Viscoelastic Properties Of Soft Materials Research Articles

A New Approach for Measuring Viscoelastic Properties of Soft Materials Using the Dynamic Response of a Spherical Object Placed at the Sample Interface

BackgroundThere are several techniques to characterize the mechanical properties of soft materials, such as the indentation method and the method based on the application of a spherical object placed inside the sample. The indentation systems usually yield the elastic properties of materials and their mathematical models do not consider the inertia of the sample involved in motion and radiation damping, while placing an object inside the sample is not practical and this procedure can alter the mechanical properties of the sample for the method based on the application of a bubble/sphere placed inside the sample.ObjectiveA new approach for the identification of the viscoelastic properties of soft materials using the dynamic response of a spherical object placed at the sample interface was proposed.MethodsThe spherical object placed at the sample interface was pressed using an electromagnet and the dynamic response of the spherical object was tracked using a high-speed camera, while the dynamic response of the spherical object placed at the sample interface was estimated using a comprehensive analytical model. The effects of the shear modulus, viscosity, Poisson’s ratio and density of the soft sample, the radius and density of the spherical object and the damping due to radiation were considered in this mathematical model. The shear modulus and viscosity of the soft sample were determined by matching the experimentally identified and theoretically estimated responses of the spherical object.ResultsThe shear moduli and viscosities of the three phantoms with the gelatin mass ratios of 0.20, 0.25 and 0.29 were measured to be 3450, 4300 and 4950 Pa and 12.5, 14.0 and 15.0 Pa⋅s, respectively. The shear modulus and viscosity of the phantom increases as the gelatin mass ratio increases. The frequency of oscillations of the hemisphere placed at the phantom interface increases as the gelatin mass ratio increases due to stiffness increase.ConclusionsAfter matching the experimental and theoretical steady-state displacements and amplitudes of oscillations of the hemisphere at the sample interface, the comparison of the experimentally identified and theoretically predicted frequency of oscillations further confirmed the identified material properties of the samples. The approach presented here is expected to provide valuable information on material properties in biomedical and industrial applications.

Characterization of visco-hyperelastic behavior of open cell polyurethane foam through transient shear testing

Soft materials find ever-growing use in load-bearing applications, and hence, there is an imminent need for accurate testing and modeling of their mechanical properties. Nonlinearities and rate-dependence are inherent in the mechanical response of such materials, thanks to their multiphasic flexible constituents. Fragility of soft materials and rapid changes in their constituents depending on the environment add further challenges in mechanical characterization. This study introduces an efficient methodology to measure nonlinear viscoelastic properties of soft materials from transient shear response and validates the methodology experimentally on flexible polyurethane foams. Direct estimation of friction forces due to the Poynting effect and independent uniaxial compression tests of the same foam are also documented, validating the characterization framework. Both methods deliver comparable results with the transient-shear-based framework. Simple sample preparation and installation further complements the reliable and quick estimation capability, and thus offers a suitable characterization methodology for fragile and time-sensitive samples such as biological tissues. While validating the proposed framework on foams, fast relaxations and sensitivity to peak strains and strain rates are observed. Buckling and failure of the foam in the vicinity of the inertial plate could be possible explanations of those observations. Besides estimating the visco-hyperelastic properties of the foams, the tester dissipation stemming from material and frictional damping is also estimated in the framework. Increases in frictional dissipation due to compromised lubrication in guide rails over a period of three days are observed to shorten transients. Proper lubrication and control of friction before each test suppresses stochastic dry friction and improves repeatability and accuracy of the material characterization methodology.

Identification of the viscoelastic properties of soft materials using a convenient dynamic indentation system and procedure

The responses of soft structures such as tissue depend on their viscoelastic properties. Therefore, the knowledge of the elastic and damping properties of soft materials is of great interest. This paper presents the identification of the viscoelastic properties of soft materials using a convenient dynamic indentation system and procedure. Using an electromagnet, a force is applied to a rigid sphere located at the soft-material interface and the dynamic response of the sphere is recorded using a high-speed camera. The recorded video is processed to identify the displacement of the sphere as a function of time. The dynamic response of the sphere located at the soft-material interface is predicted using an analytical model that considers the shear modulus and density of the soft sample, the radiation damping due to shear waves, and the radius and density of the sphere. By matching the measured and predicted steady-state displacements of the sphere, the shear modulus of the soft sample is determined. The viscous damping ratio of the soft sample is identified by using an equivalent viscous damping ratio for the soft sample in the analytical model and matching the measured and predicted oscillation amplitudes of the sphere. Experiments and analyzes are performed using gelation phantoms with different mechanical properties, spheres of different materials and sizes, and different force levels to verify the system and procedure. Three experiments are performed for each gelation phantom, sphere, and external force, and the repeatability of the results is presented. The results show that the dynamic indentation system and procedure presented in this study can be conveniently used to determine the viscoelastic properties of soft materials in practical applications.

Shape stability of a gas bubble in a soft solid.

Predicting the onset of non-spherical oscillations of bubbles in soft matter is a fundamental cavitation problem with implications to sonoprocessing, polymeric materials synthesis, and biomedical ultrasound applications. The shape stability of a bubble in a Kelvin-Voigt viscoelastic medium with nonlinear elasticity, the simplest constitutive model for soft solids, is analytically investigated and compared to experiments. Using perturbation methods, we develop a model reducing the equations of motion to two sets of evolution equations: a Rayleigh-Plesset-type equation for the mean (volume-equivalent) bubble radius and an equation for the non-spherical mode amplitudes. Parametric instability is predicted by examining the natural frequency and the Mathieu equation for the non-spherical modes, which are obtained from our model. Our theoretical results show good agreement with published experiments of the shape oscillations of a bubble in a gelatin gel. We further examine the impact of viscoelasticity on the time evolution of non-spherical mode amplitudes. In particular, we find that viscosity increases the damping rate, thus suppressing the shape instability, while shear modulus increases the natural frequency, which changes the unstable mode. We also explain the contributions of rotational and irrotational fields to the viscoelastic stresses in the surroundings and at the bubble surface, as these contributions affect the damping rate and the unstable mode. Our analysis on the role of viscoelasticity is potentially useful to measure viscoelastic properties of soft materials by experimentally observing the shape oscillations of a bubble.

Nanomechanical mapping of soft materials with the atomic force microscope: methods, theory and applications.

Fast, high-resolution, non-destructive and quantitative characterization methods are needed to develop materials with tailored properties at the nanoscale or to understand the relationship between mechanical properties and cell physiology. This review introduces the state-of-the-art force microscope-based methods to map at high-spatial resolution the elastic and viscoelastic properties of soft materials. The experimental methods are explained in terms of the theories that enable the transformation of observables into material properties. Several applications in materials science, molecular biology and mechanobiology illustrate the scope, impact and potential of nanomechanical mapping methods.

Indentation creep tests to assess the viscoelastic properties of soft materials: Theory, method and experiment

Determining time-dependent mechanical properties of soft materials is essential in understanding their deformation behaviors under various stimuli. This paper investigates the use of indentation creep tests to measure the viscoelastic properties of soft materials at local areas. A simple scaling law between the reduced creep function and the creep displacement of the indenter is revealed in this paper based on a theoretical analysis. We show that the scaling relation can be used to interpret indentation creep tests of viscoelastic soft solids with arbitrary surface profile provided that the contact area does not change. Both numerical and practical experiments have been performed to validate the theory and the analytical solution. In our experiments, a low-cost portable indentation system is proposed to measure the reduced creep function. Our results show that the low-cost instrument and the analytical solution to interpret the experimental data reported here represent a useful testing method to deduce the intrinsic viscoelastic properties of soft materials in a non-destructive manner.

Modeling ramp-hold indentation measurements based on Kelvin–Voigt fractional derivative model

Interpretation of experimental data from micro- and nano-scale indentation testing is highly dependent on the constitutive model selected to relate measurements to mechanical properties. The Kelvin–Voigt fractional derivative model (KVFD) offers a compact set of viscoelastic features appropriate for characterizing soft biological materials. This paper provides a set of KVFD solutions for converting indentation testing data acquired for different geometries and scales into viscoelastic properties of soft materials. These solutions, which are mostly in closed-form, apply to ramp-hold relaxation, load-unload and ramp-load creep-testing protocols. We report on applications of these model solutions to macro- and nano-indentation testing of hydrogels, gastric cancer cells and ex vivo breast tissue samples using an atomic force microscope (AFM). We also applied KVFD models to clinical ultrasonic breast data using a compression plate as required for elasticity imaging. Together the results show that KVFD models fit a broad range of experimental data with a correlation coefficient typically R2 > 0.99. For hydrogel samples, estimation of KVFD model parameters from test data using spherical indentation versus plate compression as well as ramp relaxation versus load-unload compression all agree within one standard deviation. Results from measurements made using macro- and nano-scale indentation agree in trend. For gastric cell and ex vivo breast tissue measurements, KVFD moduli are, respectively, 1/3–1/2 and 1/6 of the elasticity modulus found from the Sneddon model. In vivo breast tissue measurements yield model parameters consistent with literature results. The consistency of results found for a broad range of experimental parameters suggest the KVFD model is a reliable tool for exploring intrinsic features of the cell/tissue microenvironments.

New MEMS Tweezers for the Viscoelastic Characterization of Soft Materials at the Microscale.

As many studies show, there is a relation between the tissue’s mechanical characteristics and some specific diseases. Knowing this relationship would help early diagnosis or microsurgery. In this paper, a new method for measuring the viscoelastic properties of soft materials at the microscale is proposed. This approach is based on the adoption of a microsystem whose mechanical structure can be reduced to a compliant four bar linkage where the connecting rod is substituted by the tissue sample. A procedure to identify both stiffness and damping coefficients of the tissue is then applied to the developed hardware. Particularly, stiffness is calculated solving the static equations of the mechanism in a desired configuration, while the damping coefficient is inferred from the dynamic equations, which are written under the hypothesis that the sample tissue is excited by a variable compression force characterized by a suitable wave form. The whole procedure is implemented by making use of a control system.

High strain-rate soft material characterization via inertial cavitation

Mechanical characterization of soft materials at high strain-rates is challenging due to their high compliance, slow wave speeds, and non-linear viscoelasticity. Yet, knowledge of their material behavior is paramount across a spectrum of biological and engineering applications from minimizing tissue damage in ultrasound and laser surgeries to diagnosing and mitigating impact injuries. To address this significant experimental hurdle and the need to accurately measure the viscoelastic properties of soft materials at high strain-rates (103–108 s−1), we present a minimally invasive, local 3D microrheology technique based on inertial microcavitation. By combining high-speed time-lapse imaging with an appropriate theoretical cavitation framework, we demonstrate that this technique has the capability to accurately determine the general viscoelastic material properties of soft matter as compliant as a few kilopascals. Similar to commercial characterization algorithms, we provide the user with significant flexibility in evaluating several constitutive laws to determine the most appropriate physical model for the material under investigation. Given its straightforward implementation into most current microscopy setups, we anticipate that this technique can be easily adopted by anyone interested in characterizing soft material properties at high loading rates including hydrogels, tissues and various polymeric specimens.

Identification of the viscoelastic properties of soft materials at low frequency: Performance, ill-conditioning and extrapolation capabilities of fractional and exponential models

We report about the experimental identification of viscoelastic constitutive models for frequencies ranging within 0–10Hz. Dynamic moduli data are fitted forseveral materials of interest to medical applications: liver tissue (Chatelin et al., 2011), bioadhesive gel (Andrews et al., 2005), spleen tissue (Nicolle et al., 2012) and synthetic elastomer (Osanaiye, 1996). These materials actually represent a rather wide class of soft viscoelastic materials which are usually subjected to low frequencies deformations.We also provide prescriptions for the correct extrapolation of the material behavior at higher frequencies. Indeed, while experimental tests are more easily carried out at low frequency, the identified viscoelastic models are often used outside the frequency range of the actual test.We consider two different classes of models according to their relaxation function: Debye models, whose kernel decays exponentially fast, and fractional models, including Cole–Cole, Davidson–Cole, Nutting and Havriliak–Negami, characterized by a slower decay rate of the material memory. Candidate constitutive models are hence rated according to the accurateness of the identification and to their robustness to extrapolation. It is shown that all kernels whose decay rate is too fast lead to a poor fitting and high errors when the material behavior is extrapolated to broader frequency ranges.

Viscoelastic Properties of Soft Materials on a Basis of Starch Mixtures

The effect of concentration of carbohydrates on viscoelastic properties of corn and rice starch is discussed in this paper. The texture of starch pastes was estimated using a fractional rheological linear standard solid model. Rheological parameters of the model were identified and their changes for different values of sugar concentrations in the 2 tested starches were discussed. The fractional rheological models can be very useful to control the biomaterial texture for the needs of final product, which is particularly significant from the point of view of materials engineering. The proposed method for interpretation of the results of rheological measurements of these pastes allows for a comprehensive estimation of the analyzed biomaterial texture.

Experimental methods for the characterization of the frequency-dependent viscoelastic properties of soft materials

A characterization method based on Rayleigh wave propagation was developed for the quantification of the frequency-dependent viscoelastic properties of soft materials at high frequencies; i.e., up to 4 kHz. Planar harmonic surface waves were produced on the surface of silicone rubber samples. The phase and amplitude of the propagating waves were measured at different locations along the propagation direction, which allowed the calculation of the complex Rayleigh wavenumbers at each excitation frequency using a transfer function method. An inverse wave propagation problem was then solved to obtain the complex shear/elastic moduli from the measured wavenumbers. In a separate, related investigation, dynamic indentation tests using atomic force microscopy (AFM) were performed at frequencies up to 300 Hz. No systematic verification study is available for the AFM-based method, which can be used when the dimensions of the test samples are too small for other existing testing methods. The results obtained from the Rayleigh wave propagation and AFM-based indentation methods were compared with those from a well-established method, which involves the generation of standing longitudinal compression waves in rod-shaped test specimens. The results were cross validated and qualitatively confirmed theoretical expectations presented in the literature for the frequency-dependence of polymers.

Optimised determination of viscoelastic properties using compliant measurement systems

An analysis of a novel indentation model has been implemented to obtain master curves describing the optimal experimental parameters necessary to achieve the highest possible accuracy in the determination of viscoelastic properties of soft materials. The indentation model is a rigid indenter driven by a compliant measurement system, such as an atomic force microscope or optical tweezers, into a viscoelastic half space. The viscoelastic material is described as a multiple relaxation Prony series. The results have been extended via an application of a viscoelastic equivalence principle to other physical models such as poroelasticity. Optimisation of the indentation parameters has been conducted over many orders of magnitude of the velocity, viscoelastic moduli, spring stiffness, relaxation times and the duration of indentation resulting in a characteristic master curve. It is shown that using sub-optimal conditions gives the appearance of a more elastic material than is actually the case. For a two term Prony series the ideal ramp duration was found to be approximately one eighth of the relaxation. Also the ideal ramp duration for a three term Prony series was determined and shown to guarantee distinct relaxation times under specific conditions.

Examining the validity of strain-rate frequency superposition when measuring the linear viscoelastic properties of soft materials

A strain-rate frequency superposition (SRFS) technique, recently proposed to analyze the low-frequency behavior of soft materials, is evaluated. The application of SRFS to an emulsion and multiarm star polymer solution produces master curves that, while promising in appearance, do not match the anticipated linear response as they seem inconsistent with dynamic frequency sweep data and the Kramers–Kronig relation. While the raw unscaled frequency sweep data are well described using the generalized Maxwell model, strong discrepancies appear when the same procedure is applied to SRFS data. These inconsistencies appear to be generic and are observed in other materials as well. An examination of Lissajous curves obtained from large amplitude oscillatory strain shows that nonlinear contributions, such as solvent-mediated convective flow, strongly affect SRFS master curves. Based on these findings, SRFS should be approached with caution when applied to soft materials.

Frequency-dependent viscoelasticity measurement by atomic force microscopy

We demonstrate a new technique for investigating viscoelastic properties of soft materials using the atomic force microscope. A small oscillatory voltage is added to the deflection signal of the atomic force microscope causing a vertical oscillatory sample motion. Monitoring the amplitude and phase of this motion allows determination of the viscous and elastic moduli of the sample as a function of frequency during contact imaging. This technique is applied to suspended poly(vinyl alcohol) nanofibers and poly(vinyl alcohol) hydrogels, giving results similar to those measured using traditional static methods. However, the moduli of both the fibers and the hydrogels show a significant frequency dependence. The Young's modulus of the fibers increases with frequency, while for the viscoelastic hydrogels, the storage modulus dominates the mechanical response at low frequency whereas the loss modulus dominates at high frequency.

Comments on the use of the force mode in atomic force microscopy for polymer films

Atomic force microscopy (A.F.M.) was first described as a powerful technique for studying insulating, hard surfaces. Since then, it has also been considered as an appropriate technique for investigating, at a submicromic scale, elastic and viscoelastic properties of soft materials as polymer films. An attempt is made to show how macroscopic models can be fruitfully employed in order to interpret the force curves obtained in AFM on polymer films. Through an analysis of the slope variation and the way the instability occurs at the tip-sample contact, it is shown that a macroscopic approach is a useful way to explain most of the features of the force curves. Furthermore the importance is underlined of the initial conditions. It is shown that for polymer samples which have a stiffness within the range of the probe one, drastic changes of the force curve shape can occur when the initial conditions vary. Finally, this approach should allow one to clarify the conditions for which the macroscopic approach fails.

A new equipment to measure the viscoelastic properties of soft materials and its application to articular cartilage.

A new equipment to measure the viscoelastic properties of soft materials has been developed. The equipment has an advantage; the measurement of an acceleration and a collision velocity convey enough information to calculate a load and a deformation history of a specimen. The new equipment is applied to measure viscoelastic properties of articular cartilages. The specimens were taken from the proximal head of the avian tibia. In the first phase of the experiment, the specimens were filled with intercellular fluid. In the second phase, the intercellular fluid was partly exchanged with normal saline solution. The following results were obtained: (1) The collision between the impactor and the articular cartilage is not perfectly elastic, and energy is dissipated; (2) The dissipated energy is proportional to the square of the collision velocities; (3) No significant difference between the amounts of dissipated energy in the articular cartilages with the intercellular fluid and those with the normal saline solution was observed. The maximum displacement of the articular cartilages with the intercellular fluid is, however, smaller than those with the normal saline solution; (4) Intercellular fluid has the properties of a spring; (5) The articular cartilages can be approximated with a Kelvin model whose constants vary exponentially with depth; (6) The physical properties of intercellular fluid largely contribute to the physical properties of the articular cartilages.