Dispersion control in pressure-driven flow through bowed rectangular microchannels
aa r X i v : . [ phy s i c s . f l u - dyn ] F e b Journal manuscript No. (will be inserted by the editor)
Dispersion control in pressure-driven flow through bowed rectangularmicrochannels
Garam Lee · Alan Luner · Jeremy Marzuola · Daniel M. Harris ∗ Received: date / Accepted: date
Abstract
In fully-developed pressure-driven flow, the spread-ing of a dissolved solute is enhanced in the flow directiondue to transverse velocity variations in a phenomenon nowcommonly referred to as Taylor-Aris dispersion. It is wellunderstood that the characteristics of the dispersion are sen-sitive to the channel’s cross-sectional geometry. Here wedemonstrate a method for manipulation of dispersion in asingle rectangular microchannel via controlled deformationof its upper wall. Using a rapidly prototyped multi-layer mi-crochip, the channel wall is deformed by a controlled pres-sure source allowing us to characterize the dependence ofthe dispersion on the deflection of the channel wall and over-all channel aspect ratio. For a given channel aspect ratio, anoptimal deformation to minimize dispersion is found, con-sistent with prior numerical and theoretical predictions. Ourexperimental measurements are also compared directly tonumerical predictions using an idealized geometry.
Keywords
Taylor dispersion · Dispersion control · Deformable microchannel · Xurography
When a localized patch of dissolved solute or other passivetracer is introduced into a pressure-driven flow, it spreadsrapidly as it moves downstream through the subtle inter-play of fluid advection and molecular diffusion [19,3]. Thisphenomenon is now commonly referred to as Taylor-Arisdispersion and plays a critical role in many continuous mi-crofluidic devices. Depending on the application, it may be
Daniel M. HarrisE-mail: daniel [email protected] School of Engineering, Brown University, Providence, RI 02912,USA. Department of Mathematics, University of North Carolina atChapel Hill, Chapel Hill, NC 27599, USA. desirable to enhance dispersion (e.g. for mixing or reactions[17,12]) or to minimize dispersion (e.g. for separations orchromatography [10,21]). In the present work, we developa novel technique for experimental control of dispersion ona single microchip via the continuous deformation of a rect-angular channel’s side wall. We first summarize the relevantphysical quantities associated with the problem in what fol-lows, also reviewed in detail within the 2006 review paperby Dutta et al. [11].Consider steady, fully-developed, pressure-driven flowthrough a channel of uniform cross-section. At some instantin time, a localized patch of a passive solute is introducedinto the flow and transported downstream and spreads due tothe effects of advection and diffusion. On long time scales,the solute assumes a Gaussian shape and grows diffuselywith an enhanced dispersion in the flow direction that canbe described by a Taylor-Aris dispersivity KK κ = + (cid:18) Uh κ (cid:19) f = + Pe f (1)where κ is the molecular diffusivity of the solute, U is themean flow velocity, h is the characteristic channel height,and f is a scale-free parameter that depends only on thecross-sectional shape of the channel. The Peclet number Pe = Uh / κ represents a ratio of the diffusive to advective timescales. The second term on the right hand side of equation(1) represents the enhancement due to the presence of a non-uniform fluid flow and becomes dominant for large Pecletnumbers.Over the past several decades, there has been persistentinterest in understanding the role of a microchannel’s cross-sectional shape on its dispersion characteristics [7,9,11,1,5,4,6,22,2]. As defined in equation (1), the cross-sectionalshape defines the dispersion factor f through the boundary Garam Lee et al. α f wh Fig. 1
Dispersion factor f of a rectangular channel as a function ofthe channel’s aspect ratio α = w / h . The dispersion factor f is definedin equation (1) and computed numerically using the method describedin Section 2 based on a standard theoretical formulation [11]. For ref-erence, the case of infinite parallel plates corresponds to a dispersionfactor of f = conditions imposed on the flow problem. Physically, the dis-persion factor f is directly correlated to how “uniform” thevelocity profile is across a particular channel’s cross-section.A more uniform flow field will tend to lead to lower disper-sion.The dispersion factor for a rectangular channel as a func-tion of the aspect ratio α = w / h ≥ f =
1. All rectangu-lar geometries exceed this value due to the presence of thevertical side walls, even in the limit where α → ∞ (where f ≈
8) [8]. More recently, numerical and theoretical workhave focused on how practical modifications to the tradi-tional rectangular channel geometry might reduce the dis-persion factor [9,11,6]. One particular message that arisesthroughout these works is that by preferentially enlargingthe cross-section near the vertical side walls, it is possibleto reduce the dispersion factor below the value for the purerectangle. This finding is rationalized physically in the re-view by Dutta et al. [11]: since a considerable amount offlow restriction (i.e. locally slower fluid flow) arises due tothe presence of the vertical side walls, decreasing the lo-cal hydrodynamic resistance in these side regions leads toa more uniform velocity profile across the channel, therebyreducing overall dispersion. Despite the numerical and the-oretical progress in this direction, experimental results arelacking.The present work focuses on a single class of cross-sectional geometries shown in Figure 2. One wall of a rect-angular channel of overall aspect ratio α is deformed in acontinuous curve with a peak deformation d . For this class y -0.5 0.500.511.52 0 wx h d (a) (b) δ = 0.3δ = 0.6δ = 0δ = - 0.3δ = - 0.6 wh Fig. 2
Cross-sectional geometry. (a) Geometrical parameters of thechannel with deformation of one wall. (b) The deflection parameter δ is defined as the ratio of the maximum deformation d over the orig-inal undeformed channel height h as δ = d / h . Positive δ representsoutward deformation, and negative δ represents inward deformation. of shapes, we expect the dispersion factor f to depend pre-dominantly on two dimensionless geometric quantities: thebase aspect ratio α and deflection parameter δ = d / h . Notethat a positive deflection parameter δ > δ < δ = h as the height of the undeformed rectangular geometry.Dutta et al. [11] theoretically and numerically analyzeda similar “bowed” geometry (with a parabolic height profileon both top and bottom) and demonstrated that an optimalbowing exists to minimize the dispersion coefficient for agiven channel aspect ratio. In particular, in the asymptoticlimit of | δ | ≪ α ≫ δ o = − . hw = − . α . (2)This asymptotic analysis can be applied to our idealized ge-ometry as well, and yields the same result. According totheir result, the optimal deflection parameter is predicted tobe inwards ( δ o <
0) with its magnitude decreasing with thechannel’s aspect ratio.To the best of the authors’ knowledge, there remains noexperimental validation of the prediction of an optimal bow-ing. One recent experiment work relevant to the present dis-cussion was contributed by Miles et al. [16] who demon-strated the possibility of “flattening” the velocity profile in ahigh-aspect ratio channel via bowing. However only a singlechannel and deflection parameter were studied, and disper-sion coefficients were not measured as part of this work.The principal contributions of the present work are asfollows. First, we develop and describe a novel and acces-sible rapid-prototyping fabrication technique which allowsfor continuous control of a microchannel’s cross-section bydeforming one channel wall using a controlled external pres-sure source. Dispersion experiments are then performed whichdemonstrate that the dispersion factor can be directly con-trolled on a single microchip by nearly an order of magni- ispersion control in pressure-driven flow through bowed rectangular microchannels 3 tude. Finally, the measured dispersion factors are shown tobe in good agreement with numerical predictions of an ide-alized geometry and both exhibit a local minimum in thedispersion factor at an optimal deflection parameter.
The dispersion factor f can be computed as sequence of two2D Poisson problems over the cross-sectional domain of thechannel. The theoretical formulation of this problem is re-viewed in detail in prior work [11], but we review the gov-erning equations here briefly. We present the non-dimensionalformulation of the problem which is normalized by charac-teristic length scale h and characteristic velocity scale p z h / µ ,where p z is the pressure gradient in the z (flow) direction and µ is the fluid’s dynamic viscosity. The first Poisson prob-lem corresponds to the fully-developed steady flow problemthrough a channel of uniform cross-section Ω : ∇ u = − u = u ( x , y ) is the non-dimensional velocity field.From this solution we can compute the average non-dimensionalvelocity ¯ u by integrating over the cross-section and normal-izing by the cross-sectional area | Ω | . The second Poissonproblem is ∇ g = − u ¯ u and Z Ω g dA = ∇ g · ˆ n = n isthe unit vector normal to the boundary. We can then use thesolution g ( x , y ) to compute the dispersion coefficient f as f ( Ω ) = | Ω | Z Ω u ¯ u g dA . (5)In order to solve these two problems numerically, weparameterize the cross-sectional domain via a polygonal ap-proximation made with a fine discretization of the boundarycurves. Then, we solve the corresponding Poisson problemsthrough the MATLAB PDE solver package by generatinga reasonably fine triangular mesh and the resulting finite-element stiffness matrices for both Dirichlet and Neumannboundary conditions. From these, we are able to accuratelyintegrate the numerical solution over the domain and com-pute the dispersion factor f for an arbitrary cross-sectionalgeometry.In the present work, the cross-sectional geometry is de-fined by two non-dimensional parameters: the aspect ratio α = w / h and the deflection parameter δ = d / h . For the pur-poses of the numerical calculation we assume the deformedboundary to take the form of a parabola, as plotted in Figure2(b). For the idealized case of a thin membrane (i.e. negli-gible bending resistance) under a uniform pressure loading, δ α Fig. 3
Numerical calculation of the dispersion factor f as a function ofthe aspect ratio α = w / h and deflection parameter δ . The white solidline indicates a local minimum in the numerically determined disper-sion factor corresponding to an “optimal” value δ o for each aspect ratio.The dashed line represents an asymptotic analytical prediction for sucha minimum (Eq. (2), [11]). the linearized governing equation predicts a parabolic pro-file [24]. Despite this fact, for small deformations the nu-merical results are found to be relatively insensitive to theexact functional form of the smooth upper curve.The results of the numerical calculation are summarizedin Figure 3. Of particular note, is that for each aspect ratioan optimal deflection parameter δ o exists at which the dis-persion factor achieves a local minimum. These results areconsistent with the prior numerical work on bowed chan-nels wherein both the top and bottom walls are deformedsymmetrically [11]. In the figure, we have also overlaid theasymptotic prediction for the optimal deflection parameterfor a bowed channel (Eq. (2)) as predicted by Dutta [11]which shows good agreement with our numerical predic-tions, particularly for large aspect ratios, the regime in whichthis theoretical result is derived.The numerical code used to perform these calculationsis included as supplementary material. The results of thenumerical study will be compared directly to experimentalmeasurements in a following section. Garam Lee et al. advantages of this technique, the required expensive equip-ment, highly trained personnel, and cleanroom environmentare not readily accessible to all researchers. Furthermorethe resources typically required for a single mold can limitearly-stage prototyping of channel designs.For our work, we use a fabrication method that relies ona commercial desktop craft cutter. In this technique, some-times referred to as xurography, double-sided tape is cut intothe designed channel shape by a commercial craft cutter andadhered to a transparent film and an acrylic sheet as an up-per and lower boundary of the channel. Using a craft cutterfor microchannel fabrication has been proposed before [23];however, the lack of the necessary wall smoothness and res-olution for many microfluidic experiments have limited theimplementation of the technique. Recent work [18] demon-strated an optimized fabrication technique which achieveda surface roughness comparable to other leading fabrica-tion techniques and can produce channels with dimensionsas small as ∼
100 µm. Ease of use of the devices and ex-tremely short fabrication time makes prototyping of differ-ent channel designs significantly less costly. Furthermore,for controlled Taylor dispersion experiments, a channel witha much longer length than typical microchannel sizes is re-quired to faithfully reach the Taylor-Aris dispersion regime.With the craft cutter used in this work we are able to cut astraight channel with length up to 20 cm (typically limited toa few centimeters in soft lithography due to the wafer size)and thus avoid the need for a serpentine channel [5,22].Multi-layer soft lithography, which was first introducedby S.R. Quake and co-workers, is a technique to build an ac-tive microfluidic system containing microvalves and pumps[20]. The system made from this technique consists of a flowchannel and an air channel stacked perpendicular to eachother, and the two channels are separated by a thin PDMSmembrane. When pressurized air is applied in the air chan-nel, the membrane is deflected and obstructs the flow in theflow channel resulting in a flexible actuator or valve withoutthe need for manufacturing or control processes with reso-lution smaller than the scale of the channel itself.Inspired by this multi-layer soft lithography technique, anovel design of double-layer parallel channels is presentedwithin this work. Instead of placing the air channel and theflow channel perpendicular to each other, the channels aredesigned to be aligned parallel and exactly over top of oneanother. The parallel placement of two channels allows fordirect control of the channel’s cross-sectional shape by de-forming one of the channel’s walls as depicted in Figure4. The flow channel and the air channel are separated by aflexible membrane so that the membrane can bulge upwardand downward in a controlled manner in response to the ad-justable static pressure in the air channel. 3.2 Deformable microchip fabricationEach deformable microchip incorporates two channels whichare placed parallel and directly over top of each other, witha flexible membrane between the two channels. Fluids flowalong the straight flow channel and the air channel allowsfor control of the cross-sectional geometry of the flow bychanging the pressure in the channel as shown in Figure 4.For the flow layer and the air layer, double-sided poly-imide tape (Bertech PPTDE-1) is used as the base material.This tape is of total thickness 101 . . . α = .
0, 4 .
8, and 7 . w of 300 . . . µ m, although this stepcould take several attempts and became easier with prac-tice. A 3 mm thick optically clear acrylic sheet is attachedon the air layer as a chip base ensuring mechanical stabilityand flatness. Capillary tubing (Cole Parmer PTFE
30 AWGThin wall tubing) and a tape gasket are adhered to the flowand air inlets using an epoxy adhesive (PARTS Express 5Minute Quik-cure Epoxy adhesive).3.3 Membrane deformationA long and narrow rectangular membrane deformed undera uniform pressure loading takes an approximately 2D ge-ometry for the majority of its length [24]. We will hence-forth refer to this region of the membrane with a uniform ispersion control in pressure-driven flow through bowed rectangular microchannels 5 (a)(b)(c) (d)(e)
TopFlow layerAir layerMembraneBaseFlow inletDye holeFlow inlet Air inletFlow outletFlow channelAir channelNega(cid:127)ve P Posi(cid:127)ve P Flow outletAir inletDye inlet
Fig. 4
Double-layer deformable microchip design. (a - c) Schematic and sectional views of the microchip. The flow channel includes an inlet andan outlet while the air channel is a closed channel without an outlet. A flexible membrane placed between two channels is deformed via appliedpressure (or vacuum) in the air channel. (d) The complete chip is composed of five layers. Double-sided polyimide tape is used for the flow and theair layer. The top and the base layer are made of transparent PET film and acrylic, respectively. (e) Fabricated microchip. The deformed channelshape can be observed in the zoomed image; in this case, a positive pressure is applied to the air channel. Blue food dye was used here for ease ofvisualization. cross-sectional shape as the ‘flat region.’ By avoiding theends where 3D effects are observed (measured to be approx-imately 1 mm in length in the present work), the long flatregion allows us to realize the experiment in a channel witha uniform but readily tunable cross-section. In this work, theentire experiment is conducted within the flat region of 15cm length, always at least 5 mm away from the ends of themembrane.In addition to the uniform pressure from the air channel,the fluid movement in the flow channel generates a pres-sure gradient along the channel length due to its viscosity. Inthis experiment, however, the pressure applied from the flowchannel to the membrane can be safely neglected as the typi-cal pressure applied from the air channel required to deformthe channel significantly is on the order of 10 kPa whereasthe pressure drop along the flow channel is in the order of0.1 kPa (for the mean flow speeds of 1 . / s used in thiswork).For a given membrane material, a larger air pressure isneeded to deform a narrower channel to the same deflec-tion parameter. Since a high pressure difference across themembrane was observed to result in air passing through the membrane, pressure differences higher than 200 kPa werenot used in our experiments. Considering the elastic behav-ior of the material and the applicable pressure range, plasti-cized PVC film (Stretch-tite premium plastic food wrap) isused for α = . . α = . α = . . ∼ µ m/kPaand the α = . ∼ µ m/kPa.3.4 Bright field imagingFluorescence microscopy has been widely used in numer-ous fields of modern science including microfluidics due to Garam Lee et al. (e)(a) Light panel CameraFocusing railMicrochip h = 101.6 µm h = 203.2 µm h = 304.8 µm h = 406.4 µm h = 508.0 µm C o h (g/m ) I (b) (d)(c) Fig. 5
Setup and image processing for bright-field imaging. (a) Optical setup. (b) Raw image, (c) decomposed B channel, (d) inverted result from(c). (e) Intensity linearity validation. Channels of different thickness h (fabricated by changing the number of stacked double-sided tape layers) arefilled with dye solutions of known concentration C o and the intensity I of the inverted B channel is measured. The dashed line corresponds to alinear fit of I = . C o h + . I o = . C o = its superior sensitivity and the linear relationship betweenthe concentration of a fluorophore and its fluorescence in-tensity [13,14]. However, the use of this technique requiresa special excitation source such as a mercury lamp and aseries of optical filters. While the system has many advan-tages for measurements requiring high accuracy, a variety ofissues have to be considered [13]. One significant issue forthe present experiments is the photobleaching effect whichrefers to a permanent loss of the ability for a fluorescentchemical compound to fluoresce. This effect can add sig-nificant error in the calibration between the intensity signaland the concentration field. Considering the apparent disad-vantages of using fluorescence microscopy as well as overallhigher equipment costs, bright field imaging on a flat lightpanel was used in the present work.The imaging setup is shown in Figure 5(a). The microflu-dic chip is placed on 17 x 24 inch light panel (ArtographLightPad A950) to illuminate the microchip with uniformback lighting. To capture the intensity changes in the chan-nel, a camera (Nikon D500) is placed vertically over the chipmounted on a focusing rail (Oben MFR4-5 Macro Focus-ing Rail) for precise camera alignment and focusing. To as-sess contrast between the channel region and the yellow taperepresenting the channel walls, the RGB channels of chan-nel images with and without flurescein dye were decom-posed and compared. Among the three channels, the Blue(‘B’) channel demonstrated the best ability to distinguishthe channel region. In fluorescence microscopy, higher in-tensity represents a higher concentration of dye moleculesalong the light path. In the contrary, in bright field imaging,a larger number of dye molecules block the light arriving tothe camera sensors which yields lower intensity. Hence, the raw B channel intensity values are first inverted by subtract-ing them from the maximum possible intensity value 255 toarrive a final map of the intensity I . This sequence of imageprocessing steps is shown for a sample channel in Figure5(b-d).Images processed as above were obtained to verify andcharacterize a linear relationship between the measured in-tensity and the product of the channel thickness and a pre-scribed concentration of uniform dye solution in the chan-nel. This product (height times concentration) is proportionalto the number of dye molecules within the cross-sectionalarea of the channel. As shown in Figure 5(e), a robust lin-ear regime was found up to an intensity value of 160, af-ter which the relationship becomes nonlinear. By filling achannel with a known concentration of fluorescein dye solu-tion, we can now use this relationship to measure the heightprofile of a deformed channel. Furthermore, in this linearregime, we can also faithfully translate the intensity signalread by the camera into a 2D map of the dye concentrationas it moves through the chip.3.5 Dispersion experimentThe overall experimental setup is shown schematically inFigure 6. The fabricated microchip is connected to a glasssyringe installed on a syringe pump (Harvard apparatus Stan-dard Infuse-Withdraw Pump 11 Elite). A macro lens (Mi-takon Zhongyi 20 mm f/2 4.5x Super Macro Lens) is mountedon the camera to image the channel. The resolution of theimages taken in this setup is 1.04 µ m per pixel. The cam-era is centered 15 cm downstream from the dye inlet to en-sure that the solute patch reaches the long-time asymptoticregime. Prior work [9] has shown that in rectangular chan- ispersion control in pressure-driven flow through bowed rectangular microchannels 7 Pressurized airPressure gaugeCameraMicrochipWater inflowSyringe pump D y e Check valve
Fig. 6
Schematic of the overall experimental setup. The pressure inthe air channel is controlled using an air filled syringe, and monitoredwith a pressure gauge. Fluid flow in the flow channel is introduced viaa syringe pump. nel, the dispersivity approaches 95% of its long-time asymp-totic value after a time t ≈ W κ . (6)For the mean flow velocity of 1 . / s used throughout thepresent work, the system is estimated to be in the asymptoticregime after around 5 cm downstream, using equation (6) asa guide.First, the deformed profiles of the flow channel are mea-sured with various pressures applied to the air channel. Byfilling up the flow channel with dye solution of a knownconcentration, the deformation profile is measured using thelinear relationship between the intensity and the height ofthe channel. The cross-sectional area of the channel is thencalculated by numerically integrating the deformed channelprofile. After this measurement, the channel is flushed withpure water to remove any residual dye solution.Once the flow syringe and flow channel are filled withwater (and no air bubbles), the desired pressure is applied
130 140 150 160 170 t (s) Fig. 7
Curve fitting result for a sample experiment: α = . δ = − .
36. The solid line is the best fit to equation (7) whereas the datapoints represent the results from six independent experimental trials.The Taylor-Aris dispersivity K is determined by the fit and can then beused to compute the dispersion factor f using equation (8). to the air channel. Any water exiting the outlet of the flowchannel is wicked away using a lint-free cloth.Next a small droplet of fluorescein (HiMedia Laborato-ries Fluorescein sodium salt) solution with a known molecu-lar diffusivity κ = . × − cm / s [2] is placed on the dyehole using a syringe full of fluorescein dye solution. A solu-tion of 2 g / L fluorescein dye is used which was confirmed toremain in the linear measurement regime at the downstreammeasurement location while also producing enough inten-sity to maximize the signal to noise ratio in the intensitysignal. The dye hole is tape sealed again ensuring no bubbleis formed near the hole. The flow rate required to generatea flow velocity of U = . / s is set on the syringe pumpwhich is calculated based on the measured cross-sectionalarea, as described previously.With a mean flow velocity of 1 . / s in these chips,the corresponding Peclet number Pe is calculated to be ∼ ∼ . I ( t ) asthe dye plug passes through the imaging location.For each geometry, intensity profiles I ( t ) gathered fromsix independent trials are overlaid, and the collected inten-sity time series are curve fit to a translating and diffusingGaussian distribution I ( t ) − I o = B √ π Kt exp (cid:20) − ( x − Ut ) Kt (cid:21) (7)as described in prior work [18]. The measurement position x and the average velocity U are known experimental pa-rameters and so only the dispersivity K and overall ampli-tude B is fit during this process. The overall amplitude B varies between trials depending on the exact amount of dyeintroduced in the initial condition, but is otherwise irrele-vant. An example of such a fit is shown in Figure 7. In thisplot, each data set is normalized by the peak measured in-tensity value I max so the curves can be superposed directly.Now knowing K for our trial, the dispersion factor f can thusultimately be calculated by inverting equation (1) yielding f = Pe (cid:18) K κ − (cid:19) . (8) Dispersivity of fluorescein dye in a deformed microchipswith three different aspect ratios was tested. The measuredwidths of the flow channels are 300.4 µm, 488.6 µm and752.9 µm, and the corresponding aspect ratios of the unde-formed ( δ =
0) chips are α = f withdifferent deflection parameters δ in the α = . Garam Lee et al. experimental results are compared with corresponding nu-merical predictions, described in Section 2. Note that thetheoretical curve assumes an idealized geometry with a fixedside wall height whereas in experiment the sidewalls alsodeform with the applied pressure, as can be seen in the mea-sured channel geometries shown in Figure 8(b). Despite thisdifference, the overall agreement is very good. In particular,our results demonstrate that precise control of the disper-sion is achievable in a single microchannel, with the disper-sion factor varying by nearly an order of magnitude betweengeometries. Furthermore, the postulated presence of a localminimum in the dispersion coefficient has also been con-firmed, and occurs at an inward bowing of around δ ≈ − . . / sin all experiments, the trend in the dispersion factor is di-rectly correlated to the length of the solute band in the flowdirection. Figure 8(c) shows the intensity profile over timeat our measurement cross-section, corresponding to the datafrom Figure 8(a). Note that the figure is a time series of theintensity profile measured at a fixed spatial location, ratherthan an image of the solute distribution at a fixed moment intime.Figure 9 summarizes the results obtained from all threechips fabricated in the present study. In general the compar-ison to the idealized theory is good, and each shows a localminimum in the dispersion factor for a particular level ofinward bowing, as predicted numerically and in prior nu-merical and theoretical work by Dutta et al. [11]. Some dis- crepancies and limitations to the comparison do exist, andwill be detailed in the following section. While the overall trends of the numerics are well captured byour experimental results, some quantitative discrepancies doexist. In this section we will discuss some of the limitationsof the comparison and approach.The numerical values are calculated based on an ideal-ized deformed channel shape that is defined by only twoparameters: the aspect ratio and the deflection parameter,where the upper arc of the bowed wall is assumed to bestrictly parabolic. However, the channel geometry measuredfrom the experiment shows that this assumed geometry inthe numerical calculation is of course not a perfect repre-sentation. This effect leads to the largest discrepancy in ourcomparison for the smallest aspect ratio chip when δ > δ . In our experimental work we measured a cleardeformation of the side walls that was correlated with theapplied pressure in the air channel. Different material selec-tion for the flow channel, mechanical clamping of the overallchip, or a different manufacturing method could potentiallymitigate this effect. (a) (b)(c) x (µm) y ( µ m )
200 100 0 100 2000100
0δ = 0.450.290.15-0.21-0.38-0.58 t -0.6 -0.4 -0.2 0 0.2 0.4 0.6024681012141618 f δ Fig. 8 (a) Dispersion factor f versus deflection parameter δ in the α = . δ α δ α Fig. 9
Experimental results of the dispersion factor f for three dif-ferent microchips corresponding to α = . , . .
4. Results areoverlaid on the numerical predictions. Each data point is derived fromsix independent experimental trials. In this figure, f values obtained inthe simulation larger than 40 are depicted in the darkest color of thecolor map for improved contrast within the region of interest. Lastly, in long-time regime, the intensity profile of thedye patch along the channel direction should converge to-wards a simple Gaussian distribution. However, we observednotably skewed intensity profiles for cases with positive de-flection parameters δ >
0. The profile becomes increasinglyskewed with larger deflection parameter δ and is likely an-other factor contributing to the quantitative discrepancy be-tween the numerics (computed from long-time asymptoticequations) and experiments. In designing the experiment,the value computed from equation (6) was used as a criterionto determine the minimum distance the solute patch shouldtravel before measuring the intensity profile. However, thisguideline was only verified numerically in rectangular chan-nels, which is evidently different for the deformed channelshape considered in our work. Future theoretical and nu-merical work might consider other methods to better quan-tify the evolution of transient asymmetries in the distributioncaptured by higher moments (such as the skewness), as wasrecently documented for rectangular and elliptical geome-tries [2]. We have established an accessible method to control the dis-persion factor within a single multi-layer microchannel bydeforming one of the channel walls in a controlled man-ner. The fabrication technique uses a commercial craft cutterwhich allows us to achieve this goal without relying on theexpensive equipment and facilities required by other com- mon microfabrication techniques. The dispersion control tech-nique presented here does not rely on manufacturing or con-trol process with resolution beyond that of the microchannelitself. To complete dispersion measurements in our customchannels, we developed and validated a robust bright-fieldimaging setup using an artist’s tracing light pad and DSLRcamera, which further reduces the cost for performing suchexperiments.In our experimental measurements, we were able to iso-late the sensitive relationship between the dispersion factor f and the deflection parameter δ on a single microchip. Inparticular, for three microchannels of different aspect ratios,we experimentally verified the presence of a local minimumin the dispersion factor as δ is varied continuously. Thisminimum value of f was found when the upper channel wallwas deformed inward by an amount which depends on theoverall aspect ratio of the channel.Future work may include more detailed quantitative as-sessment of the early time distributions in the experimentwhere significant skewness in the distribution is present [2].Furthermore, the numerical code could be paired with an op-timization scheme to identify other channel geometries witheven lower dispersion coefficients.Given that the channel cross-section (and thus the dis-persion factor) can now be controlled on a single chip, ourcontributions open up potential for other exciting areas ofstudy. One area of interest might be in implementing real-time active control of dispersion processes in pressure-drivenflows. Alternatively, a similar device to ours could be usedto study dispersion through a time-dependent channel geom-etry [15] with potential relevance to transport in biologicalprocesses. Conflicts of interest
There are no conflicts to declare.
Data availability
The datasets generated during and/or analysed during thecurrent study are available from the corresponding authoron reasonable request.
Acknowledgements
DMH and GL gratefully acknowledge the finan-cial support of the Brown OVPR Salomon Award. JLM and AL weresupported in part by NSF CAREER Grant DMS-1352353 (2014-2020)and NSF Applied Math Grant DMS-1909035 (2019-Present). Further-more, GL and DMH would like to acknowledge A. Taylor for supportand guidance with the craft-cutter technique, K. Dalnoki-Veress foradvice on membrane selection, and K. Breuer for use of his invertedmicroscope in preliminary experiments.
References
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