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

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2.III. OBSTRUCTED DIFFUSION 45
Milner et al. (2001) but the derivation was not included in their paper. The rescaled
Rouse term (equation 2.51) is needed. Each point on the chain can be considered as
Brownian particle in a series of obstacles which disappear and re-appear with frequency
ν. The boxes are a distance a apart. It is assumed that after the removal of an obstacle
the particle has time to fully equilibrate before the obstacle reappears. The particle
experiences a force of F in the x direction at all points in space. Thus after a constraint
ν
F
x
a
Figure 2.15: Schematic showing a Brownian particle, subject to a spring force and
moving in an array of vanishing and re-appearing obstacles.
is removed the probability of accepting the hop can be found by solving the zero flux
Smoluchowski equation,
k B T ∂Ψ
∂x
− FΨ = 0 (2.53)
the solutions of which have a Boltzmann distribution
( ) Fx
Ψ(x) = A exp . (2.54)
k B T
When a constraint to the right of the particle is removed the particle is localised to the
region 0...2a, hence the normalised solution for the probability density is
where α =
right is
Ψ + (x) = e αx α
e 2αa − 1 . (2.55)
F
k B T
. Consequently, the probability that the particle accepts the hop to the
P + (a) =
=
∫ 2a
Ψ + (x)dx
a
(2.56)
1
1 + e −αa .