Experimental measurement of parameters governing flow rates and partial saturation in paper-based microfluidic devices Dharitri Rath 1, Sathishkumar N 1, Bhushan J. Toley 1* 1 Department of Chemical Engineering Indian Institute of Science Bengaluru, Karnataka 560012 India Number of pages: 11; Number of figures: 9; Number of tables: 1 * Correspondence to: Bhushan J. Toley Department of Chemical Engineering Indian Institute of Science C V Raman Avenue Bengaluru, Karnataka 560012 Phone: +91-80-2293-3114 E-mail: bhushan@iisc.ac.in S1
Electronic Supporting Information S1. Determination of permeability as a function of saturation Figure S1. Calculation of permeability as a function of saturation. A. Cross section of the porous material shown as set of parallel tubes/capillaries. B. A single tube of radius, r, shown in isolation. C. Two parallel tubes are equivalent to two parallel fluidic resistors. D. When the material is partially saturated, the smallest pores are filled with fluid (orange colour). E,F. Representations of experimentally obtained variation of capillary pressure, ψ, as a function of normalized saturation, Se (E) and of effective pore radius, r pore, as a function of Se (F). Plot in (F) represents the cumulative pore size distribution. pore A theoretical method was developed to obtain the variation of permeability, κ, with saturation, Se (note that experiments only provided κ s, i.e. permeability at Se = 1). For this purpose, the porous material is assumed to be composed of a set of parallel tubes (pores) of fixed length, L, but varying diameters (Fig. S1A). Let us now consider one tube of radius, r pore, from this set (Fig. S1B). Assuming that the pressure drop across this tube is p (Fig. 2B), the following two correlations may be written: 4 π r pore Q= p (S1) 8µ L S2
according to Hagen-Poiseuille law, and p Q= according to Darcy s law (equation 2; µ L κ A main text). Comparing equations (S1) and (2) and substituting for the area of the tube, 2 A πr pore =, yields κ = C r 2 pore (S2) where C is a constant of proportionality. This shows that permeability scales as the square of pore radius. In addition, the term µ L in equation (2; main text) represents additive fluidic κ A resistance, as described previously by Fu et al 1. Thus, for a fluid flowing through two parallel tubes of equal length, L, but different cross-sectional areas, A 1 and A 2, an equivalent fluidic resistance, R eq, can be calculated by using the rule of parallel electrical resistances as: κ A κ A κ A R µ L µ L µ L 1 eq eq 1 1 2 2 = = + (S3) eq or, κ A = κ A + κ A (S4) eq eq 1 1 2 2 Where κ eq and Aeq are effective permeability and cross-sectional area, respectively, of the two tubes placed in parallel (Fig. S1C). Substituting for κ from equation (S2) and for = π yields: 2 A r pore κ eq = C 2( 4 4 r1 r2 ) + (S5) where C 2 is a proportionality constant and r 1 and r 2 and the radii of the two pores. The variation of the permeability of the porous material as a function of saturation may then be derived by considering the material to be composed of a set of parallel tubes such that the number of parallel tubes increases with increasing saturation. For example, when the S3
membrane is partially saturated, say Se= 0.3, only the smallest 30% of the pores in the material are filled with fluid because they exert the maximum capillary pressure (Fig. S1D). The effective permeability of these 30% pores is calculated by first converting experimentally obtained data for ψ vs Se (Fig. S1E) to rpore vs Se (Fig. S1F) by using the Young-Laplace equation for capillary force generated by a thin tube: 2γ cosα r pore= (S6) ψ where γ is the liquid-air surface tension and α is the contact angle. For these calculations, the contact angle was assumed to be zero. This is a reasonable assumption because several commercial diagnostic membranes are treated using proprietary methods to render them hydrophilic. This is based both on our personal communication with membrane manufacturers and as published by Linnes et al (Discussion section) 2. The relationship between r pore and Serepresents the cumulative pore size distribution of the membrane (Fig. S1F). This graph was divided into ten equal intervals: Se = 0 0.1, 0.1 0.2 up to 0.9 1 (Fig. S1F). Now, consider the case of Se= 0.3; here the membrane is assumed to be composed of three parallel tubes of effective radii measured at the centres of the three intervals: Se = 0 0.1, 0.1 0.2, and 0.2 0.3, i.e at r m1, r m2, and r m3 respectively (Fig. S1F). The permeability of the membrane was then calculated using equation (S5) as: κ ( Se = 0.3) = C 2( 4 4 4 rm 1 rm 2 rm 3) + + (S7) or in general 4 κ = = C 2 r ( Se N /10) N i= 1 mi (S8) S4
where N is a natural number (1 N 10 ). The proportionality constant, C 2 was obtained by equating the permeability at Se= 1 obtained using equation (S8) to the experimentally obtained permeability at Se= 1, i.e. permeability vs saturation (κ vs Se) relation for the material. κ s. Equation (S8) was then used to generate the S2. Modelling protocol in COMSOL Figure S2. Boundary conditions for COMSOL modelling (A) 3D representation of a paper strip with length L. (B) 2D domain of the paper strip used for modelling with the relevant boundary conditions. Fig. S2A represents the schematic of a paper strip in 3D which has a variation of material properties along its length. Thus the 2D modelling domain is presented in Fig. S2B with the pertinent boundary conditions. Richard s equation in COMSOL is defined as follows: t ( ρε p).( ρu) + = Q m Where the first term represents the volume fraction of pores occupied by water (where ρε p θ C ρε p = SeS+ t t ρg t = ), and further defined as, ( ) m p, in which the first term on the right hand side describes the rate of change of storage of water content in the sample. This includes a term S, the storage coefficient, which describes the changes in the fluid storage in the porous matrix volume due to compression and expansion of pore spaces and the water when it is fully saturated. For the paper materials, this term is assumed to be zero. The S5
second term on the right-hand side includes C m = θ H P, the specific moisture capacity, which describes the change in the moisture content in the sample with the pressure head. Also, the capillary pressure and pressure heads are related by p= ψ = ρgh. Please note that the capillary pressure in COMSOL interface is the dependent variable denoted by p. Now p the linear velocity term in COMSOL is written as (Darcy s law), K u= p, where K is ρ g Κ κ the hydraulic conductivity which is related to permeability as =. Although, the ρ g µ equations in COMSOL does not show the saturation, θ, this is related to the capillary pressure and permeability through Van Genutchen correlations (Equations 6-7 in the main text), hence the user can obtain the spatiotemporal variation of ψ andθ after solving the equations. The input parameters used to solve the rate of imbibition for a NC FF120 paper strip are presented in Table S1. Table S1: Parameters for solving the flow problem for NC FF120 Variable Value Description Source ρ 1000 [kg/m 3 ] Density of fluid Common knowledge θ 0.75 Porosity of the membrane Experimentally measured s θ 0 Residual saturation Assumption r κ 9.5x10-7 [m/s] Hydraulic conductivity at s 100% saturation Experimentally measured S 0 Storage coefficient Assumption α 1 Van Genutchen parameter Calculated from experimental data n 2.66 Van Genutchen parameter Calculated from experimental data l 13.11 Van Genutchen parameter Calculated from experimental data S6
After entering the input parameters, the boundary conditions are specified at four edges as mentioned in the main text (also mentioned in Fig S2B). Meshing for the domain can be set to physics controlled mesh with extra fine setting, however since a rectangular domain is used in this case, and a mapped mesh is preferred. Further, the boundary layers were refined at the edges. The final mesh consisted of ~11000 elements with 44625 DOFs (degrees of freedom). S3. Pore Size Distribution Figure S3. Data points (black dots) representing pore size distribution as a function of r pore for (A) Nitrocellulose FF120HP (B) Glass fibre (GFDVA) and (C) Whatmann filter paper grade-1 with their corresponding curves fitted with guide to eye distribution (red lines). S7
After calculating the cumulative pore size distribution ( r pore vs Se; Fig. S1F), the fraction of pores in each size range can be calculated as f = Se( r ) Se( r ) r 1 r2 2 1 where f is the fraction of pores having a pore size in between r 1 and r 2, given r < 1 r 2. r 1 r2 Plots of pore size distribution curves obtained using this method are shown in Fig. S3, which shows that the distribution for NC FF 120 (Fig. S3A) is narrower as compared to the other two materials (Fig. S3B and C). This data is useful to get a rough estimation of the distribution of pores inside the various commercially available materials. S4. Data fitting to obtain Van Genutchen parameter Plots showing fits to experimental data for all three materials are shown below in Fig. S4-S5. Figure S4. Curve fitting to obtain Van Genutchen parameters. Experimental data points (black dots) and best fits (red lines) for capillary pressure as a function of saturation for (A) Nitrocellulose FF120HP (B) Glass fibre (GF/DVA) and (C) Whatman filter paper grade-1. The fits were obtained using MATLAB curve fitting tool using the nonlinear regression analysis where R 2 ~ 0.99. S8
Figure S5. Curve fitting to obtain Van Genutchen parameters. Experimental data points (black dots) and best fits (red lines) for the relative permeability as a function of saturation for (A) Nitrocellulose FF120HP (B) Glass fibre (GFDVA) and (C) Whatmann filter paper grade-1. The fits were obtained using MATLAB curve fitting tool using the nonlinear regression analysis where R 2 ~ 0.99. S5. Modelling and experimental measurement of imbibition and saturation COMSOL simulations were run using Van Genutchen parameters corresponding to GF/DVA and Whatman filter paper Grade 1 (Table 3). For both materials, the simulated results matched experimental data very well (Fig. S6 for GF/DVA and Fig. S7 for filter paper Grade 1). Figure S6. Spatiotemporal variation of saturation for GF/DVA: Comparison of experimental data (black markers) and COMSOL simulations (red line). S9
Figure S7. Spatiotemporal variation of saturation for Whatman filter paper 1: Comparison of experimental data (black markers) and COMSOL simulations (red line). S6. Flow imaging setup Figure S8. Flow imaging setup. Image showing the in-house constructed humidity chamber (flow chamber), humidity meter, and the webcam mounted on a stand. S10
Flow through paper membranes was visualized using the setup show in Figure S8. The approximate size of the humidity chamber was 20 cm (L) x 13 cm (W) x 4 cm (H). It was made from 5 rectangular pieces of acrylic bonded at the edges using dichloromethane; the base was left open. A humidity meter was introduced into the chamber through a hole in one of the side walls. The wall opposite to the one in which the humidity meter was introduced contained two rectangular windows that were cut out and covered with adhesive-backed silicone. Humidity in the chamber was maintained by placing wet paper towels within the chamber (not shown). After a relative humidity of 80% was reached, fluid was introduced into a small reservoir connected to one end of the paper strip by a syringe needle inserted through the silicone window. This ensured that humidity did not decrease during fluid introduction. S7. Estimation of pore size from SEM images Figure S9. Pores considered for determining pore radius of nitrocellulose. A. SEM image of NC FF120 (reproduced from main text). B. Pores marked for measuring size. Scale bar represents 10 µm. An SEM image of the surface of NC FF120 was used to estimate pore sizes. Bright regions in the image (Fig. S9A) were considered to be composed of nitrocellulose and the darker S11
regions were considered as pores. Ten different pores (Fig. S9B) were marked as regions of interest using ImageJ and their areas were calculated. Pore radii for each pore were calculated assuming the pores to be circular. The average pore size for these pores was 16 µm with a standard deviation of 10 µm. While this is a crude method of estimating pore size, the average size (16 µm) is very close to the average pore size estimated in Fig. S3A. This, to a certain extent, supports the assumption of zero contact angle made in equation S6. References 1. E. Fu, S. A. Ramsey, P. Kauffman, B. Lutz and P. Yager, Microfluid. Nanofluidics, 2011, 10, 29 35. 2. J. C. Linnes, Biomed. Microdevices, 2016, 1 12. S12