# A mathematical model for a copolymer in an emulsion

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J Math Chem (2010) 48:83–94 DOI 10.1007/s10910-009-9564-y ORIGINAL PAPER A mathematical model for a copolymer in an emulsion F. den Hollander · N. Pétrélis Received: 3 June 2007 / Accepted: 22 April 2009 / Published online: 16 July 2009 © The Author(s) 2009. This article is published with open access at Springerlink.com 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 ran- domly 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 diago- nally 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 {(α, β) ∈ R2 : |β| ≤ α}. We show that the phase diagram has two regimes: (1) in the supercritical regime where the oil blocks perco- late, 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. Keywords Random copolymer · Random emulsion · Localization · Delocalization · Phase transition · Percolation 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. F. den Hollander (B) Mathematical Institute, Leiden University, P.O. Box 9512,

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