Clogging sensitivity of flow distributors designed for radially elongated hexagonal pillar array columns: a computational modelling

Flow distributor located at the beginning of the micromachined pillar array column (PAC) has significant roles in uniform distribution of flow through separation channels and thus separation efficiency. Chip manufacturing artifacts, contaminated solvents, and complex matrix of samples may contribute to clogging of the microfabricated channels, affect the distribution of the sample, and alter the performance of both natural and engineered systems. An even fluid distribution must be achieved cross-sectionally through careful design of flow distributors and minimizing the sensitivity to clogging in order to reach satisfactory separation efficiency. Given the difficulty to investigate experimentally a high number of clogging conditions and geometries, this work exploits a computational fluid dynamic model to investigate the effect of various design parameters on the performance of flow distributors in equally spreading the flow along the separation channels in the presence of different degrees of clogging. An array of radially elongated hexagonal pillars was selected for the separation channel (column). The design parameters include channel width, distributor width, aspect ratio of the pillars, and number of contact zone rows. The performance of known flow distributors, including bifurcating (BF), radially interconnected (RI), and recently introduced mixed-mode (MMI) in addition to two new distributors designed in this work (MMII and MMIII) were investigated in terms of mean elution time, volumetric variance, asymmetry factors, and pressure drop between the inlet and the monitor line for each design. The results show that except for pressure drop, the channel width and aspect ratio of the pillars has no significant influence on flow distribution pattern in non-clogged distributors. However, the behavior of flow distributors in response to clogging was found to be dependent on width of the channels. Also increasing the distributor width and number of contact zone rows after the first splitting stage showed no improvement in the ability to alleviate the clogging. MMI distributor with the channel width of 3 µm, aspect ratio of the pillars equal to 20, number of exits of 8, and number of contact zones of 3 exhibited the highest stability and minimum sensitivity to different degrees of clogging.


Methods
Layouts of flow distributors. The flow distributors investigated in this study and their schematics indicating each part of the flow distributors (bifurcating stages, contact zones, smaller pillars and wedges are depicted in Fig. 1. These distributors include (a) BF design in which the flow is split through n series of bifurcating channels (purple channels), creating second channels of equal length. As numerically demonstrated by Davydova and his co-workers 25 , to decrease dispersion in the BF design, an equal width was set to all bifurcation levels; (b) RI distributor with high AR hexagonal pillar arrays, including interconnecting channels with frequent mixing nodes (yellow channels); and (c-e) mixed-mode designs (MM I -MM III ) containing different combination of BF, RI, and mixed-size distributors. The geometry of MM I distributor design, adopted from Ref 22 , starts with bifurcating channels (purple channels), followed by contact zone rows (yellow channels), and continues with another bifurcating stage. Knowing the higher values of AR used in this research (> 15) than the Ref 22 (AR = 15), the number of bifurcating stages for MM I distributor in this study is less than Ref 22 . In contrast, the geometry of MM II distributor initiates with RI design and is followed by contact zone rows in the middle of each of the splitting stage. The geometry of MM III distributor design starts with bifurcating channels, followed by rows of contact zones after the second splitting stage, and continues with a series of smaller rectangular pillars and wedges (green channels). The incorporation of contact zone rows after the splitting stages in MM designs provides more time for the fluid to redistribute after the clogged regions. The series of smaller rectangular pillars and wedges at the end of the MM III distributors were designed to provide a higher chance for the entrapped species in the clogged channel to elute very slowly even below the limit of detection. Each flow distributor was connected to an array of radially elongated hexagonal pillars as the separation channel.
Clogging study. To assess the stability of different flow distributors in response to clogging, different geometrical design parameters were considered: width of the channels throughout the entire geometry, aspect ratios of the elongated pillars (AR) in separation channel, distributor width, and number of contact zone rows consisting of interconnected channels after each of the splitting stage. Given that the smallest feature throughout the entire geometry is the distance between the two adjacent pillars, and this distance depends on the quality of etching process 26,27 , the minimum channel width was considered to be 2.5 µm. A maximum channel width of 3.5 µm was studied to consider the possible effect of dispersion in larger channels. Since the sidewall effect is no longer influential on performance of the distributor for AR > 15, a minimum AR = 15 was used throughout the study. On the other hand, a maximum AR of 25 was used in our clogging sensitivity studies to eliminate the increased pressure drop caused by an increased tortuosity.
The red box depicted in Fig. 1 was modified to reflect different clogging percentages of the 4-outlet-level channel. The chance of local clogging is higher closer to the inlet. Therefore, it induces an increased local flow resistance and a drop in velocity proportional to the clogging's cross-section. Each distributor was first studied to figure out the effect of channel width on clogging sensitivity, then the ones with minimum sensitivity to clogging were selected to investigate the effect of aspect ratio of the pillars. Furthermore, two and three channels were modified to induce the extreme degree of clogging simultaneously, and the results were shown on the selected distributor. Also, the influence of increasing the distributor's width and adding the number of contact zones were studied on the selected distributor.
Numerical methods and calculations. All simulations were performed on Dell Precision Tower 7910 XCTO Base each equipped with an Intel Xeon processor E5-2630 (2.4 GHz, 16 cores) and 32 Gb, 2400 MHz ram memory, running on Linux server. Simulations were accomplished with the COMSOL Multiphysics commercial software (COMSOL 5.3) in a 2D simulation model. Each flow distributor was divided into 500,000 and 3,000,000 triangular and free quad meshes. Mesh and time independencies were examined via decreasing the mesh size and time step to half of their initial values for the BF distributor with channel width of 3 µm integrated with 100 µm wide and 5 µm thick radially elongated hexagonal pillars rows. The difference in volumetric variances was less than 2.5%.
An incompressible laminar fluid flow, governed by the Navier-Stokes equations, was computed using the stationary solver. As the boundary condition, a linear velocity was applied at the inlet of the distributor and zero pressure at the outlet of the column. All inlet velocities were chosen such that a relevant mean velocity of 0.25 mm/s was obtained in the subsequent bed. The water with a viscosity of 1 × 10 −3 kg/ (m s) and a density of 998.2 kg/m 3 was selected as the carrier fluid. The water was also used as the tracer with the molecular diffusion coefficient of 10 −9 m 2 /s. The time dependent transport of diluted species governed by the Fick's law was computed with the same mesh density as used in the velocity field calculation. The inlet plane was set to the inflow with an arbitrary concentration of the tracer. The outlet plane was set to the outflow with zero concentration. The rest of the boundaries were set to no-slip wall. Chromatograms were obtained by integrating the tracer concentration over the monitor line (the red line in each of the distributors shown in Fig. 1). The red boxes in each distributor represent the clogging zone at the 4-outlet level to study the clogging sensitivity of the distributors. The performance of each flow distributor was evaluated based on the volumetric variance (σ v 2 ) of the tracer band exiting the distributor. To this end, peaks area (m 0 ), mean elution time (m 1 ), and time-based variance (m 2 , σ t 2 ) for each simulation were calculated based on moment analysis (Eqs. (1-3)) 28 .  www.nature.com/scientificreports/ wherein C(t), t 1 , and t 2 refer to mass fraction of the tracer as a function of time, peak start time, and peak end times, respectively 28 . Thereafter, σ v 2 is determined with Eq. (4). 22 where F is flow rate. Peak shape measurements (i.e. asymmetry factors, AsF and TF) was calculated based on the ref 28 .

Results
Calculating the performance of non-clogged flow distributors. The performance of each flow distributor was first assessed in non-clogged conditions. Figure 2 depicts the concentration profile of eluting band in each flow distributor (channel width = 3 µm, AR = 20) where the tracer band reaches the monitor line (red line) along with its corresponding chromatogram. Numerical values regarding the eluting peaks including mean elution time, volumetric variance, asymmetry factors as well as pressure drop between the inlet and the monitor line were calculated from computational simulation for each distributor and summarized in Table 1.
The results show that increasing the number of contact zone rows in distributors increases the total volume capacity of the system and therefore rises the elution time (changed from 0.032 s in BF to 0.084 in MM III ). Because the dispersion is strongly governed by the volume of distributors, the increase in this volume leads to a peak broadening and distortion. Based on the results of the corresponding chromatograms ( Fig. 2), BF design creates the narrowest peak (σ v 2 = 0.044 nL 2 , channel width = 3 µm and AR = 20) with a relatively perfect symmetry ∼ = 1.1 owing to its small volume, while the results of the RI distributor is a widest peak (σ v 2 = 0.15 nL 2 , channel width = 3 µm and AR = 20) with asymmetry response, tailing factor higher than 1.5, and minimal pressure drops (0.075-0.13 bar) due to its interconnected structure. Figure 3 represents the dependency of volumetric variance of the eluted peak and pressure drop over the distributor to the distributor length. As expected, BF distributor generated the minimum σ v 2 values due to their minimal volume and length, while RI contributed to the largest σ v 2 values. In each set of designs, increasing the length of the distributor increases σ v 2 value slightly except for RI distributor ( Fig. 3(a)). For instance, from MM I with the smallest channel width of 2.5 µm to MM I with the largest channel width of 3.5 µm, σ v 2 value increased from 0.055 to 0.058 nL 2 .
These values vary for MM II and MM III from 0.071 to 0.074 nL 2 and 0.064 to 0.068 nL 2 , respectively. On the other hand, pressure drop over the length of each set of distributor designs decreased by increasing either the channel width or tortuosity ( Fig. 3. (b)). The largest Δp values of 0.26 and 0.23 bar as a result of narrower channel width, and Δp-values of 0.30 and 0.25 bar due to increase in tortuosity were observed in MM I and MM III distributor designs, respectively. This phenomenon is due to the presence of rows of radially interconnected pillars with high value of AR in their flow distributor structure which increases the fluid path length and generates an extra pressure drop.
Among different MM flow distributors in this study, the peaks eluting from MM I and MM III have longer mean elution times (Table 1) due to the presence of more than one fluid contact zone after the flow splitting stages (Fig. 1). On the other hand, among MM-type distributors, the peak leaving the MM II distributor show the highest volumetric variance (σ v 2 = 0.074 nL 2 , channel width = 3 µm and AR = 20) which may be attributed to the RI structure at the entrance. All the MM-type distributors produce a completely symmetric peak with AsF ∼ = 1.
Overall, among all the non-clogged distributors, RI produced the widest peaks, MM I showed the maximum pressure drop, MM III generated the largest mean elution time, and MM II showed an intermediate response. Therefore, BF distributor outperforms in the absence of clogging with the lowest elution time and pressure drop, and perfect symmetry.
Effect of channel width on clogging sensitivity of distributors. The results of clogging sensitivity for different flow distributors tested under different channel widths (2.5, 3, 3.5 µm) are shown in Fig. 4. Each distributor was fed into a separation column with a channel width equal to that of the distributor, where the AR of the elongated pillars in the separation column was fixed to 20. As explained in Methods, the percentage of clogging in the red box (represented in Fig. 1) was changed in 10 incremental steps to reach the maximum 100% clogging with the σ v 2 -value for each case being calculated. The results show that there are no significant changes in the σ v 2 values at the first four clogging degrees (10, 20, 30, and 40%) for the channel width of 3 µm (Fig. 4 (b)), therefore these clogging steps were eliminated for the rest of the study. Figure 4 depicts that flow distributors with various channel width show different behavior in the presence of clogging. For instance, BF-type distributors are more sensitive to clogging compared to the other distributor types; BF with 2.5 µm channel width shows the widest peak (σ v 2 = 2.72 nL 2 ) at 90% clogging. However, for 3 µm and 3.5 µm channel widths, the widest peak occurs at the 80% clogging. Therefore, the widest peak, as an undesirable outcome for each flow distributor, may occur at different degrees of clogging. These undesirable states and the relevant degree of clogging were compared among different distributors (Fig. 5). The σ v 2 values of BF and MM II distributors are the largest and the www.nature.com/scientificreports/ most affected ones under changing the channel width, with more than 12 times higher σ v 2 value than the values in the absence of clogging. However, the eluted peaks from RI-, MM I -, and MM III -type distributors are approximately 3 or 4 times wider than the eluted peaks in the absence of clogging. Also, the minimum fluctuations in σ v 2 values occurs as a function of clogging percentages in RI, MM I , and MM III distributors (Fig. 4), which shows their ability in compensating the clogging.
The numerical measures for each undesirable state of the eluted peak and the relevant degree of clogging are summarized in Table 2. Numerical measures regarding the eluted peaks produced by the non-clogged flow distributors are compared to the widest eluted peaks, undesirable states, at the relevant degree of clogging with the data summarized in Tables 1 and 2. The results show that for channel clogging exceeding 70%, the mean elution time did not increase significantly except for the BF distributor. This is where pressure drop increased approximately 0.02 bar in each flow distributor in the presence of clogging except for the MM III distributor, showing negligible change in Δp after clogging. The σ v 2 values for all flow distributors with undesirable eluted band becomes markedly higher at the relevant degree of clogging. Figure 6 shows the concentration profile (channel width = 3 µm) and the corresponding chromatogram of the most affected band because of clogging right before the moment of elution for each flow distributor. The results show that once the clogging occurs, part of the injected sample is trapped in a region near the blockage. Given that the velocity in the clogged channel is much lower than the velocity in other channels, the amount of species www.nature.com/scientificreports/ that enters the clogged channel takes more time to elute. Therefore, when the degree of clogging exceeds 70%, the eluted peaks either show net tailing with asymmetric behavior or produce extra peaks because of fouling.
There is an unexpected drop in σ v 2 value at higher degrees of clogging (Fig. 4) which may be attributed to the small amount of sample entering the blocked region. Due to the lower fluid velocity in the clogged region, it takes more time for the sample to elute which dilute the sample below the detection limit (< 0.1% of the maximum of the peak) by the time it reaches the monitor line. The ripples or waves observed in the BF and MM II chromatograms are similarly interpreted.
Decreasing the channel width increases the Δp value noticeably regardless of the clogging percentage. Therefore, considering the best performance of the flow distributors with the channel width of 3 µm, this channel width was chosen for the rest of the study. Given that the MM I and MM III distributors outperformed under the fouled conditions, the other flow distributors were omitted from further consideration.
Effect of aspect ratio of the pillars on clogging sensitivity of distributors. The effect of aspect ratio of the pillars on volumetric variances of the tracer peak was evaluated for the MM I and MM III distributors in the presence of clogging ( Fig. 7 and Table 3). Three different ARs of 15, 20, 25 were considered. It is obvious that increasing the AR increases the total volume of the distributor. Inlet velocities were adjusted for each flow distributor based on their tortuosity to preserve the flow rate constant at 1.32 µL/min. σ v 2 and m 1 values were shown to be nearly independent from the considered ARs in the MM I distributor. MM III -type, however, produced a broad extra peak after the main peak when the AR was set to 15 or 25. These peaks were eluted below the limit of detection for channel clogging degrees beyond 90%. Additionally, when the degree of clogging increased to 100%, the eluted peaks shifted backwards in the chromatograms, resulted in a drop in the m 1 values (data not shown).
Moreover, the Δp-values over the distributors with different channel widths (same No. of outlets) in the absence and presence of clogging are given in Tables 1 and 3, respectively, indicating that the optimal value for the aspect ratio of the pillars compromise between the pressure drop and volumetric variance. For instance, in MM I designs, increasing the AR decreases σ v 2 value which means the generated peak is sharper, though the pressure drop increases significantly (the same explanation applies to MM III as well). Therefore, MM I -type distributor with the channel width of 3 µm and AR = 20 was modified to investigate the effect of the number of outlets (distributor width) and contact zone rows on the ability of the distributor to alleviate the clogging.
Given the temporal change in location of clogging, the influence of several clogged zones induced at the same time were investigated (See Supplementary Information). To this end, two and three channels were modified to induce 80% local clogging simultaneously in MM 1 distributor. As it was expected, for clogging occurred in more than one area, the volume of entrapped samples became larger, and the extra peak started increasing in intensity (See Fig S1 and S2). Moreover, with the increase in the number of clogged areas, the eluted peaks shifted backwards in the chromatograms, resulted in a drop in the m 1 values. " Also, the three-dimensional (3D) MM I distributor with 18 µm channel depth in the absence and presence of 80% degree of clogging at the 4-outlet-level channel was investigated. The concentration profile and the corresponding chromatograms are shown in Figs. S3 and S4, respectively. The results of 3D simulation showed a  22,29 . This extra band broadening is proportional to the volume of the distributor and therefore, the time the tracer spends in flow distributor. As a result, the observed differences between distributors in 2D geometry would be enhanced in 3D geometry owing to the contribution of top and bottom wall dispersions in band broadening. Thus, as it was expected, the σ v 2 values of the 3D MM I distributor in the absence (0.10 nL 2 ) and presence of 80% clogging (0.33 nL 2 ) were twice larger than the values     Effect of distributor width and number of contact zone rows on clogging sensitivity. MM Itype distributor can only have 2 n number of exits (NE) due to its intrinsic structure. MM I -type design with channel width of 3 µm and AR = 20 has the total width of 827 µm (NE = 8). Although increasing the NE to 32 leads to a very wide distributor with limited performance (distributor with the width of 1651 µm), the distributor with NE = 16 was investigated to understand the effect of distributor width on compensating the channel fouling. Also, MM I -type distributor with NR = 5 was investigated to assess the influence of NR on clogging sensitivity. On the other hand, MM I -type distributor has only three rows (NR) of contact zones after the first splitting stage. The liquid velocity was kept constant for all the studied cases and an equal clogging probability was assumed for each distributor. Since the MM I distributor showed inconsiderable difference between the numerical values in the non-clogged and less than 60% clogged conditions, the comparative simulations were performed between the non-clogged states and cases with more than 60% clogging. The results of these simulations are shown in Fig. 8 and the numerical values of the non-clogged cases are summarized in Table 4.
As the sample volume of the distributor increases through increasing the distributor width or length, the eluted peak becomes wider when the fluid travels further along the distributor toward the monitor line, resulted in a peak broadening and distortion. Therefore, the peak eluted from MM I -NE16-NR3-type distributor (σ v 2 = 0.14  www.nature.com/scientificreports/ nL 2 ) is approximately threefold wider than the MM I -NE8-NR3 one (σ v 2 = 0.053 nL 2 ) (seen in the chromatograms in Fig. 8 (a)) with the longest mean elution time. Furthermore, the peaks eluted from MM I -NE8-NR5-and MM I -NE16-NR3-type distributors exhibited a strong peak fronting with asymmetry factors about 0.87. Figure 8 (b) shows the resulting chromatograms for 80% clogging of the 4-outlet-level channel in all MM I -type distributors. Even though the MM I -type distributor exhibits maximum stability and minimum sensitivity towards clogging hitherto, increasing the distributor width (NE = 16) leads to an extra peak or peak distortion after 60% clogging. Unexpectedly, adding to the number of contact zone rows after the first bifurcating stage to overcome the clogging had no significant influence on peak shapes or volumetric variances.

Discussions and conclusion
One source of "extra-bed" band broadening in micromachined pillar array column is the analyte dispersion by flow distributors 30 . To decrease the band broadening contributed to flow distributor as little as possible, flow distributor needs to be carefully designed and evaluated 31 . Typically, a combination of small sample volume and high radial permeability flow path for the sample band to reach the actual separation channels (column) is the key to achieve the desired uniform distribution 23 . However, another issue that could affect the performance of flow distributor is the local clogging of the microfabricated channels. Therefore, the ideal flow distributor should provide a minimal sample volume and high radial permeability as well as minimal sensitivity to clogging. Given that the flow distributor is located at the entrance of the actual separation bed, the designed flow distributor needs to be integrated into the final structure of the separation bed geometry.
In this work, we employed arrays of radially elongated hexagonal pillars as the separation channels and integrated them with different flow distributor designs to investigate the performance of distributors in alleviating possible local clogging. The selected distributors include BF, RI, and MMs. To characterize the performance of flow distributors in response to channel clogging, employing the CFD model, their ability to distribute the sample band equally throughout the column were studied in the absence and presence of clogging. Since the chance of clogging is higher at the entrance of the microfabricated channels, and these clogged channels affect the performance of the system the most, the 4-outlet-level channel in each distributor was modified to reflect different clogging percentages. Furthermore, the effect of main geometrical design parameters (e.g. channel width, aspect ratio of the pillars, distributor width, and number of contact zone rows) on stability of the distributors in response to different clogging percentages were characterized. It is worth mentioning that in the presence of stationary phase, the analytes are assumed to be diffusing with a rate determined by the stationary phase diffusion  www.nature.com/scientificreports/ coefficient in parallel with that in the mobile phase. Although this phenomenon increases the mean elution time, the trend in clogging would not change. Moreover, the results of this study can be translated to other fluidic systems dealing with clogging challenges, for example in reactors, diagnostics assays, organ-on-chip platforms, and high-throughput drug testing systems. First, with the number of exits, contact zone rows, and aspect ratio of the pillars fixed, the channel width throughout the geometry varied within 2.5-3.5 µm. According to the Taylor-Aris theory in open-tubular channels, the volumetric variance associated with the dispersion can be expected to be proportional to the 3rd power of the channel width 25 . Since the channel width markedly influences the axial dispersion and the microfabrication of larger channel widths is easier, it is informative to investigate the effect of channel width on the performance of distributors. While varying the channel width has no significant difference in numerical measures of each set of distributors (for example in BF-types) in non-clogged states other than for pressure drop, the behaviour of flow distributors was different in the presence of clogging with varying the channel widths. Because of their structures, BF distributors have the minimum sample volume, therefore they produce the sharpest peaks in the absence of clogging ( σ 2 v ∼ = 0.046 nL 2 ) with minimum elution times. Nevertheless, with decreasing the dimension of flow paths via decreasing the channel width from 3.5 to 2.5 µm, and increasing the degree of clogging up to 90%, BF with 2.5 µm channel width shows the widest peak (σ v 2 = 2.72 nL 2 ). The interconnected flow paths of RI distributors lead to minimum pressure drops ( ≥ 0.13 bar) and sensitivity to different degrees of clogging in the absence or presence of clogging. A lower velocity near the sidewalls of RI distributors, however, results in asymmetric peaks with net tailing. MM distributors show an intermediate behavior in the absence of clogging. The reason of this is that they designed to be the best representatives of BF and RI distributors with minimal volume and ability to alleviate local clogging. In MM I distributor, the rows of the elongated pillars with higher aspect ratio are present in the geometry, giving the fluid more transversal path to distribute. This is the reason of higher pressure drop in MM I distributor. Under changing the channel width in the presence of 80% clogging, MM II distributors are the most affected ones, with more than 12 times higher σ v 2 values than the values in the absence of clogging. The reason for this is that the contact zones after the clogged channel are small which gives the fluid limited time to overcome band distortion. The minimum fluctuations in σ v 2 values occurs as a function of clogging percentages in MM I and MM III distributors owing to the presence of several contact zone rows after the clogged section, giving the distributors the ability to compensate the errors. Accordingly, MM I and MM III distributors with the channel width of 3 µm was selected to study the effect of aspect ratio of pillars on the performance of distributors. It is worth mentioning that no significant changes in the numerical measures at the first four clogging degrees (10, 20, 30, and 40%) was found for all distributors. However, when the degree of clogging exceeds, extra peaks or ripples generated after the main peak due to the lower velocity in the clogged channel. This gives the amount of species that enters the clogged channel more time to elute.
As stated above, one way to enhance the performance of a flow distributor is to increase the radial permeability. This can be achieved through increasing the aspect ratio of the pillars. With the other geometrical parameters fixed, the aspect ratio of the pillars in the separation bed varied within 15-25. When the pillars elongate radially, the length of fluid paths and thus the volume of the distributors increase. This explains why the pressure drop is higher in systems with AR = 25. Furthermore, when the degree of clogging increases to 80%, MM I and MM III distributors produce an extra peak after the main peak, which increases the volumetric variances. This extra peak was broader when the AR was set to 15 or 25 in MM III distributor. Therefore, MM I distributors show the highest ability to compensate the clogging. AR = 20 was chosen given its most efficiency along with suitable pressure drop. Moreover, increasing the MM I distributor's width or adding the number of contact zone rows may enhance the performance of a distributor in terms of the ability to alleviate the clogging. The increase in the total volume of the distributor, however, increased the volumetric variances, and showed no better performance.
Overall, the MM I distributor with the channel width, the aspect ratio of the pillars, the number of exits, and the number of contact zone rows set to 3, 20, 3, and 8 µm, respectively, demonstrated the highest performance when the clogging occurs.