Computer Simulation of the Dynamic Behavior of Double Polymer Brush-Solvent Systems
Hałagan Krzysztof 1, Banaszak Michał 2,3, Jung Jarosław 1, Polanowski Piotr 1, Sikorski Andrzej 4*
1 Łódź University of Technology
Department of Molecular Physics
ul. Żeromskiego 116, 90-924 Łódź, Poland2 Adam Mickiewicz University
Faculty of Physics
ul. Uniwersytetu Poznańskiego 2, 61-614 Poznań, Poland3 Adam Mickiewicz University
NanoBiomedical Centre
ul. Wszechnicy Piastowskiej 3, 61-614 Poznań, Poland4 University of Warsaw
Faculty of Chemistry
ul. Pasteura 1, 02-093 Warsaw, Poland
*E-mail: sikorski@chem.uw.edu.pl
Received:
Received: 23 November 2021; accepted: 4 December 2021; published online: 13 December 2021
DOI: 10.12921/cmst.2021.0000030
Abstract:
Opposing polymer brush systems were investigated by computer simulations. In a coarse-grained model, chains were restricted to a face-centered cubic lattice with the excluded volume interactions only. The macromolecules were grafted onto two parallel impenetrable surfaces. The dynamic properties of these systems were studied by means of Monte Carlo simulations. The Dynamic Lattice Liquid model and a highly efficient parallel machine ARUZ were employed, which enabled studying large systems at long time scales. The influence of the surface grating density on the system dynamic was shown and discussed. It was demonstrated that the self-diffusion coefficient of solvent depended strongly on the grafting density.
Key words:
dynamic lattice liquid, lattice models, Monte Carlo method, polymer brushes, polymer dynamics
References:
[1] E. Eisenriegler, Polymers Near Surfaces, World Scientific, Singapore (1993).
[2] W.-L. Chen, R. Cordero, H. Tran, C.K. Ober, 50th anniversary perspective: polymer brushes: novel surfaces for future materials, Macromolecules 50, 4089–4113 (2017).
[3] W.J. Brittain, S. Minko, A structural definition of polymer brushes, J. Polym. Sci. Part A: Polym. Chem. 45, 3505–3512 (2007).
[4] J.O. Zoppe, N.C. Ataman, P. Mocny, J. Wang, J. Moraes, H.-A. Klok, Surface-initiated controlled radical polymerization: state-of-art, opportunities, and challenges in surface and interface engineering with polymer brushes, Chem. Rev. 117, 1105–1318 (2017).
[5] J. Yan, M.R. Bockstaller, K. Matyjaszewski, Brush-modified materials: Control of molecular architecture, assembly behavior, properties and applications, Prog. Polym. Sci. 100, 101180 (2020).
[6] K. Binder, A. Milchev, Polymer Brushes on Flat and Curved Surfaces: How Computer Simulations can Help to test Theories and to Interpret Experiments, J. Polym. Sci. Part B: Polym. Phys. 50, 1516–1555 (2012).
[7] D. Reith, A. Milchev, P. Virnau, K. Binder, Computer simulation studies of chain dynamics in polymer brushes, Macromolecules 45, 4381–4393 (2012).
[8] B. Deng, E.F. Palermo, Y. Shi, Comparison of chain-growth polymerization in solution versus on surface using reactive coarse-grained simulations, Polymer 129, 105–118 (2017).
[9] I.G. Elliot, T.L. Kuhl, R. Faller, Molecular simulation study of the structure of high density polymer brushes in good solvent, Macromolecules 43, 9131–9138 (2010).
[10] T. Pakula, E.B. Zhulina, Computer simulations of polymers in thin layers. II. Structure of polymer melt layers consisting of end-to-end grafted chains, J. Chem. Phys. 95, 4691–4697 (1991).
[11] E.B. Zhulina, T. Pakula, Structure of dense polymer layers between end-grafting and end-adsorbing walls, Macromolecules 25, 754–758 (1992).
[12] J. Huang,W. Jiang, S. Han, Dynamic Monte Carlo simulation on the polymer chain with one end grafted on a flat surface, Macromol. Theory Simul. 10, 339–342 (2001).
[13] P. Polanowski, K. Hałagan, J. Pietrasik, J.K. Jeszka, K. Matyjaszewski, Growth of polymer brushes by “grafting from” via ATRP – Monte Carlo simulations, Polymer 130, 267–279 (2017).
[14] J. Genzer, In silico polymerization: computer simulation of controlled radical polymerization in bulk and on flat surfaces, Macromolecules 39, 7157–7169 (2006).
[15] S. Turgman-Cohen, J. Genzer, Computer simulation of controlled radical polymerization: effect of chain confinement due to initiator grafting density and solvent quality in “grafting from” method, Macromolecules 43, 9567–9577 (2010).
[16] S. Turgman-Cohen, J. Genzer, Computer simulation of concurrent bulk- and surface initiated living polymerization, Macromolecules 45, 2128–2137 (2012).
[17] A. Milchev, J.P. Wittmer, D.P. Landau, Formation and equilibrium properties of living polymer brushes, J. Chem. Phys. 112, 1606–1615 (2000).
[18] K. Hałagan, M. Banaszak, J. Jung, P. Polanowski, A. Sikorski, Dynamics of polymer opposing brushes. A computer simulation study, Polymers 13, 2758 (2021).
[19] K. Binder, Scaling concepts for polymer brushes and their test with computer simulation, Eur. Phys. J. E 9, 293–298 (2002).
[20] W.M. de Vos, F.A.M. Leermakers, Modeling the structure of a polydisperse polymer brush, Polymer 50, 305–316 (2009).
[21] M.W. Matsen, Field theoretic approach for block polymer melts: SCFT and FTS, J. Chem. Phys. 152, 110901 (2020).
[22] S.T. Milner, Polymer brushes, Science 251, 905–914 (1991).
[23] T. Kreer, Polymer-brush lubrication: a review of recent theoretical advances, Soft Matter 12, 3479–3501 (2016).
[24] L.I. Klushin, A.M. Skvortsov, S. Qi, T. Kreer, F. Schmid, Polydispersity effects on interpenetration in compressed brushes, Macromolecules 52, 1810–1820 (2019).
[25] A. Galuschko, L. Spirin, T. Kreer, A. Johner, C. Pastorino, J. Wittmer, J. Baschnagel, Frictional forces between strongly compressed, nonentangled polymer brushes: Molecular dynamics simulations and scaling theory, Langmuir 26, 6418–6429 (2010).
[26] P.R. Desai, S. Sinha, S. Das, Compression of polymer brushes in the weak interpenetration regime: scaling theory and molecular dynamics simulations, Soft Matter 13, 4159–4166 (2017).
[27] C.-H. Tai, G.-T. Pan, H.-Y. Yu, Entropic effects in solventfree bidisperse polymer brushes investigated using Density Functional Theories, Langmuir 35, 16835–16849 (2019).
[28] P. Romiszowski, A. Sikorski, Properties of polymer sandwich brushes, Colloid. Surface. A 321, 254–257 (2008).
[29] P. Romiszowski, A. Sikorski, The Monte Carlo dynamics of polymer chains in sandwich brushes, Rheol. Acta 47, 565–569 (2008).
[30] A. Mendonça, F. Goujon, P. Malfreyt, D.J. Tildedsley, Monte Carlo simulations of the static friction between two grafted polymer brushes, Phys. Chem. Chem. Phys. 18, 6164–6174 (2016).
[31] F. Goujon, A. Ghoufi, P. Malfreyt, D.J. Tildesley, Frictional forces in polyelectrolyte brushes: effects of sliding velocity, solvent quality and salt, Soft Matter 8, 4635–4644 (2012).
[32] F. Goujon, A. Ghoufi, P. Malfreyt, D.J. Tildesley, The kinetic friction coefficient of neutral and charged polymer brushes, Soft Matter 9, 2966–2972 (2013).
[33] O.J. Hehmeyer, M.J. Stevens, Molecular dynamics simulations of grafted polyelectrolytes on two apposing walls, J. Chem. Phys. 122, 134909 (2005).
[34] T. Pakula, Simulation on the completely occupied lattices, [In:] Simulation methods for polymers, M. Kotelyanskii, D.N. Theodorou (Eds.), Marcel Dekker, New York-Basel (2004).
[35] P. Polanowski, A. Sikorski, Simulation of diffusion in a crowded environment, Soft Matter 10, 3597–3607 (2014).
[36] H. Gao, P. Polanowski, K. Matyjaszewski, Gelation in living copolymerization of monomer and divinyl cross linker: comparison of ATRP experiments with Monte Carlo simulations, Macromolecules 42, 5925–5932 (2009).
[37] P. Polanowski, J.K Jeszka, K. Matyjaszewski, Modeling of branching and gelation in living copolymerization of monomer and divinyl cross-linker using dynamic lattice liquid model (DLL) and Flory-Stockmayer model, Polymer 51, 6084–6092 (2010).
[38] P. Polanowski, J.K. Jeszka, K. Krysiak, K. Matyjaszewski, Influence of intramolecular crosslinking on gelation in living copolymerization of monomer and divinyl cross-linker. Monte Carlo simulation studies, Polymer 79, 171–178 (2015).
[39] M. Kozanecki, K. Halagan, J. Saramak, K. Matyjaszewski, Diffusive properties of solvent molecules in the neighborhood of a polymer chain as seen by Monte-Carlo simulations, Soft Matter 12, 5519–5528 (2016).
[40] K. Matyjaszewski, H. Dong, W. Jakubowski, J. Pietrasik, A. Kusumo, Grafting from surfaces for “everyone”: ARGET ATRP in the presence of air, Langmuir 23, 4528–4531
(2007).
[41] K. Matyjaszewski, P.J. Miller, N. Shukla, B. Immaraporn, A. Gelman, B.B. Luokala, T.M. Silovan, G. Kickelbick, T. Vallant, H. Hoffmann, T. Pakula, Polymers at interfaces: using atom transfer radical polymerization in the controlled growth of homopolymers and block copolymers from silicon surfaces in the absence of untethered sacrificial initiator, Macromolecules 32, 8716–8724 (1999).
[42] Y. Tsuji, K. Ohno, S. Yamamoto, A. Goto, T. Fukuda, Structure and properties of high-density polymer brushes prepared by surface-initiated living radical polymerization, Adv. Polym. Sci. 197, 1–45 (2006).
[43] A. Khabibullin, E. Mastan, K. Matyjaszewski, S. Zhu, Surface-initiated atom transfer radical polymerization, Adv. Polym. Sci. 270, 29–76 (2016).
[44] P. Polanowski, J.K. Jeszka, K. Matyjaszewski, Polymer brush relaxation during and after polymerization – Monte Carlo simulation study, Polymer 173, 190–196 (2019).
[45] R. Kiełbik, K. Hałagan, W. Zatorski, J. Jung, J. Ulański, A. Napieralski, K. Rudnicki, P. Amrozik, G. Jabłoński, D. Stożek, P. Polanowski, Z. Mudza, J. Kupis, P. Panek, ARUZ – Large-scale, Massively parallel FPGA-based Analyzer of Real Complex Systems, Comput. Phys. Commun. 232, 22–34 (2018).
[46] J. Jung, P. Polanowski, R. Kiełbik, W. Zatorski, J. Ulański, A. Napieralski, T. Pakuła, K. Hałagan, Polish Patent no. 2016, 223795, 2017, 227249, 2017, 227250, 2018, PL/EP3079066.
[47] J. Jung, P. Polanowski, R. Kielbik, K. Halagan, W. Zatorski, J. Ulański, A. Napieralski, T. Pakula, European Patent no. 2017, EP3079066, 2018, EP307907.
[48] J. Jung, R. Kiełbik, K. Hałagan, P. Polanowski, A. Sikorski, Technology of Real-World Analyzers (TAUR) and its practical application, Comput. Methods Sci. Technol. 26, 69–75 (2020).
[49] D. Ben-Avraham, S. Havlin, Diffusion and reactions in fractals and disordered systems, Cambridge University Press, Cambridge (2000).
[50] A. Wedemeier, H. Merlitz, J. Langowski, Anomalous diffusion in the presence of mobile obstacles, Europhys. Lett. 88, 38004 (2009).
[51] E. Vilaseca, A. Isvoran, S. Madurga, L. Pastor, J.L. Garcés, F. Mas, New insights into diffusion in 3D crowded media by Monte Carlo simulations: effect of size, mobility and spatial distribution of obstacles, Phys. Chem. Chem. Phys. 13, 7396 (2011).
[52] P. Polanowski, A. Sikorski, Diffusion of small particles in polymer films, J. Chem. Phys. 147, 014902 (2017).
[53] P. Polanowski, T. Pakula, Studies of mobility, interdiffusion, and self-diffusion in two-component mixtures using the dynamic lattice liquid model, J. Chem. Phys. 118, 11139–11146 (2003).
[54] M. Lang, M. Werner, R. Dockhorn, T. Kreer, Arm retraction dynamics in dense polymer brushes, Macromolecules 49, 5190–5201 (2016).
[55] H. Yasuda, C.E. Lamaze, L.D. Ikenberry, Permeability of solutes through hydrated polymer membranes. Part I. Diffusion of sodium chloride, Makromol. Chem. 118, 19–35 (1968).
Opposing polymer brush systems were investigated by computer simulations. In a coarse-grained model, chains were restricted to a face-centered cubic lattice with the excluded volume interactions only. The macromolecules were grafted onto two parallel impenetrable surfaces. The dynamic properties of these systems were studied by means of Monte Carlo simulations. The Dynamic Lattice Liquid model and a highly efficient parallel machine ARUZ were employed, which enabled studying large systems at long time scales. The influence of the surface grating density on the system dynamic was shown and discussed. It was demonstrated that the self-diffusion coefficient of solvent depended strongly on the grafting density.
Key words:
dynamic lattice liquid, lattice models, Monte Carlo method, polymer brushes, polymer dynamics
References:
[1] E. Eisenriegler, Polymers Near Surfaces, World Scientific, Singapore (1993).
[2] W.-L. Chen, R. Cordero, H. Tran, C.K. Ober, 50th anniversary perspective: polymer brushes: novel surfaces for future materials, Macromolecules 50, 4089–4113 (2017).
[3] W.J. Brittain, S. Minko, A structural definition of polymer brushes, J. Polym. Sci. Part A: Polym. Chem. 45, 3505–3512 (2007).
[4] J.O. Zoppe, N.C. Ataman, P. Mocny, J. Wang, J. Moraes, H.-A. Klok, Surface-initiated controlled radical polymerization: state-of-art, opportunities, and challenges in surface and interface engineering with polymer brushes, Chem. Rev. 117, 1105–1318 (2017).
[5] J. Yan, M.R. Bockstaller, K. Matyjaszewski, Brush-modified materials: Control of molecular architecture, assembly behavior, properties and applications, Prog. Polym. Sci. 100, 101180 (2020).
[6] K. Binder, A. Milchev, Polymer Brushes on Flat and Curved Surfaces: How Computer Simulations can Help to test Theories and to Interpret Experiments, J. Polym. Sci. Part B: Polym. Phys. 50, 1516–1555 (2012).
[7] D. Reith, A. Milchev, P. Virnau, K. Binder, Computer simulation studies of chain dynamics in polymer brushes, Macromolecules 45, 4381–4393 (2012).
[8] B. Deng, E.F. Palermo, Y. Shi, Comparison of chain-growth polymerization in solution versus on surface using reactive coarse-grained simulations, Polymer 129, 105–118 (2017).
[9] I.G. Elliot, T.L. Kuhl, R. Faller, Molecular simulation study of the structure of high density polymer brushes in good solvent, Macromolecules 43, 9131–9138 (2010).
[10] T. Pakula, E.B. Zhulina, Computer simulations of polymers in thin layers. II. Structure of polymer melt layers consisting of end-to-end grafted chains, J. Chem. Phys. 95, 4691–4697 (1991).
[11] E.B. Zhulina, T. Pakula, Structure of dense polymer layers between end-grafting and end-adsorbing walls, Macromolecules 25, 754–758 (1992).
[12] J. Huang,W. Jiang, S. Han, Dynamic Monte Carlo simulation on the polymer chain with one end grafted on a flat surface, Macromol. Theory Simul. 10, 339–342 (2001).
[13] P. Polanowski, K. Hałagan, J. Pietrasik, J.K. Jeszka, K. Matyjaszewski, Growth of polymer brushes by “grafting from” via ATRP – Monte Carlo simulations, Polymer 130, 267–279 (2017).
[14] J. Genzer, In silico polymerization: computer simulation of controlled radical polymerization in bulk and on flat surfaces, Macromolecules 39, 7157–7169 (2006).
[15] S. Turgman-Cohen, J. Genzer, Computer simulation of controlled radical polymerization: effect of chain confinement due to initiator grafting density and solvent quality in “grafting from” method, Macromolecules 43, 9567–9577 (2010).
[16] S. Turgman-Cohen, J. Genzer, Computer simulation of concurrent bulk- and surface initiated living polymerization, Macromolecules 45, 2128–2137 (2012).
[17] A. Milchev, J.P. Wittmer, D.P. Landau, Formation and equilibrium properties of living polymer brushes, J. Chem. Phys. 112, 1606–1615 (2000).
[18] K. Hałagan, M. Banaszak, J. Jung, P. Polanowski, A. Sikorski, Dynamics of polymer opposing brushes. A computer simulation study, Polymers 13, 2758 (2021).
[19] K. Binder, Scaling concepts for polymer brushes and their test with computer simulation, Eur. Phys. J. E 9, 293–298 (2002).
[20] W.M. de Vos, F.A.M. Leermakers, Modeling the structure of a polydisperse polymer brush, Polymer 50, 305–316 (2009).
[21] M.W. Matsen, Field theoretic approach for block polymer melts: SCFT and FTS, J. Chem. Phys. 152, 110901 (2020).
[22] S.T. Milner, Polymer brushes, Science 251, 905–914 (1991).
[23] T. Kreer, Polymer-brush lubrication: a review of recent theoretical advances, Soft Matter 12, 3479–3501 (2016).
[24] L.I. Klushin, A.M. Skvortsov, S. Qi, T. Kreer, F. Schmid, Polydispersity effects on interpenetration in compressed brushes, Macromolecules 52, 1810–1820 (2019).
[25] A. Galuschko, L. Spirin, T. Kreer, A. Johner, C. Pastorino, J. Wittmer, J. Baschnagel, Frictional forces between strongly compressed, nonentangled polymer brushes: Molecular dynamics simulations and scaling theory, Langmuir 26, 6418–6429 (2010).
[26] P.R. Desai, S. Sinha, S. Das, Compression of polymer brushes in the weak interpenetration regime: scaling theory and molecular dynamics simulations, Soft Matter 13, 4159–4166 (2017).
[27] C.-H. Tai, G.-T. Pan, H.-Y. Yu, Entropic effects in solventfree bidisperse polymer brushes investigated using Density Functional Theories, Langmuir 35, 16835–16849 (2019).
[28] P. Romiszowski, A. Sikorski, Properties of polymer sandwich brushes, Colloid. Surface. A 321, 254–257 (2008).
[29] P. Romiszowski, A. Sikorski, The Monte Carlo dynamics of polymer chains in sandwich brushes, Rheol. Acta 47, 565–569 (2008).
[30] A. Mendonça, F. Goujon, P. Malfreyt, D.J. Tildedsley, Monte Carlo simulations of the static friction between two grafted polymer brushes, Phys. Chem. Chem. Phys. 18, 6164–6174 (2016).
[31] F. Goujon, A. Ghoufi, P. Malfreyt, D.J. Tildesley, Frictional forces in polyelectrolyte brushes: effects of sliding velocity, solvent quality and salt, Soft Matter 8, 4635–4644 (2012).
[32] F. Goujon, A. Ghoufi, P. Malfreyt, D.J. Tildesley, The kinetic friction coefficient of neutral and charged polymer brushes, Soft Matter 9, 2966–2972 (2013).
[33] O.J. Hehmeyer, M.J. Stevens, Molecular dynamics simulations of grafted polyelectrolytes on two apposing walls, J. Chem. Phys. 122, 134909 (2005).
[34] T. Pakula, Simulation on the completely occupied lattices, [In:] Simulation methods for polymers, M. Kotelyanskii, D.N. Theodorou (Eds.), Marcel Dekker, New York-Basel (2004).
[35] P. Polanowski, A. Sikorski, Simulation of diffusion in a crowded environment, Soft Matter 10, 3597–3607 (2014).
[36] H. Gao, P. Polanowski, K. Matyjaszewski, Gelation in living copolymerization of monomer and divinyl cross linker: comparison of ATRP experiments with Monte Carlo simulations, Macromolecules 42, 5925–5932 (2009).
[37] P. Polanowski, J.K Jeszka, K. Matyjaszewski, Modeling of branching and gelation in living copolymerization of monomer and divinyl cross-linker using dynamic lattice liquid model (DLL) and Flory-Stockmayer model, Polymer 51, 6084–6092 (2010).
[38] P. Polanowski, J.K. Jeszka, K. Krysiak, K. Matyjaszewski, Influence of intramolecular crosslinking on gelation in living copolymerization of monomer and divinyl cross-linker. Monte Carlo simulation studies, Polymer 79, 171–178 (2015).
[39] M. Kozanecki, K. Halagan, J. Saramak, K. Matyjaszewski, Diffusive properties of solvent molecules in the neighborhood of a polymer chain as seen by Monte-Carlo simulations, Soft Matter 12, 5519–5528 (2016).
[40] K. Matyjaszewski, H. Dong, W. Jakubowski, J. Pietrasik, A. Kusumo, Grafting from surfaces for “everyone”: ARGET ATRP in the presence of air, Langmuir 23, 4528–4531
(2007).
[41] K. Matyjaszewski, P.J. Miller, N. Shukla, B. Immaraporn, A. Gelman, B.B. Luokala, T.M. Silovan, G. Kickelbick, T. Vallant, H. Hoffmann, T. Pakula, Polymers at interfaces: using atom transfer radical polymerization in the controlled growth of homopolymers and block copolymers from silicon surfaces in the absence of untethered sacrificial initiator, Macromolecules 32, 8716–8724 (1999).
[42] Y. Tsuji, K. Ohno, S. Yamamoto, A. Goto, T. Fukuda, Structure and properties of high-density polymer brushes prepared by surface-initiated living radical polymerization, Adv. Polym. Sci. 197, 1–45 (2006).
[43] A. Khabibullin, E. Mastan, K. Matyjaszewski, S. Zhu, Surface-initiated atom transfer radical polymerization, Adv. Polym. Sci. 270, 29–76 (2016).
[44] P. Polanowski, J.K. Jeszka, K. Matyjaszewski, Polymer brush relaxation during and after polymerization – Monte Carlo simulation study, Polymer 173, 190–196 (2019).
[45] R. Kiełbik, K. Hałagan, W. Zatorski, J. Jung, J. Ulański, A. Napieralski, K. Rudnicki, P. Amrozik, G. Jabłoński, D. Stożek, P. Polanowski, Z. Mudza, J. Kupis, P. Panek, ARUZ – Large-scale, Massively parallel FPGA-based Analyzer of Real Complex Systems, Comput. Phys. Commun. 232, 22–34 (2018).
[46] J. Jung, P. Polanowski, R. Kiełbik, W. Zatorski, J. Ulański, A. Napieralski, T. Pakuła, K. Hałagan, Polish Patent no. 2016, 223795, 2017, 227249, 2017, 227250, 2018, PL/EP3079066.
[47] J. Jung, P. Polanowski, R. Kielbik, K. Halagan, W. Zatorski, J. Ulański, A. Napieralski, T. Pakula, European Patent no. 2017, EP3079066, 2018, EP307907.
[48] J. Jung, R. Kiełbik, K. Hałagan, P. Polanowski, A. Sikorski, Technology of Real-World Analyzers (TAUR) and its practical application, Comput. Methods Sci. Technol. 26, 69–75 (2020).
[49] D. Ben-Avraham, S. Havlin, Diffusion and reactions in fractals and disordered systems, Cambridge University Press, Cambridge (2000).
[50] A. Wedemeier, H. Merlitz, J. Langowski, Anomalous diffusion in the presence of mobile obstacles, Europhys. Lett. 88, 38004 (2009).
[51] E. Vilaseca, A. Isvoran, S. Madurga, L. Pastor, J.L. Garcés, F. Mas, New insights into diffusion in 3D crowded media by Monte Carlo simulations: effect of size, mobility and spatial distribution of obstacles, Phys. Chem. Chem. Phys. 13, 7396 (2011).
[52] P. Polanowski, A. Sikorski, Diffusion of small particles in polymer films, J. Chem. Phys. 147, 014902 (2017).
[53] P. Polanowski, T. Pakula, Studies of mobility, interdiffusion, and self-diffusion in two-component mixtures using the dynamic lattice liquid model, J. Chem. Phys. 118, 11139–11146 (2003).
[54] M. Lang, M. Werner, R. Dockhorn, T. Kreer, Arm retraction dynamics in dense polymer brushes, Macromolecules 49, 5190–5201 (2016).
[55] H. Yasuda, C.E. Lamaze, L.D. Ikenberry, Permeability of solutes through hydrated polymer membranes. Part I. Diffusion of sodium chloride, Makromol. Chem. 118, 19–35 (1968).