Abstract.
We investigate microphase separation patterns on curved surfaces in three-dimensional space by numerically solving a nonlocal Cahn-Hilliard equation for diblock copolymers. In our model, a curved surface is implicitly represented as the zero level set of a signed distance function. We employ a discrete narrow band grid that neighbors the curved surface. Using the closest point method, we apply a pseudo-Neumann boundary at the boundary of the computational domain. The boundary treatment allows us to replace the Laplace-Beltrami operator by the standard Laplacian operator. In particular, we can apply standard finite difference schemes in order to approximate the nonlocal Cahn-Hilliard equation in the discrete narrow band domain. We employ a type of unconditionally stable scheme, which was introduced by Eyre, and use the Jacobi iterative to solve the resulting implicit discrete system of equations. In addition, we use the minimum number of grid points for the discrete narrow band domain. Therefore, the algorithm is simple and fast. Numerous computational experiments are provided to study microphase separation patterns for diblock copolymers on curved surfaces in three-dimensional space.
Graphical abstract
Similar content being viewed by others
References
I.P. Campbell, G.J. Lau, J.L. Feaver, M.P. Stoykovich, Macromolecules 45, 1587 (2012)
C. Singh, M. Goulian, A.J. Liu, G.H. Fredrickson, Macromolecules 27, 2974 (1994)
F. Liu, N. Goldenfeld, Phys. Rev. A 39, 4805 (1989)
R. Choksi, M.A. Peletier, J.F. Williams, SIAM J. Appl. Math. 69, 1712 (2009)
P. Tang, F. Qiu, H. Zhang, Y. Yang, Phys. Rev. E 72, 016710 (2005)
T. Baumgart, S.T. Hess, W.W. Webb, Nature 425, 821 (2003)
A.R. Bausch, M.J. Bowick, A. Cacciuto, A.D. Dinsmore, M.F. Hsu, D.R. Nelson, M.G. Nikolaides, A. Travesset, D.A. Weitz, Science 299, 1716 (2003)
P. Sens, S.A. Safran, Eur. Phys. J. E 1, 237 (2000)
K. Binder, S. Puri, S.K. Das, J. Horbach, J. Stat. Phys. 138, 51 (2010)
G. Brown, A. Chakrabarti, J. Chem. Phys. 102, 1440 (1995)
K. Binder, J. Non-Equil. Thermodyn. 23, 1 (1998)
H. Xiang, K. Shin, T. Kim, S.I. Moon, T.J. McCarthy, T.P. Russell, Macromolecules 37, 5660 (2004)
R. Choksi, M. Maras, J.F. Williams, SIAM J. Appl. Dyn. Syst. 10, 1344 (2011)
M. Pinna, A.V. Zvelindovsky, Eur. Phys. J. B 85, 1 (2012)
D. Jeong, J. Shin, Y. Li, Y. Choi, J.-H. Jung, S. Lee, J. Kim, Curr. Appl. Phys. 14, 1263 (2014)
R. Choksi, X. Ren, J. Stat. Phys. 113, 151 (2003)
S.W. Sides, G.H. Fredrickson, Polymer 44, 5859 (2003)
S.W. Sides, B.J. Kim, E.J. Kramer, G.H. Fredrickson, Phys. Rev. Lett. 96, 250601 (2006)
K.O. Rasmussen, G. Kalosakas, J. Polym. Sci. Pol. Phys. 40, 1777 (2002)
H.D. Ceniceros, G.H. Fredrickson, Multiscale Model. Simul. 2, 452 (2004)
B. Shahriari, PhD thesis, Simon Fraser Univeristy (2010)
T.L. Chantawansri, A.W. Bosse, A. Hexemer, H.D. Ceniceros, C.J. García-Cervera, E.J. Kramer, G.H. Fredrickson, Phys. Rev. E 75, 031802 (2007)
I. Chavel, Eigenvalues in Riemannian Geometry, Vol. 115 (Academic Press, London, 1984)
C.B. Macdonald, S.J. Ruuth, J. Sci. Comput. 35, 219 (2008)
S. Osher, R.P. Fedkiw, J. Comput. Phys. 169, 463 (2001)
D.J. Eyre, in MRS Proceedings, Vol. 529 (Cambridge University Press, 1998) p. 39
I.N. Bronshtein, K.A. Semendyayev, Handbook of Mathematics, 3rd edition (Springer-Verlag, New York, 1997) p. 892
T. Ohta, K. Kawasaki, Macromolecules 19, 2621 (1986)
Y. Nishiura, I. Ohnishi, Phys. D 84, 31 (1995)
S. Puri, H.L. Frisch, J. Phys. A 27, 6027 (1994)
S. Glotzer, D. Stauffer, N. Jan, Phys. Rev. Lett. 72, 4109 (1994)
S. Glotzer, E.A. Di Marzio, M. Muthukumar, Phys. Rev. Lett. 74, 2034 (1995)
C.B. Macdonald, J. Brandman, S.J. Ruuth, J. Comput. Phys. 230, 7944 (2011)
J.B. Greer, J. Sci. Comput. 29, 321 (2006)
H.-K. Zhao, S. Osher, R. Fedkiw, in Proceedings of the IEEE Workshop on Variational and Level Set Methods, Washington, DC, 2001, edited by A.D. Williams (IEEE Comput. Soc., Los Alamitos, 2001) p. 194
H.-K. Zhao, S. Osher, B. Merriman, M. Kang, Comput. Vis. Image Und. 80, 295 (2000)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Jeong, D., Kim, J. Microphase separation patterns in diblock copolymers on curved surfaces using a nonlocal Cahn-Hilliard equation. Eur. Phys. J. E 38, 117 (2015). https://doi.org/10.1140/epje/i2015-15117-1
Received:
Accepted:
Published:
DOI: https://doi.org/10.1140/epje/i2015-15117-1