Dense suspensions shear thicken — their viscosity jumps when sheared hard enough. The standard explanation involves packing fraction: as particles are forced closer together, frictional contacts form, and the suspension transitions from a flowing to a jammed-like state. The packing fraction φ, relative to a maximum packing φ_m, is the control parameter.
This paper shows the real control variable is the constraint ratio: the number of mechanical constraints per particle.
When particles come into frictional contact, each contact imposes constraints on the relative motion of the two particles. The number of constraints per contact depends on the contact geometry — sliding friction, rolling friction, torsional friction each contribute differently. And the number of contacts per particle depends on the packing, but also on the particle shape, roughness, and the details of the interaction potential.
The constraint ratio collapses the onset and magnitude of shear thickening across different particle types, shapes, and interaction models onto a single curve. Systems that have the same packing fraction but different constraint ratios thicken differently. Systems with different packing fractions but the same constraint ratio thicken the same way.
This means packing fraction was always a proxy — it worked because constraint ratio and packing fraction are correlated for spherical particles with standard friction. But when the correlation breaks (non-spherical particles, unusual friction coefficients, particles with constrained rolling), the packing-fraction description fails and the constraint ratio succeeds.
The implication for materials design is direct. To control shear thickening, modify the contact geometry rather than the packing. Particles that roll freely (fewer constraints per contact) thicken later. Particles that are rough or angular (more constraints per contact) thicken earlier. The macroscopic rheology follows from the microscopic counting.