Depinning and dynamics of imbibition fronts in paper under increasing ambient humidity

Alexander S. Balankin, H. Zapata López, E. Pineda León, D. Morales Matamoros, L. Morales Ruiz, Dan Silva López, and M. A. Rodríguez

Phys. Rev. E 87, 014102 – Published 15 January 2013

Abstract

We study the effects of ambient air humidity on the dynamics of imbibition in a paper. We observed that a quick increase of ambient air humidity leads to depinning and non-Washburn motion of wetting fronts. Specifically, we found that after depinning the wetting front moves with decreasing velocity $v \propto {(h_{p} / h_{D})}^{γ}$ , where $h_{D}$ is the front elevation with respect to its pinned position at lower humidity $h_{p}$ , while $γ ≅ 1 / 3$ . The spatiotemporal maps of depinned front activity are established. The front motion is controlled by the dynamics of local avalanches directed at 30° to the balk flow direction. Although the roughness of the pinned wetting front is self-affine and the avalanche size distribution displays a power-law asymptotic, the roughness of the moving front becomes multiaffine a few minutes after depinning.

Received 12 October 2012

DOI:https://doi.org/10.1103/PhysRevE.87.014102

Authors & Affiliations

Alexander S. Balankin¹, H. Zapata López¹, E. Pineda León¹, D. Morales Matamoros², L. Morales Ruiz¹, Dan Silva López¹, and M. A. Rodríguez³

¹Grupo “Mecánica Fractal,” Instituto Politécnico Nacional, México D.F., Mexico 07738
²Instituto Mexicano del Petróleo (IMP), México D.F., Mexico 07730
³Instituto de Física de Cantabria (IFCA), CSIC–UC, E-39005 Santander, Spain

Article Text (Subscription Required)

Click to Expand

References (Subscription Required)

Click to Expand

Issue

Vol. 87, Iss. 1 — January 2013

Reuse & Permissions

Access Options

Author publication services for translation and copyediting assistance advertisement

Images

Figure 1
Time evolution of the depinned wetting front $z_{D} (x, t) = z (x, t) - \min_{x} z_{p} (x, t_{p})$ after vapor injection (notice that, for clarity, the front configurations are shown every 40 s, instead of the interval of 20 s used in experiments). $z_{p} (x, t_{p})$ represents the pinned configuration of the wetting front in experiment at ${RH}_{1} = 30 \pm 1 %$ and $T = 23 \pm 0.7^{\circ} C$ . Vapor injection (during 30 s) leads to an increase of air humidity ${RH}_{2} = 75 \pm 2 %$ and temperature $T = 28 \pm 2^{\circ} C$ . $z_{M} (x, t_{M})$ represents the configuration of the front pinned at high air humidity after 2 h.Reuse & Permissions
Figure 2
The log-log graphs of $h_{D} (t)$ in millimeters versus $(t - t_{0})$ in seconds for (a) experimental data shown in Fig. 1; straight line represents the best fit ( $R^{2} = 0.99$ ) with $t_{0} = 70$ s and $δ_{D} = 0.752$ for $t \geq t_{0} + τ_{0} \geq 120$ s and (b) for ten experiments performed at different air humidity; curves from bottom to top correspond to ${RH}_{2} =$ 65 $%$ , 65 $%$ , 66 $%$ , 74 $%$ , 75 $%$ , 77 $%$ , 78 $%$ , 79 $%$ , 84 $%$ , and 90 $%$ ; straight lines are a guide for the eye only. The graphs of (c) front velocity $v$ (in mm/min.) versus the dimensionless ratio $h_{D} / h_{p}$ for experiment shown in Fig. 1; straight line—data fitting by Eq. (3) with $β = (1 - δ_{D}) / δ_{D} = 1 / 3$ . (d) Fitting parameter $v_{D}$ (in mm/min.) versus ${RH}_{2}$ (in percent); straight line shows the tendency.Reuse & Permissions
Figure 3
Spatiotemporal distribution of imbibition activity obtained from the evolution of the wetting front after fast increase in ambient air humidity (for experiment shown in Fig. 1) at different thresholds $Δ_{th} =$ (a) 7, (b) 5, (c) 3, and (d) 1 pixel $=$ 0.063 mm.Reuse & Permissions
Figure 4
Log-log graph of probability density $p_{τ}$ (in 1/s) of waiting times $τ_{W}$ (in seconds). Bins—experimental data averaged over ten experiments at different air humidity in the range of $65 % \leq {RH}_{2} \leq 90 %$ ; straight line—power-law fit $p_{τ} \propto τ_{W}^{- α_{τ}}$ with $α_{τ} = 1.65$ .Reuse & Permissions
Figure 5
Time evolution of the fragment of the depinned wetting front in the period between $t_{1} = 82$ and $t_{7} = 84$ min. after vapor injection. Notice the local superpositions of front curves at different $t_{i}$ . $A_{j}$ denotes the local avalanches defined as continuously wetted areas between front configurations. The global avalanche can be defined as the set of local avalanches $A_{j}$ with $1 \leq j \leq 12$ .Reuse & Permissions
Figure 6
(a) Probability density distribution $p_{S}$ (in 1/mm $^{2}$ ) of local avalanche areas $S_{A}$ (in mm $^{2}$ ): bins—experimental data averaged over ten experiments at different air humidity; straight line—power-law fit $p_{S} \propto S_{A}^{- α_{A}}$ with $α_{A} = 1.537$ and (b) the corresponding survival function $1 - P_{S}$ , where $P_{S} = N (S_{A}) / N_{Total}$ is the dimensionless cumulative distribution, while $N_{Total} = 264$ . (c) Log-log plot of the $q$ -order height-height correlation functions $C_{q}$ versus window size $Δ x$ for different $q$ and (d) spectrum of multiaffine exponents $ζ_{q}$ versus $q$ .Reuse & Permissions

Physical Review E

covering statistical, nonlinear, biological, and soft matter physics