Reptation

Two closely related concepts are reptons and entanglement. A 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.

Theory and mechanism

Reptation theory describes the effect of polymer chain entanglements on the relationship between molecular mass and chain relaxation time. The theory predicts that, in entangled systems, the relaxation time τ is proportional to the cube of molecular mass, M: τ ∝ M ³. The prediction of the theory can be 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. 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_B T μ_tube.

By then recalling that in 1-dimension the mean squared displacement due to Brownian motion is given by

s(t)² = 2D_tube t,

we obtain

s(t)² = 2k_B T μ_tube t.

The time necessary for a polymer chain to displace the length of its original tube is then

t = L² / (2k_B T μ_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 ∝ M³).

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. Note that the polymer’s zero-shear viscosity gives an approximation of the actual observed dependency, τ ∝ M^3.4; this relaxation time has nothing to do with the reptation relaxation time.