Experimental measurement of parameters governing flow rates and

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Experimental measurement of parameters governing flow rates and partial saturation in paper-based microfluidic devices Dharitri Rath, Sathishkumar N., and Bhushan J. Toley Langmuir, Just Accepted Manuscript • DOI: 10.1021/acs.langmuir.8b01345 • Publication Date (Web): 03 Jul 2018 Downloaded from http://pubs.acs.org on July 4, 2018

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Experimental measurement of parameters governing flow rates and partial saturation in paper-based microfluidic devices

Dharitri Rath1, Sathishkumar N1, Bhushan J. Toley1*

1

Department of Chemical Engineering Indian Institute of Science Bengaluru, Karnataka 560012 India

* 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: [email protected]

ACS Paragon Plus Environment

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Abstract Paper-based microfluidic devices are rapidly becoming popular as a platform for developing point-of-care medical diagnostic tests. However, the design of these devices largely relies on trial and error, owing to a lack of proper understanding of fluid flow through porous membranes. Any porous material having pores of multiple sizes contains partially saturated regions, i.e. regions where less than 100% of the pores are filled with fluid. The capillary pressure and permeability of the material change as a function of the extent of saturation. While methods to measure these relationships have been developed in other fields of study, these methods have not yet been adapted for paper for use by the larger community of analytical chemists. In the current work, we present a set of experimental methods that can be used to measure the relationships between capillary pressure, permeability, and saturation for any commercially available paper membrane. These experiments can be performed using commonly available lab instruments. We further demonstrate the use of the Richard’s equation in modelling imbibition into 2D paper networks, thus adding new capability to the field. Predictions of spatiotemporal saturation from the model were in strong agreement with experimental measurements. To make these methods readily accessible to a wide community of chemists, biologists, and clinicians, we present the first report of a simple protocol to measure the flow rates considering the effect of partial saturation. Use of this protocol could drastically reduce the trial and error involved in designing paper-based microfluidic devices.

Keywords: Paper-based microfluidics, point-of-care diagnostics, paper analytical devices, Richard’s equation, partial saturation, permeability


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Introduction Paper networks have emerged as a popular platform for the development of point-of-care diagnostic devices since they were first introduced almost a decade ago1. Compared to traditional lateral flow assays like the pregnancy strip, patterned paper devices are advantageous because they can be used for i) detecting multiple targets on a single device2, or ii) conducting highly sensitive signal-amplified assays, e.g. two-dimensional paper networks (2DPNs)3. Such devices (referred to as paper-based microfluidic devices henceforth) have gained attention because they present several advantages such as low cost of fabrication, low reagent consumption, rapid visual signal generation, and operation with minimal or no ancillary equipment3. As a result, there is an increasing interest in the development of paperbased devices with newer capabilities, e.g. implementing sensitive detection techniques4, measuring signals simultaneously from multiple targets2, developing new flow control techniques for automatic reagent delivery5, and integrating dry storage of reagents6. As the capabilities of paper-based microfluidic devices have increased, there has been a concomitant increase in the geometric complexity of the paper networks utilized7–9. Yet, the design of paper networks for these applications is currently largely based on trial and error.

Several commercially available porous substrates have been used for fabricating diagnostic devices, e.g. traditional paper (cellulose), nitrocellulose, and glass fibre. In the field of paperbased microfluidics, the term ‘paper’ has been used to refer to all such porous materials collectively. These commercially available paper membranes differ in a number of characteristics such as material of construction, porosity, pore size distribution, and surface characteristics (contact angle)10. Obviously, these parameters determine the characteristics of flow through these membranes. However, a thorough understanding of how these parameters affect flow rates seem to be lacking in the paper microfluidics community. Instruments


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required to conduct measurements of pore size distribution in porous materials are usually not available in labs that are developing point-of-care diagnostic devices. A method for measuring these material properties in-house along with a quantitative understanding of how these properties affect fluid flow through paper membranes will be extremely useful for designing multidimensional paper networks for diagnostic devices.

Fluid flow in diagnostic membranes is driven by capillary pressure generated by the porous material, which is a function of the pore size, liquid-air surface tension, and liquid-solid contact angle. The paper microfluidics community, thus far, has largely relied on two mathematical models to model flow through such membranes11. In the first model, the paper matrix is assumed to be composed of a bundle of capillary tubes and the flow inside these is modelled by the Lucas-Washburn equation11,12 (originally proposed for a single capillary), which can be used to relate the position of the wetting front with time as follows:

L2 =

γ rpore t 4µ

(1)

where L is the distance traversed by the fluid front, rpore is the average radius of the pore, γ is the effective surface tension of the fluid-air interface (including the dependence on contact angle), and µ is the dynamic viscosity of the fluid. According to this equation, the position of the fluid front in a straight (1D) porous membrane is proportional to the square root of time13,14. A major limitation of this method, however, is that it is limited to 1D domains. The second mathematical model that has been used extensively is the Darcy’s law11,15, which is a phenomenologically-derived Ohm’s law-like model that relates the pressure gradient to the average flow rate linearly as follows:

Q=−

κA ∆p ∆p = − µL µL

(

KA

)

(2)


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where Q is the volumetric flow rate,

κ is the permeability, A is the cross sectional area of

the paper strip, and ∆ p is the pressure difference over length L . Here, µ L

κA

represents the

additive resistance to flow, equivalent to electrical resistance in an electric circuit. While Darcy’s law may be used to model flow through multidimensional porous domains, its application for modelling imbibition requires the solution of a moving boundary problem, which is difficult to implement. Mendez et al have presented a comprehensive numerical solution to such a moving boundary problem16. However, because of the mathematical and computational complexity involved in the solution, this method has not been adopted by the paper microfluidics community and the use of Darcy’s law has been restricted exclusively to modelling flow in fully wet domains.

Both Washburn equation and Darcy’s law assume that a sharp fluid front exists and that the porous material behind the wetting front is fully saturated. However, from traditional models of fluid flow in porous media like soil, it is well known that when there is a variation in pore size within the medium, the advancing fluid front is partially saturated, i.e. only a fraction of the pores is filled with fluid17. A direct consequence of partial saturation is that the capillary pressure induced by the material and the permeability of the material change with the extent of saturation. In contrast to Washburn equation and Darcy’s law, the Richard’s equation can be used to describe the motion of a fluid in partially saturated porous media18–20. This has been extensively used to model fluid flow in soil21, however has largely been ignored by the paper microfluidics community. For example, when quantifying the concentration of species in a porous material11, the species transport equation is modelled without considering the effect of partial saturation. This may overestimate or sometimes underestimate the actual sample being delivered to the detection zone in case of the lateral flow assays. In order to solve the Richard’s equation for a given porous material, the capillary pressure and


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permeability of the material as a function of saturation must be known. Recently, Perez-Cruz et al demonstrated the use of the Richard’s equation in modelling imbibition through 2D shapes of filter paper19. However, the parameters relating the permeability and capillary pressure to saturation were obtained by fitting experimental data to the mathematical model because of the complexity in obtaining such parameters experimentally. Another notable model of fluid imbibition that considers partial saturation has been developed by Cummins et al22, but they have only demonstrated its utility in modelling flow in a 1D domain.

In the current work, we, for the first time, consolidated a protocol for measuring the flow rates into the diagnostic membranes considering the effect of partial saturation. The protocol includes experimental methods for measuring i) the extent of partial saturation during imbibition into commercially available paper membranes, and ii) capillary pressure and permeability of the membranes as a function of saturation. These experiments only require a centrifuge, a weighing balance, and a webcam – instruments that are commonly available in laboratories developing paper-based microfluidic devices. The relationships between capillary pressure, permeability, and saturation were further used to solve the Richard’s equation to predict flow rates and partial saturation in arbitrary 2D geometries of paper. Rates of imbibition and extents of partial saturation predicted by Richard’s equation (solved in COMSOL) accurately matched experimental measurements. The protocol presented in this paper can be used by researchers developing paper-based microfluidic devices to predict imbibition and extent of saturation in complex multidimensional shapes of paper. These methods represent a new tool for objectively designing multidimensional paper networks without the use of trial and error.


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Experimental Paper materials and characterization Three different commercially available paper materials were used. Nitrocellulose (NC FF120HP) and glass fibre (GF/DVA) were procured from Wipro GE Healthcare Pvt. Ltd., (Bengaluru, India). Whatman filter paper Grade 1 was procured from Sigma-Aldrich (Bengaluru, India). Paper strips of desired dimensions were drawn using AutoCAD (Autodesk, San Rafael, CA, USA) and then cut using a 50W CO2 laser cutter using a VLS 3.60 laser engraver (Universal Laser Systems, Scottsdale, AZ). Porosity, θ s , of each material was obtained by measuring the difference in dry and wet weight (after saturation with DI water) of 2 cm x 0.4 cm pieces of membranes. Scanning electron microscopy (SEM) was performed on gold-sputtered paper membranes using a Zeiss Ultra 55 field emission SEM (Oberkochen, Germany) at 4000x magnification.

Measurement of capillary pressure as a function of saturation Rectangular paper strips (2 cm x 0.4 cm) were saturated with DI water and placed in 200 µl centrifuge tubes with holes at their bottoms. These 200 µl centrifuge tubes were then placed in 1.5 ml centrifuge tubes, which were then placed in a centrifuge and spun at progressively higher speeds starting from a minimum of 500 RPM with increments of 100 RPM (Fig. 1). After spinning at each speed for 30 seconds, strips were removed from the centrifuge and their weights were measured, which were used to calculate the saturation, fraction of the total volume of the strip occupied by fluid (0 ≤

θ

θ , defined as the

≤ θs , where θs is the

porosity). The value of saturation is normalized by defining: Se = (θ − θ r ) (θ s − θ r ) , where θr is the residual moisture content in the strips. While the value of θr was not known, the residual moisture content was assumed to be included in the “dry weight” of the strips. The


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value of Se was thus calculated as: Se = (W − Wdry ) (Wsat − Wdry ) , where W is the weight after each spin, Wdry is the dry weight, and Wsat is the weight at full saturation. At the end of each spin, the system was assumed to be at equilibrium at which the centrifugal force pushing the fluid out of the strip is equal to the capillary pressure pulling the fluid into the strip. Thus, the capillary pressure, ψ , induced by the paper on the fluid can be calculated by equating it to the pressure induced by centrifugal force, Fc , at a given angular speed of rotation, ω , which can be calculated using the equation17: 2 2 2 Fc ∆ρω ( r2 − r1 ) ψ= = A 2

(3)

where ∆ ρ is difference in the densities of wetting (water) and non-wetting (air) fluids,

r1 and

r2 are the distances of the near and far edges of the paper strip, respectively, from the axis of rotation (Fig. 1).

Measurement of permeability at 100% saturation Permeability at 100% saturation ( κ s i.e.

κ at Se =1) was obtained experimentally by

applying the Darcy’s law over a short flow domain, assuming that the material is fully saturated for short wicking lengths. According to the Darcy’s law (Eq 2), if the flow rate, Q , of a fluid of viscosity, µ , flowing through a length, L , of a porous material of cross sectional area, A , is known, then the permeability of the material,

κ , can be calculated given

the pressure drop, ∆ p , across the porous domain. To obtain the flow rate, Q , flow was visualized through 1 cm-wide strips of the three materials laid horizontally on a flat surface. Water containing orange food colouring was introduced into each strip through a reservoir attached to its one end. Time lapse images of each strip were acquired at intervals of 1s using a Logitech webcam C525 (Logitech, Newark, CA) operated using HandyAvi (AZcendant,


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Tempe, AZ). The time required for the visible fluid front to progress 1 cm along the length of each strip was noted and used to calculate the flow rate, Q . Because flow is driven by wicking, the pressure drop, ∆ p , over this 1 cm length is the capillary pressure, ψ , induced by the material. Using available data for ψ vs Se for each material, ∆ p for this flow was calculated as a weighted average:

∆p =

∑ Se.ψ ∑ Se

(4)

where the summation was over all available experimental data points for ψ vs Se . Values of

∆ p and Q thus obtained were used in Equation (2) to calculate the permeability at 100% saturation, κ s .

Visualization of flow and measurement of saturation A setup similar to the one used to determine κ s was used. Rectangular strips, 1 cm x 12 cm, of the three paper materials were cut with a 50W CO2 laser cutter and laminated with pressure sensitive adhesive (PSA; 3M double sided tape 9731) on one side. The PSA was graduated with marks every 1 cm made using the laser cutter. Water containing orange food dye was introduced into each strip using a reservoir connected to one end and time-lapse images were acquired every 10s using a Logitech webcam. Experiments were conducted inside a custom-built humidity chamber with a relative humidity of 80% or above (maintained by introducing wet paper towels in the chamber) to minimize evaporation. The humidity chamber was built in-house and consisted simply of 5 sheets of transparent acrylic bonded at their edges using dichloromethane (See ESI; Section S6). A relative humidity probe was inserted into the chamber through a drilled hole. Flow was stalled at a predetermined time point by disconnecting the fluid reservoir. The disconnected strip was cut immediately at the graduations and the weight of each 1cm-long segment was measured.


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These weights were used to calculate the saturation of the strip as a function of distance from the fluid source. The procedure was then repeated for multiple time points and three materials. At least three replicates were conducted at each time point and the error bars represent one standard deviation around the mean values. A similar method was used to visualize and acquire time-lapse images of flow through 2D shapes of NC FF120.

Theoretical Determination of permeability as a function of saturation 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). A detailed procedure is presented in the Electronic Supporting Information (ESI; Section S1). Briefly, the paper strip is assumed to be composed of a set of parallel capillaries of equal length and varying diameters. A comparison of Darcy’s law and Hagen-Poiseuille law applied to this system shows that permeability, κ , scales as the square of pore radius for a single capillary, as previously shown by Cummins et al22. Further, assuming a set of parallel capillaries to act as a set of parallel fluidic resistors, κ can be calculated as a function of Se by making the assumption that the number of parallel capillaries increase as Se increases (See ESI; Section S1).

Modelling fluid imbibition using Richard’s equation The modelling domain was a rectangle, 1 cm x 10 cm, representing paper strips used to monitor flow and measure saturation experimentally. Richard’s equation19,20, which describes the motion of liquid in porous materials, can be written as:

 κ (θ )  ∂θ ∂ψ = ∇.  ∇(ψ (θ ) + ρ gz )  ∂ψ ∂t  µ 


(5)

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where ρ gz is the gravitational head, which can be neglected for paper strips placed on horizontal surfaces. Equation (5) was solved for ψ as a function of time and space numerically in COMSOL Multiphysics 5.1 using the “Richards’s equation” interface in the ‘Subsurface flow’ module. The functional relationship, ψ (θ ) , obtained using experiments can then be used to back calculate θ as a function of space and time.

Equation (5) is notoriously difficult to solve because of its nonlinear nature, given that both

ψ (θ ) and κ (θ ) are nonlinear functions. Several correlations have been proposed to describe the functional relationships, ψ (θ ) and κ (θ ) 20,23. One such set of relations that has been used extensively is called the Van Genutchen formulation24, a standard notation of which is represented by the following equations:

Se = [1+[α H p ]n ]−m

(6)

κr = Sel [1 − (1 − Se1/ m )m ]2

(7)

where Se is normalized saturation, H p =

ψ , and κ r = κ . The parameters, α and n , can κs ρg

be obtained by fitting the available data for

ψ

vs Se to equation (6) and the parameter, l ,

can be obtained by fitting data for κ vs Se to equation (7). Data were fit using the ‘Curve Fitting Tool’ in MATLAB using nonlinear least squares regression analysis. Note that m and n are related as m = 1 −

1 . n

The following initial and boundary conditions were used for the solution of equation (5). The edge of the domain connected to the fluid reservoir (along the width) was assumed to be fully saturated, i.e. Se = 1 . A Dirichlet boundary condition, ψ edge = ψ ( Se = 1) , was therefore set for this edge using the available experimental data for ψ ( Se) . A non-flux Neumann boundary


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condition was set for the other three edges because fluid could not escape out of the strip from those edges. The selection of initial condition for this problem required some thought. Because the domain is dry initially, the normalized saturation, Se , equals zero. It is tempting to set the initial condition as ψ init = ψ ( Se = 0) by extrapolating the available experimental data for ψ ( Se) . However, this would be incorrect because the initial condition must hold for the entire rectangular domain over which the equation is being solved. As Se decreases and nears zero, this represents a condition where a very small fraction of the pores are filled with fluid. The capillary pressure associated with such a small fraction of pores cannot be set as an initial condition for the entire domain. Instead, the domain can be represented by an average capillary pressure as defined in Equation (4). The initial condition for ψ was thus set to:

ψ init = ∑

Seψ

(8)

∑ Se

A consequence of using this initial condition is that the domain is artificially set to a non-zero initial value of Se = Seinit , because the functional relationship ψ ( Se) is fixed and given as an input to the model. In order to extract true values of saturation that match experimental data,

Se values obtained after solving the Richard’s equation were rescaled as: Sescaled =

Se − Seinit 1 − Seinit

(9)

Values of Sescaled thus obtained were used to compare with experimentally obtained values of normalized saturation, Se . A detailed procedure for COMSOL simulations is provided in the Electronic Supporting Information (ESI; Section S2).

Results and discussion Characterization of paper materials


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The basic characterization of paper materials included measurement of porosity and acquisition of SEM images. Porosity values measured for NC FF120, GF/DVA, and Grade 1 filter paper were 0.75, 0.9, and 0.7, respectively. NC FF120 had similar porosity as that of Whatman Grade 1 filter paper whereas GF/DVA had a higher porosity. These numbers were corroborated by SEM images of the three materials (Fig. 2) – compared to NC FF120 (Fig. 2A) and Grade 1 filter paper (Fig. 2C), GF/DVA (Fig. 2B) appears to be more loosely packed with more volume available for fluid absorption. Further, an SEM image of NC FF120 was used to roughly estimate the average pore size (ESI Section S7).

Capillary pressure measurements The capillary pressure,ψ , as a function of normalized saturation, Se , was measured using equations (3). For all three materials, ψ decreased with increasing Se , as expected. The relationships were nonlinear (Fig. 3A-C) and the trends for ψ vs Se matched those measured for other porous materials20,23,25. These curves provide the following information: let us assume a point at Se = 0.3 for NC FF120 (dotted lines; Fig. 3A), for which the value of ψ is ~175 kPa. This means that the smallest 30% of the pores of the material are filled with fluid and they induce an average capillary pressure of ~175 kPa. The maximum capillary pressures induced by nitrocellulose (Fig. 3A) and filter paper (Fig. 3C) were above 600 kPa, whereas by GF/DVA (Fig. 3B) was less than 100 kPa. This is a general trend observed in most diagnostic membranes, i.e. more porous membranes tend to have larger pores and hence lower capillary pressures. Note that for Whatman filter paper, it was not possible to obtain data for Se < 0.3 because the maximum speed on the centrifuge had been reached and no further fluid could be centrifuged out of the membrane. These ψ vs Se relationships can further be utilized to calculate an approximate pore size distribution in these membranes (ESI Section S3). While the centrifugation technique used here for capillary pressure


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measurements has been used for similar measurements of soil earlier26, to the best of our knowledge, this is the first use of this technique to calculate capillary pressure vs saturation relationships for diagnostic membranes. Although other methods for making such measurements exist, e.g. long-column method20, and quasi-steady microscale modelling27, the method proposed here is comparatively simple as it only requires a centrifuge and a weighing machine.

Permeability measurements Permeability values at 100% saturation ( κ s , at Se = 1 ) for NC FF120, GF/DVA, and Grade 1 filter paper were 9.4 x 10-14, 1.5 x 10-12, and 1.7 x 10-13 m2, respectively. It may be observed that the κ s of GF/DVA is an order of magnitude higher than that of nitrocellulose and filter paper; κ s for the latter two materials are of similar orders of magnitude. This is in accordance with capillary pressure measurements (Fig. 3A-C), which show lower capillary pressures for GF/DVA compared to the other two materials. A lower capillary pressure suggests higher pore sizes, and hence higher permeability.

Permeability, κ , as a function of normalized saturation, Se , was further calculated using Equation (S8; ESI). For all three materials, the permeability increased with Se (Fig. 3D-F; logarithmic Y-axis is used for ease of visualization) and the relationship was nonlinear. This can be interpreted as follows: at any value of Se < 1, say Se = 0.3, only 30% of the pore volume is filled with fluid and hence available for flow. In addition, this 30% volume comprises of the smallest available pores. As Se increases, a larger fraction of the pores as well as larger sized pores become available for flow. As a result, more parallel paths are available for flow, which reduces the net resistance, amounting to an increase in permeability.


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The trends observed for κ vs Se (Fig. 3D-F) matched those reported previously by Jaganathan et.al20, measured using digital volumetric imaging (DVI).

Parameter estimation by data fitting Van Genutchen parameters, n , α , and l , calculated by fitting data to equations (6) and (7) are shown in Table 1. These parameters were used for solving the Richard’s equation to model imbibition of fluid into the three materials. Plots showing fits to experimental data for all three materials are provided in the ESI (Section S4).

Modelling and experimental measurement of imbibition and saturation The movement of the visible wetting front of a coloured fluid imbibing into a paper strip laid on a horizontal surface is shown as a movie in the Supporting Information (ESI; Movie S1). This flow was modelled by solving the Richard’s equation in COMSOL (ESI; Movie S2). A comparison of snapshots of flow in 1 cm-wide, 10 cm-long strips, and the corresponding results from the mathematical model shows strong agreement for all three materials (Fig. 4; heat maps show values of Sescaled using Eq. 13). For example, for NC FF120, the experimentally observed fluid front at 10 min lies at ~ 7 cm (Fig. 4A). By visual comparison, the fluid front predicted by the model, defined by the location at which Sescaled reaches zero, also lies at ~ 7 cm (Fig. 4A). Similar observations are made for GF/DVA (Fig. 4B) and filter paper (Fig. 4C). For filter paper, the apparent discrepancy between experiments and model results (Fig. 4C) may be due to lack of a coloured dye, but other experiments confirmed that the extent of partial saturation were predicted correctly for this material (ESI Fig. S7). Note that even though the strip is 2D in this case, theoretically there is no variation in capillary


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forces along the width of the strip, and so this could be considered a 1D model. Slight variations in the location of the fluid front along the width of the strip in Fig. 4A, 4B can be attributed to edge effects arising from the damages inflicted at the edges by the method of cutting used. Further, estimation of the location of the fluid front in a 1D domain could easily have been accomplished by using the Washburn equation (Eq. 1). The real value of using Richard’s equation is in its ability to estimate partial saturation as a function of space and time, as displayed in the heat maps in Fig. 4.

A comparison of experimentally measured partial saturation values and those predicted by the COMSOL model was further made at different time points for the three materials. For nitrocellulose, saturation at different locations on the 10 cm strip were measured at intervals of 2 min (Fig. 5). It was observed that at 5 min (Fig. 5A), the strip was fully saturated only up to a length of ~2 cm and partially saturated up to a length of ~7 cm. The extent of saturation increased as time progressed, and at the end of 15 min (Fig. 5F), Se at the end of the strip was around 30% wherein the fully saturated portion spans up to a length of around 3 cm out of 10 cm. The reason for partial saturation in such kind of porous membranes is the nonuniform pore size distribution – a fraction of pores having smaller radii exerts larger capillary pressure and gets filled with fluid, whereas the larger pores exert a lower pressure and remain empty. This phenomenon of a large length of a paper strip being partially saturated has largely been ignored by the paper microfluidics community. It has been common assumption that all pores in the paper up until the apparent fluid front are filled with fluid.

Use of the Richard’s equation to model fluid flow through commercially available diagnostic membranes represents a new tool to predict not only the location of the fluid front but also the extent of partial saturation. COMSOL simulations were used to model flow through the


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NC FF120 strips and the saturation levels predicted by the model as a function of time matched experimental measurements extremely well (Fig. 5). Similar experiments were performed on 10 cm-long strips of GF/DVA and Whatman filter paper Grade 1; time intervals for obtaining saturation data for GF/DVA and filter paper were 0.5 min and 2 min, respectively. Because the flow rate of fluid through GF/DVA was higher than in the other two materials, more frequent measurements were made. COMSOL simulations were run using Van Genutchen parameters corresponding to each material (Table 1). For both materials, the simulated results matched experimental data very well (ESI Section S5).

We acknowledge that while for GF/DVA and Whatman filter paper 1, Van Genutchen parameters used to solve the model were exactly as experimentally measured and shown in Table 1, for nitrocellulose, the parameter α had to be changed to a value of 1 from its measured value of 0.1 for the results to match experimental data accurately. Similar experiments will need to be conducted on a library of materials to determine whether mismatch in parameter α is an exception or the rule. However, for two out of the three materials we used in this study, measured parameters produced simulation results that matched experimental results accurately. We are currently building a library of materials in our lab and plan to conduct a larger study as a part of our future research.

Imbibition into complex 2D geometries The method was then validated for modelling imbibition in 2D geometries. Three different 2D shapes of NC FF120 were selected inspired by the work reported by Mendez et al.16 These shapes comprise of 1D paper strips that are connected to 2D fans of various arc lengths (Fig. 6). Three such fan-shaped geometries with central angles 90°, 180°, and 270° were selected. Mendez et al used a moving boundary Darcy’s law solution to predict the location


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of the fluid front as a function of time in such geometries. Though their simulation results matched the experimental data well, the method was complex and required some mathematical training. Here, we demonstrate the applicability of the Richard’s equation in obtaining the location of the fluid front as a function of time as well as in predicting the extent of partial saturation. Solution of this problem in COMSOL makes it accessible to a wide research community with minimal mathematical training. Time lapse images of coloured fluid introduced through reservoirs into the three different shapes were acquired. The location of the fluid front along the centre of the shape (dotted lines, insets; Fig. 6) was recorded over time. The COMSOL model was then used to model flow for all three shapes. The location of the fluid front was extracted from the model, defined as the locus of all points at 20% saturation. Predictions of the model matched experimental results well for all three shapes (Fig. 6). In addition, the simulations could be used to predict the saturation levels over the 2D domain (Fig. 6C; inset). It should be understood that by using a different strategy, the domains in Fig. 6 could be considered a combination of two 1D domains, i.e. by using cartesian coordinates in the rectangular strip and radial coordinates in the circular portion. However, here we have used finite element modelling in COMSOL, which treats these domains as true 2D domains, and hence our method can be used for any 2D domain.

The methods presented in this paper can be consolidated into a protocol to model imbibition of fluid into any paper material of arbitrary 2D shape. In theory, the method can be used to model flow in 3D domains as well; in COMSOL, extension to 3D domains will be trivial. The protocol involves the following steps: Step 1: Conduct experiments to measure capillary pressure of the paper as a function of saturation (Fig. 3A-C); Step 2: Conduct experiments to measure the permeability of paper at 100% saturation; Step 3: Calculate the permeability of the membrane as a function of saturation using equation S8 (ESI) (Fig. 3D-F); Step 4: Fit the


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relationships between capillary pressure, permeability, and saturation to the Van Genutchen formulations (Eqs 10-11) to obtain parameters α , n , m , and l (Table 1); and Step 5: Model fluid imbibition in COMSOL Multiphysics 5.1 using the “Richards’s equation” interface in the ‘Subsurface flow’ module (Fig. 4-6). A word of caution is that the PSA attached to the bottom of paper strips in this method could alter flow rates at the paper-PSA interface (except for NC FF120, which is prelaminated from one side) but the results of our model matched experimental data well, so we conclude that the effects were negligible. Slight delamination of the PSA was observed for Whatman filter paper Grade 1. Attention should be give to the integrity of this lamination while adopting this protocol for other materials.

The use of paper based microfluidic devices for developing point-of-care diagnostics has garnered a lot of attention in recent years. A few notable types of devices that have been used are lateral flow assays, microPADs, and 2DPNs. In addition several devices based on more complex 3D geometries have also been developed2,8,14. In contrast to traditional PDMS-based microfluidic devices, a critical feature of these paper-based microfluidic devices is imbibition, i.e. the presence of a moving fluid front. A formal protocol for measuring imbibition into such complex shapes has been lacking because of i) the lack of appreciation of the phenomenon of partial saturation in porous media, ii) the mathematical complexity in modelling fluid flow under such a scenario, and iii) the lack of available data on the detailed material properties required to realistically model imbibition. In this paper, we have addressed all three points. We have shed light on how different pore sizes in a paper material lead to partial saturation and varying capillary pressures and permeability, and designed experiments to measure capillary pressure and permeability as a function of saturation for any paper material. Finally, we have shown how these measurements can enable precise modelling of imbibition of fluids into complex geometries using COMSOL, a modelling tool


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that is now rapidly becoming accessible to many. We believe that these methods bridge a large gap that has existed in the design of paper networks for diagnostics.

Conclusion In contrast to what most literature on the topic of paper-based microfluidics suggests, a paper material having pores of multiple sizes cannot be represented by a single capillary pressure and permeability. In fact, these properties of the material change depending on the extent of moisture content (saturation) in these materials. The different methods presented in this paper constitute a novel protocol to: a) measure the relationships between capillary pressure, permeability, and the moisture content, b) measure the flow rate and partial saturation, and c) model fluid imbibition and predict the spatiotemporal variation of the moisture content in a multidimensional paper membrane with an advancing fluid front. These protocols can easily be carried out in a moderately-equipped laboratory. The strong match between experimental and simulation results demonstrated in this paper validate the method as an important tool in designing paper-based microfluidic devices. A notable result from our studies is that a large fraction of pores in a paper strip that appears saturated to the naked eye, might in reality be lacking fluid, i.e. filled with air. This phenomenon might have a significant effect on the sensing chemistries conducted in such strips, e.g. in lateral flow assays. Our methods provide a tool to understand these phenomena better and ultimately design better and more sensitive paper-based diagnostic devices.


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Supporting Information Calculations performed to obtain permeability as a function of saturation; the modelling protocol used in COMSOL describing input parameters and other details; calculation of pore size distribution from the capillary pressure vs. saturation data; plots showing the fits to the experimental data to obtain Van Genutchen parameters; modelling and experimental measurement of imbibition and partial saturation for GF/DVA and Whatman filter paper 1; details of the in-house imaging setup used for flow measurements; methodology used for estimation of pore size from SEM images.

Conflict of interest We have none to declare.

Acknowledgement This work was supported by the Department of Science and Technology (DST), India, in the form of an Extramural Research Grant (EMR) to BT and a National Postdoctoral Fellowship (NPDF) to DR; and by the Indian Institute of Science (IISc Bangalore) in the form of a generous start-up grant to BT.


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Table 1: Van Genutchen parameters

Parameters

NC FF120

GF/DVA

Whatman filter paper

n

2.66

2.05

1.30

α

0.10

1

1

l

13.11

1.50

0.03


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Figure 1. Schematic of the setup used for capillary pressure measurements. The paper material was placed inside a small centrifuge tube with a hole at the bottom, which was then placed inside a larger centrifuge tube.

Figure 2. SEM images of the surfaces of three paper materials (A) NC FF120, (B) GF/DVA and (C) Whatman Grade 1 at 4000x magnification. Scale bars represent 10 µm.


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Figure 3. Capillary pressure (A-C) and permeability (D-F) as a function of saturation for NC FF120 (A,D); GF/DVA (B,E) and Whatman Grade 1 filter paper (C,F). A schematic of the cross section of the flow channel is shown above (A). Error bars represent one standard deviation (N=3).

Figure 4. Flow measurements and simulations. Comparison of experimentally observed and COMSOL-modelled fluid distribution in 1 cm-wide, 10 cm-long strips of A. NC FF120, B. GF/DVA, and C. Whatman filter paper 1. Model results match experimental observations well.


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Figure 5. Spatiotemporal variation of saturation for NC FF120. Comparison of experimental data (black markers) and COMSOL simulations showing Sescaled (red line) for different time points. Error bars represent one standard deviation (N=3).

Figure 6. Modelling imbibition in complex 2D geometries. A comparison of the location of the fluid front along the central dotted line measured experimentally (black markers) and predicted using COMSOL simulations (red markers) for three different 2D geometries of NC FF120. The central angles of the arcs are 90° (A), 180° (B), and 270° (C). Insets show shapes of the domain. Simulated saturation values, Sescaled , for the 270° fan are shown in the inset in (C). Error bars represent one standard deviation (N=3; note that error bars are smaller than the size of the marker and not visible).


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TOC Graphic


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A protocol for measuring capillary pressure and permeability and using those to mathematically model fluid imbibition into paper microfluidic networks. 77x40mm (300 x 300 DPI)


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Experimental measurement of parameters governing flow rates and

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