Modeling the Transition between Localized and Extended Deposition in Flow Networks through Packings of Glass Beads

We use a theoretical model to explore how fluid dynamics, in particular, the pressure gradient and wall shear stress in a channel, affect the deposition of particles flowing in a microfluidic network. Experiments on transport of colloidal particles in pressure-driven systems of packed beads have shown that at lower pressure drop, particles deposit locally at the inlet, while at higher pressure drop, they deposit uniformly along the direction of flow. We develop a mathematical model and use agent-based simulations to capture these essential qualitative features observed in experiments. We explore the deposition profile over a two-dimensional phase diagram defined in terms of the pressure and shear stress threshold, and show that two distinct phases exist. We explain this apparent phase transition by drawing an analogy to simple one-dimensional mass-aggregation models in which the phase transition is calculated analytically.

Figure 1

We use a network approach to model the bead packing here shown in the absence of particles. (a) We skeletonize the image of the packing, and then generate a network. The edges of the network represent the channels through which fluid may flow in the packing and the nodes represent the junctions where these channels meet. (b) We obtain the flow rates in the channels by applying Kirchhoff’s laws [30]. (c) Zoomed-out view showing the network as a whole. The color in (b) and (c) shows the magnitude of the channel flow rates in SI units (${\mathrm{m}}^3/\mathrm{s}$). (a) and (b) have the same scale bar. The gray background shows the experimental micrograph of the beads.

Figure 2

The cumulative probability distribution function $F(x)$ of positions of particles deposited along the flow direction for the localized and the extended case obtained by simulation and experiment show a similar qualitative behavior. Darker colors indicate later times. The position along the direction of the flow $x$ is normalized by the total length of the medium. The labels indicate the normalized pressure gradient.

Figure 3

The final frames of the simulation over a range of values of shear stress threshold $\widehat{\tau }$ and applied pressure $\mathrm{\Delta }\widehat{P}$ show a clear separation between the localized and extended deposition regimes similar to experimental observations in [15]. The dashed line serves to guide the eye.

Figure 4

In the parameter space of normalized pressure $\mathrm{\Delta }\widehat{P}$ and shear stress threshold $\widehat{\tau }$, a boundary separates the two regimes of localized and extended deposition, reminiscent of the phases in [21]. The solid circles show the critical values of pressure $P_c$ at which the transition to the localized phase occurs in simulations. The solid line corresponds to the best fit $\mathrm{\Delta }\widehat{P}=1.04{\widehat{\tau }}^{0.71}-0.04$. The hat notation used here denotes normalization by the maximum value.