Reptation

Reptation is a specific (snake-like) thermal motion of very long linear macromolecules (polymer chains) in the melts and concentrated solutions of polymers. These systems, called also entangled polymers, are characterized with effective internal scale, commonly known as ‘the length of macromolecule between adjacent entanglements’ M_e . The concept of reptation was introduced into polymer physics by Pierre-Gilles de Gennes to explain the dependence of the mobility of a macromolecule on its length. It is used as a mechanism to explain the viscous flow in an amorphous polymer.

Mechanism

File:Reptation- blob model.svg

The reptation 'blob' model for viscous flow in amorphous polymers.

Effective cross-links from entanglements with other polymer chains restrict polymer chain motion to a 'tube' within these restrictions. Since polymer chains would have to be broken to allow the restricted chain to pass through them, the mechanism for movement (flow) of this restricted chain is reptation. In the 'blob' model, the polymer chain is made up of <math>n</math> Kuhn lengths of individual length <math>l</math>. The chain is assumed to form tangled 'blobs' between each effective cross-link, containing <math>n_{e}</math> Kuhn length segments in each. The mathematics of random walks can show that the average end-to-end length of a polymer chain, made up of <math>n_{e}</math> Kuhn lengths is <math>d=l n_{e}^{1/2}</math>. Therefore if there are <math>n</math> total Kuhn lengths, and <math>A</math> blobs on a particular chain:
<math>A= \dfrac{n}{n_{e}}</math>
The total end-to-end length of the restricted chain <math>L</math> is then:
<math>L=Ad= \dfrac{nl}{n_{e}^{1/2}}</math>
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 <math>t</math>:
<math>t=\dfrac{l^2 n^3 \mu}{n_{e} k T}</math>
where <math>\mu</math> is cofficient of friction on a particular polymer chain, <math>k</math> is Boltzmann's constant, and <math>T</math> is the temperature.
The linear macromolecules reptate, if the length of macromolecule M is bigger than ten times ‘the length of macromolecule between adjacent entanglements’ M_e . There is no reptation motion for polymers with M<10 M_e, so that the point 10 M_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 M³, 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.