We study the phase transitions of a random copolymer chain with quenched disorder. We apply a replica variational approach based on a Gaussian trial Hamiltonian in terms of the correlation functions of monomer Fourier coordinates. This allows us to study collapse, phase separation and freezing transitions within the same mean field theory. The effective free energy of the system is derived analytically and analysed numerically. Such quantities as the radius of gyration or the average value of the overlap between different replicas are treated as observables and evaluated by introducing appropriate external fields to the Hamiltonian. We obtain the phase diagram and show that this system exhibits a scale dependent freezing transition. The correlations between replicas appear at different length scales as the temperature decreases. This indicates the existence of the topological frustration.
We present the results of our study of the freezing transition of an amphiphilic random copolymer. We here confirm that a replica variational approach predicts a "scale"dependent freezing transition due to the connectivity of the chain. In addition we suggest that two systems, a random copolymer and an Ising spin-glass, can be directly related to each other on the mean-field level in the vicinity of the freezing transition. Both systems have the same type of effective free energy. The properties of replica symmetrical (RS) and replica symmetry broken (RSB) solutions are discussed. The latter has larger radius of gyration and effective free energy, and is less phase separated. It might be related to a globule with more than one hydrophobic core.
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