pubs.acs.org/Langmuir © 2009 American Chemical Society
Imbibition in Porous Membranes of Complex Shape: Quasi-stationary Flow in Thin Rectangular Segments Sergio Mendez,†,^ Erin M. Fenton,† Gil R. Gallegos,‡ Dimiter N. Petsev,† Scott S. Sibbett,*,† Howard A. Stone,§ Yi Zhang,† and Gabriel P. Lopez*,† †
Center for Biomedical Engineering, and Department of Chemical and Nuclear Engineering, University of New Mexico, Albuquerque, New Mexico 87131, ‡Department of Computer Science, New Mexico Highlands University, Las Vegas, New Mexico 87701, and §School of Engineering and Applied Sciences, Harvard University, Cambridge, Massachusetts 02138. ^ Current address: Department of Chemical Engineering, California State University, Long Beach, CA. Received July 8, 2009. Revised Manuscript Received October 6, 2009
The sustained liquid flow of a typical lateral flow assay can be mimicked by two-dimensional shaped, thin porous membranes, specifically rectangular membranes appended to circular sectors. In designing these fan-shaped devices, we have been aided by analytical equations and finite-element simulations. We show both mathematically and experimentally how a continuous increase in unwetted pore volume causes a deviation from traditional imbibition, and leads to quasi-stationary flow in the rectangular element. These results are both theoretically and practically important because they indicate how medical diagnostic test strips may be fabricated without incorporating an absorbent pad.
Introduction Membrane-based lateral flow immunoassay tests provide quick and low-cost detection of various important physiological analytes. Common urine-based tests include those for glucose, human chorionic gonadotropin (pregnancy hormone), and 9-tetrahydrocannabinol (pharmacological agent of marijuana). Blood-based kits include those for cholesterol, diabetes, hepatitis C, human influenza, and human immunodeficiency virus type 1. These tests are used widely in health care and home settings.1 Most lateral flow tests are performed using a nitrocellulose membrane (a thin porous film), upon which a liquid sample flows by capillary action to a reagent-containing line or spot, where analyte within the sample reacts with the reagent to produce a detectable signal. In the case of home tests to date, the signal is always colorimetric, and is interpreted by the unaided eye. The flow of sample to reagent is driven by capillary action within pores of the film. First, the membrane is contacted with an aqueous sample and held in place, thereby filling all submerged pores and creating a wetted region. For times t > 0, the liquid-air interface within the membrane migrates toward dry regions as a consequence of a surface-tension induced pressure differential at the interface. There is a qualitative analogy between such flows through porous media, and the capillary action of an array of dry, hydrophilic capillaries dipped in fluid: for each capillary in the array, curvature at the air-liquid interface creates a force that drives migration of the interface toward dry regions. In a porous membrane of constant cross-section, liquid moves according to Darcy0 s Law Æus æ ¼
ks ΔP μLc
ð1Þ
*Corresponding authors. E-mail:
[email protected],
[email protected]. (1) Warsinke, A. Anal. Bioanal. Chem. 2009, 393, 1393. (2) Bird, R. B.; Stewart, W. E., Lightfoot, E. N. Transport Phenomena, revision 2; John Wiley & Sons, Inc.: New York, 2007; p 189.
1380 DOI: 10.1021/la902470b
where Æusæ is the superficial fluid velocity,2 ks is the superficial permeability of the porous medium, 4P is the pressure difference over the length Lc of the liquid-filled region, and μ is the viscosity. Liquid flows toward dry regions, whether the interface is one among many in an array of geometrically wellordered capillaries, or in a torturous network of interconnected pores.3 The advent of medical lateral flow assays in the 1950s capped a series of developments in paper-based spot tests and chromatographic separations dating back to the 1830s.4,5 In recent years, interest in the medical utility of paper substrates has resulted in the development of two-dimensional lateral flow assays4,6 and three-dimensional microfluidics.7 The closely related technique of thin layer chromatography continues to receive significant attention,8-13 as do experimental and theoretical studies of imbibition.14-17 (3) Williams, R. J. Colloid Interface Sci. 1981, 79, 287. (4) Fenton, E. M.; Mascarenas, M. R.; Lopez, G. P.; Sibbett, S. S. ACS Appl. Mater., Interfaces 2009, 1, 124. (5) Weil, H. Colloid Polym. Sci. 1953, 132, 149. (6) Martinez, A. W.; Phillips, S. T.; Butte, M. J.; Whitesides, G. M. Angew. Chem., Int. Ed. 2007, 14, 1364. (7) Martinez, A. W.; Phillips, S. T.; Whitesides, G. M. Proc. Natl. Acad. Sci. U.S.A 2008, 105, 19606. (8) Poole, C. F. J. Chromatogr., A 2003, 1000, 963. (9) Nyiredy, Sz. J. Chromatogr., A 2003, 1000, 985. (10) Nurok, D. J. Chromatogr., A 2004, 1044, 83. (11) Berezkin, V. G.; Litvin, E. F.; Balushkin, A. O.; Roz_ yzo, J. K.; Malinowska, I. Chem. Anal. (Warsaw) 2005, 50, 349. (12) Novotny, A. L.; Nurok, D.; Replogle, R. W.; Hawkins, G. L.; Santini, R. E. Anal. Chem. 2006, 78, 2823. (13) Bakry, R.; Bonn, G. K.; Mair, D.; Svec, F. Anal. Chem. 2007, 79, 486. (14) Krishnamoorthy, S.; Makhijani, V.; Lei, M.; Giridharan, M.; Tisone, T. In Proceedings of the 2000 International Conference on Modeling and Simulation of Microsystems [Online]; Nano Science and Technology Institute: Cambridge, MA, 2000. http://www.nsti.org/publications/MSM/2000/pdf/T42.08.pdf (accessed March 4, 2009). (15) Hyv€aluoma, J.; Raiskinm€aki, P.; J€asberg, A.; Koponen, A.; Kataja, M.; Timonen, J. Phys. Rev. E 2006, 73, 036705-1. (16) Reyssat, M.; Sangne, L. Y.; van Nierop, E. A.; Stone, H. A. Europhys. Lett. 2009, 86, 56002-1. (17) Medina, A.; Perez-Rosales, C.; Pineda, A. Rev. Mex. Fı´sic. 2001, 47, 537.
Published on Web 10/21/2009
Langmuir 2010, 26(2), 1380–1385
Mendez et al.
Article
model of a porous rectangular strip, it is well-known that the wetted area covers a distance l(t): sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ks γ cos θpffiffi t lðtÞ ¼ 2 φμrm
Figure 1. Two-dimensional shapes of thin porous membranes, where ω = central angle.
Our particular interest is in lateral flow devices that are fabricated to meet the needs of users in resource-poor areas. Typically, these users want devices that are (i) low-cost; (ii) small, lightweight, and easily handled; (iii) impervious to ambient contaminants and humidity; (iv) operate without electrical power; (v) operate without special fluids such as buffer or filtered water; (vi) are not prone to operator error; and (vii) generate results in a few minutes or less. Additional attractive features are high sensitivity, multiplex capability, semiquantitative readout, and an indefinite shelf life at temperatures between 0 and 55 C. In a previous paper, we showed how lateral flow devices can be fabricated to meet the needs of all the above except high sensitivity, semiquantitation, and indefinite storage.4 Toward addressing the problem of high sensitivity, in this paper we investigate various complex shapes (Figure 1) and their effect on imbibition-driven flow. Most commercial medical lateral flow assays achieve high sensitivity by imbibing a relatively high volume of sample across one or more reaction zones. High volume flow is important for two reasons: (i) the number of analyte molecules captured at a reaction zone depends on the total volume flow of sample across the reaction zone, and (ii) thorough washing of unbound detector molecules from the reaction zone lowers the background and enhances sensitivity. Hence, sensitivity increases with increasing flow volume, up to the limit of binding saturation. Typically, the generation of large and sustained flow on medical lateral flow assays is accomplished by attaching a dry absorbent pad on the distal end of a rectangular strip of porous media such as polyester-backed nitrocellulose. When the advancing liquid front reaches the pad, the liquid exits a region only ∼100 to 200 μm thick, consisting solely of nitrocellulose, and enters a second porous region that is typically >1000 μm thick-the combined thickness of both the nitrocellulose membrane and pad. This design substantially increases the available unwetted pore volume, and results in relatively large and sustained flow across the reaction zone of the strip. The driving force for the imbibition is the capillary suction pressure Pc given by the equation Pc ¼
2γ cos θ rm
ð2Þ
where γ = surface tension, θ = contact angle of the liquid with the material, and rm = mean pore radius.18 For porous media, the capillary pressure is established by the mean pore radius, which, in the case of the nitrocellulose membranes used here, is essentially constant along the paper strip. For the simplest one-dimensional (18) Adamson, A. W. Physical Chemistry of Surfaces, 3rd ed.; John Wiley & Sons: New York, 1976; p 461.
Langmuir 2010, 26(2), 1380–1385
ð3Þ
where φ is porosity of the material and l(0) = 0. This result is known as the Lucas-Washburn equation.19 It predicts that flow velocity diminishes with increasing time, and assumes one-dimensional flow of a single-phase liquid through homogeneous material of constant cross-sectional area. Additional assumptions and a derivation are provided in the Supporting Information. In the presence of the absorbent pad, flow is sustained over time because liquid in the thin membrane (i) contacts the porous pad, (ii) imbibes into a porous space of widening cross-section, and (iii) encounters a continuous increase in unwetted pore volume as it advances. Hence, the constant cross-section assumed by LucasWashburn dynamics does not apply, causing flow to deviate from eq 3. In the research described here we present two-dimensional analogues of the absorbent pad (Figure 1). Also, we show experimentally, analytically, and numerically how a continuous increase in pore volume causes a clear deviation from LucasWashburn dynamics, namely quasi-stationary flow. Finally, we present an asymptotic analysis and the resulting key parameters governing flow in fan-shaped lateral flow devices.
Experimental Section The membranes used were Millipore Hi-Flow HF135 nitrocellulose (Millipore Corp., Billerica, MA). This membrane is composed of a thin film of porous nitrocellulose on a substrate of polyester. Membranes were cut into two-dimensional shapes (Figure 1) by a computer-controlled cutting machine.4 In some experiments, the nitrocellulose side of HF135 was capped with vinyl cover tape (gift of G&L Precision Die Cutting, Inc., San Jose, CA) to form a laminar composite. No evaporation of fluid occurs within these capped devices except along the peripheral edge where a thin layer of nitrocellulose is exposed. Capped devices are of interest because (i) they remain clean, (ii) they remain dry, (iii) evaporation is negligible, (iv) experimental results are easily obtained, (v) models do not require an evaporation term, and (vi) capped devices are probably better suited than noncapped devices for use in resource-poor areas. The reported thickness of HF-135 nitrocellulose is 135 ( 15 μm,20 hence the absolute amount of liquid lost to the ambient by evaporation from capped devices is small: we have measured it to be
