Rheology control by branching modeling - TU/e · Introduction Mechanism of Relaxation of Entangled...
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IntroductionMechanism of Relaxation of Entangled Polymers
Generalized Models for Rheology of EBPModeling
Rheology control by branching modeling
Volha ShchetnikavaJ.J.M. Slot
Department of Mathematics and Computer ScienceTU EINDHOVEN
April 11, 2012
Volha Shchetnikava J.J.M. Slot Rheology control by branching modeling
IntroductionMechanism of Relaxation of Entangled Polymers
Generalized Models for Rheology of EBPModeling
Outline
Introduction
Mechanism of Relaxation of Entangled Polymers
relaxation of linear polymersdynamic tube dilationactivated primitive path fluctuationsbranch point diffusionhierarchical relaxation
Generalized Models for Rheology of EntangledBranched Polymers
Modeling
Topological structure of LDPETime-marching model
Presentation Outline
• Introduction
• Open Problems in Molecular Rheology
– Complex Architectures
– Nonlinear Flows
• DYNACOP Progress on Theory and Simulation
• Conclusions
Presentation Outline
• Introduction
• Open Problems in Molecular Rheology
– Complex Architectures
– Nonlinear Flows
• DYNACOP Progress on Theory and Simulation
• Conclusions
Volha Shchetnikava J.J.M. Slot Rheology control by branching modeling
IntroductionMechanism of Relaxation of Entangled Polymers
Generalized Models for Rheology of EBPModeling
Introduction
LDPE molecules have a highly branched structure characterized by:
Broad molecular weight distribution
Both long and short side chains are present
Irregularly spaced branches
Transition from short to long chain branching at Me
Exhibit ”strain hardening” in uniaxial extensional flow
Exhibit ”strain softening” in shear flow
Volha Shchetnikava J.J.M. Slot Rheology control by branching modeling
IntroductionMechanism of Relaxation of Entangled Polymers
Generalized Models for Rheology of EBPModeling
Introduction
There is an intimate relationship between polymer structure, rheologyand processing
There is an intimate relationship between polymer structure, rheology and processing.
Typical film blowing operation in polymer companies
Uniaxial extensional viscosity and of various PE melts (Laun, 1984)
There is an intimate relationship between polymer structure, rheology and processing.
Typical film blowing operation in polymer companies
Uniaxial extensional viscosity and of various PE melts (Laun, 1984)
Typical film blowing operation in polymer companies
Uniaxial extensional viscosity of various PE melts
Volha Shchetnikava J.J.M. Slot Rheology control by branching modeling
IntroductionMechanism of Relaxation of Entangled Polymers
Generalized Models for Rheology of EBPModeling
Introduction
LCB characterization
Branching structureAffect
−−−−−−−−−→ Rheological behavior
Rheological behavior (Experiments):
sensitive to LCBcannot determine the LCB structure quantitatively
Molecular rheological theory (Modeling):
number of brancheslength of branchesposition of branch point
Volha Shchetnikava J.J.M. Slot Rheology control by branching modeling
IntroductionMechanism of Relaxation of Entangled Polymers
Generalized Models for Rheology of EBPModeling
Relaxation of linear polymersDynamic tube dilationActivated primitive path fluctuationsBranch point diffusionHierarchical relaxation
Relaxation modulus of linear polymers
Shear strain
σ(t) = G(t)γ0
Step Strain ExperimentSTRESS RELAXATION
Sample is initially at restAt time t = 0, apply instantaneous shear strain γ0The shear relaxation modulus
G(t, γ0) ≡ σ(t)/γ0 (2-1)
For small strains, the modulus does not depend on strain. Linear vis-coelasticity corresponds to this small strain regime. Linear response meansthat stress is proportional to the strain, and thus the modulus is independentof strain.
σ(t) ≡ G(t)γ0 (2-3)
Figure 1: Stress Relaxation modulus of linear polymers. A is monodispersewith Mw < MC , B is monodisperse with M � MC , and C is polydispersewith Mw � MC . Linear polymers are viscoelastic liquids.
1
Oscillatory shear
σ(t) = G(t)γ0[G ′(ω)sin(ωt)+G ′′(ω)cos(ωt)]
Parameters:
The Plateau modulus, GN0
The molecular weight between two entanglements, Me,0
The Rouse time of a segment between two entanglements, τe
Volha Shchetnikava J.J.M. Slot Rheology control by branching modeling
IntroductionMechanism of Relaxation of Entangled Polymers
Generalized Models for Rheology of EBPModeling
Relaxation of linear polymersDynamic tube dilationActivated primitive path fluctuationsBranch point diffusionHierarchical relaxation
The ”Tube” framework
The ”Tube” Model - de Gennes, Edwards, Doi (1970’s)The tube model offers a simple framework for understanding entangled polymer behavior.
The existence of other chains constraints motion of a test chain
to a tube-like region.
The test chain can only escape by diffusing along the tube axis. This
process is called reptation.
Entanglements of one polymer with its neighborscreates a ”tube-like” region that confines the polymer toa quasi-one-dimensional motion on a short time scale
Sketch of a tube, where b is a Kunh length, ais the tube diameter, L is the contour lengthof the polymer itself and Ltube is the lengthof the tube
Volha Shchetnikava J.J.M. Slot Rheology control by branching modeling
IntroductionMechanism of Relaxation of Entangled Polymers
Generalized Models for Rheology of EBPModeling
Relaxation of linear polymersDynamic tube dilationActivated primitive path fluctuationsBranch point diffusionHierarchical relaxation
Basic processes of relaxation
Reptation
Relaxation Mechanisms (by motion of the chain)
d L3
Pierre de Gennes
The test chain can only escape by diffusionalong the tube axis (reptation)
Primitive path fluctuations, in which the endsof the chain randomly pull away from the endsof the tube
Constraint release where portion of a chain can be relaxed locally
Volha Shchetnikava J.J.M. Slot Rheology control by branching modeling
IntroductionMechanism of Relaxation of Entangled Polymers
Generalized Models for Rheology of EBPModeling
Relaxation of linear polymersDynamic tube dilationActivated primitive path fluctuationsBranch point diffusionHierarchical relaxation
Dynamic tube dilation
The polymer fraction alreadyrelaxed = solvent
Marrucci, 1985
Global effect
New ”equilibrium” state:
Increase of the tubediameter a and of the Me
Decrease of Leq
⇓Speeds up the polymer relaxation
Local effect
On the reptation and the fluctuationprocesses
Inter-relationship between allthe relaxation mechanisms
Volha Shchetnikava J.J.M. Slot Rheology control by branching modeling
IntroductionMechanism of Relaxation of Entangled Polymers
Generalized Models for Rheology of EBPModeling
Relaxation of linear polymersDynamic tube dilationActivated primitive path fluctuationsBranch point diffusionHierarchical relaxation
Tube model for branched polymers
A star polymer cannot reptate -instead it must relax by deep armretraction
Becomes exponentially more difficultto relax segments closer to branchpoint.
Branched Polymer Dynamics
”shoulder” in loss modulus = primitivepath fluctuation modes → broad range ofrelaxation times
Star polymers – “breathing modes”
“shoulder” in loss modulus = primtive path fluctuation
modes -> broad range of relaxation times
A star polymer cannot
reptate – instead it must
relax by deep arm
retraction
Becomes exponentially
more difficult to relax
segments closer to branch
point.
Slide from Daniel Read, Univ. of Leeds
Linear Polymers
a()R exp [U()]
Rouse
modes
inside the
tube
Linear polymers
Star polymers – “breathing modes”
“shoulder” in loss modulus = primtive path fluctuation
modes -> broad range of relaxation times
A star polymer cannot
reptate – instead it must
relax by deep arm
retraction
Becomes exponentially
more difficult to relax
segments closer to branch
point.
Slide from Daniel Read, Univ. of Leeds
Linear Polymers
a()R exp [U()]
Rouse
modes
inside the
tube
Volha Shchetnikava J.J.M. Slot Rheology control by branching modeling
IntroductionMechanism of Relaxation of Entangled Polymers
Generalized Models for Rheology of EBPModeling
Relaxation of linear polymersDynamic tube dilationActivated primitive path fluctuationsBranch point diffusionHierarchical relaxation
Tube dilution of stars
At long times the outer parts of the arms act as solvent. This means that the numberof entanglement constraints effective during relaxation of star arms diminishes withtime.
Φ = unrelaxed volumefraction
Φ = 1− ξMe(Φ) = Me/Φα
a(Φ) = a/Φα/2
α is a dilution exponent
α = 1, 4/3
Dynamic Tube Dilution Applied
to Monodisperse Melt of Stars(Ball and McLeish 1989)
“dilution exponent”
= 1, 4/3
F = unrelaxed volume fraction
Me (F)=Me/F
F 1
Volha Shchetnikava J.J.M. Slot Rheology control by branching modeling
IntroductionMechanism of Relaxation of Entangled Polymers
Generalized Models for Rheology of EBPModeling
Relaxation of linear polymersDynamic tube dilationActivated primitive path fluctuationsBranch point diffusionHierarchical relaxation
Relaxation of asymmetric star
When t < τa, all arms retractwhile the branch point remainsanchored
When t = τa, the short arm hasrelaxed and the branch pointmakes a random hop within theconfining tube
When t > τa, the whole polymerreptates with the branch pointacting as a ”fat” friction bead
Hierarchical Relaxation of Asymmetric Star
Asymmetric Star: Hierarchical Relaxation Processes
1. When t<a,, all arms retract while the branch
point remains anchored.
kBT
brDbr
p2a2
2a
Branch Point Motion:
McLeish et al., Macromolecules, 32, 1999.
Arm Retraction Time:
a 0Za1.5
exp(Za )
a
entanglement points
2. When t=a, the short arm has relaxed and
branch point takes a random hop along the
confining tube.
3. When t>a, the whole polymer reptates with
the branch point as a ``fat’’ friction bead.
Hierarchical Relaxation of Asymmetric Star
Asymmetric Star: Hierarchical Relaxation Processes
1. When t<a,, all arms retract while the branch
point remains anchored.
kBT
brDbr
p2a2
2a
Branch Point Motion:
McLeish et al., Macromolecules, 32, 1999.
Arm Retraction Time:
a 0Za1.5
exp(Za )
a
entanglement points
2. When t=a, the short arm has relaxed and
branch point takes a random hop along the
confining tube.
3. When t>a, the whole polymer reptates with
the branch point as a ``fat’’ friction bead.
Branch Point Motion:
kBT
ζbr= Dbr =
p2a2
2τa
with p2 = 1/12; smaller for short arms
Arm retraction Time:
τa = τ0Z1.5a exp(νZa)
McLeish et al., Macromolecules, 32, 1999.
Volha Shchetnikava J.J.M. Slot Rheology control by branching modeling
IntroductionMechanism of Relaxation of Entangled Polymers
Generalized Models for Rheology of EBPModeling
Relaxation of linear polymersDynamic tube dilationActivated primitive path fluctuationsBranch point diffusionHierarchical relaxation
Relaxation of H polymer
First, the arms relax by star-likebreathing mode
Then, the backbone relaxes by”reptation” - but with frictionconcentrated at the ends of thechain
In general not always leading tosuccessful predictions of theexperimental data
Not So Successful Prediction:Polyisoprene H polymer H110B20A
We take Dbr = p2a2/2qa, with p2 = 1/12
Branched Polymer Dynamics
Volha Shchetnikava J.J.M. Slot Rheology control by branching modeling
IntroductionMechanism of Relaxation of Entangled Polymers
Generalized Models for Rheology of EBPModeling
Relaxation of linear polymersDynamic tube dilationActivated primitive path fluctuationsBranch point diffusionHierarchical relaxation
Relaxation of arbitrary branched polymer
Occurs from the outside of the polymer towards the inside
Linear rheology of arbitrarily branched polymers
Relaxation of a branched polymer
Occurs from the outside of the polymer towards
the inside
Slide from Daniel Read, Univ. of LeedsSlide from Daniel Read, Univ. of Leeds
Volha Shchetnikava J.J.M. Slot Rheology control by branching modeling
IntroductionMechanism of Relaxation of Entangled Polymers
Generalized Models for Rheology of EBPModeling
Relaxation of linear polymersDynamic tube dilationActivated primitive path fluctuationsBranch point diffusionHierarchical relaxation
Relaxation of arbitrary branched polymer
Occurs from the outside of the polymer towards the inside
Linear rheology of arbitrarily branched polymers
Relaxation of a branched polymer
Occurs from the outside of the polymer towards
the inside
Slide from Daniel Read, Univ. of Leeds
Volha Shchetnikava J.J.M. Slot Rheology control by branching modeling
IntroductionMechanism of Relaxation of Entangled Polymers
Generalized Models for Rheology of EBPModeling
Relaxation of linear polymersDynamic tube dilationActivated primitive path fluctuationsBranch point diffusionHierarchical relaxation
Relaxation of arbitrary branched polymer
Sometimes side arms relax - relaxation cannot proceed further until themain arm ”catches up”. Side arms give extra ”friction”.
Linear rheology of arbitrarily branched polymers
Relaxation of a branched polymer
Sometimes side arms relax – relaxation cannot
proceed further until the main arm “catches up”.
Side arms give extra “friction”Slide from Daniel Read, Univ. of Leeds
Volha Shchetnikava J.J.M. Slot Rheology control by branching modeling
IntroductionMechanism of Relaxation of Entangled Polymers
Generalized Models for Rheology of EBPModeling
Relaxation of linear polymersDynamic tube dilationActivated primitive path fluctuationsBranch point diffusionHierarchical relaxation
Relaxation of arbitrary branched polymer
Sometimes side arms relax - relaxation cannot proceed further until themain arm ”catches up”. Side arms give extra ”friction”.
Linear rheology of arbitrarily branched polymers
Relaxation of a branched polymer
Sometimes side arms relax – relaxation cannot
proceed further until the main arm “catches up”.
Side arms give extra “friction”Slide from Daniel Read, Univ. of Leeds
Volha Shchetnikava J.J.M. Slot Rheology control by branching modeling
IntroductionMechanism of Relaxation of Entangled Polymers
Generalized Models for Rheology of EBPModeling
Relaxation of linear polymersDynamic tube dilationActivated primitive path fluctuationsBranch point diffusionHierarchical relaxation
Relaxation of arbitrary branched polymer
Eventually there is an effectively linear section which relaxes viareptation, with side-arms providing the friction
Linear rheology of arbitrarily branched polymers
Relaxation of a branched polymer
Eventually there is an effectively linear section
which relaxes via reptation, with side-arms
providing the friction.
c.f. H-polymer terminal relaxationSlide from Daniel Read, Univ. of Leeds
Volha Shchetnikava J.J.M. Slot Rheology control by branching modeling
IntroductionMechanism of Relaxation of Entangled Polymers
Generalized Models for Rheology of EBPModeling
Relaxation of linear polymersDynamic tube dilationActivated primitive path fluctuationsBranch point diffusionHierarchical relaxation
Relaxation of arbitrary branched polymer
And finally relaxed!
Linear rheology of arbitrarily branched polymers
Relaxation of a branched polymer
and….. relax!
Slide from Daniel Read, Univ. of Leeds
Volha Shchetnikava J.J.M. Slot Rheology control by branching modeling
IntroductionMechanism of Relaxation of Entangled Polymers
Generalized Models for Rheology of EBPModeling
Models
Hierarchical Model (Larson, Park, Wang; 2001, 2005, 2010)
Linear, Star, H, CombStar-linear blends
BOB Model (Das et al., 2006, 2008)
Linear, Star, H, CombStar-linear blendsCommercial polyolefins
van Ruymbeke (2005, 2006, 2007, 2008)
Linear, Star, H, Comb, Pom-Pom, Caylee-treeStar-linear blends
Volha Shchetnikava J.J.M. Slot Rheology control by branching modeling
IntroductionMechanism of Relaxation of Entangled Polymers
Generalized Models for Rheology of EBPModeling
Differences between models
eral refinements of the relaxation mechanisms and by employing a logarithmic algorithmfor calculating the time evolution of the arm retraction coordinate � to replace the linearmethod used before �Park et al. �2005��. This logarithmic integration method is similar tothat employed in the original hierarchical model �Larson �2001�� and the bob model �Daset al. �2006�� and has been found to significantly increase the computational speed of thehierarchical model �see Appendix A�. Modifications of the hierarchical-3.0 model overthe hierarchical-2.0 model �Park et al. �2005�� are described in Appendix B. All predic-tions of the hierarchical model presented in the current work were obtained using thehierarchical-3.0 simulation code.
Table I summarizes the main physical and computational differences between thehierarchical-3.0 and bob models that will be discussed in more details below. The bobcode can also handle branch-on-branch architectures, which are not yet considered in thehierarchical code. In following discussions, we employ the terminology of Larson �2001�.The fractional arm coordinate � is used to represent the depth of the arm retraction. Itsvalue runs from zero to unity when the free end of the arm retracts along the arm contourfrom its initial location at time t=0 to the other immobile end of the arm which could beeither a branch point or the middle point of a linear polymer, since a linear chain iseffectively treated as a symmetric two-arm star. The determination of � values of thecompound arms that are formed due to the collapse of side branch arms is described inSec. II C. The lengths of the arms, Sa�=Ma /Me�, and the backbone segments between thebranch points, Sb�=Mb /Me�, are measured in units of the entanglement spacing Me, whereMa and Mb are the molecular weights of the arms and backbone segments, respectively.The absolute arm coordinate, z, used in the bob model �Das et al. �2006�� is related to �simply by z=�Sa. The equilibration time �e, tube diameter a, and entanglement spacingMe are defined in the original undilated tube. We use the “G” definition of the entangle-ment spacing as given in Larson et al. �2003�, which is related to the plateau modulus GN
0
by
Me =4
5
�RT
GN0 , �1�
where � is the density of the polymers, R is the gas constant, and T is the absolutetemperature. In the dynamic dilution theory �Marrucci �1985�; Ball and McLeish �1989��,
TABLE I. Main differences between the hierarchical model �Larson �2001�; Park et al. �2005�; current work�and the bob model �Das et al. �2006��.
Element of algorithm Hierarchical Bob
Arm retraction potential Ueff
and relaxation time �late
Analytical formulas fromMilner and McLeish �1997;1998�
Numerical evaluation of Taylorexpansion at each time step
Compound arm fluctuation Entire compound arm fluctuateswith lumped branch point frictions
Growing portion of compoundarm fluctuates
Arm retraction in CR-Rouse regime No arm retraction �Park et al. �2005��;arm retraction in fat �Larson �2001��
and thin tubes �this work�
Arm retraction in thin tube
Branch point friction Time independent Time dependentReptation In a partly dilated tube In an undilated tubeDisentanglement Yes NoDilution exponent � �=1 �Larson �2001��;
�=4 /3 �Park et al. �2005���=1
Branch point friction p2 p2=1 /12 �Park et al. �2005�� p2=1 /40
226 WANG, CHEN, AND LARSON
Volha Shchetnikava J.J.M. Slot Rheology control by branching modeling
IntroductionMechanism of Relaxation of Entangled Polymers
Generalized Models for Rheology of EBPModeling
Problems
”The hierarchical and bob models quantitatively predict the linearrheology of a wide range of branched polymer melts but alsoindicate that there is still no unique solution to cover all types ofbranched polymers without case-by-case adjustment of parameterssuch as the dilution exponent α and the factor p2 which defines thehopping distance of a branch point relative to the tube diameter.”
Z. Wang and R. Larson, 2010
Volha Shchetnikava J.J.M. Slot Rheology control by branching modeling
IntroductionMechanism of Relaxation of Entangled Polymers
Generalized Models for Rheology of EBPModeling
Topological structure of LDPETime-marching algorithm
Our approach
We want to:
Understand the role of each generation of segments within molecules in therelaxation of the total ensemble
Consider the effect of taking a limited number of generations into account
Assume that the rest of the ensemble will relax automatically due to dynamictube dilation (disentanglement relaxation)
We need to:
Choose a representative ensemble of molecules
Analyse the distribution of generations of segments in the ensemble
Find all topologically different architectures belonging to a given generation
Extend the time-marching model to treat the relaxation of high enoughgenerations (up to 6 ?)
Volha Shchetnikava J.J.M. Slot Rheology control by branching modeling
IntroductionMechanism of Relaxation of Entangled Polymers
Generalized Models for Rheology of EBPModeling
Topological structure of LDPETime-marching algorithm
Branching in CSTR
Simulation conditions were chosen to achieve monomer conversion and
physical properties of alpha IUPAC LDPE (Tackx and Tacx, 1998), where
polydispersity is 26 and Mn = 29kg/mol.
D.-M. Kim, P.D. Iedema / Chemical Engineering Science 63 (2008) 2035–2046 2041
Table 2Reactor configuration and reaction conditions (a) CSTR and (b) tubular reactors
Reactor condition Value
(a)Pressure 1850 barTemperature 260 ◦CVolume 0.3 × 10−3 m3
Feed condition 16.75 kmol/m3
Monomer 16.75 kmol/m3
Chain transfer agent 1.00 × 10−2 kmol/m3
Initiator 3.50 × 10−5 kmol/m3
Pre-exponential factor of scission rate, krs0 7 × 106 m3/(kmol s)
(b)Reaction parameters ValuePressure (bar) 1850Temperature (◦C) T0 170
T1 220T2 220T3 220
Reactor Dimension (m) Diameter 0.059Length 1800Additional feeding positions 560, 960, 1360
Feed condition (kg/s) Monomer 16.75CTA 0.12Initiator, S 7 × 10−3
Initiator, C 2.5 × 10−3
Pre-exponential factor of scission rate (m3/(kmol s)) 3.7 × 105 s−1
Notes: Subscript 0 for temperature means reactor inlet position, while subscripts 1, 2, 3 mean the additional injection points. Initiator C is evenly injected atadditional feeding location.
0
0.2
0.4
0.6
100 102 104 106 108
dW/d
{log
(n)}
Chain Length
Fig. 2. Comparative MWD Plot between experimental and simulation resultin CSTR (scission paradox). Alpha IUPAC ldPE has Mn of 29.0 kg/kmoland D of 26.0. Simulation results with topological scission model and linearscission model have Mn of 28.0 kg/kmol, 29.0 kg/kmol and D of 27.8 and26.0, respectively (solid line, alpha IUPAC ldPE; dash-dot line, simulationresult with topological scission model; dashed line, simulation result withlinear scission model).
for CSTR as proposed by Graessley (1965):
�n = ktrp
kp
{(M0 − M)
M
}(34)
100 102 104 106 1080
0.4
1.2
0.8
5.0
1.0
1.5
0
Chain Length
Ψn1
x106
(km
ol/m
3 )
Ψn2 x
105(k
mol
/m3 )
Fig. 3. First and second branching moments in CSTR. Overall concentrationsof first and second moments are 1.001×10−2 and 2.005 kmol/m3, respectively(solid line, first branching moment; dashed line, second branching moment).
gives 3.346 × 10−3, which is close to the value from the directcomputation. First and second branching moments are shown inFig. 3. The second branching moment has a similar concentra-tion profile as the first branching moment but shifted to longerchain length region. Due to the definition of branching mo-ments, the second moment has a 100 times higher value than thefirst moment. The overall concentration of the first branchingmoment is 1.001×10−2 kmol/m3, while the LCB concentrationas directly calculated from Eq. (17) is 1.030 × 10−2 kmol/m3,proving that the first moment calculation is performed in a con-sistent manner.
Simulation results with topological scission (dash-dot line) and with linear scission (dashed line)
Volha Shchetnikava J.J.M. Slot Rheology control by branching modeling
IntroductionMechanism of Relaxation of Entangled Polymers
Generalized Models for Rheology of EBPModeling
Topological structure of LDPETime-marching algorithm
Branching in CSTR
Input data:
Chain length Concentration Branching densityn1 c1 b1
.
.
.
.
.
.
.
.
.nn cn bn
Number of branch points in a molecule follow a binomial distributionP(n) with respect to the chain length n
⇓Bimodal chain length\degree of branching distribution
⇓Computational ”synthesis” of architectures for a given combination ofchain length and number of branch points (n,N) by a conditional MonteCarlo method
Volha Shchetnikava J.J.M. Slot Rheology control by branching modeling
IntroductionMechanism of Relaxation of Entangled Polymers
Generalized Models for Rheology of EBPModeling
Topological structure of LDPETime-marching algorithm
Representative ensembleBinominal distribution P(n) determines the range of chain lengths nu , . . . , nv which can have N number ofbranch points. Parameters which control the generation of molecules:
binsbranch - the grid on the branch point number axisbinslength - the grid on the chain length axis
binfractionsN - fractional contribution of each bin
Each bin is represented by a pair n,N. Number of architectures generated in bin depends on the size of a bin.
01
23
45
67
0
1
2
3
4
5
−12
−10
−8
−6
−4
−2
0
log10(N)
log10(n)
log1
0(bi
nfra
ctio
nsnN
)
Volha Shchetnikava J.J.M. Slot Rheology control by branching modeling
IntroductionMechanism of Relaxation of Entangled Polymers
Generalized Models for Rheology of EBPModeling
Topological structure of LDPETime-marching algorithm
Representative ensemble
Macromolecules are described by graphs (trees) and represented by:
Vertices - branch points and arm ends
Weight of the edge - molecular weight of the strand
The adjacency matrix of a weighted graph
We specify an ensemble of a large number of branched molecules byintroducing the following parameters:
α labels the molecular species 1, . . . ,Ns
cα indicates the concentration of particular species
Volha Shchetnikava J.J.M. Slot Rheology control by branching modeling
IntroductionMechanism of Relaxation of Entangled Polymers
Generalized Models for Rheology of EBPModeling
Topological structure of LDPETime-marching algorithm
Seniority
Seniority is a property of an interior segment of a branched molecule and
is simply the number of segments (chain portions between branch points)
that connects it to the retracting chain end responsible for its relaxation.
More generalization is added if the molecularweight (Mbi) of the segments of seniority i isallowed to vary. Then, the mass concentration
Cwi ¼ Ni
Mbi
Mw
is defined, Mw being the weight-average molecu-lar weight of the molecule. The set of parametersCi
n and Ciw is called the seniority distribution, as
it represents the probability of existence of a seg-ment with the seniority i in the molecule.
In the case of the regular Cayley tree, Mbi
being identical for each seniority, Cin and Ci
w
are essentially the same, but this seniority dis-tribution is not really consistent with the ran-domly branching mechanism observed in freeradical polymerization. The more realistic pic-ture is the irregular Cayley tree for which, onone hand, the weight of the segments can be dif-ferent for each level, and on the other hand,there is a possibility to generate uncompletedlevels. This means that the number of segmentswith seniority i is not twice the number of seg-ments with seniority i þ 1.
Starting from this description of the molecule,it is possible to derive the expression of the cor-responding relaxation modulus G(t). After a sud-den step strain, the whole molecule undergoesstress and starts to relax. Mcleish20,24 andRubinstein et al.25 postulate that there is a hier-archization of the relaxation of the different lev-els according to their relaxation times. One levelcan only start relaxing just after the relaxation
of the previous one is completed. Thus, theseniority represents the range of time at whicha segment relaxes rather than its position in themolecule. Segments of seniority equal to unitystart to relax first.
The return to an equilibrium configuration of asegment of seniority 1 is achieved by fluctuationof the free end, as for star polymers.8,26 Segmentsof seniority 2 also behave like retracting armsbecause on the time scale of their own dynamics,the segments of seniority 1 are rapidly and contin-ually reconfiguring and do not contribute to theentanglement network. The only distinction isthat the effective drag on any currently relaxingsegments is imposed on its outer end, and it isarising from the dissipation of segments withlower seniority attached at this end.
The effective tube diameter also increaseswith seniority. The difference of time scale inthe relaxation times according to the seniorityimplies that the relaxed levels play the role ofsolvent for the unrelaxed ones. As the effectiveentanglement molecular weight Me dilutes withconcentration as c�1, it should vary with theseniority, as given in the following equation:
Mei ¼ Me
Ci
ð1Þ
where Me is the equilibrium value of the massbetween entanglements, Ci is the dynamic con-centration of the level i, and represents the frac-tion of unrelaxed material in the sample. It isexpressed as a function of the seniority distribu-tion by:
Ci ¼XNc
k¼i
Ck ð2Þ
Equation 1 shows that the mass betweenentanglements increases as the seniorityincreases. The first consequence is a decrease ofthe relaxation time of the molecule in agreementwith the experience.
The relaxation time of a segment of seniority1 can be cast in the following form:
�1ð�Þ ¼ �0 � exp � �Mb1
Me1�2
� �ð3Þ
where �o is a time given by Pearson and Hel-fand4 as:
Figure 3. The seniority distribution in a branchedmolecule. To calculate this parameter for a given seg-ment, one needs to count the number of strands tothe furthest chain end on each side of the segment.The seniority is the smallest of the two values.
MODELING OF LINEAR VISCOELASTIC BEHAVIOR OF LOW-DENSITY POLYETHYLENE 1977
Volha Shchetnikava J.J.M. Slot Rheology control by branching modeling
IntroductionMechanism of Relaxation of Entangled Polymers
Generalized Models for Rheology of EBPModeling
Topological structure of LDPETime-marching algorithm
Distribution of seniorities
Seniority Mass fraction % Number fraction %0 3.67 2.911 63.52 56.322 14.57 15.043 6.73 8.034 3.81 4.955 2.33 3.266 1.49 2.247 0.99 1.59...
......
20 0.031 0.078...
......
66 0.00004 0.00016
Volha Shchetnikava J.J.M. Slot Rheology control by branching modeling
IntroductionMechanism of Relaxation of Entangled Polymers
Generalized Models for Rheology of EBPModeling
Topological structure of LDPETime-marching algorithm
MWD of seniorities
0 5000 10000 150000
1
2
x 10−4 0
0 1 2 3
x 104
0
1
x 10−4 1
0 1 2 3
x 104
0
1
2
x 10−4 2
0 0.5 1 1.5 2
x 104
0
1
2
x 10−4 3
0 1 2 3
x 104
0
1
2
x 10−4 4
0 5000 10000 150000
0.5
1
1.5
2
2.5
3
3.5x 10
−4 5
Volha Shchetnikava J.J.M. Slot Rheology control by branching modeling
IntroductionMechanism of Relaxation of Entangled Polymers
Generalized Models for Rheology of EBPModeling
Topological structure of LDPETime-marching algorithm
Topologies of seniorities
Seniority Number of topologies1 12 13 24 75 566 22127 2447513
Seniority 4 Seniority 3
Volha Shchetnikava J.J.M. Slot Rheology control by branching modeling
IntroductionMechanism of Relaxation of Entangled Polymers
Generalized Models for Rheology of EBPModeling
Topological structure of LDPETime-marching algorithm
Topologies of seniorities
Seniority 5
44.39 % 11.56 %
11.68 % 6.62 %
5.13 %
Volha Shchetnikava J.J.M. Slot Rheology control by branching modeling
IntroductionMechanism of Relaxation of Entangled Polymers
Generalized Models for Rheology of EBPModeling
Topological structure of LDPETime-marching algorithm
Relaxation of a branched polymer
Reptation and contour length fluctuations are considered as simultaneousprocesses
Survival probability of oriented segments iscalculated by summing up all contributionsover types of arms and positions along thearms
Relaxation of a branched polymer
x 0
1
x x
0 1
0
Survival probability of oriented segments:
F(t) = ϕ i prept (xi ,t).p fluct (xi ,t).penvir (xi ,t)( )dxi
0
1
∫
i
∑0
)(
NGtG
All types of arms (fractions ϕi)
xi not relaxed by reptation
xi not relaxed by fluctuations
xi not relaxed by the environment if not relaxed otherwise
Reptation and contour length fluctuations are considered as simultaneous processes
calculated by summing up all contributions over types of arms
and positions along the arms
Volha Shchetnikava J.J.M. Slot Rheology control by branching modeling
IntroductionMechanism of Relaxation of Entangled Polymers
Generalized Models for Rheology of EBPModeling
Topological structure of LDPETime-marching algorithm
Time-marching algorithm
Reptation
Fluctuations
Polymer « solvent »
G(t)
Molecules
G’(ω),G’’(ω)
Time t
No analytical function can be found for complex polymers
t ti-1 ti
psurvival (x, ti) = psurvival (x, ti-1) . psurvival (x, between ti-1 and ti )
Explicit time-marching algorithm:
Φ(ti-1) τreptation (x,ti) τfluctuation (x,ti)
psurvival (x, between ti-1 and ti )
Φ(ti) G(ti)
E. van Ruymbeke, R. Keunings, C. Bailly, J.N. N. F. M., 128 (June 2005)
Time-marching algorithm
Volha Shchetnikava J.J.M. Slot Rheology control by branching modeling
IntroductionMechanism of Relaxation of Entangled Polymers
Generalized Models for Rheology of EBPModeling
Topological structure of LDPETime-marching algorithm
Relaxation of a branched polymer
How does this molecular section relax?
Additional friction
Relaxation of a branched polymer
Reptation Contour length Fluctuations
xb=0 xb=1
Leq xb=0 xb=1
xb=xbr
Leq
Relaxed branches
xbranch=0
xbranch=1
U(x)
x
U(x)
2 Fluctuations modes:
Coordinate system:
EVR et al., Macromol. 06
Volha Shchetnikava J.J.M. Slot Rheology control by branching modeling
IntroductionMechanism of Relaxation of Entangled Polymers
Generalized Models for Rheology of EBPModeling
Topological structure of LDPETime-marching algorithm
PolydispersityPolydispersity
Log(M)
105
1
2
3
4
G’, G”(ω)
[Pa]
ω (1/sec)
a
G’, G”(ω) [Pa]
ω (1/sec)
b
G’(ω) [Pa]
ω (1/sec)
c
G”(ω) [Pa]
ω (1/sec)
d
10-4 10-2 100 102 104
103
104
105
106
10-5 100 105
103
104
105
10-4 10-2 100 102 104
102
104
106
10-5 100 105102
103
104
105
106
)log()(
MdMdw Polydispersity fixed to 1.05
Explicit time-marching algorithm: ΦDTD (t) calculated at each time step
( )1
0
( ) , )( (, .) fluct iDTD i ip ii
re t p x tt p x dxtϕ
Φ =
∑ ∫
Backbone: H=1 to 1.2
Arms: H=1 to 1.2
Backbone and arms: H=1 to 1.2
Volha Shchetnikava J.J.M. Slot Rheology control by branching modeling