Under pressure: Hydrogel swelling in a granular medium

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Science Advances  12 Feb 2021:
Vol. 7, no. 7, eabd2711
DOI: 10.1126/sciadv.abd2711
  • Fig. 1 Swelling of a hydrogel confined in a 3D granular medium.

    (A) Image of apparatus for testing hydrogel swelling in confinement. A hydrogel (orange) is embedded within a granular medium composed of glass beads (hazy transparent circles) packed within a transparent acrylic chamber with an overlying loaded piston. (B) Image of a hydrogel swollen within a medium in the absence of an applied load, showing that it swells freely and retains its spherical shape. (C) Image of a hydrogel swollen within a medium under a strong applied load, corresponding to a confining stress of 22 kPa, showing that it deforms strongly and exhibits hindered swelling. The white dotted circle shows an inscribed circle, representing the size of the region of the hydrogel in contact with surrounding beads, for which swelling is hindered. The black dotted circle shows the outline of the hydrogel under no applied load from (B). As shown by the white space in between the black dashed outline and the projection of the hydrogel, its projected area is smaller under strong applied load. Photo credit: Jean-François Louf, Princeton University.

  • Fig. 2 Characterization of hydrogel swelling in confinement over time.

    Measurements of (A) normalized circularity Ψ and (B) fractional change in projected area A show that hydrogels under a small confining stress of 0.2 kPa (dark symbols) remain spherical and swell more, while hydrogels under a strong confining stress of 22 kPa (light symbols) deform into a nonspherical shape and swell less. Both quantities are normalized by their initial value. To perform these measurements, we binarize our images of hydrogel swelling inside the granular media using a given threshold value; to assess the uncertainty in these measurements, we vary this threshold value by ±10%, for which the binarized images still closely approximate the shape of the imaged hydrogel. The resulting SD in the measurements of the normalized Ψ and A is represented by the error bars in the plot. When not shown, error bars are smaller than the symbol size.

  • Fig. 3 Hydrogel swelling is determined by the competition between osmotic swelling and local confinement.

    (A) Fractional change in hydrogel volume ~ R3 decreases with increasing stress σ, in media with different mean bead radii Rb. Curves show the prediction of Eq. 4; colors show Rb, and different sets of curves show ±1 SD in Ri from the measured mean. (B) Net osmotic swelling force (blue curve) decreases as the hydrogel swells and is eventually balanced by the confining force transmitted by beads (red curve). Right schematic shows a hydrogel of initial radius Ri (inner circle), hindered swollen radius Rf (dark orange with dashed circle), and equilibrium unconfined swollen radius Rf,u (light orange) surrounded by beads (gray). Inset illustrates the force exerted by beads (red) and the force exerted by the swelling hydrogel at the contact (blue). (C) Hydrogel swelling in confinement is described by the balance between the net osmotic swelling force Πfaf2 and the confining force transmitted by the medium σRb2, each computed from independent measurements, as shown by the dashed line. Gray points show the results of our calculation based on measurements of water absorption by hydrogels in soil from (49, 50).

  • Fig. 4 Unbalanced hydrogel swelling restructures the surrounding medium.

    (A) Images show a swelling hydrogel (green) restructuring the surrounding medium, as indicated by the motion of dyed tracer beads (blue), for an experiment with small confining stress σ = 0.2 kPa. The dashed orange line indicates the initial location of the bead showing the maximal displacement Δ. (B) Maximal bead displacement as a function of time for an experiment with small confining stress σ = 0.2 kPa. (C) Measurements of the plateau value of the maximal bead displacement Δf as a function of confining stress σ, for experiments with hydrogels confined in media with different mean bead radii Rb. The medium is restructured by hydrogel swelling at low confining stress but is restructured less as confining stress increases. (D) The onset of medium restructuring, quantified by the normalized bead displacement, arises when the swelling number Ns ≡ Πi(ai/Rb)2/σ(1 + μ) becomes sufficiently large.

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