where ϵ 0 is the vacuum permittivity. Eq. 1 applies only if two charges are within a cut-off distance of r C = 0.15 nm. Beyond the cut-off distance, a PPPM Ewald solver with an accuracy of 10 -5 is used to calculate the long-range electrostatics . Initially, the solvated hydrogel system, including chain monomers, ions, and water molecules, is treated by the steepest descent minimization to remove the nonphysical contacts between the atoms. After minimization, a τ R ≈ 60-ns long canonical-ensemble simulation is performed to relax the hydrogel structure with a timestep of 2 fs at T = 300 K and P = P z ≈ P x ≈ P y ≈ 1.0 atm. The Nose-Hover algorithm is used to maintain the temperature and pressure. Given that the Kuhn size of PAA chain is b ≈ 0.6 nm (i.e., roughly 2 AA units) , the Zimm relaxation time of a chain segment composed of N/2 Kuhn segments in good solvent could be approximated as τ Z ≈ (N/2) 3/2 τ 0 ≈ 10 ns, where the self-diffusion time of a Kuhn monomer of size a ≈ 1 mm to a distance equivalent to its own size is taken as the upper limit for the monomeric relaxation time, τ 0 ≈ 1 ns. Thus, our relaxation simulations are sufficiently long to avoid any effect due to initial chain configurations. The relaxed hydrogel configuration is used for uni-axial stretching and compression simulations.