Swelling-induced finite bending of functionally graded pH-responsive hydrogels: a semi-analytical method

doi:10.1007/s10483-019-2478-6

摘要/Abstract

摘要： Recently, pH-sensitive hydrogels have been utilized in the diverse applications including sensors, switches, and actuators. In order to have continuous stress and deformation fields, a new semi-analytical approach is developed to predict the swelling induced finite bending for a functionally graded (FG) layer composed of a pH-sensitive hydrogel, in which the cross-link density is continuously distributed along the thickness direction under the plane strain condition. Without considering the intermediary virtual reference, the initial state is mapped into the deformed configuration in a circular shape by utilizing a total deformation gradient tensor stemming from the inhomogeneous swelling of an FG layer in response to the variation of the pH value of the solvent. To enlighten the capability of the presented analytical method, the finite element method (FEM) is used to verify the accuracy of the analytical results in some case studies. The perfect agreement confirms the accuracy of the presented method. Due to the applicability of FG pH-sensitive hydrogels, some design factors such as the semi-angle, the bending curvature, the aspect ratio, and the distributions of deformation and stress fields are studied. Furthermore, the tangential free-stress axes are illustrated in deformed configuration.

关键词: convex metric spaces, (sub) compatible mapping, set-valued generalized nonexpansive mapping, common fixed points, pH-sensitive hydrogel, finite element method (FEM), functionally graded (FG) layer, finite bending, semianalytical solution

Abstract: Recently, pH-sensitive hydrogels have been utilized in the diverse applications including sensors, switches, and actuators. In order to have continuous stress and deformation fields, a new semi-analytical approach is developed to predict the swelling induced finite bending for a functionally graded (FG) layer composed of a pH-sensitive hydrogel, in which the cross-link density is continuously distributed along the thickness direction under the plane strain condition. Without considering the intermediary virtual reference, the initial state is mapped into the deformed configuration in a circular shape by utilizing a total deformation gradient tensor stemming from the inhomogeneous swelling of an FG layer in response to the variation of the pH value of the solvent. To enlighten the capability of the presented analytical method, the finite element method (FEM) is used to verify the accuracy of the analytical results in some case studies. The perfect agreement confirms the accuracy of the presented method. Due to the applicability of FG pH-sensitive hydrogels, some design factors such as the semi-angle, the bending curvature, the aspect ratio, and the distributions of deformation and stress fields are studied. Furthermore, the tangential free-stress axes are illustrated in deformed configuration.

Key words: convex metric spaces, (sub) compatible mapping, set-valued generalized nonexpansive mapping, common fixed points, semianalytical solution, finite element method (FEM), pH-sensitive hydrogel, finite bending, functionally graded (FG) layer

中图分类号:

65M06
65M12

M. SHOJAEIFARD, M. R. BAYAT, M. BAGHANI. Swelling-induced finite bending of functionally graded pH-responsive hydrogels: a semi-analytical method[J]. Applied Mathematics and Mechanics (English Edition), 2019, 40(5): 679-694.

参考文献

[1] MARCOMBE, R., CAI, S., HONG, W., ZHAO, X., LAPUSTA, Y., and SUO, Z. A theory of constrained swelling of a pH-sensitive hydrogel. Soft Matter, 6, 784-793(2010)
[2] DROZDOV, A. D., DECLAVILLE-CHRISTIANSEN, J., and SANPOREAN, C. G. Inhomogeneous swelling of pH-responsive gels. International Journal of Solids and Structures, 87, 11-25(2016)
[3] DROZDOV, A. and DECLAVILLE-CHRISTIANSEN, J. Modeling the effects of pH and ionic strength on swelling of anionic polyelectrolyte gels. Modelling and Simulation in Materials Science and Engineering, 23, 055005(2015)
[4] DROZDOV, A. and DECLAVILLE-CHRISTIANSEN, J. Swelling of pH-sensitive hydrogels. Physical Review E, 91, 022305(2015)
[5] LI, H., GO, G., KO, S. Y., PARK, J. O., and PARK, S. Magnetic actuated pH-responsive hydrogelbased soft micro-robot for targeted drug delivery. Smart Materials and Structures, 25, 027001(2016)
[6] LI, P., KIM, N. H., and LEE, J. H. Swelling behavior of polyacrylamide/laponite clay nanocomposite hydrogels:pH-sensitive property. Composites Part B:Engineering, 40, 275-283(2009)
[7] MAZAHERI, H., BAGHANI, M., NAGHDABADI, R., and SOHRABPOUR, S. Coupling behavior of the pH/temperature sensitive hydrogels for the inhomogeneous and homogeneous swelling. Smart Materials and Structures, 25, 085034(2016)
[8] MAZAHERI, H., BAGHANI, M., and NAGHDABADI, R. Inhomogeneous and homogeneous swelling behavior of temperature-sensitive poly-(N-isopropylacrylamide) hydrogels. Journal of Intelligent Material Systems and Structures, 27, 324-336(2016)
[9] CHESTER, S. A. and ANAND, L. A thermo-mechanically coupled theory for fluid permeation in elastomeric materials:application to thermally responsive gels. Journal of the Mechanics and Physics of Solids, 59, 1978-2006(2011)
[10] MORIMOTO, T. and ASHIDA, F. Temperature-responsive bending of a bilayer gel. International Journal of Solids and Structures, 56, 20-28(2015)
[11] DEHGHANY, M., ZHANG, H., NAGHDABADI, R., and HU, Y. A thermodynamicallyconsistent large deformation theory coupling photochemical reaction and electrochemistry for light-responsive gels. Journal of the Mechanics and Physics of Solids, 116, 239-266(2018)
[12] TOH, W., NG, T. Y., HU, J., and LIU, Z. Mechanics of inhomogeneous large deformation of photo-thermal sensitive hydrogels. International Journal of Solids and Structures, 51, 4440-4451(2014)
[13] ATTARAN, A., BRUMMUND, J., and WALLMERSPERGER, T. Modeling and simulation of the bending behavior of electrically-stimulated cantilevered hydrogels. Smart Materials and Structures, 24, 035021(2015)
[14] LIU, T. Y., HU, S. H., LIU, T. Y., LIU, D. M., and CHEN, S. Y. Magnetic-sensitive behavior of intelligent ferrogels for controlled release of drug. Langmuir, 22, 5974-5978(2006)
[15] IONOV, L. Biomimetic hydrogel-based actuating systems. Advanced Functional Materials, 23, 4555-4570(2013)
[16] 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, 588-590(2000)
[17] WONG, A. P., PEREZ-CASTILLEJOS, R., LOVE, J. C., and WHITESIDES, G. M. Partitioning microfluidic channels with hydrogel to construct tunable 3-D cellular microenvironments. Biomaterials, 29, 1853-1861(2008)
[18] NIKOLOV, S., FERNANDEZ-NIEVES, A., and ALEXEEV, A. Mesoscale modeling of microgel mechanics and kinetics through the swelling transition. Applied Mathematics and Mechanics (English Edition), 39(1), 47-62(2018) http://doi.org/10.1007/s10483-018-2259-6
[19] SU, H., YAN, H., and JIN, B. Finite element method for coupled diffusion-deformation theory in polymeric gel based on slip-link model. Applied Mathematics and Mechanics (English Edition), 39(4), 581-596(2018) http://doi.org/10.1007/s10483-018-2315-7
[20] ZHANG, H. Strain-stress relation in macromolecular microsphere composite hydrogel. Applied Mathematics and Mechanics (English Edition), 37(11), 1539-1550(2016) http://doi.org/10.1007/s10483-016-2110-9
[21] FLORY, P. J. Thermodynamics of high polymer solutions. The Journal of Chemical Physics, 10, 51-61(1942)
[22] CAI, S. and SUO, Z. Mechanics and chemical thermodynamics of phase transition in temperaturesensitive hydrogels. Journal of the Mechanics and Physics of Solids, 59, 2259-2278(2011)
[23] 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, 1779-1793(2008)
[24] LUCANTONIO, A., NARDINOCCHI, P., and TERESI, L. Transient analysis of swelling-induced large deformations in polymer gels. Journal of the Mechanics and Physics of Solids, 61, 205-218(2013)
[25] ZHANG, Y., LIU, Z., SWADDIWUDHIPONG, S., MIAO, H., DING, Z., and YAND, Z. pHsensitive hydrogel for micro-fluidic valve. Journal of Functional Biomaterials, 3, 464-479(2012)
[26] ZALACHAS, N., CAI, S., SUO, Z., and LAPUSTA, Y. Crease in a ring of a pH-sensitive hydrogel swelling under constraint. International Journal of Solids and Structures, 50, 920-927(2013)
[27] HE, T., LI, M., and ZHOU, J. Modeling deformation and contacts of pH sensitive hydrogels for microfluidic flow control. Soft Matter, 8, 3083-3089(2012)
[28] ARBABI, N., BAGHANI, M., ABDOLAHI, J., MAZAHERI, H., and MASHHADI, M. M. Finite bending of bilayer pH-responsive hydrogels:a novel analytic method and finite element analysis. Composites Part B:Engineering, 110, 116-123(2017)
[29] ABDOLAHI, J., BAGHANI, M., ARBABI, N., and MAZAHERI, H. Analytical and numerical analysis of swelling-induced large bending of thermally-activated hydrogel bilayers. International Journal of Solids and Structures, 99, 1-11(2016)
[30] HU, Z., ZHANG, X., and LI, Y. Synthesis and application of modulated polymer gels. Science, 269, 525-527(1995)
[31] TIMOSHENKO, S. Analysis of bi-metal thermostats. Journal of the Optical Society of America, 11, 233-255(1925)
[32] ABDOLAHI, J., BAGHANI, M., ARBABI, N., and MAZAHERI, H. Finite bending of a temperature-sensitive hydrogel tri-layer:an analytical and finite element analysis. Composite Structures, 164, 219-228(2017)
[33] ZHANG, J., WU, J., SUN, J., and ZHOU, Q. Temperature-sensitive bending of bigel strip bonded by macroscopic molecular recognition. Soft Matter, 8, 5750-5752(2012)
[34] GUVENDIREN, M., BURDICK, J. A., and YANG, S. Kinetic study of swelling-induced surface pattern formation and ordering in hydrogel films with depth-wise crosslinking gradient. Soft Matter, 6, 2044-2049(2010)
[35] GUVENDIREN, M., BURDICK, J. A., and YANG, S. Solvent induced transition from wrinkles to creases in thin film gels with depth-wise crosslinking gradients. Soft Matter, 6, 5795-5801(2010)
[36] GUVENDIREN, M., YANG, S., and BURDICK, J. A. Swelling-induced surface patterns in hydrogels with gradient crosslinking density. Advanced Functional Materials, 19, 3038-3045(2009)
[37] WU, Z., BOUKLAS, N., and HUANG, R. Swell-induced surface instability of hydrogel layers with material properties varying in thickness direction. International Journal of Solids and Structures, 50, 578-587(2013)
[38] WU, Z., BOUKLAS, N., LIU, Y., and HUANG, R. Onset of swell-induced surface instability of hydrogel layers with depth-wise graded material properties. Mechanics of Materials, 105, 138-147(2017)
[39] WU, Z., MENG, J., LIU, Y., LI, H., and HUANG, R. A state space method for surface instability of elastic layers with material properties varying in thickness direction. Journal of Applied Mechanics, 81, 081003(2014)
[40] ROCCABIANCA, S., GEI, M., and BIGONI, D. Plane strain bifurcations of elastic layered structures subject to finite bending:theory versus experiments. IMA Journal of Applied Mathematics, 75, 525-548(2010)
[41] LUCANTONIO, A., NARDINOCCHI, P., and PEZZULLA, M. Swelling-induced and controlled curving in layered gel beams. Proceedings of the Royal Society of London A:Mathematical, Physical and Engineering Sciences, 470, 20140467(2014)
[42] KIERZENKA, J. and SHAMPINE, L. F. A BVP solver that controls residual and error. Journal of Numerical Analysis Industrial and Applied Mathematics, 3, 27-41(2008)
[43] BYRD, R. H., GILBERT, J. C., and NOCEDAL, J. A trust region method based on interior point techniques for nonlinear programming. Mathematical Programming, 89, 149-185(2000)
[44] ALMASI, A., BAGHANI, M., MOALLEMI, A., BANIASSADI, M., and FARAJI, G. Investigation on thermal stresses in FGM hyperelastic thick-walled cylinders. Journal of Thermal Stresses, 41, 204-221(2018)