Nonlinear elasticity of microsphere heaps

Thermal fluctuations, geometric exclusion, and external driving all govern the mechanical response of dense particulate suspensions. Here, we measure the stress-strain response of quasi-two-dimensional flow-stabilized microsphere heaps in a regime in which all three effects are present using a microfluidic device. We observe that the elastic modulus and the mean interparticle separation of the heaps are tunable via the confining stress provided by the fluid flow. Furthermore, the measured stress-strain curves exhibit a universal nonlinear shape, which can be predicted from a thermal van der Waals equation of state with excluded volume. This analysis indicates that many-body interactions contribute a significant fraction of the stress supported by the heap.

Figure 1

Overview of deformation experiment. (a) Microfluidic experimental setup. (b) Grayscale fluorescence microscopy image depicting steady-state microsphere heap with suspension flowing around it. (c) Top-view and side-view diagram of Hele-Shaw microfluidic device geometry with fluid flow shown by blue streamlines. $W\times H\times L=1\mathrm{mm}\times 0.94\mu \mathrm{m}\times 14\mathrm{mm}$, not to scale), showing a single microsphere flowing toward the flat-topped barrier ($w\times h\times \ell =563\mu \mathrm{m}\times 0.84\mu \mathrm{m}\times 10\mu \mathrm{m}$. (a) Schematic microchip geometry steady-state raw fluorescence images, (d) preperturbation and (e) 2.0 s after a perturbation of amplitude $\Delta P=-3.5$ kPa. Linear grayscale represents fluorescence intensity, and blue line is the detected heap boundary. Fluid flow direction is from top to bottom. (f) Image-difference of the same sequence.

Figure 2

Observation of uniform strain. (a) Parametric plot of $h_{\mathrm{post}}\left(x\right)$ versus $h_{\mathrm{pre}}\left(x\right)$ with inset local strain $\gamma \left(x\right)=h_{\mathrm{post}}\left(x\right)/h_{\mathrm{pre}}\left(x\right)$. In both plots, the dashed line corresponds to a mean strain of 12%, measured from the slope of (a).

Figure 3

(a) Fluorescence microscopy image of a microsphere heap composed of bidisperse particles under steady confining stress ${\sigma }_0=0.1$ Pa. Diameter of dim particles is 600 nm and of bright particles is 710 nm. (b) Intensity contour lines showing particle centroids in microsphere heap, detected using Wiener deconvolution. Light contours with $+$ (red) correspond to bright regions, and dark contours with $\times$ (black) correspond to dim regions. (c) Pair radial distribution functions, $g(r/d)$, measured at three different ${\sigma }_0$: 0.1 Pa, 1.1 Pa, 3.9 Pa. (d) Peak position as a function of the confining stress ${\sigma }_0$, measured from a parabolic fit to the peak, with 1-$\sigma$ (68.3%) confidence intervals. (e) Schematic of possible configurations of microspheres in the channel and their apparent separations when observed from above.

Figure 4

Stress-strain measurement protocol. (a) Heap growth to steady state at $P_0=1$ kPa (${\sigma }_0=0.7$ Pa) during two hours of particle accumulation, followed by a sequence of 80 randomized deformations, and then repeated for a larger value of $P_0=2$ kPa. (b) Inset: Zoom of $A\left(t\right)$, for which the extracted values of $\Delta A$ (black) and $\Delta \sigma$ (green) are shown in (c).

Figure 5

Raw stress-strain measurements. (a) Scatter plot of perturbation amplitude $\Delta \sigma$ vs. deformation amplitude $\Delta A$ for seven difference values of ${\sigma }_0$ indicated by symbol shape. Lines are fits to Eq. (10). The measurement range is largest at intermediate values of ${\sigma }_0$, truncated for small values by the lower limit of heap formation at $\mathrm{Pe}\approx 2$ [25] and at large values by the pressure regulator. (b) Stress-strain measurements for the same data. Each line is a linear fit to determine the elastic modulus $K$ [see Eq. (1)]. (c) Inset: Measured values of $K$ from (b) as a function of ${\sigma }_0$, with error bars indicating one standard deviation estimates obtained from the fits. The dashed line is the linear fit to Eq. (2) with $\alpha =23.3\pm 0.74$.

Figure 6

Scaled stress-strain measurements. (a) Strain $\Delta A/A_0$ as a function of the scaled perturbation amplitude $\Delta \sigma /{\sigma }_0$, showing data collapse. Dashed solid line is from Eq. (10) with $\tilde{\alpha }=22.5$; thin dotted line is Eq. (1) with $K=\alpha {\sigma }_0$ and $\alpha =23.3$. Symbol legend is the same as described in the legend of Fig. 5. (b) Inset: Log-log plot of $\Delta A/A_0$ vs. $\left|\Delta \sigma \right|/{\sigma }_0$, where compressions are upward (green) triangles and decompressions are downward (brown) triangles. Dashed solid line is Eq. (1) with $K=\alpha {\sigma }_0$ and $\alpha =23.3$. (c) The same data, with the curves vertically shifted by 0.1 to allow for comparison. Each curve is fit to Eq. (10) to find a best-fit value of $\tilde{\alpha }$; solid lines denote the fit region and the dashed lines are continuations beyond that region. (d) Inset: Values of $\tilde{\alpha }$ for each dashed curve fit in (c). Error bars are one-standard-deviation parameter-variance estimates for each regression obtained from adding the parameter variance from the Hessian of the error function and systematic errors. Dashed line is a linear fit: $\tilde{\alpha }=(23.4\pm 2.3)+(-0.24\pm 0.17){\sigma }_0$.

Figure 7

(a)–(d) Processed images showing amplitude of intensity fluctuations at different inflow velocities. Each pixel is given an RGB color proportional to the standard deviation measured over 8 s (80 images) after steady state was reached. The colormap is reported in arbitrary units, with purple indicating low fluctuations and red indicating high fluctuations. The large purple areas are the excluded zones. Tracks of single particles are yellow trails. The strongly fluctuating (fluid-like) areas at the surface of each heap are deep red, while the weakly fluctuating (solid-like) interior of the heap is light red. From left to right, the inflow velocities are 3.6 $\mu \mathrm{m}$/s, 7.1 $\mu \mathrm{m}$/s, 14.2 $\mu \mathrm{m}$/s, 35.5 $\mu \mathrm{m}$/s. Microsphere areal density far from the heap is constant ${\rho }_{\infty }=110/{\left(100\mu \mathrm{m}\right)}^2$. (e) Schematic drawing showing streamlines ${\psi }_L$ and ${\psi }_R$ bounding the excluded zone of width $\xi$.

Figure 8

(a) Simulated depth-averaged velocity field. The two solid lines are the streamlines that define $\xi$. (b) Solid line shows plot of ${\xi }_0/w$ in simulation versus the value of $h/H$. Dashed lines show the range of experimental measurements of $h/H$ and ${\xi }_0/w$. (c) Simulated relative heap permeability versus heap packing fraction $\varphi$ at seven values of $P_0$ (left is low $P_0$, right is high). Solid line given by Eq. (A1). (d) Final calibration of ${\sigma }_0$ [pressure on heap surface calculated by Eq. (A2)] as a function imposed pressure $P_0$ for each of the seven experimental conditions.