Is the free energy of hydrogel the sum of elastic energy and ionic energy?

Saturated ionized porous media are profusely present in both biology and geology. To learn more about the constitutive behavior of these media, we use hydrogel as a model material. A picture of hydrogel is shown in Fig. 1. Hydrogel is an ionized polymer network with a large water content, and its electromechanical behavior is not fully understood. The main problem is caused by a lack of knowledge of the interplay between its free energy components: • W_el, the energy change caused by deformation of network chains to a more elongated state; • W_ion, the energy change caused by electrostatic interactions between fixed charges and mobile ions; • W _mix, the energy change caused by polymer-solvent mixing. Typically, the strain of the gel and the apparent density of the components are the state variables of the mixture. But, because the elastic stiffness of the gel is orders of magnitude lower than the volumetric modulus of water, it is reasonable to assume incompressibility of all mixture components. Hence, we may replace the apparent densities by volume fractions N^β, which is the volume fraction of the component β per unit initial mixture, i.e. N ^β = V^β/V₀, (1) where the superscript β denotes solid (s), fluid (f), cations (+) and anions (-), V ^β is the current volume associated with the component β and V₀ is the initial mixture volume. The strain is described by the deformation gradient tensor (F), which defines the deformation with respect to the initial configuration. It is convenient to choose the stress free state for this initial configuration. In hydrogel the stress free state is not unambiguously defined, but in this paper we use the situation of the hydrogel just after polymerization because it seems reasonable to assume that the elastic stress in the polymer network is then at a minimum. In addition, the volumetric variation with respect to the initial configuration is described by J, the third invariant of the deformation gradient tensor (det (F)). Thus, J = V/V ₀, (2) where V is the mixture volume (m³), and saturation requires that J = N^s + N^f + N⁺ + N^-. (3) Okay and Sariisik (1999) assumed that the total free energy (Jm ^-3) is the sum of all these separate parts (W_el, W _ion and W_mix), which is an extension of the Flory-Rehner Hypothesis (Flory and Rehner Jr. (1943)) to ionized porous media. Furthermore, they assumed that W_el is only a function of deformation, W _ion depends only on the apparent densities of the constituents of the gel, and W_mix is a function of the volume fraction of the polymer network Ns: W(J, N^s, N^f, N⁺, N^-) = W_el(J) + W_mix(N^s) + W_ion(N ^f, N⁺, N^-). (4) The state variables in (4) are not independent. It seems likely that the volume contribution of the ions is negligible: N⁺, N^- ≪ N^f, N⁺, N^- ≪ Ns. (5) Therefore, the saturation condition (3) simplifies to N^f = J - N^s, (6) where, in turn, the solid volume fraction depends on the deformation and the initial solid fraction N ₀^s. Thus, N^s = NN₀^s/J. (7) We can further reduce the number of state variables using electro-neutrality, i.e. N⁺/V̄⁺ - N ^-/V̄^- + c₀^fc = 0, (8) which relates N^- to N⁺ and c₀^fc, the concentration of 'fixed charge' with respect to the initial mixture volume, and V̄^β is the molar volume (m³ mol^-1). Because c₀^fc 0 is constant in time, the state variables in the general equation for the free energy (4) can be reduced to J and N+ by using (6)-(8): W(J, N⁺) = W_el(J) +W_mix(J) + W_ion(J, N⁺). (9) Finally, we can write the sum of W _el(J) and W_mix(J) as W_def(J), which accounts for changes in free energy due to deformation: W(J, N⁺) = W _def (J) + W_ion(J, N⁺). (10) It is common practice to use Donnan osmosis for W_ion (Donnan (1924)): W _ion(N^f, N⁺, N^-) = μ₀^fN^f + μ₀⁺N ⁺ + μ₀N^- - RTΓ_g(N ⁺/V̄⁺ + N^-/V̄^-)lnN ^f +RTN⁺/V̄⁺[ln(f⁺N ⁺/V̄⁺) - 1] +RTN^-/V̄ ^-[ln(f^-N^-/V̄^-) - 1], (11) where R is the universal gas constant (J mol^-1K^-1), T is the absolute temperature (K) and fβ is the activity coefficient (-) of component β. Γ_g is the osmotic coefficient inside the gel (-), which is a measure for electric screening. Now, it follows that ∂W_ion/∂N^f = -RTΓ_g(N ⁺/V̄⁺N^f + N^-/V̄ ^-N^f) (12) is minus the Donnan osmotic pressure of the gel. The right-hand side of eq. (12) is the classical Van 't Hoff's expression for osmotic pressure. The classical expression for the electrochemical potentials μ⁺ and μ^- of the ionic species is recovered from expression (11) by μ⁺ = Fξ/V̄⁺ + ∂W _ion/∂N⁺ = Fξ/V̄⁺ + RT/V̄⁺ ln(f⁺N⁺/V̄⁺(N ^f)^Γg), μ^- = Fξ/V̄^- + ∂W_ion/∂N^- = Fξ/V̄^- + RT/V̄^- ln(f^-N^-/V̄^-(N ^f)^Γg), (13) in which ξ is the Donnan electrical potential (V) and F the Faraday constant (C mol^-1). In this article, we discuss the results of confined swelling and compression experiments performed by Frijns et al. (2003). In these experiments, an Acrylic Acid Acrylamid copolymer gel was allowed to deform in one direction under the influence of a combination of a mechanical load and an ionic load using a set-up as shown in Fig. 2. While the samples were loaded by the protocol of Table 1, the corresponding height in the equilibrium situation was measured.