Supersymmetry theory of microphase separation in homopolymer-oligomer mixtures

Alexander Olemskoi; Ivan Krakovsky; Alexey Savelyev

doi:10.1103/PhysRevE.69.021803

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Supersymmetry theory of microphase separation in homopolymer-oligomer mixtures

Alexander Olemskoi^*, Ivan Krakovsky^†, and Alexey Savelyev^‡

Physical and Technical Department, Sumy State University, Rimskii-Korsakov Street 2, 40007 Sumy, Ukraine
Department of Macromolecular Physics, Charles University V Holešovičkách 2, 180 00 Prague 8, Czech Republic

^*Electronic address: olemskoi@ssu.sumy.ua
^†Electronic address: ivank@kmf.troja.mff.cuni.cz
^‡Electronic address: alexsav@kmf.troja.mff.cuni.cz

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Phys. Rev. E 69, 021803 – Published 17 February, 2004

DOI: https://doi.org/10.1103/PhysRevE.69.021803

Abstract

The mesoscopic structure of periodically alternating layers of stretched homopolymer chains surrounded by perpendicularly oriented oligomeric tails is studied for systems with both strong (ionic) and weak (hydrogen) interactions. We focus on the consideration of the distribution of oligomers along the homopolymer chains that is described by the effective equation of motion with the segment number playing the role of imaginary time. The supersymmetry technique is developed to consider associative hydrogen bonding, self-action effects, inhomogeneity, and temperature fluctuations in the oligomer distribution. Making use of the self-consistent approach allows one to explain experimentally observed temperature dependence of the structure period and the order-disorder transition temperature and period as functions of the oligomeric fraction for systems with different bonding strengths. A whole set of parameters of the model used is found for strong, intermediate, and weak coupled systems being Poly (4-vinyl pyridine)–dodecyl benzene sulfonic acid [P4VP- $(DBSA)_{x}],$ P4VP- $[Zn (DBS)_{2}]_{x},$ and P4VP- 3-pentadecyl ${Phenol}_{x},$ respectively. A passage from the former two to the latter is shown to cause a crucial decrease in the magnitude of both parameters of hydrogen bonding and self-action, as well as the order-disorder transition temperature.

References (23)

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Strictly speaking, the above definition $δ c (n) \equiv c (n) - \bar{c}$ of fluctuations is applicable to the disordered phase only $(T > T_{c}) .$ Below the critical temperature $(T < T_{c}),$ one needs to pass to the frequency representation $δ c (ν) \equiv c (ν) - \bar{c} δ (ν - ν_{0})$ where the proper frequency $ν_{0} \equiv 2 π (b / λ)$ is relevant to the period λ of the oligomer alternation in the ordered phase.

[c1] M. Antonietti, A. Wenzel, and A. Thunemann, Langmuir 12, 2111 (1996).

[c2] O. Ikkala, J. Ruokolainen, G. Ten Brinke, M. Torkkeli, and R. Serimaa, Macromolecules 28, 7088 (1995).

[c3] J. Ruokolainen, G. Ten Brinke, O.T. Ikkala, M. Torkkeli, and R. Serimaa, Macromolecules 29, 3409 (1996).

[c4] J. Ruokolainen et al., Phys. Rev. E 54, 6646 (1996).

[c5] G. Ten Brinke, J. Ruokolainen, and O. Ikkala, Europhys. Lett. 35, 91 (1996).

[c6] F. Tanaka and M. Ishida, Macromolecules 30, 1836 (1997).

[c7] E. Dormidontova and G. Ten Brinke, Macromolecules 31, 2649 (1998).

[c8] L. Leibler, Macromolecules 13, 1602 (1980).

[c9] G.N. Fredrickson, S.T. Milner, and L. Leibler, Macromolecules 25, 6341 (1992).

[c10] C.D. Sfatos and E.I. Shakhnovich, Phys. Rep. 288, 77 (1997).

[c11] J. Zinn-Justin, Quantum Field Theory and Critical Phenomena (Clarendon Press, Oxford, 1993).

[c12] A.I. Olemskoi, Physica A 270, 444 (1999).

[c13] A.I. Olemskoi and V.A. Brazhnyi, Physica A 273, 368 (1999).

[c14] A.I. Olemskoi, Usp. Fiz. Nauk. 171, 503 (2001) [Phys. Usp. 44, 479 (2001)].

[c15] T.A. Vilgis, J. Phys. A 24, 5321 (1991).

[c16] P.C. Martin, E.D. Siggia, and H.A. Rose, Phys. Rev. A 8, 423 (1973).

[c17] J. Kurchan, J. Phys. I 2, 1333 (1992).

[c18] L. D. Landau and E. M. Lifshitz, Statistical Physics, Part 1 (Pergamon Press, Oxford, 1980).

[c19] A. I. Olemskoi, Theory of Structure Transformations in Non-Equilibrium Condensed Matter (NOVA Science, New York, 1999).

[c20] K.R. Shull, Macromolecules 25, 2122 (1992).

[c21] E. Helfand and Y. Tagani, J. Polym. Sci., Part B: Polym. Lett. 9, 799 (1971).

[c22] R. Remmert, Theory of Complex Functions (Springer, New York, 1991).

[c23] Strictly speaking, the above definition $δ c (n) \equiv c (n) - \bar{c}$ of fluctuations is applicable to the disordered phase only $(T > T_{c}) .$ Below the critical temperature $(T < T_{c}),$ one needs to pass to the frequency representation $δ c (ν) \equiv c (ν) - \bar{c} δ (ν - ν_{0})$ where the proper frequency $ν_{0} \equiv 2 π (b / λ)$ is relevant to the period λ of the oligomer alternation in the ordered phase.

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Supersymmetry theory of microphase separation in homopolymer-oligomer mixtures

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References (23)

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