Network Viscoelasticity from Brillouin Spectroscopy

Abstract

Even though the physical nature of shear and longitudinal moduli are different, empirical correlations between them have been reported in several biological systems. This correlation is of fundamental interest and immense practical value in biomedicine due to the importance of the shear modulus and the possibility to map the longitudinal modulus at high-resolution with all-optical spectroscopy. We investigate the origin of such a correlation in hydrogels. We hypothesize that both moduli are influenced in the same direction by underlying physicochemical properties, which leads to the observed material-dependent correlation. Matching theoretical models with experimental data, we quantify the scenarios in which the correlation holds. For polymerized hydrogels, a correlation was found across different hydrogels through a common dependence on the effective polymer volume fraction. For hydrogels swollen to equilibrium, the correlation is valid only within a given hydrogel system, as the moduli are found to have different scalings on the swelling ratio. The observed correlation allows one to extract one modulus from another in relevant scenarios.

Figure 1

Schematic illustration of the frequency dependence of the elastic modulus in hydrogels. The low frequency plateau typically varies by about 2 orders of magnitude (from few kHz to 0.1 MHz), whereas at GHz frequencies, the modulus is about a few hundred MPa (for a Poisson’s ratio ∼ 0.48) with much smaller variations (∼30%) over the same composition range. The blue line denotes a hydrogel with a low cross-linked density, while the red line corresponds to a high cross-linked density hydrogel, with the latter displaying a higher elastic modulus than the former.

Figure 2

Shear and longitudinal moduli vs polymer volume fraction ϕ. The monomer concentration and hydrogel type are represented by color and shape, respectively. Different points with the same color and shape represent different percentages of cross-linker. (A) Log–log plot of shear modulus G_r vs experimental polymer volume fraction ϕ of AA hydrogels in relaxed state. The solid line represents G_r ∼ ϕ^2.2±0.1, R² = 0.85. (B) Longitudinal modulus M_r vs experimental polymer volume fraction ϕ of the same hydrogels. The red and blue dotted lines respectively denote the predictions by the linear and inverse laws of mixtures, using M_p = 16.35 GPa and M_w = 2.22 GPa. In both A and B, the data points are scattered for the different cross-linking agent contents at a constant AA concentration.

Figure 3

Longitudinal modulus M vs effective polymer volume fraction ϕ_eff of AA hydrogels in relaxed state. M_p = 16.35 GPa and M_w = 2.22 GPa are fixed values in the representation of M(ϕ_eff) by eq 4 (blue dotted line). The red dotted line denotes the linear dependence of M(ϕ_eff).

Figure 4

Longitudinal modulus vs shear modulus. Log–log plot of M_r(G_r) of the experimental moduli (symbols) and computed correlation (dashed line, eq 5) between the two moduli.

Figure 5

Shear and longitudinal moduli vs polymer volume fraction ϕ_Sw. (A) Log–log plot of the shear modulus G_s vs ϕ_Sw of the AA hydrogels that are swollen to equilibrium. Each condition of the hydrogels shows distinct behavior. The slopes obtained from the power law representation for the swollen neutral, swollen cationic charged condition, and swollen low molecular weight conditions are 2.43 (solid line), 1.22 (dotted line), and 2.44 (dashed line), respectively. (B) Longitudinal modulus M_s vs ϕ_Sw of hydrogels swollen to equilibrium exhibiting large scattering of the data points. No trend following the inverse rule of mixtures can be assumed.

Figure 6

Shear and longitudinal moduli for swollen hydrogels. (A) Comparison of the shear modulus measured in relaxed gels G_r affected by the swelling ratio, vs the actual shear modulus measured on these gels G_s. The black line represents G_s = G_rQ^–1/3. (B) The behavior of M_s seems a better fit following the inverse law of mixtures when considering ϕ_Seff = (ϕ_eff × Q^–1).

Figure 7

Prediction of the relationship for swollen hydrogels. Log–log plot of experimental M_s vs experimental G_s. Symbols represent the experimental data, while the theoretical prediction of eq 9 assuming different swelling ratios for all the samples is represented with dashed lines [Q = 1 (black, for the relaxed state); Q = 2 (red); Q = 10 (green); Q = 20 (blue)]. The experimental data do not follow the same trend because the samples have different Q values.