Universality of crossover scaling for the adsorption transition of lattice polymers

Abstract

Recently, it has been proposed that the adsorption transition for a single polymer in dilute solution, modeled by lattice walks in three dimensions, is not universal with respect to intermonomer interactions. Moreover, it has been conjectured that key critical exponents ϕ, measuring the growth of the contacts with the surface at the adsorption point, and 1/δ, which measures the finite-size shift of the critical temperature, are not the same. However, applying standard scaling arguments the two key critical exponents should rather be identical, hence pointing to a potential breakdown of these standard scaling arguments. Both of these conjectures are in contrast to the well-studied situation in two dimensions, where there are exact results from conformal field theory: these exponents are both accepted to be 1/2 and universal. We use the flatPERM algorithm to simulate self-avoiding walks and trails on the hexagonal, square, and simple cubic lattices up to length 1024 to investigate these claims. Walks can be seen as a repulsive limit of intermonomer interaction for trails, allowing us to probe the universality of adsorption. For each lattice model we analyze several thermodynamic properties to produce different methods of estimating the critical temperature and the key exponents. We test our methodology on the two-dimensional cases, and the resulting spread in values for ϕ and 1/δ indicates that there is a systematic error which can far exceed the statistical error usually reported. We further suggest a methodology for consistent estimation of the key adsorption exponents which gives ϕ=1/δ=0.484(4) in three dimensions. Hence, we conclude that in three dimensions these critical exponents indeed differ from the mean-field value of 1/2, as had previously been calculated, but cannot find evidence that they differ from each other. Importantly, we also find no substantive evidence of any nonuniversality in the polymer adsorption transition.

Phase transitions in solvent-dependent polymer adsorption in three dimensions

“…For polymers fully adsorbed to the surface φ = 1, but at the adsorption transition it is expected that φ = 1/2, from mean-field predictions. The three-dimensional case is slightly different with numerical evidence suggesting a slight deviation from the meanfield value; recent Monte Carlo studies converge near Grassberger's value φ ≈ 0.48 [6,9,17,18] but other values are possible [28]. Further, there is some disagreement over the value of φ as the bulk interaction is changed from non-interacting (ω = 1) to strongly repulsive (ω = 0) [20].…”

“…In a good solvent, φ becomes a crossover exponent at the critical temperature T ads controlling critical behavior and it has been proposed [15,16] that φ = 1/2 for SAWs in any dimension, making φ superuniversal at the adsorption transition. However, recent consensus due to numerical simulation is that φ is not super universal * chris.bradly@unimelb.edu.au † owczarek@unimelb.edu.au ‡ t.prellberg@qmul.ac.uk [6,8,9,17]. Numerical simulation is thus a useful tool in this field and can be applied to other questions.…”

“…Since the transition is both a surface and bulk transition, it is not obvious what is the best signature of the transition. The methods we have explored for the adsorption transition [17] do not work as well here. The complexity of the phase diagram limits the range of parameters that are sure to hit only the SAG-AE boundary without probing other transitions in the system.…”

Monte Carlo simulation of 3-star lattice polymers pulled from an adsorbing surface

“…Lastly, we mention that the phase boundaries meet at the point (a c , 1). Accurate determination of a c is difficult, even for SAWs [8]. In principle, we can estimate a c by restricting the analysis to the fixed value y = 1.…”

“…The self-avoiding walk is the canonical model for linear polymers in a dilute solution [2]. Of the many variations to this model, the adsorption of polymers to a surface has attracted considerable attention [3][4][5][6][7][8]. For a walk with one end tethered to an attractive surface the configuration of the polymer is that of an extended coil dominated by entropic repulsion at high temperatures.…”