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

18 CHAPTER 2. INTRODUCTION TO MOLECULAR RHEOLOGY
small enough that all quantities can be considered to be homogeneous inside the cube.
This situation is illustrated in figure 2.3. A subchain inside such a cube has its end to
end vector denoted by r n where n labels the bead number. This subchain will cross a
plane with normal in the α direction with probability r nα /L (α and β denote Cartesian
components). If the chain carries a tension F n then the force in the β direction caused
by this chain is F nβ . Note that both the end to end vector and the tension depend
upon bead number. The total polymer contribution to stress is obtained by summing
over all subchains inside the cube. This is achieved by summing over all chains and all
bead numbers,
σ αβ = 1 L 3
∑
chains,n
r nα F nβ . (2.10)
If the cube contains enough chains the sum over chains can be replaced by an ensemble
average 〈...〉. This average is over all subchains with the same bead position within the
same course-graining volume. An average over bead positions is not taken. The cube
will contain cL 3 /N chains (c is the monomer density) thus the polymer contribution
to stress inside this cube is
σ αβ = c ∑
〈r nα F nβ 〉 . (2.11)
N
n
The tension, F n , is a known function of the local chain configuration (equation 2.7).
Taking the continuous limit allows the sum over beads to be replaced by an integral
along the chain contour and r n by the first derivative of the chain space-curve. This
leads to the final expression for the stress
σ αβ = c N
3k B T
b 2
∫ N
0
〈 ∂Rα
∂n
〉
∂R β
dn. (2.12)
∂n
2.2.7 Non-linear constitutive equation
A non-linear constitutive equation can be constructed from the Rouse model in the
following way [Larson (1988)]. Beginning with the continuous representation of equation
2.8
( )
∂R
ζ 0
∂t − κ .R = 3k BT
≈ b 2
∂ 2 R
+ f(n, t). (2.13)
∂n2 The quantity f(n, t) represents the local Brownian forces acting on an individual beads,
which are assumed to be random and isotropically distributed. Thus the forces act
equally in all directions and are uncorrelated in time and position along the chain.
The size of the force is fixed by insisting that all points on the chain are in thermal