Rheology control by branching modeling - TU/e · Introduction Mechanism of Relaxation of Entangled Polymers Generalized Models for Rheology of EBP Modeling Rheology control by branching

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



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




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




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.


Uniaxial extensional viscosity and of various PE melts (Laun, 1984)


Uniaxial extensional viscosity of various PE melts




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




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





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





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





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





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.


Linear Polymers

a()R exp [U()]

Rouse

modes

inside the

tube





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





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



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.


kBT

ζbr= Dbr =

p2a2

2τa

with p2 = 1/12; smaller for short arms

Arm retraction Time:

τa = τ0Z1.5a exp(νZa)






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





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






Occurs from the outside of the polymer towards the inside



Occurs from the outside of the polymer towards

the inside







Sometimes side arms relax - relaxation cannot proceed further until themain arm ”catches up”. Side arms give extra ”friction”.



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






Sometimes side arms relax - relaxation cannot proceed further until themain arm ”catches up”. Side arms give extra ”friction”.



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






Eventually there is an effectively linear section which relaxes viareptation, with side-arms providing the friction



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






And finally relaxed!



and….. relax!





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




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




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




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 ?)





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)





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





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

)





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





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





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





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





Topologies of seniorities

Seniority Number of topologies1 12 13 24 75 566 22127 2447513

Seniority 4 Seniority 3





Topologies of seniorities

Seniority 5

44.39 % 11.56 %

11.68 % 6.62 %

5.13 %






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


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





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






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





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





Thank You!


Documents

Rheology control by branching modeling - TU/e · Introduction Mechanism of Relaxation of Entangled Polymers Generalized Models for Rheology of EBP Modeling Rheology control by branching