Cates, M. E. (Michael E.) and Ball, R. C.. (1988) Statistics of a polymer in a random potential, with implications for a nonlinear interfacial growth model. Journal de Physique, Vol.49 (No.12). pp. 2009-2018. ISSN 0302-0738

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Abstract

We examine and extend recent results for the statistics of a gaussian polymer chain of length t in a quenched random potential μ (r). This problem can be mapped onto one involving the nonlinear evolution equation h (r, t) = ∇2h + (∇h)2 - μ(r) for a growing interface in the presence of a time-independent, but spatially varying, random flux - μ (r). [The corresponding problem for a directed polymer, equivalent to a flux random in both time and space, was studied by M. Kardar, G. Parisi and Y.-C. Zhang, Phys. Rev. Lett. 56 (1986) 889.] A free chain in a true white-noise potential, in space dimensions d = 2, 3, is predicted to collapse to a linear size R ˜ v1 /(d - 4)[ln V] 1/(d - 4) where v is the mean-square potential fluctuation (presumed small) and V the volume of the system. This agrees with a recent replica calculation of S. F. Edwards and M. Muthukumar (J. Chem. Phys., 89 (1988) 2435), but with an extra logarithmic dependence on V. The difference between this result and that for an annealed potential is commented upon, with reference to the phenomenon of localization as it applies to polymers in a quenched environment ; the effect of a saturating potential is discussed. In contrast, a chain with one end fixed at a particular place (say at r) is typically arranged as a tadpole consisting of a collapsed head region (occupying a deep minimum of μ) connected to r by an extended tail. The free energy surface of such a chain F (r, t ) consists asymptotically of a series of conical valleys, separated by sharp ridges. Several results for the statistics of this surface are obtained. The evolved profile in the equivalent interfacial growth model is obtained from h (r, t ) = - F (r, t ), and hence corresponds to a series of conical mountains.

Item Type:	Journal Article
Subjects:	Q Science > QA Mathematics Q Science > QC Physics Q Science > QP Physiology
Divisions:	Faculty of Science > Physics
Library of Congress Subject Headings (LCSH):	Molecular dynamics, Polymers -- Statistical methods, Nonlinear mechanics
Journal or Publication Title:	Journal de Physique
Publisher:	Editions Physique
ISSN:	0302-0738
Date:	1988
Volume:	Vol.49
Number:	No.12
Page Range:	pp. 2009-2018
Identification Number:	10.1051/jphys:0198800490120200900
Status:	Peer Reviewed
Publication Status:	Published
URI:	http://wrap.warwick.ac.uk/id/eprint/39600

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