Mathematical Modeling of Hydrogels Swelling Based on the Finite Element Method

DOI: 10.4236/am.2013.48A022   PDF   HTML   XML   7,258 Downloads   10,414 Views   Citations


In recent years, hydrogels have been introduced as new materials suitable for applications in areas such as biomedical engineering, agriculture, etc. The rate and degree of hydrogel swelling are important parameters that control the diffusion of drugs or solvents inside a polymer network. Therefore, the description of the dynamic swelling process of the hydrogels is very important in applications that require precise control of the absorption of solvents inside the hydrogel structure. To date, most of the numerical models developed for describing the swelling process are based in the finite difference methods. Even though numerical models supported in finite differences can be easily implemented, their use is limited to samples with very simple shapes. In this paper, a new model based on the finite element method is proposed. The diffusion equation is solved in a time-deformable grid. An original procedure is proposed to numerically solve the non-linear algebraic equation system that permits computing a new grid for each time-step. Hydrogel samples of different shapes were prepared in order to conduct experimental tests to validate the numerical proposed model. Numerical results show that the new model is able to describe the mass and shape changes in the hydrogel samples in time. An application of the numerical model to determine the relation between diffusion coefficients and density in Poly-acrylamide samples allows verifying the versatility of the model.

Share and Cite:

A. Blanco, G. González, E. Casanova, M. Pirela and A. Briceño, "Mathematical Modeling of Hydrogels Swelling Based on the Finite Element Method," Applied Mathematics, Vol. 4 No. 8A, 2013, pp. 161-170. doi: 10.4236/am.2013.48A022.

Conflicts of Interest

The authors declare no conflicts of interest.


[1] [1] F. Ganji, S. Vasheghani-Farahani E. and Vasheghani-Fa rahani, “Theoretical Description of Hydrogel Swelling: A Review,” Iranian Polymer Journal, Vol. 19, No. 5, 2010, pp. 375-398.
[2] A. Singh, P. K. Sharma, V. K. Garg and G. Garg, “Hy drogels: A Review,” International Journal of Pharma ceutical Sciences Review and Research, Vol. 4, No. 2, 2010, pp. 97-105.
[3] T. L. Porter, R. Stewart, J. Reed and K. Morton, “Models of Hydrogel Swelling with Applications to Hydration Sensing,” Sensors, Vol. 7, No. 9, 2007, pp. 1980-1991. doi:10.3390/s7091980
[4] P. J. Flory and J. Rehner, “Statistical Mechanics of Cross Linked Polymer Networks I,” Journal of Chemical Physics, Vol. 11, No. 11, 1943, pp. 512-520. doi:10.1063/1.1723791
[5] P. J. Flory and J. Rehner, “Statistical Mechanics of Cross Linked Polymer Networks II,” Journal of Chemical Phy sics, Vol. 11, No. 11, 1943, pp. 521-526. doi:10.1063/1.1723792
[6] C. S. Brazel and N. A. Peppas, “Dimensionless Analysis of Swelling of Hydrophilic Glassy Polymers with Subse quent Drug Release from Relaxing Structures,” Biomaterials, Vol. 20, No. 8, 1999, pp. 721-732. doi:10.1016/S0142-9612(98)00215-4
[7] P. A. Sandoval, Y. Baena, M. Aragón, J. Rosas and L. Ponce, “Overall Mechanisms That Rule the Active Phar maceutical Ingredient’s Delivery Process from Hydro philic Matrices Elaborated with Ether Cellulose,” Revista Colombiana de Ciencias Químico-Farmacéuticas, Vol. 37, No. 2, 2008, pp. 105-121.
[8] N. A. Peppas and R. W. Korsmeyer, “Dynamically Swell ing Hydrogels in Controlled Release Applications,” In: N. A. Peppas, Ed., Hydrogels in Medicine and Pharmacy, CRC Press, Boca Raton, 1987.
[9] D. De Kee, Q. Liu and J. Hinestroza, “Viscoelastic (Non Fickian) Diffusion,” The Canadian Journal of Chemical Engineering, Vol. 83, No. 6, 2005, pp. 913-929. doi:10.1002/cjce.5450830601
[10] N. A. Peppas and J. J. Sahlin, “A Simple Equation for the Description of Solute Release. III. Coupling of Diffusion and Relaxation,” International Journal of Pharmaceutics, Vol. 57, No. 2, 1989, pp. 169-172. doi:10.1016/0378-5173(89)90306-2
[11] J. Zhang, X. Zhao, Z. Suo and H. Jiang, “A Finite Element Method for Transient Analysis of Concurrent Large Deformation and Mass Transport in Gels,” Journal of Applied Physics, Vol. 105, 2009, Article ID: 092532.
[12] M. Kang and R. Huang, “A Variational Approach and Finite Element Implementation for Swelling of Polymeric Hydrogels under Geometric Constraints,” Journal of Ap plied Mechanics, Vol. 77, 2010, Article ID: 061004, 12 Pages.
[13] W. Hong, X. Zhao, J. Zhou and Z. Suo, “A Theory of Coupled Diffusion and Large Deformation in Polymeric Gels,” Journal of the Mechanics and Physics of Solids, Vol. 56, No. 5, 2008, pp. 1779-1793 doi:10.1016/j.jmps.2007.11.010
[14] J. Crank, “The Mathematic of Diffusion,” 2nd Edition, Clarendon Press, Oxford, 1979.
[15] G. Rossi and K. A. Mazich, “Kinetics of Swelling for a Cross-Linked Elastomer or a Gel in the Presence of a Good Solvent,” Physical Review A, Vol. 44, No. 8, 1991, pp. R4793-R4796. doi:10.1103/PhysRevA.44.R4793
[16] G. Rossi and K. A. Mazich, “Macroscopic Description of the Kinetics of Swelling for a Cross-Linked Elastomer or a Gel,” Physical Review E, Vol. 48, No. 2, 1993, pp. 1182-1191. doi:10.1103/PhysRevE.48.1182
[17] K. A. Mazich, G. Rossi and C. A. Smith, “Kinetics of So lvent Diffusion and Swelling in a Model Elastomeric System,” Macromolecules, Vol. 25, No. 25, 1992, pp. 6929-6933. doi:10.1021/ma00051a032
[18] J. Siepmann, K. Podual, M. Sriwongjanya, N. A. Peppas and R. Bodmeier, “A New Model describing the Swelling and Drug Release Kinetics from Hydroxypropyl Methyl cellulose Tablets,” Journal of Pharmaceutical Sciences, Vol. 88, No. 1, 1999, pp. 65-72. doi:10.1021/js9802291
[19] J. Siepmann, H. Kranz, R. Bodmeier and N. A. Peppas, “HPMC Matrices for Controlled Drug Delivery: A New Model Combining Diffusion, Swelling and Dissolution Mechanisms and Predicting the Release Kinetics,” Phar maceutical Research, Vol. 16, No. 11, 1999, pp. 1748-1756. doi:10.1023/A:1018914301328
[20] H. Fujita, “Diffusion in Polymer-Diluent Systems,” For tschritte Der Hochpolymeren-Forschung, Advances in Polymer Science, Vol. 3, No. 1, 1961, pp. 1-47.
[21] E. C. Achilleos, R. K. Prudhomme, K. N. Christodoulou, K. R. Gee and I. G. Kevrekidis, “Dynamic Deformation Visualization in Swelling of Polymer Gels,” Chemical Engineering Science, Vol. 55, No. 17, 2000, pp. 3335-3340. doi:10.1016/S0009-2509(00)00002-6
[22] O. C. Zienkiewicz, R. L. Taylor and J. Z. Zhu, “The Fi nite Element Method: Its Basis and Fundamentals,” 6th Edition, Elsevier, Oxford, 2005.

comments powered by Disqus

Copyright © 2020 by authors and Scientific Research Publishing Inc.

Creative Commons License

This work and the related PDF file are licensed under a Creative Commons Attribution 4.0 International License.