Rigid spheres moving through soft solids

TL;DR

The paper probes how buoyant rigid spheres move through dense hydrogel suspensions, uncovering a constant-velocity rise toward a free surface that contrasts with the material’s thixotropic characteristics. By varying end boundaries (free surface, lid, and mesh) and analyzing both motion and flow with time-lapse imaging and PIV, the authors show that the intruder kinetics are governed by boundary-induced confinement, with an effective viscosity η_eff that decays exponentially with buoyancy stress: $η_{\rm eff} \propto \exp(-σ_S/σ_0)$. Temperature and diffusion studies further reveal a stronger-than-water Arrhenius dependence and highly localized flow around the intruder, indicating non-Newtonian rheology beyond simple lubrication. The discussion links the exponential stress sensitivity to boundary conditions via a simple viscoelastic model, suggesting that confinement stresses near rigid end boundaries slow motion and produce sublinear scaling, thereby motivating theoretical work to fully describe boundary-driven rheology in hydrogel suspensions.

Abstract

We present the results of an experimental investigation into buoyant rigid spheres rising through highly concentrated collections of hydrated hydrogel particles. The volume fraction of particles is such that the mechanical properties of the material are intermediate between a very viscous fluid and a soft solid. Despite the established time dependent, non-Newtonian character of hydrogels, we find that when the surface of the material is free, an immersed buoyant sphere rises with a constant speed. The effects of the motion are observed to be highly localized around the sphere. When the stress exerted on the material is changed by varying the mass of the sphere, its terminal velocity is found to depend exponentially on its buoyancy. Qualitatively distinct behavior is found when a solid lid is placed on the surface of the material. In this case, a seemingly thixotropic, sublinear time-dependent motion is found. It is observed that linear motion of the sphere is accompanied by flow at the surface of the material whereas fluid movement is suppressed when a lid is present. We use these observations to provide a hypothesis which links the exponential stress dependence of the rheology of the material to the effects of the boundary conditions on the kinematics of the intruder.

Figures (15)

• Figure 1: Sketch (left) and front view of the apparatus with time lapse imaging (right). The sample comprised $12.7$ g of dry hydrogel beads in $2$ Litres of triply boiled tap water. The hollow sphere is $40\pm0.01$mm diameter and weighed $3.47$ g. The mid-plane of the sphere appears wider owing to lensing effects from the round container. The images are captured at $40$ second intervals. The equidistant separation of the time lapse images of the sphere highlights its constant speed.
• Figure 2: Plot of the distance $\delta$ travelled by a ping-pong ball as a function of time for a sphere moving towards a free surface. The buoyancy stress is $\sigma_S \sim 210$ Pa. Error bars indicate a 5% error due to imaging resolution limitations. The linear dependence of distance with time (solid line) is clear. For comparison with previous work we also indicate a square root time dependence (blue dash-dotted line). The inset (a) has the same axes but in double logarithmic scaling, uses the same symbols and highlights the square root $\delta\propto t^{1/2}$ motion and exponent of the blue dashed line.
• Figure 3: (a) Examples of distance versus time for ping pong balls rising in a sample with a free surface. The rise speed depends on the weight of the partially filled sphere where the weight ranges from $0$ to $17.3$ g. This corresponds to a buoyancy stress $\sigma_S$ of $\approx 240$ to $\approx 130$ Pa. The lines indicate straight line fits to the data. (b) The displacement data can be rescaled by dividing out the effective velocity $U$ to collapse all data on a linear master curve with slope 1 (dash-dotted line) over three orders of magnitude. (c) Graph of the slope of the fit to each measured time history plotted as a function of the buoyancy stress $\sigma_S$. Note the plot is presented on a log-lin scale. The dash-dotted line represents an exponential decay with a decay constant of $27$ Pa. The error indicates the fitting error.
• Figure 4: (a) Time histories obtained with ping-pong balls scaled using the fitted velocity $U$ for samples of hydrogels with $C = 6.35, 6.15$ and $6.0$ g/L with a free surfaces. (b) Log-linear plot of the effective viscosity based on $U$ of the hydrogel sample versus buoyancy stress $\sigma_S$ for cases with lid-free surface $\bullet$. The dash-dotted line indicates an exponential dependence on $\sigma_S$ with stress factor $\sigma_0$, which depends on the sample. We used $\sigma_0 = 27,17,12$ Pa for the different samples respectively.
• Figure 5: Log-linear plot of the effective viscosity of the hydrogel sample versus buoyancy stress $\sigma_S$ for cases with a free surface. The open circles represent the ping pong ball data shown previously. Black $\triangle$: 19 mm sphere; green $\triangleleft$: 25 mm sphere; blue $\square$ 20 mm sphere. The dash-dotted line indicates an exponential dependence on $\sigma_S$.