A variational approach to the localization transition of heteropolymers at interfaces

Trovato, Antonio; Maritan, Amos

doi:10.1209/epl/i1999-00260-6

Cited by 18 publications

(21 citation statements)

References 22 publications

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“…[20] and [26]) and that h(·) = h c (·) (cf. [27]) are set forth. The approaches are non rigorous, mostly based on replica computations, with the exception of [20] whose method is the real space renormalization technique for one-dimensional disordered systems first proposed in [13] in the context of quantum Ising model with transverse field and then applied with remarkably precise results to random walk in random environment, see e. g. [17].…”

Section: 4mentioning

confidence: 99%

A Numerical Approach to Copolymers at Selective Interfaces

2006

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Abstract. We consider a model of a random copolymer at a selective interface which undergoes a localization/delocalization transition. In spite of the several rigorous results available for this model, the theoretical characterization of the phase transition has remained elusive and there is still no agreement about several important issues, for example the behavior of the polymer near the phase transition line. From a rigorous viewpoint non coinciding upper and lower bounds on the critical line are known.In this paper we combine numerical computations with rigorous arguments to get to a better understanding of the phase diagram. Our main results include: -Various numerical observations that suggest that the critical line lies strictly in between the two bounds.-A rigorous statistical test based on concentration inequalities and super-additivity, for determining whether a given point of the phase diagram is in the localized phase. This is applied in particular to show that, with a very low level of error, the lower bound does not coincide with the critical line. -An analysis of the precise asymptotic behavior of the partition function in the delocalized phase, with particular attention to the effect of rare atypical stretches in the disorder sequence and on whether or not in the delocalized regime the polymer path has a Brownian scaling. -A new proof of the lower bound on the critical line. This proof relies on a characterization of the localized regime which is more appealing for interpreting the numerical data.

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Section: 4mentioning

confidence: 99%

A Numerical Approach to Copolymers at Selective Interfaces

2006

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“…The problem has been set forth, at least in the terms that we are stating here, in [9], where replica techniques are used to derive a phase diagram like the one depicted in Figure 1. In [9] there is a hint to the possibility that h c (0) = 1 and such a viewpoint is taken up more seriously in [15]. However in [14] the value of 2/3 appears: the analysis is still based on replicas and ω 1 is standard Gaussian.…”

Section: 2mentioning

confidence: 99%

“…It is widely believed (e.g. [9], [11], [14], [15]) that the transition is of order larger than one, which implies that the length of the polymer excursions diverges approaching the critical curve from inside L. Deriving a lower bound on the free energy boils down to guess the typical behavior of the polymer close to the critical curve: a precise control on the free energy would require the proper scaling of the excursions. Below we present two localization strategies which lead to different lower bounds.…”

Section: 2mentioning

confidence: 99%

On the Localization Transition of Random Copolymers Near Selective Interfaces

Bodineau¹,

Giacomin²

2004

J Stat Phys

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Abstract. In this note we consider the (de)localization transition for random directed (1 + 1)-dimensional copolymers in the proximity of an interface separating selective solvents. We derive a rigorous lower bound on the free energy. This yields a substantial improvement on the bounds from below on the critical line that were known so far. Our result implies that the critical curve does not lie below the critical curve conjectured by Monthus [11] on the base of a renormalization group analysis. We discuss this result in the light of the (rigorous and non rigorous) approaches present in the literature and, by making an analogy with a particular asymptotics of a disordered wetting model, we propose a simplified framework in which the question of identifying the critical curve, as well as understanding the nature of the transition, may be approached.

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“…It is quite obvious, from the form of the Hamiltonian H n , equation (25), that there is no discrete ground state eigenvalue, since there is no bounded "well" in the energy. However, following Garel et al [7], we investigate what a ground state dominance approximation predicts, assuming that a ground state eigenvector can be defined.…”

Section: Hartree Approximation 41 Principle Of the Approximationmentioning

confidence: 99%

“…We believe that this failure comes at least partly from the absence of a ground state, as mentioned above. It has also been shown recently that the replica symmetry assumption can be invalid [25]. In particular, this method cannot predict the concentration profile of the monomers at the interface.…”

Section: Ground State In the Hartree Approximationmentioning

confidence: 99%

Adsorption of a Gaussian random copolymer chain at an interface

Châtellier

Joanny

2000

The European Physical Journal E - Soft Matter

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We consider the adsorption of an isolated, Gaussian, random, and quenched copolymer chain at an interface. We first propose a simple analytical method to obtain the adsorption/depletion transition, by averaging over the disorder the partition function instead of the free energy. The adsorption thresholds obtained by previous authors at a solid/liquid and at a liquid/liquid interface for multicopolymer chains can be rederived using this method. We also compare the adsorption thresholds obtained for bimodal and for Gaussian disorder; they only agree for small disorder. We focus on the specific case of an ideally flat asymmetric liquid/liquid interface, and consider the situation where the chain is composed of monomers of two different chemical species A and B. The replica method is developed for this case. We show that the Hartree approximation, coupled to a replica symmetry assumption, leads to the same adsorption thresholds as obtained from our general method. In order to describe the properties of the adsorbed (or depleted) chain, we develop a new approximation for long chains, within the framework of the replica theory. In most cases, the behavior of a random copolymer chain can be mapped onto that of a homopolymer chain at an asymmetric attractive interface. The values of the effective adsorption energy are different for a random and a periodic copolymer chain. Finally, we consider the case of uncorrelated annealed disorder. The behavior of an annealed chain can be mapped onto that of a homopolymer chain at an asymmetric non attractive interface; hence, an annealed chain cannot adsorb at an asymmetric interface. PACS. 05.90.+m Other topics in statistical physics, thermodynamics, and nonlinear dynamical systems -68.10.-m Fluid surfaces and fluid-fluid interfaces -82.65.Dp Thermodynamics of surfaces and interfaces List of notations a size of a monomer, D length defined by equation (A.10), D eff length defined by D eff ≡ a/ 6|∆χ eff |, f fraction of charged monomers, i discrete curvilinear coordinate along the chain, s continuous curvilinear coordinate along the chain, u operator defined by u ≡ n α=1 θ(z α ), u(s) real number defined by u(s) ≡ n α=1 θ z α (s) , U energy of adsorption per monomer,potential acting on a monomer located at a position z, z position with respect to the interface (z < 0 in the water phase), δ length defined by δ ≡ a/ ∆χ/6, θ(z) Heaviside function, defined by θ(z < 0) = 0 and θ(z > 0) = 1, ξ(s) parameter describing the chemical nature of the sth monomer: ξ(s) = χ A in the case of a A monomer and ξ(s) = −χ B for a B monomer, a

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A variational approach to the localization transition of heteropolymers at interfaces

Cited by 18 publications

References 22 publications

A Numerical Approach to Copolymers at Selective Interfaces

A Numerical Approach to Copolymers at Selective Interfaces

On the Localization Transition of Random Copolymers Near Selective Interfaces

Adsorption of a Gaussian random copolymer chain at an interface

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