Research ArticleMATERIALS SCIENCE

Direct link between mechanical stability in gels and percolation of isostatic particles

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Science Advances  31 May 2019:
Vol. 5, no. 5, eaav6090
DOI: 10.1126/sciadv.aav6090
  • Fig. 1 Snapshots from the entire gelation process reconstructed via particle tracking of a typical sample close to the cluster-gel line (ϕ = 7.5, cp = 1 mg/g) using the salt injection protocol.

    Particles are colored according to the size of the cluster they belong to, going from blue for monomers to red for the percolated cluster. See also movie S1.

  • Fig. 2 Different regimes of gelation.

    (A) Phase diagram with respect to colloid volume fraction ϕ and polymer concentration cp. Black symbols represent experimental points categorized from their final state obtained in the reservoir cell. The spinodal line (solid curve) is obtained from free-volume theory in polymer dilute regime, extended beyond the polymer overlap concentration as a guide for the eye (dotted curve). (B) Comparison of system evolution in terms of largest cluster extent (lmax/L) and of mean coordination number (N¯C). Symbols ◊, Δ, □, and ○ correspond to (ϕ, cp) = (4.2%, 1 mg/g), (8%, 1.5 mg/g), (16%, 1.2 mg/g), and (27%, 1 mg/g), respectively, as highlighted in (A).

  • Fig. 3 Evolution of space-spanning microstructure and mechanical response.

    (A and B) Percolation processes for a dilute (ϕ = 8%, cp = 1.5 mg/g) and a dense (ϕ = 27%, cp = 1 mg/g) sample. The processes of isotropic and directed percolation of all particles are plotted as thin orange curve and orange symbols, respectively. The processes of isotropic and directed percolation of isostatic particles are plotted as thick purple curve and purple symbols, respectively. (C and D) Mechanical response for the same samples. Elastic (G′) and viscous (G′′) shear moduli at the highest available frequency (f=0.1τB1), obtained by two-particle microrheology, are drawn as filled and open circles, respectively. Error bars are obtained following (71, 72). The thick gray curve is the internal stress Σ obtained from the measure of bond-breaking probability (73). The thin orange and thick purple vertical lines show the isotropic percolation times for all particles (τITall) and isostatic particles (τITIS), respectively. The orange dashed vertical line in (D) shows the directed percolation time for all particles (τDall). The gray vertical band shows the possible range of mechanical gelation time τgel (see the Supplementary Materials).

  • Fig. 4 Directed percolation and isostaticity percolation.

    (A) Ratio of the time of isostaticity percolation τITIS to that of directed percolation τDall, as a function of colloid volume fraction. Horizontal dashed line shows when both times are equal. (B) Detail of a reconstruction from confocal coordinates around the percolation time in a dilute sample (ϕ = 8%, cp = 1.5 mg/g). Isostatic particles are drawn to scale; non-isostatic ones are drawn smaller for clarity. The bond network is displayed in orange. (C and D) Same as (B) at later times. (E) Increment of Euclidean distance between two isostatic clusters, averaged over all such pairs initially connected by a non-isostatic network strand. The reference time is the percolation time tperco.

  • Fig. 5 Sketch of the two possible paths to mechanically stable gel.

    (A) Dilute path. (B) Dense path. Isostatic particles are shown in purple; non-isostatic particles are in gray.

  • Fig. 6 Breakup of the network by internal stress.

    (A) Reconstruction from experimental coordinates (ϕ = 29%, cp = 0.7 mg/g) of a strand rupture event. Particles are drawn to scale and colored by a measure of twofold symmetry q2 (74, 75) (see the Supplementary Materials for its definition) from blue (low) to red (high). We note that q2 is a measure of the degree of local stretching. (B) Same event from a topological point of view. The red line indicates the shortest on-graph path between the two particles of interest, whose drastic change indicates the breakup event. The meshed surface is a Gaussian coarse graining of the network pattern.

Supplementary Materials

  • Supplementary material for this article is available at http://advances.sciencemag.org/cgi/content/full/5/5/eaav6090/DC1

    Supplementary Text

    Fig. S1. Reservoir cell.

    Fig. S2. Temporal change of the structure factor.

    Fig. S3. Temporal change in the characteristic wave number 〈q〉.

    Fig. S4. Robustness of the bond network used in percolation analysis.

    Fig. S5. Coordination number analysis for a dilute and a dense sample.

    Fig. S6. Gelation path dependence on polymer concentration.

    Fig. S7. Cluster phase formation observed by our method.

    Fig. S8. Role of hydrodynamics on colloidal phase separation.

    Fig. S9. Temporal change in loss angle.

    Fig. S10. Cluster size distributions at respective percolation times.

    Movie S1. Reconstructions from confocal coordinates of the whole process of gelation at ϕ = 7.7%, cp = 1 mg/g.

    Movie S2. Reconstructions in a thin slice from confocal coordinates of the whole process of gelation at ϕ = 27%, cp = 1 mg/g.

  • Supplementary Materials

    The PDF file includes:

    • Supplementary Text
    • Fig. S1. Reservoir cell.
    • Fig. S2. Temporal change of the structure factor.
    • Fig. S3. Temporal change in the characteristic wave number 〈q〉.
    • Fig. S4. Robustness of the bond network used in percolation analysis.
    • Fig. S5. Coordination number analysis for a dilute and a dense sample.
    • Fig. S6. Gelation path dependence on polymer concentration.
    • Fig. S7. Cluster phase formation observed by our method.
    • Fig. S8. Role of hydrodynamics on colloidal phase separation.
    • Fig. S9. Temporal change in loss angle.
    • Fig. S10. Cluster size distributions at respective percolation times.
    • Legends for movies S1 and S2

    Download PDF

    Other Supplementary Material for this manuscript includes the following:

    • Movie S1 (.mp4 format). Reconstructions from confocal coordinates of the whole process of gelation at ϕ = 7.7%, cp = 1 mg/g.
    • Movie S2 (.mp4 format). Reconstructions in a thin slice from confocal coordinates of the whole process of gelation at ϕ = 27%, cp = 1 mg/g.

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