Contact measurement of internal fluid flow within poly(n-isopropylacrylamide) gels

Stress relaxation in network materials: the contribution of the network

Stress relaxation in network materials with permanent crosslinks is due to the transport of fluid within the network (poroelasticity), the viscoelasticity of the matrix and the viscoelasticity of the network. While relaxation associated with the matrix was studied extensively, the contribution of the network remains unexplored. In this work we consider two and three-dimensional stochastic fiber networks with viscoelastic fibers and explore the dependence of stress relaxation on network structure. We observe that relaxation has two regimes - an initial exponential regime, followed by a stretched exponential regime - similar to the situation in other disordered materials. The stretch exponent is a function of density, fiber diameter and the network structure, and has a minimum at the transition between the affine and non-affine regimes of network behavior. The relaxation time constant of the first, exponential regime is similar to the relaxation time constant of individual fibers and is independent of network density and fiber diameter. The relaxation time constant of the second, stretched exponential regime is a weak function of network parameters. The stretched exponential emerges from the heterogeneity of relaxation dynamics on scales comparable with the mesh size, with higher heterogeneity leading to smaller stretch exponents. In composite networks of fibers whose relaxation time constant is selected from a distribution with set mean, the stretch exponent decreases with increasing the coefficient of variation of the fiber time constant distribution. As opposed to thermal glass formers and colloids, in these athermal systems the dynamic heterogeneity is introduced by the network structure and does not evolve during relaxation. While in thermal systems the control parameter is the temperature, in this athermal case the control parameter is a non-dimensional structural parameter which describes the degree of non-affinity of the network.

Poroelastic shape relaxation of hydrogel particles

Hydrogels are commonly used in research and energy, manufacturing, agriculture, and biomedical applications. These uses typically require hydrogel mechanics and internal water transport, described by the poroelastic diffusion coefficient, to be characterized. Sophisticated indentation-based approaches are typically used for this purpose, but they require expensive instrumentation and are often limited to planar samples. Here, we present Shape Relaxation (SHARE), an alternative way to assess the poroelastic diffusion coefficient of hydrogel particles that is cost-effective, straightforward, and versatile. This approach relies on first indenting a hydrogel particle via swelling within a granular packing, and then monitoring how the indented shape of the hydrogel relaxes after it is removed from the packing. We validate this approach using experiments in packings with varying grain sizes and confining stresses; these yield measurements of the poroelastic diffusion coefficient of polyacrylamide hydrogels that are in good agreement with those previously obtained using indentation approaches. We therefore anticipate that the SHARE approach will find broad use in a range of applications of hydrogels and other swellable soft materials.

Characterizing poroelasticity of biological tissues by spherical indentation: an improved theory for large relaxation

Flow of fluids within biological tissues often meets with resistance that causes a rate- and size-dependent material behavior known as poroelasticity. Characterizing poroelasticity can provide insight into a broad range of physiological functions, and is done qualitatively in the clinic by palpation. Indentation has been widely used for characterizing poroelasticity of soft materials, where quantitative interpretation of indentation requires a model of the underlying physics, and such existing models are well established for cases of small strain and modest force relaxation. We showed here that existing models are inadequate for large relaxation, where the force on the indenter at a prescribed depth at long-time scale drops to below half of the initially peak force (i.e., F(0)/F(∞) > 2). We developed an indentation theory for such cases of large relaxation, based on Biot theory and a generalized Hertz contact model. We demonstrated that our proposed theory is suitable for biological tissues (e.g., spleen, kidney, skin and human cirrhosis liver) with both small and large relaxations. The proposed method would be a powerful tool to characterize poroelastic properties of biological materials for various applications such as pathological study and disease diagnosis.

A new framework for characterization of poroelastic materials using indentation

To characterize a poroelastic material, typically an indenter is pressed onto the surface of the material with a ramp of a finite approach velocity followed by a hold where the indenter displacement is kept constant. This leads to deformation of the porous matrix, pressurization of the interstitial fluid and relaxation due to redistribution of fluid through the pores. In most studies the poroelastic properties, including elastic modulus, Poisson ratio and poroelastic diffusion coefficient, are extracted by assuming an instantaneous step indentation. However, exerting step like indentation is not experimentally possible and usually a ramp indentation with a finite approach velocity is applied. Moreover, the poroelastic relaxation time highly depends on the approach velocity in addition to the poroelastic diffusion coefficient and the contact area. Here, we extensively studied the effect of indentation velocity using finite element simulations which has enabled the formulation of a new framework based on a master curve that incorporates the finite rise time. To verify our novel framework, the poroelastic properties of two types of hydrogels were extracted experimentally using indentation tests at both macro and micro scales. Our new framework that is based on consideration of finite approach velocity is experimentally easy to implement and provides a more accurate estimation of poroelastic properties. STATEMENT OF SIGNIFICANCE: Hydrogels, tissues and living cells are constituted of a sponge-like porous elastic matrix bathed in an interstitial fluid. It has been shown that these materials behave according to the theory of 'poroelasticity' when mechanically stimulated in a way similar to that experienced in organs within the body. In this theory, the rate at which the fluid-filled sponge can be deformed is limited by how fast interstitial fluid can redistribute within the sponge in response to deformation. Here, we simulated indentation experiments at different rates and formulated a new framework that inherently captures the effects of stimulation speed on the mechanical response of poroelastic materials. We validated our framework by conducting experiments at different length-scales on agarose and polyacrylamide hydrogels.

The shape of hanging elastic cylinders

Deformations of heavy elastic cylinders with their axis in the direction of earth's gravity field are investigated. The specimens, made of polyacrylamide hydrogels, are attached from their top circular cross section to a rigid plate. An equilibrium configuration results from the interplay between gravity that tends to deform the cylinders downwards under their own weight, and elasticity that resists these distortions. The corresponding steady state exhibits fascinating shapes which are measured with lab-based micro-tomography. For any given initial radius to height ratio, the deformed cylinders are no longer axially symmetric beyond a critical value of a control parameter that depends on the volume force, the height and the elastic modulus: self-similar wrinkling hierarchies develop, and dimples appear at the bottom surface of the shallowest samples. We show that these patterns are the consequences of elastic instabilities.

Friction of Poroelastic Contacts with Thin Hydrogel Films

We report on the frictional behavior of thin poly(dimethylacrylamide) hydrogel films grafted on glass substrates in sliding contact with a glass spherical probe. Friction experiments are carried out at various velocities and normal loads applied with the contact fully immersed in water. In addition to friction force measurements, a novel optical setup is designed to image the shape of the contact under steady-state sliding. The velocity dependence of both friction force F_t and contact shape is found to be controlled by a Péclet number, Pe, defined as the ratio of the time τ needed to drain the water out of the contact region to a contact time a/ v, where v is the sliding velocity and a is the contact radius. When Pe < 1, the equilibrium circular contact achieved under static normal indentation remains unchanged during sliding. Conversely, for Pe > 1, a decrease in the contact area is observed together with the development of a contact asymmetry when the sliding velocity is increased. A maximum in F_t is also observed at Pe ≈1. These experimental observations are discussed in the light of a poroelastic contact model based on a thin-film approximation. This model indicates that the observed changes in contact geometry are due to the development of a pore pressure imbalance when Pe > 1. An order-of-magnitude estimate of the friction force and its dependence on normal load and velocity are also provided under the assumption that most of the frictional energy is dissipated by poroelastic flow at the leading and trailing edges of the sliding contact.

Probing the swelling-dependent mechanical and transport properties of polyacrylamide hydrogels through AFM-based dynamic nanoindentation

Hydrogels are composed of a crosslinked polymer network and water. The constitutive behaviors of hydrogels have been modeled based on Flory-Huggins theory. Within this model, the thermodynamic and kinetic parameters are assumed to be of constant values and are typically characterized through swelling tests. Since most hydrogels can absorb a large amount of solvent from the dry state to the swollen state, and the network size and solvent concentration of the hydrogels change significantly, the assumption of constant values of the thermodynamic and kinetic properties as the network swells is questionable. In this work, we have experimentally shown that even for the simple neutral polyacrylamide (PAAm) hydrogels, their mechanical responses cannot be fully described by the Flory-Huggins theory with constant thermodynamic parameters: N (number of chains per unit volume of dry polymers) and χ (polymer-solvent interaction parameter). For a more complete and precise characterization of the hydrogels, we measure the evolving properties of the gels as the network swells. Here, we use dynamic indentation to measure the poroelastic properties (shear modulus G, Poisson's ratio ν and diffusivity D) of the hydrogels under a wide range of swelling ratios. We also use linear perturbation to build the link between G, ν and N, χ, and plot the thermodynamic parameters in the Flory-Huggins theory as a function of the hydrogel swelling ratio. Consequently, the validity of the hydrogel models based on Flory-Huggins theory can be quantitatively examined.

High-frequency linear rheology of hydrogels probed by ultrasound-driven microbubble dynamics

Ultrasound-driven microbubble dynamics are central to biomedical applications, from diagnostic imaging to drug delivery and therapy. In therapeutic applications, the bubbles are typically embedded in tissue, and their dynamics are strongly affected by the viscoelastic properties of the soft solid medium. While the behaviour of bubbles in Newtonian fluids is well characterised, a fundamental understanding of the effect on ultrasound-driven bubble dynamics of a soft viscoelastic medium is still being developed. We characterised the resonant behaviour in ultrasound of isolated microbubbles embedded in agarose gels, commonly used as tissue-mimicking phantoms. Gels with different viscoelastic properties were obtained by tuning agarose concentration, and were characterised by standard rheological tests. Isolated bubbles (100-200 μm) were excited by ultrasound (10-50 kHz) at small pressure amplitudes (<1 kPa), to ensure that the deformation of the material and the bubble dynamics remained in the linear regime. The radial dynamics of the bubbles were recorded by high-speed video microscopy. Resonance curves were measured experimentally and fitted to a model combining the Rayleigh-Plesset equation governing bubble dynamics, with the Kelvin-Voigt model for the viscoelastic medium. The resonance frequency of the bubbles was found to increase with increasing shear modulus of the medium, with implications for optimisation of imaging and therapeutic ultrasound protocols. In addition, the viscoelastic properties inferred from ultrasound-driven bubble dynamics differ significantly from those measured at low frequency with the rheometer. Hence, rheological characterisation of biomaterials for medical ultrasound applications requires particular attention to the strain rate applied.

Unified solution for poroelastic oscillation indentation on gels for spherical, conical and cylindrical indenters

An oscillation indentation method is developed for characterizing the local poroelastic properties of soft and hydrated materials such as hydrogels and biological tissues. In the dynamic oscillation indentation measurement, an indenter is pressed into the material to a certain depth and held for a period of time. After a plateau of force is reached, an oscillation of small depth is superimposed sweeping through a range of frequencies. The shift between the force and displacement spectra is denoted as the phase lag that characterizes the energy dissipative behavior of the soft hydrated materials due to solvent migration. A unified solution is obtained for the three widely used shapes of indenters for soft materials: cylindrical punch, spherical indenter and conical indenter. The solutions are summarized in remarkably simple forms allowing for easy extraction of material parameters including shear modulus, Poisson's ratio and diffusivity from the oscillation indentation measurements. The oscillation indentation measurement was demonstrated on a polyacrylamide (PAAm) gel using an atomic force microscope. It is shown that the time-dependent behavior of the PAAm gel at the micron scale is dominated by poroelasticity and the properties can be accurately extracted from the explicit expressions derived in this work. This method has great potential to be applied on heterogeneous biological tissues where local properties are of interest.

Poroelasticity-driven lubrication in hydrogel interfaces

It is widely accepted that hydrogel surfaces are slippery, and have low friction, but dynamic applied stresses alter the hydrogel composition at the interface as water is displaced. The induced osmotic imbalance of compressed hydrogel which cannot swell to equilibrium should drive the resistance to slip against it. This paper demonstrates the driving role of poroelasticity in the friction of hydrogel-glass interfaces, specifically how poroelastic relaxation of hydrogels increases adhesion. We translate the work of adhesion into an effective surface energy density that increases with the duration of applied pressure from 10 to 50 mJ m^-2, as measured by micro-indentation. A model of static friction coefficient is derived from an area-based rules of mixture for the surface energies, and predicts the friction coefficient changes upon initiation of slip. For kinetic friction, the competition between duration of contact and relaxation time is quantified by a contacting Péclet number, Pe_C. A single length parameter on the scale of micrometers fits these two models to experimental micro-friction data. These models predict how short durations of applied pressure and faster sliding speeds, do not disrupt interfacial hydration; this prevailing water maintains low friction. At low speeds where interface drainage dominates, the osmotic suction works against slip for higher friction. The prediction of friction coefficients after adhesion characterization by micro-indentation makes use of the interplay between poroelasticity, adhesion, and friction. This approach provides a starting point for prediction of, and design for, hydrogel interfacial friction.

Poroelastic indentation of mechanically confined hydrogel layers

We report on the poroelastic indentation response of hydrogel thin films geometrically confined within contacts with rigid spherical probes of radii in the millimeter range. Poly(PEGMA) (poly(ethylene glycol) methyl ether methacrylate), poly(DMA) (dimethylacrylamide) and poly(NIPAM) (N-isopropylacrylamide) gel films with thickness less than 15 μm were grafted onto glass substrates using a thiol-ene click chemistry route. Changes in the indentation depth under constant applied load were monitored over time as a function of the film thickness and the radius of curvature of the probe using an interferometric method. In addition, shear properties of the indented films were measured using a lateral contact method. In the case of poly(PEGMA) films, we show that poroelastic indentation behavior is adequately described within the framework of an approximate contact model derived within the limits of confined contact geometries. This model provides simple scaling laws for the characteristic poroelastic time and the equilibrium indentation depth. Conversely, deviations from this model are evidenced for poly(DMA) and poly(NIPAM) films. From lateral contact experiments, these deviations are found to result from strong changes in the shear properties as a result of glass transition (poly(DMA)) or phase separation (poly(NIPAM)) phenomena induced by the drainage of the confined films squeezed between the rigid substrates.

Mechanical measurements of heterogeneity and length scale effects in PEG-based hydrogels

Colloidal-probe spherical indentation load-relaxation experiments with a probe radius of 3 μm are conducted on poly(ethylene glycol) (PEG) hydrogel materials to quantify their steady-state mechanical properties and time-dependent transport properties via a single experiment. PEG-based hydrogels are shown to be heterogeneous in both morphology and mechanical stiffness at this scale; a linear-harmonic interpolation of hyperelastic Mooney-Rivlin and Boussinesq flat-punch indentation models was used to describe the steady-state response of the hydrogels and determine upper and lower bounds for indentation moduli. Analysis of the transient load-relaxation response during displacement-controlled hold periods provides a means of extracting two time constants τ1 and τ2, where τ1 and τ2 are assigned to the viscoelastic and poroelastic properties, respectively. Large τ2 values at small indentation depths provide evidence of a non-equilibrium state characterized by a phenomenon that restricts poroelastic fluid flow through the material; for larger indentations, the variability in τ2 values decreases and pore sizes estimated from τ2via indentation approach those measured via macroscopic swelling experiments. The contact probe methodology developed here provides a means of assessing hydrogel heterogeneity, including time-dependent mechanical and transport properties, and has potential implications in hydrogel biomedical and engineering applications.

Using Indentation to Quantify Transport Properties of Nanophase-Segregated Polymer Thin Films

Indentation of hydrated Nafion thin films reveals that both the in-plane diffusivity of water and the intrinsic permeability of the phase-segregated network decrease dramatically with decreasing film thickness. Using pore-network theory, this decrease in diffusivity is attributed to both an increase in ionic-domain heterogeneity and a reduction in ionic-domain connectivity upon confinement.

Deswelling of ultrathin molecular layer-by-layer polyamide water desalination membranes

The selective layer of pressure-induced water desalination membranes is an ultrathin and highly crosslinked aromatic polyamide (PA) film that separates salt from water based on differences in permeability, which is a product of diffusivity and solubility. Characterizing the transport properties of the selective layer is necessary in understanding its permselective performance. However, measuring transport of ultrathin films in general is nontrivial. Here, Poroelastic Relaxation Indentation (PRI) is employed as a simple deswelling technique for measuring the transport properties of these ultrathin selective layers.

Centrifugal compression of soft particle packings: theory and experiment

An exact method is developed for computing the height of an elastic medium subjected to centrifugal compression, for arbitrary constitutive relation between stress and strain. Example solutions are obtained for power-law media and for cases where the stress diverges at a critical strain--for example as required by packings composed of deformable but incompressible particles. Experimental data are presented for the centrifugal compression of thermo-responsive N-isopropylacrylamide (NIPA) microgel beads in water. For small radial acceleration, the results are consistent with Hertzian elasticity, and are analyzed in terms of the Young elastic modulus of the bead material. For large radial acceleration, the sample compression asymptotes to a value corresponding to a space-filling particle volume fraction of unity. Therefore we conclude that the gel beads are incompressible, and deform without deswelling. In addition, we find that the Young elastic modulus of the particulate gel material scales with cross-link density raised to the power 3.3±0.8, somewhat larger than the Flory expectation.