Research ArticleBIOMATERIALS

Hierarchical wrinkling in a confined permeable biogel

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Science Advances  16 Oct 2015:
Vol. 1, no. 9, e1500608
DOI: 10.1126/sciadv.1500608
  • Fig. 1 Dynamics of pattern formation in a confined film of casein gel (4 wt % caseinate and 4 wt % GDL in water).

    (A) Sketch of the cell where adhesion to both top and bottom walls is turned off. The cell is sealed, and the gel is only anchored to the four sides. Typical dimensions are L > 1 cm and e ≈ 100 μm. (B) Light transmission microscopy. Successive generations of patterns are highlighted in color to stress the absence of coarsening after formation. The successive wavelengths are λ = 1.5 mm (yellow), λ = 0.75 mm (orange), and λ = 0.32 mm (brown). (C) Three-dimensional reconstruction from fluorescent confocal microscopy, which highlights that the patterns observed in (B) correspond to wrinkles. The contrast in (B) is not due to thickness inhomogeneities but is due to altitude gradients, as indicated by yellow vertical lines in (A). Scale bars, 1 mm.

  • Fig. 2 Three-dimensional analysis of the wrinkling process.

    (A) Confocal (x, z) cuts showing syneresis, swelling, wrinkling, and cascade buckling. Scale bar, 100 μm (real size ratio). (B to D) Confocal microscopy measurements of the evolution of the volume of the gel phase relative to cell volume, excess area, and velocity along the z-direction. Crosses correspond to times in (A).

  • Fig. 3 The wrinkling experiment.

    (A) Initial configuration where the sealed cell contains a homogeneous protein solution. (B) Around the isoelectric pH, the gel forms and immediately expels the solvent, leading to (C). If tensile stresses were not released, a reversible swelling-back mechanism would occur, leading to (D), a flat swollen gel layer. If tensile stresses were released, the swelling-back mechanism would lead either to buckling (E) or to wrinkling (F), depending on whether bending is free or hindered by a transverse load.

  • Fig. 4 Acid-induced protein gel properties behave nonmonotonically with pH.

    (A) pH decreases over time in a 4 wt % sodium caseinate solution acidified by 4 wt % GDL in water. The horizontal line indicates the isoelectric pH of caseins. (B) Corresponding evolution of the fraction of free caseins. Symbols are direct measures upon addition of GDL. The continuous line is deduced from xfree at the same pH obtained by addition of 1 M HCl and upon waiting for equilibration (inset). (C) Evolution of the elastic modulus G′ measured in a rheometer with full adhesion to the cone-plane geometry. (D) Evolution of pore size measured by confocal microscopy in a slit geometry with either adhesion to all walls (black line) or no adhesion to the top wall [that is, allowed syneresis and swelling (orange dashed line)].

  • Fig. 5 Sketch of the two limit scenarios for wavelength selection.

    In both cases, the gel film of thickness h is destabilized by the excess area, which gives rise to wrinkles of amplitude A(t) and wavelength λ. A single wavelength is represented, and the longitudinal dimension L of the system is much larger than λ. The value of λ is set by the interplay between the bending rigidity B and a transverse load σ due to a pressure gradient p2p1 in the solvent of viscosity η. (A) Darcy scenario: the gel initially sits without sticking to the bottom wall. σ is due to the flow of the solvent through the porous gel of permeability α to fill the growing blister. (B) Poiseuille scenario: the gel film of negligible porosity initially lies in the middle of the cell, separated by a distance H from each wall. σ is due to the lubrication flow in the top and bottom solvent layers.

  • Fig. 6 Comparing model predictions λD, λP, and λD + P with measured wavelengths λexp.

    Dots, primary pattern; squares, secondary blisters. Lines are the best linear fits through the origin, taking into account (A) only the points that should be in Darcy mode H < H*, (B) only the points that should be in Poiseuille mode H > H*, and (C) all points. The prefactors are 0.63, 0.69, and 0.67 respectively. The dashed line in (B) is the best affine fit (λexp = 0.52λP + 0.33 mm) to all data points.

Supplementary Materials

  • Supplementary material for this article is available at http://advances.sciencemag.org/cgi/content/full/1/9/e1500608/DC1

    Fig. S1. Fourier spectra of the binary altitudes of Fig. 1A corresponding to the primary, secondary, and tertiary patterns.

    Wavelength measurements

    Processing of confocal images

    Characteristic time measurements

    Fig. S2. Evidence for the ripples generated by the junction to flat patches (30).

    Fraction of free caseins as a function of pH

    Microstructure measurements

    Fig. S3. Patterns corresponding to the samples in table S1.

    Fig. S4. Microstructure evolution of a 4 wt % sodium caseinate solution acidified with 4 wt % GDL in water.

    Permeability measurements

    Mixed Darcy-Poiseuille model

    Fig. S5. Permeability measurements.

    Fig. S6. Schematic side view of the slit.

    Table S1. Characteristics of the samples used in Fig. 6.

    Video S1. Transmitted light microscopy images of pattern formation in a casein (4 wt %)–GDL (4 wt %) aqueous dispersion.

    Video S2. Three-dimensional reconstruction of pattern formation using confocal microscopy in a casein (4 wt %)–GDL (4 wt %) aqueous dispersion (table S1C).

    Video S3. Confocal microscopy (x, z) cut of the evolution of the gel film deflection of a casein (4 wt %)–GDL (4 wt %) aqueous dispersion when both top and bottom walls are coated with acrylamide brushes (table S1A).

    Video S4. Confocal microscopy (x, z) cut of the evolution of the gel film deflection of a casein (4 wt %)–GDL (4 wt %) aqueous dispersion when both top and bottom walls are coated with acrylamide brushes (table S1C).

    Video S5. Titration of a dispersion with 4 wt % casein by 1 M HCl.

    Video S6. Confocal microscopy (x, y) cut of the evolution of the local structure of a casein (4 wt %)–GDL (4 wt %) aqueous dispersion in sticky boundaries.

    Video S7. Confocal microscopy (x, y) cut of the evolution of the local structure of a casein (4 wt %)–GDL (4 wt %)aqueous dispersion when the top wall is coated with acrylamide brushes and the bottom wall remains sticky.

  • Supplementary Materials

    This PDF file includes:

    • Fig. S1. Fourier spectra of the binary altitudes of Fig. 1A corresponding to the primary, secondary, and tertiary patterns.
    • Wavelength measurements
    • Processing of confocal images
    • Characteristic time measurements
    • Fig. S2. Evidence for the ripples generated by the junction to flat patches (30).
    • Fraction of free caseins as a function of pH
    • Microstructure measurements
    • Fig. S3. Patterns corresponding to the samples in table S1.
    • Fig. S4. Microstructure evolution of a 4 wt % sodium caseinate solution acidified with 4 wt % GDL in water.
    • Permeability measurements
    • Mixed Darcy-Poiseuille model
    • Fig. S5. Permeability measurements.
    • Fig. S6. Schematic side view of the slit.
    • Table S1. Characteristics of the samples used in Fig. 6.
    • Legends for videos S1 to S7

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    Other Supplementary Material for this manuscript includes the following:

    • Video S1 (.avi format). Transmitted light microscopy images of pattern formation in a casein (4 wt %)–GDL (4 wt %) aqueous dispersion.
    • Video S2 (.avi format). Three-dimensional reconstruction of pattern formation using confocal microscopy in a casein (4 wt %)–GDL (4 wt %) aqueous dispersion (table S1C).
    • Video S3 (.avi format). Confocal microscopy (x, z) cut of the evolution of the gel film deflection of a casein (4 wt %)–GDL (4 wt %) aqueous dispersion when both top and bottom walls are coated with acrylamide brushes (table S1A).
    • Video S4 (.avi format). Confocal microscopy (x, z) cut of the evolution of the gel film deflection of a casein (4 wt %)–GDL (4 wt %) aqueous dispersion when both top and bottom walls are coated with acrylamide brushes (table S1C).
    • Video S5 (.mp4 format). Titration of a dispersion with 4 wt % casein by 1 M HCl.
    • Video S6 (.mp4 format). Confocal microscopy (x, y) cut of the evolution of the local structure of a casein (4 wt %)–GDL (4 wt %) aqueous dispersion in sticky boundaries.
    • Video S7 (.mp4 format). Confocal microscopy (x, y) cut of the evolution of the local structure of a casein (4 wt %)–GDL (4 wt %)aqueous dispersion when the top wall is coated with acrylamide brushes and the bottom wall remains sticky.

    Files in this Data Supplement:

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