Abstract
The self-assembly of amphiphilic homopolymers tightly grafted to the spherical nanoparticle and immersed in a selective solvent is studied by the computer experiment method. Conditions under which macromolecules form thin membrane-like layers surrounding the nanoparticle are determined. It is first shown that the emerging polymer structures may be approximated by complete embedded minimal surfaces satisfying the Weierstrass representation, namely, helicoid, catenoid, and Enneper and Costa surfaces. Mathematical constructions defining these minimal surfaces highlight a new type of ordering of polymer structures and determine its symmetry classification similar to crystal classification by Fedorov groups. Calculations for the two considered sets of parameters show that structures approximated by a helicoid are energetically more favorable than structures approximated by other minimal surfaces.
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REFERENCES
F. S. Bates and G. H. Fredrickson, Annu. Rev. Phys. Chem. 41, 525 (1990).
T. Lodge, Microchim. Acta 116, 1 (1994).
I. W. Hamley, K. A. Koppi, J. H. Rosedale, F. S. Bates, K. Almdal, and K. Mortensen, Macromolecules 26, 5959 (1993).
Block Copolymers in Nanoscience, Ed. by M. Lazzari, G. Liu, and S. Lecommandoux (Wiley, Darmstadt, 2006).
T. P. Lodge, Macromol. Chem. Phys. 204, 265 (2003).
L. Leibler, Macromolecules 13, 1602 (1980).
A. N. Semenov, Sov. Phys. JETP 61, 733 (1985).
I. Ya. Erukhimovich and A. R. Khokhlov, Vysokomol. Soedin. A 35, 1808 (1993).
G. Floudas, N. Hadjichristidis, Y. Tselikas, and I. Erukhimovich, Macromolecules 30, 3090 (1997).
V. Abetz and R. Stadler, Macromolecules 30, 7435 (1997).
I. Ya. Erukhimovich, Yu. G. Smirnova, and V. Abetz, Polym. Sci., Ser. A 45, 1093 (2003).
Y. G. Smirnova, G. Brinke, and I. Ya. Erukhimovich, J. Chem. Phys. 124, 054907 (2006).
I. Ya. Erukhimovich, Eur. Phys. J. E 18, 383 (2005).
R. Nap, N. Sushko, I. Erukhimovich, and G. Brinke, Macromolecules 39, 6765 (2006).
Y. A. Kriksin, P. G. Khalatur, I. Ya. Erukhimovich, G. Brinke, and A. R. Khokhlov, Soft Matter 5, 2896 (2009).
A. Glagoleva, I. Erukhimovich, and V. Vasilevskaya, Macromol. Theory Simul. 22, 31 (2013).
I. Erukhimovich, Polym. Sci., Ser. C 60, 49(2018).
S. Lee, M. J. Bluemle, and F. S. Bates, Science 330, 349 (2010).
D. A. Hajduk, P. E. Harper, S. M. Gruner, C. C. Honeker, G. Kim, E. L. Thomas, and L. J. Fetters, Macromolecules 27, 4063 (1994).
E. L. Thomas, D. B. Alward, D. J. Kinning, D. C. Martin, D. L. Handlin, and L. J. Fetters, Macromolecules 19, 2197 (1986).
A. K. Khandpur, S. Foerster, F. S. Bates, I. W. Hamley, A. J. Ryan, W. Bras, K. Almdal, and K. Mortensen, Macromolecules 28, 8796 (1995).
A. Reddy, X. Feng, E. L. Thomas, and G. M. Grason, Macromolecules 54, 9223 (2021).
R. Mosseri and J. F. Sadoc, J. Phys. Colloques 51, C7–C257 (1990).
A. Talis, A. Everstov, and V. Kraposhin, Acta Crystallogr., Sect. A 77, 7 (2021).
T. Castle, M. E. Evans, S. T. Hyde, S. Ramsden, and V. Robins, Interface Focus 2, 555 (2012).
B. K. Vainshtein, Modern Crystallography, Vol. 1 (Nauka, Moscow, 1979; Springer, Berlin, 1994).
A. A. Tuzhilin and A. T. Fomenko, Elements of Geometry and Topology of Minimal Surfaces (URSS, Moscow, 2022; American Mathematical Society, 2023).
A. T. Fomenko, Variational Methods in Topology (Nauka, Moscow, 1982; CRC Press, 2020).
L. Anetor, Minimal Surfaces Embedded in Euclidean Space (Balkan Press, Bucharest, 2016).
W.-F. Pu, A. Ushakova, R. Liu, A. A. Lazutin, and V. V. Vasilevskaya, J. Chem. Phys. 152, 234903 (2020).
A. S. Ushakova, A. A. Lazutin, and V. V. Vasilevskaya, Macromolecules 54, 6285 (2021).
A. S. Ushakova and V. V. Vasilevskaya, Polymers 14, 4358 (2022).
Z. R. Saraev, A. A. Lazutin, and V. V. Vasilevskaya, Molecules 27, 8535 (2022).
A. A. Lazutin, V. V. Vasilevskaya, Polymer 255, 125172 (2022).
D. Khoffman and G. Karkher, Results of Science and Technology in the series Modern Problems of Mathematics. Fundamental Directions, Vol. 90 (Fizmatlit, Moscow, 2003) [in Russian].
T. I. Löbling, J. S. Haataja, C. V. Synatschke, F. H. Schacher, M. Müller, A. Hanisch, A. H. Groschel, and A. H. E. Müller, ACS Nano 8, 11330 (2014).
S. J. Plimpton, Computat. Phys 117, 1 (1995).
J. D. Weeks, D. Chandler, and H. C. Andersen, J. Chem. Phys. 54, 5237 (1971).
L. Verlet, Phys. Rev. 159, 98 (1967).
J. Smith, Compos. Sci. Technol. 63, 1599 (2003).
M. Bishop, M. H. Kalos, and H. L. Frisch, J. Chem. Phys. 70, 1299 (1979).
G. S. Grest and K. Kremer, Phys. Rev. A 33, 3628 (1986).
J. Cho and Y. Ogata, J. Geom. 108, 463 (2017).
Funding
This work was supported by the Russian Science Foundation (project no. 19-73-20104) and carried out using the equipment of the shared research facilities of HPC computing resources at the Lomonosov Moscow State University. The data processing was performed using facilities of the Interlaboratory Computer Center at the Nesmeyanov Institute of Organoelement Compounds, Russian Academy of Sciences.
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Mitkovskiy, D.A., Lazutin, A.A., Ushakova, A.S. et al. Geometric Features of Structuring of Amphiphilic Macromolecules on the Surface of a Spherical Nanoparticle. Polym. Sci. Ser. C 65, 3–10 (2023). https://doi.org/10.1134/S1811238223700297
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DOI: https://doi.org/10.1134/S1811238223700297