles), 0.5 $\mu \mathrm{l}{\min }^{-1}$ (pentagons).

Figure 13

Normalized longitudinal $[D_L\left(\tau \right)/D_0$ (a, b)] and transverse $[D_T\left(\tau \right)/D_0$ (c, d)] dispersion coefficients as a function of the normalized lag time $\tau =t〈v_x〉/d_c$ for 2-$\mu \mathrm{m}$ particles in 90% glycerol-water (a, c) or 0.1% HPAM (b, d) solutions in a square $0^{\circ }$ array. Symbols indicate flow rates: 0.1 $\mu \mathrm{l}{\min }^{-1}$ (squares), 0.3 $\mu \mathrm{l}{\min }^{-1}$ (triangles), 0.5 $\mu \mathrm{l}{\min }^{-1}$ (pentagons).

Figure 14

Normalized longitudinal $[D_L\left(\tau \right)/D_0$ (a, b)] and transverse $[D_T\left(\tau \right)/D_0$ (c, d)] dispersion coefficients as a function of the normalized lag time $\tau =t〈v_x〉/d_c$ for 2-$\mu \mathrm{m}$ particles in 90% glycerol-water (a, c) or 0.1% HPAM (b, d) solutions in a hexagonal array. Symbols indicate flow rates: 0.1 $\mu \mathrm{l}{\min }^{-1}$ (squares), 0.3 $\mu \mathrm{l}{\min }^{-1}$ (triangles), 0.5 $\mu \mathrm{l}{\min }^{-1}$ (pentagons).