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Reptation is the thermal motion of very long linear, entangled macromolecules in polymer melts or concentrated polymer solutions. Derived from the word reptile, reptation suggests the movement of entangled polymer chains as being analogous to snakes slithering through one another.

"The forward and backward diffusion is reminiscent of the motion of a snake and for this reason the motion was given the name reptation by de Gennes, after the Latin word reptare, which means to creep." Hunt, Thomas A., Theory and Simulation of Polymer Liquids Under Extensional and Shear Flows, Diss, Swinburne University of Technology, Hawthorn, Australia 2008, Swinburne Search Engine 2013.

You can find confirmation of the Latin translation of the word reptare here, http://en.wiktionary.org/wiki/reptar.

Repton is a mobile point residing in the cells of a lattice, connected by bonds.

Entanglement means the topological restriction of molecular motion by other chains.

The concept of reptation was introduced into polymer physics in 1971 by Pierre-Gilles de Gennes to explain the dependence of the mobility of a macromolecule on its length. Reptation is used as a mechanism to explain the viscous flow in an amorphous polymer. Sir Sam Edwards and Masao Doi later refined reptation theory.

Theory and mechanism

Reptation theory describes the effect of polymer chain entanglements on the relationship between molecular mass and chain relaxation time (or similarly, the polymer’s zero-shear viscosity). The theory predicts that, in entangled systems, the relaxation time τ is proportional to the cube of molecular mass. This is a reasonable approximation of the actual observed relationship, τ~(Molecular Mass)^3.4.

The prediction of the theory is arrived at by a relatively simple argument. First, each polymer chain is envisioned as occupying a tube of length L, through which it may move with snake-like motion (creating new sections of tube as it moves). Furthermore, if we consider a time scale comparable to τ, we may focus on the overall, global motion of the chain. Thus, we define the tube mobility as μ_tube=v/f, where v is the velocity of the chain when it is pulled by a force f. Note also that μ_tube will be inversely proportional to the degree of polymerization (and thus also inversely proportional to chain weight).

The diffusivity of the chain through the tube may then be written as D_tube=k_BTμ_tube. By then recalling that in 1-dimension the mean squared displacement due to Brownian motion is given by s(t)² = 2D_tubet, we obtain s(t)²=2k_BTμ_tubet. The time necessary for a polymer chain to displace the length of its original tube is then t=L²/(2k_BTμ_tube). By noting that this time is comparable to the relaxation time, we establish that τ~L²/μ_tube. Since the length of the tube is proportional to the degree of polymerization, and μ_tube is inversely proportional to the degree of polymerization, we observe that τ~(DP_n)³ (and so τ~(Molecular Mass)³).

From the preceding analysis, we see that molecular mass has a very strong effect on relaxation time in entangled polymer systems. Indeed, this is significantly different from the untangled case, where relaxation time is observed to be proportional to molecular mass. This strong effect can be understood by recognizing that, as chain length increases, the number of tangles present will dramatically increase. These tangles serve to reduce chain mobility. The corresponding increase in relaxation time can result in viscoelastic behavior, which is often observed in polymer melts.

Models

The blob model, explaining the entanglement of long polymer chains.

The tube model, explaining the basically one-dimensional mobility of long polymer chains.

Entangled polymers are characterized with effective internal scale, commonly known as the length of macromolecule between adjacent entanglements $M_e$.

Entanglements with other polymer chains restrict polymer chain motion to a thin virtual tube passing through the restrictions. Without breaking polymer chains to allow the restricted chain to pass through it, the chain must be pulled or flow through the restrictions. The mechanism for movement of the chain through these restrictions is called reptation.

In the blob model, the polymer chain is made up of $n$ Kuhn lengths of individual length $l$. The chain is assumed to form blobs between each entanglement, containing $n_e$ Kuhn length segments in each. The mathematics of random walks can show that the average end-to-end distance of a section of a polymer chain, made up of $n_e$ Kuhn lengths is $d=l\sqrt{n_e}$. Therefore if there are $n$ total Kuhn lengths, and $A$ blobs on a particular chain:

$A=\frac{n}{n_e}$

The total end-to-end length of the restricted chain $L$ is then:

$L=Ad=\frac{nl\sqrt{n_e}}{n_e}=\frac{nl}{\sqrt{n_e}}$

This is the average length a polymer molecule must diffuse to escape from its particular tube, and so the characteristic time for this to happen can be calculated using diffusive equations. A classical derivation gives the reptation time $t$:

$t=\frac{l^2{n}^3\mu }{n_{e}kT}$

where $\mu$ is cofficient of friction on a particular polymer chain, $k$ is Boltzmann's constant, and $T$ is the absolute temperature.

The linear macromolecules reptate, if the length of macromolecule $M$ is bigger than ten times $M_e$. There is no reptation motion for polymers with $M<10M_e$, so that the point $10M_e$ is a point of dynamic phase transition.

Due to the reptation motion the coefficient of self-diffusion and conformational relaxation times of macromolecules depend on the length of macromolecule as $M^{-2}$ and $M^3$, correspondingly. The conditions of existence of reptation in the thermal motion of macromolecules of complex architecture (macromolecules in the form of branch, star, comb and others) have not been established yet.

The dynamics of shorter chains or of long chains at short times is usually described by the Rouse model.

References

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