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Direct observation of impact propagation and absorption in dense colloidal monolayers

  1. Lucio Isaa,3
  1. aLaboratory for Interfaces, Soft Matter and Assembly, Department of Materials, ETH Zurich, 8093 Zurich, Switzerland;
  2. bDepartment of Mechanical and Process Engineering, ETH Zurich, 8092 Zurich, Switzerland;
  3. cEngineering and Applied Science, California Institute of Technology, Pasadena, CA 91125;
  4. dLaboratoire Quartz, Unité de Recherche EA-7393, Institut Supérieur de Mécanique de Paris - Supméca, 93400 Saint-Ouen, France
  1. Edited by David A. Weitz, Harvard University, Cambridge, MA, and approved October 3, 2017 (received for review July 10, 2017)

Significance

Single-particle characterization of the impact response has unveiled design principles to focus and control stress propagation in macroscopic granular crystalline arrays. We demonstrate that similar principles apply to aqueous monolayers of microparticles excited by localized mechanical pulses. By inducing extreme local deformation rates and tracking the motion of each particle with velocities that reach up to few meters per second, we reveal that a regime of elastic collisions, typically forbidden due to overdamping, becomes accessible. This provides insights on the stress propagation and energy absorption of dense suspensions upon fast deformation rates.

Abstract

Dense colloidal suspensions can propagate and absorb large mechanical stresses, including impacts and shocks. The wave transport stems from the delicate interplay between the spatial arrangement of the structural units and solvent-mediated effects. For dynamic microscopic systems, elastic deformations of the colloids are usually disregarded due to the damping imposed by the surrounding fluid. Here, we study the propagation of localized mechanical pulses in aqueous monolayers of micron-sized particles of controlled microstructure. We generate extreme localized deformation rates by exciting a target particle via pulsed-laser ablation. In crystalline monolayers, stress propagation fronts take place, where fast-moving particles (V approximately a few meters per second) are aligned along the symmetry axes of the lattice. Conversely, more viscous solvents and disordered structures lead to faster and isotropic energy absorption. Our results demonstrate the accessibility of a regime where elastic collisions also become relevant for suspensions of microscopic particles, behaving as “billiard balls” in a liquid, in analogy with regular packings of macroscopic spheres. We furthermore quantify the scattering of an impact as a function of the local structural disorder.

The mechanisms of propagation and absorption of large stresses in particulate materials can be very different depending on the size of the particles and their arrangement. For regular packings of macroscopic spheres, e.g., in a Newton’s cradle, stress pulses, including impacts and shocks, are conveyed through elastic contacts (Hertzian interactions) (1?3). This makes it possible to direct and focus them, if the material provides specific paths [e.g., linear chains (4, 5) or lattices (6)] for the stress propagation (7, 8). The presence of a dispersing fluid does not alter this physics, provided that two neighboring grains/particles gain enough relative inertia to perform elastic scattering, mediated by hydrodynamic interactions (9?11).

Instead, dense disordered packings of macroscopic grains are materials whose energy absorption is controlled by local structural rearrangements and dissipation (12, 13). Absence of inertia in dense suspensions of microparticles and nanoparticles typically prevents elastic collisions and provides similar routes for energy dissipation (14??17). At high Peclét numbers, shear dominates the structural response of the material and Brownian diffusion becomes negligible. For example, at sufficiently high strain rates (up to <mml:math><mml:mrow><mml:mrow><mml:mover accent="true"><mml:mi>γ</mml:mi><mml:mo>˙</mml:mo></mml:mover></mml:mrow><mml:mo>≈</mml:mo></mml:mrow></mml:math>γ˙≈ 103 s?1) and volume fractions, shear establishes highly dissipative particle chains, where lubrication films break down and the response is dominated by frictional contacts (18??21), leading to a viscosity increase (discontinuous shear thickening). Additionally, during impacts, “snow-plough” jammed fronts of nondeformable spheres propagate through the material and efficiently absorb energy (14, 22, 23).

Here, we investigate the mechanism of localized stress propagation in 2D crystalline lattices and disordered ensembles of microparticles in a liquid. Using pulsed laser ablation (PLA) to excite localized mechanical pulses (24), we access a regime of extremely high local shear rates, sufficient to induce interparticle Hertzian contacts and therefore a response analogous to the one of regular and disordered collections of macroscopic spheres. We study the effects of fluid viscosity and microstructural order on the stress propagation at the single-particle level.

Colloidal monolayers are prepared using a suspension of light-absorbing (SiO2 half-covered with 50 nm of gold) and light-transparent (SiO2) particles (radius R = 3.69 μm) that sediment toward the bottom glass surface of an observation cell (SI Materials and Methods and Fig. 1A). The coated particles function as “shock initiators” (SIs). When a metallic surface is illuminated by pulsed laser light, heat is not dissipated quickly enough, and some material is ablated from the surface. The expansion of high-pressure plasma generates an isotropic pressure wave (Movie S1) that travels away from the ablated material (25, 26). SIs that are confined and have the axis that links the Au-coated and the uncoated hemispheres oriented perpendicular to the substrate (Fig. 1A) behave in the same fashion. Under illumination with pulsed laser light [laser energy (LE) 0.09 μJ < LE < 0.25 μJ, λ = 532 nm, pulse duration tpulse = 4 ns, radius of the laser spot ≈ 3R], the ablation of the gold coating triggers an ultrashort (<mml:math><mml:mrow><mml:mi>f</mml:mi><mml:mo>=</mml:mo><mml:mrow><mml:mn>1</mml:mn><mml:mo>/</mml:mo><mml:mrow><mml:mn>2</mml:mn><mml:mi>π</mml:mi><mml:msub><mml:mi>t</mml:mi><mml:mrow><mml:mtext>pulse</mml:mtext></mml:mrow></mml:msub></mml:mrow></mml:mrow><mml:mo>=</mml:mo><mml:mn>40</mml:mn></mml:mrow></mml:math>f=1/2πtpulse=40 MHz) pressure wave. This effect is illustrated in Fig. 1B. Upon PLA of the gold cap, the SI does not move, but a radial pressure wave develops and pushes the surrounding particles outward (red arrows). At radial distances larger than a few particle diameters, no motion is observed, suggesting that the pressure wave, and consequently the particle velocities, have decayed to zero. Despite this fact, in a semidiluted monolayer (Fig. 1C), some particles that are located far away from the SI are also set into motion (red circles). Remarkably, the stress propagates only where the particles are close enough to contact and form chains. This observation rules out any possible long-distance displacement caused by the pressure wave or fluid flows associated to the ejected Au plasma, which are strictly isotropic (Fig. 1B and Movie S1). Particle–particle interactions are responsible for the propagation, instead. Further details on PLA are in SI Materials and Methods.

To study these interactions in well-defined structures, we assemble highly ordered (mean hexagonal order parameter ψ6 = 0.95 ± 0.05) 2D colloidal lattices of light-transparent particles (Fig. 1D, white particles) with a single light-absorbing inclusion (Fig. 1D, black particle). Under illumination of the SI by a laser pulse, the pressure wave sets into motion the first surrounding layer of particles. These particles travel radially due to inertia (Fig. 1B) and transmit the stress to the rest of the lattice (Movie S2). Here, the particles located along the symmetry axes of the monolayer move more efficiently than the others in the lattice (Fig. 1E, red arrows).

We monitor the propagation of strain through the colloidal crystals by recording images at ~300 kHz and by measuring the global velocity V of the particles from their overall displacements (SI Materials and Methods and Fig. 2 AD). We reach velocities up to few meters per second, far beyond the typical velocities of colloidal particles tracked in shear experiments (27?29). The laser pulse energy determines the initial velocity V0 of the first layer of particles around the SI, as discussed below and in SI Materials and Methods. The lattice efficiently absorbs small stresses (e.g., Fig. 2A), whereas more intense perturbations propagate further in the crystals, traveling primarily along the symmetry axes (e.g., Fig. 2D). When the viscosity of the surrounding fluid is increased, at similar maximum propagation distances, the propagation becomes less directional, and all particles, including those not on the crystalline axes, undergo measurable displacements (compare Fig. 2E, η = 1 mPa?s and Fig. 2F, η = 4 mPa?s).

Fig. 2.

Propagation of strain waves in 2D colloidal crystals at different excitation levels. (AD) Experimental data in water (η = 1 mPa?s). The stress is transmitted along the six symmetry axes of the crystals. The global velocities V and the propagation distance increase with the laser pulse energy (LE). LE is (A) 0.09, (B) 0.14, (C) 0.15, and (D) 0.16 μJ. The red crosses correspond to the SIs. The surrounding white particles were not tracked because of their out-of-plane displacement (buckling). The size of the dots does not match the physical size of the particles. (E and F) In fluids with larger viscosity η, the stress propagation becomes more isotropic. LE is (E) 0.17 and (F) 0.25 μJ; η is (E) 1 and (F) 4 mPa?s. (G and H) Numerical simulations (drawn using Voronoi tessellation; see SI Materials and Methods) show the same behavior; η is (G) 1 and (H) 4 mPa?s. Excitation velocity V0 is (G) 12 and (H) 26 m/s. (I and J) Decay of the instantaneous velocity Vp along a particle chain; η is (I) 1 and (J) 4 mPa?s. The colors correspond to different particles, from black (first particle next to the SI) to red.

To understand the physics behind the propagation/absorption of local strains, we perform 2D numerical simulations of discrete particle lattices (SI Materials and Methods). We model the particles’ motion accounting for Stokes’ drag, Hertzian contacts (30), and hydrodynamic lubrication forces (31). We also set values for the average interparticle gap before excitation (d = 400 nm) and for the particle surface roughness (ξ = 8.5 nm) that match the experimental conditions (SI Materials and Methods and Fig. S1 A and B). We do not include contact friction between the particles. Friction between the particles and the substrate is also neglected, since a thin fluid film is always present and prevents direct contact (SI Materials and Methods and Fig. S1C). The simulations are initialized by setting the excitation velocity V0. Numerical results for η = 1 mPa?s and η = 4 mPa?s (Fig. 2 G and H) faithfully reproduce the experiments shown in Fig. 2 E and F and reveal that the more isotropic propagation at large η stems from increasing lubrication forces, through which moving neighboring particles drag each other. The propagation depth of the strain waves into the colloidal lattice is estimated by studying the wave decay within the directions of maximum propagation, i.e., the symmetry axes around the SI. Numerical simulations allow the perturbation to be monitored by looking at the instantaneous particle velocities Vp (Fig. 2 I and J) along the alignment direction, from the excitation spot (Vp = V0) to the periphery (Vp ? V0). The signature of interparticle contacts is revealed by a steep wavefront radiated from the SI, as long as the Hertzian elastic potential is involved in the interactions between the colloids (SI Materials and Methods and SI Discussion). The data confirms that larger fluid viscosity causes faster dissipation along symmetry axes. As soon as the momentum is too weak to lead to contact and elastic deformation, the steep front disappears in favor of a smoother decay (Fig. 2J, for t > 0.2 μs) driven by the diffusion of particle inertia, mediated by the fluid viscosity (SI Materials and Methods and SI Discussion). The speed of the steep wavefronts (Fig. 2I, cw ≈ 165 ± 25 m/s and Fig. 2J, cw ≈ 275 ± 95 m/s) is at least one order of magnitude larger than the velocity of the colloids and one order of magnitude smaller than the wave speed in the bulk material of the particles (c0) or in the solvent (cwater), i.e., V0 ? cw ? (c0, cwater). This distinct separation of timescales rules out any effect of flow advection or wave propagation in the fluid on the elastic wave radiation through the particle network. At these wave speeds, the wavelength λ = cw/f in the lattice is similar to the size of the particles and the size of the laser spot (~10 μm, SI Materials and Methods).

Fig. S1.

Characterization of the colloidal monolayers. (A) A 3D profile of the particle surface roughness, as measured from AFM data (Inset) after subtracting the hemispherical profile. (B) Statistical distribution of the interparticle gap d; p(d) is extracted directly from the first peak of the pair correlation function g(r) of the lattice. Small negative values in the distribution of surface-to-surface gaps come from minor particle size poyldispersity. (C) Force?distance curve measured between a SiO2 particle (R = 3.69 μm) and a glass surface in DI water. Black and red data correspond to the approach and the retraction mode, respectively. (Inset) Frictional force versus normal load for sliding velocity V = 2 (black), 4 (red), and 8 (blue) μm/s.

Experimentally, resolving the instantaneous particle velocity or the wavefront speed requires accuracy far beyond the capacity of high-speed optical imaging. Instead, the energy absorption properties and the full acoustic features (32) of the monolayers can be quantified from the decay of the global velocity V of the particles, and compared with the simulations. First, we fix the initial conditions, i.e., the LE in the experiments (Fig. 3 AC, LE = 0.17 μJ) and the excitation velocity V0 in the numerical simulations (Fig. 3 DF, V0 = 12 m/s), and then increase the viscosity of the medium (SI Materials and Methods). The quantitative agreement between simulations and experiments indirectly supports the hypothesis that the laser intensity determines the initial particle velocity. After excitation, the wave propagation is strongly affected by the solvent viscosity due to the fluid flow induced by the motion of the particles. This is further quantified by Fig. 3G, which shows how the global particle velocity decays along the symmetry axes of the crystal, i.e., along chains of particles j = 1, 2, ..., 6, as a function of their distance L from a given particle i with velocity Vi (SI Materials and Methods). A semilog plot of the data reveals an exponential decay <mml:math><mml:mrow><mml:msub><mml:mi>V</mml:mi><mml:mrow><mml:mi>j</mml:mi><mml:mo>></mml:mo><mml:mi>i</mml:mi></mml:mrow></mml:msub><mml:mo>=</mml:mo><mml:msub><mml:mi>V</mml:mi><mml:mi>i</mml:mi></mml:msub><mml:mo>?</mml:mo><mml:mi>exp</mml:mi><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:mo>?</mml:mo><mml:mrow><mml:mi>L</mml:mi><mml:mo>/</mml:mo><mml:mrow><mml:msub><mml:mi>l</mml:mi><mml:mrow><mml:mtext>att</mml:mtext></mml:mrow></mml:msub></mml:mrow></mml:mrow></mml:mrow><mml:mo>)</mml:mo></mml:mrow></mml:mrow></mml:math>Vj>i=Vi?exp(?L/latt), where the attenuation length latt measures the penetration depth of the mechanical perturbation. On average, latt decreases with the viscosity of the dispersing fluid (Fig. 3G, Inset). Experiments (solid symbols) and simulations (empty symbols and solid line) reveal a similar response of the material to the applied pulse, in agreement with the velocity maps shown in Fig. 3 AF. In the simulations, the attenuation length is robustly extracted from the decay of the energy field E, <mml:math><mml:mrow><mml:msub><mml:mrow><mml:mover accent="true"><mml:mi>l</mml:mi><mml:mo>~</mml:mo></mml:mover></mml:mrow><mml:mrow><mml:mtext>att</mml:mtext></mml:mrow></mml:msub></mml:mrow></mml:math>l~att, compatibly with the decay of the velocity field, latt = 2<mml:math><mml:mrow><mml:msub><mml:mrow><mml:mover accent="true"><mml:mi>l</mml:mi><mml:mo>~</mml:mo></mml:mover></mml:mrow><mml:mrow><mml:mtext>att</mml:mtext></mml:mrow></mml:msub></mml:mrow></mml:math>l~att [SI Materials and Methods: E is proportional to the kinetic energy of the particles, E(r,t) ∝ exp(?r/<mml:math><mml:mrow><mml:msub><mml:mrow><mml:mover accent="true"><mml:mi>l</mml:mi><mml:mo>~</mml:mo></mml:mover></mml:mrow><mml:mrow><mml:mtext>att</mml:mtext></mml:mrow></mml:msub></mml:mrow></mml:math>l~att) ∝ V2(r) ∝ exp(?2r/latt), where r is the distance from the source].

Fig. 3.

Average strain wave penetration depth in 2D colloidal crystals dispersed in fluids with different viscosity. A fast dissipation in highly viscous fluids is revealed by the (AC) experimental and (DF) numerical velocity maps for η = (A and D) 1, (B and E) 4, and (C and F) 10 mPa?s. Initial conditions are fixed LE (experiments, LE = 0.17 μJ) and fixed instantaneous initial velocity (simulations, V0 = 12 m/s). The average value of ψ6, calculated over a circular region with a four-lattice-constants radius centered on the Sis, is (A?C) 0.98 ± 0.02 for the experiments and (DF) 1 for the simulations. This dissipation is quantified by the (G) decay of the global velocity V for any initial velocity Vi of particles aligned along the symmetry axes of the crystal, plotted versus the distance from particle i (in units of 2R). Solid symbols correspond to experimental data obtained by averaging over ~10 chains. Dashed lines are fits to the experimental data by an exponential law <mml:math><mml:mrow><mml:msub><mml:mi>V</mml:mi><mml:mrow><mml:mi>j</mml:mi><mml:mo>></mml:mo><mml:mi>i</mml:mi></mml:mrow></mml:msub><mml:mo>=</mml:mo><mml:msub><mml:mi>V</mml:mi><mml:mi>i</mml:mi></mml:msub><mml:mo>?</mml:mo><mml:mi>exp</mml:mi><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:mo>?</mml:mo><mml:mrow><mml:mi>L</mml:mi><mml:mo>/</mml:mo><mml:mrow><mml:msub><mml:mi>l</mml:mi><mml:mrow><mml:mtext>att</mml:mtext></mml:mrow></mml:msub></mml:mrow></mml:mrow></mml:mrow><mml:mo>)</mml:mo></mml:mrow></mml:mrow></mml:math>Vj>i=Vi?exp(?L/latt) with a characteristic attenuation length latt. The colors correspond to η = 1 (black), 4 (red), and 10 (blue) mPa?s. (Inset) Experimental (solid symbols) and numerical (empty symbols) attenuation lengths as a function of the viscosity of the dispersing medium. The solid line shows the trend of the numerical data.

The radial motion of the particles away from the SI is the consequence of two distinct mechanisms: (i) a radial expansion, driven by inertia and normal lubrication forces with diffusive momentum transfer and (ii) a weakly attenuated propagation, triggered by elastic deformations of the particles (9?11) (SI Materials and Methods). The first regime involves dissipation due to Stokes’ drag and tangential lubrication interactions. In the second regime, when two colloids reach sufficiently small separation distances, the strain rate and the stress in the interstitial fluid diverge: the fluid clamped by its viscosity (11) within the surface roughness behaves in a “solid-like” manner (9, 33), and the particles deform elastically (strain > 10?4) against the confined liquid layer. The particle surface roughness identifies the critical cutoff distance for occurrence of elastic deformation (9) (Fig. S1A). The behavior of latt with η can be, in fact, only explained by taking into account elastic deformation of the particles, as captured by 2D numerical simulations (SI Materials and Methods) and compatibly with an elementary 1D description (SI Discussion). These conditions (V ≈ 1 m/s, <mml:math><mml:mrow><mml:msup><mml:mrow><mml:mover accent="true"><mml:mi>γ</mml:mi><mml:mo>˙</mml:mo></mml:mover></mml:mrow><mml:mrow><mml:mo>?</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msup><mml:mo>≈</mml:mo></mml:mrow></mml:math>γ˙?1≈ 10?6 s to 10?9 s) are in contrast to the case of jamming suspensions under shear flows (e.g., shear thickening <mml:math><mml:mrow><mml:msup><mml:mrow><mml:mover accent="true"><mml:mi>γ</mml:mi><mml:mo>˙</mml:mo></mml:mover></mml:mrow><mml:mrow><mml:mo>?</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msup><mml:mo>≈</mml:mo></mml:mrow></mml:math>γ˙?1≈ 10?1 s to 10?3 s), in which the fluid has time to escape upon particle?particle contact (20, 21). These shear rates are also two to three orders of magnitude larger than macroscopic shear rates observed for impact protection materials employing shear-thickening fluid, but may become relevant for higher-energy projectiles, e.g., in spacecraft shielding (34).

The data presented were obtained using perfect hexagonal lattices. However, the stress propagation is drastically affected by particle misalignments (Fig. S3) and by the presence of structural defects in the lattices. We report the velocity maps of monolayers that include local defects, such as a dislocation (Fig. 4A) or a vacancy (Fig. 4B). In both cases, the stress propagation is abruptly arrested at the defect. In the extreme case of disordered (glassy) monolayers (SI Materials and Methods and Fig. 4C), the wave propagation becomes very short-ranged, even when the SI is illuminated at high power (LE = 0.16 μJ). Numerical simulations of aqueous monolayers (η = 1 mPa?s, V0 = 12 m/s) with a controlled degree of disorder (SI Materials and Methods) highlight the propagation depth of stress pulses as a function of the hexagonal order parameter ψ6 (Fig. 4D). The attenuation length <mml:math><mml:mrow><mml:msub><mml:mrow><mml:mover accent="true"><mml:mi>l</mml:mi><mml:mo>~</mml:mo></mml:mover></mml:mrow><mml:mrow><mml:mtext>att</mml:mtext></mml:mrow></mml:msub></mml:mrow></mml:math>l~att of the ballistic coherent (35) field (SI Materials and Methods) swiftly drops to ~2R, because of multiple scattering (35), within 0.85 < ψ6 < 1 (Fig. 4D, Inset, red), while the packing (area) fraction ? of the material remains constant (Fig. 4D, Inset, blue). This indicates that local order dictates the propagation of strain. Values of ψ6 < 0.85 unavoidably lower the packing fraction and increase the initial separation d between the colloids (Fig. 4E). Fig. 4F shows the dependence of <mml:math><mml:mrow><mml:msub><mml:mrow><mml:mover accent="true"><mml:mi>l</mml:mi><mml:mo>~</mml:mo></mml:mover></mml:mrow><mml:mrow><mml:mtext>att</mml:mtext></mml:mrow></mml:msub></mml:mrow></mml:math>l~att on d in disordered (any ψ6 < 1, red) and crystalline (ψ6 = 1, black) monolayers. In the crystalline case, <mml:math><mml:mrow><mml:msub><mml:mrow><mml:mover accent="true"><mml:mi>l</mml:mi><mml:mo>~</mml:mo></mml:mover></mml:mrow><mml:mrow><mml:mtext>att</mml:mtext></mml:mrow></mml:msub></mml:mrow></mml:math>l~att depends weakly on d, whereas disordered structures cause stronger attenuations. This observation, in conjunction with Fig. 4E (ψ6 vs. d), demonstrates that the decay of strain pulses in samples with randomness is due to multiple scattering rather than to the packing density.

Fig. 4.

Strain propagation in the presence of local disorder. (AC) The pulse induced by the SI rapidly fades when traveling through (A) dislocations, (B) vacancies, or (C) glassy structures. The circles corresponding to particles are drawn with a size proportional to their value of ψ6. (D) Attenuation length <mml:math><mml:mrow><mml:msub><mml:mrow><mml:mover accent="true"><mml:mi>l</mml:mi><mml:mo>~</mml:mo></mml:mover></mml:mrow><mml:mrow><mml:mtext>att</mml:mtext></mml:mrow></mml:msub></mml:mrow></mml:math>l~att (in units of 2R, numerical data) extracted from the energy decay of the ballistic pulse (SI Materials and Methods) in colloidal monolayers excited in water at V0 = 12 m/s. Each point is an average of 20 configurations. (Inset) At 0.85 < ψ6 < 1, the packing (area) fraction ? is constant (blue data) and <mml:math><mml:mrow><mml:msub><mml:mrow><mml:mover accent="true"><mml:mi>l</mml:mi><mml:mo>~</mml:mo></mml:mover></mml:mrow><mml:mrow><mml:mtext>att</mml:mtext></mml:mrow></mml:msub></mml:mrow></mml:math>l~att shows a rapid decrease (red data); ? and ψ6 are computed within a disk of radius 50 μm around the SI (i.e., the center of simulating box). (E) At ψ6 < 0.85, the hexagonal order parameter ψ6 and the initial mean interparticle distance d start to be strongly correlated. (F) Attenuation length <mml:math><mml:mrow><mml:msub><mml:mrow><mml:mover accent="true"><mml:mi>l</mml:mi><mml:mo>~</mml:mo></mml:mover></mml:mrow><mml:mrow><mml:mtext>att</mml:mtext></mml:mrow></mml:msub></mml:mrow></mml:math>l~att as a function of d for a perfect crystal (black circles) and for disordered structures (red circles) prepared as described in SI Materials and Methods. The effect of the interparticle distance on <mml:math><mml:mrow><mml:msub><mml:mrow><mml:mover accent="true"><mml:mi>l</mml:mi><mml:mo>~</mml:mo></mml:mover></mml:mrow><mml:mrow><mml:mtext>att</mml:mtext></mml:mrow></mml:msub></mml:mrow></mml:math>l~att is weak (black data), and the energy decay is mostly due to an increase of disorder.

Fig. S3.

Numerical simulations of the momentum transfer in single units with misalignments. (A) Momentum transfer p to a steady colloid (red) due to the collision with a striking particle (gray) at an initial gap d with initial velocity V0 = 5 m/s and misalignment φ. Inset refers to the initial condition, before the striker is set in motion. (B) Momentum transfer after collision of the same striker with two target colloids; φ is defined with respect to the red particle (see Inset in A). (C) Energy transfer after a three-particle collision. K0 is the kinetic energy of the striker, <mml:math><mml:mrow><mml:msub><mml:mi>K</mml:mi><mml:mi>T</mml:mi></mml:msub><mml:mo>=</mml:mo><mml:mstyle displaystyle="true"><mml:msubsup><mml:mo>∑</mml:mo><mml:mrow><mml:mi>i</mml:mi><mml:mo>=</mml:mo><mml:mn>1</mml:mn></mml:mrow><mml:mn>2</mml:mn></mml:msubsup><mml:mrow><mml:mrow><mml:mo>(</mml:mo><mml:mn>1</mml:mn><mml:mo>/</mml:mo><mml:mn>2</mml:mn><mml:mo>)</mml:mo></mml:mrow><mml:mi>m</mml:mi><mml:msubsup><mml:mi>V</mml:mi><mml:mi>i</mml:mi><mml:mn>2</mml:mn></mml:msubsup></mml:mrow></mml:mstyle></mml:mrow></mml:math>KT=∑i=12(1/2)mVi2 and <mml:math><mml:mrow><mml:msub><mml:mi>K</mml:mi><mml:mi>R</mml:mi></mml:msub><mml:mo>=</mml:mo><mml:mstyle displaystyle="true"><mml:msubsup><mml:mo>∑</mml:mo><mml:mrow><mml:mi>i</mml:mi><mml:mo>=</mml:mo><mml:mn>1</mml:mn></mml:mrow><mml:mn>2</mml:mn></mml:msubsup><mml:mrow><mml:mrow><mml:mo>(</mml:mo><mml:mn>1</mml:mn><mml:mo>/</mml:mo><mml:mn>2</mml:mn><mml:mo>)</mml:mo></mml:mrow><mml:mi>I</mml:mi><mml:msubsup><mml:mrow><mml:mover accent="true"><mml:mi>θ</mml:mi><mml:mo>˙</mml:mo></mml:mover></mml:mrow><mml:mi>i</mml:mi><mml:mn>2</mml:mn></mml:msubsup></mml:mrow></mml:mstyle></mml:mrow></mml:math>KR=∑i=12(1/2)Iθ˙i2 are the translational and rotational kinetic energies of the targets, where m and I are the mass and the moment of inertia of the particles, and Vi and <mml:math><mml:mrow><mml:msub><mml:mrow><mml:mover accent="true"><mml:mi>θ</mml:mi><mml:mo>˙</mml:mo></mml:mover></mml:mrow><mml:mi>i</mml:mi></mml:msub></mml:mrow></mml:math>θ˙i denote the translational and rotational velocities; d = 400 nm, and η = 1 mPa·s. Here, dissipation from Stokes drag is not included in the equations of motion; we focus only on the analysis of particle interactions.

All of our observations unambiguously show how the propagation of localized “extreme” strain waves depends on the excitation energy, the local particle arrangement, and the solvent viscosity. This mechanism is qualitatively different from direct (frictional) and indirect (hydrodynamic) contact-based models describing fluids jamming at lower shear rates. Instead, it sheds light on the mechanical response to much faster deformation rates, e.g., during impact and shocks, offering insights on the stress propagation and energy absorption of dense suspensions where elastic contacts can be specifically designed, e.g., by introducing local defects or by changing the solvent viscosity.

SI Materials and Methods

Preparation and Characterization of Colloidal Monolayers.

Our model materials consist of SiO2 colloidal particles dispersed in aqueous solvents. We employ a mixture of light-transparent and light-absorbing particles with the same size (R = 3.69 μm). The light-transparent particles are commercial SiO2 microbeads (SiO2-R-1076-2, in aqueous solution; Microparticles GmbH). The particles have ~3% polydispersity, corresponding to ±100 nm in the particle’s radius. We have performed atomic force microscopy (AFM) measurements to estimate the surface roughness ξ of the colloids (Fig. S1A), which is also a key input parameter of our numerical simulations: SiO2 particles are first dried on a silicon substrate and then imaged with an AFM to extract their surface topography (Fig. S1A, Inset); ξ = 8.5 nm is hereby defined as the peak-to-valley height of the roughness distribution and constitutes the cutoff distance employed in the numerical simulations. At small separation distances, 2ξ = 17 nm, the interstitial fluid is trapped between asperities and hardly flows; beyond this limit, the particles elastically deform.

Light-absorbing particles are metal-coated Janus spheres prepared from the same SiO2 beads. The colloids are first deposited on a microscope slide. After drying, we evaporate 5 nm of chromium (adhesion layer) followed by 50 nm of gold (light-absorbing layer) with a 90° glancing angle, thus achieving the coating of one hemisphere. The Janus particles are then redispersed via sonication in deionized (DI) water (MilliQ), in which the uncoated beads are also added. Suspensions with larger viscosity are prepared by replacing water with a mixture of DI water/glycerin (η = 4 mPa·s, Glyc. 40%wt, η = 10 mPa·s, Glyc. 60%wt) (36).

The mixture of light-absorbing and light-transparent particles is pipetted into a 1-mm-thick closed glass sample cell (cell 137-QS; Hellma Analytics) where the particles sediment onto the lower surface. Close-packed monolayers are finally obtained by tilting the cell by 10°; the colloids slide on the substrate and accumulate near a lateral wall forming hexagonal 2D crystals.

To outline the typical interparticle distance of the crystalline monolayer at rest, we measure the position of the first peak of the radial distribution function g(r), which describes the probability of finding two particles at relative center-to-center distance r. We report the probability distribution of the interparticle gap d = r ? 2R showing a mean value of ~400 nm (Fig. S1B), which also constitutes an input for the numerical simulations.

When 2D disordered structures are investigated (Fig. 4C), we use a 50:50 combination of smaller (R1 = 3.14 μm) and larger (R2 = 3.69 μm) light-transparent particles, instead. The presence of polydispersity prevents the formation of ordered lattices. The degree of hexagonal order is quantified by looking at the local hexagonal order parameter, which, for a given particle m, is defined as (37)<mml:math display="block"><mml:mrow><mml:msubsup><mml:mi>ψ</mml:mi><mml:mn>6</mml:mn><mml:mi>l</mml:mi></mml:msubsup><mml:mrow><mml:mo>(</mml:mo><mml:mi>m</mml:mi><mml:mo>)</mml:mo></mml:mrow><mml:mo>=</mml:mo><mml:mfrac><mml:mn>1</mml:mn><mml:mi>N</mml:mi></mml:mfrac><mml:mstyle displaystyle="true"><mml:munderover><mml:mo>∑</mml:mo><mml:mrow><mml:mi>n</mml:mi><mml:mo>=</mml:mo><mml:mn>1</mml:mn></mml:mrow><mml:mi>N</mml:mi></mml:munderover><mml:mrow><mml:mi>exp</mml:mi><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:mn>6</mml:mn><mml:mi>j</mml:mi><mml:msubsup><mml:mi>θ</mml:mi><mml:mi>n</mml:mi><mml:mi>m</mml:mi></mml:msubsup></mml:mrow><mml:mo>)</mml:mo></mml:mrow></mml:mrow></mml:mstyle><mml:mo>,</mml:mo></mml:mrow></mml:math>ψ6l(m)=1N∑n=1Nexp(6jθnm),where N is the number of n nearest neighbors of m and <mml:math><mml:mrow><mml:msubsup><mml:mi>θ</mml:mi><mml:mi>n</mml:mi><mml:mi>m</mml:mi></mml:msubsup></mml:mrow></mml:math>θnm is the angle between the horizontal axis and the bond linking m to n. The hexagonal order parameter of a single monolayer, <mml:math><mml:mrow><mml:msubsup><mml:mi>ψ</mml:mi><mml:mn>6</mml:mn><mml:mi>L</mml:mi></mml:msubsup><mml:mo>=</mml:mo><mml:mrow><mml:mo>〈</mml:mo><mml:mrow><mml:msubsup><mml:mi>ψ</mml:mi><mml:mn>6</mml:mn><mml:mi>l</mml:mi></mml:msubsup></mml:mrow><mml:mo>〉</mml:mo></mml:mrow></mml:mrow></mml:math>ψ6L=〈ψ6l〉, is computed as the average of the local parameter of the particles lying inside a circular region with a 50-μm radius around the SI; this region contains more than 100 particles. When several configurations with similar disorder are considered, <mml:math><mml:mrow><mml:msub><mml:mi>ψ</mml:mi><mml:mn>6</mml:mn></mml:msub></mml:mrow></mml:math>ψ6 is the mathematical average. Everywhere in the manuscript, we only consider the absolute value of the order parameters introduced above.

To observe the dynamic behavior of the aqueous mixture of colloidal particles, we employ a high-speed microscope imaging system, which consists of a high-speed camera (Vision Research Phantom v1610) and a long working-distance microscope (Infinity K2/SC). The particle mixtures are imaged at a maximum of 300 kHz, at a spatial resolution of 0.90 μm per pixel, with a total resolution of 128 × 128 pixels. The trajectories of the particles are finally obtained by analyzing the acquired high-speed videos with a Matlab image processing toolbox and custom tracking routines.

Friction Between the Particles and the Glass Substrate.

The presence of a glass slide underneath the colloidal monolayer raises natural questions about possible friction between the particles and the substrate. We glue a colloidal particle onto a tipless AFM cantilever and performed colloidal AFM tests to probe any possible adhesion between our microparticles and the glass substrate. Fig. S1C shows the force?distance curves during an approach?retraction (black?red) cycle using a 2-nN load. The overlap of the two curves indicates the absence of adhesion between the colloid and the glass slide. Plots obtained using other loads (0.5, 10, and 50 nN) show identical behavior. In addition to the static adhesion test, Fig. S1C, Inset shows the frictional force versus normal load for three values of lateral velocity/scan rate (V = 2, 4, and 8 μm/s), as measured by AFM. The curves cross the origin for all scan rates and applied normal loads, thus suggesting that the motion of the particle on the substrate always happens within the regime of hydrodynamic lubrication. The normal load corresponding to the buoyant weight of one of the particles (~2 pN) is significantly lower than the normal loads applied in the AFM tests. This implies that a lubricating fluid layer is always present between the particles and the glass substrate. Moreover, at larger velocities, such as the ones characteristic of our experiments, the lubricating film becomes even thicker (5).

Data Processing.

In Fig. 3G, we describe the global velocity decay of the particles located in the six symmetry axes of the colloidal lattice. We consider monolayers immersed in fluids with three different viscosities: η = 1 mPa?s (black data), η = 4 mPa?s (red data), and η = 10 mPa?s (blue data). Experimental and numerical results are processed as follows. For the first six colloids j = 1, 2,.., 6 that belong to a chain (symmetry axis), we plot the global velocity Vj > i of the particles farther down the chain as a function of the relative distance L = j ? i. Data points that correspond to the same L are finally averaged to highlight the mean wave decay in the crystal. Fig. 3G shows excellent agreement between numerical (dashed lines) and experimental (symbols) data; we observe that the larger the fluid viscosity is, the steeper is the decay of the stress pulse.

PLA.

The mechanical excitation of Janus, Au-coated, microparticles (or SIs) is achieved using a focused pulsed laser beam that targets the surface of selected particles. The laser is a 532-nm wavelength, Q-switched Nd:YAG pulsed laser with a pulse duration of 4 ns and diameter of the spot of ~10 μm. The pulse energy is controlled through a half-wave plate mounted on a motorized stage. The beam comes from below the glass sample cell (Fig. 1A) and is focused on a single SI. The temperature rise that is induced by the focused pulsed energy results in the vaporization and removal of material from the particles’ surface. If the SI is not confined, the momentum of the ejected Au plasma causes an abrupt motion of the colloid in the opposite direction (Fig. S2 A and B and Movie S3, before and after illumination, respectively). The particle displacement, Δr, after PLA grows with the laser pulse energy (Fig. S2C). We report a similar increase of displacements regardless of the solvent viscosity (albeit the absolute values of Δr become smaller at larger η). No motion is observed when the SIs are capless (Movie S4). As a consequence of the fast expansion of plasma, a radial pressure wave develops from the ablated surface (Movie S1, obtained using a free-standing gold piece). Even when the SI is anchored to the substrate, this pressure wave is sufficient to push the neighboring particles outward at velocities that decrease with the distance from the SI, as described in the main text (Fig. 1B).

Fig. S2.

Local excitation of Au-coated colloidal particles (SI) via PLA. (A) Au-coated Janus particles (SIs) on a glass surface are (B) selectively excited by a green pulsed laser beam. The illuminated particle (red dashed circle and sketch in Inset) is propelled in one direction (red arrows) within a few microseconds due to the ablation of gold plasma in the opposite direction. (C) Plot of the distance Δr, normalized by the particle radius R, traveled by the SI after illumination at various the laser energies LE for η = 1 (black), 4 (red), and 10 (blue) mPa?s.

When laser ablation is induced on the surface of a particle, the energy absorbed by the surface is expected to be less than 5% of the laser pulse energy, due to the ~95% reflectivity of gold at 532 nm and other experimental losses. The absorbed energy is converted into the kinetic energy of the ablated plasma and the particle. The ablated plasma has highly directional momentum (pointing outward from the particle surface) and carries most of the energy, because of the small mass compared to the particle (Δm/m < 0.1% in this case). For dry particles, it has been observed that the kinetic energy gained by the particles is much less than the laser energy (38). In this work, we employ a less direct mechanism of excitation where a pressure wave (generated due to the ablated mass) is used to drive particles next to SI. We typically deliver less than 0.5 μJ of energy within a time frame of 4 ns, and a much smaller fraction of energy obtained (~0.01%) is acquired by the particles. In this system, the majority of the absorbed laser energy is carried by the pressure wave and dispersed away (in the bulk) within short distances from the SI.

PLA is a complex phenomenon, and we recognize the presence of potential experimental pitfalls in the description of physical events occurring in close proximity to the ablated particle. For example, the temperature increase affects the medium viscosities and leads to a local burst. This effect is evident in Movie S1, in which we use a thick free-standing gold piece to highlight the pressure wave. We also observe buckling of the monolayers when large illumination powers are used (e.g., see Movie S1). Finally, in relation to the excitation mechanisms described in the main text, the pressure shock wave (Fig. 1B and Movie S1) is responsible for the isotropic strain wave propagation, while the propulsion of the SI following PLA (Fig. S2 A and B and Movie S3) may lead to asymmetries along specific directions in the experiments. The isotropic pressure wave is, in any case, two orders of magnitude faster than particle motion, and its contribution occurs earlier; hence this can be considered to be the dominant mechanism. A detailed and exhaustive description of these phenomena goes beyond the scope of this manuscript. Fig. 1 B and C of the main text reveals that the shock wave decays very fast (within one to three particle diameters) in the medium and that long-range strain propagations in the monolayer are only due to particle?particle interactions (Hertzian contact and lubrication forces), which also occur on much shorter time scales. The analysis based on the velocity V of the first “available” layer around the SI and the excellent quantitative agreement with numerical simulations leave no ambiguity and allow us to neglect, in first approximation, the details of local phenomena around the SI.

Numerical Simulations.

We employ a discrete element method to simulate the wave propagation in the colloidal monolayers. The particles interact via hydrodynamic lubrication and Hertzian elastic forces (9, 10, 30, 31). We solve equations of motion depending on the interparticle distance dij = |rj-ri|-2R, where ri and rj are the absolute positions of two colliding particles and R is the particle radius. At large distances, only lubrication forces are computed. When dij is smaller than a cutoff distance dcut = 2ξ, elastic interactions are also included. Here, ξ = 8.5 nm is the peak-to-valley surface roughness of a particle measured by AFM (9) (Fig. S1A).

Each particle interacts only with its six nearest neighbors. Our minimal model neglects long-range hydrodynamic forces that play a minor role when a dense system is confined next to a surface, i.e., the glass slide underneath the monolayer (39).

When the interparticle distance dij is larger than the cutoff distance, the equations of motion for the translational and rotational motion of the ith particle with mass mi and inertia Ii are given by<mml:math display="block"><mml:mrow><mml:msub><mml:mi>m</mml:mi><mml:mi>i</mml:mi></mml:msub><mml:mfrac><mml:mrow><mml:msup><mml:mi>d</mml:mi><mml:mn>2</mml:mn></mml:msup><mml:msub><mml:mi mathvariant="bold-italic">r</mml:mi><mml:mi>i</mml:mi></mml:msub></mml:mrow><mml:mrow><mml:mi>d</mml:mi><mml:msup><mml:mi>t</mml:mi><mml:mn>2</mml:mn></mml:msup></mml:mrow></mml:mfrac><mml:mo>=</mml:mo><mml:mstyle displaystyle="true"><mml:munderover><mml:mo>∑</mml:mo><mml:mi>j</mml:mi><mml:mrow><mml:mi>n</mml:mi><mml:mi>n</mml:mi></mml:mrow></mml:munderover><mml:mrow><mml:msub><mml:mi mathvariant="bold-italic">F</mml:mi><mml:mrow><mml:mtext>lub</mml:mtext></mml:mrow></mml:msub><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:msub><mml:mi>d</mml:mi><mml:mrow><mml:mi>i</mml:mi><mml:mi>j</mml:mi></mml:mrow></mml:msub><mml:mo>,</mml:mo><mml:msub><mml:mi mathvariant="bold-italic">V</mml:mi><mml:mrow><mml:mi>i</mml:mi><mml:mi>j</mml:mi></mml:mrow></mml:msub></mml:mrow><mml:mo>)</mml:mo></mml:mrow><mml:mo>+</mml:mo><mml:msub><mml:mi mathvariant="bold-italic">F</mml:mi><mml:mrow><mml:mtext>drag</mml:mtext></mml:mrow></mml:msub><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:msub><mml:mi mathvariant="bold-italic">V</mml:mi><mml:mi>i</mml:mi></mml:msub></mml:mrow><mml:mo>)</mml:mo></mml:mrow></mml:mrow></mml:mstyle><mml:mo>,</mml:mo></mml:mrow></mml:math>mid2ridt2=∑jnnFlub(dij,Vij)+Fdrag(Vi),<mml:math display="block"><mml:mrow><mml:msub><mml:mi>I</mml:mi><mml:mi>i</mml:mi></mml:msub><mml:mfrac><mml:mrow><mml:msup><mml:mi>d</mml:mi><mml:mn>2</mml:mn></mml:msup><mml:msub><mml:mi mathvariant="bold-italic">θ</mml:mi><mml:mi>i</mml:mi></mml:msub></mml:mrow><mml:mrow><mml:mi>d</mml:mi><mml:msup><mml:mi>t</mml:mi><mml:mn>2</mml:mn></mml:msup></mml:mrow></mml:mfrac><mml:mo>=</mml:mo><mml:mstyle displaystyle="true"><mml:munderover><mml:mo>∑</mml:mo><mml:mi>j</mml:mi><mml:mrow><mml:mi>n</mml:mi><mml:mi>n</mml:mi></mml:mrow></mml:munderover><mml:mrow><mml:msub><mml:mi mathvariant="bold-italic">T</mml:mi><mml:mrow><mml:mtext>lub</mml:mtext></mml:mrow></mml:msub><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:msub><mml:mi>d</mml:mi><mml:mrow><mml:mi>i</mml:mi><mml:mi>j</mml:mi></mml:mrow></mml:msub><mml:mo>,</mml:mo><mml:msub><mml:mi mathvariant="bold-italic">V</mml:mi><mml:mrow><mml:mi>i</mml:mi><mml:mi>j</mml:mi></mml:mrow></mml:msub><mml:mo>,</mml:mo><mml:msub><mml:mi mathvariant="bold-italic">ω</mml:mi><mml:mrow><mml:mi>i</mml:mi><mml:mi>j</mml:mi></mml:mrow></mml:msub></mml:mrow><mml:mo>)</mml:mo></mml:mrow></mml:mrow></mml:mstyle><mml:mo>.</mml:mo></mml:mrow></mml:math>Iid2θidt2=∑jnnTlub(dij,Vij,ωij).Here, θi is the rotation angle of the ith particle, and ωij is the relative angular velocity.

The short-range hydrodynamic force and torque (lubrication regime) are defined as (20, 39)<mml:math display="block"><mml:mrow><mml:msub><mml:mi mathvariant="bold-italic">F</mml:mi><mml:mrow><mml:mtext>lub</mml:mtext></mml:mrow></mml:msub><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:msub><mml:mi mathvariant="italic">d</mml:mi><mml:mrow><mml:mi>i</mml:mi><mml:mi>j</mml:mi></mml:mrow></mml:msub><mml:mo>,</mml:mo><mml:msub><mml:mi mathvariant="bold-italic">V</mml:mi><mml:mrow><mml:mi>i</mml:mi><mml:mi>j</mml:mi></mml:mrow></mml:msub></mml:mrow><mml:mo>)</mml:mo></mml:mrow><mml:mo>=</mml:mo><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:mtable><mml:mtr><mml:mtd><mml:mrow><mml:mfrac><mml:mrow><mml:msub><mml:mrow><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:msub><mml:mi mathvariant="bold-italic">r</mml:mi><mml:mi>i</mml:mi></mml:msub><mml:mo>-</mml:mo><mml:msub><mml:mi 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mathvariant="normal">i</mml:mi></mml:msub><mml:mo>-</mml:mo><mml:msub><mml:mi mathvariant="bold-italic">r</mml:mi><mml:mi mathvariant="normal">j</mml:mi></mml:msub></mml:mrow><mml:mo>|</mml:mo></mml:mrow></mml:mrow></mml:mfrac></mml:mrow></mml:mtd></mml:mtr><mml:mtr><mml:mtd><mml:mrow><mml:mfrac><mml:mrow><mml:msub><mml:mrow><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:msub><mml:mi mathvariant="bold-italic">r</mml:mi><mml:mi>i</mml:mi></mml:msub><mml:mo>-</mml:mo><mml:msub><mml:mi mathvariant="bold-italic">r</mml:mi><mml:mi>j</mml:mi></mml:msub></mml:mrow><mml:mo>)</mml:mo></mml:mrow></mml:mrow><mml:mi>y</mml:mi></mml:msub></mml:mrow><mml:mrow><mml:mrow><mml:mo>|</mml:mo><mml:mrow><mml:msub><mml:mi mathvariant="bold-italic">r</mml:mi><mml:mi>i</mml:mi></mml:msub><mml:mo>-</mml:mo><mml:msub><mml:mi mathvariant="bold-italic">r</mml:mi><mml:mi>j</mml:mi></mml:msub></mml:mrow><mml:mo>|</mml:mo></mml:mrow></mml:mrow></mml:mfrac></mml:mrow></mml:mtd><mml:mtd><mml:mrow><mml:mfrac><mml:mrow><mml:mo>-</mml:mo><mml:msub><mml:mrow><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:msub><mml:mi mathvariant="bold-italic">r</mml:mi><mml:mi>i</mml:mi></mml:msub><mml:mo>-</mml:mo><mml:msub><mml:mi mathvariant="bold-italic">r</mml:mi><mml:mi>j</mml:mi></mml:msub></mml:mrow><mml:mo>)</mml:mo></mml:mrow></mml:mrow><mml:mi>x</mml:mi></mml:msub></mml:mrow><mml:mrow><mml:mrow><mml:mo>|</mml:mo><mml:mrow><mml:msub><mml:mi mathvariant="bold-italic">r</mml:mi><mml:mi>i</mml:mi></mml:msub><mml:mo>-</mml:mo><mml:msub><mml:mi mathvariant="bold-italic">r</mml:mi><mml:mi>j</mml:mi></mml:msub></mml:mrow><mml:mo>|</mml:mo></mml:mrow></mml:mrow></mml:mfrac></mml:mrow></mml:mtd></mml:mtr></mml:mtable></mml:mrow><mml:mo>)</mml:mo></mml:mrow><mml:mrow><mml:mo>[</mml:mo><mml:mrow><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:mtable><mml:mtr><mml:mtd><mml:mrow><mml:mfrac><mml:mrow><mml:mo>-</mml:mo><mml:mn>3</mml:mn><mml:mi>π</mml:mi><mml:mi>η</mml:mi><mml:msup><mml:mi>R</mml:mi><mml:mn>2</mml:mn></mml:msup></mml:mrow><mml:mrow><mml:mn>2</mml:mn><mml:msub><mml:mi>d</mml:mi><mml:mrow><mml:mi>i</mml:mi><mml:mi>j</mml:mi></mml:mrow></mml:msub></mml:mrow></mml:mfrac></mml:mrow></mml:mtd><mml:mtd><mml:mrow><mml:mfrac><mml:mrow><mml:mn>12</mml:mn><mml:mi>π</mml:mi><mml:mi>η</mml:mi><mml:msqrt><mml:mn>2</mml:mn></mml:msqrt><mml:msup><mml:mi>R</mml:mi><mml:mrow><mml:mrow><mml:mn>3</mml:mn><mml:mo>/</mml:mo><mml:mn>2</mml:mn></mml:mrow></mml:mrow></mml:msup></mml:mrow><mml:mrow><mml:mn>5</mml:mn><mml:msqrt><mml:mrow><mml:msub><mml:mi>d</mml:mi><mml:mrow><mml:mi>i</mml:mi><mml:mi>j</mml:mi></mml:mrow></mml:msub></mml:mrow></mml:msqrt></mml:mrow></mml:mfrac></mml:mrow></mml:mtd></mml:mtr><mml:mtr><mml:mtd><mml:mn>0</mml:mn></mml:mtd><mml:mtd><mml:mrow><mml:mi>π</mml:mi><mml:mi>η</mml:mi><mml:mi>R</mml:mi><mml:mo>?</mml:mo><mml:mi>ln</mml:mi><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:mfrac><mml:mrow><mml:msub><mml:mi>d</mml:mi><mml:mrow><mml:mi>i</mml:mi><mml:mi>j</mml:mi></mml:mrow></mml:msub></mml:mrow><mml:mi>R</mml:mi></mml:mfrac></mml:mrow><mml:mo>)</mml:mo></mml:mrow></mml:mrow></mml:mtd></mml:mtr></mml:mtable></mml:mrow><mml:mo>)</mml:mo></mml:mrow><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:mtable><mml:mtr><mml:mtd><mml:mrow><mml:msub><mml:mi mathvariant="italic">V</mml:mi><mml:mrow><mml:mi>N</mml:mi><mml:mo>,</mml:mo><mml:mi>i</mml:mi><mml:mi>j</mml:mi></mml:mrow></mml:msub></mml:mrow></mml:mtd></mml:mtr><mml:mtr><mml:mtd><mml:mrow><mml:msub><mml:mi mathvariant="italic">V</mml:mi><mml:mrow><mml:mi>T</mml:mi><mml:mo>,</mml:mo><mml:mi>i</mml:mi><mml:mi>j</mml:mi></mml:mrow></mml:msub></mml:mrow></mml:mtd></mml:mtr></mml:mtable></mml:mrow><mml:mo>)</mml:mo></mml:mrow><mml:mo>+</mml:mo><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:mtable><mml:mtr><mml:mtd><mml:mn>0</mml:mn></mml:mtd><mml:mtd><mml:mn>0</mml:mn></mml:mtd></mml:mtr><mml:mtr><mml:mtd><mml:mn>0</mml:mn></mml:mtd><mml:mtd><mml:mrow><mml:mo>?</mml:mo><mml:mi>π</mml:mi><mml:mi>η</mml:mi><mml:msup><mml:mi>R</mml:mi><mml:mn>2</mml:mn></mml:msup><mml:mo>?</mml:mo><mml:mi>ln</mml:mi><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:mfrac><mml:mrow><mml:msub><mml:mi>d</mml:mi><mml:mrow><mml:mi>i</mml:mi><mml:mi>j</mml:mi></mml:mrow></mml:msub></mml:mrow><mml:mi>R</mml:mi></mml:mfrac></mml:mrow><mml:mo>)</mml:mo></mml:mrow></mml:mrow></mml:mtd></mml:mtr></mml:mtable></mml:mrow><mml:mo>)</mml:mo></mml:mrow><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:mtable><mml:mtr><mml:mtd><mml:mn>0</mml:mn></mml:mtd></mml:mtr><mml:mtr><mml:mtd><mml:mrow><mml:msub><mml:mi>ω</mml:mi><mml:mrow><mml:mi>z</mml:mi><mml:mo>,</mml:mo><mml:mi>i</mml:mi><mml:mi>j</mml:mi></mml:mrow></mml:msub></mml:mrow></mml:mtd></mml:mtr></mml:mtable></mml:mrow><mml:mo>)</mml:mo></mml:mrow></mml:mrow><mml:mo>]</mml:mo></mml:mrow><mml:mo>,</mml:mo></mml:mrow></mml:math>Flub(dij,Vij)=((ri-rj)x|ri-rj|(ri-rj)y|ri-rj|(ri-rj)y|ri-rj|-(ri-rj)x|ri-rj|)[(-3πηR22dij12πη2R3/25dij0πηR?ln(dijR))(VN,ijVT,ij)+(000?πηR2?ln(dijR))(0ωz,ij)],<mml:math display="block"><mml:mrow><mml:msub><mml:mi mathvariant="bold-italic">T</mml:mi><mml:mrow><mml:mtext>lub</mml:mtext></mml:mrow></mml:msub><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:msub><mml:mi>d</mml:mi><mml:mrow><mml:mi>i</mml:mi><mml:mi>j</mml:mi></mml:mrow></mml:msub><mml:mo>,</mml:mo><mml:msub><mml:mi mathvariant="bold-italic">V</mml:mi><mml:mrow><mml:mi>i</mml:mi><mml:mi>j</mml:mi></mml:mrow></mml:msub></mml:mrow><mml:mo>)</mml:mo></mml:mrow><mml:mo>=</mml:mo><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:mtable><mml:mtr><mml:mtd><mml:mrow><mml:mo>?</mml:mo><mml:mi>π</mml:mi><mml:mi>η</mml:mi><mml:msup><mml:mi>R</mml:mi><mml:mn>2</mml:mn></mml:msup><mml:mo>?</mml:mo><mml:mi>ln</mml:mi><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:mfrac><mml:mrow><mml:msub><mml:mi>d</mml:mi><mml:mrow><mml:mi>i</mml:mi><mml:mi>j</mml:mi></mml:mrow></mml:msub></mml:mrow><mml:mi>R</mml:mi></mml:mfrac></mml:mrow><mml:mo>)</mml:mo></mml:mrow></mml:mrow></mml:mtd><mml:mtd><mml:mrow><mml:mfrac><mml:mrow><mml:mn>8</mml:mn><mml:mi>π</mml:mi><mml:mi>η</mml:mi><mml:msup><mml:mi>R</mml:mi><mml:mn>3</mml:mn></mml:msup></mml:mrow><mml:mn>5</mml:mn></mml:mfrac><mml:mi>ln</mml:mi><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:mfrac><mml:mrow><mml:msub><mml:mi>d</mml:mi><mml:mrow><mml:mi>i</mml:mi><mml:mi>j</mml:mi></mml:mrow></mml:msub></mml:mrow><mml:mi>R</mml:mi></mml:mfrac></mml:mrow><mml:mo>)</mml:mo></mml:mrow></mml:mrow></mml:mtd></mml:mtr></mml:mtable></mml:mrow><mml:mo>)</mml:mo></mml:mrow><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:mtable><mml:mtr><mml:mtd><mml:mrow><mml:msub><mml:mi>V</mml:mi><mml:mrow><mml:mi>T</mml:mi><mml:mo>,</mml:mo><mml:mi>i</mml:mi><mml:mi>j</mml:mi></mml:mrow></mml:msub></mml:mrow></mml:mtd></mml:mtr><mml:mtr><mml:mtd><mml:mrow><mml:msub><mml:mi>ω</mml:mi><mml:mrow><mml:mi>z</mml:mi><mml:mo>,</mml:mo><mml:mi>i</mml:mi><mml:mi>j</mml:mi></mml:mrow></mml:msub></mml:mrow></mml:mtd></mml:mtr></mml:mtable></mml:mrow><mml:mo>)</mml:mo></mml:mrow><mml:mo>,</mml:mo></mml:mrow></mml:math>Tlub(dij,Vij)=(?πηR2?ln(dijR)8πηR35ln(dijR))(VT,ijωz,ij),where η denotes the fluid viscosity and <mml:math><mml:mrow><mml:msub><mml:mi mathvariant="italic">V</mml:mi><mml:mrow><mml:mi>N</mml:mi><mml:mo>,</mml:mo><mml:mi>i</mml:mi><mml:mi>j</mml:mi></mml:mrow></mml:msub><mml:mo>=</mml:mo><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:msub><mml:mi mathvariant="bold-italic">V</mml:mi><mml:mi>i</mml:mi></mml:msub><mml:mo>-</mml:mo><mml:msub><mml:mi mathvariant="bold-italic">V</mml:mi><mml:mi>j</mml:mi></mml:msub></mml:mrow><mml:mo>)</mml:mo></mml:mrow><mml:mo>?</mml:mo><mml:msub><mml:mi mathvariant="bold-italic">n</mml:mi><mml:mrow><mml:mi>i</mml:mi><mml:mi>j</mml:mi></mml:mrow></mml:msub></mml:mrow></mml:math>VN,ij=(Vi-Vj)?nij and <mml:math><mml:mrow><mml:msub><mml:mi mathvariant="italic">V</mml:mi><mml:mrow><mml:mi>T</mml:mi><mml:mo>,</mml:mo><mml:mi>i</mml:mi><mml:mi>j</mml:mi></mml:mrow></mml:msub><mml:mo>=</mml:mo><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:msub><mml:mi mathvariant="bold-italic">V</mml:mi><mml:mi>i</mml:mi></mml:msub><mml:mo>-</mml:mo><mml:msub><mml:mi mathvariant="bold-italic">V</mml:mi><mml:mi>j</mml:mi></mml:msub></mml:mrow><mml:mo>)</mml:mo></mml:mrow><mml:mo>?</mml:mo><mml:msub><mml:mi mathvariant="bold-italic">t</mml:mi><mml:mrow><mml:mi>i</mml:mi><mml:mi>j</mml:mi></mml:mrow></mml:msub></mml:mrow></mml:math>VT,ij=(Vi-Vj)?tij are the normal and tangential components of the relative velocity <mml:math><mml:mrow><mml:msub><mml:mi mathvariant="bold-italic">V</mml:mi><mml:mrow><mml:mi>i</mml:mi><mml:mi>j</mml:mi></mml:mrow></mml:msub></mml:mrow></mml:math>Vij in the local xy coordinate system; <mml:math><mml:mrow><mml:msub><mml:mi mathvariant="bold-italic">n</mml:mi><mml:mrow><mml:mi>i</mml:mi><mml:mi>j</mml:mi></mml:mrow></mml:msub></mml:mrow></mml:math>nij and <mml:math><mml:mrow><mml:msub><mml:mi mathvariant="bold-italic">t</mml:mi><mml:mrow><mml:mi>i</mml:mi><mml:mi>j</mml:mi></mml:mrow></mml:msub></mml:mrow></mml:math>tij are the normal and tangential unit vectors in the local coordinates and are defined as <mml:math><mml:mrow><mml:mrow><mml:mrow><mml:mo>(</mml:mo><mml:msub><mml:mi mathvariant="bold-italic">r</mml:mi><mml:mi>i</mml:mi></mml:msub><mml:mo>?</mml:mo><mml:msub><mml:mi mathvariant="bold-italic">r</mml:mi><mml:mi>j</mml:mi></mml:msub><mml:mo>)</mml:mo></mml:mrow><mml:mo>/</mml:mo><mml:mrow><mml:mrow><mml:mo>|</mml:mo><mml:mrow><mml:msub><mml:mi mathvariant="bold-italic">r</mml:mi><mml:mi>i</mml:mi></mml:msub><mml:mo>?</mml:mo><mml:msub><mml:mi mathvariant="bold-italic">r</mml:mi><mml:mi>j</mml:mi></mml:msub></mml:mrow><mml:mo>|</mml:mo></mml:mrow></mml:mrow></mml:mrow></mml:mrow></mml:math>(ri?rj)/|ri?rj| <mml:math><mml:mrow><mml:mrow><mml:mrow><mml:mrow><mml:mo>[</mml:mo><mml:mrow><mml:msub><mml:mrow><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:msub><mml:mi mathvariant="bold-italic">r</mml:mi><mml:mi>i</mml:mi></mml:msub><mml:mo>-</mml:mo><mml:msub><mml:mi mathvariant="bold-italic">r</mml:mi><mml:mi>j</mml:mi></mml:msub></mml:mrow><mml:mo>)</mml:mo></mml:mrow></mml:mrow><mml:mi>y</mml:mi></mml:msub><mml:mtext>,</mml:mtext><mml:mo>-</mml:mo><mml:msub><mml:mrow><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:msub><mml:mi mathvariant="bold-italic">r</mml:mi><mml:mi>i</mml:mi></mml:msub><mml:mo>-</mml:mo><mml:msub><mml:mi mathvariant="bold-italic">r</mml:mi><mml:mi>j</mml:mi></mml:msub></mml:mrow><mml:mo>)</mml:mo></mml:mrow></mml:mrow><mml:mi>x</mml:mi></mml:msub></mml:mrow><mml:mo>]</mml:mo></mml:mrow></mml:mrow><mml:mo>/</mml:mo><mml:mrow><mml:mrow><mml:mo>|</mml:mo><mml:mrow><mml:msub><mml:mi mathvariant="bold-italic">r</mml:mi><mml:mi>i</mml:mi></mml:msub><mml:mo>-</mml:mo><mml:msub><mml:mi mathvariant="bold-italic">r</mml:mi><mml:mi>j</mml:mi></mml:msub></mml:mrow><mml:mo>|</mml:mo></mml:mrow></mml:mrow></mml:mrow></mml:mrow></mml:math>[(ri-rj)y,-(ri-rj)x]/|ri-rj|.

Our model rigorously accounts for the particles’ rotation. However, it represents a negligible contribution: the rotational to translational kinetic energy ratio is always small, <mml:math><mml:mrow><mml:msub><mml:mi>K</mml:mi><mml:mi>R</mml:mi></mml:msub><mml:mo>=</mml:mo><mml:mrow><mml:mo>(</mml:mo><mml:mn>1</mml:mn><mml:mo>/</mml:mo><mml:mn>2</mml:mn><mml:mo>)</mml:mo></mml:mrow><mml:msub><mml:mi>I</mml:mi><mml:mi>i</mml:mi></mml:msub><mml:msup><mml:mrow><mml:mover accent="true"><mml:mi>θ</mml:mi><mml:mo>˙</mml:mo></mml:mover></mml:mrow><mml:mn>2</mml:mn></mml:msup><mml:mo>?</mml:mo><mml:msub><mml:mi>K</mml:mi><mml:mi>T</mml:mi></mml:msub><mml:mo>=</mml:mo><mml:mrow><mml:mo>(</mml:mo><mml:mn>1</mml:mn><mml:mo>/</mml:mo><mml:mn>2</mml:mn><mml:mo>)</mml:mo></mml:mrow><mml:msub><mml:mi>m</mml:mi><mml:mi>i</mml:mi></mml:msub><mml:msubsup><mml:mi>V</mml:mi><mml:mi>i</mml:mi><mml:mn>2</mml:mn></mml:msubsup></mml:mrow></mml:math>KR=(1/2)Iiθ˙2?KT=(1/2)miVi2 (Fig. S3C), since the particle?particle interaction is much faster than rotational inertia and the colloids do not gain significant spin. The friction force on a single particle due to the presence of the fluid is given by Stokes’ drag formula,<mml:math display="block"><mml:mrow><mml:msub><mml:mi mathvariant="bold-italic">F</mml:mi><mml:mrow><mml:mtext>drag</mml:mtext></mml:mrow></mml:msub><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:msub><mml:mi mathvariant="bold-italic">V</mml:mi><mml:mi>i</mml:mi></mml:msub></mml:mrow><mml:mo>)</mml:mo></mml:mrow><mml:mo>=</mml:mo><mml:mo>?</mml:mo><mml:mn>6</mml:mn><mml:mi>π</mml:mi><mml:mi>η</mml:mi><mml:mi>R</mml:mi><mml:msub><mml:mi mathvariant="bold-italic">V</mml:mi><mml:mi>i</mml:mi></mml:msub><mml:mo>.</mml:mo></mml:mrow></mml:math>Fdrag(Vi)=?6πηRVi.When the interparticle distance dij becomes smaller than the cutoff distance dcut, the two spheres start to deform elastically. The equation of motion becomes <mml:math display="block"><mml:mrow><mml:msub><mml:mi>m</mml:mi><mml:mi>i</mml:mi></mml:msub><mml:mfrac><mml:mrow><mml:msup><mml:mi mathvariant="normal">d</mml:mi><mml:mn>2</mml:mn></mml:msup><mml:msub><mml:mi mathvariant="bold-italic">r</mml:mi><mml:mi>i</mml:mi></mml:msub></mml:mrow><mml:mrow><mml:mi mathvariant="normal">d</mml:mi><mml:msup><mml:mi>t</mml:mi><mml:mn>2</mml:mn></mml:msup></mml:mrow></mml:mfrac><mml:mo>=</mml:mo><mml:mstyle displaystyle="true"><mml:munderover><mml:mo>∑</mml:mo><mml:mi>j</mml:mi><mml:mrow><mml:mi>n</mml:mi><mml:mi>n</mml:mi></mml:mrow></mml:munderover><mml:mrow><mml:msub><mml:mi mathvariant="bold-italic">F</mml:mi><mml:mrow><mml:mtext>contact</mml:mtext></mml:mrow></mml:msub><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:msub><mml:mi>δ</mml:mi><mml:mrow><mml:mi>i</mml:mi><mml:mi>j</mml:mi></mml:mrow></mml:msub></mml:mrow><mml:mo>)</mml:mo></mml:mrow></mml:mrow></mml:mstyle><mml:mo>+</mml:mo><mml:mstyle displaystyle="true"><mml:munderover><mml:mo>∑</mml:mo><mml:mi>j</mml:mi><mml:mrow><mml:mi>n</mml:mi><mml:mi>n</mml:mi></mml:mrow></mml:munderover><mml:mrow><mml:msub><mml:mi mathvariant="bold-italic">F</mml:mi><mml:mrow><mml:mtext>visc</mml:mtext></mml:mrow></mml:msub><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:msub><mml:mi>δ</mml:mi><mml:mrow><mml:mi>i</mml:mi><mml:mi>j</mml:mi></mml:mrow></mml:msub><mml:mo>,</mml:mo><mml:msub><mml:mi mathvariant="bold-italic">V</mml:mi><mml:mrow><mml:mi>i</mml:mi><mml:mi>j</mml:mi></mml:mrow></mml:msub></mml:mrow><mml:mo>)</mml:mo></mml:mrow></mml:mrow></mml:mstyle><mml:mo>+</mml:mo><mml:msub><mml:mi mathvariant="bold-italic">F</mml:mi><mml:mrow><mml:mtext>drag</mml:mtext></mml:mrow></mml:msub><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:msub><mml:mi mathvariant="bold-italic">V</mml:mi><mml:mi>i</mml:mi></mml:msub></mml:mrow><mml:mo>)</mml:mo></mml:mrow><mml:mo>,</mml:mo><mml:mo>?</mml:mo><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:msub><mml:mi mathvariant="italic">d</mml:mi><mml:mrow><mml:mi>i</mml:mi><mml:mi>j</mml:mi></mml:mrow></mml:msub><mml:mo>≤</mml:mo><mml:msub><mml:mi>d</mml:mi><mml:mrow><mml:mtext>cut</mml:mtext></mml:mrow></mml:msub></mml:mrow><mml:mo>)</mml:mo></mml:mrow><mml:mo>,</mml:mo></mml:mrow></mml:math>mid2ridt2=∑jnnFcontact(δij)+∑jnnFvisc(δij,Vij)+Fdrag(Vi),?(dij≤dcut),where <mml:math><mml:mrow><mml:msub><mml:mi mathvariant="italic">δ</mml:mi><mml:mrow><mml:mi>i</mml:mi><mml:mi>j</mml:mi></mml:mrow></mml:msub><mml:mo>=</mml:mo><mml:mn>2</mml:mn><mml:mi mathvariant="italic">R</mml:mi><mml:mo>+</mml:mo><mml:msub><mml:mi mathvariant="italic">d</mml:mi><mml:mrow><mml:mtext>cut</mml:mtext></mml:mrow></mml:msub><mml:mo>-</mml:mo><mml:mrow><mml:mo>|</mml:mo><mml:mrow><mml:msub><mml:mi mathvariant="bold-italic">r</mml:mi><mml:mi>i</mml:mi></mml:msub><mml:mo>-</mml:mo><mml:msub><mml:mi mathvariant="bold-italic">r</mml:mi><mml:mi>j</mml:mi></mml:msub></mml:mrow><mml:mo>|</mml:mo></mml:mrow></mml:mrow></mml:math>δij=2R+dcut-|ri-rj|. The elastic deformation of the particles is related to the presence of two different forces, Fcontact and Fvisc. The contact force follows Hertzian mechanics and is given by<mml:math display="block"><mml:mrow><mml:msub><mml:mi mathvariant="bold-italic">F</mml:mi><mml:mrow><mml:mtext>contact</mml:mtext></mml:mrow></mml:msub><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:msub><mml:mi>δ</mml:mi><mml:mrow><mml:mi>i</mml:mi><mml:mi>j</mml:mi></mml:mrow></mml:msub></mml:mrow><mml:mo>)</mml:mo></mml:mrow><mml:mo>=</mml:mo><mml:mfrac><mml:mi mathvariant="italic">E</mml:mi><mml:mrow><mml:mn>3</mml:mn><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:mn>1</mml:mn><mml:mo>-</mml:mo><mml:msup><mml:mi mathvariant="italic">ν</mml:mi><mml:mn>2</mml:mn></mml:msup></mml:mrow><mml:mo>)</mml:mo></mml:mrow></mml:mrow></mml:mfrac><mml:msqrt><mml:mrow><mml:mn>2</mml:mn><mml:mi mathvariant="italic">R</mml:mi></mml:mrow></mml:msqrt><mml:msubsup><mml:mi mathvariant="italic">δ</mml:mi><mml:mrow><mml:mi>i</mml:mi><mml:mi>j</mml:mi></mml:mrow><mml:mrow><mml:mrow><mml:mn>3</mml:mn><mml:mo>/</mml:mo><mml:mn>2</mml:mn></mml:mrow></mml:mrow></mml:msubsup><mml:msub><mml:mi mathvariant="bold-italic">n</mml:mi><mml:mrow><mml:mi>i</mml:mi><mml:mi>j</mml:mi></mml:mrow></mml:msub><mml:mo>.</mml:mo></mml:mrow></mml:math>Fcontact(δij)=E3(1-ν2)2Rδij3/2nij.Here, E and ν are the Young’s modulus and the Poisson’s ratio of silica. We do not account for a tangential elastic force [e.g., Hertz–Mindlin (30)]: The lubrication film between particles prevents dry friction, and the duration of the elastic interaction is shorter than the one stemming from purely viscous repulsion (see above), leading to a contribution that does not provide relevant rotational accelerations (Fig. S3C). In turn, the viscous force arises from the squeeze-out of fluid from the outside of the contact region (9?11, 39) and is given by <mml:math display="block"><mml:mrow><mml:msub><mml:mi mathvariant="bold-italic">F</mml:mi><mml:mrow><mml:mtext>visc</mml:mtext></mml:mrow></mml:msub><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:msub><mml:mi>δ</mml:mi><mml:mrow><mml:mi>i</mml:mi><mml:mi>j</mml:mi></mml:mrow></mml:msub><mml:mo>,</mml:mo><mml:msub><mml:mrow><mml:mover accent="true"><mml:mi>δ</mml:mi><mml:mo>˙</mml:mo></mml:mover></mml:mrow><mml:mrow><mml:mi>i</mml:mi><mml:mi>j</mml:mi></mml:mrow></mml:msub></mml:mrow><mml:mo>)</mml:mo></mml:mrow><mml:mo>=</mml:mo><mml:mfrac><mml:mn>3</mml:mn><mml:mn>2</mml:mn></mml:mfrac><mml:mi>π</mml:mi><mml:mi>η</mml:mi><mml:msup><mml:mi mathvariant="italic">R</mml:mi><mml:mn>2</mml:mn></mml:msup><mml:mfrac><mml:mrow><mml:msub><mml:mrow><mml:mover accent="true"><mml:mi>δ</mml:mi><mml:mo>˙</mml:mo></mml:mover></mml:mrow><mml:mrow><mml:mi>i</mml:mi><mml:mi>j</mml:mi></mml:mrow></mml:msub></mml:mrow><mml:mrow><mml:msub><mml:mi mathvariant="italic">d</mml:mi><mml:mrow><mml:mtext>cut</mml:mtext></mml:mrow></mml:msub></mml:mrow></mml:mfrac><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:mn>1</mml:mn><mml:mo>+</mml:mo><mml:mfrac><mml:mrow><mml:msub><mml:mi mathvariant="italic">δ</mml:mi><mml:mrow><mml:mi>i</mml:mi><mml:mi>j</mml:mi></mml:mrow></mml:msub></mml:mrow><mml:mrow><mml:msub><mml:mi mathvariant="italic">d</mml:mi><mml:mrow><mml:mtext>cut</mml:mtext></mml:mrow></mml:msub></mml:mrow></mml:mfrac></mml:mrow><mml:mo>)</mml:mo></mml:mrow><mml:msub><mml:mi mathvariant="bold-italic">n</mml:mi><mml:mrow><mml:mi>i</mml:mi><mml:mi>j</mml:mi></mml:mrow></mml:msub><mml:mo>.</mml:mo></mml:mrow></mml:math>Fvisc(δij,δ˙ij)=32πηR2δ˙ijdcut(1+δijdcut)nij.We integrate the equations of motion using the fourth-order Runge?Kutta method with the following system parameters: ρ = 1,850 kg/m3, R = 3.69 μm, d = 400 nm, dcut = 17 nm, E = 73.1 GPa, ν = 0.17 (bulk silica values), and Tsimulation = 1 per camera frame rate = 3.214 μs. The convergence is controlled by choosing an integration time step Δt = 0.5 ns equal to one-tenth of the fastest event in simulations, i.e., the collision duration (30) <mml:math><mml:mrow><mml:msub><mml:mi>τ</mml:mi><mml:mi>c</mml:mi></mml:msub><mml:mo>≈</mml:mo><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:mrow><mml:mrow><mml:mn>2</mml:mn><mml:mi>R</mml:mi></mml:mrow><mml:mo>/</mml:mo><mml:mrow><mml:msub><mml:mi>c</mml:mi><mml:mn>0</mml:mn></mml:msub></mml:mrow></mml:mrow></mml:mrow><mml:mo>)</mml:mo></mml:mrow><mml:msup><mml:mrow><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:mrow><mml:mrow><mml:msub><mml:mi>c</mml:mi><mml:mn>0</mml:mn></mml:msub></mml:mrow><mml:mo>/</mml:mo><mml:mrow><mml:msub><mml:mi>V</mml:mi><mml:mn>0</mml:mn></mml:msub></mml:mrow></mml:mrow></mml:mrow><mml:mo>)</mml:mo></mml:mrow></mml:mrow><mml:mrow><mml:mrow><mml:mn>1</mml:mn><mml:mo>/</mml:mo><mml:mn>5</mml:mn></mml:mrow></mml:mrow></mml:msup><mml:mo>≈</mml:mo><mml:mn>5</mml:mn></mml:mrow></mml:math>τc≈(2R/c0)(c0/V0)1/5≈5 ns, where <mml:math><mml:mrow><mml:msub><mml:mi mathvariant="italic">c</mml:mi><mml:mn>0</mml:mn></mml:msub><mml:mo>≈</mml:mo><mml:msqrt><mml:mrow><mml:mrow><mml:mi>E</mml:mi><mml:mo>/</mml:mo><mml:mi>ρ</mml:mi></mml:mrow></mml:mrow></mml:msqrt></mml:mrow></mml:math>c0≈E/ρ is the longitudinal wave speed inside the particles. Energy conservation is fulfilled with a 10?8 relative error.

The assumptions made in our model are supported by a quantification of dimensionless numbers that identify the experimental conditions. The typical velocity of the fluid, <mml:math><mml:mrow><mml:msub><mml:mi>V</mml:mi><mml:mi>f</mml:mi></mml:msub><mml:mo>≈</mml:mo><mml:msqrt><mml:mrow><mml:mrow><mml:mi>R</mml:mi><mml:mo>/</mml:mo><mml:mi>d</mml:mi></mml:mrow></mml:mrow></mml:msqrt><mml:mo>?</mml:mo><mml:msub><mml:mi>V</mml:mi><mml:mi>p</mml:mi></mml:msub></mml:mrow></mml:math>Vf≈R/d?Vp, is defined by the squeeze-out of liquid from the interparticle gap (9). The particle velocity Vp goes up to 1 m/s to 10 m/s, whereas d ranges between 17 and 400 nm. From Vf, we estimate the following conditions.

  • i) The first condition is fluid incompressibility, where the Mach number of the fluid is <mml:math><mml:mrow><mml:mi>M</mml:mi><mml:mi>a</mml:mi><mml:mo>=</mml:mo><mml:mrow><mml:mrow><mml:msub><mml:mi>V</mml:mi><mml:mi>f</mml:mi></mml:msub></mml:mrow><mml:mo>/</mml:mo><mml:mrow><mml:msub><mml:mi>c</mml:mi><mml:mrow><mml:mtext>water</mml:mtext></mml:mrow></mml:msub></mml:mrow></mml:mrow></mml:mrow></mml:math>Ma=Vf/cwater, where cwater is the sound velocity in water. We find that Ma is always lower than 0.1.

  • ii) We then verified the absence of fluid turbulence by estimating the Reynolds number of the fluid, <mml:math><mml:mrow><mml:mi>R</mml:mi><mml:msub><mml:mi>e</mml:mi><mml:mi>f</mml:mi></mml:msub><mml:mo>=</mml:mo><mml:mrow><mml:mrow><mml:msub><mml:mi>ρ</mml:mi><mml:mi>f</mml:mi></mml:msub><mml:msub><mml:mi>V</mml:mi><mml:mi>f</mml:mi></mml:msub><mml:mi>d</mml:mi></mml:mrow><mml:mo>/</mml:mo><mml:mi>η</mml:mi></mml:mrow></mml:mrow></mml:math>Ref=ρfVfd/η, which goes up to ~20, far below the turbulence threshold.

  • iii) Finally, particle inertia can be quantified by looking at the Stokes number (<mml:math><mml:mrow><mml:mi>S</mml:mi><mml:mi>t</mml:mi><mml:mo>=</mml:mo><mml:mrow><mml:mrow><mml:msub><mml:mi>ρ</mml:mi><mml:mi>p</mml:mi></mml:msub><mml:msub><mml:mi>V</mml:mi><mml:mi>p</mml:mi></mml:msub><mml:mi>R</mml:mi></mml:mrow><mml:mo>/</mml:mo><mml:mi>η</mml:mi></mml:mrow></mml:mrow></mml:math>St=ρpVpR/η), which reaches values up 102 for the largest particle velocity (in water). This indicates that inertia effects are dominant over viscous drag in the early stages of the propagation. In turn, also, the Reynolds number of the particle can be larger than 1 [<mml:math><mml:mrow><mml:mi>R</mml:mi><mml:mi>e</mml:mi><mml:mo>=</mml:mo><mml:mi>S</mml:mi><mml:mi>t</mml:mi><mml:mo>?</mml:mo><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:mrow><mml:mrow><mml:msub><mml:mi>ρ</mml:mi><mml:mi>f</mml:mi></mml:msub></mml:mrow><mml:mo>/</mml:mo><mml:mrow><mml:msub><mml:mi>ρ</mml:mi><mml:mi>p</mml:mi></mml:msub></mml:mrow></mml:mrow></mml:mrow><mml:mo>)</mml:mo></mml:mrow></mml:mrow></mml:math>Re=St?(ρf/ρp)]. This leads to deviations from the expression used for the viscous drag (Stokes flow) at the highest velocities. However, the contribution of the viscous drag is weak, and more accurate models do not affect the details of the propagation mechanisms (see also SI Discussion).

Additional numerical simulations, which explicitly account for colloidal interaction forces, show that the wave propagation very weakly depends on the static pair potential of the particles, which can therefore be safely neglected. In particular, the data reported in Fig. S4 show the full wave propagation for two viscosity values with and without the colloidal interaction potential, which we approximate potential using the Derjaguin–Landau–Verwey–Overbeek (DLVO) theory (40),<mml:math display="block"><mml:mrow><mml:msub><mml:mi>F</mml:mi><mml:mrow><mml:mi>i</mml:mi><mml:mi>j</mml:mi></mml:mrow></mml:msub><mml:mo>=</mml:mo><mml:mi>k</mml:mi><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:mfrac><mml:mi>R</mml:mi><mml:mn>2</mml:mn></mml:mfrac></mml:mrow><mml:mo>)</mml:mo></mml:mrow><mml:mi>Z</mml:mi><mml:msup><mml:mi>e</mml:mi><mml:mrow><mml:mo>?</mml:mo><mml:mi>k</mml:mi><mml:msub><mml:mi>d</mml:mi><mml:mrow><mml:mi>i</mml:mi><mml:mi>j</mml:mi></mml:mrow></mml:msub></mml:mrow></mml:msup><mml:mo>?</mml:mo><mml:mfrac><mml:mrow><mml:mi>H</mml:mi><mml:mi>R</mml:mi></mml:mrow><mml:mrow><mml:mn>12</mml:mn><mml:msubsup><mml:mi>d</mml:mi><mml:mrow><mml:mi>i</mml:mi><mml:mi>j</mml:mi></mml:mrow><mml:mn>2</mml:mn></mml:msubsup></mml:mrow></mml:mfrac><mml:mo>,</mml:mo></mml:mrow></mml:math>Fij=k(R2)Ze?kdij?HR12dij2,where <mml:math><mml:mrow><mml:mi>Z</mml:mi><mml:mo>=</mml:mo><mml:mn>64</mml:mn><mml:mi>π</mml:mi><mml:msub><mml:mi>ε</mml:mi><mml:mn>0</mml:mn></mml:msub><mml:mi>ε</mml:mi><mml:msup><mml:mrow><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:mrow><mml:mrow><mml:msub><mml:mi>k</mml:mi><mml:mi>B</mml:mi></mml:msub><mml:mi>T</mml:mi></mml:mrow><mml:mo>/</mml:mo><mml:mi>e</mml:mi></mml:mrow></mml:mrow><mml:mo>)</mml:mo></mml:mrow></mml:mrow><mml:mn>2</mml:mn></mml:msup><mml:mo>?</mml:mo><mml:msup><mml:mrow><mml:mi>tanh</mml:mi></mml:mrow><mml:mn>2</mml:mn></mml:msup><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:mrow><mml:mrow><mml:mi>e</mml:mi><mml:msub><mml:mi>?</mml:mi><mml:mn>0</mml:mn></mml:msub></mml:mrow><mml:mo>/</mml:mo><mml:mrow><mml:mn>4</mml:mn><mml:msub><mml:mi>k</mml:mi><mml:mi>B</mml:mi></mml:msub><mml:mi>T</mml:mi></mml:mrow></mml:mrow></mml:mrow><mml:mo>)</mml:mo></mml:mrow></mml:mrow></mml:math>Z=64πε0ε(kBT/e)2?tanh2(e?0/4kBT) is an interaction constant, <mml:math><mml:mrow><mml:msup><mml:mi>k</mml:mi><mml:mrow><mml:mo>?</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msup></mml:mrow></mml:math>k?1 = 35 nm (Debye length), <mml:math><mml:mrow><mml:msub><mml:mi>?</mml:mi><mml:mn>0</mml:mn></mml:msub></mml:mrow></mml:math>?0 = ?76 mV (surface potential), and H = 2.1·10?21 J (Hamaker constant).

Fig. S4.

Weak dependence on the interparticle electrostatic potential. Decay of the instantaneous velocity Vp along a particle chain; η is (A) 1 and (B) 4 mPa?s. V0 is (A) 12 and (B) 26 m/s. The black and red curves are computed by including and by neglecting the electrostatic interactions between the particles, respectively. The electrostatic interaction is modeled using DLVO theory. The resulting velocity decays overlap, showing no detectable difference. (C) Maximum particle velocity plotted as a function of the normalized distance L/2R from the SI for the sets of data in A and B. Filled red triangles correspond to the data without electrostatic potential, while the open back triangles are the results of the simulations including electrostatic effects.

Strain Wave Simulations in Monolayers with Controlled Degree of Disorder.

Colloidal crystalline monolayers are prepared by positioning spherical particles (radius R = 3.69 μm) separated by a uniform distance (d = 0.4 μm) onto a 2D regular hexagonal lattice (<mml:math><mml:mrow><mml:msubsup><mml:mi>ψ</mml:mi><mml:mn>6</mml:mn><mml:mi>L</mml:mi></mml:msubsup></mml:mrow></mml:math>ψ6L = 1) in water (η = 1 mPa·s). Disordered monolayers are obtained as perturbations of crystalline configurations. For this, we apply a random initial velocity (the magnitudes of x and y components of each velocity vector are extracted from two independent normal distributions with 0 m/s mean and 1 m/s SD) to every particle of the crystal, except the six neighbors of the SI (these degrees of freedom are frozen). This process is iterated to achieve configurations with different degrees of local order. We ensure that, after randomization, the colloidal monolayer is at rest: the velocity of all particles is zero and the particles do not overlap.

The Voronoi tessellation of the structures allows for the local computation of the particle separation d (average distance between a given particle and its nearest neighbors), the packing (area) fraction ? (ratio between the cross-section of a particle to the area of a Voronoi cell), and the local order parameter <mml:math><mml:mrow><mml:msubsup><mml:mi>ψ</mml:mi><mml:mn>6</mml:mn><mml:mi>l</mml:mi></mml:msubsup></mml:mrow></mml:math>ψ6l. These parameters are averaged within a circular region of radius 50 μm (corresponding to ~6 times the lattice constant) centered on the SI to obtain the average structural features of the monolayers. Fig. S5 A and H shows two examples: a perfect crystal (? = 81%, <mml:math><mml:mrow><mml:msubsup><mml:mi>ψ</mml:mi><mml:mn>6</mml:mn><mml:mi>L</mml:mi></mml:msubsup></mml:mrow></mml:math>ψ6L = 1) and an amorphous monolayer (? = 75%, <mml:math><mml:mrow><mml:msubsup><mml:mi>ψ</mml:mi><mml:mn>6</mml:mn><mml:mi>L</mml:mi></mml:msubsup></mml:mrow></mml:math>ψ6L = 0.525). The average <mml:math><mml:mrow><mml:msub><mml:mi>ψ</mml:mi><mml:mn>6</mml:mn></mml:msub></mml:mrow></mml:math>ψ6 of the random configurations monotonically decreases with the duration of the randomization of the initial configurations and is statistically reproducible. Twenty configurations are generated for each average value of <mml:math><mml:mrow><mml:msub><mml:mi>ψ</mml:mi><mml:mn>6</mml:mn></mml:msub></mml:mrow></mml:math>ψ6, from 0.998 (crystal, 0.02% SD) to 0.514 (amorphous, 16.6% SD).

Fig. S5.

Strain waves simulated in monolayers without and with local disorder. (AG) Crystalline monolayer (<mml:math><mml:mrow><mml:msubsup><mml:mi>ψ</mml:mi><mml:mn>6</mml:mn><mml:mi>L</mml:mi></mml:msubsup></mml:mrow></mml:math>ψ6L = 1). (A) Initial configurations illustrated using the Voronoi tessellation of the lattice. The dots indicate the particle centers and the gray cells correspond to the SI (dark gray) and its six nearest neighbors (light gray) to which an initial radial velocity is applied (red arrows). (B) Global velocity field V and (C) maximum of the total energy field E in the crystal lattice. All velocity fields (energy fields) here refer to the Lagrangian (Eulerian) description. (D) Maximum of the potential energy field and (E) ratio of the potential to the total energy along one symmetry axis; the red solid line is the Virial theorem prediction for a Hertzian potential. (F) Total energy averaged over all radial directions and plotted as a function of time and distance. The layers correspond to the circular rings shown in A. (G) Magnitude (in decibels re max) of E (red) or V (blue) as a function of the nondimensional distance r/2R; V is arbitrarily shifted by 10 dB for clarity; the solid lines highlight the exponential fits. (HN) Disordered monolayer (<mml:math><mml:mrow><mml:msubsup><mml:mi>ψ</mml:mi><mml:mn>6</mml:mn><mml:mi>L</mml:mi></mml:msubsup></mml:mrow></mml:math>ψ6L = 0.525). (H) Initial configurations illustrated using the Voronoi tessellation of the lattice, same as in A. (I) Global velocity field V and (J) maximum of the total energy field E for one numerical configuration. (K) Global velocity field and (L) maximum of the total energy field averaged over 20 numerical configurations. (M) Total energy averaged over all radial directions and plotted as a function of time and distance. The layers correspond to the circular rings shown in A. (N) Magnitude (in decibels re max) of E (red) or V (blue) as a function of r/2R; V is arbitrarily shifted by 10 dB for clarity; the solid lines highlight the exponential fits.

We mimic PLA by applying an initial radial velocity (red arrows in Fig. S5 A and H) to the six neighbors of the SI. We simulate the dynamics (Numerical Simulations) of all samples over a time corresponding to one frame of the movies recorded in the experiments (3.214 μs). This allows us to determine how the structural features of the monolayers affect the attenuation of the mechanical perturbation. The postprocessing of the data is done in three steps.

In step 1, we estimate the global velocity V of each particle (from the overall displacement over the duration of the simulation; Fig. S5 B and I), their translation and rotational kinetic energy, <mml:math><mml:mrow><mml:msub><mml:mi>K</mml:mi><mml:mi>T</mml:mi></mml:msub><mml:mo>=</mml:mo><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:mrow><mml:mn>1</mml:mn><mml:mo>/</mml:mo><mml:mn>2</mml:mn></mml:mrow></mml:mrow><mml:mo>)</mml:mo></mml:mrow><mml:mi>m</mml:mi><mml:msubsup><mml:mi>V</mml:mi><mml:mi>p</mml:mi><mml:mn>2</mml:mn></mml:msubsup></mml:mrow></mml:math>KT=(1/2)mVp2 and <mml:math><mml:mrow><mml:msub><mml:mi>K</mml:mi><mml:mi>R</mml:mi></mml:msub><mml:mo>=</mml:mo><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:mrow><mml:mn>1</mml:mn><mml:mo>/</mml:mo><mml:mn>2</mml:mn></mml:mrow></mml:mrow><mml:mo>)</mml:mo></mml:mrow><mml:mi>I</mml:mi><mml:msup><mml:mrow><mml:mover accent="true"><mml:mi>θ</mml:mi><mml:mo>˙</mml:mo></mml:mover></mml:mrow><mml:mn>2</mml:mn></mml:msup></mml:mrow></mml:math>KR=(1/2)Iθ˙2, as a function of time (from the instantaneous particles’ velocity Vp and rotational motion θ), and the Hertzian elastic potential energy U stored at each contact when they occur (from the elastic deformation of the particles against the interstitial fluid). The kinetic energy is computed at the center of mass of every particle, and the potential energy is assigned at the middle position between two particles. These operations provide the particle trajectories and a Lagrangian description of the energies inside the lattice. In the following, we neglect KR, which typically provides an insignificant contribution (Fig. S3C) to the energy balance in the lattice, KR/KT < 10?4, within the analyzed timeframe (Fig. S5 F or M).

In step 2, we map the energies KT and U (Lagrangian description) of all configurations onto the Voronoi lattice of the original crystal (Fig. S5A). This provides an Eulerian description of the energy field at the particle scale and a common frame to all our lattices; the energy flux is revealed by plotting the maximum of the total energy E = KT + U (Eulerian description) over time as a function of the position in the lattice, as shown in Fig. S5 C and J. In crystalline samples, a plot of the potential energy versus position in the crystal lattice (Fig. S5D) shows that the long-range propagation along symmetry axes is closely related to the elastic deformation of the particles, i.e., to the propagation of longitudinal elastic waves. Along one symmetry axis of the crystal lattice, the ratio between the potential and total energy (Fig. S5E) confirms the nonlinear nature of the propagation: The data fulfill the Virial theorem (41) U/E = 4/9 (red line). In disordered samples, a fraction of the input perturbation is randomly scattered (Fig. S3) and generates an incoherent response. Averaging over the 20 configurations with similar disorder (Fig. S5 K and L) cancels out the incoherent contribution and reveals the underlying symmetry of the ballistic coherent pulse (35). The wave propagation in disordered monolayers appears hence more isotropic.

In step 3, we extract a 1D description of the total energy E, as a function of the distance r from the SI and time t, by integrating the mean energy field over the angular coordinates (Fig. S5 F and M). The plot of the maximum of E over t, as a function of r, reveals an exponential decay (red points in Fig. S5 G and N), from which we extract an attenuation length <mml:math><mml:mrow><mml:msub><mml:mrow><mml:mover accent="true"><mml:mi>l</mml:mi><mml:mo stretchy="true">~</mml:mo></mml:mover></mml:mrow><mml:mrow><mml:mtext>att</mml:mtext></mml:mrow></mml:msub></mml:mrow></mml:math>l~att, calculated over the first six particles from the SI. The <mml:math><mml:mrow><mml:msub><mml:mrow><mml:mover accent="true"><mml:mi>l</mml:mi><mml:mo>~</mml:mo></mml:mover></mml:mrow><mml:mrow><mml:mtext>att</mml:mtext></mml:mrow></mml:msub></mml:mrow></mml:math>l~att is smaller in disordered samples (<mml:math><mml:mrow><mml:msub><mml:mrow><mml:mover accent="true"><mml:mi>l</mml:mi><mml:mo>~</mml:mo></mml:mover></mml:mrow><mml:mrow><mml:mtext>att</mml:mtext></mml:mrow></mml:msub></mml:mrow></mml:math>l~att/2R = 1.124 at <mml:math><mml:mrow><mml:msub><mml:mi>ψ</mml:mi><mml:mn>6</mml:mn></mml:msub></mml:mrow></mml:math>ψ6 = 0.514) than in the perfect crystals (<mml:math><mml:mrow><mml:msub><mml:mrow><mml:mover accent="true"><mml:mi>l</mml:mi><mml:mo>~</mml:mo></mml:mover></mml:mrow><mml:mrow><mml:mtext>att</mml:mtext></mml:mrow></mml:msub></mml:mrow></mml:math>l~att/2R = 3.395 at <mml:math><mml:mrow><mml:msub><mml:mi>ψ</mml:mi><mml:mn>6</mml:mn></mml:msub></mml:mrow></mml:math>ψ6 = 1). In amorphous samples, the attenuation length depends both on the viscous dissipation mechanisms previously described and on the scattering mean free path associated to the multiple scattering (35) of the elastic perturbation; the later plateaus at the particle size 2R in highly disordered monolayers. The blue points in Fig. S5 G and N correspond to the magnitude (in decibels) of the particle’s global velocity V along one symmetry axis of the monolayer (in a disordered sample, they correspond to the global velocity in the direction of a symmetry axis of the original crystal); the slope is consistent with the decay of the total energy (<mml:math><mml:mrow><mml:mi>V</mml:mi><mml:mo>∝</mml:mo><mml:msup><mml:mi>E</mml:mi><mml:mrow><mml:mrow><mml:mn>1</mml:mn><mml:mo>/</mml:mo><mml:mn>2</mml:mn></mml:mrow></mml:mrow></mml:msup></mml:mrow></mml:math>V∝E1/2, i.e., <mml:math><mml:mrow><mml:msub><mml:mi>l</mml:mi><mml:mrow><mml:mtext>att</mml:mtext></mml:mrow></mml:msub><mml:mo>=</mml:mo><mml:mn>2</mml:mn><mml:mo>?</mml:mo><mml:msub><mml:mrow><mml:mover accent="true"><mml:mi>l</mml:mi><mml:mo>~</mml:mo></mml:mover></mml:mrow><mml:mrow><mml:mtext>att</mml:mtext></mml:mrow></mml:msub></mml:mrow></mml:math>latt=2?l~att), demonstrating that measuring the attenuation length from the global velocity, as done for the experimental data in the main text (Fig. 3G), provides reliable information.

SI Discussion

Here, we elucidate the effect of particles’ elasticity starting from the derivation of the linearized wave equations in alignments of particles (along a single symmetry axis x), in the long-wavelength approximation. The following 1D description does not include the full features of the experimental configurations, for which we use numerical simulations. Nonetheless, it illustrates and unravels the role of the main ingredients on strain wave propagation.

Upon particle approach, when the interstitial fluid thickness is large enough, e.g., for <mml:math><mml:mi>d</mml:mi></mml:math>d <mml:math><mml:mrow><mml:mo>?</mml:mo><mml:msub><mml:mi>d</mml:mi><mml:mrow><mml:mtext>cut</mml:mtext></mml:mrow></mml:msub></mml:mrow></mml:math>?dcut, the shear rates associated with fluid squeeze-out are not sufficiently large to generate strong enough hydrodynamic pressure to cause elastic deformation of the particles (9?11). Here, d is the initial surface-to-surface distance between two spheres, and dcut is the cutoff length determined, for instance, from the particle surface roughness (Fig. S1 A and B). In this regime, particle inertia is balanced exclusively by lubrication forces and Stokes’ drag, i.e.,<mml:math display="block"><mml:mrow><mml:msub><mml:mi mathvariant="bold-italic">F</mml:mi><mml:mrow><mml:mtext>lub</mml:mtext></mml:mrow></mml:msub><mml:mo>?</mml:mo><mml:mi mathvariant="bold-italic">x</mml:mi><mml:mo>=</mml:mo><mml:mo>?</mml:mo><mml:mfrac><mml:mrow><mml:mn>3</mml:mn><mml:mi>π</mml:mi><mml:mi>η</mml:mi><mml:msup><mml:mi>R</mml:mi><mml:mn>2</mml:mn></mml:msup><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:msub><mml:mi>V</mml:mi><mml:mi>i</mml:mi></mml:msub><mml:mo>?</mml:mo><mml:msub><mml:mi>V</mml:mi><mml:mrow><mml:mi>i</mml:mi><mml:mo>?</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msub></mml:mrow><mml:mo>)</mml:mo></mml:mrow></mml:mrow><mml:mrow><mml:mn>2</mml:mn><mml:msub><mml:mi>d</mml:mi><mml:mrow><mml:mi>i</mml:mi><mml:mo>,</mml:mo><mml:mi>i</mml:mi><mml:mo>?</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msub></mml:mrow></mml:mfrac><mml:mo>,</mml:mo></mml:mrow></mml:math>Flub?x=?3πηR2(Vi?Vi?1)2di,i?1,<mml:math display="block"><mml:mrow><mml:msub><mml:mi mathvariant="bold-italic">F</mml:mi><mml:mrow><mml:mtext>drag</mml:mtext></mml:mrow></mml:msub><mml:mo>?</mml:mo><mml:mi mathvariant="bold-italic">x</mml:mi><mml:mo>=</mml:mo><mml:mo>?</mml:mo><mml:mn>6</mml:mn><mml:mi>π</mml:mi><mml:mi>η</mml:mi><mml:mi>R</mml:mi><mml:msub><mml:mi>V</mml:mi><mml:mi>i</mml:mi></mml:msub><mml:mo>,</mml:mo></mml:mrow></mml:math>Fdrag?x=?6πηRVi,and the 1D equation of translational motion is<mml:math display="block"><mml:mrow><mml:mi>m</mml:mi><mml:msub><mml:mrow><mml:mover accent="true"><mml:mi>u</mml:mi><mml:mo>¨</mml:mo></mml:mover></mml:mrow><mml:mi>i</mml:mi></mml:msub><mml:mo>=</mml:mo><mml:msub><mml:mi>F</mml:mi><mml:mrow><mml:mtext>lub</mml:mtext></mml:mrow></mml:msub><mml:mrow><mml:mo>(</mml:mo><mml:mi>i</mml:mi><mml:mo>)</mml:mo></mml:mrow><mml:mo>-</mml:mo><mml:msub><mml:mi>F</mml:mi><mml:mrow><mml:mtext>lub</mml:mtext></mml:mrow></mml:msub><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:mi>i</mml:mi><mml:mo>+</mml:mo><mml:mn>1</mml:mn></mml:mrow><mml:mo>)</mml:mo></mml:mrow><mml:mo>-</mml:mo><mml:mn>6</mml:mn><mml:mi>π</mml:mi><mml:mi>η</mml:mi><mml:mi>R</mml:mi><mml:msub><mml:mrow><mml:mover accent="true"><mml:mi>u</mml:mi><mml:mo>˙</mml:mo></mml:mover></mml:mrow><mml:mi>i</mml:mi></mml:msub><mml:mo>.</mml:mo></mml:mrow></mml:math>mu¨i=Flub(i)-Flub(i+1)-6πηRu˙i.Here, <mml:math><mml:mrow><mml:msub><mml:mi>u</mml:mi><mml:mi>i</mml:mi></mml:msub><mml:mo>=</mml:mo><mml:msub><mml:mi>x</mml:mi><mml:mi>i</mml:mi></mml:msub><mml:mrow><mml:mo>(</mml:mo><mml:mi>t</mml:mi><mml:mo>)</mml:mo></mml:mrow><mml:mo>?</mml:mo><mml:msub><mml:mi>x</mml:mi><mml:mi>i</mml:mi></mml:msub><mml:mrow><mml:mo>(</mml:mo><mml:mn>0</mml:mn><mml:mo>)</mml:mo></mml:mrow></mml:mrow></mml:math>ui=xi(t)?xi(0) is the displacement of the sphere <mml:math><mml:mi>i</mml:mi></mml:math>i, <mml:math><mml:mrow><mml:msub><mml:mi>d</mml:mi><mml:mrow><mml:mi>i</mml:mi><mml:mo>,</mml:mo><mml:mi>i</mml:mi><mml:mo>?</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msub><mml:mo>=</mml:mo><mml:mi>d</mml:mi><mml:mo>+</mml:mo><mml:msub><mml:mi>u</mml:mi><mml:mi>i</mml:mi></mml:msub><mml:mo>?</mml:mo><mml:msub><mml:mi>u</mml:mi><mml:mrow><mml:mi>i</mml:mi><mml:mo>?</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msub></mml:mrow></mml:math>di,i?1=d+ui?ui?1 is the fluid thickness, and <mml:math><mml:mrow><mml:msub><mml:mrow><mml:mover accent="true"><mml:mi>u</mml:mi><mml:mo>˙</mml:mo></mml:mover></mml:mrow><mml:mi>i</mml:mi></mml:msub><mml:mo>=</mml:mo><mml:msub><mml:mi>V</mml:mi><mml:mi>i</mml:mi></mml:msub></mml:mrow></mml:math>u˙i=Vi denotes the time derivative. We introduce the strain <mml:math><mml:mrow><mml:msub><mml:mi mathvariant="italic">?</mml:mi><mml:mi>i</mml:mi></mml:msub><mml:mo>=</mml:mo><mml:mo>?</mml:mo><mml:msubsup><mml:mi>u</mml:mi><mml:mi>i</mml:mi><mml:mo>′</mml:mo></mml:msubsup></mml:mrow></mml:math>?i=?ui′ (derivative over the axial coordinate) and the nondimensional initial fluid thickness <mml:math><mml:mrow><mml:msub><mml:mi mathvariant="italic">?</mml:mi><mml:mn>0</mml:mn></mml:msub><mml:mo>=</mml:mo><mml:mrow><mml:mi>d</mml:mi><mml:mo>/</mml:mo><mml:mrow><mml:mn>2</mml:mn><mml:mi>R</mml:mi></mml:mrow></mml:mrow></mml:mrow></mml:math>?0=d/2R. In the long-wavelength approximation, the strain is <mml:math><mml:mrow><mml:msub><mml:mi mathvariant="italic">?</mml:mi><mml:mi>i</mml:mi></mml:msub><mml:mo>≈</mml:mo><mml:mrow><mml:mrow><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:msub><mml:mi>u</mml:mi><mml:mrow><mml:mi>i</mml:mi><mml:mo>?</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msub><mml:mo>?</mml:mo><mml:msub><mml:mi>u</mml:mi><mml:mi>i</mml:mi></mml:msub></mml:mrow><mml:mo>)</mml:mo></mml:mrow></mml:mrow><mml:mo>/</mml:mo><mml:mrow><mml:mn>2</mml:mn><mml:mi>R</mml:mi></mml:mrow></mml:mrow></mml:mrow></mml:math>?i≈(ui?1?ui)/2R, and a Taylor expansion of the lubrication force <mml:math><mml:mrow><mml:msub><mml:mi>F</mml:mi><mml:mrow><mml:mtext>lub</mml:mtext></mml:mrow></mml:msub><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:mi>i</mml:mi><mml:mo>±</mml:mo><mml:mn>1</mml:mn></mml:mrow><mml:mo>)</mml:mo></mml:mrow><mml:mo>≈</mml:mo><mml:msub><mml:mi>F</mml:mi><mml:mrow><mml:mtext>lub</mml:mtext></mml:mrow></mml:msub><mml:mrow><mml:mo>(</mml:mo><mml:mi>i</mml:mi><mml:mo>)</mml:mo></mml:mrow><mml:mo>±</mml:mo><mml:mn>2</mml:mn><mml:mi>R</mml:mi><mml:msub><mml:mrow><mml:mi>F</mml:mi><mml:mo>′</mml:mo></mml:mrow><mml:mtext>lub</mml:mtext></mml:msub><mml:mrow><mml:mo>(</mml:mo><mml:mi>i</mml:mi><mml:mo>)</mml:mo></mml:mrow><mml:mo>+</mml:mo><mml:mn>2</mml:mn><mml:msup><mml:mi>R</mml:mi><mml:mn>2</mml:mn></mml:msup><mml:msub><mml:msup><mml:mi>F</mml:mi><mml:mo>″</mml:mo></mml:msup><mml:mrow><mml:mtext>lub</mml:mtext></mml:mrow></mml:msub><mml:mrow><mml:mo>(</mml:mo><mml:mi>i</mml:mi><mml:mo>)</mml:mo></mml:mrow></mml:mrow></mml:math>Flub(i±1)≈Flub(i)±2RF′lub(i)+2R2F″lub(i) yields<mml:math display="block"><mml:mrow><mml:mi>m</mml:mi><mml:msub><mml:mrow><mml:mover accent="true"><mml:mi mathvariant="italic">?</mml:mi><mml:mo>¨</mml:mo></mml:mover></mml:mrow><mml:mi>i</mml:mi></mml:msub><mml:mo>≈</mml:mo><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:mn>2</mml:mn><mml:mi>R</mml:mi></mml:mrow><mml:mo>)</mml:mo></mml:mrow><mml:msup><mml:mrow><mml:mrow><mml:mo>[</mml:mo><mml:mrow><mml:mfrac><mml:mrow><mml:mn>3</mml:mn><mml:mi>π</mml:mi><mml:mi>η</mml:mi><mml:msup><mml:mi>R</mml:mi><mml:mn>2</mml:mn></mml:msup><mml:msub><mml:mrow><mml:mover accent="true"><mml:mi mathvariant="italic">?</mml:mi><mml:mo>˙</mml:mo></mml:mover></mml:mrow><mml:mi>i</mml:mi></mml:msub></mml:mrow><mml:mrow><mml:mn>2</mml:mn><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:msub><mml:mi mathvariant="italic">?</mml:mi><mml:mn>0</mml:mn></mml:msub><mml:mo>?</mml:mo><mml:msub><mml:mi mathvariant="italic">?</mml:mi><mml:mi>i</mml:mi></mml:msub></mml:mrow><mml:mo>)</mml:mo></mml:mrow></mml:mrow></mml:mfrac></mml:mrow><mml:mo>]</mml:mo></mml:mrow></mml:mrow><mml:mo>″</mml:mo></mml:msup><mml:mo>?</mml:mo><mml:mn>6</mml:mn><mml:mi>π</mml:mi><mml:mi>η</mml:mi><mml:mi>R</mml:mi><mml:msub><mml:mrow><mml:mover accent="true"><mml:mi mathvariant="italic">?</mml:mi><mml:mo>˙</mml:mo></mml:mover></mml:mrow><mml:mi>i</mml:mi></mml:msub><mml:mo>.</mml:mo></mml:mrow></mml:math>m?¨i≈(2R)[3πηR2?˙i2(?0??i)]″?6πηR?˙i.Given the diffusion coefficient <mml:math><mml:mrow><mml:mi>D</mml:mi><mml:mo>=</mml:mo><mml:mrow><mml:mrow><mml:mn>9</mml:mn><mml:mi>η</mml:mi><mml:mi>R</mml:mi></mml:mrow><mml:mo>/</mml:mo><mml:mrow><mml:mn>2</mml:mn><mml:mi>ρ</mml:mi><mml:mi>d</mml:mi></mml:mrow></mml:mrow></mml:mrow></mml:math>D=9ηR/2ρd and the relaxation time <mml:math><mml:mrow><mml:mi>τ</mml:mi><mml:mo>=</mml:mo><mml:mrow><mml:mrow><mml:mn>2</mml:mn><mml:mi>ρ</mml:mi><mml:msup><mml:mi>R</mml:mi><mml:mn>2</mml:mn></mml:msup></mml:mrow><mml:mo>/</mml:mo><mml:mrow><mml:mn>9</mml:mn><mml:mi>η</mml:mi></mml:mrow></mml:mrow></mml:mrow></mml:math>τ=2ρR2/9η, the linearized (<mml:math><mml:mrow><mml:mi mathvariant="italic">?</mml:mi><mml:mo>?</mml:mo><mml:msub><mml:mi mathvariant="italic">?</mml:mi><mml:mn>0</mml:mn></mml:msub></mml:mrow></mml:math>???0) equation for the strain field <mml:math><mml:mrow><mml:mi mathvariant="italic">?</mml:mi><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:mi>x</mml:mi><mml:mo>=</mml:mo><mml:mn>2</mml:mn><mml:mi>i</mml:mi><mml:mi>R</mml:mi><mml:mo>,</mml:mo><mml:mi>t</mml:mi></mml:mrow><mml:mo>)</mml:mo></mml:mrow><mml:mo>=</mml:mo><mml:msub><mml:mi mathvariant="italic">?</mml:mi><mml:mi>i</mml:mi></mml:msub><mml:mrow><mml:mo>(</mml:mo><mml:mi>t</mml:mi><mml:mo>)</mml:mo></mml:mrow></mml:mrow></mml:math>?(x=2iR,t)=?i(t)<mml:math display="block"><mml:mrow><mml:mrow><mml:mover accent="true"><mml:mi mathvariant="italic">?</mml:mi><mml:mo>¨</mml:mo></mml:mover></mml:mrow><mml:mo>≈</mml:mo><mml:mi>D</mml:mi><mml:mrow><mml:mover accent="true"><mml:mi mathvariant="italic">?</mml:mi><mml:mo>˙</mml:mo></mml:mover></mml:mrow><mml:mo>″</mml:mo><mml:mo>?</mml:mo><mml:mfrac><mml:mrow><mml:mover accent="true"><mml:mi mathvariant="italic">?</mml:mi><mml:mo>˙</mml:mo></mml:mover></mml:mrow><mml:mi>τ</mml:mi></mml:mfrac></mml:mrow></mml:math>?¨≈D?˙″??˙τdescribes the diffusion and the dissipation of an initially localized perturbation<mml:math display="block"><mml:mrow><mml:mrow><mml:mover accent="true"><mml:mi mathvariant="italic">?</mml:mi><mml:mo>˙</mml:mo></mml:mover></mml:mrow><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:mi>x</mml:mi><mml:mo>,</mml:mo><mml:mi>t</mml:mi></mml:mrow><mml:mo>)</mml:mo></mml:mrow><mml:mo>∝</mml:mo><mml:mfrac><mml:mrow><mml:mi>exp</mml:mi><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:mrow><mml:mrow><mml:mo>?</mml:mo><mml:msup><mml:mi>x</mml:mi><mml:mn>2</mml:mn></mml:msup></mml:mrow><mml:mo>/</mml:mo><mml:mrow><mml:mn>4</mml:mn><mml:mi>D</mml:mi><mml:mi>t</mml:mi></mml:mrow></mml:mrow><mml:mo>?</mml:mo><mml:mrow><mml:mi>t</mml:mi><mml:mo>/</mml:mo><mml:mi>τ</mml:mi></mml:mrow></mml:mrow><mml:mo>)</mml:mo></mml:mrow></mml:mrow><mml:mrow><mml:msqrt><mml:mrow><mml:mn>4</mml:mn><mml:mi>π</mml:mi><mml:mi>D</mml:mi><mml:mi>t</mml:mi></mml:mrow></mml:msqrt></mml:mrow></mml:mfrac><mml:mo>.</mml:mo></mml:mrow></mml:math>?˙(x,t)∝exp(?x2/4Dt?t/τ)4πDt.In this regime, the attenuation is not exponential, as observed in experiments (Fig. 3G), but diffusive; the penetration depth of a perturbation, <mml:math><mml:mrow><mml:msub><mml:mi>l</mml:mi><mml:mi>D</mml:mi></mml:msub><mml:mo>=</mml:mo><mml:msup><mml:mrow><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:mn>4</mml:mn><mml:mi>D</mml:mi><mml:mi>t</mml:mi></mml:mrow><mml:mo>)</mml:mo></mml:mrow></mml:mrow><mml:mrow><mml:mrow><mml:mn>1</mml:mn><mml:mo>/</mml:mo><mml:mn>2</mml:mn></mml:mrow></mml:mrow></mml:msup><mml:mo>∝</mml:mo><mml:msup><mml:mi>η</mml:mi><mml:mrow><mml:mrow><mml:mn>1</mml:mn><mml:mo>/</mml:mo><mml:mn>2</mml:mn></mml:mrow></mml:mrow></mml:msup></mml:mrow></mml:math>lD=(4Dt)1/2∝η1/2, would therefore increase with the viscosity of the fluid.

The situation is, instead, different in the case where two particles are sufficiently close (<mml:math><mml:mrow><mml:mi>d</mml:mi><mml:mo>≈</mml:mo><mml:msub><mml:mi>d</mml:mi><mml:mrow><mml:mtext>cut</mml:mtext></mml:mrow></mml:msub></mml:mrow></mml:math>d≈dcut) to generate large enough shear rates leading to elastic deformations of the particles (9?11). This case is probed from the analysis of a two-body impact dynamics. For a given separation <mml:math><mml:mrow><mml:msub><mml:mi>d</mml:mi><mml:mrow><mml:mn>12</mml:mn></mml:mrow></mml:msub><mml:mo>=</mml:mo><mml:mi>d</mml:mi><mml:mo>+</mml:mo><mml:msub><mml:mi>u</mml:mi><mml:mn>2</mml:mn></mml:msub><mml:mo>?</mml:mo><mml:msub><mml:mi>u</mml:mi><mml:mn>1</mml:mn></mml:msub></mml:mrow></mml:math>d12=d+u2?u1 between particles 1 and 2, the time integration of the equation of translational motion in the normal directions gives<mml:math display="block"><mml:mrow><mml:mi>m</mml:mi><mml:mrow><mml:mo>[</mml:mo><mml:mrow><mml:msub><mml:mrow><mml:mover accent="true"><mml:mi>d</mml:mi><mml:mo>˙</mml:mo></mml:mover></mml:mrow><mml:mrow><mml:mn>21</mml:mn></mml:mrow></mml:msub><mml:mrow><mml:mo>(</mml:mo><mml:mi>t</mml:mi><mml:mo>)</mml:mo></mml:mrow><mml:mo>?</mml:mo><mml:msub><mml:mrow><mml:mover accent="true"><mml:mi>d</mml:mi><mml:mo>˙</mml:mo></mml:mover></mml:mrow><mml:mrow><mml:mn>21</mml:mn></mml:mrow></mml:msub><mml:mrow><mml:mo>(</mml:mo><mml:mn>0</mml:mn><mml:mo>)</mml:mo></mml:mrow></mml:mrow><mml:mo>]</mml:mo></mml:mrow><mml:mo>=</mml:mo><mml:mo>?</mml:mo><mml:mn>3</mml:mn><mml:mi>π</mml:mi><mml:mi>η</mml:mi><mml:msup><mml:mi>R</mml:mi><mml:mn>2</mml:mn></mml:msup><mml:mo>?</mml:mo><mml:mi>ln</mml:mi><mml:mrow><mml:mo>[</mml:mo><mml:mrow><mml:mfrac><mml:mrow><mml:msub><mml:mi>d</mml:mi><mml:mrow><mml:mn>21</mml:mn></mml:mrow></mml:msub><mml:mrow><mml:mo>(</mml:mo><mml:mi>t</mml:mi><mml:mo>)</mml:mo></mml:mrow></mml:mrow><mml:mrow><mml:msub><mml:mi>d</mml:mi><mml:mrow><mml:mn>21</mml:mn></mml:mrow></mml:msub><mml:mrow><mml:mo>(</mml:mo><mml:mn>0</mml:mn><mml:mo>)</mml:mo></mml:mrow></mml:mrow></mml:mfrac></mml:mrow><mml:mo>]</mml:mo></mml:mrow><mml:mo>?</mml:mo><mml:mn>6</mml:mn><mml:mi>π</mml:mi><mml:mi>η</mml:mi><mml:mi>R</mml:mi><mml:mrow><mml:mo>[</mml:mo><mml:mrow><mml:msub><mml:mi>d</mml:mi><mml:mrow><mml:mn>21</mml:mn></mml:mrow></mml:msub><mml:mrow><mml:mo>(</mml:mo><mml:mi>t</mml:mi><mml:mo>)</mml:mo></mml:mrow><mml:mo>?</mml:mo><mml:msub><mml:mi>d</mml:mi><mml:mrow><mml:mn>21</mml:mn></mml:mrow></mml:msub><mml:mrow><mml:mo>(</mml:mo><mml:mn>0</mml:mn><mml:mo>)</mml:mo></mml:mrow></mml:mrow><mml:mo>]</mml:mo></mml:mrow><mml:mo>.</mml:mo></mml:mrow></mml:math>m[d˙21(t)?d˙21(0)]=?3πηR2?ln[d21(t)d21(0)]?6πηR[d21(t)?d21(0)].The initial separation is <mml:math><mml:mrow><mml:msub><mml:mi>d</mml:mi><mml:mrow><mml:mn>21</mml:mn></mml:mrow></mml:msub><mml:mrow><mml:mo>(</mml:mo><mml:mn>0</mml:mn><mml:mo>)</mml:mo></mml:mrow><mml:mo>=</mml:mo><mml:mi>d</mml:mi><mml:mo>=</mml:mo><mml:mn>400</mml:mn></mml:mrow></mml:math>d21(0)=d=400 nm, and the collision velocity is <mml:math><mml:mrow><mml:msub><mml:mi>V</mml:mi><mml:mn>0</mml:mn></mml:msub><mml:mo>=</mml:mo><mml:mo>?</mml:mo><mml:msub><mml:mrow><mml:mover accent="true"><mml:mi>d</mml:mi><mml:mo>˙</mml:mo></mml:mover></mml:mrow><mml:mrow><mml:mn>21</mml:mn></mml:mrow></mml:msub><mml:mrow><mml:mo>(</mml:mo><mml:mn>0</mml:mn><mml:mo>)</mml:mo></mml:mrow></mml:mrow></mml:math>V0=?d˙21(0). The minimal separation is reached at time tcut, when <mml:math><mml:mrow><mml:msub><mml:mi>d</mml:mi><mml:mrow><mml:mn>21</mml:mn></mml:mrow></mml:msub><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:msub><mml:mi>t</mml:mi><mml:mrow><mml:mtext>cut</mml:mtext></mml:mrow></mml:msub></mml:mrow><mml:mo>)</mml:mo></mml:mrow><mml:mo>=</mml:mo><mml:msub><mml:mi>d</mml:mi><mml:mrow><mml:mtext>cut</mml:mtext></mml:mrow></mml:msub><mml:mo>=</mml:mo><mml:mn>17</mml:mn></mml:mrow></mml:math>d21(tcut)=dcut=17nm and <mml:math><mml:mrow><mml:msub><mml:mrow><mml:mover accent="true"><mml:mi>d</mml:mi><mml:mo>˙</mml:mo></mml:mover></mml:mrow><mml:mrow><mml:mn>21</mml:mn></mml:mrow></mml:msub><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:msub><mml:mi>t</mml:mi><mml:mrow><mml:mtext>cut</mml:mtext></mml:mrow></mml:msub></mml:mrow><mml:mo>)</mml:mo></mml:mrow><mml:mo>=</mml:mo><mml:mn>0</mml:mn></mml:mrow></mml:math>d˙21(tcut)=0 upon rebound, if<mml:math display="block"><mml:mrow><mml:msub><mml:mi>V</mml:mi><mml:mn>0</mml:mn></mml:msub><mml:mo>></mml:mo><mml:msub><mml:mi>V</mml:mi><mml:mrow><mml:mi>c</mml:mi><mml:mi>r</mml:mi></mml:mrow></mml:msub><mml:mo>=</mml:mo><mml:mfrac><mml:mi>R</mml:mi><mml:mrow><mml:mn>2</mml:mn><mml:mi>τ</mml:mi></mml:mrow></mml:mfrac><mml:mi>ln</mml:mi><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:mfrac><mml:mi>d</mml:mi><mml:mrow><mml:msub><mml:mi>d</mml:mi><mml:mrow><mml:mtext>cut</mml:mtext></mml:mrow></mml:msub></mml:mrow></mml:mfrac></mml:mrow><mml:mo>)</mml:mo></mml:mrow><mml:mo>+</mml:mo><mml:mfrac><mml:mrow><mml:mi>d</mml:mi><mml:mo>?</mml:mo><mml:msub><mml:mi>d</mml:mi><mml:mrow><mml:mtext>cut</mml:mtext></mml:mrow></mml:msub></mml:mrow><mml:mi>τ</mml:mi></mml:mfrac><mml:mo>.</mml:mo></mml:mrow></mml:math>V0>Vcr=R2τln(ddcut)+d?dcutτ.The critical velocity is of the order of 1 m/s in water, suggesting that collisions between particles, mediated by the interstitial fluid, occur in the range of excitations probed in our experiments. As a consequence, the interaction between particles through the interstitial fluid (10, 11) relies on the elastohydrodynamic repulsion introduced above and given by (9)<mml:math display="block"><mml:mrow><mml:msub><mml:mi mathvariant="bold-italic">F</mml:mi><mml:mrow><mml:mtext>ehd</mml:mtext></mml:mrow></mml:msub><mml:mrow><mml:mo>(</mml:mo><mml:mi>i</mml:mi><mml:mo>)</mml:mo></mml:mrow><mml:mo>?</mml:mo><mml:mi mathvariant="bold-italic">x</mml:mi><mml:mo>=</mml:mo><mml:mfrac><mml:mn>3</mml:mn><mml:mn>2</mml:mn></mml:mfrac><mml:mi>π</mml:mi><mml:mi>η</mml:mi><mml:msup><mml:mi>R</mml:mi><mml:mn>2</mml:mn></mml:msup><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:mfrac><mml:mrow><mml:msub><mml:mrow><mml:mover accent="true"><mml:mi>δ</mml:mi><mml:mo>˙</mml:mo></mml:mover></mml:mrow><mml:mrow><mml:mi>i</mml:mi><mml:mo>,</mml:mo><mml:mi>i</mml:mi><mml:mo>?</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msub></mml:mrow><mml:mrow><mml:msub><mml:mi>d</mml:mi><mml:mrow><mml:mtext>cut</mml:mtext></mml:mrow></mml:msub></mml:mrow></mml:mfrac></mml:mrow><mml:mo>)</mml:mo></mml:mrow><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:mn>1</mml:mn><mml:mo>+</mml:mo><mml:mfrac><mml:mrow><mml:msub><mml:mi>δ</mml:mi><mml:mrow><mml:mi>i</mml:mi><mml:mo>,</mml:mo><mml:mi>i</mml:mi><mml:mo>?</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msub></mml:mrow><mml:mrow><mml:msub><mml:mi>d</mml:mi><mml:mrow><mml:mtext>cut</mml:mtext></mml:mrow></mml:msub></mml:mrow></mml:mfrac></mml:mrow><mml:mo>)</mml:mo></mml:mrow><mml:mo>+</mml:mo><mml:mfrac><mml:mrow><mml:mi>E</mml:mi><mml:msqrt><mml:mrow><mml:mn>2</mml:mn><mml:mi>R</mml:mi></mml:mrow></mml:msqrt><mml:msubsup><mml:mi>δ</mml:mi><mml:mrow><mml:mi>i</mml:mi><mml:mo>,</mml:mo><mml:mi>i</mml:mi><mml:mo>?</mml:mo><mml:mn>1</mml:mn></mml:mrow><mml:mrow><mml:mrow><mml:mn>3</mml:mn><mml:mo>/</mml:mo><mml:mn>2</mml:mn></mml:mrow></mml:mrow></mml:msubsup></mml:mrow><mml:mrow><mml:mn>3</mml:mn><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:mn>1</mml:mn><mml:mo>?</mml:mo><mml:msup><mml:mi>ν</mml:mi><mml:mn>2</mml:mn></mml:msup></mml:mrow><mml:mo>)</mml:mo></mml:mrow></mml:mrow></mml:mfrac><mml:mo>,</mml:mo></mml:mrow></mml:math>Fehd(i)?x=32πηR2(δ˙i,i?1dcut)(1+δi,i?1dcut)+E2Rδi,i?13/23(1?ν2),where <mml:math><mml:mrow><mml:msub><mml:mi>δ</mml:mi><mml:mrow><mml:mi>i</mml:mi><mml:mo>,</mml:mo><mml:mi>i</mml:mi><mml:mo>?</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msub><mml:mo>=</mml:mo><mml:msub><mml:mi>u</mml:mi><mml:mrow><mml:mi>i</mml:mi><mml:mo>?</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msub><mml:mo>?</mml:mo><mml:msub><mml:mi>u</mml:mi><mml:mi>i</mml:mi></mml:msub><mml:mo>≈</mml:mo><mml:mn>2</mml:mn><mml:mi>R</mml:mi><mml:msub><mml:mi mathvariant="italic">?</mml:mi><mml:mi>i</mml:mi></mml:msub></mml:mrow></mml:math>δi,i?1=ui?1?ui≈2R?i is the overlap distance between particles, and E and ν are the elastic modulus and the Poisson’s ratio of silica. At moderate strains, <mml:math><mml:mrow><mml:msub><mml:mi mathvariant="italic">?</mml:mi><mml:mi>m</mml:mi></mml:msub><mml:mo>≈</mml:mo><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:mrow><mml:mrow><mml:msub><mml:mi>δ</mml:mi><mml:mi>m</mml:mi></mml:msub></mml:mrow><mml:mo>/</mml:mo><mml:mrow><mml:mn>2</mml:mn><mml:mi>R</mml:mi></mml:mrow></mml:mrow></mml:mrow><mml:mo>)</mml:mo></mml:mrow><mml:mo>?</mml:mo><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:mrow><mml:mrow><mml:msub><mml:mi>d</mml:mi><mml:mrow><mml:mtext>cut</mml:mtext></mml:mrow></mml:msub></mml:mrow><mml:mo>/</mml:mo><mml:mrow><mml:mn>2</mml:mn><mml:mi>R</mml:mi></mml:mrow></mml:mrow></mml:mrow><mml:mo>)</mml:mo></mml:mrow></mml:mrow></mml:math>?m≈(δm/2R)?(dcut/2R), the linearized equation of motion becomes<mml:math display="block"><mml:mrow><mml:mrow><mml:mover accent="true"><mml:mi mathvariant="italic">?</mml:mi><mml:mo>¨</mml:mo></mml:mover></mml:mrow><mml:mo>≈</mml:mo><mml:msup><mml:mrow><mml:mrow><mml:mo>[</mml:mo><mml:mrow><mml:msub><mml:mi>D</mml:mi><mml:mrow><mml:mtext>cut</mml:mtext></mml:mrow></mml:msub><mml:mrow><mml:mover accent="true"><mml:mi mathvariant="italic">?</mml:mi><mml:mo>˙</mml:mo></mml:mover></mml:mrow><mml:mo>+</mml:mo><mml:mtext>const</mml:mtext><mml:mo>?</mml:mo><mml:msubsup><mml:mi>c</mml:mi><mml:mn>0</mml:mn><mml:mn>2</mml:mn></mml:msubsup><mml:msup><mml:mi mathvariant="italic">?</mml:mi><mml:mrow><mml:mrow><mml:mn>3</mml:mn><mml:mo>/</mml:mo><mml:mn>2</mml:mn></mml:mrow></mml:mrow></mml:msup></mml:mrow><mml:mo>]</mml:mo></mml:mrow></mml:mrow><mml:mo>″</mml:mo></mml:msup><mml:mo>?</mml:mo><mml:mfrac><mml:mrow><mml:mover accent="true"><mml:mi mathvariant="italic">?</mml:mi><mml:mo>˙</mml:mo></mml:mover></mml:mrow><mml:mi>τ</mml:mi></mml:mfrac><mml:mo>≈</mml:mo><mml:msup><mml:mrow><mml:mrow><mml:mo>[</mml:mo><mml:mrow><mml:msub><mml:mi>D</mml:mi><mml:mrow><mml:mtext>cut</mml:mtext></mml:mrow></mml:msub><mml:mrow><mml:mover accent="true"><mml:mi mathvariant="italic">?</mml:mi><mml:mo>˙</mml:mo></mml:mover></mml:mrow><mml:mo>+</mml:mo><mml:msubsup><mml:mi>c</mml:mi><mml:mi>w</mml:mi><mml:mn>2</mml:mn></mml:msubsup><mml:mi mathvariant="italic">?</mml:mi></mml:mrow><mml:mo>]</mml:mo></mml:mrow></mml:mrow><mml:mo>″</mml:mo></mml:msup><mml:mo>?</mml:mo><mml:mfrac><mml:mrow><mml:mover accent="true"><mml:mi mathvariant="italic">?</mml:mi><mml:mo>˙</mml:mo></mml:mover></mml:mrow><mml:mi>τ</mml:mi></mml:mfrac><mml:mo>,</mml:mo></mml:mrow></mml:math>?¨≈[Dcut?˙+const?c02?3/2]″??˙τ≈[Dcut?˙+cw2?]″??˙τ,in which <mml:math><mml:mrow><mml:msub><mml:mi>D</mml:mi><mml:mrow><mml:mtext>cut</mml:mtext></mml:mrow></mml:msub><mml:mo>=</mml:mo><mml:mrow><mml:mrow><mml:mn>9</mml:mn><mml:mi>η</mml:mi><mml:mi>R</mml:mi></mml:mrow><mml:mo>/</mml:mo><mml:mrow><mml:mn>2</mml:mn><mml:mi>ρ</mml:mi><mml:msub><mml:mi>d</mml:mi><mml:mrow><mml:mtext>cut</mml:mtext></mml:mrow></mml:msub></mml:mrow></mml:mrow></mml:mrow></mml:math>Dcut=9ηR/2ρdcut, <mml:math><mml:mrow><mml:msub><mml:mi>c</mml:mi><mml:mn>0</mml:mn></mml:msub><mml:mo>=</mml:mo><mml:msqrt><mml:mrow><mml:mrow><mml:mi>E</mml:mi><mml:mo>/</mml:mo><mml:mi>ρ</mml:mi></mml:mrow></mml:mrow></mml:msqrt></mml:mrow></mml:math>c0=E/ρ is the speed of sound in the bulk material of the particles, where const ~1 is a numerical constant, and <mml:math><mml:mrow><mml:msub><mml:mi>c</mml:mi><mml:mi>w</mml:mi></mml:msub><mml:mo>∝</mml:mo><mml:msub><mml:mi>c</mml:mi><mml:mn>0</mml:mn></mml:msub><mml:msubsup><mml:mi mathvariant="italic">?</mml:mi><mml:mi>m</mml:mi><mml:mrow><mml:mrow><mml:mn>1</mml:mn><mml:mo>/</mml:mo><mml:mn>2</mml:mn></mml:mrow></mml:mrow></mml:msubsup></mml:mrow></mml:math>cw∝c0?m1/2 stands for a linearized approximation of the amplitude-dependent speed (7) of elastic strain waves that propagate along alignments of particles. In this regime, the mechanical response of the colloid depends on the ratio between the lubrication and the elastic contributions <mml:math><mml:mrow><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:mrow><mml:mrow><mml:msub><mml:mi>D</mml:mi><mml:mrow><mml:mtext>cut</mml:mtext></mml:mrow></mml:msub><mml:mrow><mml:mover accent="true"><mml:mi mathvariant="italic">?</mml:mi><mml:mo>˙</mml:mo></mml:mover></mml:mrow></mml:mrow><mml:mo>/</mml:mo><mml:mrow><mml:msubsup><mml:mi>c</mml:mi><mml:mi>w</mml:mi><mml:mn>2</mml:mn></mml:msubsup><mml:mi mathvariant="italic">?</mml:mi></mml:mrow></mml:mrow></mml:mrow><mml:mo>)</mml:mo></mml:mrow><mml:mo>∝</mml:mo><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:mrow><mml:mrow><mml:mi>η</mml:mi><mml:mi>ω</mml:mi></mml:mrow><mml:mo>/</mml:mo><mml:mi>E</mml:mi></mml:mrow></mml:mrow><mml:mo>)</mml:mo></mml:mrow><mml:mo>?</mml:mo><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:mrow><mml:mi>R</mml:mi><mml:mo>/</mml:mo><mml:mrow><mml:msub><mml:mi>d</mml:mi><mml:mrow><mml:mtext>cut</mml:mtext></mml:mrow></mml:msub></mml:mrow></mml:mrow></mml:mrow><mml:mo>)</mml:mo></mml:mrow><mml:mo>?</mml:mo><mml:msup><mml:mrow><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:mrow><mml:mi>R</mml:mi><mml:mo>/</mml:mo><mml:mrow><mml:msub><mml:mi>δ</mml:mi><mml:mi>m</mml:mi></mml:msub></mml:mrow></mml:mrow></mml:mrow><mml:mo>)</mml:mo></mml:mrow></mml:mrow><mml:mrow><mml:mrow><mml:mn>1</mml:mn><mml:mo>/</mml:mo><mml:mn>2</mml:mn></mml:mrow></mml:mrow></mml:msup></mml:mrow></mml:math>(Dcut?˙/cw2?)∝(ηω/E)?(R/dcut)?(R/δm)1/2, where <mml:math><mml:mrow><mml:mi>ω</mml:mi><mml:mo>=</mml:mo><mml:mn>2</mml:mn><mml:mi>π</mml:mi><mml:mi>f</mml:mi></mml:mrow></mml:math>ω=2πf is the angular frequency. At high viscosity or low initial energy (i.e., at small strain <mml:math><mml:mrow><mml:msub><mml:mi mathvariant="italic">?</mml:mi><mml:mi>m</mml:mi></mml:msub></mml:mrow></mml:math>?m), viscous lubrication dominates, and the diffusion of particle momentum occurs as described above. Conversely, at low viscosity or high initial impact velocity, the elasticity prevails: Weakly attenuated elastic waves propagate via the network of particles. In this case, the dispersion relation of a harmonic wave, <mml:math><mml:mrow><mml:mi mathvariant="italic">?</mml:mi><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:mi>x</mml:mi><mml:mo>,</mml:mo><mml:mi>t</mml:mi></mml:mrow><mml:mo>)</mml:mo></mml:mrow><mml:mo>=</mml:mo><mml:msup><mml:mi mathvariant="normal">e</mml:mi><mml:mrow><mml:mi>j</mml:mi><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:mi>ω</mml:mi><mml:mi>t</mml:mi><mml:mo>?</mml:mo><mml:mi>q</mml:mi><mml:mi>x</mml:mi></mml:mrow><mml:mo>)</mml:mo></mml:mrow></mml:mrow></mml:msup></mml:mrow></mml:math>?(x,t)=ej(ωt?qx), shows that the imaginary part of the wavenumber <mml:math><mml:mi>q</mml:mi></mml:math>q (the inverse of the attenuation length) is asymptotically proportional to the viscosity, <mml:math><mml:mrow><mml:msub><mml:mi>l</mml:mi><mml:mrow><mml:mtext>att</mml:mtext></mml:mrow></mml:msub><mml:mo>∝</mml:mo><mml:msup><mml:mi>η</mml:mi><mml:mrow><mml:mo>?</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msup></mml:mrow></mml:math>latt∝η?1, in agreement with our observations (see the data at low viscosities, in Fig. 3G, Inset). Both the magnitude <mml:math><mml:mrow><mml:msub><mml:mi mathvariant="italic">?</mml:mi><mml:mi>m</mml:mi></mml:msub></mml:mrow></mml:math>?m and the speed of the pulse <mml:math><mml:mrow><mml:msub><mml:mi>c</mml:mi><mml:mi>w</mml:mi></mml:msub></mml:mrow></mml:math>cw thus decrease as the perturbation propagates far from the SI (Fig. 2 I and J); at long distances, the transport finally results in the diffusive regime described above, i.e., the momentum of the particles is not sufficient to induce collisions. In this regime, the curve of the penetration depth as a function of the viscosity bends (see the inflection of data at highest viscosities, in Fig. 3G, Inset).

Acknowledgments

We thank Ramakrishna Shivaprakash Narve for the Atomic Force Microscopy friction and adhesion data and Michele Zanini and Svetoslav Anachov for particle roughness measurement and analysis. L.I. and I.B. acknowledge financial support from Swiss National Science Foundation Grant PP00P2_144646/1 and ETH Postdoctoral Fellowship FEL-02 14-1. S.J. acknowledges financial support from the Agence Nationale de la Recherche and the Fondation de Recherche pour l'Aéronautique et l'Espace, Project METAUDIBLE ANR-13-BS09-0003-01. C.D. acknowledges Air Force Office of Scientific Research Center of Excellence Grant FA9550-12-1-0091.

Footnotes

  • ?1Present address: Physical and Theoretical Chemistry Laboratory, Department of Chemistry, University of Oxford, Oxford, OX1 3QZ, UK.

  • ?2I.B., J.C., W.-H.L., and S.J. contributed equally to this work.

  • ?3To whom correspondence may be addressed. Email: daraio{at}caltech.edu or lucio.isa{at}mat.ethz.ch.
  • Author contributions: I.B., C.D., and L.I. designed research; I.B., J.C., W.-H.L., and S.J. performed research; I.B., J.C., W.-H.L., S.J., and L.I. analyzed data; and I.B., J.C., S.J., C.D., and L.I. wrote the paper.

  • The authors declare no conflict of interest.

  • This article is a PNAS Direct Submission.

  • This article contains supporting information online at www.danielhellerman.com/lookup/suppl/doi:10.1073/pnas.1712266114/-/DCSupplemental.

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Online Impact

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