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Hierarchical amorphous ordering in colloidal gelation

Nature Physics volume 19, pages 1171–1177 (2023)Cite this article

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Abstract

Amorphous gels are formed in various soft matter and biomatter when phase separation is dynamically arrested without crystallization. The dynamic arrest in gelation has been attributed to glass transition, but a microscopic foundation is lacking. To address this issue, we experimentally study the gelation of a sticky colloid model using the single-particle-level dynamic analysis of in situ confocal microscopy observations. We show that, during gelation, individual colloids first aggregate into tetrahedra, which then grow to form poly-tetrahedral clusters. Subsequently, pentagonal bipyramids are formed as sets of five tetrahedra, and finally, these pentagonal bipyramids self-catalyse secondary growth to form medium-range amorphous order and arrest dynamics. This hierarchical ordering is primarily driven by local potential energy, not free energy. Thus, the amorphous ordering and arrest mechanism fundamentally differ between gels formed by phase separation and glasses formed homogeneously. These findings will deepen our understanding of two types of amorphous solid, namely, gel and glass.

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Fig. 1: Our experimental setup.
Fig. 2: Pentagonal bipyramids in a dilute gel.
Fig. 3: Hierarchical ordering of pentagonal bipyramids in gelation.
Fig. 4: Elementary process forming a pentagonal bipyramid cluster and its impact on dynamical arrest.

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Data availability

The data supporting the findings of this study are available from the corresponding authors upon reasonable request.

Code availability

The computer codes used in this study are available from the corresponding authors upon reasonable request.

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Acknowledgements

H. Tanaka acknowledges receipt of Grants-in-Aid for Specially Promoted Research (JSPS KAKENHI grant no. JP20H05619) and Scientific Research (A) (JSPS KAKENHI grant no. JP18H03675) from the Japan Society for the Promotion of Science (JSPS).

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Authors and Affiliations

  1. Institute of Industrial Science, University of Tokyo, Tokyo, Japan

    Hideyo Tsurusawa & Hajime Tanaka

  2. Research Center for Advanced Science and Technology, University of Tokyo, Tokyo, Japan

    Hajime Tanaka

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  1. Hideyo Tsurusawa
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Contributions

H. Tsurusawa and H. Tanaka conceived the project. H. Tanaka supervised the project. H. Tsurusawa performed the experiments and analysed the data. Both authors discussed the results and wrote the manuscript.

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Correspondence to Hideyo Tsurusawa or Hajime Tanaka.

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Extended data

Extended Data Fig. 1 Confirmation of ‘stress-free’ gelation.

In-situ data of a dilute gel sample with ϕc = 0.053 and cp = 2.2 mg/g is analysed. a, The averaged contact number, 〈Nc〉, is plotted as a function of time. The dotted line indicates the isostatic condition of 〈Nc〉 = 6. This sample became isostatic at t = 2070 s. The region of t ≥ 2070 s is coloured in yellow. b, The radius gyration of the largest cluster (\({L}_{\max }\)) is plotted as a function of time. Lsys is half the system’s size. If we set the percolation threshold as \({L}_{\max }/{L}_{{{{\rm{sys}}}}}=0.8\), this sample percolated at t = 5990 s (magenta arrow). If we employ a more severe threshold of \({L}_{\max }/{L}_{{{{\rm{sys}}}}}=0.95\), the percolation time shifts to 12000 s. c, The edge-to-edge distances of the biggest cluster along X, Y, or Z directions are plotted as a function of time. These distances are normalised by the full length of the system (2Lsys). The biggest cluster spanned the system along the X, Y, and Z direction at t = 5000 s, 4640 s, and 7250 s, respectively. Due to the finite size effect, the percolation time has arbitrariness in an approximate range from 5000 s to 12000 s. Regardless of the percolation threshold, percolation occurs ‘after’ 〈Nc〉 overwhelms 6. Thus, this gel sample is classified as a stress-free gel25.

Extended Data Fig. 2 Three-dimensional visualisations of the early stage of gelation.

Typical 3D images reconstructed from a gel sample with ϕc = 0.053 and cp = 2.2 mg/g. In the visualisations by cluster length, small dots show the particles in small clusters with their mass below 15. For each cluster, cluster length is calculated as its radius gyration. P is the fraction of the symmetric order in all the particles (see Methods). At t = 0 s, colloids start phase separation. At t = 180 s, the first pentagonal bipyramid is formed. At t = 420 s, the fraction of tetrahedra overwhelms 50%. The late stage is shown in the following figure.

Extended Data Fig. 3 Three-dimensional visualisations of the late stage of gelation.

At t = 1380 s, the fraction of pentagonal bipyramid clusters overwhelms the fraction of isolated pentagonal bipyramids. At t = 5990 s, the radius gyration of the largest cluster reaches the percolation threshold of \({L}_{\max }/{L}_{{{{\rm{sys}}}}}=0.8\).

Extended Data Fig. 4 Optimisation of the threshold parameters to identify pentagonal bipyramids.

a, Radial distribution function, g(r), calculated from a gel sample. The plot is obtained by averaging over 10 timesteps (from t = 9230 s to t = 10040 s). We experimentally determine the diameter of our colloids (σ) as 2150 nm such that g(r) becomes maximum at r/σ = 1. b, g(r) zoomed near the first minimum. We set the bond threshold as r/σ = 1.295. c, The histogram of the angles between two connecting bonds (θbond). d, The number of regular pentagons detected by our algorithm. e, The number of pentagonal bipyramids detected by our algorithm. In d and e, we fix as δθplane = 10∘ and δθrot = 10∘. The bond threshold, dbond, and the angular threshold, δθedge, are varied. See Supplementary Methods about the parameters’ optimisation. A gel sample with ϕc = 0.053 and cp = 2.2 mg/g at t = 10040 s is analysed.

Extended Data Fig. 5 Impact of the planar and angular criteria on detecting five-fold symmetry.

To evaluate the net contribution of the planar and angular criteria, we directly compare our method with the topological cluster classification (TCC) method. For the ‘TCC-like’ analysis, the planar and angular criteria are disabled by the following settings: δθplane = δθrot = δθedge = 180∘. For our method, the optimal thresholds are set as follows: δθplane = 10∘, δθrot = 10∘, and δθedge = 12∘. a, The number of the regular pentagons detected by the two methods. dbond is the upper threshold to define a bond. b, The number of pentagonal bipyramids detected by the two methods. c, The distributions of the planar, rotation, and edge angles detected by the two methods. In c, the data is analysed in the pentagonal bipyramids that are detected by the threshold of dbond/σ = 1.295 (see the arrows in b). A gel sample with ϕc = 0.053 and cp = 2.2 mg/g at t = 10040 s is analysed.

Extended Data Fig. 6 Formation of pentagonal mono-pyramids.

a, 3D models of pentagonal pyramids. b, The number of each pyramid type is plotted as a function of time. More than 98% of pentagonal planes form bipyramids during the gelation process. An alternative model against Fig. 3b is the structural evolution between (i) a pentagon (not-pyramid), (ii) a pentagonal mono-pyramid, and (iii) a pentagonal bipyramid. However, the experimental data rejects this alternative model. A gel sample with ϕc = 0.053 and cp = 2.2 mg/g at t = 10040 s is analysed.

Extended Data Fig. 7 The role of poly-tetrahedra in increasing the contact number of a gel.

For each time step, the fractions of the tetrahedron, 2-tetrahedra, and 3-tetrahedra are plotted as functions of 〈Nc〉. The fractions of the poly-tetrahedral orders monotonically increase as 〈Nc〉. These fractions are saturated before 〈Nc〉 approaches 6. A gel sample with ϕc = 0.053 and cp = 2.2 mg/g at t = 10040 s is analysed.

Extended Data Fig. 8 High-speed microscopy to access elementary process of the local ordering.

High-speed microscopy acquired time-lapse 3D images every 1.5 s during gelation. The temporal resolution of the high-speed microscopy is even faster than the Brownian motion time of our colloids (τB ~ 2 s). We applied the high-speed microscopy to a gel sample with ϕc = 0.045 and cp = 0.96 mg/g. Since high-speed microscopy limits the scanning volume, the number of particles in the limited 3D volume is typically 400. a, Temporal evolution of the fractions of the tetrahedron, 2-tetrahedra, 3-tetrahedra, a pentagonal bipyramid, and an octahedron in the gel. b, 3D visualisations of the high-speed microscopy at t = − 135 s, 0 s, 157.5 s, 252 s, 390 s, 750 s, and 990 s. Blue particles form pentagonal bipyramids. A small cluster is marked in magenta at t = 157.5 s and 252 s. Figure 4a-f analyse the transformation of this marked cluster. Figure 4g analyses the trajectory data of 400 time steps from t = 390 s to 990 s.

Extended Data Fig. 9 Universality of the hierarchical ordering in the arrested cluster phase separation.

a and b, Temporal changes of the fractions of tetrahedra, 2-tetrahedra, 3-tetrahedra, octahedra, and pentagonal in the formation of non-percolating clusters (a) and dilute gels (b). c, Averaged contact number 〈Nc〉 and the fraction of pentagonal bipyramids are plotted for all time steps and all the phase separation samples in our study.

Supplementary information

Supplementary Information

Supplementary Video 1

Elementary process forming a pentagonal bipyramid cluster. This is the full video for Fig. 4a–f. The video starts at t = 157.5 s and ends at t = 252.0 s. One frame corresponds to 1.5 s. To improve visibility, we correct both drift and rotation of the cluster so that the centre of mass and direction of the rotational axis (the red one) are fixed. A pentagonal ring is coloured blue only when it satisfies all the topological, planar and rotational criteria.

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Tsurusawa, H., Tanaka, H. Hierarchical amorphous ordering in colloidal gelation. Nat. Phys. 19, 1171–1177 (2023). https://doi.org/10.1038/s41567-023-02063-x

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