A unique route of colloidal phase separation yields stress-free gels

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Science Advances  07 Oct 2020:
Vol. 6, no. 41, eabb8107
DOI: 10.1126/sciadv.abb8107
  • Fig. 1 Phase diagram and a typical kinetic pathway to a dilute gel.

    (A) Schematic explanation of depletion attractions in colloid-polymer mixtures. U(r) is the interaction potential, r is the distance between the centers of mass of the two colloids, and β = 1/kBT is the inverse of the thermal energy. (B) Phase diagram of a colloid-polymer mixture with the attraction range of ξ = 7% on the plane of the polymer concentration cp and the colloidal volume fraction ϕc. Triangles, circles, and squares represent states of homogeneous fluids, clusters, and gels, respectively. See fig. S1 for the phase diagram at ξ = 2%. (C) 3D reconstructions of the gelation process observed in a sample with ϕc = 0.046, cp = 2.2 mg/g, and ξ = 7%. Clusters are colored according to the radius gyration of each cluster Rg (see Materials and Methods) normalized by a half of system length, Lsys;Rgnorm=Rg/Lsys. Isolated colloids and colloids in very small clusters are rendered as small points. The movies of the full processes are available in movies S1 and S2.

  • Fig. 2 Two distinct pathways of gelation: SF-G and VPS-G.

    (A) Comparison of the characteristic time scales of percolation τperc and folding τopen as a function of ϕc. (B) Schematic explanation on the bond angle (θbond) analysis. Initially, the center particle (purple) is bonded with its two neighbors (blue) with a rather open central angle, θbond. The local free energy is controlled by the particle configurations. The bond angle should be open under tension, whereas it should be closed in a stress-free stable state. Thus, the bond angle may be used as a measure of mechanical tension. (C and D) Distributions of θbond for samples undergoing gelation with ϕc = 0.083 and 0.17, respectively (cp = 2.2 mg/g and ξ = 7% in both samples). The borders between the different ranges of the bond angle (open, closed, and stretched) are shown by vertical dashed lines in (C) and (D). (E) Temporal evolution of ϕopen, the fraction of particles with open angles [70 ° < θbond < 180°; see (B), left] in the group of colloids with NC = 2, for various colloid volume fractions ϕc, various polymer concentrations cp, and the two polymer/colloid size ratios ξ. We can see the two groups of curves (ϕc ≥ 0.1 and ϕc < 0.1).

  • Fig. 3 Folding of chain-like clusters to compact structures.

    (A) Schematic picture of aggregation to form a chain-like cluster and its transformation to a compact, rigid structure. The inset shows the temporal change of the fraction of open bonds (70 ° < θbond < 180°) in FPD and BD simulations (see Supplementary Text for the details) for ϕc = 0.060 and the scaled potential depth Δ = βU0 = 7.0. (B) A folding process from a single-strand to a multiple-strand configuration observed by a high-speed scan at ϕc = 0.048, cp = 0.96 mg/g, and ξ = 7%. Red particles (single-stranded) have open bond angles, whereas blue particles (multiple-stranded) have closed bond angles (or NC ≥ 3). The aggregation process, followed by a high-speed scan, can be seen in movie S3, and the movie corresponding to (B) is movie S4. (C) Shapes of the potentials used in simulations. (D to F) Numerical simulations of a folding process. Numerical simulations of the folding process of a single-stranded chain cluster. Initially, 10 colloids are arranged in contact on a line. (D) BD simulation without hydrodynamic interactions and with a short-range Asakura-Oosawa (AO) attraction. (E) FPD simulation with hydrodynamic interactions and with a short-range AO attraction [see (D)]. (F) FPD simulation with hydrodynamic interactions and with a Lennard-Jones (LJ) potential [see (D)]. (G) Temporal evolution of the aspect ratio σ31, which is defined as the ratio between the size along the long axis and the one along the short axis (see Supplementary Text for the definition), for simulations (D to F).

  • Fig. 4 Temporal evolution of the solidity of clusters.

    (A) Temporal change in the averaged contact number 〈NC〉 (red) and the averaged cooperativity angle 〈θcoop〉 (blue) for the gelation process. (B) Temporal change in the averaged relative displacement of bonded pairs of colloids Δd scaled by the particle radius a, Δd/a, inside clusters (see Supplementary Text). We set t0 at 174 s (early stage) and 331.5 s (middle stage) as the reference times. The dotted line is the threshold distance for bonds. (C) Histogram of the cooperativity angle, θcoop, (see Supplementary Text) from the early (t = 30 s) to intermediate (t = 990 s) stage. In (A) to (C), the structural and dynamical analyses of the accurate datasets of (x, y, z, dx, dy, dz) for all particles in the high-speed observation of a SF-G sample (ϕc = 0.048, cp = 0.96 mg/g, and ξ = 7%) (see movie S3).

  • Fig. 5 Difference between SF-G and VPS-G.

    (A to C) Comparison between the percolation time and the characteristic time of local compaction for SF-G and VPS-G. (A) ϕc = 0.046 (SF-G). (B) ϕc = 0.083 (SF-G). (C) ϕc = 0.17 (VPS-G). All samples have cp = 2.2 mg/g and ξ = 7%. ϕss represents the fraction of single-stranded segments. The blue and yellow vertical dotted lines indicate τopen and τperc, respectively (see also Fig. 2A). In SF-G (A and B), colloids percolate after the decay of ϕss. In VPS-G (C), on the other hand, the percolation time coincides with the time when the fraction of single-stranded domains becomes 50% of all particles. (D to F) Analysis of stretched segments in VPS-G. (D) Temporal change of ϕstretch, the fraction of colloids with 140 ° < θbond < 180° in colloids with NC = 2 (see Fig. 2B, right bottom). The symbols in (D) are the same as in Fig. 2E. This result for stretched bonds is similar to the one for open bonds in Fig. 2E, but the threshold angle is different (see Fig. 2B). We can see that the fraction of stretched bonds does not decrease with time but that of open bonds initially decreases. (E and F) 3D images of stretched segments during a process of VPS-G formation. The images are taken at t = 2400 s during gelation at ϕc = 0.17, cp = 2.2 mg/g, and ξ = 7%. Red particles represent strongly stretched segments with NC = 2 and 140 ° < θbond < 180°. Blue particles form closed angles (or have NC ≥ 3). In (E), the image is sliced from the full volume for the better visibility of network structures. In (F), only stretched segments are visualized in the entire volume of scanning.

  • Fig. 6 Two universal pathways of gelation.

    (A) Schematic explanation of SF-G and VPS-G pathways to gels. See the main text for detailed explanations. The kinetic pathway to gas + liquid phase separation (the equilibrium state) is blocked (indicated by ×) by both the presence of hydrodynamic interactions (in the initial stage) and mechanical stabilization by the formation of rigid isostatic structures (in the postfolding regime). (B) 2D projection of all time steps of all gel samples as functions of the averaged contact number 〈NC〉 and the normalized averaged cluster mass (M*) (see Materials and Methods and fig. S4). The normalization is made such that M* = 1 when all colloids are involved in a network.

Supplementary Materials

  • Supplementary Materials

    A unique route of colloidal phase separation yields stress-free gels

    Hideyo Tsurusawa, Shunto Arai, Hajime Tanaka

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