Abstract
The prediction of the imbibition into two-dimensional geometries is extremely important to develop new paper-based microfluidics design principles. In this regard, a two-dimensional model using Richard’s equation, which has been extensively applied in soil mechanics, is applied in this work to model the imbibition into paper-based networks. Compared to capillary-based models, the developed model is capable of predicting the imbibition into two-dimensional domains. The numerical solution of the proposed model shows a good agreement with the experimental measurements of water imbibition into different chromatography paper-based designs. It is expected that this framework can be applied to develop new design rules for controlling the flow in paper-based microfluidics devices.
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References
Barry DA, Parlange JY, Sander GC, Sivapalan M (1993) A class of exact-solutions for Richards equation. J Hydrol 142:29–46. doi:10.1016/0022-1694(93)90003-r
Barry DA, Wissmeier L, Parlange JY, Sander GC, Lockington DA (2009) Comment on “An analytic solution of capillary rise restrained by gravity” by N. Fries and M. Dreyer. J Colloid Interface Sci 338:293–295. doi:10.1016/j.jcis.2009.06.015
Cai JC, Perfect E, Cheng CL, Hu XY (2014) Generalized modeling of spontaneous imbibition based on Hagen–Poiseuille flow in tortuous capillaries with variably shaped apertures. Langmuir 30:5142–5151. doi:10.1021/la5007204
Camporese M, Ferraris S, Putti M, Salandin P, Teatini P (2006) Hydrological modeling in swelling/shrinking peat soils. Water Resour Res. doi:10.1029/2005wr004495
Cate DM, Adkins JA, Mettakoonpitak J, Henry CS (2015) Recent developments in paper-based microfluidic devices. Anal Chem 87:19–41. doi:10.1021/ac503968p
Chen X, Dai Y (2015) An approximate analytical solution of Richards’ equation. Int J Nonlinear Sci Numer Simul 16:239–247. doi:10.1515/ijnsns-2015-0034
Elizalde E, Urteaga R, Berli CLA (2015) Rational design of capillary-driven flows for paper-based microfluidics. Lab Chip 15:2173–2180. doi:10.1039/c4lc01487a
Fries N, Dreyer M (2008) An analytic solution of capillary rise restrained by gravity. J Colloid Interface Sci 320:259–263. doi:10.1016/j.jcis.2008.01.009
Fu EL, Ramsey S, Kauffman P, Lutz B, Yager P (2011) Transport in two-dimensional paper networks. Microfluid Nanofluid 10:29–35. doi:10.1007/s10404-010-0643-y
Hall C (2012a) Comment on “Imbibition in layered systems of packed beads” by Reyssat M. et al. Epl. doi:10.1209/0295-5075/98/56003
Hall C (2012b) Comment on source-like solution for radial imbibition into a homogeneous semi-infinite porous medium. Langmuir 28:8587. doi:10.1021/la3009305
Hall C, Green K, Hoff WD, Wilson MA (1996) A sharp wet front analysis of capillary absorption into n-layer composite. J Phys D Appl Phys 29:2947–2950. doi:10.1088/0022-3727/29/12/002
Hamraoui A, Nylander T (2002) Analytical approach for the Lucas–Washburn equation. J Colloid Interface Sci 250:415–421
Hayek M (2015) An analytical model for steady vertical flux through unsaturated soils with special hydraulic properties. J Hydrol 527:1153–1160. doi:10.1016/j.jhydrol.2015.06.010
Hayek M (2016) An exact explicit solution for one-dimensional, transient, nonlinear Richards’ equation for modeling infiltration with special hydraulic functions. J Hydrol 535:662–670. doi:10.1016/j.jhydrol.2016.02.021
He Y, Wu Y, Fu J-Z, Wu W-B (2015) Fabrication of paper-based microfluidic analysis devices: a review. Rsc Adv 5:78109–78127. doi:10.1039/c5ra09188h
Huinink H (2016) Fluids in Porous Media. Morgan & Claypool Publishers, San Rafael, CA. doi:10.1088/978-1-6817-4297-7
Kalinka F, Ahrens B (2011) A modification of the mixed form of Richards equation and its application in vertically inhomogeneous soils. Adv Sci Res 6:123–127. doi:10.5194/asr-6-123-2011
Liana DD, Raguse B, Gooding JJ, Chow E (2012) Recent advances in paper-based sensors. Sensors 12:11505–11526. doi:10.3390/s120911505
Lucas R (1918) Ueber das Zeitgesetz des kapillaren Aufstiegs von Flüssigkeiten. Kolloid-Zeitschrift 23:15–22. doi:10.1007/bf01461107
Masoodi R, Tan H, Pillai KM (2012) Numerical simulation of liquid absorption in paper-like swelling porous media. Aiche J 58:2536–2544. doi:10.1002/aic.12759
Medina A, Perez-Rosales C, Pineda A, Higuera FJ (2001) Imbibition in pieces of paper with different shapes. Revista Mexicana De Fisica 47:537–541
Mendez S et al (2010) Imbibition in porous membranes of complex shape: quasi-stationary flow in thin rectangular segments. Langmuir 26:1380–1385. doi:10.1021/la902470b
Mullins BJ, Braddock RD (2012) Capillary rise in porous, fibrous media during liquid immersion. Int J Heat Mass Transf 55:6222–6230. doi:10.1016/j.ijheatmasstransfer.2012.06.046
Nery E, Kubota L (2013) Sensing approaches on paper-based devices: a review. Anal Bioanal Chem 405:7573–7595. doi:10.1007/s00216-013-6911-4
Nia SF, Jessen K (2015) Theoretical analysis of capillary rise in porous media. Transp Porous Media 110:141–155. doi:10.1007/s11242-015-0562-1
Philip JR, Ven Te C (1969) Theory of infiltration. In: Advances in hydroscience, vol 5. Elsevier, pp 215–296. doi:http://dx.doi.org/10.1016/B978-1-4831-9936-8.50010-6
Praticò FG, Moro A (2008) Flow of water in rigid solids: development and experimental validation of models for tests on asphalts. Comput Math Appl 55:235–244. doi:10.1016/j.camwa.2007.04.014
Reyssat M, Courbin L, Reyssat E, Stone HA (2008) Imbibition in geometries with axial variations. J Fluid Mech 615:335–344. doi:10.1017/s0022112008003996
Reyssat M, Sangne LY, van Nierop EA, Stone HA (2009) Imbibition in layered systems of packed beads. Epl. doi:10.1209/0295-5075/86/56002
Richards LA (1931) Capillary conduction of liquids through porous mediums. J Appl Phys 1:318–333. doi:10.1063/1.1745010
Washburn EW (1921) The dynamics of capillary flow. Phys Rev 17:273
Wijaya M, Leong EC, Rahardjo H (2015) Effect of shrinkage on air-entry value of soils. Soils Found 55:166–180. doi:10.1016/j.sandf.2014.12.013
Witelski TP (2003) Intermediate asymptotics for Richards’ equation in a finite layer. J Eng Math 45:379–399. doi:10.1023/a:1022609020200
Xiao JF, Stone HA, Attinger D (2012) Source-like solution for radial imbibition into a homogeneous semi-infinite porous medium. Langmuir 28:4208–4212. doi:10.1021/la204474f
Zhmud BV, Tiberg F, Hallstensson K (2000) Dynamics of capillary rise. J Colloid Interface Sci 228:263–269. doi:10.1006/jcis.2000.6951
Acknowledgements
This work was partially supported by the National Council of Science and Technology of Mexico (CONACYT) through the scholarship 312230.
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Perez-Cruz, A., Stiharu, I. & Dominguez-Gonzalez, A. Two-dimensional model of imbibition into paper-based networks using Richards’ equation. Microfluid Nanofluid 21, 98 (2017). https://doi.org/10.1007/s10404-017-1937-0
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DOI: https://doi.org/10.1007/s10404-017-1937-0