Extracting non-linear viscoelastic material properties from violently-collapsing cavitation bubbles

Abstract

Determination of high strain-rate constitutive material parameters from highly compliant, non-linear materials including hydrogels and soft tissues remains a significant challenge. To address this challenge we recently developed a laser-based Inertial Microcavitation Rheometry (IMR) technique, capable of constitutively characterizing soft materials at ultra-high strain-rates ($O\left(10^3\right)\sim O\left(10^8\right)s^{-1}$). This technique utilizes spatially-focused, and temporally-resolved, single cavitation bubbles in conjunction with a theoretical framework for determining the non-linear viscoelastic material properties of the surrounding soft material.

Here, by advancing our spatiotemporal imaging capability and tracking the motion of native material features close to the inertially cavitating bubble, we find significant deviation from the IMR predicted material displacements at a critical material Mach number of 0.08, marking the transition to violent collapse. Interestingly, this critical Mach number appears to be almost independent of the material properties of the surrounding hydrogel. We show that through proper temporal resolution of the cavitation dynamics, all of the highly nonlinear viscoelastic properties of the surrounding material can still be uniquely determined from just the first expansion and collapse cycle alone, as long as the local Mach number is below the critical threshold. Furthermore, we show that the inclusion of a higher order strain stiffening term on the rate-dependent non-linear spring can lead to better estimation of the material properties and full recovery of the equilibrium shear modulus, which was not possible with previous nonlinear neo-Hookean Kelvin–Voigt model formulations. Thus, the developments presented here significantly expand the applicability and robustness of IMR for accurately determining the rate-dependent, finite deformation constitutive behavior of soft materials.

Introduction

Cavitation is a common phenomenon found in many biological and engineering systems, from the prey-stunning capacity of the snapping shrimp to cavitation erosion on metallic pumps and propellers, to cavitation induced cell and tissue damage [1], [2], [3], [4], [5], [6]. When harnessed carefully, focused, energy-driven cavitation can be a very beneficial tool in a wide array of surgical, medical, and materials applications including: tissue phantom, laser surgery, lithotripsy, DNA injection and more recently, high to ultra-high strain-rates soft material characterization [7], [8], [9], [10], [11], [12], [13], [14], [15].

In particular, laser-based Inertial Microcavitation Rheometry (IMR), a new technique we recently developed, can provide a robust experimental approach for characterizing highly compliant materials at large strains (up to 1000% strain) and ultra-high strain-rates ($O\left(10^3\right)\sim O\left(10^8\right){\text{s}}^{-1}$) [15]. Here, a single isolated spherical bubble is induced within a viscoelastic medium through a spatially-focused, pulsed laser [16] while real-time, ultra high-speed videography is employed for recording the subsequent bubble expansion–collapse cycles. Due to the dependency of the bubble dynamics on the mechanical properties of the surrounding medium, the viscoelastic properties of the material can be determined by analyzing the temporal evolution of the bubble radius within a relevant framework of associated governing equations of inertial cavitation [15], [17], [18], [19], [20], [21].

While at small to moderate material stretches (e.g., 1$<$ $\lambda$ $<$6) the material surrounding the cavitating bubble remains primarily viscoelastic, higher stretch ratios can introduce a new host of additional physical phenomena. These include, but are not limited to, significant strain stiffening and inelastic material effects (e.g., material damage, plasticity, combustion, etc.) as the local, material Mach number exceeds $M\sim 0.1$, a condition of violent collapse.

The natural occurrence and physics of violent bubble collapse have been extensively discussed within the fluid mechanics community [7], [22], [23], [24], while here we extend this notion and definition to observations in solid, albeit highly compliant, materials. While it is generally understood that violent collapses bear significant changes in the material’s compressibility and stability of the contracting bubble and surrounding material, direct experimental evidence thereof still remains sparse.

Here by employing ultra-high speed imaging of up to 2 million frames per second, we provide direct experimental evidence that significant inelastic effects can occur during the collapse process in a model system of soft and stiff polyacrylamide (PA) hydrogels with nominal equilibrium shear moduli ranging from a few hundred to a few thousand pascals. By closely tracking native, inherent material features, we find a significant departure in the motion of our fiducial material markers from viscoelastic behavior at a critical collapse Mach number of around 0.08, consistent with the notion of a violent collapse. Yet, we find that as long as the temporal sampling frequency is high enough, i.e., greater than one million frames per second, the non-linear viscoelastic material properties can still be uniquely determined from just the first collapse regime until close to the bubble collapse point.

Furthermore, we show that through the incorporation of an explicit, higher order strain stiffening term in the constitutive response of the surrounding material, the accuracy in the estimated material parameters can be improved while also recovering the quasi-statically determined equilibrium shear modulus.

In sum, this paper provides a new approach for accurately deducing non-linear viscoelastic material properties from violently collapsing cavitation bubbles while providing direct experimental evidence of significant inelastic behavior during and after the initial bubble collapse.