Flow dynamics through discontinuous clogs of rigid particles in tapered microchannels

Suspended particles flowing through complex porous spaces exhibit clogging mechanisms determined by factors including their size, deformability, and the geometry of the confinement. This study describes the clogging of rigid particles in a microfluidic device made up of parallel microchannels that taper from the inlet to the outlet, where the constriction width is approximately equal to the particle size. This converging geometry summarizes the dynamics of clogging in flow channels with constrictions that narrow over multiple length scales. Our novel approach allows the investigation of suspension flow dynamics in confined systems where clogs are formed both by sieving and bridging mechanisms simultaneously. Here, flow tests are conducted at constant driving pressures for different particle volume fractions, and a power-law decay which appears to be peculiar to the channels’ tapered geometry is observed in all cases. Compared to non-tapered channels, the power-law behavior shows flowrate decay is significantly weaker in tapered channels. This weaker flowrate decay is explained by the formation of discontinuous clogs within each channel. Micrographs of the clogged channels reveal clogs do not grow continuously from their initial positions around the channels’ outlet. Rather, new clogs spanning the width of the channel at their points of inception are successively formed as the cake grows toward the inlet area in each microchannel. The results show changes in particle volume fraction at constant driving pressure affect the clogging rate without impacting the underlying dynamics. Unexpectedly, analyses of the particles packing behavior in the microchannels, and post-clogging permeability of the microfluidic devices, reveal the presence of two distinct regimes of driving pressure, though only a small portion of the total device volume and channels surface area are occupied by clogs, regardless of the particle volume fraction. This novel investigation of discontinuous clogging over multiple particle diameters provides unique insights into additional mechanisms to control flow losses in filtration and other confined systems.


Repeatability of Flow Tests
Although new microfluidic devices are prepared for each flow test, device-to-device variation in hydraulic resistance (R H ) does not impact the consistency of the results. Presented in Fig. S2 are flowrate decay curves obtained for different conditions of φ when ∆P = 1000 mbar. Within each of the three plots shown of the raw data, in Fig. S2a-c, the difference in R H between the two trials is reflected in differences in Q 0 : Q 0 R H = ∆P in each case. For each condition of φ and ∆P, the two normalized flowrate decay curves follow the same trend and decay to a similar final value ( Fig. S2d-f). The comparison shows the experiments are repeatable, as the decay start (τ d ) and end times (τ f ), and power-law decay exponent (n) are approximately the same in the compared cases, with n varying ≈ 8%. Consequently, the result of only one flow test is reported and included in the subsequent analyses, for each condition of φ and ∆P investigated. R H ≈ 80 mbar·min·µL -1 for the flow tests reported in the manuscript. Also, the flow meter can accurately measure water flowrates in the range 0 -80 µL·min -1 with a resolution of 0.06 µL·min -1 . The accuracy is 5% of the measured value above 2.4 µL·min -1 and 0.12 µL·min -1 for values below 2.4 µL·min -1 . In the flow tests conducted in this study, the minimum and maximum flowrates are ∼1 µL·min -1 and ∼30 µL·min -1 respectively, which are well within the limits of the flow meter. Fig. S3b shows the variability of flowrate measurements, beyond the decay end time (τ f ) for flow tests conducted at ∆P and φ = 0.05, 0.10 and 0.25%. These are examples of cases where a final plateau was reached. The fluctuations in Q are no more than 2%, despite the low value of Q itself. When Q(t) features a plateau at early times in the flow tests, fluctuations in Q are less than 1%.

Number of Clogging Events
Fig . S4 shows the number of clogging events, N, varies linearly as φ for constant ∆P. Also, it suggests there is a minimum number of clogs required for the device to be "fully clogged" -which is ∼500 in these cases. By extension, it implies a minimum number of particles is also required for the device to be "fully clogged."

Suspension Flowrate Decay
Presented in Fig. S5 is the flowrate decay curves at different ∆P when φ = 0.01 and 0.10%. It shows the flowrate decay is faster at φ = 0.10% for each ∆P. Also, each decay curve features three distinct timescales: a power-law flowrate decay preceded by an initial plateau which ends at t = τ d , and succeeded by a final plateau or slower decay rate depending on φ . A final plateau is reached at t ≡ τ f ≈ 400 s when φ = 0.10%. It signifies the end of the flowrate decay. However, a final plateau is not observed at φ = 0.01%: instead, the cake continued to grow but at a much slower rate.

Evolution of Fluorescence Intensity
Changes in the intensity of fluorescence signals (I/I m ) at the midpoint of the channel exit area were analyzed for some flow tests to estimate the end of the initial plateau (τ d ) in the flowrate decay curves. The result when ∆P = 2000 mbar and φ = 0.05% is presented in Fig. S6. It indicates significant increases in I/I m in the first second of the flow test, which is less than the time it takes to fill the device with pure water (

Effect of Microchannel Depth
Although the result presented in this study is based on 10 µm deep microchannels, it is important to investigate any potential change in flow dynamics and clogging pattern when the channel depth is changed. Fig. S7 shows the results when a suspension with φ = 0.10% is flowed at ∆P = 500 mbar through a device with 22.5 µm deep channels. The R H of the device is ∼25 mbar·min·µL -1 . This value is ∼3 times less than that of devices with 10 µm deep microchannels (∼80 mbar·min·µL -1 ). Fig. S5(b) includes a flowrate decay curve at the same conditions, φ = 0.10% flowed at ∆P = 500 mbar, in a device with 10 µm deep channels, where Q 0 ≈ 6 µL·min -1 . In the 22.5 µm channel, Q 0 ≈ 16 µL·min -1 , ∼3 times greater, because R H is ∼3 times less in the deeper channels. Despite this difference in R H , however, similar flow dynamics are observed compared to results obtained with φ = 0.10% and ∆P = 500 mbar in 10 µm deep channels. The initial plateau persists until τ d ∼ 10s, corresponding to the clogging by sieving step. A power-law decay, corresponding to the cake growth, is observed after the initial plateau (Fig. S7a). The exponent, n, of the power-law decay is ∼0.2, which is within ∼30% of that reported in the devices with 10 µm deep channels (Table S1).
A notable difference in the flow dynamics between the 22.5 µm and 10 µm deep channels is the delay in the onset of the final plateau. For 22.5 µm deep channels, no plateau is observed even up to 1500 s, which is due to lower R H and more filtrate flow through the interstices in the clogs.
Micrographs of the clogged 22.5 µm deep channels show the clogging pattern is similar to the observation in 10 µm deep channels. The clogs grow discontinuously from the initial points, beginning where w c ≈ d p , towards the channel inlet: multiple distinct clogs are observed in each channel (Fig. S7b). Unlike the 10 µm deep channels where only two layers of particles can fit within the depth of the channels, up to five layers of particles can fit in the 22.5 µm deep channels.
In short, both the clogging behavior and flowrate decay are robust despite changes in the depth of the channels. This result further strengthens the conclusion that the power-law decay is peculiar to the tapered geometry and the presence of discontinuous clogging.

Decay Timescales and Power-law Exponents
The timescales and exponents of the power-law fits observed both when the decay curve reaches a final plateau and otherwise are presented in Table S1 for all conditions of ∆P and φ examined in this study. R H » 80 mbar min mL -1 Figure S1. Linear ramp of ∆P with pure water to estimate R H of the microfluidic device (inlet reservoir and channels combined) and inlet reservoir (device without channels).  0.21 a The power-law exponent in Q ∽ t −n . b The power-law exponent, n, corresponding to the clog growth timescale. c The power-law exponent, n, associated with the point of inflection on the decay curve when a final plateau was not reached. If a plateau is reached beyond τ f , '-' is shown in lieu of a value for n.