Imbibition in Porous Membranes of Complex Shape: Quasi-stationary Flow in Thin Rectangular Segments

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S. Mendez, E. M. Fenton, G. R. Gallegos, D. N. Petsev, S. S. Sibbett, H. A. Stone, Y. Zhang, and G. P. Lopez

"Imbibition in Porous Membranes of Complex Shape: Quasi-stationary Flow in Thin Rectangular Segments"

Langmuir 26 (2), 1380-1385 (2010)

Entry by Meredith Duffy, AP 225, Fall 2011

Keywords: capillary pressure, imbibition, porosity, Lucas-Washburn equation, finite element method


Paper-based devices are highly useful for simple medical diagnostics, such as pregnancy and drug tests, as well as for scientific assays in general in resource-poor areas due to their low cost, ease of production and use, minimal resource requirements, powerless operation, and rapidity of results generation. Lateral flow assays such as porous rectangular nitrocellulose membranes are by far the most popular type, drawing analyte-containing solution from the dipped end to the reagent zone at the distal end by capillary action. However, achieving high sensitivity with these devices can be difficult and is typically resolved simply by using a large fluid volume. Greater sample volume both increases the amount of analyte present to bind the (usually) colorimetric reagent at the reaction zone as well as washes away more of the unbound reagent, serving to simultaneously enhance signal and reduce background noise. When increasing sample volume is not an option, this steady flow across the reaction zone can still be maintained through the addition of an absorbent pad at the far end of the strip, which works by increasing the pore volume available for the advancing liquid front to wet, thereby increasing the capillary suction pressure driving its flow and counteracting the continual decrease in fluid velocity with time exhibited for a constant paper thickness.

Here, Mendez et al. present fan-shaped membrane geometries as a two-dimensional alternative to the thick absorbent pad. Several geometries (Figure 1) are contemplated, and the 270 degree fan in particular is explored experimentally, analytically and numerically to investigate the quasi-static flow achievable with this shape.

Methods and Results

The authors cut membranes made of thin films of porous nitrocellulose with a polyester backing into various shapes using a computer-controlled machine. The membranes were rouhly 135 microns thick, and some were capped with vinyl cover tape on the nitrocellulose side to eliminate evaporation except around the device edges, which they deemed negligible. Uncapped devices were tested in humidity-controlled environments.

The first assays used simply involved dipping a membrane in a dish of food coloring, holding it out of solution, and measuring the time it took for the liquid front to reach markers drawn on the membrane. Imbibition is driven by capillary suction pressure, P_c:
where gamma is surface tension, theta is the liquid-solid contact angle, and r_m is mean pore radius. For a fan-shaped membrane, the authors define two phases of imbibition based on the location of the advancing fluid front. In Phase 1, before the front reaches the circular sector of the fan, it must travel up the rectangular section. The authors assert that during this period, flow is governed by the Lucas-Washburn equation below (Eqn 3), in which the distance l(t) covered by the front scales as the square root of time:

where k_s is the superficial permeability of the porous medium, phi is porosity and mu is viscosity.

In Phase 2, however, when the front reaches the fan, it distributes itself radially and accesses a continuously increasing pore volume, creating quasi-stationary flow within the rectangular segment. This deviates from Lucas-Washburn flow, unsurprisingly since their equation is intended for fixed cross-sectional area. To demonstrate this phenomenon, the authors alternately dipped a fan with a 270-degree circular sector in red-dyed water and pure water, dipping once every 30 seconds and taking an image after every 60 seconds (Figure S2). Both phases of imbibition can be qualitatively observed in the images. From frame 0 to 3, fluid velocity slows and band width decreases. But starting with frame 4, three stripes of each color remain on the rectangular portion of the fan in each image, appearing never to move position due to the almost-constant velocity in that region (nor alternate position, since the picture is only taken after dipping in red).


To quantify this data, the authors traced the vertical displacement of each band's centerpoint over time and estimated the velocity of each band as it reached heights of 2cm and 3.3cm (near the middle and at the top of the rectangular portion, respectively). From this they plotted fluid velocity at those two heights as a function of time. To compare the results to a finite-element simulation, the authors employed COMSOL with a triangular mesh and arbitrary Lagrangean-Eulerian methods to simultaneously solve Darcy's Law and the mass balance equation (Eqns 4 and 5). Darcy's Law describes fluid velocity v as a function of the pressure gradient in the liquid-filled region, the viscosity, and the inerstitial permeability k_i = k_s/phi. In the mass-balance equation, rho is fluid density and F is an evaporative term set to zero for capped devices and computed from Knudsen's equation for uncapped devices.

Unfortunately, experimental limitations prevented collection of data on short time scales, so Phase 1 agreement between experiment and theory was not determinable. Phase 2 agreement was less than stellar (Figure 5), as the experimental data exhibited several deviations from the almost zero-slope computed lines, but the idea of quasi-stationary flow is apparent ("quasi" perhaps emphasized, since although the slopes are roughly zero the flow at P6, or 2cm height, is faster than that at P7, or 3.4 cm) and moreover, deviation from the power-law decay of the Lucas-Washburn equation (curve c) is highly apparent. Finally, the authors provide a mathematical analysis deriving a formula for the quasi-steady flow rate q as a function of capillary pressure and the dimensions (d=width, L_r=length) of the rectangular membrane segment:


Although they unfortunately do not graph this equation with the computational and experimental data, the authors calculate a velocity of 1.6 cm/min for the fan geometry above and a rectangular section length of 3.1 cm, which does compare fairly well to the experimentally observed 1.8 cm/min.


The authors accomplished what they set out to do, namely to provide an alternative to the thick absorbent pads appended to paper test strips to maintain a steady flow. That said, it is unclear what advantages their fans geometry have, given that in a practical sense whatever decrease in thickness they achieved, they more than compensated for with width and a fairly inconvenient shape for packing and storing. Nonetheless, their design is clever in its simplicity and ease of production, and their efforts to tie together experiments, computation and mathematical analysis are commendable. Generating more (or better) supporting data in the future may aid the authors in proving their model, as well as in finding ways to improve further upon their paper-based diagnostic technologies.