Phase Diagram of Diblock Copolymer Melt in Dimension d = 5
Dzięcielski Michał, Lewandowski Krzysztof, Banaszak Michał
Faculty of Physics, A. Mickiewicz University
ul. Umultowska 85, 61-614 Poznań, Poland
e-mail: mbanasz@amu.edu.pl
Received:
Received: 21 November 2011; revised: 05 December 2011; accepted: 09 December 2011; published on-line: 15 December 2011
DOI: 10.12921/cmst.2011.17.01.17-23
OAI: oai:lib.psnc.pl:735
Abstract:
Using the self-consistent field theory (SCFT) in spherical unit cells of various dimensionalities, D, a phase diagram of a diblock, A-b-B, is calculated in 5 dimensional space, d = 5. This is an extension of a previuos work for d = 4. The phase diagram is parameterized by the chain composition, f, and incompatibility between A and B, quantified by the product χN. We predict 5 stable nanophases: layers, cylinders, 3D spherical cells, 4D spherical cells, and 5D spherical cells. In the strong segregation limit, that is for large χN, the order-order transition compositions are determined by the strong segregation theory (SST) in its simplest form. While the predictions of the SST theory are close to the corresponding SCFT extrapolations for d = 4, the extrapolations for d = 5 significantly differ from them. We find that the S5 nanophase is stable in a narrow strip between the ordered S4 nanophase and the disordered phase. The calculated orderdisorder transition lines depend weakly on d, as expected
Key words:
diblock copolymer melt, field theory, order-disorder transition, order-order transition, phase diagram
References:
[1] I.W. Hamley, Developments in Block Copolymer Science and Technology (John Wiley & Sons, Berlin, 2004).
[2] G.H. Fredrickson, The Equlibrium Theory of Inhomogeneous Polymers (Clarendon Press, Oxford, 2006).
[3] T.S. Bailey, C.M. Hardy, T.H. Epps, F.S. Bates, Macromolecules 35, 7007 (2002).
[4] M. Takenaka, T. Wakada, S. Akasaka, S. Nishisuji, K. Saijo, H. Shimizu, M.I. Kim, H. Hasegawa, Macromolecules 40, 4399 (2007).
[5] L. Leibler, Macromolecules 13, 1602 (1980).
[6] M. Banaszak, M.D. Whitmore, Macromolecules 25, 3406 (1992).
[7] J.D. Vavasour, M.D. Whitmore, Macromolecules 25, 5477 (1992).
[8] M.W. Matsen, M. Schick, Phys. Rev. Lett. 72, 2660 (1994).
[9] M.W. Matsen, M.D. Whitmore, J. Chem. Phys. 105, 9698 (1996).
[10] M.W. Matsen, J. Chem. Phys. 114, 10528 (2001).
[11] E.M. Lennon, K. Katsov, G.H. Fredrickson, Phys. Rev. Lett. 101, 138302 (2008).
[12] T. Taniguchi, Journal of the Physical Society of Japan 78, 041009 (2009).
[13] I.W. Hamley, Progress in Polymer Science 34, 1161 (2009).
[14] M.W. Matsen, in Soft Condensed Matter, Vol. 1, edited by G. Gompper and M. Schick (John Wiley & Sons, Berlin, 2005).
[15] E.W. Cochran, C.J. Garcia-Cervera, G.H. Fredrickson, Macromolecules 39, 2449 (2006).
[16] A.K. Khandpur, S. Forster, F.S. Bates, I.W. Hamley, A.J. Ryan, W. Bras, K. Almdal, K. Mortensen, Macromolecules 28, 8796 (1995).
[17] P.G. de Gennes, Scaling Concepts in Polymer Physics (Cornell University Press, Ithaca, 1979).
[18] M.W. Matsen, J. Phys.: Condens. Matter 14, R21 (2002).
[19] A.N. Semenov, Sov. Phys. JETP 61, 733 (1985).
[20] M. Banaszak, A. Koper, K. Lewandowski, P. Knychala, Submitted.
Using the self-consistent field theory (SCFT) in spherical unit cells of various dimensionalities, D, a phase diagram of a diblock, A-b-B, is calculated in 5 dimensional space, d = 5. This is an extension of a previuos work for d = 4. The phase diagram is parameterized by the chain composition, f, and incompatibility between A and B, quantified by the product χN. We predict 5 stable nanophases: layers, cylinders, 3D spherical cells, 4D spherical cells, and 5D spherical cells. In the strong segregation limit, that is for large χN, the order-order transition compositions are determined by the strong segregation theory (SST) in its simplest form. While the predictions of the SST theory are close to the corresponding SCFT extrapolations for d = 4, the extrapolations for d = 5 significantly differ from them. We find that the S5 nanophase is stable in a narrow strip between the ordered S4 nanophase and the disordered phase. The calculated orderdisorder transition lines depend weakly on d, as expected
Key words:
diblock copolymer melt, field theory, order-disorder transition, order-order transition, phase diagram
References:
[1] I.W. Hamley, Developments in Block Copolymer Science and Technology (John Wiley & Sons, Berlin, 2004).
[2] G.H. Fredrickson, The Equlibrium Theory of Inhomogeneous Polymers (Clarendon Press, Oxford, 2006).
[3] T.S. Bailey, C.M. Hardy, T.H. Epps, F.S. Bates, Macromolecules 35, 7007 (2002).
[4] M. Takenaka, T. Wakada, S. Akasaka, S. Nishisuji, K. Saijo, H. Shimizu, M.I. Kim, H. Hasegawa, Macromolecules 40, 4399 (2007).
[5] L. Leibler, Macromolecules 13, 1602 (1980).
[6] M. Banaszak, M.D. Whitmore, Macromolecules 25, 3406 (1992).
[7] J.D. Vavasour, M.D. Whitmore, Macromolecules 25, 5477 (1992).
[8] M.W. Matsen, M. Schick, Phys. Rev. Lett. 72, 2660 (1994).
[9] M.W. Matsen, M.D. Whitmore, J. Chem. Phys. 105, 9698 (1996).
[10] M.W. Matsen, J. Chem. Phys. 114, 10528 (2001).
[11] E.M. Lennon, K. Katsov, G.H. Fredrickson, Phys. Rev. Lett. 101, 138302 (2008).
[12] T. Taniguchi, Journal of the Physical Society of Japan 78, 041009 (2009).
[13] I.W. Hamley, Progress in Polymer Science 34, 1161 (2009).
[14] M.W. Matsen, in Soft Condensed Matter, Vol. 1, edited by G. Gompper and M. Schick (John Wiley & Sons, Berlin, 2005).
[15] E.W. Cochran, C.J. Garcia-Cervera, G.H. Fredrickson, Macromolecules 39, 2449 (2006).
[16] A.K. Khandpur, S. Forster, F.S. Bates, I.W. Hamley, A.J. Ryan, W. Bras, K. Almdal, K. Mortensen, Macromolecules 28, 8796 (1995).
[17] P.G. de Gennes, Scaling Concepts in Polymer Physics (Cornell University Press, Ithaca, 1979).
[18] M.W. Matsen, J. Phys.: Condens. Matter 14, R21 (2002).
[19] A.N. Semenov, Sov. Phys. JETP 61, 733 (1985).
[20] M. Banaszak, A. Koper, K. Lewandowski, P. Knychala, Submitted.
