Four-dimensional imaging and free-energy analysis of sudden pore-filling events in wicking of yarns

What are the mechanisms at play in the spontaneous imbibition dynamics in polyethylene terephthalate filament yarns at pore scale? Processes at pore scale such as waiting times between the filling of two neighboring pores, as observed in special irregular porous media, like yarns, may overrule the predicted behavior by well-known laws such as Washburn's law. While the imbibition physics are well known, classic models like Washburn's law cannot explain the dynamics observed for yarns. The stepwise dynamics is discussed in terms of the interplay of thermodynamic free energy and viscous dissipation. Time-resolved synchrotron x-ray microtomography documents water filling at pore scale. Spontaneous imbibition in yarns is characterized by a series of fast pore-filling events separated by long periods of low flux. Four-dimensional imaging allows the extraction of interface areas at the boundaries between water, air, and polymer and the calculation of free-energy evolution. It is found that the waiting periods correspond to quasistable water configurations of almost vanishing free-energy gradient. The distributions of pore filling event sizes and waiting times spread over several orders of magnitude, resulting in the pronounced stepwise uptake dynamics.

Figure 1

Comparison of different normalized uptake curves depending on water-air interface area change and external upstream resistance: Washburn [1] $R_p\gg R_e$, $d{A}_{wa}=0$ and $d{A}_{ws}=\text{const}>0$; Washburn [1] $R_p\ll R_e$, $d{A}_{wa}=0$ and $d{A}_{ws}=\text{const}>0$; sinusoidal [39] $R_p\gg R_e$, $d{A}_{wa}\ne d{A}_{wa}\ne \text{const}$; sinusoidal [39] $R_p\ll R_e$, $d{A}_{wa}\ne d{A}_{wa}\ne \text{const}$; this study, experimental example for a yarn in this study.

Figure 2

Experimental setup showing (a) the sample holder mounted to the water-filled reservoir and (b) the whole ensemble on the rotating stage in front of the detector system. (c) Drawing of the holder with the yarn position as the dashed line. (d) Close-up of the sample holder.

Figure 3

Illustration of image processing: (a) raw reconstructed image without water, (b) corresponding machine learning enhanced phase contrast image, (c) fiber-air segmented image, (d) raw reconstructed image with water, (e) water-fiber-air segmented image, (f) time encoded segmented image for voxel air to water transition, (g) water segmentation in a voxel by maximal gray value gradient (black, gray value over time; red, low-pass filtered; and blue, gradient); and (h)–(j) 3D view of the procedure to obtain triangulated water interface mesh, (h) voxel data, (i) Leviner mesh, and (j) smooth mesh (white, fiber; blue, water; and red, water interface).

Figure 4

Visualization of the yarn pore structure: (a) grayscale tomographic horizontal slice, with the corresponding labeled pore space in the (b) horizontal, (c) longitudinal, and (d) frontal planes. (e) The 3D projection. Note that, for the sake of improved readability, label colors do not necessarily correspond to the same pores.

Figure 5

Pore property distributions obtained from XTM void images segmented with a restricted watershed: (a) median shape factor, (b) aspect ratio, (c) relative area variance, (d) connectivity, (e) median cross-section area, and (f) pore arc length. Distributions have been fitted to a log-normal distribution.

Figure 6

(a) Example of a regular pore, (b) example of an irregular pore, and (c) example of a pore connection displayed on a 22-$\mu \mathrm{m}$ (eight-pixel) grid.

Figure 7

(a) Volume uptake over time for all analyzed samples and (b) volume flux corresponding to three exemplary samples [highlighted as dashed lines in (a)]: black, 3/III, with 2.5 mN/tex; red, 7/III, with 10 mN/tex; and blue, 8/III, with 30 mN/tex.

Figure 8

The 3D false color visualization of water configuration inside the yarn at selected time steps (in seconds). The height of the displayed is 4 mm.

Figure 9

(a) Volume flux $Q$ (black line) compared to the evolution of the wet surface $A_{ws}$ (blue line) and water-air interface $A_{wa}$ (red line) with the free energy $F$ (green line). (b) Volume flux $Q$ with interface and free-energy evolution rates. Colored arrows point to the respective axes.

Figure 10

Distribution of the number of filling events per pore for all examined samples and pores combined.

Figure 11

(a) Water volume flux (black) and evolution rates for the wet surface (blue) and water-air interface (red) for the pore with a single step. (b)–(f) Visualization of the filling process with water in false color. Pore cross sections spaced by 27.5 $\mu \text{m}$ are in dark green.

Figure 12

(a) Comparison of the interface area evolution (blue, wet surface; red, water-air interface) to the flux (black) for a pore with three filling events. (b)–(e) Water configuration in false color before and after the flux peaks at selected time steps in seconds. The green shows the outline of the pore spaced by 27.5 $\mu \mathrm{m}$.

Figure 13

(a) Evolution of interface areas (blue, wet surface; red, water-air interface) with corresponding free energy (green). (b) Visualization of the corresponding water configuration at selected time steps. The dark green slices in the 3D projection outline the pore geometry and are spaced by 27.5 $\mu \text{m}$ (ten pixels). The bright green slice highlights the position of the displayed cross section. The raw image is shown on top in grayscale. The color coding denotes the time when water reaches a location: purple labeled water entered the pore before $t=500$ s and orange to yellow capture the filling event. Dark green outlines the pore geometry.

Figure 14

(a) Comparison of volume flux to interface and free-energy evolution of two neighboring pores combined (solid lines). Dashed (dash-dotted) lines correspond to the dark green (red) labeled pore. The top of the green pore is cropped to better visualize the contact area. Black shows the flux, red the water-air interface, blue the wet surface, and green the free energy. (b)–(e) Visualization of selected time steps with false colored water.

Figure 15

(a)–(c) Waiting time distribution (black solid line) with fitted earthquake $\mathrm{\Gamma }$ (red solid line) and generalized $\mathrm{\Gamma }$ distribution fit (red dashed line) and (d)–(f) maximum volume flux distribution (black line) and fitted Pareto function (red line) for samples (a) and (d) 3/III, with 2.5 mN/tex, (b) and (e) 10/III, with 10 mN/tex, and (c) and (f) 8/III, with 30 mN/tex.