The Triangle That Thickens

Dense suspensions — cornstarch in water is the classic example — exhibit shear thickening: apply force, and the fluid gets more viscous. Push gently and it flows. Push hard and it resists like a solid. At extreme stress, the thickening becomes discontinuous — the viscosity jumps abruptly, as if the suspension has been switched from fluid to solid.

D’Amico, Tu, and Singh (arXiv:2501.02062) identify what controls the switch: triangular loops in the frictional contact network.

When stress is low and thickening is mild, the contact network between particles consists of quasilinear chains — strings of particles transmitting force along the compression axis. The structure is one-dimensional: chains, not meshes. As stress increases and contacts become frictional, the network develops loops. Specifically, third-order loops — triangles of three mutually contacting particles — proliferate.

The relationship between triangle count and viscosity is tight: the number of edges participating in triangular loops collapses all the data — across different packing fractions, applied stresses, and friction coefficients — onto a single curve. The triangle count is the universal order parameter for shear thickening.

The mechanism: a linear chain of contacts can slide. A triangular loop cannot — it’s geometrically rigid. When enough triangles form, the contact network locks, and the suspension cannot deform without breaking contacts. The transition from continuous to discontinuous shear thickening is the percolation of triangular rigidity through the contact network.

The structural insight: the macroscopic rheology of dense suspensions is not governed by packing geometry, friction, or stress individually. It’s governed by mesoscale network topology — specifically, the simplest possible structural motif (the triangle) acting as a rigidity switch. The physics is in the connectivity, not the components.

D’Amico, Tu, & Singh, “Topological insights into dense frictional suspension rheology: Third order loops drive discontinuous shear thickening,” arXiv:2501.02062 (2025).