October 16, 2021

Bazar Lead

Just Law & Legal

Multiscale dynamics of colloidal deposition and erosion in porous media

INTRODUCTION

Colloidal transport underlies a wide array of processes that affect our everyday lives. It can be beneficial, helping to degrade contaminants in groundwater aquifers (13), mobilize oil from underground reservoirs (4), and carry therapeutics through gels and tissues in the body (5, 6). It can also be harmful, enabling the distribution of microplastics, contaminants, and pathogens throughout soils, sediments, and groundwater aquifers (79). Being able to predict and control colloidal transport is therefore critically important. However, while transport is well studied in bulk liquids, all of these examples involve the transport of colloidal particles in a disordered, three-dimensional (3D) porous medium. In this case, not only do confinement and tortuosity imposed by the medium alter particle transport but also the particles, in turn, can alter the medium by depositing onto its solid matrix as they are transported (1014), yielding coupled dynamics that pose a challenge to current understanding.

Even basic studies of particle transport and deposition often yield conflicting results. For example, in many cases, colloids have been found to enhance the transport of chemicals through a porous medium (1517), while other studies report the opposite, claiming that colloids suppress transport by reducing the permeability of the medium (1821). This discrepancy is thought to be rooted in the different flow conditions and colloidal chemistries explored, which can strongly influence the interactions between flowing fluid, particles, and the medium. However, systematic investigation of these interactions remains challenging; typical 3D media are opaque, precluding direct characterization of flow and particle transport within the pore space. Thus, experiments often investigate flow around isolated obstacles (22, 23) or through 2D arrays of pores (2428). While these studies provide tremendous insight into particle transport and deposition, they do not fully capture the connectivity and complexity of a 3D pore space. Recent work has extended these investigations to transparent 3D media, but only investigated single-particle behavior (29), did not investigate different colloidal chemistries (30, 31), or only focused on particle deposition near the inlet of the medium (32). Furthermore, magnetic resonance imaging and X-ray microtomography yield additional insights into the spatial distribution of particles in the pore space—for example, indicating that the distribution of deposited particles is sensitive to the particle charge (33, 34), size (35), and imposed flow conditions (36). However, such approaches typically do not capture pore-scale dynamics of colloidal transport and deposition due to limitations in spatial or temporal resolution.

In lieu of direct microscopic observations, studies often use a combination of continuum modeling and filtration theory (37) to describe particle transport and deposition. In this hybrid approach, a single parameter—the collector efficiency—quantifies all particle interactions with the medium (e.g., due to electrostatics). While in many cases this approach provides a useful way to model colloidal transport and deposition, it often does not reliably predict experimentally observed deposition profiles without the use of additional fitting parameters, reflecting the combined influence of many different factors on particle transport and deposition (3739). To shed further light on this problem, various computational schemes are being developed, generating intriguing predictions of particle deposition and erosion (4046). However, in the absence of experimental studies connecting the dynamics of colloidal deposition and erosion at the pore scale to flow and transport at the scale of the overall porous medium, accurate prediction or control of particle transport and deposition remains elusive.

Here, we address this gap in knowledge by directly visualizing the dynamics of colloidal particle transport in transparent, 3D porous media. Our experiments probe length scales ranging from single pores to thousands of pores and time scales ranging from the injection duration of one pore volume (PV) to thousands of PVs, enabling us to connect pore-scale dynamics to macroscopic transport. As particles are transported through a pore space, they deposit on the surrounding solid matrix of the medium. At high injection pressures, hydrodynamic stresses also continually erode the particles from the solid matrix, causing deposited particles to be distributed through the entire porous medium. Conversely, at low injection pressures, hydrodynamic stresses are weaker and the relative influence of erosion is suppressed, causing deposited particles to be localized only near the inlet of the medium. Unexpectedly, these macroscopic distribution behaviors are tuned by the elevated imposed pressure in similar ways for particles of different charges, although the pore-scale distribution of deposited particles is sensitive to particle charge. Thus, our results reveal the rich pore-scale dynamics of colloidal transport in a porous medium, deepening understanding of how the multiscale interactions between flowing fluid, particles, and the solid matrix impact colloidal transport and deposition.

RESULTS

Experimental platform for visualization of colloidal dynamics

We prepare rigid 3D porous media by lightly sintering dense, disordered packings of hydrophilic glass beads, with diameters d between 38 and 45 μm, in thin-walled square quartz capillaries of cross-sectional area A = 1 mm2. The experimental geometry is schematized in Fig. 1. The packings have lengths L ranging from 5 mm to 2 cm and porosity ϕ0 ≈ 0.41 as measured previously using confocal microscopy (47). Scattering of light from the surfaces of the beads typically precludes direct observation of flow and transport within the medium. Following our previous work (47, 48), we overcome this limitation by formulating a fluid mixture with a refractive index matching that of the glass beads—enabling full characterization of pore space structure and subsequent visualization of colloidal deposition using confocal microscopy (Materials and Methods).

Fig. 1 Schematic of experimental setup.

Our experiments are performed using 3D porous media composed of glass beads of diameter d between 38 and 45 μm, densely packed within thin-walled square quartz capillaries with cross-sectional length w = 1 mm. A dilute colloidal suspension containing particles of diameter dp = 1 μm is injected through the pore space at a fixed imposed pressure drop. The fluorescent particles are visualized at scales ranging from that of individual pores to the overall porous medium using confocal microscopy.

Before each experiment, we map the pore space by saturating the medium with the fluorescently dyed fluid and acquiring a cross-sectional image of the full pore space at a fixed depth in the medium. We identify the glass beads by their contrast with the dyed fluid, with cross sections shown by the black circles in Fig. 2, and the fluid-saturated pore space by the bright region between the beads. Thus, this protocol enables us to characterize the pore space structure at subpore resolution before colloidal injection.

Fig. 2 Macroscopic deposition profile of positively charged colloidal particles is tuned by injection pressure.

Confocal micrographs show extended (A) and localized (C) deposition profiles after t = 3.1 and 1.9 hours for amine-functionalized polystyrene particles injected at 260 or 80 kPa, respectively. Black circles show cross sections through the beads making up the porous media, white space shows pore space, and red shows deposited colloidal particles. Corresponding line traces in (B) and (D) show the amount of deposition at each position along the flow direction, quantified by Ad/Apore,0, where Ad is the laterally averaged area occupied by deposited particles and Apore,0 is the pore space area before colloidal injection. Traces show Ad with a running average over every 100 μm along the length of the medium to minimize noise; different traces are shown for images obtained every 2 min in (B) and 3 min in (D), which we represent by the total number of suspension PVs injected. The traces exhibit some spatial fluctuations that likely reflect the influence of packing heterogeneities in the disordered media. In some cases, Ad/Apore,0 slightly exceeds one due to the nonzero thickness of the optical slice. For clarity, we only show the deposition profile for the first 4000 μm of the media, shorter than their full lengths. Flow direction is from left to right, and the scale bars represent 200 μm.

We then inject the undyed fluid, laden with fluorescent colloidal particles of diameter dp = 1 μm at a concentration of ∼109 particles/ml = 0.05 volume %, at a fixed pressure drop ΔP across the overall medium. The magnitude of the characteristic interstitial flow velocity ranges from ∼0.5 to 5 cm/min, or ∼7 to 70 m/day, comparable to that of forced-gradient groundwater flow in a sand aquifer. We denote the time t at which particles begin to enter the medium as t = 0 and represent subsequent times by the total number of suspension PVs injected,

0tQ(t)dt/[ALϕ(t)]

, where Q(t) and ϕ(t) are the time-dependent volumetric flow rate and porosity, respectively. To characterize particle deposition in the pore space, we continually acquire successive images spanning the entire cross section of the medium; in parallel, we measure the effluent mass, providing a direct measure of the flow rate. The Reynolds number characterizing our experiments is ∼10−4 to 10−3, indicating that the flow is laminar. The particle Péclet number Pe, quantifying the importance of advection relative to diffusion in determining the particle motion, is >104; hence, particle transport is primarily due to fluid advection. Because our experiments explore the injection of up to thousands of PVs of particle suspension, they range from “clean bed” conditions characterized by minimal prior deposition to nearly clogged conditions. Furthermore, our experiments test two different colloidal particle chemistries characterized by “favorable” attractive electrostatic interactions or “unfavorable” repulsive interactions with the solid matrix of the medium. Thus, our work complements previous characterization of these distinct conditions and modes of interaction using transport measurements, magnetic resonance imaging, X-ray microtomography, optical imaging, and static light scattering (13, 14, 2936).

Dynamics of positively charged colloidal particles

We first investigate the injection of a dilute suspension of positively charged, amine-functionalized polystyrene particles into the porous medium at a large imposed pressure drop, ΔP = 260 kPa. In this case, we anticipate particle deposition to be localized near the inlet of the medium due to strong electrostatic attraction between the positively charged particles and negatively charged beads (17, 48), as suggested by the Derjaguin-Landau-Verwey-Overbeek (DLVO) calculations detailed in the Supplementary Materials and summarized in fig. S1. Unexpectedly, we instead observe the formation of an extended deposition profile of particles that spans the entire length of the porous medium, as shown in Fig. 2A and movie S1. This profile persists over the course of injection: the amount of deposition at each position along the flow direction—quantified by the fraction of the initial pore space area Apore,0 occupied by deposited particles having total area Ad—steadily grows in time, as shown in Fig. 2B and fig. S2.

Inspection of particle deposition at the pore scale provides a clue to the mechanism underlying extended deposition. Because Pe ≫1 and particle-bead interactions are attractive, we expect fluid flow to advect particles toward the upstream faces of the beads, forcing them to initially attach. Consistent with this expectation, some particles initially deposit on the upstream surfaces of the pristine beads, as shown in the top panel of Fig. 3A. As particle injection continues, these deposits continue to grow, as shown in movie S2, suggesting that the hydrodynamic stresses due to fluid flow are sufficient to overcome any possible electrostatic repulsion between like-charged particles, detailed further in the Supplementary Materials, although nanoscale heterogeneity in the colloidal interactions may also play a role (49). This observation is consistent with previous findings in 2D media (2428, 32). The deposits do not all grow monotonically, however. In many cases, single- and multiparticle deposits are eroded—abruptly removed by fluid flow, as predicted by recent simulations (42); an example is shown in the bottom panel of Fig. 3A and in movie S3. The eroded particles are then redeposited in other pores downstream. This process of deposition, erosion, and redeposition progresses continuously over the course of injection, enabling particles to be deposited throughout the extent of the medium. Moreover, while substantial, erosion does not completely balance deposition: the instantaneous rate of pore-scale deposition is slightly larger than the rate of erosion, as shown by the solid points in Fig. 3B. This difference results in net deposition throughout the entire medium that progressively increases over time, as quantified in the top panel of Fig. 3C. Therefore, we hypothesize that pore-scale erosion generates the extended macroscopic deposition profile shown in Fig. 2A.

Fig. 3 Deposition and erosion of positively charged colloidal particles at the pore scale.

(A) Arrows in the upper and lower sequences of micrographs show monotonic deposition and cyclic deposition–erosion of particles upstream of a bead, respectively, at 260 kPa. (i) to (iii) show micrographs after the injection of 675, 903, and 1074 total PVs of the colloidal suspension. Black circles show cross sections through the beads making up the porous media, white space shows pore space, and red shows deposited colloidal particles. Scale bar, 20 μm. (B) Rates of particle deposition (squares) and erosion (diamonds), calculated by subtracting successive images, are shown as a function of total PVs injected for 260 and 80 kPa, indicated by the filled and open symbols, respectively. (C) Top, middle, and bottom panels show the fraction of the pore space area occupied by deposited particles Ad/Apore,0, the volumetric flow rate Q normalized by its initial value (1.5 ml/hour for 260 kPa and 0.15 ml/hour for 80 kPa), and the interstitial flow speed, respectively, as they vary with the total PVs injected. Experiments at 260 and 80 kPa are indicated by filled and open symbols, respectively. All data are for amine-functionalized polystyrene particles. Error bars in (C) reflect uncertainty arising from binarizing the micrographs or from variation in flow rate measurements, as detailed in the Supplementary Materials. Error bars that are not observable in (C) are smaller than the symbol size.

To quantify this hypothesis, we analyze the stresses on colloidal particles at the pore scale (Materials and Methods). Because the imposed pressure drop is so large, we expect that hydrodynamic stresses are large enough to drive erosion. As a first step toward quantifying this expectation, we use Darcy’s law to estimate the characteristic viscous pressure drop across an isolated deposit of diameter D, ΔPD = μQD/(Ak), at t = 0; here, μ is the fluid dynamic shear viscosity and k is the permeability of the porous medium, measured previously for a pristine medium (50). For deposits of size D ranging from one to four particle diameters, as seen in movie S3, ΔPD ranges from ∼10 to 60 Pa. We conjecture that this characteristic pressure drop is larger than the yield stress required to fluidize dense colloidal aggregates with slight interparticle attraction, σy, which we estimate to be between ∼1 and 10 Pa based on previous shear rheology measurements (51). Hence, we expect that fluid flow drives erosion, as observed in the experiments. This balance of pressures neglects spatiotemporal variations in flow as well as the full details of colloid-colloid and colloid-bead interactions; it therefore provides an order-of-magnitude estimate of the onset of erosion. To more rigorously test our expectation, we perform mechanistic particle trajectory simulations explicitly incorporating the particle forces and torques that arise from the interplay between hydrodynamics and colloidal interactions (49). The simulations confirm our expectation: they reveal that multiparticle clusters are both deposited and eroded from the surfaces of the beads under these conditions, as shown in fig. S3 and detailed in the Supplementary Materials. As particles continue to deposit, the permeability of the medium and the volumetric flow rate decrease, as shown in the middle panel of Fig. 3C. The interstitial flow speed v(t) ≡ Q(t)/[ϕ(t)A], calculated by directly estimating the time-dependent porosity ϕ(t) ≡ ϕ0[1 – Ad(t)/Apore,0] from our micrographs, concomitantly decreases as shown in the bottom panel of Fig. 3C; this quantity provides a measure of the magnitude of the characteristic interstitial flow velocity. However, even at these lower flow rates and flow speeds, ΔPD ≳ σy, enabling erosion to continue to progress. Thus, particles continue to deposit, erode, and redeposit in the entire porous medium over the course of the experiment, driving extended deposition throughout.

Our analysis also suggests that the relative contribution of erosion can be tuned by the imposed pressure drop. To test this prediction, we repeat our experiments, but at a lower ΔP = 80 kPa. Under these conditions, we expect hydrodynamic stresses and hence the relative magnitude of erosion to be less dominant, likely resulting in different deposition behavior. Consistent with this expectation, we observe starkly different deposition behavior: instead of particles depositing throughout the entire porous medium, as in the high pressure case, particles only deposit locally near the inlet, as shown in Fig. 2C and movie S4. We quantify this observation in Fig. 2D, which shows that the fraction of the pore space that is occupied by particles no longer spans the entire length of the medium but instead appears to decay exponentially, as shown in fig. S2, approaching zero at a distance Ld ∼ 7% of its length. The deposition profiles at different times collapse when rescaled by the number of injected particles, as shown in fig. S2, again indicating that the deposition process is consistent over time (35, 52, 53).

As expected, these differences in macroscopic deposition reflect strong differences in pore-scale deposition and erosion, in addition to previously documented differences in particle transport at different pressures (36). For ΔP = 80 kPa, the instantaneous rates of particle deposition and erosion—as well as the difference between them—are one order of magnitude larger than at ΔP = 260 kPa, as shown by the open points in Fig. 3B. This difference results in a larger fraction of particles depositing in the medium over a shorter amount of time, shown in the top panel of Fig. 3C. As a result, the volumetric flow rate decreases, indicating that localized deposition at the inlet “chokes off” the flow, as shown in the middle panel of Fig. 3C. The corresponding interstitial flow speed is nearly an order of magnitude smaller than that of the high pressure case, as shown in the bottom panel of Fig. 3C. Consequently, the characteristic viscous pressure drop across a deposit, ΔPD ∼ 1 to 6 Pa, is insufficient to strongly overcome the deposit yield stress, estimated as σy ∼ 1 to 10 Pa, and erosion is suppressed. Mechanistic particle trajectory simulations of particle dynamics confirm this expectation, indicating that erosion is suppressed under these conditions, as shown in fig. S4. Therefore, particle deposition is localized to near the inlet of the medium.

Dynamics of negatively charged colloidal particles

Our experiments thus far explored the case of positively charged particles. How does deposition change when the particle charge is altered? To answer this question, we next investigate the injection of a dilute suspension of negatively charged carboxyl-functionalized polystyrene particles. Intriguingly, at a large ΔP = 170 kPa, we again observe the formation of an extended deposition profile spanning the length of the porous medium, as shown in Fig. 4A, fig. S2, and movie S5. However, microscopy reveals that pore-scale deposition is markedly altered by the change in particle charge. In this case, because the particles have the same charge as the beads composing the medium, the electrostatic interactions between particles and beads are repulsive. Hence, carboxyl-functionalized particles do not deposit on the upstream surfaces of the pristine beads; instead, they are strained in the tight pore throats between beads, where they continue to grow and form loose deposits, exemplified by Fig. 5A. This observation corroborates similar findings in experiments using larger particles (33, 34).

Fig. 4 Macroscopic deposition profile of negatively charged colloidal particles is tuned by injection pressure.

Confocal micrographs show extended (A) and localized (C) deposition profiles after t = 5.6 and 16.7 hours for carboxyl-functionalized polystyrene particles injected at 170 or 80 kPa, respectively. Black circles show cross sections through the beads making up the porous media, white space shows pore space, and red shows deposited colloidal particles. Corresponding line traces in (B) and (D) show the amount of deposition at each position along the flow direction, quantified by Ad/Apore,0, where Ad is the laterally averaged area occupied by deposited particles and Apore,0 is the pore space area before colloidal injection. Traces show Ad with a running average over every 100 μm along the length of the medium to minimize noise; different traces are shown for images obtained every 3 min in (B) and 2 min in (D), which we represent by the total number of suspension PVs injected. The traces exhibit some spatial fluctuations that likely reflect the influence of packing heterogeneities in the disordered media. In some cases, Ad/Apore,0 slightly exceeds one due to the nonzero thickness of the optical slice. For clarity, we only show the deposition profile for the first 6000 μm of the media, shorter than their full lengths. Flow direction is from left to right, and the scale bars represent 200 μm.

Fig. 5 Deposition and erosion of negatively charged colloidal particles at the pore scale.

(A) Arrows in the upper and lower sequences of micrographs show monotonic deposition and cyclic deposition–erosion of particles upstream of a bead, respectively, at 170 kPa. (i) to (iii) show micrographs after the injection of 610, 1376, and 2603 total PVs of colloidal suspension. Black circles show cross sections through the beads making up the porous media, white space shows pore space, and red shows deposited colloidal particles. Scale bar, 20 μm. (B) Rates of particle deposition (squares) and erosion (diamonds), calculated by subtracting successive images, are shown as a function of total PVs injected for 170 and 80 kPa, indicated by the filled and open symbols, respectively. (C) Top, middle, and bottom panels show the fraction of the pore space area occupied by deposited particles Ad/Apore,0, the volumetric flow rate Q normalized by its initial value (1.2 ml/hour for 170 kPa and 0.16 ml/hour for 80 kPa), and the interstitial flow speed, respectively, as they vary with the total PVs injected. Experiments at 170 and 80 kPa are indicated by filled and open symbols, respectively. All data are for carboxyl-functionalized polystyrene particles. Error bars in (C) reflect uncertainty arising from binarizing the micrographs or from variation in flow rate measurements, as detailed in the Supplementary Materials. Error bars that are not observable in (C) are smaller than the symbol size.

Despite this difference in where particles deposit, we observe similar deposition dynamics to the amine-functionalized case: some deposits grow monotonically, while in many other cases, single- and multiparticle deposits are eroded and redeposited downstream, as shown in the bottom panel of Fig. 5A and in movies S6 and S7. Erosion does not completely balance deposition: the instantaneous rate of pore-scale deposition is slightly larger than the rate of erosion, indicated by the solid points in Fig. 5B. This difference again results in net deposition in the entire medium that progressively increases over time, as shown in the top panel of Fig. 5C. The volumetric flow rate and interstitial flow speed, shown in the middle and bottom panels of Fig. 5C, respectively, decrease concomitantly. However, hydrodynamic stresses are still sufficient to strongly drive erosion: The characteristic viscous pressure drop across a deposit, ΔPD ∼ 10 to 60 Pa, is larger than the deposit yield stress, again estimated as σy ∼ 1 to 10 Pa. Particle trajectory simulations of particle dynamics again confirm this expectation, revealing that clusters of particles are both deposited and eroded from the surfaces of the beads under these conditions, as shown in fig. S5. Therefore, a similar process of continual flow-driven deposition, erosion, and redeposition again results in extended deposition of particles throughout the medium.

Consistent with this picture, at a smaller ΔP = 80 kPa, we again find that deposition is localized—in this case, to the first ∼20% of the length of the medium, as shown in Fig. 4C and movie S8. The deposition profiles at different times also collapse when rescaled by the number of injected particles, as shown in fig. S2, again indicating that the deposition process is consistent over time (35, 52, 53). At this lower pressure, the instantaneous rates of particle deposition and erosion—as well as the difference between them—are over one order of magnitude larger than at ΔP = 170 kPa, shown by the open points in Fig. 5B. This difference again results in a larger fraction of particles depositing in the medium over a shorter amount of time for the lower pressure drop condition, shown in the top panel of Fig. 5C. The volumetric flow rate decreases concurrently, resulting in slow interstitial flow that cannot drive erosion as strongly, as shown by the middle and bottom panels of Fig. 5C, respectively: the characteristic viscous pressure drop ΔPD ∼ 1 to 6 Pa is much smaller than in the 170-kPa case and is insufficient to overcome the deposit yield stress, estimated as σy ∼ 1 to 10 Pa. Mechanistic particle trajectory simulations of particle dynamics additionally confirm this expectation, indicating that erosion is suppressed under these conditions, as shown in fig. S6. Together, our experiments reveal that macroscopic deposition of colloidal particles is tuned by imposed pressure in similar ways, independent of particle charge.

Changes in the overall permeability of the medium

Having established how hydrodynamics affect macroscopic deposition of particles, we now ask how deposition, in turn, affects fluid flow. Our observations of localized deposition—primarily in a region spanning a length Ld < L along the porous medium—suggest that the overall permeability of the medium can be calculated by considering flow in the deposited and pristine, particle-free regions separately. In particular, for a given time t > 0 after the initiation of particle injection, we use Darcy’s law to describe the pressure drop across the overall medium as

ΔP=μQ(t)Ld(t)Akd(t)Deposited region+μQ(t)[LLd(t)]Ak0Pristine region

(1)where kd and k0 are the permeabilities of the particle-deposited and pristine particle-free regions, respectively; in the case of extended deposition, Ld = L in this equation. This pressure drop can also be related to the initial volumetric flow rate Q0 before the initiation of particle injection: ΔP = μQ0L/(Ak0). Because ΔP is fixed in our experiments, combining this equation with Eq. 1 yields a prediction for the overall permeability of the medium, k:

k(t)k(t)k0=Q(t)Q(t)Q0=11+Ld(t)[1/kd(t)1]

(2)where

Ld(t)Ld(t)/L

and

kd(t)kd(t)/k0

are the normalized deposition length and permeability of the deposit-filled region, respectively. As shown in Fig. 6A, this prediction quantifies the intuition that the overall permeability of a medium with minimal colloidal deposition (

Ld1

,

kd1

) is only minimally altered, while a medium with substantial deposition (

Ld1

,

kd1

) becomes clogged and impermeable to further flow. Thus, processes that seek to control the permeability of the medium during colloidal injection should focus on controlling the two parameters

Ld

and

kd

.

Fig. 6 Pore-scale and macroscopic features of colloidal deposition control overall flow behavior.

(A) Color map showing the normalized overall permeability of the medium,

k˜k/k0

, calculated using Eq. 2 of the main text for different values of the normalized permeability and length of the particle-deposited region,

kdkd/k0

and

LdLd/L

, respectively. Yellow represents

k˜=1

, while pink represents

k˜=0

. (B) Measurements of the overall permeability of the medium, represented by the vertical axis, are in reasonable agreement with the prediction of Eq. 2, represented by the horizontal axis. Diagonal line shows a one-to-one relation. The deviation of the measurements from this relation as deposition progresses and

k˜

decreases likely arises from the simplifying assumption that the deposited and pristine regions can be treated separately, with a homogeneous permeability in each; in reality, the amount of deposition varies spatially as well. Different points in both panels represent different experiments carried out for amine- or carboxyl-functionalized particles (orange or blue), at different imposed pressures, and using different particle volume fractions, as indicated by the legend underneath. Multiple points show different time points of each experiment. Error bars reflect uncertainty arising from binarizing the micrographs, defining a threshold for Ld, or from variation in flow rate measurements, as detailed in the Supplementary Materials.

Our experiments enable a direct test of this prediction: Our transport measurements shown in Figs. 3C and 5C directly yield

k(t)=Q(t)

, while our macroscopic and pore-scale visualization yield

Ld(t)

and

kd(t)

, respectively, enabling us to independently compute

k(t)

via Eq. 2. To determine

Ld(t)

, we use the micrographs shown in Figs. 2 and 4 to determine the position along the medium at which Ad/Apore,0 falls below a threshold value, as detailed in the Supplementary Materials. To determine

kd(t)

, we use the confocal micrographs to determine the pore space area Apore(t), the pore space perimeter Ppore(t), and the porosity ϕ(t), all of which are time dependent. We then estimate

kd(t)

by modeling the pore space as a parallel bundle of capillary tubes (Materials and Methods):

kd(t)=[ϕ(t)/ϕ0][dh(t)/dh,0]2

, where the hydraulic diameter dh(t) ∼ Apore(t)/Ppore(t). Last, we compute

k(t)

via Eq. 2. Despite the simplifying assumptions made, this model yields reasonable agreement—within a factor of ∼2—between the measured and computed

k

for all of the previously described experiments, as well as additional experiments performed at intermediate pressure drops and higher inlet particle concentrations, as shown in Fig. 6B. By connecting colloidal deposition at the pore scale to changes in macroscopic transport, this model helps to confirm the consistency of our data.

DISCUSSION

Our work directly connects the dynamic processes of colloidal deposition and erosion at the pore scale, deposition profiles at the macroscopic scale, and bulk fluid transport. We find that particles can deposit throughout the entire medium at large pressures via continual erosive bursts. Erosion was previously theorized to occur using pore-scale simulations (42), and indirect signatures of erosion have been detected using bulk transport measurements (54); our results provide a connection between these pore-scale events and macroscopic deposition behavior. Moreover, while the pore-scale characteristics of deposition depend on the interactions between particles and the solid matrix, we find that the macroscopic characteristics of deposition are tuned by imposed pressure in unexpectedly similar ways for particles with different surface properties—highlighting the importance of hydrodynamic interactions in determining colloidal transport and deposition. Specifically, our results suggest that erosion plays a dominant role in distributing particles throughout the pore space when the viscous pressure drop ΔPD across deposited particles exceeds a threshold value, which we conjecture is given by the deposit yield stress σy. The transition between localized and extended deposition is therefore likely to be abrupt, tuned by the imposed pressure drop, unlike the flow rate–controlled case that has been found to yield more gradual deposition behavior (32). Elucidating this transition will be a valuable direction for future work. Furthermore, it will be interesting to explore whether collective clogging effects (35) manifest for both positively and negatively charged particles, given the similarity of our results for both cases.

To facilitate visualization using confocal microscopy, our experiments use an aqueous refractive index–matched mixture as the fluid phase. However, though this fluid mixture has different hydrodynamic, dielectric, and refractive properties than pure water—resulting in differences in colloidal interactions—we expect that our results are also applicable to pure water–based colloids. This expectation is supported by the DLVO calculations shown in fig. S1, which indicate that, despite the quantitative differences in colloidal interactions between the two different fluid phases, the nature of colloid-colloid and colloid-bead interactions is qualitatively similar. This expectation is further confirmed by mechanistic particle trajectory simulations explicitly incorporating the different particle forces and torques in a pure water–based system; we find similar behaviors as those obtained using the refractive index–matched fluid mixture, but at different values of the imposed fluid pressure, as detailed in the Supplementary Materials and summarized in figs. S3 to S6. Hence, our results are likely generalizable to a broader range of aqueous-based colloids.

Because diverse applications require control over colloidal deposition in porous media, we anticipate that our findings will be broadly relevant. In some cases, localized deposition—which we find arises at low injection pressures—is essential. One important example is the deposition of iron nanoparticles within groundwater aquifers for in situ immobilization of heavy metal contaminants (55); in this case, nanoparticle deposition is required, but only at specific locations of the medium. Another example is filtration or containment of pathogens and waste materials, which requires effective capture near the inlet of the medium (7). By shedding light on the conditions under which the relative influence of erosion is promoted or suppressed, our work may assist in the determination of optimal injection pressures and medium lengths to promote localized deposition in these cases. In other cases, particles must be able to traverse long distances as they are advected through the pore space. Key examples are enhanced oil recovery (4), in which injected particles must be able to reach residual oil in a reservoir; nanoparticle-assisted groundwater remediation (13), in which injected particles must be able to reach trapped contaminants in an aquifer; and drug delivery through tissues in the body (5, 6). Our results suggest that, even if particles deposit in the pore space, erosion at sufficiently large injection pressures can be leveraged to distribute the particles through the entire medium. Thus, our work may provide guidelines for more effective colloidal transport in environmental, energy, and biomedical settings.

MATERIALS AND METHODS

Experimental setup

We prepare rigid 3D porous media by lightly sintering dense, disordered packings of hydrophilic glass beads, with diameters d between 38 and 45 μm, in thin-walled square quartz capillaries of cross-sectional area A = 1 mm2 for under a minute at 900C. The packings have lengths L ranging from 5 mm to 2 cm and porosity ϕ0 ≈ 0.41 as measured previously using confocal microscopy (47). Before each experiment, the pore space is saturated with the particle-free fluid before the homogenized colloidal suspension is injected into the medium. We impose a constant pressure drop across the medium using a Teledyne ISCO LC-5000 syringe pump, and use an OMEGA differential pressure sensor to independently verify the pressure drop across the medium. We continually image the pore space, as detailed below; in parallel, we measure the volumetric flow rate by measuring the effluent mass over time using a Mettler Toledo balance with an accuracy of 1 mg and converting to a volume using a density ρ ≈ 1.23 g/cm3, calculated as a weighted average of the densities of the different fluid components. The experiments last until a filter cake of particles begins to form at the inlet of the medium.

Fluid and colloid properties

We formulate an index-matched aqueous fluid composed of 82 weight % (wt %) glycerol (Sigma-Aldrich), 12 wt % dimethyl sulfoxide (Sigma-Aldrich), and 6 wt % ultrapure water. This fluid has a density ρ ≈ 1.23 g/cm3, calculated as a weighted average of the densities of the different fluid components, and a dynamic shear viscosity μ ≈ 60 mPa-s, as previously determined using a shear rheometer (47). The colloids used are fluorescent amine-functionalized (Sigma-Aldrich) and carboxyl-functionalized polystyrene particles (FluoSpheres, Thermo Fisher Scientific) with a mean diameter of dp = 1 μm; DLVO analysis of the colloidal interactions is given in the Supplementary Materials. Stock suspensions are sonicated to uniformly disperse the particles, diluted to 0.05 volume % (for the experiments described in Figs. 2 to 5) or 0.1 volume % (for additional experiments shown in Fig. 6) in the fluid mixture, and further homogenized by vortexing and additional sonication. The zeta potentials of the amine- and carboxyl-functionalized particles are +25 ± 8 and −34 ± 5 mV, respectively, as measured in the fluid mixture itself using a Malvern Zetasizer Nano ZS. We verify that the magnitudes of these zeta potentials are sufficiently large to maintain particle stability in the suspension over the entire course of injection, as detailed in the Supplementary Materials. We also do not observe any noticeable deformation or swelling of the polystyrene particles in the presence of glycerol and dimethyl sulfoxide at the ratios used in our tests; this observation is consistent with previous reports indicating that polystyrene does not swell or soften when exposed to these solvents (5658). The Reynolds number characterizing our experiments is ρvdb/μ ≤ 3 × 10−4, where db is the diameter of a pore body, indicating that the flow is laminar. The particle Péclet number Pe = 6πμv(dp/2)2/(kBT) ∼ 104 to 105, where kB is Boltzmann’s constant and T is temperature, and the ratio of viscous forces to gravitational forces on the particles is given by 9μv/[2Δρg(dp/2)2] ∼ 105 to 106, where Δρ ≈ 0.17 g/cm3 is the density difference between particles and fluid and g is gravitational acceleration, indicating that particle transport is primarily due to advection by the fluid.

Confocal microscopy

Before each experiment, we saturate the pore space with the particle-free fluid, dyed with rhodamine 6G (Sigma-Aldrich). We use a laser scanning Nikon A1R+ confocal microscope to acquire high-resolution optical slices at a fixed depth within the pore space; these span the entire width and length of the medium and thus provide a full cross-sectional image. The acquisition time for the entire cross section is ∼2 min. We identify the glass beads by their contrast with the dyed fluid, with cross sections shown by the black circles in Fig. 2, and the fluid-saturated pore space by the bright region between the beads. Hence, this protocol enables us to characterize the pore space structure before colloidal injection. We then inject the colloidal suspension, composed of fluorescent particles dispersed in the same but undyed fluid, and acquire successive cross-sectional images of particles in the pore space. We adjust laser power and gain amplitude to avoid pixel saturation for each experiment. The time required to raster across the width of each optical slice is longer than the time required for individual particles to be advected across; as a result, the images primarily reflect deposited particles, not particles dispersed in the suspension. The focal depth in each experiment is 7 μm except for tests with the carboxyl-functionalized polystyrene particles at 80 kPa, where focal depth is 38 μm due to the use of a lower-magnification objective lens to scan the longer porous medium; the pore space area Apore and area of deposited particles Ad determined from the images thus correspond to a 2D projection of a volume spanning ∼1 pore body to ∼1 bead in depth. For all experiments, we restrict our analysis of the images to a distance ∼60 μm away from the transverse boundaries of the capillary to minimize edge effects.

Simulations of particle deposition and erosion

To explore the influence of hydrodynamic and colloidal interactions on particle deposition and erosion, we use the Parti-Suite software package (59) to examine the trajectories of single particles or clusters of particles as they flow through the pore space. Specifically, we simulate Lagrangian trajectories that explicitly incorporate the forces and torques on particles (49) in a Happel sphere-in-cell model of the pore space surrounding an individual bead (37). Previous work has established the ability of this framework to quantitatively model particle deposition and erosion from surfaces under the influence of imposed flow (49). Specifically, the particles are initialized at random points at the entrance of the pore upstream of the bead. We choose the flow velocity to match that of the experiments. Before particle-bead contact, the particle velocity is computed from the action of hydrodynamic forces exerted by the imposed flow, computed using the parameter values given in table S2. Upon contact with the bead surface, particle movement is dictated by the balance between hydrodynamic and attachment torques (49), computed using the parameter values given in tables S1 and S2, respectively. Multiparticle clusters are treated as larger 4-μm-diameter particles and are simulated in the same way, but using physicochemical parameters that describe the relevant colloid-colloid interactions. We use these simulations to assess deposition and erosion of amine- and carboxyl-functionalized particles and particle clusters, at high and low imposed pressures, and both in the refractive index–matched fluid mixture and in pure water. These results are detailed in the Supplementary Materials.

Calculation of permeability of deposit-filled region

To determine

kd(t)kd/k0

, we use the images in a square region near the inlet of the medium approximately 400 μm × 400 μm across to determine the time-dependent pore space area Apore(t), the deposited particle cross-sectional area Ad(t), the pore space perimeter Ppore(t), and the porosity ϕ(t) in the particle deposit-filled region. We then calculate

kd(t)

by modeling the pore space as a parallel bundle of cylindrical capillary tubes, each of which has a hydraulic diameter dh and length l. The pressure drop across each tube is then given by the Hagen-Poiseuille equation as

32μlv/dh2

, where v = Q/(ϕA) is the interstitial flow speed in the medium. This pressure drop is also given by Darcy’s law as μvϕl/kd. Equating these two relations for pressure drop yields

kd=ϕdh2/32

. Hence,

kd(t)=[ϕ(t)/ϕ0][dh(t)/dh,0]2

. We follow typical convention (60) in defining the hydraulic diameter dh(t) ∼ Apore(t)/Ppore(t), which we directly measure using confocal micrographs in a square region near the inlet of the medium approximately 400 μm × 400 μm across.

Acknowledgments: It is a pleasure to acknowledge T. Bhattacharjee for assistance with data analysis, M. Derakhshandeh for assistance with a preliminary version of the experimental platform, R.K. Prud’homme for stimulating discussions, the developers of the Parti-Suite software package (https://wpjohnsongroup.utah.edu/trajectoryCodes.html) for providing this open-access resource, and the anonymous reviewers for constructive and insightful feedback. Funding: This work was supported by the Grand Challenges Initiative of the Princeton Environmental Institute, the Alfred Rheinstein Faculty Award, and a postdoctoral fellowship from the Princeton Center for Complex Materials (PCCM) to N.B. We also thank the Princeton University Library Open Access Fund Program for support in helping make this publication open access. Author contributions: N.B. and S.S.D. designed the experiments; N.B. performed experiments; N.B., J.S., and S.S.D. analyzed the data; N.B. and S.S.D. developed the theoretical model; S.S.D. designed and supervised the overall project. All authors discussed the results and wrote the manuscript. Competing interests: The authors declare that they have no competing interests. Data and materials availability: All data needed to evaluate the conclusions in the paper are present in the paper and/or the Supplementary Materials. Additional data related to this paper may be requested from the authors.