Supplementary Information Microfluidic quadrupole and floating concentration gradient Mohammad A. Qasaimeh, Thomas Gervais, and David Juncker

Size: px

Start display at page:

Download "Supplementary Information Microfluidic quadrupole and floating concentration gradient Mohammad A. Qasaimeh, Thomas Gervais, and David Juncker"

Constance McKenzie
5 years ago
Views:

1 Mohammad A. Qasaimeh, Thomas Gerais, and Daid Juncker Supplementary Figure S1 The microfluidic quadrupole (MQ is modeled as two source (Q inj and two drain (Q asp points arranged in the classical quardupolar form. Using the superposition principle to produce an in plane elocity pattern. The elocity profile at any gien point is the sum of the contribution of the i th inlet located at a distance ρ i. Supplementary Figure S2 Numerical analysis of the microfluidic quadrupole (MQ and the calculated shear stress on the bottom substrate. (a The calculated wall shear stress on the substrate (z0 around the stagnation point (SP conerges to zero as the flow elocity tends to zero as well (see Fig. 4b. The shear stress profile along the line Y- Y is shown in Fig. 4c. Aspiration and injection flow rates used are 100 nl/s and 10 nl/s respectiely. (b By keeping the same injection flow rate while changing the aspiration flow rate each time, the local maxima of shear stresses at the inner edge of each aspiration aperture increases linearly with respect to the aspiration s flow rate. Supplementary Information 1/7

Mohammad A. Qasaimeh, Thomas Gerais, and Daid Juncker Supplementary Figure S3 3D numerical simulations of gradients in the microfluidic quadrupole.

A close-up iew of the gradient corresponding to the white rectangle is shown as inset.

2 Mohammad A. Qasaimeh, Thomas Gerais, and Daid Juncker Supplementary Figure S3 3D numerical simulations of gradients in the microfluidic quadrupole. (a The relatie concentration of fluorescein when 50 nl/s and 10 nl/s were used as aspiration and injection flow rate, respectiely. A close-up iew of the gradient corresponding to the white rectangle is shown as inset. (b Comparison between the analytical solution, numerical simulation, and experimental measurements of the gradient profiles along the line a-b in (a. Supplementary Figure S4 Constant gradient size with different flow confinements. The constant shape is achieed when keeping constant aspiration flow rates (Q asp 100 nl/s and changing the injection flow rates (Q inj each time. (a Q inj 50 nl/s (Q asp /Q inj 2. (b Q inj 20 nl/s (Q asp /Q inj 5. (c Q inj 10 nl/s (Q asp /Q inj 10. Closes up iews of the respectie gradient area defined by the rectangle are shown in insets. Scale bar is 200 µm in the oeriews and 100 µm in the close-up iews. (d Fluorescence intensity profiles along the central line across each gradient shown in the insets. The gradients hae almost the same shape when the aspiration flow rates are constant. Supplementary Information 2/7

3 Mohammad A. Qasaimeh, Thomas Gerais, and Daid Juncker Supplementary Figure S5 Schematic illustrating measurement of gradient length. The gradient length was defined as the distance between 90% and 10% of the maximum normalized intensity. Supplementary Table S1: Parameters used to compute the full elocity profile and stream functions Ψ of the model shown in the schematics aboe. i Q i r ' i 4 Qi i ae ( r 2 i 1 r r ' i ( r r ' ψ i, ψ 4 i1 ψ i ψ i (series expansion for x<<d/2, y<<d/2 1 + Q inj d /2 ˆ x 2 -Q asp d /2 ˆ y 3 + Q inj d /2 ˆ x 4 -Q asp d /2 ˆ y Q inj Q asp Q inj Q asp (x d /2 ˆ x + yˆ y ((x d /2 2 + y 2 xˆ x + (y d /2 y ˆ ( x 2 + (y d /2 2 (x + d /2 ˆ x + yˆ y ((x + d /2 2 + y 2 xˆ x + (y + d /2 y ˆ ( x 2 + (y + d /2 2 Q inj x d /2 Q inj π 2y 4xy arctan O( x, y y 2 H 2 d 2 π d Q asp arctan x y d /2 Q inj Q asp 2x d 4xy + O(x 3,y 3 d 2 x + d /2 Q inj π 2y 4xy 3 3 arctan + + O( x, y y 2π H 2 d 2 d Q asp arctan x y + d /2 Q asp 2x d 4xy + O(x 3, y 3 d 2 Supplementary Information 3/7

4 Mohammad A. Qasaimeh, Thomas Gerais, and Daid Juncker Supplementary Methods Calculation of the flow elocity profile for a MQ. The elocity profile of a microfluidic quadrupole can be determined using equation (3 from the main paper with the alues presented in Supplementary Table S1 by modeling the MQ in 2D as shown in the schematics aboe. Instead of the elocity potential φ, we use here the common stream function ψ, with the usual relations describing potential flow 14 : x ae φ x ψ y, y ae φ y ψ x, (S1 To obtain the stream functions found in Supplementary Table S1, we integrate the x and y components of the elocity profile (dropping the integration constant. Calculation of the confinement radius R. The confinement radius R corresponds to the point on the x axis where ae ( x, y 0along the y0 line, the elocity profile introduced in Table 1 reduces to: ae Qinj ( x,0 xˆ Q asp xxˆ d / 2ˆ y Q + xxˆ + d / 2ˆ y, (S2 ( x d / / 4 2 H ( x d / 2 2 H 2 2 x + d π + π x + d / 4 inj xˆ Q asp By setting x R and ae (R,0 0, the expression is reduced to 0 Q inj 2Rxˆ 2Rxˆ Q 2 2 asp 2 2 ( R d / 4 R + d / 4, (S3 And when soling for R: d Qasp / Qinj + 1 R, (S4 2 Q / Q 1 asp inj Calculations of the elocity profile and stream function around the stagnation point. The stream function close to the stagnation point (x<<d, y<<d can be computed by summing the second order series expansions found in Supplementary Table S1. Supplementary Information 4/7

5 Mohammad A. Qasaimeh, Thomas Gerais, and Daid Juncker ψ Q inj 8xy Q asp 8xy Q 8xy, d 2 d 2 πh d 2 (S5 This stream function can be represented as the imaginary part of a power law conformal map of the form: w 4 Q πhd 2 z2, (S6 Where w φ + i ψ and z x + iy. This function has been known to describe the flow of an irrotational fluid around a corner 14. In this particular case, each quadrant in the schematics aboe can be seen as a corner with the x and y symmetry axes forming irtual walls. The elocity profile corresponding to that particular flow profile is then found using the definition of the stream function as in equation (S1 ae ψ ψ xˆ yˆ y x 8 Q 2 πhd ( xxˆ + yyˆ, (S7 Flow profile near the inlets. The exact solution of the flow profile along the XX and YY axes can be obtained by setting respectiely x0 and y0 in the elocity profile presented in Supplementary Table S1. 4 Q (2x / d XX ' ( x,0 xˆ πhd 4 1 (2x / d 4 Q (2 y / d y y YY ' (0, ˆ πhd 4 1 (2y / d, (S8 This result demonstrates how equation (S7 greatly underestimates the flow elocity when ery close to the inlets. Neertheless, the fourth power dependence in the denominator also shows that equation (S7 is still ery accurate (6% error up to half way between the stagnation point and the inlet along the XX and YY axes. Reynolds number estimations inside the MQ. In the MQ, the flow profile aries from 0 at the stagnation point to maximum alues near the apertures. For a flow rate of 100 nl/s, equation (S8 gies a maximum elocity 2 mm/s on the YY axis at y (d a /2. Using the definition of the Reynolds number in parallel plate geometry using maximal elocity 45, the maximum Reynolds number in the MQ is on the order of 2 ρh Re (S9 ηd The flow is purely drien by iscosity eerywhere under the microfluidic probe. Supplementary Information 5/7

6 Mohammad A. Qasaimeh, Thomas Gerais, and Daid Juncker Assessment of the shear stress near the stagnation point. Shear stress can also be coneniently calculated in a Hele-Shaw cell due to the fact that the flow remains parabolic in nature een at length scales on the order of H, the gap between the plates. Combining equations (1 and (2 of the main paper, the Hele-Shaw elocity profile can be expressed as z z H S ( x, y, z 6 1 ae( x, y (S10 H H The shear stress at the surfaces comprising the MQ can thus be calculated to be: H S ( x, y, z 6η τ η ae ( x, y, (S11 z z 0 H Near the stagnation point, using equations (S7 and (S11, the magnitude of the shear stress yields: τ 6η 8 Q x 2 + y 2 48η Q r (S12 H πhd 2 πh 2 d d where r is the absolute distance from the SP in any direction. When replacing the constants by their experimental alues (using Q inj 10 nl/s and Q asp 100 nl/s, the shear stress inside the MQ is estimated to: r τ 0,31 [ Pa], (S13 d For example, by considering the flow rates mentioned aboe, the shear stress at a point located 50 µm away from the SP is Pa (calculated using equation (S13. This alue matches ery well with the results from the 3D FEM simulations, Fig. 4c. Maximal shear stress inside the MQ. The maximal shear stress inside the MQ occurs at the point of maximum elocity (i.e right at the aspiration aperture edge. Under the same experimental conditions mentioned in the preious section, and by using equation (S8 for the aerage elocity profile, the maximum shear stress is estimated to be: τ 48η Q πh 2 d (1 a/d ˆ y 1 (1 a/d 4, (S14 where a is the inlet s diameter (a 360 µm in the present case and η (10-3 Pa s is the iscosity of water at 20 C. Using these alues and Q 55nL /s, we obtain a shear stress Supplementary Information 6/7

7 Mohammad A. Qasaimeh, Thomas Gerais, and Daid Juncker equals 0.26 Pa. This alue is close to the magnitude of the shear stress found numerically around the aspiration apertures ( Pa, Fig. 4c. The slight oerestimation of the shear in equation (S14 on the inside edge of the aperture (0.26 Pa instead of 0.15 Pa might be ascribed to the limitation of the 2D analytical model (with point sources at the center of the actual probe apertures when compared to the full 3D model with apertures with a finite size. Indeed, equation (S13 accurately predicts the slope of the shear stress except in close proximity of the aspiration apertures. Uniformity of the concentration profile along the YY interface. The full partial differential equation describing steady-state conection and diffusion close to the SP is obtained by combining equations (5 and (7 from the main text to yield D 2 C x Q πhd x C 2 x + D 2 C y 8 Q 2 πhd y C 2 y 0. (S15 Using the dimensionless groups Equation (S15 can be normalized to Pé 8 Q πhd, x x Pé, y y Pé, (S16 d d 2 C x 2 + x C x + 2 C y 2 y C y 0. (S17 From the scale of this problem, we can see that mass transport is dominated by diffusion when x <1, y <1. Using Q 55 nl / s, D 500 µm 2 /s, H 50 µm and d 1075 µm yields a Pé number around The typical size of the diffusion-dominated region in d d the problem is therefore x < 15 µ m, y < 15µ m. Beyond that region, which Pé Pé can be conceied as the effectie dimensions of the stagnation point, transport is dominated by conection. Along the y-axis, the high Pé number means that the diffusion profile deeloped in the stagnation point will be stretched along the y axis. Since the initial concentration profile is translationally inariant along the axis y ( C y 0, so will be the final concentration profile. Supplementary reference 45. Bruus, H. Theoretical microfluidics (Oxford Uniersity Press 2008, p. 80. Supplementary Information 7/7

PHY121 Physics for the Life Sciences I

PHY121 Physics for the Life Sciences I PHY Physics for the Life Sciences I Lecture 0. Fluid flow: kinematics describing the motion. Fluid flow: dynamics causes and effects, Bernoulli s Equation 3. Viscosity and Poiseuille s Law for narrow tubes