Effect of the polymer network on the diffusion of foreign particles in hydrogels

Yuri Zabashta; Yaroslav Lazarenko; Oleksandr Alekseev; Lena Vergun; Kateryna Hnatiuk; Dmytro Andrusenko; Kateryna Yablochkova; Roman Dinzhos; Neli Koseva; Maksym Lazarenko

doi:10.17721/1812-5409.2026/1.37

Authors

Yuri Zabashta Taras Shevchenko National University of Kyiv
Yaroslav Lazarenko Buchanan Lyceum No. 4, Bucha, Kyiv region, Ukraine
Oleksandr Alekseev Taras Shevchenko National University of Kyiv
Lena Vergun Taras Shevchenko National University of Kyiv
Kateryna Hnatiuk Taras Shevchenko National University of Kyiv
Dmytro Andrusenko Taras Shevchenko National University of Kyiv
Kateryna Yablochkova Taras Shevchenko National University of Kyiv https://orcid.org/0000-0001-8719-5776
Roman Dinzhos Petro Mohyla Black Sea National University, Mykolayiv, Ukraine; Institute of Polymers, Bulgarian Academy of Sciences, Sofia, Bulgaria
Neli Koseva Institute of Polymers, Bulgarian Academy of Sciences, Sofia, Bulgaria
Maksym Lazarenko Taras Shevchenko National University of Kyiv https://orcid.org/0000-0003-0719-3522

DOI:

https://doi.org/10.17721/1812-5409.2026/1.37

Keywords:

hydrogels, agarose, gelatine, rhodamine, polymer network, diffusion

Abstract

In the present work, we introduce a physical model of foreign particle diffusion in polymer hydrogels that accounts for the network architecture of these systems. The model employs a classical diffusion theory framework in condensed media to derive a theoretical dependence of the diffusion coefficient on the volume occupied by the hydrogel’s network. We demonstrate that the changes in the diffusion of foreign particles can be attributed to the changes in the free activation energy due to the presence of the polymer network. Using optical image analysis, we obtained the dependence of the diffusion coefficient in agarose and gelatine hydrogels on the concentration of the gelling agent. The experimental data obtained are in good agreement with the theoretical dependences predicted by the proposed physical model. Furthermore, when the size of the diffusing particles is much smaller than the hydrogel pore size, the diffusion coefficient appears to be independent of the nature of the hydrogel and determined solely by the gelling agent concentration.

Pages of the article in the issue: 285 - 292

Language of the article: English

References

Alekseev, O. M., Zabashta, Y. F., Kovalchuk, V. I., Lazarenko, M. M., & Bulavin, L. A. (2019). The structure of polymer clusters in aqueous solutions of hydroxypropyl Cellulose. Ukrainian journal of physics, 64(3), 238–238. 10.15407/ujpe64.3.238

Armstrong, M., Walker, S., Mains, J., & Wilson, C. G. (2014). Drug delivery across the nasal mucosa. Mucoadhesive Materials and Drug Delivery Systems, 61–82. https://doi.org/10.1002/9781118794203.ch03

Bagchi, B. (2023). Nonequilibrium statistical mechanics: an introduction with applications. Chapman and Hall/CRC. https://doi.org/10.1201/9781003157601

Birkhoff, G. (2015). Hydrodynamics. Princeton University Press.

Clegg, J. R., Adebowale, K., Zhao, Z., & Mitragotri, S. (2024). Hydrogels in the clinic: An update. Bioengineering & translational medicine, 9(6), e10680. https://doi.org/10.1002/btm2.10680

Crank, J. (1979). The mathematics of diffusion. Oxford university press.

Davydiuk, A. L., Andrusenko, D. A., Lazarenko, M. M., Burbelo, R. M., Kuzmich, A. G., Yablochkova, K. S., Dinzhos R. V., & Alekseev, O. M. (2024). A new photoacoustic method to study diffusion in soft matter. Molecular Crystals and Liquid Crystals, 768(16), 858–867. https://doi.org/10.1080/15421406.2024.2372126

Frenkel, J. (1955). Kinetic Theory of Liquids. Dover Publications. Inc., New York.

Hnatiuk, K. I., Zabashta, Y. F., Lazarenko, M. M., Alekseev, S. A., Yablochkova, K. S., Ushcats, M. V., Dinzhos, R.V., Vergun, L. Yu., Andrusenko, D. A., Alekseev, A. N., & Bulavin, L. A. (2023). Dielectric Relaxation in Nanocrystals: A Scale Effect. The Journal of Physical Chemistry C, 127(25), 12107–12117. https://doi.org/10.1021/acs.jpcc.3c01523

Ho, T. C., Chang, C. C., Chan, H. P., Chung, T. W., Shu, C. W., Chuang, K. P., Duh, T. H., Yang, M. H., & Tyan, Y. C. (2022). Hydrogels: properties and applications in biomedicine. Molecules, 27(9), 2902. https://doi.org/10.3390/molecules27092902

Jiang, F., Xu, X. W., Chen, F. Q., Weng, H. F., Chen, J., Ru, Y., Xiao, Q., & Xiao, A. F. (2023). Extraction, modification and biomedical application of agarose hydrogels: a review. Marine Drugs, 21(5), 299. https://doi.org/10.3390/md21050299

Lazarenko, M. M., Zabashta, Y. F., Alekseev, O. M., Alekseev, S. A., Yablochkova, K. S., Vergun, L. Y., Andrusenko. D. A., Cherevko, K. V., Shevchenko, V. B., Dinzhos, R. V., & Bulavin, L. A. (2024). Thermally activated processes: the underlying mechanism of activated state formation. RSC advances, 14(53), 39242-39252. 1 https://doi.org/0.1039/D4RA06983H

Lazarenko, M. M., Zabashta, Y. F., Honcharuk, D. K., Alekseev, O. M., Yablochkova, K. S., Vergun, L. Y., Andrusenko, D. A., Lazarenko, M. V., & Bulavin, L. A. (2025). Determining hydrogel porosity through dielectric relaxation intensity ratios between water and hydrogel. Soft Matter, 21(21), 4298–4305. https://doi.org/10.1039/D5SM00077G

Majer, G., & Melchior, J. P. (2014). Characterization of the fluorescence correlation spectroscopy (FCS) standard Rhodamine 6G and calibration of its diffusion coefficient in aqueous solutions. The Journal of Chemical Physics, 140(9). https://doi.org/10.1063/1.4867096

Mehta, P., Sharma, M., & Devi, M. (2023). Hydrogels: An overview of its classifications, properties, and applications. Journal of the Mechanical Behavior of Biomedical Materials, 147, 106145. https://doi.org/10.1016/j.jmbbm.2023.106145

Milano, F., Masi, A., Madaghiele, M., Sannino, A., Salvatore, L., & Gallo, N. (2023). Current trends in gelatin-based drug delivery systems. Pharmaceutics, 15(5), 1499. https://doi.org/10.3390/pharmaceutics15051499

Rahbani, J., Behzad, A. R., Khashab, N. M., & Al‑Ghoul, M. (2013). Characterization of internal structure of hydrated agar and gelatin matrices by cryo‑SEM. Electrophoresis, 34(3), 405–408. https://doi.org/10.1002/elps.201200434

Richtering, W., Potemkin, I. I., Rudov, A. A., Sellge, G., & Trautwein, C. (2016). Could multiresponsive hollow shell–shell nanocontainers offer an improved strategy for drug delivery? Nanomedicine, 11(22), 2879–2883. https://doi.org/10.2217/nnm-2016-0327

Roca-Arroyo, A. F., Gutierrez-Rivera, J. A., Morton, L. D., & Castilla-Casadiego, D. A. (2025). Hydrogel network architecture design space: impact on mechanical and viscoelastic properties. Gels, 11(8), 588. https://doi.org/10.3390/gels11080588

Smeets, N. M., & Hoare, T. (2013). Designing responsive microgels for drug delivery applications. Journal of Polymer Science Part A: Polymer Chemistry, 51(14), 3027–3043. https://doi.org/10.1002/pola.26707

Velasco, D., Danoux, C. B., Redondo, J. A., Elvira, C., San Roman, J., Wray, P. S., & Kazarian, S. G. (2011). pH-sensitive polymer hydrogels derived from morpholine to prevent the crystallization of ibuprofen. Journal of controlled release, 149(2), 140–145. https://doi.org/10.1016/j.jconrel.2010.10.015

Wang, S., Liu, R., Fu, Y., & Kao, W. J. (2020). Release mechanisms and applications of drug delivery systems for extended-release. Expert Opinion on Drug Delivery, 17(9), 1289–1304. https://doi.org/10.1080/17425247.2020.1788541

Wolde-Kidan, A., Herrmann, A., Prause, A., Gradzielski, M., Haag, R., Block, S., & Netz, R. R. (2021). Particle diffusivity and free-energy profiles in hydrogels from time-resolved penetration data. Biophysical Journal, 120(3), 463–475. https://doi.org/10.1016/j.bpj.2020.12.020

Yang, D. (2022). Recent advances in hydrogels. Chemistry of Materials, 34(5), 1987–1989. https://doi.org/10.1021/acs.chemmater.2c00188

Yuan, Y., Zhang, Q., Lin, S., & Li, J. (2025). Water: The soul of hydrogels. Progress in Materials Science, 148, 101378. https://doi.org/10.1016/j.pmatsci.2024.101378

Zhu, T., Ni, Y., Biesold, G. M., Cheng, Y., Ge, M., Li, H., Huang, J., Lin, Z., & Lai, Y. (2023). Recent advances in conductive hydrogels: classifications, properties, and applications. Chemical Society Reviews, 52(2), 473–509. https://doi.org/10.1039/D2CS00173J

Zhu, Y., Zhang, H., Tan, Q., Wu, Y., Du, M., Hadjichristidis, N., Jiang Yu., & Zhang, D. (2025). Strong and ultra-tough hydrogel with hierarchical cross-linking network architecture constructed by a hyperbranched topological structure. Macromolecules, 58(13), 6512–6523 https://doi.org/10.1021/acs.macromol.5c00513.