prediction of the rheological behavior of ldpe · both long and short side chains are present ......

IntroductionModeling

Numerical experiments

Prediction of the Rheological Behavior of LDPE

Volha Shchetnikava1

J.J.M. Slot1, E. van Ruymbeke2

1Department of Mathematics and Computer Science,TU Eindhoven2Bio and Soft Matter, IMCN, Universite Catholique de Louvain

April 09, 2014

Volha Shchetnikava J.J.M. Slot, E. van Ruymbeke Prediction of the Rheological Behavior of LDPE



LDPE resins are made by high pressure autoclave or tubular process and mainly usedfor blown film extrusion and injection molding applications.

They are widely used for

Shrink film for books, bundling and pallets

Overwrap film for towels, tissues

Film for bakery goods, meat, coffee, frozen foods

Liquid packaging (milk cartons and bag-in-box applications)

Liners, bags, shoppers and foams

Greenhouse and tunnels

Insulation and semiconductive layers

Presentation Outline

• Introduction

• Open Problems in Molecular Rheology

– Complex Architectures

– Nonlinear Flows

• DYNACOP Progress on Theory and Simulation

• Conclusions




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




QuantumMechanics

Molecular Dynamics

Mesoscale Dynamics

Finite Elem.Analysis

Engineering

years

hours

minutes

seconds

microsec

nanosec

picosec

femtosec

1 A 1 nm 10 nm micron mm m

Time

Distance

Electrons Atoms Clusters Continuum (grids)

Population balances

with Monte Carlo for

stochastic nature

SEC/MALLS

Validation

G’(),

G’’()

DPI #674 Rheology Control by Branching Modelling

UNIVERSITEIT VAN AMSTERDAM

Modeling of LDPE polymerization process is based on the original algorithm of Tobita and produce a datastructure by Monte Carlo simulation. The algorithm takes into account all processes occurring in a batch reactorduring free-radical polymerization:

initiation of free radicals

propagation or polymerization

termination by disproportionation and combination

chain transfer by long-chain branching to small molecules and by scission




Modeling and Simulation of Polymers Across Various Scales

Atomistic Molecular Dynamics

Coarse-Grained Molecular Dynamics

• Kremer-Grest Model

Coarse-Grained Stochastic Dynamics

• Twentanglement

Slip-link Models

• NAPLES

Tube Models

• Rolie-Poly

• Pom-Pom

• Time-Marching Algorithm

NAPLES

NAPLES

SAN SEBASTIAN

TWENTE

LEEDS, LOUVAIN




The degree of branching and average molecular weight of the strand of themacromolecules determine the linear rheology and extension hardening of LDPE.

calculated the nonlinear transient response instrong shear and extension of the test materials.Theoretical predictions and experimental dataare given over a wide range of deformation ratesin Fig. 3 for the first three samples. Using thenew algorithm for assigning effective segmentpriorities, for all three melts the technologicallyessential extensional response is predicted withremarkable accuracy. The onset, slope, and max-imum of the extensional hardening are consistentwith the data in each case. [Note that for the highermolecular weight (LDPE 2 and 3) materials, theextensional sample always breaks before the max-imum stress is reached.] Notably, we also capturein the case of LDPE 1 the rate at which harden-ing sets in (as well as its much reduced scale).In shear, the model predicts the existence andposition of a transient stress maximum in eachcase, and the qualitatively different thinning be-havior in contrast to the hardening in extension.

The successful nonlinear predictions are un-expected. The phenomenon is highly sensitive todetails of the long-chain branching still under-determined by the solution measurements andlinear rheology. Might the nonlinear predictionsbe fortuitous and other numerical ensembles equal-ly consistent with the linear measurements giveradically different nonlinear predictions?We testedthis by constructing examples of these alternativedistributions. The dashed curves in Fig. 3 forLDPE 2 show the extensional and shear pre-dictions for an ensemble constructed from a blendof two, rather than three, degrees of conversion

(parameter values in SOM). MWD, g(M), andlinear rheology are essentially identical, but thereaction parameters in the two fractions differmarkedly from any of the three in the first mod-el. Yet we see that the nonlinear predictions arerobust. The reason for this commercially vitalfeature is subtle, however; it is sensitive to thebranched structure only though the relaxationtime/priority distribution. Providing that this iscorrect, variations in structure within that ensem-ble will not result in variation of rheologicalresponse. The additional constraints from thepolymerization scheme are sufficient to ensurethat the ensemble belongs to the correct region ofrelaxation time/priority space. Figure S1 showsan example of a time-dependent correlation mapof relaxation time and priority (for LDPE 1). Italso indicates the two extreme structures of per-fect combs and perfect Cayley trees that consti-tute bounds for such maps. At the latest time, theouter structures and lowest molecular weightsresemble comblike topologies, but at longer timesthe larger structures acquire a more ramifiedtopology of branching, although an importantfinding is that the ensemble is always very farfrom being accurately represented by Cayleytrees (20).

We are now in a position to start exploringhypothetical variations in reaction conditionswith a view toward molecular design of newmelts. We chose to tackle the important ques-tion of independent tuning of the linear andnonlinear rheology and created different single-

batch ensembles (Table 1) with near-identicallinear rheology.

The predicted MWD, branching structure,and linear rheology spectrum for the two meltsare shown in Fig. 4. Also shown is the predictedtransient response in strong extension of two resins.The first is predicted to show much strongerextension hardening (similar in magnitude toLDPE 2 and 3) than the second (similar toLDPE 1). This is a result of its higher degree ofbranching (largerCb), compensated in the secondby a smaller strand molecular weight (the pa-rameter t is also larger for melt 1). This exampleserves to illustrate that, by separately controlling thedegree of branching and strandmolecular weight,independent control can in principle be exercisedover the linear rheology and extension hardeningof LDPE resins. This is a vital principle for thedesign of custom materials.

References and Notes1. P.-G. de Gennes, Scaling Concepts in Polymer Physics

(Cornell Univ. Press, Ithaca, NY, 1980).2. P. A. Small, Adv. Polym. Sci. 18, 1 (1975).3. J. M. Dealy, R. G. Larson, Structure and Rheology of

Molten Polymers (Hanser, Munich, 2006).4. H. Tobita, Journal of Polymer Science Part B 39, 391

(2001).5. M. Doi, S. F. Edwards, The Theory of Polymer Dynamics

(Oxford Univ. Press, Oxford, 1986)6. P.-G. de Gennes, J. Chem. Phys. 55, 572 (1971).7. P.-G. de Gennes, J. Phys. (Paris) 36, 1199 (1975).8. J. Klein, D. Fletcher, L. J. Fetters, Nature 304, 526

(1983).9. K. R. Shull, E. J. Kramer, L. J. Fetters, Nature 345, 790

(1990).10. D. S. Pearson, E. Helfand, Macromolecules 17, 888

(1984).11. T. C. B. McLeish et al., Macromolecules 32, 6734

(1999).12. N. J. Inkson, R. S. Graham, T. C. B. McLeish, D. J. Groves,

C. M. Fernyhough, Macromolecules 39, 4217(2006).

13. J. Juliani, L. A. Archer, Macromolecules 35, 10048(2002).

14. E. van Ruymbeke et al., Macromolecules 40, 5941(2007).

15. T. C. B. McLeish, Adv. Phys. 51, 1379 (2002).16. C. Das, N. J. Inkson, D. J. Read, M. A. Kelmanson,

T. C. B. McLeish, J. Rheol. 50, 207 (2006).17. D. K. Bick, T. C. B. McLeish, Phys. Rev. Lett. 76, 2587

(1996).18. T. C. B. McLeish, R. G. Larson, J. Rheol. 42, 81 (1998).19. R. J. Blackwell, T. C. B. McLeish, O. G. Harlen, J. Rheol.

44, 121 (2000).20. P. Stanescu, J. C. Majesté, C. Carrot, J. Polym. Sci. B

Polym. Phys. 43, 1973 (2005).Acknowledgments: We thank the Engineering and Physical

Sciences Research Council (UK) for funding under the“Microscale Polymer Processing” (MuPP) consortiumand the European Union Framework Programme 7 PeopleProgramme for funding under the Marie Curie InitialTraining Network Dynamics of Architecturally ComplexPolymers. The data described in this work are archived onthe MuPP project online materials database at www.irc.leeds.ac.uk/mupp2.

Supporting Online Materialwww.sciencemag.org/cgi/content/full/333/6051/1871/DC1Materials and MethodsFig. S1Tables S1 to S3

15 April 2011; accepted 27 July 201110.1126/science.1207060

Table 1. The reaction model parameters of the hypothetical materials. xs1, the degree of conversion ofcomponent 1; w1, the weight fraction of component 1 (there is only one component used in this case).

Resin t b Cb Cs xs1 w1

Batch 1 1 × 10−3 8 × 10−5 2 × 10−2 0 0.15 1Batch 2 8.5 × 10−4 8 × 10−5 5 × 10−3 0 0.15 1

Fig. 4. Two reaction models (Batch 1 and Batch 2)with parameters as in Table 1 were used to calculate(A) molecular weight (and branching; g) distributions,(B) linear rheology (solid curves are G′′; dashed curvesare G′), and (C) extensional rheology at a range ofrates (black curves are Batch 1; red curves are Batch2). All rheology calculations use the parameters fromtable S2. Although they have very similar linear rhe-ology, this has been achieved through different com-

binations of branching and molecular weight distributions. As a result, the degree of extension hardeningfor the two hypothetical resins is quite different.

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calculated the nonlinear transient response instrong shear and extension of the test materials.Theoretical predictions and experimental dataare given over a wide range of deformation ratesin Fig. 3 for the first three samples. Using thenew algorithm for assigning effective segmentpriorities, for all three melts the technologicallyessential extensional response is predicted withremarkable accuracy. The onset, slope, and max-imum of the extensional hardening are consistentwith the data in each case. [Note that for the highermolecular weight (LDPE 2 and 3) materials, theextensional sample always breaks before the max-imum stress is reached.] Notably, we also capturein the case of LDPE 1 the rate at which harden-ing sets in (as well as its much reduced scale).In shear, the model predicts the existence andposition of a transient stress maximum in eachcase, and the qualitatively different thinning be-havior in contrast to the hardening in extension.

The successful nonlinear predictions are un-expected. The phenomenon is highly sensitive todetails of the long-chain branching still under-determined by the solution measurements andlinear rheology. Might the nonlinear predictionsbe fortuitous and other numerical ensembles equal-ly consistent with the linear measurements giveradically different nonlinear predictions?We testedthis by constructing examples of these alternativedistributions. The dashed curves in Fig. 3 forLDPE 2 show the extensional and shear pre-dictions for an ensemble constructed from a blendof two, rather than three, degrees of conversion

(parameter values in SOM). MWD, g(M), andlinear rheology are essentially identical, but thereaction parameters in the two fractions differmarkedly from any of the three in the first mod-el. Yet we see that the nonlinear predictions arerobust. The reason for this commercially vitalfeature is subtle, however; it is sensitive to thebranched structure only though the relaxationtime/priority distribution. Providing that this iscorrect, variations in structure within that ensem-ble will not result in variation of rheologicalresponse. The additional constraints from thepolymerization scheme are sufficient to ensurethat the ensemble belongs to the correct region ofrelaxation time/priority space. Figure S1 showsan example of a time-dependent correlation mapof relaxation time and priority (for LDPE 1). Italso indicates the two extreme structures of per-fect combs and perfect Cayley trees that consti-tute bounds for such maps. At the latest time, theouter structures and lowest molecular weightsresemble comblike topologies, but at longer timesthe larger structures acquire a more ramifiedtopology of branching, although an importantfinding is that the ensemble is always very farfrom being accurately represented by Cayleytrees (20).

We are now in a position to start exploringhypothetical variations in reaction conditionswith a view toward molecular design of newmelts. We chose to tackle the important ques-tion of independent tuning of the linear andnonlinear rheology and created different single-

batch ensembles (Table 1) with near-identicallinear rheology.

The predicted MWD, branching structure,and linear rheology spectrum for the two meltsare shown in Fig. 4. Also shown is the predictedtransient response in strong extension of two resins.The first is predicted to show much strongerextension hardening (similar in magnitude toLDPE 2 and 3) than the second (similar toLDPE 1). This is a result of its higher degree ofbranching (largerCb), compensated in the secondby a smaller strand molecular weight (the pa-rameter t is also larger for melt 1). This exampleserves to illustrate that, by separately controlling thedegree of branching and strandmolecular weight,independent control can in principle be exercisedover the linear rheology and extension hardeningof LDPE resins. This is a vital principle for thedesign of custom materials.

References and Notes1. P.-G. de Gennes, Scaling Concepts in Polymer Physics

(Cornell Univ. Press, Ithaca, NY, 1980).2. P. A. Small, Adv. Polym. Sci. 18, 1 (1975).3. J. M. Dealy, R. G. Larson, Structure and Rheology of

Molten Polymers (Hanser, Munich, 2006).4. H. Tobita, Journal of Polymer Science Part B 39, 391

(2001).5. M. Doi, S. F. Edwards, The Theory of Polymer Dynamics

(Oxford Univ. Press, Oxford, 1986)6. P.-G. de Gennes, J. Chem. Phys. 55, 572 (1971).7. P.-G. de Gennes, J. Phys. (Paris) 36, 1199 (1975).8. J. Klein, D. Fletcher, L. J. Fetters, Nature 304, 526

(1983).9. K. R. Shull, E. J. Kramer, L. J. Fetters, Nature 345, 790

(1990).10. D. S. Pearson, E. Helfand, Macromolecules 17, 888

(1984).11. T. C. B. McLeish et al., Macromolecules 32, 6734

(1999).12. N. J. Inkson, R. S. Graham, T. C. B. McLeish, D. J. Groves,

C. M. Fernyhough, Macromolecules 39, 4217(2006).

13. J. Juliani, L. A. Archer, Macromolecules 35, 10048(2002).

14. E. van Ruymbeke et al., Macromolecules 40, 5941(2007).

15. T. C. B. McLeish, Adv. Phys. 51, 1379 (2002).16. C. Das, N. J. Inkson, D. J. Read, M. A. Kelmanson,

T. C. B. McLeish, J. Rheol. 50, 207 (2006).17. D. K. Bick, T. C. B. McLeish, Phys. Rev. Lett. 76, 2587

(1996).18. T. C. B. McLeish, R. G. Larson, J. Rheol. 42, 81 (1998).19. R. J. Blackwell, T. C. B. McLeish, O. G. Harlen, J. Rheol.

44, 121 (2000).20. P. Stanescu, J. C. Majesté, C. Carrot, J. Polym. Sci. B

Polym. Phys. 43, 1973 (2005).Acknowledgments: We thank the Engineering and Physical

Sciences Research Council (UK) for funding under the“Microscale Polymer Processing” (MuPP) consortiumand the European Union Framework Programme 7 PeopleProgramme for funding under the Marie Curie InitialTraining Network Dynamics of Architecturally ComplexPolymers. The data described in this work are archived onthe MuPP project online materials database at www.irc.leeds.ac.uk/mupp2.

Supporting Online Materialwww.sciencemag.org/cgi/content/full/333/6051/1871/DC1Materials and MethodsFig. S1Tables S1 to S3

15 April 2011; accepted 27 July 201110.1126/science.1207060

Table 1. The reaction model parameters of the hypothetical materials. xs1, the degree of conversion ofcomponent 1; w1, the weight fraction of component 1 (there is only one component used in this case).

Resin t b Cb Cs xs1 w1

Batch 1 1 × 10−3 8 × 10−5 2 × 10−2 0 0.15 1Batch 2 8.5 × 10−4 8 × 10−5 5 × 10−3 0 0.15 1

Fig. 4. Two reaction models (Batch 1 and Batch 2)with parameters as in Table 1 were used to calculate(A) molecular weight (and branching; g) distributions,(B) linear rheology (solid curves are G′′; dashed curvesare G′), and (C) extensional rheology at a range ofrates (black curves are Batch 1; red curves are Batch2). All rheology calculations use the parameters fromtable S2. Although they have very similar linear rhe-ology, this has been achieved through different com-

binations of branching and molecular weight distributions. As a result, the degree of extension hardeningfor the two hypothetical resins is quite different.

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Batch1∗ Mn Batch2∗ MnMax seniority 53 14

Weight fr. % g/mol Weight fr. % g/molLinears 14 6221 39 16030

Seniority 1 56 6389 48.5 16101Seniority 2 13.8 6499 9 16491Seniority 3 6.4 6639 2.4 16971Seniority 4 3.6 6568 0.7 17992Seniority 5 2.19 6867 0.2 15180

Total 96.17 99.9

*Read, D. J.; Auhl, D.; Das, C.; den Doelder, J.; Kapnistos, M.; Vittorias, I.; McLeish. T. C. B. ”Linking Models of Polymerization and Dynamics to Predict Branched Polymer Structure and Flow” Science, 333, 1871-1874 (2011)




Our approachLDPE analysisRheological behavior of LDPETime-marching algorithm

What we want to do:

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,because hierarchical relaxation is only active down to a certain molecular depth

Assume that the rest of the ensemble will relax automatically due to dynamictube dilation (disentanglement relaxation)

What we need to do for that:

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

Pruning the ensemble of molecules by considering only fractions of senioritiesand several molecular weights in MWD

Extend the time-marching model to treat the relaxation of high enoughgenerations (up to 6 ?)

Find a total stress relaxation which is a sum of the contributions from thesegments of increasing seniority





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





Distribution of seniorities

LDPE1∗ Mn LDPE2∗ Mn

Max seniority 47 91

Weight fr.% g/mol Weight fr.% g/mol

Linears 28.6 9797 21 12882Seniority 1 48 6025 48.8 5803Seniority 2 10.7 5437 12.5 5738Seniority 3 4.5 4418 5.9 4748Seniority 4 2.6 4208 3.8 5112Seniority 5 1.7 4190 2.2 4784

Total 96.9 94.6

LDPE3∗ Mn LDPE6∗ Mn

Max seniority 99 83


Linears 28.9 20607 12 5506Seniority 1 46 10020 54 5475Seniority 2 10.8 8960 14.2 5748Seniority 3 5 6884 6.9 5845Seniority 4 2.7 5967 4.1 5956Seniority 5 1.9 6360 2.5 5953

Total 95.4 94.2

*Read, D. J.; Auhl, D.; Das, C.; den Doelder, J.; Kapnistos, M.; Vittorias, I.; McLeish. T. C. B. ”Linking Models of Polymerization and Dynamics to Predict Branched Polymer Structure and Flow” Science, 333, 1871-1874 (2011)





MWD of seniorities

0 2 4 6

x 104

0

2

4

6

8x 10

−5 Linears

M [g/mol]

W(M

)

DataFlory

0 2 4 6 8

x 104

0

2

4

6

8x 10

−5 Seniority 2

M [g/mol]W

(M)

DataFlory

0 2 4 6 8

x 104

0

2

4

6

8x 10

−5 Seniority 3

M [g/mol]

W(M

)

DataFlory

0 5 10

x 104

0

2

4

6

8x 10

−5 Seniority 4

M [g/mol]

W(M

)

DataFlory

0 2 4 6 8

x 104

0

2

4

6

8x 10

−5 Seniority 5

M [g/mol]

W(M

)

DataFlory

0 2 4 6 8

x 104

0

2

4

6

8x 10

−5 Seniority 6

M [g/mol]

W(M

)

DataFlory





Analysis of topologies

Seniority Number of topologies1 12 13 24 75 566 22127 2447513





M

W(M)

M

τ(M)

Molecular weight distribution Relaxation time distribution

Seniority relaxation

Relaxation timeSeniority 1

Relaxation timeSeniority 2

Relaxation timeSeniority 3Topology 1

Relaxation timeSeniority 3Topology 2

... ... ... ... ... ... ...





LDPE2 Mn LDPE6 Mn

Max seniority 91 83


Linears 21 12882 12 5506Seniority 1 48.8 5803 54 5475Seniority 2 12.5 5738 14.2 5748Seniority 3 5.9 4748 6.9 5845Seniority 4 3.8 5112 4.1 5956Seniority 5 2.2 4784 2.5 5953

Total 94.5 94.2





LDPE6 Mn

Max seniority 83

Weight fr.% g/mol

Linears 12 5506Seniority 1 54 5475Seniority 2 14.2 5748Seniority 3 6.9 5845Seniority 4 4.1 5956Seniority 5 2.5 5953

Total 94.2





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 canbe relaxed locally

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

kB T

ζbr

= Dbr =p2a2

2τa





Modified approach

G(t) = G 0N ∗ F (t) + FLR + FFR ,

F (t) = Ψ(t) ∗ ΨCRR (t),

where, contributions coming from

Ψ(t) - primitive-path fluctuation and reptation

ΨCRR (t) - constraint release

FLR - longitudinal Rouse modes

FFR - unconstrained fast Rouse motions

Ψ(t) =m∑i

ϕi

∫ 1

0prept (xi , t) · pfluct (xi , t)dxi ,

where pfluct (xi , t) and prept (xi , t) are the probabilities that segment xi survives from relaxation by fluctuation orreptation, correspondingly

pfluct (xi , t) = exp

(−t

τfluct (xi , t)

).





Relaxation of stars

For t < τe the chain doesn’t feel the tube and stress isrelaxed by fast Rouse motion inside the tube, at slightlylonger time stress is relaxed by redistribution of chainsegments along the tube via longitudinal Rouse motion.

FFR =∑

i

5ϕi

4Za,i

N∑j=Za,i

exp

(−2j2t

τRouse,i

),

FLR =∑

i

ϕi

4Za,i

Za,i −1∑j=1

exp

(−j2t

τRouse,i

).

For t > τe star chains relax their stress by deep armretractions that can be analyzed as a thermally activatedprocess in an effective potential U(x). For shallowfluctuations where U(x) < kT , the potential is irrelevant

τearly (xi ) =9π3

16τe Z 4x4

i ,

Z = MMe

-number of entanglements in the star arm.

The tube theory doesn’t take into account that for t < τe the chain partially relaxes its stress via a Rouse process.Modification of the coordinate system is needed!

x=1x=0

xeq=0

x=xstart

x =1eq

τearly (xstart ) = τe and xeq =x−xstart1−xstart

.





Relaxation of stars

Fixed tube

The entropic barrier for the deeper fluctuations is givenby:

U(xi ) =3kT

2Zx2

i .

Such fluctuations require exponentially increasing time

τlate (x) = τ0exp(U(x)

kbT)

The transition between shallow and deep fluctuationshappens at xtr , such that U(xtr ) = kT

τfluct (xi ) = τearly (xi ) xi ≤ xtr ,

τfluct (xi ) = τearly (xtr ) exp

(U(xi ) − U(xtr )

kT

)xi > xtr

Tube dilation(DTD)

At long times the outer parts of the arms act as”solvent”. This means that the number of entanglementconstraints effective during relaxation of star armsdiminishes with time.

Leq (t) = Leq (0) ∗ (Φ(t))α/2, a(t) =

a(0)

(Φ(t))α/2,

Me (t) =Me (0)

(Φ(t))α,

U(xi , t) =3kT

2Zx2

i Φ(t)α xi > 1 − Φ(t)α,

where Φ(t) is the unrelaxed polymer fraction whichdetermines the speed of tube dilation.

The global effect of CRR process is to limit the maximumrate at which the entanglement density can decrease:

ΨCRR (ti ) = Φ(ti )

(ti−1

ti

)1/2

,

Φ(ti ) = max(Ψ(ti−1),ΨCRR (ti−1)).




PI viscoelastic data

100

105

102

104

106

ω [rad/s]

G’ (

ω)

[Pa]

TMA(old)

100

105

102

104

106

ω [rad/s]

G’ (

ω)

[Pa]

BoB

100

105

102

104

106

ω [rad/s]

G’ (

ω)

[Pa]

TMA(new)

100

105

102

104

106

ω [rad/s]

G"

(ω)

[Pa]

TMA(new)




PBD viscoelastic data

100

105

102

104

106

ω [rad/s]

G’ (

ω)

[Pa]

TMA

100

105

102

104

106

ω [rad/s]

G"

(ω)

[Pa]

TMA

100

105

102

104

106

ω [rad/s]

G’ (

ω)

[Pa]

BoB

100

105

102

104

106

ω [rad/s]

G"

(ω)

[Pa]

BoB




PS viscoelastic data

100

105

102

104

ω [rad/s]

G’ (

ω)

[Pa]

TMA

100

105

102

104

ω [rad/s]

G’,G

" (ω

) [P

a]

TMA

100

105

102

104

ω [rad/s]

G’(ω

) [P

a]

BoB

100

105

102

104

ω [rad/s]

G’,G

" (ω

) [P

a]

BoB




PS viscoelastic data

100

105

102

104

ω [rad/s]

G’ (

ω)

[Pa]

TMA

100

105

102

104

ω [rad/s]

G"

(ω)

[Pa]

TMA

100

105

102

104

ω [rad/s]

G’ (

ω)

[Pa]

BoB

100

105

102

104

ω [rad/s]

G"

(ω)

[Pa]

BoB




Relaxation of a branched polymer

How does this molecular section relax?

Additional friction


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





10−5

100

105

100

101

102

103

104

105

106

107

ω [rad/s]

G’,G

" (ω

) [P

a]

Sample A2 (BoB (red) vs TMA (blue))

10−5

100

105

100

101

102

103

104

105

106

107

ω [rad/s]

G’,G

" (ω

) [P

a]

Sample A3Data from Roovers,J. Macromalecules 1984, 17, 1196




Relaxation of PBD H-polymer (TGIC)

10−5

100

105

100

101

102

103

104

105

106

107

ω [rad/s]

G’,G

" (ω

) [P

a]

H12B100

10−5

100

105

100

101

102

103

104

105

106

107

ω [rad/s]

G’,G

" (ω

) [P

a]

H12B40

10−5

100

105

100

101

102

103

104

105

106

107

ω [rad/s]

G’,G

" (ω

) [P

a]

H40B40

10−5

100

105

100

101

102

103

104

105

106

107

ω [rad/s]

G’,G

" (ω

) [P

a]

H30B40




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prediction of the rheological behavior of ldpe · both long and short side chains are present ......

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