Hyperuniformity with no fine tuning in sheared sedimenting suspensions

Particle suspensions, present in many natural and industrial settings, typically contain aggregates or other microstructures that can complicate macroscopic flow behaviors and damage processing equipment. Recent work found that applying uniform periodic shear near a critical transition can reduce fluctuations in the particle concentration across all length scales, leading to a hyperuniform state. However, this strategy for homogenization requires fine tuning of the strain amplitude. Here we show that in a model of sedimenting particles under periodic shear, there is a well-defined regime at low sedimentation speed where hyperuniform scaling automatically occurs. Our simulations and theoretical arguments show that the homogenization extends up to a finite length scale that diverges as the sedimentation speed approaches zero.

I. SUPPLEMENTARY FIGURES κ=1.6, γ=10 κ=4.0, γ=10 κ=8.0, γ=10 κ=8.0, γ=0.5 κ=11.3, γ=0.5 κ=11.3, γ=3 κ=16.0, γ=10 κ=19.5, γ=0.5 κ=31.9, γ=10  Variance of the number density, σ 2 ρ , versus for a square system of size L = 200 and subsystems of size L < L. The largest system has N = 10186, φ = 0.2, and is sheared cyclically at amplitude γ = 3.01 ≈ γ c until it reaches a reversible state. We thus obtain a hyperuniform scaling with λ = 2.60. Main: By scaling the x and y axes, the data are collapsed onto a master curve, which falls off as approaches the system size. The shaded region at position y with thickness dy is mapped to position y with thickness dy that conserves its mass. Note that the height, h, of the initial and final profiles is constrained to be the same. ρ ( ) is shallower than 1. As in our 2D systems, we measure the variance over the bottom 99% of particles, and we consider windows of size < h ∞ /5 to avoid system size effects. Dashed line: Phase boundary from our theory with no free parameters, Supplementary Eq. 6. As in 2D, hyperuniform scaling emerges below a finite threshold value of A √ φ c and extends to longer lengthscales at smaller A √ φ c .

II. SUPPLEMENTARY NOTES
Supplementary Note 1: Dependence of self-organized criticality on collision rule. The simulation model for cyclically-sheared viscous suspensions that we use in this work was originally developed by Ref. [2] to study self-organized reversible states. Variants of this model have been studied in recent years. Reference [3] studied a wide range of driving and collision rules to test for the robustness of results on memory formation; Refs. [4,5] used isotropic swelling in place of shear as a simpler method for studying the critical transition, and Refs. [3,6] used center-of-mass conserving collisions to suppress long-range diffusion.
Here we probe one aspect of these kinematics in order to make a more precise comparison to previous results [1]. In a dense portion of the sample, a particle can encounter multiple other particles during a single cycle. In the present work we give just one kick to such a particle, whereas Corté et al. [1] gave one kick for each particle encountered, which increases diffusion rates in dense regions.
We find that the scaling for obtaining the critical concentration under sedimentation does not change between these two models. Supplementary Fig. 1 shows φ ∞ /φ c for simulations we performed with the "multiple kick" rule. The data only approximately collapse when plotted versus A, but they collapse cleanly when plotted versus A. To contrast the two expressions, in 5 of these simulations we kept the product κv s constant while varying κ from 1.6 to 31.9 (with γ = 10), so that A ∝ κv s is fixed but A ∝ κ 2 v s varies. Those points show that the data are better collapsed by A, as we found for single kicks in the main text. This test also serves as a further systematic check on our results, as the two simulation codes were written independently by two of us (J.W. and J.D.P.).
Supplementary Note 2: Finite size effects. In Fig. 4 in the main text, the variance of the number density is observed to fall off rapidly at large . This occurs when the window size, , is a significant fraction of the narrowest system dimension, causing the sampling windows to overlap. The samples are therefore not statistically independent, so the variation in the number density is suppressed. This effect has been investigated previously for small variations in system size [7].
Here we study this effect by probing a large range of system sizes in simulations without sedimentation. We produce hyperuniform distributions of particles by shearing a square system of side length L near the critical amplitude γ c . As in the main text, we measure the variance of the number density, which decays as −λ with λ ≈ 2.60. We then cut subsystems of side length L < L out of the original system, and we measure the variance in the same manner. This is repeated for a total of 6 system sizes, with ratios L /L from 0.022 to 1. Each curve is averaged over 42 systems to suppress noise.
The results are shown in the inset to Supplementary Fig. 2. Each curve follows the same scaling with and falls off rapidly when approaches L . We shift the curves by rescaling the x axis by L /L and the y axis by (L /L) −2.60 . Remarkably, all the data fall onto a single master curve, which shows that the effect is only sensitive to the ratio /L. Supplementary Note 3: Introducing a concentration gradient into a hyperuniform system. Here we derive the mapping used in Fig. 4c of the main text to produce artificial systems with constant concentration gradients. As stated in the main text, the mapping is uniquely determined by requiring a concentration map φ 0 → φ(y) = φ 0 + |∂φ/∂y|(h − 2y)/2 on a continuum system with uniform concentration φ 0 , where 0 < y < h.
We denote the initial uniform distribution by φ L (y) = φ 0 and the target distribution by φ R (y) = ay + b, as drawn in Supplementary Fig. 3. The total area under these curves must be identical, so b = φ 0 − ah/2, where h is the height of the system. The mapping must also conserve mass; for a region from the bottom of the system up to a height y, we have: where the point at height y maps to height y . Plugging in the expressions for φ L and φ R and performing the integral, we get: Solving for y , we retrieve the mapping: where a = ∂φ/∂y is the target vertical concentration gradient.
Supplementary Note 4: One-dimensional model. We can predict the loss of hyperuniform scaling in the 1D system by tailoring the arguments in the main text to this dimension. First, the nondimensional A 1D is given by: Equation 5 in the main text (for the effect of a concentration gradient on the variance of the number density) is modified to: σ 2 ρ ( ) grad = (∂φ/∂y) 2 (h − ) 2 /12. Equation 7 in the main text (for the scaling for vertical concentration gradients) is modified to: where we obtain the numerical prefactor by fitting to the data shown in Supplementary Fig. 4a. Adding the variance due to the vertical concentration gradient to the variance in the critical state without sedimentation, we get for the total variance in 1D: σ 2 ρ ( ) total ≈ σ 2 ρ ( ) c + 0.17A 4 1D φ 2 c . We then solve for the lengthscale H where the local scaling exponent is equal to λ = 1, where we use σ 2 ρ ( ) c ≈ 0.15 −1.44 from our measurements in 1D (shown in Fig. 4b in the main text). This computation yields: We plot the 1D phase diagram in Supplementary Fig. 4b, where the data are obtained in the same fashion as in 2D. The prediction is in good agreement with the data.