Exact curvilinear diffusion coefficients in the repton model

A) A two dimensional representation of a polymer chain with 7 reptons and the underlaying cell structure in the repton model.
B) The curvilinear representation of the same chain with horizontal lines linking two reptons in the same cell and with inclined lines for two reptons in different cells. The arrows represent the possible moves of the reptons with their respective rates.
C) The representation of a polymer chain in the model with hardcore reptons. The existence of a gap between two reptons represents a hole. The possible moves of the reptons and their respective rates are represented by arrows. The identical curvilinear dynamics of both models is evident comparing the rates in B) and C).

The Rubinstein-Duke or repton model is one of the simplest lattice model of reptation for the diffusion of a polymer in a gel or a melt. Recently, a slightly modified model with hardcore interactions between the reptons has been introduced. The curvilinear diffusion coefficients of both models are exactly determined for all chain lengths. The case of periodic boundary conditions is also considered.

The reptation of a polymer in an entangled melt was studied long ago by De Gennes who predicted the polymer length dependences of the curvilinear and self diffusion coefficients as well as the viscosity and the relaxation time. Later, Rubinstein introduced a lattice model for the polymer motion incorporating most of De Gennes' ideas of reptation. The so called repton model was then generalized by Duke to take into account the case of charged polymers during gel electrophoresis allowing the determination of the drift velocity. This repton model seems particularly well adapted for DNA gel electrophoresis when the pore size is comparable to the persistence length.

The theoretical prediction of the viscosity dependence with the polymer length is in apparent conflict with the experimental observations. This discrepancy is also observed in numerical simulations of the repton model. Furthermore, a lot of interest in the calculation of the self-diffusion in the repton model focussed  on the long polymer limit. The next to leading order term of the self-diffusion was long debated due to discrepancy between analytical and numerical results.

At the same time, the diffusion of a polymer chain in small channels attracts an increasing interest since it applies to a great range of experimental situations.  Brochard and De Gennes considered the case of a flexible polymer in a channel large compared to the monomer size but small compared to the polymer length whereas Odijk studied the case of stiff polymers with a persistence length larger than the channel width. The recent experimental access to nanometer scale channels allows to study the crossover behaviour between both regimes. The transport of long flexible polymer chains through Carbon nanotubes bring the interest to channels of width comparable to the monomer size allowing the determination of the curvilinear diffusion coefficients. Furthermore, the case of polymer diffusion in porous media with nanometer scale holes has been recently studied with a slightly modified repton model. This model presents identical dynamical rules for the curvilinear motion of the chain than the repton model and motivated the present interest on the analytical determination of the curvilinear diffusion coefficients as function of the chain length.

In the following paper, we have determined exactly the curvilinear diffusion coefficients of the repton model. The cases of open and periodic boundary conditions were considered. The inverse curvilinear diffusion coefficients present a linear behavior with the number of reptons in both situations. The next to leading order term in the curvilinear diffusion coefficient  shows interesting properties.

Exact curvilinear diffusion coefficients in the repton model A. Buhot Eur. Phys. J. E 18, 239-244 (2005).


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