Molecular modelling of entangled polymer fluids under flow The ...

44 CHAPTER 2. INTRODUCTION TO MOLECULAR RHEOLOGY
N e segments of length b, leading to N e b 2 = a 2 . As long as N e is reasonably large the
fluctuations in the length a will be small. Under this rescaling, the probability density
of the end to end vector R can be shown to be (see Doi and Edwards (1986) equation
2.36)
Ψ(R) =
This is has the same form as equation 2.2 with
( ) 3 3/2 )
2πZa 2 exp
(− 3R2
2Za 2 . (2.48)
b → a
N → Z.
(2.49)
Hence all of the results presented in this chapter can be re-derived for a Gaussian chain
which is coarse grained on the tube diameter, a, by using the above result.
For example, the Rouse spring force term can be rewritten for a chain which is
course grained to step length a.
This is needed later on see appendix 2.III and in
chapter 4 where I derive a generalised version of the MMcL model. Each bead now
contains N e monomers and so the friction per bead becomes ζ 0 N e . Using the above
result equation 2.13 rescales to
∂R
ζ 0 N e
∂t = 3k BT
a 2
∂ 2 R
+ ... (2.50)
∂s2 where R continues to ( measure a real ) distance. Introducing the Rouse time of an
entanglement segment τ e = ζ 0b 2 Ne
2
3π 2 k B
and using the relation a 2 = N
T
e b 2 gives
∂R
∂t = 1 ∂ 2 R
+ ... (2.51)
πτ e ∂s2 Alternatively the same result can be obtained by the following method.
Equation
2.13 can be rewritten as
ζ 0
∂R
∂t = 3k BT
b 2 N 2 e
which is consistent with equation 2.51.
∂ 2 R
∂(n/N e ) 2 + ...
= 1
πτ e
∂ 2 R
∂s 2 + ... (2.52)
Appendix 2.III
Obstructed diffusion
In this appendix I will derive the Rouse-like Langevin equation for a Rouse chain in
a tube experiencing constraint release at a single rate.
The problem was solved by