Structural Architecture of
Branched Polymers and Process
Modelling
Nathanael Inkson, Tom McLeish, Oliver Harlen,
Daniel Read.
School of applied mathematics, IRC in polymer science and
technology and School of Physics,
The University of Leeds, UK
Introduction – polymers and topology
Branched polymer molecules have different topologies
Very different relaxation mechanisms than linear (non-branched) polymers.
Modern chemistry has enabled the synthesis
of specific controlled polymer topologies. This
has enabled the development of theories to
describe the dynamics of these molecules,
and to develop models for more complex
structures like commercial LDPE
1. Model rheology and predict flow behaviour
2. Predict structure, predict rheology
Comb
H
Pom-pom
LDPE
Molecular theory for non-linear flow
In order to model the non-linear rheology we make use of two
constitutive models derived from Doi and Edwards tube theory:
Doi and Edwards tube model
•Orientation
•Degree of
entanglement
Pom-pom model
McLeish
and
Larson
Rolie-poly model
Likhtman,
Graham and
McLeish
•Number of branches
•Backbone Stretch
•Orientation
•Convective constraint
release
•Stretch
•Retraction and reptation
Polymer structure and rheology
Branched polymer structure produces strain hardening in extension
Resulting in large recirculating vortices in a planar contraction die
geometry.
Extensional hardening is attributed
to the stretch of chain between
branch points
Linear polymer
Long chain branched polymer
Extension
shear
e.g. HDPE, PS
e.g. LDPE
f2
f1
f1 > f 2 + f 3
1
1
1
2
2
3
5
1
1
5
2
1
1
4
1
2
1
f3
2
1
1
1
Molecules with more
branches
(on branches) are
expected to be able to
sustain a higher force
until the branch points
collapse in to the tube
This will lead to increased
strain hardening
Pom-pom equations
n
σ = ∑ 3g i λ2i (t )S i (t )
Stress
i =1
Orientation
S(t ) =
A( t )
trace( A(t ))
DA i (t )
1
= K ⋅ A i + A i ⋅ K T − ( A i − I)
τ bi
Dt
Stretch
Dλi
1
−ν * ( λ −1)
= λi (K : S i ) − (λi − 1)e
τ Si
Dt
Separation of stretch and orientational
relaxation times
for λ<q
The pom-pom model for branched polymer melts
–LDPE melt is approximated by an addition of stresses of
several decoupled pom-pom melts of unknown arm number
+
q=3
q=5
+
+
q=2
q=1
• Maximum stretch of pom-pom molecule ∝ the number of arms q
• q is found by fitting the maximum of the extensional viscosity data.
Which is the highly extended state of the material before failure, all
other rheology predicted from this
LDPE rheology and pom-pom model
7
Transient Viscosity /Pa.s
10
A linear relaxation spectrum is
taken and each mode is
assigned a pom-pom with a
variable number of arms q.
6
10
Uniaxial Extension
5
10
4
10
Shear
3
10
-1
10
0
10
1
10
Time /s
2
10
3
10
Transient extensional viscosity at differing
strain rates
4
10
From fitting the maximum
transient extensional
viscosity at different strain
rates, q values of each mode
are found
Shear and planar viscosities
are then predicted accurately
by the model.
Comb polymer extensional and shear rheology
Pom-pom parameters
Polybutadiene comb 9 measured on RME by Dr. David Groves
Model agrees well with data.
Priority (q) no more than 2 were required to fit the extensional viscosity
Comb Polymer Birefringence
Monodisperse polybutadiene comb, modelled with the pom-pom model in the
multipass rheometer, in planar contraction flow.
velocity = 0.01 mms-1
Experiment by Dr. Mark Collis, Dr. Ashish Lele, Cambridge University.
•Flow induced birefringence reveal contours of stress difference
•Transient prediction of stress
•Good agreement in stress, “stress Island” feature, fangs
•The stress optical coefficient is in agreement with the experimental value
•Left side is the experiment, right is the simulation
Comb Polymer Birefringence
t=5
t=7
Comb Polymer Birefringence
t = 10
t = 14
FlowSolve – Contraction geometry
Stretch
Stretchof
oflargest
largestpom-pom,
pom-pom,λλ
Vortex is bounded
by a region of
material that is
highly stretched
Shear orientation
upstream orients the
molecules for to
experience higher
degree of stretch
in a characteristic
wineglass shape
At steady state
LDPE melt 1 – IUPAC X
Branched polymer stress
birefringence, v=1 mm/s
Pom-pom model simulation
Data from Collis and MackleyCambridge University
LDPE non-linear rheology – pom-pom model and
flowsolve vortex simulation
LDPE in 14:1 contraction Laser Doppler velocimetry Data from Schwetz
and Munstedt - Erlangen University
Summary
• Branched polymer – Pom-pom model
–
–
–
–
Separate equations for stretch and orientation of polymer chains.
Parameters fitted from linear spectrum and uniaxial extension.
Strain hardening in extension but shear-thinning.
Vortex growth upstream of contraction.
• flowSolve is successful at modelling branched and linear polymer
melt flow
–
–
–
–
Transient time resolved features observed
Prediction of transient stress “fangs” on exit of contraction from start-up
Realistic growth and decay of vortex size with input velocity
Realistic extensional viscosity of pom-pom model predicts correct vortex
behaviour and correct quantitative stress