Replica field theory for a polymer in random media

Area of Science:

• Statistical Mechanics
• Polymer Physics
• Disordered Systems

Background:

• Revisiting the problem of a polymer chain in a random medium, previously studied by Edwards and Muthukumar.
• Addressing a discrepancy noted by Cates and Ball between replica calculations and theoretical expectations for polymer behavior in infinite media.
• Investigating the concept of a 'localization transition' from stretched to collapsed polymer conformations in finite volumes.

Purpose of the Study:

• To analytically derive polymer behavior in a random medium using replica calculations with a confining potential.
• To reconcile existing theoretical predictions regarding polymer collapse in infinite versus finite volumes.
• To explore the conditions under which a polymer chain transitions between stretched and collapsed states.

Main Methods:

• Employing replica calculation techniques in the presence of a confining harmonic potential.
• Utilizing a variational scheme with five variational parameters for analytical derivation.
• Applying a one-step replica symmetry breaking solution.

Main Results:

• Derived an analytical result for the radius of gyration (R) in d<4 dimensions: R ~ (g ln V)^(-1/(4-d)).
• Confirmed that in the infinite volume limit, polymers always collapse.
• Showed that for finite volumes, a transition from stretched to collapsed states is possible as a function of disorder strength (g).

Conclusions:

• The study reconciles previous findings, showing that the Edwards and Muthukumar result is recovered with an effective volume term.
• Demonstrated that polymers collapse in the infinite volume limit and can exhibit transitions in finite volumes.
• Highlighted the critical role of replica symmetry breaking in obtaining these results for polymer chains in random media.