## Reptation and Tube Model

The theory of reptation was first proposed by Piere Gill deGennes in 1971 and later extended to the tube model by Maasai Doi and Sam Edwards. This model describes the thermal motion of long polymer chains in concentrated solutions and melts. Doi and Edwards postulated that the motion of an entangled polymer molecule resembles the motion of a polymer in a tube, that is, they assumed that the long range motion of a polymer molecule is only allowed essentially along its own pass. Since the diffusion path resembles a wiggling snake deGennes termed the movement reptation which is the latin word for creeping. It is also the root for reptile.

The virtual tube is formed by the surrounding and entwining polymer molecules. Although each molecule is constantly moving, the virtual tube
stays about the same for the entire time it takes the molecules to pass through it. This time is known as the relaxation time, *τ _{rep}*.
Furthermore, the movement of the confined chains along the contour
of the tube is not affected by the surrounding molecules. Thus, the
tube model is based on two assumptions: (a) the movement of each molecule is independent of
those of neighboring molecules, meaning no cooperate motion of polymer molecules takes place
and (b) the lateral motion of the molecules can be neglected, that is, the molecules stay entirely within the
virtual tube formed by the surrounding chains.

The primitive path describes the shortest path between the end groups
of the polymer chain which coincides with the average positions of the monomers along the tube.
It is seen from the figure above that this path constitutes the trajectory of a hypothetical chain
of thickness *ξ*. To simplify the calculations, we replace
the real chain with a Kuhn chain that consists of *N* statistical segments of length *
a*. We further assume that the chain forms "*blobs*" of
diameter *ξ*. Both the actual Kuhn chain of contour length,
*N a*, and the primitive
path, *L*, can be treated as random coils in the melt. Thus the contour
length of the primitive path equals the number of blobs N / * N _{e}*
times its average diameter

*ξ*:

*L* ∼ *N ξ* / * N _{e}* ∼

*N a / N*

_{e}^{1/2}

where * N _{e}* is
assumed to be equal to the number of segments between two successive
entanglements, and

*ξ*is equal to the average end-to-end distance of the subchain,

*ξ*∼

*a N*

_{e}^{1/2}.

The relaxation or disentanglement time
corresponds to the time the chain requires to creep (diffuse) out of the
inital tube. As will be shown below, this time is proportional to *N*^{3}.
Assuming random (Brownian) motion the mean squared displacement is given by

*L*^{2} ∼ *D _{t} τ *

Where *D _{t}* is the diffusion
coefficient which can be calculated with the Einstein relation

*D _{t}* =

*kT*/

*μ*

_{t}and *μ _{t}* is the
coefficient of friction of the chain creeping along the tube. This
coefficient is

*N*times higher than that of an individual link,

*μ*=

_{t}*N μ*. The time necessary for the Kuhn chain to displace the length of its original tube is then

*τ* ∼ *L ^{2}* /

*D*=

_{t}*L*

^{2}*μ*/

_{t}*kT*=

*L*

^{2}*N*

*μ*/

*kT*

Recalling that *L* ∼ *N a / N _{e}*

^{1/2 }we obtain

*τ* ∼ *a ^{2}N^{3}*

*μ*/

*N*∼

_{e }T*N*

^{3}*μ*

Since the relaxation time determines the
viscosity (*η*), the reptation model predicts

*η* ∼ *N ^{3}* ∼

*M*

^{3}

where *M* is the molecular weight of
the polymer chain. This finding is in good agreement with the
experimental result, *τ* ∼ *η* ∼ *M*^{3.4}.

Since the reptation model describes the motion of chains through
entanglements, it is only valid for long chains. In practice, this model is only applicable to polymers with
*M* >> *M*_{e}. For shorter (unentangled)
chains, the viscosity is linear proportional to the molecular weight:

*η* ∼ *N* ∼ *M*

Thus, according to the reptation model, the
transition from non-entangled to entangled polymer melt leads
to a change in the power law exponent from *ν* = 1 to *ν* = 3.

##### References

P. G. De Gennes,

*J. Chem. Phys.*55, 572 (1971).P.G. DeGennes,

*Scaling Concepts in Polymer Physics*, Cornell University Press, New York, 1979M. Doi and S. F. Edwards,

*The Theory of Polymer Dynamics*, Oxford University Press, New York 1986