Self-avoiding walks interacting with a surface

Hammersley, J. M.; Torrie, G. M.; Whittington, S G

doi:10.1088/0305-4470/15/2/023

Cited by 162 publications

(235 citation statements)

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“…This result is consistent with the idea that the chain at the CAP behaves in the same manner as a chain in bulk solution. 14 This provides an alternative method to locate e c .…”

Section: Resultsmentioning

confidence: 99%

“…[13][14][15][16][17] Every walk step that contacts the surface is assigned an interaction energy e (in units of k B T, where k B is the Boltzmann constant and T is the temperature). It is well accepted that a single chain adsorbed on an attractive surface exhibits a phase transition from a desorbed state to an adsorbed state when the adsorption strength increases beyond a critical value.…”

Section: Introductionmentioning

confidence: 99%

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Conformational properties of a polymer tethered to an interacting flat surface

Qian

Sun

et al. 2010

Polym J

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The conformational properties of a lattice self-avoiding polymer chain tethered to an interacting and impenetrable flat surface were simulated using a dynamic Monte Carlo method. The results show that the conformational size reaches a minimum at the critical adsorption point (CAP) and that the scaling behavior of the polymer at the CAP is the same as that in the bulk solution. The results provide a new method to determine the CAP of polymer chains. Polymer Journal (2010) 42, 383-385; doi:10.1038/pj.2010.9; published online 17 March 2010Keywords: critical adsorption point; Monte Carlo simulation; self-avoiding chain INTRODUCTIONThe adsorption of polymer chains on surfaces by means of physical or chemical interactions is an important subject in polymer and biological sciences. A polymer chain may adsorb or desorb, depending on interactions with the surface. This phenomenon is relevant to many technological applications such as in polymer compatibilizers, colloid stabilizers and polymeric surfactants. [1][2][3] In many biological systems, ligands are attached to a surface through flexible tether chains. 4,5 The conformation of the attached tether chains will affect the binding of the ligand to a receptor and will thus influence the whole biological process. 6 The adsorption phenomenon is also a part of many physical systems; for example, polymer chains grafted to colloid particles and block copolymers at liquid-air interfaces. 7 The adsorption of polymers has attracted a large number of theoretical and experimental studies. [7][8][9][10][11][12] The growing interest in polymers interacting with substrates warrants a thorough understanding of the static and dynamic properties of the tethered chain.A mathematical model often used for studying the adsorption of tethered polymers on surfaces is the self-avoiding walk (SAW) chain in a three-dimensional (3D) simple cubic lattice that interacts with a flat surface and is restricted to lie on one side of the surface. 13-17 Every walk step that contacts the surface is assigned an interaction energy e (in units of k B T, where k B is the Boltzmann constant and T is the temperature). It is well accepted that a single chain adsorbed on an attractive surface exhibits a phase transition from a desorbed state to an adsorbed state when the adsorption strength increases beyond a critical value. The CAP e c was estimated to be about À0.29 for polymers on the simple cubic lattice. [9][10][11][12]16,18,19 In this work, we studied how the conformational properties of the chain change from a desorbed state for e4e c to an adsorbed state for eoe c . We found that both the mean square end-to-end distance /R 2 S

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“…This result is consistent with the idea that the chain at the CAP behaves in the same manner as a chain in bulk solution. 14 This provides an alternative method to locate e c .…”

Section: Resultsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Conformational properties of a polymer tethered to an interacting flat surface

Qian

Sun

et al. 2010

Polym J

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“…More recently the h ℓ+1,3 result has been conjectured to be correct by Fendley and Saleur [26] who argued that the boundary operator Φ ℓ+1, 3 propagates down the strip at the special point. In general, our exact results lend further weight to their claim that the spin degrees of freedom of the Kondo problem can be considered as the n = 2 limit of the special transition of the O(n) model [26].…”

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confidence: 92%

“…A long flexible polymer in a good solvent with an attractive short-range force between the polymer and the container wall is known to undergo an adsorption transition [1][2][3][4].…”

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confidence: 99%

Exact Results for the Adsorption of a Flexible Self-Avoiding Polymer Chain in Two Dimensions

Batchelor

Yung

1995

Phys. Rev. Lett.

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We derive the exact critical couplings (x * , y * a ), where y * a /x * = 1 + √ 2 = 1.533 . . . , for the polymer adsorption transition on the honeycomb lattice, along with the universal critical exponents, from the Bethe Ansatz solution of the O(n) loop model at the special transition. Our result for the thermal scaling dimension, and thus the crossover exponent φ = 1 2 , is in agreement with an earlier result based on conformal invariance arguments. Our result for the geometric scaling dimensions confirms recent conjectures that they aregiven by h ℓ+1,3 in the Kac formula.61. 41.+e, 64.60.Cn, 64.60.Kw, 64.60.Fr Typeset using REVT E X 1 A long flexible polymer in a good solvent with an attractive short-range force between the polymer and the container wall is known to undergo an adsorption transition [1][2][3][4].A standard model for this phenomenon is a self-avoiding walk (SAW) on a d-dimensional lattice interacting with a (d − 1)-dimensional substrate. In the lattice model, the SAW has a Boltzmann weight x per monomer (in the bulk), with weight y per adsorbed monomer (on the substrate). At the adsorption transition, y * a , the number of adsorbed monomers scales with the total length L as L a ∼ L φ , where φ is a crossover exponent. The polymer is in the adsorbed phase for y > y * a and the desorbed phase for y < y * a , where the surface attractions are not effective. In the language of surface critical phenomena, the adsorption transition is a special transition [5]. where the sum is over all configurations of closed and nonintersecting loops. Here P is the total number of closed loops of fugacity n in a given configuration. In the limit n → 0 this reduces to the required SAW generating function, with x the fugacity of a step in the bulk and y the fugacity of a step along the surface. Here L is the length of a walk in the bulk and L s is the length of a walk along the surface of the strip.The partition function can be conveniently rewritten in terms of the Boltzmann weights of the empty vertices. To do this we need to distinguish between three classes of vertices: − and − which appear (i) in the bulk and (ii) on the surface, and (iii) and on the surface.For each class we define the weights t b , tb and t s , respectively. We then considerwhere N b , Nb and N s are the total numbers of vertices (either full or empty) of class (i),(ii) and (iii). Apart from harmless normalisation factors, the two partition functions are equivalent if t b = tb, along with the identificationwhere In particular, we found that the known diagonal reflection matrices for the Izergin-Korepin model [24] lead to two integrable sets of boundary weights which preserve the O(n) symmetry [22,25]. For each case t b = tb = 2 cos λ. Thus from (3) the critical bulk fugacity iswhich is the well known bulk critical value [16]. The SAW point occurs at λ = π/8, whereThe two integrable sets of boundary weights are [22] (A) t s = sin 2λ sin λ and (B) t s = cos 2λ cos λ .Thus from (3) case (A) gives the critical surface fugacity y * o = x...

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“…If we set y = 1 (turning off the force) we have the pure adsorption problem and we write ψ + (a, 1) = κ(a). There is a critical value of a, a c > 1, such that κ(a) = κ(1) = log µ d for a ≤ a c and κ(a) > log µ d for a > a c [5]. Here µ d is the growth constant for self-avoiding walks on Z d [4,14].…”

Section: Introductionmentioning

confidence: 99%

Self-avoiding walks adsorbed at a surface and subject to a force

Rensburg¹,

Whittington²

2016

J. Phys. A: Math. Theor.

Self Cite

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Abstract. We consider self-avoiding walks terminally attached to an impenetrable surface at which they can adsorb. We call the vertices farthest away from this plane the top vertices and we consider applying a force at the plane containing the top vertices. This force can be directed away from the adsorbing surface or towards it. In both cases we prove that the free energy (in the thermodynamic limit) is identical to the free energy when the force is applied at the last vertex. This means that the criterion determining the critical force -temperature curve is identical for the two ways in which the force is applied and the response to pushing the walk is also the same in the two cases.

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Self-avoiding walks interacting with a surface

Cited by 162 publications

References 16 publications

Conformational properties of a polymer tethered to an interacting flat surface

Conformational properties of a polymer tethered to an interacting flat surface

Exact Results for the Adsorption of a Flexible Self-Avoiding Polymer Chain in Two Dimensions

Self-avoiding walks adsorbed at a surface and subject to a force

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