[Paper Review] Measuring elastic properties of granular hydrogels: Effects of capillary interaction and ionic conditions

Jiayin Zhao, Haiyi Zhong+1 more|arXiv (Cornell University)|Feb 25, 2026

Hydrogels: synthesis, properties, applications0 citations

TL;DR

The paper develops an experimental setup to quantify elastic responses of swollen granular hydrogels under wet conditions, separating capillary (elastocapillary) forces from mechanical load, and analyzes how capillarity and ionic conditions affect apparent vs effective modulus and swelling-driven stiffness.

ABSTRACT

The elastic properties of granular hydrogels are commonly characterised under wet conditions, yet the influence of capillary interactions remains unclear. In practical applications, hydrogels operate in aqueous environments containing dissolved ionic species, where swelling and elastic behaviour depend sensitively on ionic conditions. In this study, an experimental setup is developed to measure elastic responses of granular hydrogels under wet conditions. This setup directly observes liquid bridges formation and its evolution during compression. Our results show that neglecting capillary contributions leads to a systematic underestimation of the Young's modulus of hydrogels. Such an underestimation due to the capillary interaction increases as the sample size or its intrinsic stiffness decreases. In addition to the swelling ratio, the tested samples were also prepared under controlled salinity levels. The experimentally observed dependence of stiffness on swelling and salinity conditions is well captured by a modified constitutive model. The development of this study offers a robust testing protocol for measuring elastic properties of hydrogels under various environmental conditions.

Motivation & Objective

• Characterize how capillary forces affect elastic measurements of swollen hydrogel particles under compression.
• Quantify the capillary contribution to measured forces and its size dependence.
• Investigate how ionic conditions (NaCl concentration) influence swelling and elasticity.
• Validate a constitutive model that links swelling, ionic strength, and modulus.
• Demonstrate a procedure to distinguish capillary effects from pure elastic response.

Proposed method

• Use a custom loading setup to compress a single hydrogel sphere against a rigid glass plate while observing a liquid bridge.
• Compute capillary force F_cap from liquid-bridge geometry via F_cap = π R2^2 Δp + 2π R2 γ with Δp = γ(1/R1 + 1/R2) and γ = 72.8 mN/m.
• Obtain measured force F_m from the scale and define contact force F_c = F_m + F_cap.
• Fit Hertzian contact law F = (4/3) E R^{1/2} (δ − δ0)^{3/2} to obtain apparent and effective moduli.
• Swelling and water content W_c tracked under five ionic environments (0, 1, 2, 5, 10 g/L NaCl) and modeled swelling kinetics W_c(t) = W_c,max(c)(1 − exp[−k(c)t]).
• Compare experimental E* with a Brighenti et al.-type constitutive model for hydrogel stiffness under swelling and ionic conditions with fitted parameters G, J_c, γ_T.

Experimental results

Research questions

• RQ1 How do capillary forces in a liquid bridge modify the measured elastic response of swollen granular hydrogels during compression?
• RQ2 How does particle size influence the magnitude of capillary effects on the inferred Young’s modulus?
• RQ3 How do ionic conditions (NaCl concentration) and swelling level affect the effective modulus and its evolution under deformation?
• RQ4 Can a constitutive model incorporating swelling and ionic effects predict the observed modulus trends across conditions?

Key findings

• Capillary forces bias the measured modulus, with the bias being larger for smaller particles and leading to a size-dependent underestimation of the true modulus.
• Capillary force magnitudes are 2–8 mN across saline conditions, slightly decreasing with particle size.
• The effective modulus E* increases with particle radius in low-salinity groups and shows steeper softening and faster stiffening with higher salinity.
• A constitutive model incorporating swelling, hydrogen-bond interactions, and ionic strength reproduces the modulus evolution with less than 15% error across 0–2 g/L NaCl datasets.
• Higher salinity corresponds to higher G and γ_T and a smaller J_c, indicating stiffer and less extensible networks under ionic screening.
• The model captures the shift in swelling-driven stiffness behavior and aligns with observed trends as ionic strength increases.