Source: Ann. Appl. Probab. Volume 15, Number 1B
(2005), 963-983.
We analyze a (1+1)-dimension directed random walk model of a polymer dipped in a medium constituted by two immiscible solvents separated by a flat interface. The polymer chain is heterogeneous in the sense that a single monomer may energetically favor one or the other solvent. We focus on the case in which the polymer types are periodically distributed along the chain or, in other words, the polymer is constituted of identical stretches of fixed length. The phenomenon that one wants to analyze is the localization at the interface: energetically favored configurations place most of the monomers in the preferred solvent and this can be done only if the polymer sticks close to the interface.
We investigate, by means of large deviations, the energy–entropy competition that may lead, according to the value of the parameters (the strength of the coupling between monomers and solvents and an asymmetry parameter), to localization. We express the free energy of the system in terms of a variational formula that we can solve. We then use the result to analyze the phase diagram.
References
Biskup, M. and den Hollander, F. (1999). A heteropolymer near a linear interface. Ann. Appl. Probab. 9 668--687.
Bolthausen, E. (1987). Markov process large deviations in $\tau$-topology. Stochastic Process. Appl. 25 95--108.
Mathematical Reviews (MathSciNet):
MR904267
Bolthausen, E. and den Hollander, F. (1997). Localization transition for a polymer near an interface. Ann. Probab. 25 1334--1366.
Chatelin, F. (1993). Eigenvalues of Matrices. Wiley, Chichester.
Dembo, A. and Zeitouni, O. (1998). Large Deviations Techniques and Applications, 2nd ed. Springer, New York.
Feller, W. (1968). An Introduction to Probability Theory and its Applications 1, 3rd ed. Wiley, New York.
Mathematical Reviews (MathSciNet):
MR228020
Giacomin, G. (2003). Localization phenomena for random polymer models. Course lecture notes. Available at http://felix.proba.jussieu.fr/pageperso/giacomin/GBpage.html.
Grosberg, A. Yu., Izrailev, S. and Nechaev, S. (1994). Phase transition in a heteropolymer chain at a selective interface. Phys. Rev. E 50 1912--1921.
Isozaki, Y. and Yoshida, N. (2001). Weakly pinned random walk on the wall: Pathwise descriptions of the phase transition. Stochastic Process. Appl. 96 261--284.
Minc, H. (1988). Nonnegative Matrices. Wiley, New York.
Mathematical Reviews (MathSciNet):
MR932967
Monthus, C. (2000). On the localization of random heteropolymers at the interface between two selective solvents. European Phys. J. B 13 111--130.
Monthus, C., Garel, T. and Orland, H. (2000). Copolymer at a selective interface and two dimensional wetting: A grand canonical approach. European Phys. J. B 17 121--130.
Janse van Rensburg, E. J. and Rechnitzer, A. (2001). Exchange relations, dyck paths and copolymer adsorption. Preprint.
Sinai, Ya. G. (1993). A random walk with a random potential. Theory Probab. Appl. 38 382--385.
Sinai, Ya. G. and Spohn, H. (1996). Remarks on the delocalization transition for heteropolymers. In Topics in Statistical and Theoretical Physics (R. L. Dobrushin, R. A. Minlos, M. A. Shubin and A. M. Vershik, eds.) 219--223. Amer. Math. Soc., Providence, RI.
Sommer, J.-U. and Daoud, M. (1995). Copolymers at selective interfaces. Europhys. Lett. 32 407--412.
