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Finite element method for coupled diffusion-deformation theory in polymeric gel based on slip-link model

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Abstract

A polymeric gel is an aggregate of polymers and solvent molecules, which can retain its shape after a large deformation. The deformation behavior of polymeric gels was often described based on the Flory-Rehner free energy function without considering the influence of chain entanglements on the mechanical behavior of gels. In this paper, a new hybrid free energy function for gels is formulated by combining the Edwards-Vilgis slip-link model and the Flory-Huggins mixing model to quantify the time-dependent concurrent process of large deformation and mass transport. The finite element method is developed to analyze examples of swelling-induced deformation. Simulation results are compared with available experimental data and show good agreement. The influence of entanglements on the time-dependent deformation behavior of gels is also demonstrated. The study of large deformation kinetics of polymeric gel is useful for diverse applications.

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References

  1. Wichterle, O. and Lim, D. Hydrophilic gels for biological use. nature, 185(4706), 117–118 (1960)

    Article  Google Scholar 

  2. Peppas, N. A., Bures, P., Leobandung, W., and Ichikawa, H. Hydrogels in pharmaceutical formu-lations. European Journal of Pharmaceutics and Biopharmaceutics, 50(1), 27–46 (2000)

    Article  Google Scholar 

  3. Luo, Y. and Shoichet, M. S. A photolabile hydrogel for guided three-dimensional cell growth and migration. Nature Materials, 3(4), 249–254 (2004)

    Article  Google Scholar 

  4. Beebe, D. J., Moore, J. S., Bauer, J. M., Yu, Q., Liu, R. H., Devadoss, C., and Jo, B. H. Functional hydrogel structures for autonomous flow control inside microfluidic channels. nature, 404(6778), 588–590 (2000)

    Article  Google Scholar 

  5. Dong, L., Agarwal, A. K., Beebe, D. J., and Jiang, H. Adaptive liquid microlenses activated by stimuli-responsive hydrogels. nature, 442(7102), 551–554 (2006)

    Article  Google Scholar 

  6. Gibbs, J. W. and Bumstead, H. A. The Scientific Papers of J. Willard Gibbs, Longmans, London, 184–184 (1906)

    Google Scholar 

  7. Biot, M. A. General theory of three-dimensional consolidation. Journal of Applied Physics, 12(2), 155–164 (1941)

    Article  MATH  Google Scholar 

  8. Flory, P. J. and Rehner, J. Statistical mechanics of cross-linked polymer networks II: swelling. The Journal of Chemical Physics, 11(11), 521–526 (1943)

    Article  Google Scholar 

  9. Baek, S. and Pence, T. Inhomogeneous deformation of elastomer gels in equilibrium under satu-rated and unsaturated conditions. Journal of the Mechanics and Physics of Solids, 59(3), 561–582 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  10. Hong, W., Zhao, X., Zhou, J., and Suo, Z. A theory of coupled diffusion and large deformation in polymeric gels. Journal of the Mechanics and Physics of Solids, 56(5), 1779–1793 (2008)

    Article  MATH  Google Scholar 

  11. Zhang, H. Strain-stress relation in macromolecular microsphere composite hydro-gel. Applied Mathematics and Mechanics (English Edition), 37(11), 1539–1550 (2016) https://doi.org/10.1007/s10483-016-2110-9

    Article  MathSciNet  Google Scholar 

  12. Cai, S. and Suo, Z. Mechanics and chemical thermodynamics of phase transition in temperature-sensitive hydrogels. Journal of the Mechanics and Physics of Solids, 59(11), 2259–2278 (2011)

    Article  MATH  Google Scholar 

  13. Hong, W., Liu, Z., and Suo, Z. Inhomogeneous swelling of a gel in equilibrium with a solvent and mechanical load. International Journal of Solids and Structures, 46(17), 3282–3289 (2009)

    Article  MATH  Google Scholar 

  14. Hong, W., Zhao, X., and Suo, Z. Large deformation and electrochemistry of polyelectrolyte gels. Journal of the Mechanics and Physics of Solids, 58(4), 558–577 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  15. Liu, Z. S., Swaddiwudhipong, S., Cui, F. S., Hong, W., Suo, Z., and Zhang, Y. W. Analytical solutions of polymeric gel structures under buckling and wrinkle. International Journal of Applied Mechanics, 3(2), 235–257 (2012)

    Article  Google Scholar 

  16. Wineman, A. and Rajagopal, K. R. Shear induced redistribution of fluid within a uniformly swollen nonlinear elastic cylinder. International Journal of Engineering Science, 30(11), 1583–1595 (1992)

    Article  MATH  Google Scholar 

  17. Chester, S. A. and Anand, L. A coupled theory of fluid permeation and large deformations for elastomeric materials. Journal of the Mechanics and Physics of Solids, 58(11), 1879–1906 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  18. Mergell, B. and Everaers, R. Tube models for rubber-elastic systems. Macromolecules, 34(16), 5675–5686 (2001)

    Article  Google Scholar 

  19. Flory, P. J. Theory of elasticity of polymer networks-effect of local constraints on junctions. Journal of Chemical Physics, 66(12), 5720–5729 (1977)

    Article  Google Scholar 

  20. Ronca, G. and Allegra, G. Approach to rubber elasticity with internal constraints. Journal of Chemical Physics, 63(11), 4990–4997 (1975)

    Article  Google Scholar 

  21. Edwards, S. F. Theory of rubber elasticity. British Polymer Journal, 9(2), 140–143 (1977)

    Article  Google Scholar 

  22. Kloczkowski, A., Mark, J. E., and Erman, B. A diffused-constraint theory for the elasticity of amorphous polymer networks I: fundamentals and stress-strain isotherms in elongation. Macro-molecules, 28(14), 5089–5096 (1995)

    Article  Google Scholar 

  23. Edwards, S. and Vilgis, T. The effect of entanglements in rubber elasticity. Polymer, 27(4), 483–492 (1986)

    Article  Google Scholar 

  24. Higgs, P. G. and Gaylord, R. J. Slip-links, hoops and tubes: tests of entanglement models of rubber elasticity. Polymer, 31(1), 70–74 (1990)

    Article  Google Scholar 

  25. Urayama, K. Network topology-mechanical properties relationships of model elastomers. Polymer Journal, 40(8), 669–678 (2008)

    Article  Google Scholar 

  26. Meissner, B. and Matejka, L. Comparison of recent rubber-elasticity theories with biaxial stress-strain data: the slip-link theory of Edwards and Vilgis. Polymer, 43(13), 3803–3809 (2002)

    Article  Google Scholar 

  27. Yan, H. X. and Jin, B. Influence of microstructural parameters on mechanical behavior of polymer gels. International Journal of Solids and Structures, 49(3), 436–444 (2012)

    Article  Google Scholar 

  28. Yan, H. X. and Jin, B. Influence of environmental solution pH and microstructural parameters on mechanical behavior of amphoteric pH-sensitive hydrogels. European Physical Journal E, 35(5), 36–46 (2012)

    Article  Google Scholar 

  29. Yan, H. X. and Jin, B. Equilibrium swelling of a polyampholytic pH-sensitive hydrogel. European Physical Journal E, 36(3), 27–33 (2013)

    Article  MathSciNet  Google Scholar 

  30. Yan, H. X., Jin, B., Gao, S. H., and Chen, L. W. Equilibrium swelling and electrochemistry of polyampholytic pH-sensitive hydrogel. International Journal of Solids and Structures, 51(23/24), 4149–4156 (2014)

    Article  Google Scholar 

  31. Chester, S. A., Di Leo, C. V., and Anand, L. A finite element implementation of a coupled diffusion-deformation theory for elastomeric gels. International Journal of Solids and Structures, 52, 1–18 (2015)

    Article  Google Scholar 

  32. Flory, P. J. Thermodynamics of high polymer solutions. The Journal of Chemical Physics, 10(1), 51–61 (1942)

    Article  Google Scholar 

  33. Huggins, M. L. Solutions of long chain compounds. The Journal of Chemical Physics, 9(5), 440–440 (1941)

    Article  Google Scholar 

  34. Zienkiewicz, O. C., Taylor, R. L., and Fox, D. The Finite Element Method for Solid and Structural Mechanics, Butterworth-Heinemann Elsevier Ltd., Oxford, 17–19 (2014)

    Google Scholar 

  35. Zienkiewicz, O. C., Taylor, R. L., and Zhu, J. Z. The Finite Element Method: Its Basis and Fundamentals, Butterworth-Heinemann Elsevier Ltd., Oxford, 64–66 (2013)

    Google Scholar 

  36. Yoon, J., Cai, S., Suo, Z., and Hayward, R. C. Poroelastic swelling kinetics of thin hydrogel layers: comparison of theory and experiment. Soft Matter, 6(23), 6004–6012 (2010)

    Article  Google Scholar 

  37. Bouklas, N. and Huang, R. Swelling kinetics of polymer gels: comparison of linear and nonlinear theories. Soft Matter, 8(31), 8194–8203 (2012)

    Article  Google Scholar 

  38. Achilleos, E. C., Prud’homme, R. K., Christodoulou, K. N., Gee, K. R., and Kevrekidis, I. G. Dynamic deformation visualization in swelling of polymer gels. Chemical Engineering Science, 55(17), 3335–3340 (2000)

    Article  Google Scholar 

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Authors and Affiliations

  1. School of Aerospace Engineering and Applied Mechanics, Tongji University, Shanghai, 200092, China

    Hengdi Su & Bo Jin

  2. School of Mechanical and Electrical Engineering, Sanming University, Sanming, Fujian Province, 365004, China

    Huixian Yan

Authors
  1. Hengdi Su
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  2. Huixian Yan
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  3. Bo Jin
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Corresponding author

Correspondence to Bo Jin.

Additional information

Project supported by the National Natural Science Foundation of China (Nos. 11272237 and 11502131), the Natural Science Foundation of Fujian Province (No. 2016J05019), and the Foundation of the Higher Education Institutions of Fujian Education Department for Distinguished Young Scholar (No. [2016] 23)

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Su, H., Yan, H. & Jin, B. Finite element method for coupled diffusion-deformation theory in polymeric gel based on slip-link model. Appl. Math. Mech.-Engl. Ed. 39, 581–596 (2018). https://doi.org/10.1007/s10483-018-2315-7

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  • DOI: https://doi.org/10.1007/s10483-018-2315-7

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