TY - BOOK
T1 - Phase diagram for a copolymer in a micro-emulsion
AU - Hollander, den, W.Th.F.
AU - Petrelis, N.R.
PY - 2013
Y1 - 2013
N2 - In this paper we study a model describing a copolymer in a micro-emulsion. The copolymer consists of a random concatenation of hydrophobic and hydrophilic monomers, the micro-emulsion consists of large blocks of oil and water arranged in a percolation-type fashion. The interaction Hamiltonian assigns energy –a to hydrophobic monomers in oil and energy –ß to hydrophilic monomers in water, where a,ß are parameters that without loss of generality are taken to lie in the cone {(a,ß) ¿ R^2 : a = |ß|}. Depending on the values of these parameters, the copolymer either stays close to the oil-water interface (localization) or wanders off into the oil and/or the water (delocalization). We derive two variational formulas for the quenched free energy per monomer, one that is "column-based" and one that is "slope-based". Using these variational formulas we identify the phase diagram in the (a,ß)-cone. There are two regimes: supercritical (the oil blocks percolate) and subcritical (the oil blocks do not percolate). The supercritical and the subcritical phase diagram each have two localized phases and two delocalized phases, separated by four critical curves meeting at a quadruple critical point. The different phases correspond to the different ways in which the copolymer can move through the micro-emulsion.
Keywords: Random copolymer, random micro-emulsion, free energy, percolation, variational formula, large deviations, concentration of measure.
AB - In this paper we study a model describing a copolymer in a micro-emulsion. The copolymer consists of a random concatenation of hydrophobic and hydrophilic monomers, the micro-emulsion consists of large blocks of oil and water arranged in a percolation-type fashion. The interaction Hamiltonian assigns energy –a to hydrophobic monomers in oil and energy –ß to hydrophilic monomers in water, where a,ß are parameters that without loss of generality are taken to lie in the cone {(a,ß) ¿ R^2 : a = |ß|}. Depending on the values of these parameters, the copolymer either stays close to the oil-water interface (localization) or wanders off into the oil and/or the water (delocalization). We derive two variational formulas for the quenched free energy per monomer, one that is "column-based" and one that is "slope-based". Using these variational formulas we identify the phase diagram in the (a,ß)-cone. There are two regimes: supercritical (the oil blocks percolate) and subcritical (the oil blocks do not percolate). The supercritical and the subcritical phase diagram each have two localized phases and two delocalized phases, separated by four critical curves meeting at a quadruple critical point. The different phases correspond to the different ways in which the copolymer can move through the micro-emulsion.
Keywords: Random copolymer, random micro-emulsion, free energy, percolation, variational formula, large deviations, concentration of measure.
M3 - Report
T3 - Report Eurandom
BT - Phase diagram for a copolymer in a micro-emulsion
PB - Eurandom
CY - Eindhoven
ER -