Constitutive modeling of solvent plasticization and physical aging in glassy polymers

doi:10.1007/s10483-026-3355-9

摘要/Abstract

Abstract:

Glassy polymers are widely used in biomedical applications in a solvent environment, yet their long-term performance is governed by the competing effects of physical aging and solvent-induced plasticization. Here, we develop a constitutive model that explicitly couples the solvent concentration, structural relaxation, and mechanical response. This framework is built on a multiplicative decomposition of deformation and an Eyring-type flow rule, with structural evolution described by an effective temperature. A generalized shift factor is introduced to quantify how the solvent concentration and effective temperature jointly affect the relaxation time, thereby integrating physical aging and plasticization. The model is subsequently applied to methacrylate (MA)-based copolymer networks immersed in phosphate-buffered saline for up to nine months. Simulations accurately capture key experimental features, including the strong softening of highly swellable networks, the partial recovery due to aging, and the mitigating role of hydrophobic crosslinking in reducing solvent uptake. While the current single-mode description cannot reproduce the full relaxation spectrum, it establishes an efficient framework for predicting the long-term mechanical performance under coupled environmental and mechanical loading. This study provides a constitutive description of solvent-swollen glassy polymers, offering mechanistic insight into the interplay between plasticization and aging. Beyond biomedical MA networks, this framework establishes a foundation for predicting the long-term performance of polymer glasses under coupled aqueous environmental and mechanical loading.

Key words: glassy polymer, swelling, physical aging, constitutive model

中图分类号:

. [J]. Applied Mathematics and Mechanics (English Edition), 2026, 47(3): 535-554.

Xu CAO, Kerong WU, Ji LIN, Rui XIAO. Constitutive modeling of solvent plasticization and physical aging in glassy polymers[J]. Applied Mathematics and Mechanics (English Edition), 2026, 47(3): 535-554.

图/表 13

Parameter	Physical meaning	Value
Parameter	Physical meaning	29MA	36MA	50MA-DDDA
$μ 0$ /MPa	Shear modulus of equilibrium part	5	2	5
$λ L$	Stretch limit of equilibrium network	6	6	6
$μ neq$ /MPa	Non-equilibrium shear modulus	270	200	200
κ/MPa	Bulk modulus	810	600	600
$τ s 0$ /s	Basic stress relaxation time	10	2.5	2.5
$τ e 0$ /s	Basic structural relaxation time	$2 × 108$	$2 × 109$	$5 × 108$
$Q s / s y / (K ⋅ (MPa) − 1)$	Activation parameter of Eyring model	100	100	100
B₀	Activation energy factor	5 500	5 500	5 500
C₀	Heat capacity factor	1 720	1 720	1 600
T₂/K	Kauzmann temperature	275	275	275
T₀/K	Reference effective temperature	313	313	313
$κ rej 0 / (K ⋅ (MPa) − 1)$	Rejuvenation rate	0.2	0.2	0.2

参考文献 39

[1]	ARALEKALLU, S., BODDULA, R., and SINGH, V. Development of glass-based microfluidic devices: a review on its fabrication and biologic applications. Materials & Design, 225, 111517 (2023)
[2]	XIA, S. and HE, L. Buckling morphology of glassy nematic films with staggered director field. Applied Mathematics and Mechanics (English Edition), 44(11), 1841–1852 (2023) https://doi.org/10.1007/s10483-023-3052-7
[3]	HALL, J. L., PEREZ, A., KYNASTON, E. L., LINDSAY, C., and KEDDIE, J. L. Effects of environmental conditions on the micro-mechanical properties of formulated waterborne coatings. Progress in Organic Coatings, 163, 106657 (2022)
[4]	XU, S., SUN, C., YUAN, W., ZHOU, J., XU, W., ZHENG, Y., YU, C., and PAN, P. Evolution of thermal behavior, mechanical properties, and microstructure in stereo complexable poly(lactic acid) during physical ageing. Polymer, 249, 124840 (2022)
[5]	EDERA, P., BANTAWA, M., AIME, S., BONNECAZE, R. T., and CLOITRE, M. Mechanical tuning of residual stress, memory, and aging in soft glassy materials. Physical Review X, 15(1), 011043 (2025)
[6]	HUTCHINSON, J. M. Physical aging of polymers. Progress in Polymer Science, 20(4), 703–760 (1995)
[7]	ODEGARD, G. M. and BANDYOPADHYAY, A. Physical aging of epoxy polymers and their composites. Journal of Polymer Science Part B-Polymer Physics, 49(24), 1695–1716 (2011)
[8]	DE NONI, L., ZHU, X., ANDENA, L., NOH, K., LI, Y., VOLLENBERG, P., and SUE, H. J. Effect of physical aging on scratch behavior of polycarbonate. Journal of Applied Polymer Science, 141(43), 56124 (2024)
[9]	MADHUSUDANAN, M. and CHOWDHURY, M. Relaxation and entropy generation in dewetting thin glassy polymer films trapped far from equilibrium. Journal of Polymer Science, 62(22), 5052–5076 (2024)
[10]	GAYOT, S. F., KLAVZER, N., GUILLET, A., BAILLY, C., GERARD, P., PARDOEN, T., and NYSTEN, B. Influence of physical ageing and fibre proximity on the local mechanical response of the Elium thermoplastic composite matrix. Composites Part A-Applied Science and Manufacturing, 181, 108141 (2024)
[11]	DUPAS, J., VERNEUIL, E., VAN LANDEGHEM, M., BRESSON, B., FORNY, L., RAMAIOLI, M., LEQUEUX, F., and TALINI, L. Glass transition accelerates the spreading of polar solvents on a soluble polymer. Physical Review Letters, 112(18), 188302 (2014)
[12]	RINO, J. J., SUYAMBULINGAM, I., DIVAKARAN, D., SUNESH, N. P., SINGH, M. K., VISHNUVARTHANAN, M., SANJAY, M. R., and SIENGCHIN, S. Facile exfoliation and physicochemical characterization of Thespesia populnea plant leaves based bioplasticizer macromolecules reinforced with polylactic acid biofilms for packaging applications. International Journal of Biological Macromolecules, 261, 129771 (2024)
[13]	BALDANZA, A., LOIANNO, V., MENSITIERI, G., and SCHERILLO, G. Modelling changes in glass transition temperature in polymer matrices exposed to low molecular weight penetrants. Philosophical Transactions of the Royal Society A, Mathematical, Physical, and Engineering Sciences, 381, 20210216 (2023)
[14]	LIAO, D., GU, T., YAN, J., YU, Z., DOU, J., LIU, J., ZHAO, F., and WANG, J. Effect of thermal aging on the microscale mechanical response behavior of glass fiber/epoxy composites. Journal of Materials Science, 59(32), 15298–15314 (2024)
[15]	GUO, R., LI, C., and XIAN, G. Water absorption and long-term thermal and mechanical properties of carbon/glass hybrid rod for bridge cable. Engineering Structures, 274, 115176 (2023)
[16]	LE GUEN-GEFFROY, A., LE GAC, P. Y., HABERT, B., and DAVIES, P. Physical ageing of epoxy in a wet environment: coupling between plasticization and physical ageing. Polymer Degradation and Stability, 168, 108947 (2019)
[17]	ADAM, G. and GIBBS, J. H. On the temperature dependence of cooperative relaxation properties in glass-forming liquids. The Journal of Chemical Physics, 43(1), 139–146 (1965)
[18]	TURNBULL, D. and COHEN, M. H. Free-volume model of the amorphous phase: glass transition. The Journal of Chemical Physics, 34(1), 120–125 (1961)
[19]	CHEN, K. and SCHWEIZER, K. S. Molecular theory of physical aging in polymer glasses. Physical Review Letters, 98(16), 167802 (2007)
[20]	CHEN, K. and SCHWEIZER, K. S. Theory of physical aging in polymer glasses. Physical Review E, 78(3), 031802 (2008)
[21]	VOYIADJIS, G. Z. and SAMADI-DOOKI, A. Constitutive modeling of large inelastic deformation of amorphous polymers: free volume and shear transformation zone dynamics. Journal of Applied Physics, 119(22), 225104 (2016)
[22]	XING, Z. An associative rheological model for the glass transition of glassy hydrogels undergoing bound and confined interactions. Polymer International, 74(8), 727–738 (2025)
[23]	DAI, L. and XIAO, R. A thermodynamic-consistent model for the thermo-chemo-mechanical couplings in amorphous shape-memory polymers. International Journal of Applied Mechanics, 13(2), 2150022 (2021)
[24]	WILLIAMS, M. L., LANDEL, R. F., and FERRY, J. D. The temperature dependence of relaxation mechanisms in amorphous polymers and other glass-forming liquids. Journal of the American Chemical Society, 77(14), 3701–3707 (1955)
[25]	ANGELL, C. A. Formation of glasses from liquids and biopolymers. Science, 267(5206), 1924–1935 (1995)
[26]	GU, J., ZHANG, Y., DUAN, H., BAI, X., CHEN, Z., ZENG, H., and SUN, H. Constitutive model for hygro-thermo-mechanical responses of carbon fiber reinforced shape memory polymer composites: thermodynamic formulation and bending/buckling applications. Mechanics of Materials, 210, 105476 (2025)
[27]	SHI, J., LU, H., and FU, Y. Q. A bridging model of a water-triggered shape-memory effect in an amorphous polymer undergoing multiple glass transitions. Journal of Physics D-Applied Physics, 56(40), 405304 (2023)
[28]	SMITH, K. E., TRUSTY, P., WAN, B., and GALL, K. Long-term toughness of photopolymerizable (meth)acrylate networks in aqueous environments. Acta Biomaterialia, 7(2), 558–567 (2011)
[29]	SEESALA, V. S., SHEIKH, L., BASU, B., and MUKHERJEE, S. Mechanical and bioactive properties of PMMA bone cement: a review. ACS Biomaterials Science and Engineering, 10(10), 5939–5959 (2024)
[30]	LIN, J., ZHU, P., TIAN, C., ZHOU, H., and XIAO, R. Physically-based interpretation of abnormal stress relaxation response in glassy polymers. Extreme Mechanics Letters, 52, 101667 (2022)
[31]	LIN, J., QIAN, J., XIE, Y., WANG, J., and XIAO, R. A mean-field shear transformation zone theory for amorphous polymers. International Journal of Plasticity, 163, 103556 (2023)
[32]	XIAO, R. and NGUYEN, T. D. An effective temperature theory for the nonequilibrium behavior of amorphous polymers. Journal of the Mechanics and Physics of Solids, 82, 62–81 (2015)
[33]	GUO, J., LI, Z. Y., LONG, J. M., and XIAO, R. Modeling the effect of physical aging on the stress response of amorphous polymers based on a two-temperature continuum theory. Mechanics of Materials, 143, 103335 (2020)
[34]	DUAN, H., GU, J., SUN, H., ZENG, H., and RODRIGUEZ-MORALES, J. A. A thermodynamic constitutive model based on uncoupled physical mechanisms for polymer-based shape memory composites and its application in 4D printing. Applied Mathematical Modelling, 141, 115926 (2025)
[35]	XIAO, R. and NGUYEN, T. D. Modeling the solvent-induced shape-memory behavior of glassy polymers. Soft Matter, 9(39), 9455–9464 (2013)
[36]	SONG, Z., LI, X., WU, K., and CAI, S. Nonequilibrium thermodynamic modeling of case II diffusion in glassy polymers. Journal of the Mechanics and Physics of Solids, 179, 105395 (2023)
[37]	ARRUDA, E. M. and BOYCE, M. C. A three-dimensional constitutive model for the large stretch behavior of rubber elastic materials. Journal of the Mechanics and Physics of Solids, 41(2), 389–412 (1993)
[38]	REESE, S. and GOVINDJEE, S. A theory of finite viscoelasticity and numerical aspects. International Journal of Solids and Structures, 35(26), 3455–3482 (1998)
[39]	REE, T. and EYRING, H. Theory of non-Newtonian flow. I. solid plastic system. Journal of Applied Physics, 26(7), 793–800 (1955)

Time	29MA	36MA	50MA-DDDA
1 day	6.3	7.5	4.8
7 days	8.3	12.3	6.6
2 months	10.5	12.8	7.2
9 months	8.8	12	7.2