The translocation dynamics of a polymer chain through a nanopore in the absence of an external driving force is analyzed by means of scaling arguments, fractional calculus, and computer simulations. The problem at hand is mapped on a one-dimensional anomalous diffusion process in terms of the reaction coordinate s ͑i.e., the translocated number of segments at time t͒ and shown to be governed by a universal exponent ␣ =2/͑2 +2−␥ 1 ͒, where is the Flory exponent and ␥ 1 is the surface exponent. Remarkably, it turns out that the value of ␣ is nearly the same in two and three dimensions. The process is described by a fractional diffusion equation which is solved exactly in the interval 0 Ͻ s Ͻ N with appropriate boundary and initial conditions. The solution gives the probability distribution of translocation times as well as the variation with time of the statistical moments ͗s͑t͒͘ and ͗s 2 ͑t͒͘ − ͗s͑t͒͘ 2 , which provide a full description of the diffusion process. The comparison of the analytic results with data derived from extensive Monte Carlo simulations reveals very good agreement and proves that the diffusion dynamics of unbiased translocation through a nanopore is anomalous in its nature. The dynamics of polymer translocation through a pore has recently received a lot of attention and appears highly relevant in both chemical and biological processes ͓1͔. The theoretical cosideration is usually based on the assumption ͓2-4͔ that the problem can be mapped onto a onedimensional diffusion process. The so-called translocation coordinate ͑i.e., reaction coordinate s͒ is considered as the only relevant dynamic variable. The whole polymer chain of length N is assumed to be in equilibrium with a corresponding free energy F͑s͒ of an entropic nature. The onedimensional ͑1D͒ dynamics of the translocation coordinate then follows the conventional Brownian motion, and the onedimensional Smoluchowski equation ͓5͔ can be used with the free energy F͑s͒ playing the role of an external potential. In the absence of external driving force ͑unbiased translocation͒, the corresponding average first-passage time follows the law ͑N͒ ϰ a 2 N 2 / D, where a is the length of a polymer Kuhn segment and D stands for the proper diffusion coefficient. The question of the choice of the proper diffusion coefficient D, and the nature of the diffusion process, is controversial. Some authors ͓2,3͔ adopt D ϰ N −1 , as for Rouse diffusion, which yields ϰ N 3 as for polymer reptation ͓8͔, albeit the short pore constraint is less severe than that for a tube of length N. In Ref. ͓4͔ it is assumed that D is not the diffusion coefficient of the whole chain but rather that of the monomer just passing through the pore. The unbiased translocation time is then predicted to vary as ϰ N 2 . The latter assumption has been questioned ͓6,7͔ too. Indeed, on the one hand, the mean translocation time scales ͓4͔ as ϳ N 2 , but on the other hand the characteristic Rouse time ͑i.e., the time it takes for a free polymer to diffuse a distance of the order of its gyrati...

-We study the translocation dynamics of a polymer chain threaded through a nanopore by an external force. By means of diverse methods (scaling arguments, fractional calculus and Monte Carlo simulation) we show that the relevant dynamic variable, the translocated number of segments s(t), displays an anomalous diffusive behavior even in the presence of an external force. The anomalous dynamics of the translocation process is governed by the same universal exponent α = 2/(2ν + 2 − γ1), where ν is the Flory exponent and γ1 the surface exponent, which was established recently for the case of non-driven polymer chain threading through a nanopore. A closed analytic expression for the probability distribution function W (s, t), which follows from the relevant fractional Fokker-Planck equation, is derived in terms of the polymer chain length N and the applied drag force f . It is found that the average translocation time scales as τ ∝ f −1 N 2 α −1 . Also the corresponding time-dependent statistical moments, s(t) ∝ t α and s(t) 2 ∝ t 2α reveal unambiguously the anomalous nature of the translocation dynamics and permit direct measurement of α in experiments. These findings are tested and found to be in perfect agreement with extensive Monte Carlo (MC) simulations.

We suggest a theoretical description of the force-induced translocation dynamics of a polymer chain through a nanopore. Our consideration is based on the tensile (Pincus) blob picture of a pulled chain and the notion of a propagating front of tensile force along the chain backbone, suggested by Sakaue [Phys. Rev. E 76, 021803 (2007); Phys. Rev. E 81, 041808 (2010); Eur. Phys. J. E 34, 135 (2011)]. The driving force is associated with a chemical potential gradient that acts on each chain segment inside the pore. Depending on its strength, different regimes of polymer motion (named after the typical chain conformation: trumpet, stem-trumpet, etc.) occur. Assuming that the local driving and drag forces are equal (i.e., in a quasistatic approximation), we derive an equation of motion for the tensile front position X(t). We show that the scaling law for the average translocation time 〈τ〉 changes from <τ> ∼ N2ν/f1/ν to <τ> ∼ N^1+ν/f (for the free-draining case) as the dimensionless force f[over ̃]R=aNνf/T (where a, N, ν, f, and T are the Kuhn segment length, the chain length, the Flory exponent, the driving force, and the temperature, respectively) increases. These and other predictions are tested by molecular-dynamics simulation. Data from our computer experiment indicate indeed that the translocation scaling exponent α grows with the pulling force f[over ̃]R, albeit the observed exponent α stays systematically smaller than the theoretically predicted value. This might be associated with fluctuations that are neglected in the quasistatic approximation.

We study the structural and dynamic properties of a polymer melt in the vicinity of an adhesive solid substrate by means of Molecular Dynamics simulation at various degrees of surface adhesion. The properties of the individual polymer chains are examined as a function of the distance to the interface and found to agree favorably with theoretical predictions. Thus, the adsorbed amount at the adhesive surface is found to scale with the macromolecule length as Γ is proportional to √N, regardless of the adsorption strength. For chains within the range of adsorption we analyze in detail the probability size distributions of the various building blocks: loops, tails and trains, and find that loops and tails sizes follow power laws while train lengths decay exponentially thus confirming some recent theoretical results. The chain dynamics as well as the monomer mobility are also investigated and found to depend significantly on the proximity of a given layer to the solid adhesive surface with onset of vitrification for sufficiently strong adsorption.

We consider the properties of a self-avoiding polymer chain, adsorbed on a solid attractive substrate which is attached with one end to a pulling force. The conformational properties of such chain and its phase behavior are treated within a Grand Canonical Ensemble (GCE) approach. We derive theoretical expressions for the mean size of loops, trains, and tails of an adsorbed chain under pulling as well as values for the universal exponents which describe their probability distribution functions. A central result of the theoretical analysis is the derivation of an expression for the crossover exponent φ, characterizing polymer adsorption at criticality, φ = α − 1, which relates the precise value of φ to the exponent α, describing polymer loop statistics. We demonstrate that 1 − γ11 < α < 1 + ν, depending on the possibility of a single loop to interact with neighboring loops in the adsorbed polymer. The universal surface loop exponent γ11 ≈ −0.39 and the Flory exponent ν ≈ 0.59.We present the adsorption-desorption phase diagram of a polymer chain under pulling and demonstrate that the relevant phase transformation becomes first order whereas in the absence of external force it is known to be a continuous one. The nature of this transformation turns to be dichotomic, i.e., coexistence of different phase states is not possible. These novel theoretical predictions are verified by means of extensive Monte Carlo simulations.