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

Abstract

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 τ2 via 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.

Hypertension-linked mechanical changes of rat gut

“…5), similar to the stress relaxation modulus at t = ∞. Our thus-named steady-state modulus (SSM) is a common feature of many complex models though it goes by various titles, including the “aggregate modulus” of poroelastic model [34] and “indentation modulus” in an AFM study similar to ours [22,35]. …”

“…(1) (See derivation and discussion by Vriend [21]): $$Ffalse(tfalse)=\frac{4E_{\mathit{\text{Hertz}}}·\sqrt{R}·\delta {false(tfalse)}^{\frac{3}{2}}}{3false(1-{\nu }^{2}false)}$$ where F = force as calculated by tip displacement multiplied by calibrated stiffness; R = radius of spherical indenter tip; δ = indentation depth calculated as stage movement minus deflection of tip; and ν = Poisson’s Ratio. This formulation has also been used for a variety of other soft matter indentation studies [22,23]. …”

“…4) [27]. The radius constraint is often operationalized as less than 10% [22], though original Hertz calculations suggest that less than 1% is appropriate [27]. Similar to a viscoelastic “stress relaxation modulus” that looks at instantaneous changes in effective modulus during stress relaxation [31], we rearranged the Hertz contact model for parabolic contact area to determine the effective modulus at a given point in time: $$E_{\text{effective}}false(tfalse)=\frac{3·Ffalse(tfalse)·false(1-{\nu }^{2}false)}{4·\sqrt{R}·\delta {false(tfalse)}^{\frac{3}{2}}}$$ where F = force as calculated by tip displacement multiplied by calibrated stiffness; R = radius of spherical indenter tip; δ = indentation depth calculated as stage movement minus deflection of tip; and ν = Poisson’s Ratio.…”

“…The figure shows a schematic of our MSI (right, described in Section 2.2) and a graphical representation of important restrictions on the Hertz model. Among other restrictions, the Hertz model will only be valid for small strains and indentation displacements such that δ ≪ R and δ ≤ 10% of the thickness of the sample [22,27]. …”

Micromechanical anisotropy and heterogeneity of the meniscus extracellular matrix

“…The instantaneous indentation modulus, E 0 , was estimated as, $$E_0=E_{\infty }+E_1+E_2.$$We therefore used the ratio of equilibrium versus instantaneous modulus, E ∞ / E 0 , as an indicator of the degree of elasticity. This method has been used to reveal the time-dependent mechanical properties of costal cartilage [35], the insertional zone of meniscal attachments [36], as well as poly(ethylene glycol) (PEG)-based hydrogels [37]. In this study, the implement of RCF k resulted in minor changes to E 1 and E 2 (e.g., RCF 1 = 1.30 ± 0.06, RCF 2 = 1.00 ± 0.01, mean ± 95% CI, for the outer zone horizontal section of circumferential fibers), and did not affect the observed trends or conclusions.…”