Abstract
In this paper we review some recent results, obtained jointly with Stu Whittington, for a mathematical model describing a copolymer in an emulsion. The copolymer consists of hydrophobic and hydrophilic monomers, concatenated randomly with equal density. The emulsion consists of large blocks of oil and water, arranged in a percolation-type fashion. To make the model mathematically tractable, the copolymer is allowed to enter and exit a neighboring pair of blocks only at diagonally opposite corners. The energy of the copolymer in the emulsion is minus α times the number of hydrophobic monomers in oil minus β times the number of hydrophilic monomers in water. Without loss of generality we may assume that the interaction parameters are restricted to the cone \({\{(\alpha,\beta)\in \mathbb{R}^2\colon\,|\beta|\leq\alpha\}}\). We show that the phase diagram has two regimes: (1) in the supercritical regime where the oil blocks percolate, there is a single critical curve in the cone separating a localized and a delocalized phase; (2) in the subcritical regime where the oil blocks do not percolate, there are three critical curves in the cone separating two localized phases and two delocalized phases, and meeting at two tricritical points. The different phases are characterized by different behavior of the copolymer inside the four neighboring pairs of blocks.
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References
Giacomin G.: Random Polymer Models. Imperial College Press, London (2007)
den Hollander F.: Random Polymers, Lecture Notes in Mathematics 1974. Springer, Berlin (2009)
den Hollander F., Whittington S.G.: Localization transition for a copolymer in an emulsion. Theor. Prob. Appl. 51, 193–240 (2006)
den Hollander F., Pétrélis N.: On the localized phase of a copolymer in an emulsion: supercritical percolation regime. Commun. Math. Phys. 285, 825–871 (2009)
den Hollander F., Pétrélis N.: On the localized phase of a copolymer in an emulsion: subcritical percolation regime. J. Stat. Phys. 134, 209–241 (2009)
Acknowledgments
NP was supported by a postdoctoral fellowship from the Netherlands Organization for Scientific Research (grant 613.000.438).
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Invited paper on the occasion of the 65th birthdays of Ray Kapral and Stu Whittington from the Department of Chemistry at the University of Toronto.
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Open Access This is an open access article distributed under the terms of the Creative Commons Attribution Noncommercial License (https://creativecommons.org/licenses/by-nc/2.0), which permits any noncommercial use, distribution, and reproduction in any medium, provided the original author(s) and source are credited.
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den Hollander, F., Pétrélis, N. A mathematical model for a copolymer in an emulsion. J Math Chem 48, 83–94 (2010). https://doi.org/10.1007/s10910-009-9564-y
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DOI: https://doi.org/10.1007/s10910-009-9564-y