Polymer Physics

Imagining that the frictional barrier for the motion of each blob is integrated into
that for the motion of the whole chain, thus

zt¼z

n

ne

(5.46)

From the Einstein relationshipD¼kT/z, one obtains

Dt¼

kT

zt

¼

kTne

zn

¼

Dne

n

(5.47)

Therefore, the characteristic time for the reptation model becomes

tt

n^2

Dt

n^3 (5.48)

The reptation time exhibits stronger chain-length dependence than the Rouse
time (tR~n^2 ). Accordingly, the diffusion coefficient is

Drep¼

R^2

tt

¼

nb^2

tt

n^2 (5.49)

The diffusion coefficient for the reptation chain shows weaker chain-length
dependence than that for the Rouse chain (DR~n^1 ). Both (5.48) and (5.49)
imply that the tube makes an additional confinement to the self-diffusion of the
polymer chains. Therefore, the long chains diffuse slower, and the diffusion
coefficient becomes more sensitive to the chain length.
The primitive path of the long chain reveals the characteristic feature of the
reptation model for the diffusion of an ideal chain. SincetR<tt, polymer chains
perform the Rouse-chain motions along the tube before they diffuse out of the tube.
Therefore, within the time window between the relaxation time of critically
entangled chainteand the relaxation time of the reptation tubett, monomers
diffuse through a certain length of the tube. Meanwhile, the tube itself can be
regarded as a contour of an ideal chain, and the tube length is proportional to the
mean-square end-to-end distance, which corresponds to the total mean-square
displacement of each monomer. Note that the tube length is the sliding diffusion
distance of the Rouse chain, then

<½rðtÞrð 0 ފ^2 >reptube length¼<rðtÞrð 0 Þ>R<½rðtÞrð 0 ފ^2 >^1 R=^2

(5.50)

In other words, within the time period fromtetott, the polymer chain diffuses
along the tube with the Rouse mode. Owning to the tube confinement, the scaling
exponent of the original Rouse chain has been reduced into half. Therefore, the

86 5 Scaling Analysis of Polymer Dynamics