2 Experiments

2.2 Capillary rise experiments

To measure the rate of capillary rise, we brought a cellulose sponge into contact with a liquid reservoir and took time sequential images with a CMOS camera (Photron SA 1.1), as shown in figure 2(a). As reservoir liquids, we used 99 wt% ethylene glycol (1 wt% water), turpentine (Nghe An Container Joint Stock Company) and four kinds of silicone oil (10, 100, 350 and 1000 cSt), whose properties are listed in table 1. The vapour pressures of the liquids are very low at atmospheric conditions, so that the effects of evaporation during our capillary rise experiments can be neglected. We avoided water in this work, as it causes swelling of cellulose sponge in the course of capillary imbibition, to focus on the flow of liquid alone. Simultaneous swelling and wicking of hygroexpansive materials including cellulose sponges (Siddique et al. Reference Siddique, Anderson and Bondarev2009), paper (Reyssat & Mahadevan Reference Reyssat and Mahadevan2009; Lee et al. Reference Lee, Kim, Kim and Mahadevan2016b ) and hydrogels (Yoon et al. Reference Yoon, Cai, Suo and Hayward2010) due to water is a challenging problem worth pursuing based on the results of this work.

Figure 2.

Capillary rise dynamics in a cellulose sponge. (a) Wetting front of silicone oil IV propagating against gravity with time. (b) Experimental image of silicone oil IV propagating within a horizontally situated sponge. (c) Experimentally measured rise height of different liquids in a pristine (uncompressed) sponge with time. (d) Capillary rise height in the early stages and horizontal imbibition length within sponges with different average sizes of macro voids. Inset: images of pristine and compressed sponges. (e) Symbols used in figures 2, 3 and 6.

Table 1.

Liquid properties at $23\,^{\circ }\text{C}$ .

Figure 2(c) shows the temporal evolution of the rise height, $h$ , of the different liquids in pristine cellulose sponges. We see that $h$ grows like $t^{1/2}$ initially but the power law changes to $h\sim t^{1/4}$ later. This is a result clearly distinguishable from the capillary imbibition in a horizontally situated sponge, which exhibits persistent Washburn dynamics under negligible effects of evaporation. Moreover, our result is distinct from the aforementioned non-diffusive dynamics in sharp corners (Ponomarenko et al. Reference Ponomarenko, Quéré and Clanet2011; Weislogel Reference Weislogel2012) or rounded-edge pillar arrays (Obara & Okumura Reference Obara and Okumura2012), in that the exponent of the present power law changes from $1/2$ to $1/4$ rather than being a constant of $1/3$ . We explain the physical origins of the observed power laws and derive them in the following.

3 Early stages of capillary rise

We start with the early stages where the rise height grows like $t^{1/2}$ . This diffusive behaviour is readily explained via Darcy’s law, which is now written as ${\dot{h}}=-(k/\unicode[STIX]{x1D707})(\text{d}p/\text{d}z+\unicode[STIX]{x1D70C}g)$ . The pressure gradient is provided by the capillary action, thus $|\text{d}p/\text{d}z|\sim (\unicode[STIX]{x1D70E}/r)/h$ . In the early stages where $h\ll \unicode[STIX]{x1D70E}/(\unicode[STIX]{x1D70C}gr)$ , we ignore the gravitational effect, to obtain $h{\dot{h}}\sim (k/\unicode[STIX]{x1D707})(\unicode[STIX]{x1D70E}/r)$ . Integration of this relation is possible upon identifying the permeability $k$ and the radius of curvature of the advancing meniscus in the sponge, $r$ . Comparison of Darcy’s law with Washburn’s equation reveals that $k$ corresponds to the cross-sectional area of the fluid conduit in a porous medium. When both large and small voids are present, viscous liquids would preferably fill large voids due to lower frictional resistance as long as capillary actions, which are inversely proportional to the meniscus radius, are strong enough to overcome gravity. Hence, we take $k\sim R^{2}$ and $r\sim R$ , with $R$ being the characteristic radius of macro voids. Then, we obtain the following scaling law for $h$ :

(3.1) $$\begin{eqnarray}h\sim \left(\frac{\unicode[STIX]{x1D70E}Rt}{\unicode[STIX]{x1D707}}\right)^{1/2}.\end{eqnarray}$$

Although this form is identical to Washburn’s equation, a remarkable feature here is that we use only the characteristic length scale of macro voids to describe the capillary rise dynamics of the heterogeneous porous media. To verify our assumption that the wicking through sponges in the early stages is dominated by capillary flows through macro voids, we measured the rise speeds of various liquids within sponges with different sizes of macro voids. The measurement results for early-stage rise heights for different liquids and sponges of varying $R$ are plotted versus time in figure 2(c) as filled symbols. Since the theory neglects the gravitational effect, (3.1) must also hold for horizontal imbibition. Thus, we measured the imbibition length in the horizontally situated sponge as shown in figure 2(b), where a bar of uncompressed sponge touches a liquid vessel at one end. Figure 2(d) shows that the early-stage data scattered in figure 2(b) and the measurement results for horizontal imbibition are indeed collapsed onto a single straight line with a slope of $1/2$ when plotted against $\unicode[STIX]{x1D70E}Rt/\unicode[STIX]{x1D707}$ , consistent with our scaling law (3.1). The only difference in the vertical and horizontal imbibition is that (3.1) holds for the entire duration of horizontal wicking, unlike vertical wicking.

4 Late stages of capillary rise

To investigate the fluid flow beyond the early stages, we first compare the optical images of macro voids near the wet–dry interface in the early and late stages in figure 3(a,b). While the macro voids in the early stages are completely filled with liquid, those in the late stages are only wet in their walls and corners. We have also found that the transition from the complete filling (early) stage to the partial filling (late) stage occurs at a higher elevation for liquids with a higher surface tension, while the transition height is independent of the liquid viscosity. These observations indicate that liquid fails to fill the macro voids completely from a certain elevation due to gravitational effects. Then, the liquid menisci should hang in the corners of the macro voids, as depicted in figure 3(b), a physical picture consistent with previous studies dealing with corner flows (Ponomarenko et al. Reference Ponomarenko, Quéré and Clanet2011; Obara & Okumura Reference Obara and Okumura2012; Weislogel Reference Weislogel2012). Since the driving curvature force and the cross-sectional area of the fluid conduit are altogether different from those of the early stages, $k$ and $r$ in Darcy’s law must change in the late stages.

Figure 3.

Effects of gravity on liquid-filling behaviour of voids and late-stage rise dynamics. (a) Left: optical image of a sponge soaked with liquid in the early stages. Right: a schematic of a macro void completely filled with liquid. (b) Left: optical image of a sponge near its wet–dry interface in the late stages. Right: a schematic of a macro void wetted at the corner with microscale wall pores saturated with liquid. Here, $\unicode[STIX]{x1D706}$ indicates the radius of curvature of the meniscus at the wet corner. (c) A schematic of a liquid entering microscale wall pores from a wet corner of macro voids and estimates of pressure. Here, $p_{c}$ and $p_{m}$ correspond to the pressures in the corner liquid and micropore respectively. (d) Liquid rise height in the late stages plotted according to the scaling law (4.1). ( $e$ ) The experimentally measured liquid rise height scaled by Jurin’s height, $h_{J}$ , of macro voids in each case. The horizontal line indicates the boundary between the early and late stages.

If the walls of the corners are impermeable (Ponomarenko et al. Reference Ponomarenko, Quéré and Clanet2011; Obara & Okumura Reference Obara and Okumura2012; Weislogel Reference Weislogel2012), the liquid hanging in the corner is responsible for both capillary pressure ( $\unicode[STIX]{x1D70E}/r$ ) and area of conduit ( $k$ ), which leads to the $t^{1/3}$ law as explained in § 1. The critical difference of the sponges from the substrates of previous studies is that the sheets surrounding the macro voids are permeable, with a number of micropores of radius $r_{0}\sim 1$   $\unicode[STIX]{x03BC}\text{m}$ . The permeable sheets absorb liquid as it gradually wets the macro voids only in the corners. We still estimate the characteristic radius of curvature, $\unicode[STIX]{x1D706}$ , of the wet corners of macro pores at the height $h$ as $\unicode[STIX]{x1D706}\sim \unicode[STIX]{x1D70E}/(\unicode[STIX]{x1D70C}gh)$ based on the balance of capillary and gravitational effects. As the pressure difference in the micropores at the rising front, $p_{a}-p_{m}\sim \unicode[STIX]{x1D70E}/r_{0}$ , is greater than that in the corner liquid, $p_{a}-p_{c}\sim \unicode[STIX]{x1D70E}/\unicode[STIX]{x1D706}$ , with $p_{a}$ being the atmospheric pressure, the liquid flow can be driven from the wet corner to the wall pores; see figure 3(c). In the late stages, therefore, liquid rise is driven by capillary pressure provided by microscale wall pores, leading to $r\sim r_{0}$ , where $r$ is the radius of curvature responsible for the dominant driving force in Darcy’s law. Now, the remaining question is what path the liquid takes from the reservoir to the micropores at the rising front, which determines $k$ . As depicted in figure 3(c), the dominant flow path is provided by the liquid hanging in the macro voids rather than microscale wall pores with excessively small cross-sectional area causing too much viscous dissipation. Therefore, the permeability $k$ can be scaled as the cross-sectional area of a wet corner, which is a function of the rise height $h$ and the compression ratio $\unicode[STIX]{x1D716}=R/R_{0}$ , with $R_{0}$ being the macro pore size of the pristine sponge.

Figure 4 shows a schematic of a corner of an individual macro void which is compressed by external pressure in the lateral ( $x$ ) direction. We are interested in the change of fluid conduit area, $A_{0}$ to $A$ , while the meniscus curvature $\unicode[STIX]{x1D706}$ is maintained. Assuming that the void size changes insignificantly in the $y$ -direction, which is confirmed by negligible extension of the sponge in the direction perpendicular to that of pressure, we write $R/\tan \unicode[STIX]{x1D6FC}\sim R_{0}/\tan \unicode[STIX]{x1D6FC}_{0}$ , where $\unicode[STIX]{x1D6FC}_{0}$ and $\unicode[STIX]{x1D6FC}$ are the corner angles of uncompressed and compressed voids respectively. Since $A_{0}\sim \unicode[STIX]{x1D706}^{2}/\tan \unicode[STIX]{x1D6FC}_{0}$ and $A\sim \unicode[STIX]{x1D706}^{2}/\tan \unicode[STIX]{x1D6FC}$ , we obtain $A/A_{0}\sim \tan \unicode[STIX]{x1D6FC}_{0}/\tan \unicode[STIX]{x1D6FC}\sim R_{0}/R$ . The permeability of the uncompressed sponge is given by $k_{0}\sim A_{0}\sim \unicode[STIX]{x1D706}^{2}$ , and thus the permeability of the compressed sponge $k\sim A\sim \unicode[STIX]{x1D706}^{2}/\unicode[STIX]{x1D716}\sim \unicode[STIX]{x1D716}^{-1}[\unicode[STIX]{x1D70E}/(\unicode[STIX]{x1D70C}gh)]^{2}$ , where $\unicode[STIX]{x1D716}=R/R_{0}$ is the compression ratio. The permeability decreases with the rise height because the radius of curvature of the meniscus should decrease to overcome gravity. Highly compressed sponges with a low $\unicode[STIX]{x1D716}$ have a high $k$ because the corner area occupied by a meniscus is larger than the wet corner area of less compressed sponges with the same meniscus curvature.

Figure 4.

Deformation of a wet corner of macro voids by the external pressure in (a) the uncompressed sponge and (b) the compressed sponge. Given the same radius of the meniscus curvature $\unicode[STIX]{x1D706}\sim \unicode[STIX]{x1D70E}/(\unicode[STIX]{x1D70C}gh)$ , the cross-sectional area of the liquid conduit increases with the compression of the macro void.

Darcy’s law, which is written as ${\dot{h}}\sim (k/\unicode[STIX]{x1D707})(\unicode[STIX]{x0394}p/h)$ , with $\unicode[STIX]{x0394}p\sim \unicode[STIX]{x1D70E}/r_{0}$ , allows us to obtain the scaling law for $h$ in the late stages:

(4.1) $$\begin{eqnarray}h\sim \left[\frac{\unicode[STIX]{x1D70E}^{3}}{\unicode[STIX]{x1D707}(\unicode[STIX]{x1D70C}g)^{2}\unicode[STIX]{x1D716}r_{0}}\right]^{1/4}t^{1/4}.\end{eqnarray}$$

In figure 3(d), we plot the experimentally measured rise height in the late stages according to the scaling law (4.1). The data, which are the open symbols scattered in figure 2(c), are collapsed onto a single straight line regardless of changes in $\unicode[STIX]{x1D70E}$ , $\unicode[STIX]{x1D707}$ , $\unicode[STIX]{x1D70C}$ and $\unicode[STIX]{x1D716}$ by our theory. We further show that the flow in the late stages is indeed driven by the capillary pressure in the micropores rather than that in the wet corners of macro pores using (4.1) in the appendix. Figure 3(e) reveals that the transition height at which the power law changes from $h\sim t^{1/2}$ (filled symbols) to $h\sim t^{1/4}$ (empty symbols) corresponds to Jurin’s height (Jurin Reference Jurin1718) of macro voids for each sponge: $h_{J}\sim \unicode[STIX]{x1D70E}/(\unicode[STIX]{x1D70C}gR)$ . This result confirms that liquid fails to completely fill the macro voids due to gravitational effects beyond Jurin’s height, causing a qualitative difference in the rise dynamics before and after the transition height.