Imbibition of Newtonian Fluids in Paper-like Materials with the Infinitesimal Control Volume Method

Abstract

Paper-based microfluidic devices are widely used in point-of-care testing applications. Imbibition study of paper porous media is important for fluid controlling, and then significant to the applications of paper-based microfluidic devices. Here we propose an analytical approach based on the infinitesimal control volume method to study the imbibition of Newtonian fluids in commonly used paper-like materials. Three common paper shapes (rectangular paper strips, fan-shaped and circular paper sheets) are investigated with three modeling methods (corresponding to equivalent tiny pores with circle, square and regular triangle cross section respectively). A model is derived for liquid imbibition in rectangular paper strips, and the control equations for liquid imbibition in fan-shaped and circular paper sheets are also derived. The model is verified by imbibition experiments done using the mixed cellulose ester filter paper and pure water. The relation of imbibition distance and time is similar to that of the Lucas-Washburn (L-W) model. In addition, a new porosity measurement method based on the imbibition in circular paper sheets is proposed and verified. Finally, the flow rates are investigated. This study can provide guidance for the design of different shapes of paper, and for better applications of paper-based microfluidic devices.

Keywords: imbibition; infinitesimal control volume method; microfluidics; paper-based microfluidic devices; paper-like material.

Figure 1

(a) Imbibition schematic of three shapes of paper: rectangular paper strips, fan-shaped paper sheets and circular paper sheets. (b) Schematic of the infinitesimal control volume. (c) A large number of small circles, squares or regular triangles on the cross section of the infinitesimal control volume, which corresponds to three modeling methods. n and N₁ are the number of small circle layers in the thickness and width direction respectively.

Figure 2

The scanning electron microscope (SEM) results of the mixed cellulose esters (MCE) paper surface. The left and right images have a magnification of 2000 and 10,000 times, respectively.

Figure 3

(a) Schematics of the experiment and the initial condition (t = 0 s). (b–d) shows the imbibition area of a rectangular paper strip, a fan-shaped paper sheet and a circular paper sheet at different times, respectively, where the scale bar is 2 mm.

Figure 4

Comparisons of the model results and numerical solutions for the three modeling methods: (a) the small circle cross-sectional pore; (b) the small square cross-sectional pore; and (c) the small regular triangle cross-sectional pore. The illustration in (a) is a schematic of a rectangular paper strip with width W and a fan-shaped paper sheet with an angle α. (d) Comparisons of the model results for the three modeling methods. Without loss of generality, all the initial imbibition lengths are 2 mm.

Figure 5

The results of imbibition distance x as a function of time t for different initial imbibition lengths.

Figure 6

The f-t comparison results of the model and experiment: (a) the rectangular paper strip; (b) the fan-shaped paper sheet; and (c) the circular paper sheet. The marked x²~t rule shows that the relation of imbibition distance x and time t is similar to the L−W model.

Figure 7

The results of porosity η as a function of mass m. Four results are marked with the corresponding experimental images. η changes around the given parameter 0.79 with m.

Figure 8

The results of volumetric flow rate Q as a function of time t for the three paper shapes.