Equilibrium swelling of multi-stimuli-responsive superabsorbent hydrogels


Superabsorbent gels (lightly cross-linked copolymer gels with water absorption capacity up to several thousands g/g) have recently attracted substantial attention due to their novel applications in agriculture, environmental management, and civil engineering. As superabsorbent hydrogels (SAHs) are conventionally prepared by copolymerization of polyelectrolyte and temperature-sensitive neutral monomers, their response is strongly affected by temperature, pH, and ionic strength of solutions. A characteristic feature of SAHs is an anomalous increase in their elastic modulus with degree of swelling.

Constitutive equations are derived for the mechanical response and equilibrium swelling of SAHs. An advantage of the model is that it involves only six material constants. These quantities are found by fitting experimental data in equilibrium swelling tests and uniaxial compressive tests on a series of N-isopropylacrylamide-co-2-acrylamido-2-methylpropane sulfonic acid gels with various molar fractions of ionic comonomers. An acceptable agreement is demonstrated between the observations and results of simulation. The model is applied to examine the effects of temperature, pH, and molar fraction of monovalent salt in aqueous solutions on equilibrium swelling of SAHs with various molar fractions of comonomers in the feed.

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  1. 1.

    Jayakumar, A., Jose, V.K., Lee, J.-M.: Hydrogels for medical and environmental applications. Small Methods 4, 1900735 (2020)

    Article  Google Scholar 

  2. 2.

    Varaprasad, K., Raghavendra, G.M., Jayaramudu, T., Yallapu, M.M., Sadiku, R.: A mini review on hydrogels classification and recent developments in miscellaneous applications. Mater. Sci. Eng. C 79, 958–971 (2017)

    Article  Google Scholar 

  3. 3.

    Kabiri, K., Omidian, H., Zohuriaan-Mehr, M.J., Doroudiani, S.: Superabsorbent hydrogel composites and nanocomposites: a review. Polym. Compos. 32, 277–289 (2011)

    Article  Google Scholar 

  4. 4.

    Mignon, A., De Belie, N., Dubruel, P., Van Vlierberghe, S.: Superabsorbent polymers: a review on the characteristics and applications of synthetic, polysaccharide-based, semi-synthetic and smart derivatives. Eur. Polym. J 117, 165–178 (2019)

    Article  Google Scholar 

  5. 5.

    Saha, A., Rattan, B., Sekharan, S., Manna, U.: Superabsorbent hydrogel (SAH) as a soil amendment for drought management: a review. Soil Till. Res. 204, 104736 (2020)

    Article  Google Scholar 

  6. 6.

    Guha, T., Gopal, G., Kundu, R., Mukherjee, A.: Nanocomposites for delivering agrochemicals: a comprehensive review. J. Agric. Food Chem. 68, 3691–3702 (2020)

    Article  Google Scholar 

  7. 7.

    Samaddar, P., Kumar, S., Kim, K.-H.: Polymer hydrogels and their applications toward sorptive removal of potential aqueous pollutants. Polym. Rev. 59, 418–464 (2019)

    Article  Google Scholar 

  8. 8.

    Guo, Y., Bae, J., Fang, Z., Li, P., Zhao, F., Yu, G.: Hydrogels and hydrogel-derived materials for energy and water sustainability. Chem. Rev. 120, 7642–7707 (2020)

    Article  Google Scholar 

  9. 9.

    Zhao, F., Zhou, X., Liu, Y., Shi, Y., Dai, Y., Yu, G.: Super moisture-absorbent gels for all-weather atmospheric water harvesting. Adv. Mater. 31, 1806446 (2019)

    Article  Google Scholar 

  10. 10.

    He, Z., Shen, A., Guo, Y., Lyu, Z., Li, D., Qin, X., Zhao, M., Wang, Z.: Cement-based materials modified with superabsorbent polymers: a review. Constr. Build Mater. 225, 569–590 (2019)

    Article  Google Scholar 

  11. 11.

    Shibayama, M., Shirotani, Y., Hirose, H., Nomura, S.: Simple scaling rules on swollen and shrunken polymer gels. Macromolecules 30, 7307–7312 (1997)

    Article  Google Scholar 

  12. 12.

    Flory, P.J.: Molecular theory of rubber elasticity. Polymer 20, 1317–1320 (1979)

    Article  Google Scholar 

  13. 13.

    Flory, P.J.: Theory of elasticity of polymer networks. The effect of local constraints on junctions. J. Chem. Phys. 66, 5720–5729 (1977)

    Article  Google Scholar 

  14. 14.

    Drozdov, A.D., Christiansen, J.: Constitutive equations in finite elasticity of swollen elastomers. Int. J. Solids Struct. 50, 1494–1504 (2013a)

    Article  Google Scholar 

  15. 15.

    Drozdov, A.D., Christiansen, J.: Stress–strain relations for hydrogels under multiaxial deformation. Int. J. Solids Struct. 50, 3570–3585 (2013b)

    Article  Google Scholar 

  16. 16.

    Bromberg, L., Grosberg, A.Y., Matsuo, E.S., Suzuki, Y., Tanaka, T.: Dependency of swelling on the length of subchain in poly(N,N-dimethylacrylamide)-based gels. J. Chem. Phys. 106, 2906–2910 (1997)

    Article  Google Scholar 

  17. 17.

    Skouri, R., Schosseler, F., Munch, J.P., Candau, S.J.: Swelling and elastic properties of polyelectrolyte gels. Macromolecules 28, 197–210 (1995)

    Article  Google Scholar 

  18. 18.

    Dubrovskii, S.A., Rakova, G.V.: Elastic and osmotic behavior and network imperfections of nonionic and weakly ionized acrylamide-based hydrogels. Macromolecules 30, 7478–7486 (1997)

    Article  Google Scholar 

  19. 19.

    Zaroslov, Y.D., Philippova, O.E., Khokhlov, A.R.: Change of elastic modulus of strongly charged hydrogels at the collapse transition. Macromolecules 32, 1508–1513 (1999)

    Article  Google Scholar 

  20. 20.

    Gundogan, N., Melekaslan, D., Okay, O.: Rubber elasticity of poly(N-isopropylacrylamide) gels at various charge densities. Macromolecules 35, 5616–5622 (2002)

    Article  Google Scholar 

  21. 21.

    Okay, O., Durmaz, S.: Charge density dependence of elastic modulus of strong polyelectrolyte hydrogels. Polymer 43, 1215–1221 (2002)

    Article  Google Scholar 

  22. 22.

    Morozova, S., Muthukumar, M.: Elasticity at swelling equilibrium of ultrasoft polyelectrolyte gels: Comparisons of theory and experiments. Macromolecules 50, 2456–2466 (2017)

    Article  Google Scholar 

  23. 23.

    Orakdogen, N., Boyaci, T.: Finite extensibility and deviation from Gaussian elasticity of dimethylacrylamide-based gels with different charge density: Insight into pH/solvent-dependent swelling and surfactant interactions. Polymer 132, 306–324 (2017)

    Article  Google Scholar 

  24. 24.

    Orakdogen, N., Boyaci, T.: Non-gaussian elasticity and charge density dependent swelling of strong polyelectrolyte poly(N-isopropylacrylamide-co-sodium acrylate) hydrogels. Soft. Matter. 13, 9046–9059 (2017)

    Article  Google Scholar 

  25. 25.

    Hoshino, K., Nakajima, T., Matsuda, T., Sakai, T., Gong, J.P.: Network elasticity of a model hydrogel as a function of swelling ratio: from shrinking to extreme swelling states. Soft Matter 14, 9693–9701 (2018)

    Article  Google Scholar 

  26. 26.

    Rubinstein, M., Colby, R.H., Dobrynin, A.V., Joanny, J.-F.: Elastic modulus and equilibrium swelling of polyelectrolyte gels. Macromolecules 29, 398–406 (1996)

    Article  Google Scholar 

  27. 27.

    Vilgis, T.A., Johner, A., Joanny, J.-F.: Polyelectrolyte gels in poor solvent: Elastic moduli. Eur. Phys. J. E. 3, 237–244 (2000)

    Article  Google Scholar 

  28. 28.

    Ricka, J., Tanaka, T.: Swelling of ionic gels: Quantitative performance of the Donnan theory. Macromolecules 17, 2916–2921 (1984)

    Article  Google Scholar 

  29. 29.

    Dobrynin, A.V., Rubinstein, M.: Theory of polyelectrolytes in solutions and at surfaces. Prog. Polym. Sci. 30, 1049–1118 (2005)

    Article  Google Scholar 

  30. 30.

    Katchalsky, A., Lifson, S., Mazur, J.: The electrostatic free energy of polyelectrolyte solutions. I. Randomly kinked macromolecules. J. Polym. Sci. 11, 409–423 (1953)

    Article  Google Scholar 

  31. 31.

    Lifson, S., Katchalsky, A.: The electrostatic free energy of polyelectrolyte solutions. II. Fully stretched macromolecules. J. Polym. Sci. 13, 43–55 (1954)

    Article  Google Scholar 

  32. 32.

    Hasa, J., Ilavsky, M., Dusek, K.: Deformational, swelling, and potentiometric behavior of ionized poly(methacrylic acid) gels. I. Theory. J. Polym. Sci. Polym. Phys. Ed. 13, 253–262 (1975)

    Article  Google Scholar 

  33. 33.

    Ilavsky, M.: Effect of electrostatic interactions on phase transition in the swollen polymeric network. Polymer 22, 1687–1691 (1981)

    Article  Google Scholar 

  34. 34.

    Muthukumar, M.: Double screening in polyelectrolyte solutions: Limiting laws and crossover formulas. J. Chem. Phys. 105, 5183–5199 (1996)

    Article  Google Scholar 

  35. 35.

    Jia, D., Muthukumar, M.: Interplay between microscopic and macroscopic properties of charged hydrogels. Macromolecules 53, 90–101 (2020)

    Article  Google Scholar 

  36. 36.

    Barrat, J.-L., Joanny, J.-F., Pincus, P.: On the scattering properties of polyelectrolyte gels. J. Phys. II France 2, 1531–1544 (1992)

    Article  Google Scholar 

  37. 37.

    Kokufuta, E.: Polyelectrolyte gel transitions: Experimental aspects of charge inhomogeneity in the swelling and segmental attractions in the shrinking. Langmuir 21, 10004–10015 (2005)

    Article  Google Scholar 

  38. 38.

    Lopez, C.G., Lohmeier, T., Wong, J.E., Richtering, W.: Electrostatic expansion of polyelectrolyte microgels: Effect of solvent quality and added salt. J. Colloid Interface Sci. 558, 200–210 (2020)

    Article  Google Scholar 

  39. 39.

    Mussel, M., Horkay, F.: Experimental evidence for universal behavior of ion-induced volume phase transition in sodium polyacrylate gels. J. Phys. Chem. Lett. 10, 7831–7835 (2019)

    Article  Google Scholar 

  40. 40.

    Hong, W., Zhao, X., Suo, Z.: Large deformation and electrochemistry of polyelectrolyte gels. J. Mech. Phys. Solids 58, 558 (2010)

    MathSciNet  MATH  Article  Google Scholar 

  41. 41.

    Wallmersperger, T., Keller, K., Kroplin, B., Gunther, M., Gerlach, G.: Modeling and simulation of pH-sensitive hydrogels. Colloid. Polym. Sci. 289, 535–544 (2011)

    Article  Google Scholar 

  42. 42.

    Kurnia, J.C., Birgersson, E., Mujumdar, A.S.: Analysis of a model for pH-sensitive hydrogels. Polymer 53, 613–622 (2012)

    Article  Google Scholar 

  43. 43.

    Li, J., Suo, Z., Vlassak, J.J.: A model of ideal elastomeric gels for polyelectrolyte gels. Soft. Matter 10, 2582–2590 (2014)

    Article  Google Scholar 

  44. 44.

    Drozdov, A.D., deClaville Christiansen, J.: Swelling of pH–sensitive hydrogels. Phys. Rev. E. 91, 022305 (2015)

    Article  Google Scholar 

  45. 45.

    Drozdov, A.D., deClaville Christiansen, J.: Modeling the effects of pH and ionic strength on swelling of polyelectrolyte gels. J. Chem. Phys. 142, 114904 (2015)

    Article  Google Scholar 

  46. 46.

    Atta, A.M., Ismail, H.S., Elsaaed, A.M.: Application of anionic acrylamide-based hydrogels in the removal of heavy metals from waste water. J. Appl. Polym. Sci. 123, 2500–2510 (2012)

    Article  Google Scholar 

  47. 47.

    Gent, A.N.: A new constitutive relation for rubber. Rubber Chem. Technol. 69, 59–61 (1996)

    Article  Google Scholar 

  48. 48.

    Okumura, D., Chester, S.A.: Ultimate swelling described by limiting chain extensibility of swollen elastomers. Int. J. Mech. Sci. 144, 531–539 (2018)

    Article  Google Scholar 

  49. 49.

    Okumura, D., Kawabata, H., Chester, S.A.: A general expression for linearized properties of swollen elastomers undergoing large deformations. J. Mech. Phys. Solids 135, 103805 (2020)

    MathSciNet  Article  Google Scholar 

  50. 50.

    Nardinocchi, P., Pezzula, M., Placidi, L.: Thermodynamically based multiphysics modeling of ionic polymer metal composites. J. Intell. Mater. Syst. Struct. 22, 1887–1897 (2011)

    Article  Google Scholar 

  51. 51.

    Flory, P.J., Rehner, J.: Statistical mechanics of cross-linked polymer networks II Swelling. J. Chem. Phys. 11, 521–526 (1943)

    Article  Google Scholar 

  52. 52.

    Khokhlov, A.R.: On the collapse of weakly charged polyelectrolytes. J. Phys. A 13, 979–987 (1980)

    Article  Google Scholar 

  53. 53.

    Petrovic, Z.S., MacKnight, W.J., Koningsveld, R., Dusek, K.: Swelling of model networks. Macromolecules 20, 1088–1096 (1987)

    Article  Google Scholar 

  54. 54.

    Zhang, Y., Furyk, S., Sagle, L.B., Cho, Y., Bergbreiter, D.E., Cremer, P.S.: Effects of Hofmeister anions on the LCST of PNIPAM as a function of molecular weight. J. Phys. Chem. C. 111, 8916–8924 (2007)

    Article  Google Scholar 

  55. 55.

    Quesada-Perez, M., Maroto-Centeno, J.A., Forcada, J., Hidalgo-Alvarez, R.: Gel swelling theories: the classical formalism and recent approaches. Soft Matter 7, 10536–10547 (2011)

    Article  Google Scholar 

  56. 56.

    Drozdov, A.D.: Mechanical behavior of temperature-sensitive gels under equilibrium and transient swelling. Int. J. Eng. Sci. 128, 79–100 (2018)

    MathSciNet  MATH  Article  Google Scholar 

  57. 57.

    Norisuye, T., Kida, Y., Masui, N., Tran-Cong-Miyata, Q., Maekawa, Y., Yoshida, M., Shibayama, M.: Studies on two types of built-in inhomogeneities for polymer gels: Frozen segmental concentration fluctuations and spatial distribution of cross-links. Macromolecules 36, 6202–6212 (2003)

    Article  Google Scholar 

  58. 58.

    Manning, G.S.: Limiting laws and counterion condensation in polyelectrolyte solutions. I. Colligative properties. J. Chem. Phys. 51, 924–933 (1969)

    Article  Google Scholar 

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Financial support was provided by Innovationsfonden (Innovation Fund Denmark, project 9091-00010B).

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Appendix : 1

The specific free energy of a gel is determined by Eq. 37. Differentiation of Eq. 33 for the specific mechanical energy W with respect to time implies that

$$ \dot{W} = W_{,1}\dot{I}_{\text{e 1}}+W_{,2}\dot{I}_{\text{e 2}} +W_{,3}\dot{I}_{\text{e 1}}, $$


$$ W_{,m}=\frac{\partial W}{\partial I_{\mathrm{e}m}} \qquad (m=1,2,3). $$

The derivatives of the principal invariants Ie1, Ie2, and Ie3 of the Cauchy–Green tensor Be with respect to time are given by

$$ \dot{I}_{\text{e1}}=2 \mathbf{B}_{\mathrm{e}}: \mathbf{D},\qquad \dot{I}_{\text{e2}}=2 \left( I_{\text{e2}} \mathbf{I}-I_{\text{e3}} \mathbf{B}_{\mathrm{e}}^{-1}\right): \mathbf{D},\qquad \dot{I}_{\text{e3}}=2 I_{\text{e3}} \mathbf{I}: \mathbf{D}, $$

where the rate-of-strain tensor D is determined by Eq. 40. Combination of Eqs. A-1 and A-3 results in

$$ \dot{W}=2\mathbf{K}_{\text{mech}}:\mathbf{D} $$


$$ \mathbf{K}_{\text{mech}} = W_{,1}\mathbf{B}_{\mathrm{e}} -I_{\text{e3}} W_{,2}\mathbf{B}_{\mathrm{e}}^{-1} +\left( I_{\text{e2}} W_{,2}+I_{\text{e3}} W_{,3}\right)\mathbf{I}. $$

Differentiation of Eq. 20 for the specific energy of the electric field Wel with respect to time implies that

$$ \dot{W}_{\text{el}}=\frac{1}{2\epsilon J}\left( 2{\mathbf{H}}\cdot {\mathbf{C}}\cdot \dot{\mathbf{H}} +{\mathbf{H}}\cdot \dot{\mathbf{C}}\cdot {\mathbf{H}} -\frac{\dot{J}}{J}{\mathbf{H}}\cdot {\mathbf{C}}\cdot {\mathbf{H}}\right). $$

This relation together with Eq. 16 yields

$$ \dot{W}_{\text{el}}={\mathbf{E}} \cdot \dot{\mathbf{H}} +\frac{1}{2\epsilon J}\left( {\mathbf{H}}\cdot \dot{\mathbf{C}}\cdot {\mathbf{H}} -\frac{\dot{J}}{J}{\mathbf{H}}\cdot {\mathbf{C}}\cdot {\mathbf{H}}\right). $$

It follows from Eqs. 1740, and 41 that

$$ \dot{\mathbf{C}}=2\mathbf{F}^{\top}\cdot{\mathbf{D}}\cdot{\mathbf{F}}. $$

Combination of this equality with Eq. 15 yields

$$ \frac{1}{2\epsilon J} {\mathbf{H}}\cdot \dot{\mathbf{C}}\cdot {\mathbf{H}} =\frac{1}{\epsilon J} {\mathbf{H}}\cdot {\mathbf{F}}^{\top}\cdot {\mathbf{D}} \cdot {\mathbf{F}}\cdot {\mathbf{H}} =\frac{J}{\epsilon}{\mathbf{h}}\cdot {\mathbf{D}}\cdot {\mathbf{h}} =\frac{J}{\epsilon}(\mathbf{h}\otimes \mathbf{h}): \mathbf{D}, $$

where ⊗ stands for the tensor product. Keeping in mind that

$$ \dot{J}=J \mathbf{I}:\mathbf{D}, $$

and using Eq. 17, we conclude that

$$ \frac{\dot{J}}{2\epsilon J^{2}}\mathbf{H}\cdot {\mathbf{C}}\cdot {\mathbf{H}} =\frac{1}{2\epsilon J}{\mathbf{H}}\cdot \mathbf{F}^{\top}\cdot {\mathbf{F}}\cdot {\mathbf{H}} (\mathbf{I}:\mathbf{D}) =\frac{J}{2\epsilon} (\mathbf{h}\cdot \mathbf{h})\mathbf{I}:\mathbf{D}. $$

Substitution of Eqs. A-7 and A-9 into Eq. A-6 implies that

$$ \dot{W}_{\text{el}}={\mathbf{E}} \cdot \dot{\mathbf{H}} +2\mathbf{K}_{\text{el}}:\mathbf{D}, $$


$$ \mathbf{K}_{\text{el}}=\frac{J}{2\epsilon} \left[ (\mathbf{h}\otimes \mathbf{h})-\frac{1}{2} (\mathbf{h}\cdot \mathbf{h})\mathbf{I}\right]. $$

Differentiation of the other terms in Eq. 37 is straightforward. The derivative of expression Eq. 37 with respect to time is determined by Eq. 43 with

$$ \begin{array}{@{}rcl@{}} && {\Theta}_{C} = \mu^{0}+k_{\mathrm{B}}T \left[ \ln \frac{C v}{1+C v}+\frac{1}{1+C v} +\frac{\chi}{(1+C v)^{2}} -\frac{C_{\mathrm{H}^{+}}+C_{\text{Na}^{+}}+C_{\text{OH}^{-}}+C_{\text{Cl}^{-}}}{C}\right],\\ && {\Theta}_{\mathrm{H}^{+}} = {\mu}_{\mathrm{H}^{+}}^{0} +k_{\mathrm{B}}T \ln \frac{C_{\mathrm{H}^{+}}}{C},\qquad {\Theta}_{\text{Na}^{+}} = {\mu}_{\text{Na}^{+}}^{0} +k_{\mathrm{B}}T \ln \frac{C_{\text{Na}^{+}}}{C},\\ && {\Theta}_{\text{OH}^{-}} = {\mu}_{\text{OH}^{-}}^{0} +k_{\mathrm{B}}T \ln \frac{C_{\text{OH}^{-}}}{C},\qquad {\Theta}_{\text{Cl}^{-}} = {\mu}_{\text{Cl}^{-}}^{0} +k_{\mathrm{B}}T \ln \frac{C_{\text{Cl}^{-}}}{C}. \end{array} $$

Appendix : 2

Under unconstrained swelling of a gel, the deformation gradient for macrodeformation reads

$$ \mathbf{F}=(1+C v)^{\frac{1}{3}} \mathbf{I}. $$

Substitution of Eqs. B-1 and 10 into Eq. 11 implies that

$$ \mathbf{F}_{\mathrm{e}}=\left( \frac{1+C v}{1+C_{0} v}\right)^{\frac{1}{3}} \mathbf{I}. $$

It follows from Eqs. B-2 and 12 that

$$ \mathbf{B}_{\mathrm{e}}=\left( \frac{1+C v}{1+C_{0} v}\right)^{\frac{2}{3}} \mathbf{I}. $$

Insertion of this relation into Eq. 54 yields

$$ \mathbf{T}=T\mathbf{I}, $$


$$ T=-{\Pi}+\frac{G}{1+C v} \left[\frac{1}{V} \left( \frac{1+C v}{1+C_{0}v} \right)^{\frac{2}{3}}-1\right] $$


$$ V=1+\frac{3}{K} \left[ \left( \frac{1+C v}{1+C_{0}v} \right)^{\frac{2}{3}}-1\right]. $$

When pressure \(\bar {\Pi }\) is the bath is disregarded, the surface of a gel is traction-free. It follows from this condition and the equilibrium equation for the Cauchy stress tensor that T = 0. Combining this equality with Eq. B-3 and using Eq. B-4, we find that

$$ {\Pi}=\frac{G}{1+C v} \left[ \frac{1}{V} \left( \frac{1+C v}{1+C_{0}v} \right)^{\frac{2}{3}}-1\right]. $$

Substitution of Eqs. 48 and 55 for the chemical potentials of mobile ions into Eq. 56 results in

$$ \begin{array}{@{}rcl@{}} && \ln \frac{C_{\mathrm{H}^{+}}}{C} = \ln \frac{\bar{c}_{\mathrm{H}^{+}}}{\bar{c}} -\frac{e}{k_{\mathrm{B}}T}\left( {\Phi}-\bar{{\Phi}}\right),\quad \ln \frac{C_{\text{Na}^{+}}}{C} = \ln \frac{\bar{c}_{\text{Na}^{+}}}{\bar{c}} -\frac{e}{k_{\mathrm{B}}T}\left( {\Phi}-\bar{{\Phi}}\right),\\ && \ln \frac{C_{\text{OH}^{-}}}{C} = \ln \frac{\bar{c}_{\text{OH}^{-}}}{\bar{c}} +\frac{e}{k_{\mathrm{B}}T}\left( {\Phi}-\bar{\Phi}\right),\quad \ln \frac{C_{\text{Cl}^{-}}}{C} = \ln \frac{\bar{c}_{\text{Cl}^{-}}}{\bar{c}} +\frac{e}{k_{\mathrm{B}}T}\left( {\Phi}-\bar{\Phi}\right). \end{array} $$

It follows from Eq. B-7 that

$$ \begin{array}{@{}rcl@{}} && \frac{C_{\mathrm{H}^{+}}}{C} \frac{C_{\text{OH}^{-}}+C_{\text{Cl}^{-}}}{C} =\frac{\bar{c}_{\mathrm{H}^{+}}}{\bar{c}} \frac{\bar{c}_{\text{OH}^{-}}+\bar{c}_{\text{Cl}^{-}}}{\bar{c}}, \end{array} $$
$$ \begin{array}{@{}rcl@{}} && \frac{C_{\mathrm{H}^{+}}+C_{\text{Na}^{+}}}{C} \frac{C_{\text{OH}^{-}}+C_{\text{Cl}^{-}}}{C} =\frac{\bar{c}_{\mathrm{H}^{+}}+\bar{c}_{\text{Na}^{+}}}{\bar{c}} \frac{\bar{c}_{\text{OH}^{-}}+\bar{c}_{\text{Cl}^{-}}}{\bar{c}}. \end{array} $$

The electro-neutrality conditions for the gel and for the bath read

$$ \begin{array}{@{}rcl@{}} C_{\text{OH}^{-}}+C_{\text{Cl}^{-}} &=& C_{\mathrm{H}^{+}}+C_{\text{Na}^{+}}-C_{\text{b}^{-}}, \end{array} $$
$$ \begin{array}{@{}rcl@{}} \bar{c}_{\text{OH}^{-}}+\bar{c}_{\text{Cl}^{-}} &=& \bar{c}_{\mathrm{H}^{+}}+\bar{c}_{\text{Na}^{+}}. \end{array} $$

Inserting expression Eq. B-11 into Eq. B-9 and using Eqs. 35, and 6, we find that

$$ \frac{C_{\mathrm{H}^{+}}+C_{\text{Na}^{+}}}{C} \frac{C_{\text{OH}^{-}}+C_{\text{Cl}^{-}}}{C} =\frac{1}{\kappa^{2}}\left( 10^{-\text{pH}}+\theta\right)^{2}. $$

Using Eq. B-10, we present Eq. B-12 in the form

$$ X\left( X-\frac{C_{\text{b}^{-}}}{C}\right) =\frac{1}{\kappa^{2}}({10}^{-\text{pH}}+\theta)^{2}, $$


$$ X=\frac{C_{\mathrm{H}^{+}}+C_{\text{Na}^{+}}}{C}. $$

Inserting Eqs. B-10 and B-11 into Eq. B-8, we find that

$$ \frac{C_{\mathrm{H}^{+}}}{C} \frac{C_{\mathrm{H}^{+}}+C_{\text{Na}^{+}}-C_{\text{b}^{-}}}{C} =\frac{\bar{c}_{\mathrm{H}^{+}}}{\bar{c}} \frac{\bar{c}_{\mathrm{H}^{+}}+\bar{c}_{\text{Na}^{+}}}{\bar{c}}. $$

Using Eqs. 356, and B-14 and introducing the notation

$$ X_{1}=\frac{C_{\mathrm{H}^{+}}}{C}, $$

we present this relation in the form

$$ X_{1}\left( X-\frac{C_{\text{b}^{-}}}{C}\right) =\frac{1}{\kappa^{2}}10^{-\text{pH}}\left( 10^{-\text{pH}}+\theta\right). $$

Dividing Eq. B-16 by Eq. B-13, we find that

$$ X_{1}=\frac{10^{-\text{pH}}}{{10}^{-\text{pH}}+\theta} X. $$

Introducing pH in the gel by means of Eqs. 3 and 5,

$$ \text{pH}_{\mathrm{g}}=-\log \left( \kappa \frac{C_{\mathrm{H}^{+}}}{C}\right), $$

and using Eqs. B-15 and B-17, we arrive at Eq. 61.

Substitution of expression Eq. B-6 into Eqs. 48 and 49 for the chemical potential of water molecules in the gel results in

$$ \begin{array}{@{}rcl@{}} \mu &=& \mu^{0}+k_{\mathrm{B}}T \left[ \ln \frac{C v}{1+C v}+\frac{1}{1+C v} +\frac{\chi}{(1+C v)^{2}} +\frac{g}{1+C v} \left( \frac{1}{V} \left( \frac{1+C v}{1+C_{0}v} \right)^{\frac{2}{3}}-1\right)\right.\\ &&\left. -\frac{C_{\mathrm{H}^{+}}+C_{\text{Na}^{+}}+C_{\text{OH}^{-}}+C_{\text{Cl}^{-}}}{C}\right], \end{array} $$

where g = Gv/(kBT) is the dimensionless shear modulus. Using Eqs. B-10 and B-14, we present Eq. B-19 in the form

$$ \begin{array}{@{}rcl@{}} \mu &=& \mu^{0}+k_{\mathrm{B}}T \left[ \ln \frac{C v}{1+C v}+\frac{1}{1+C v} +\frac{\chi}{(1+C v)^{2}} +\frac{g}{1+C v} \left( \frac{1}{V} \left( \frac{1+C v}{1+C_{0}v} \right)^{\frac{2}{3}}-1\right)\right.\\ &&\left.-\left( 2X -\frac{C_{\mathrm{b}^{-}}}{C}\right)\right]. \end{array} $$

Applying Eqs. 35, and B-11, we transform Eq. 55 for the chemical potential of water molecules in the bath as follows:

$$ \bar{\mu} = \mu^{0}-\frac{2k_{\mathrm{B}}T}{\kappa} \left( 10^{-\text{pH}}+\theta\right). $$

Substitution of Eqs. B-20 and B-21 into Eq. 56 implies that

$$ \begin{array}{@{}rcl@{}} && \ln \frac{C v}{1+C v}+\frac{1}{1+C v} +\frac{\chi}{(1+C v)^{2}} +\frac{g}{1+C v} \left( \frac{1}{V} \left( \frac{1+C v}{1+C_{0}v} \right)^{\frac{2}{3}}-1\right)\\ && +\frac{C_{\text{b}^{-}}}{C} -2 \left[ X-\frac{1}{\kappa}\left( 10^{-\text{pH}}+\theta \right)\right]=0. \end{array} $$

It follows from Eq. B-13 that

$$ \left( X-\frac{1}{\kappa}\left( 10^{-\text{pH}}+\theta\right)\right) \left( X+\frac{1}{\kappa}\left( 10^{-\text{pH}}+\theta\right) \right) =\frac{C_{\text{b}^{-}}}{C}X , $$

which implies that

$$ X-\frac{1}{\kappa}\left( 10^{-\text{pH}}+\theta\right) =\frac{C_{\text{b}^{-}}}{C}X \left( X+\frac{1}{\kappa}\left( 10^{-\text{pH}}+\theta\right) \right)^{-1}. $$

Combining Eqs. B-22 and B-23 and using Eq. 59, we arrive at Eqs. 57 and 58.

Appendix : 3

To calculate the Young modulus of a gel in the fully swollen state, we analyze rapid uniaxial tension of a sample, where the strain rate under loading exceeds strongly the rates of diffusion for water molecules and mobile ions. The deformation gradient F for macrodeformation reads

$$ \mathbf{F}=\mathbf{F}_{2}\cdot{ \mathbf{F}}_{1}, $$


$$ \mathbf{F}_{1}=(1+Q)^{\frac{1}{3}}\mathbf{I} $$

describes isotropic extension of the sample under swelling, and

$$ \mathbf{F}_{2}=\lambda \boldsymbol{i}_{1}\otimes\boldsymbol{i}_{1} +\lambda^{-\frac{1}{2}}(\boldsymbol{i}_{2}\otimes\boldsymbol{i}_{2}+\boldsymbol{i}_{3}\otimes\boldsymbol{i}_{3}) $$

characterizes uniaxial tension along the axis i1 with elongation ratio λ (i1,i2,i3 denote unit vectors of a Cartesian frame, and the coefficient \(\lambda ^{-\frac {1}{2}}\) is calculated from Eq. 8).

Inserting expressions Eq. C-2 and C-3 into Eq. C-1 and using Eq. 12, we find that

$$ \mathbf{B}_{\mathrm{e}}=\left( \frac{1+Q}{1+Q_{0}}\right)^{\frac{2}{3}} \left[ \lambda^{2} \boldsymbol{i}_{1}\otimes\boldsymbol{i}_{1} +\lambda^{-1}(\boldsymbol{i}_{2}\otimes\boldsymbol{i}_{2}+\boldsymbol{i}_{3}\otimes\boldsymbol{i}_{3})\right]. $$

Equations (C-4) and (54) result in the following expression for the Cauchy stress tensor:

$$ \mathbf{T}= T_{1} \boldsymbol{i}_{1}\otimes\boldsymbol{i}_{1} +T_{2}(\boldsymbol{i}_{2}\otimes\boldsymbol{i}_{2}+\boldsymbol{i}_{3}\otimes\boldsymbol{i}_{3}), $$


$$ T_{1}=-{\Pi}+\frac{G}{1+Q}\left[ \frac{1}{V} \left( \frac{1+Q}{1+Q_{0}}\right)^{\frac{2}{3}}\lambda^{2} -1 \right], \quad T_{2}=-{\Pi}+\frac{G}{1+Q}\left[ \frac{1}{V} \left( \frac{1+Q}{1+Q_{0}}\right)^{\frac{2}{3}}\frac{1}{\lambda}-1 \right] $$


$$ V=1-\frac{1}{K} \left[ \left( \frac{1+Q}{1+Q_{0}}\right)^{\frac{2}{3}}\left( \lambda^{2}+\frac{2}{\lambda}\right) -3 \right]. $$

Bearing in mind that the lateral surface of a sample is traction-free, we set T2 = 0. Excluding π from this equality and Eq. C-5, we conclude that

$$ T_{1}=\frac{G}{(1+Q)V}\left( \frac{1+Q}{1+Q_{0}}\right)^{\frac{2}{3}} \left( \lambda^{2} -\frac{1}{\lambda} \right). $$

The engineering tensile stress σ = T1/λ is given by

$$ \sigma=\frac{G}{(1+Q)V}\left( \frac{1+Q}{1+Q_{0}}\right)^{\frac{2}{3}} \left( \lambda -\frac{1}{\lambda^{2}} \right). $$

At small strains, when λ = 1 + 𝜖 and 𝜖 ≪ 1, Eq. C-8 is simplified,

$$ \sigma=\alpha E \epsilon, $$

where E = 3G stands for the Young modulus of the polymer network, and

$$ \alpha=\frac{1}{(1+Q)^{\frac{1}{3}}(1+Q_{0})^{\frac{2}{3}}} \left[1-\frac{3}{K}\left( \left( \frac{1+Q}{1+Q_{0}}\right)^{\frac{2}{3}}-1\right)\right]^{-1}. $$

When results in tensile tests are compared on fully swollen samples (with equilibrium degree of swelling Q) and as-prepared samples (with degree of swelling Qprep), it follows from Eqs. C-9 and C-10 that the Young moduli measured in these tests, E and Eprep, are connected by Eq. 62.

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Drozdov, A. Equilibrium swelling of multi-stimuli-responsive superabsorbent hydrogels. Mech Soft Mater 3, 1 (2021). https://doi.org/10.1007/s42558-020-00032-5

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  • Superabsorbent hydrogels
  • Stimuli-responsive polymers
  • Equilibrium swelling
  • Modeling