mcleish 98 molecular

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Molecular constitutive equations for a class of branchedpolymers: The pom-pom polymer

T. C. B. McLeisha)

IRC in Polymer Science and Technology, Department of Physics andAstronomy, University of Leeds, Leeds LS2 9JT, United Kingdom

R. G. Larson

Department of Chemical Engineering, 2300 Hayward,University of Michigan, Ann Arbor, Michigan 48109-2136

(Received 27 May 1997; final revision received 15 September 1997)

Synopsis

Polymer melts with long-chain side branches and more than one junction point, such as commercial

low density polyethylene LDPE , have extensional rheology characterized by extreme strain

hardening, while the shear rheology is very shear thinning, much like that of unbranched polymers.

Working with the tube model for entangled polymer melts, we propose a molecular constitutive

equation for an idealized polymer architecture, which, like LDPE, has multiple branch points per

molecule. The idealized molecule, called a pom-pom, has a single backbone with multiple

branches emerging from each end. Because these branches are entangled with the surrounding

molecules, the backbone can readily be stretched in an extensional flow, producing strain hardening.

In start-up of shear, however, the backbone stretches only temporarily, and eventually collapses as

the molecule is aligned, producing strain softening. Here we develop a differential/integral

constitutive equation for this architecture, and show that it predicts rheology in both shear and

extension that is qualitatively like that of LDPE, much more so than is possible with, for example,

the K-BKZ integral constitutive equation. 1998 The Society of Rheology.

S0148-6055 98 00401-5

I. INTRODUCTION

Polymer melts with long-chain branching have rheological properties that differ dis-

tinctly from those of linear polymers or polymers with side branches too short to entangle

with surrounding polymers Meissner 1972 ; Laun 1984 ; McLeish 1995 ; Larson

1988 . For commercial polymer melts, such as polyethylene, these differences show up

most strikingly in extensional flows. Melts such as low density polyethylene LDPE ,

which have multiple, irregularly spaced, long side branches, show a strain-hardening

phenomenon in uniaxial extensional flow that differs qualitatively from the behavior of

unbranched melts in similar flows. In shearing flows, however, the behavior of LDPE is

highly strain-softening, not qualitatively different from the behavior of ordinary un-branched melts. The terms strain-hardening and strain-softening here refer to the behavior

of the transient viscosities after start-up of flow. In this context, strain-hardening means

that for strain rates in the nonlinear regime, the viscosity during the start-up rises above

a Corresponding author.

1998 by The Society of Rheology, Inc.J. Rheol. 42 1 , January/February 1998 810148-6055/98/42 1 /81/30/$10.00


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the linear viscoelastic low-strain-rate response, while for strain-softening, the transient

viscosity falls below that observed in the linear viscoelastic limit. Examples of strain-

hardening and strain-softening will be displayed in Secs. III and IV.Ordinary linear polymers typically show predominantly strain-softening characteristics

in both shear and extensional flows, while long-chain branched melts are strain-hardening

in uniaxial extensional flow and strain-softening in shear. The strain-hardening behavior

of long-chain-branched polyethylenes is a desirable feature in certain processing flows

dominated by extensional flow, such as fiber spinning or film blowing. If the melt is

strain hardening, then regions of the melt that happen to become thinner, and therefore

more highly strained, than surrounding regions, are able to support a higher force per unit

area without undergoing run-away thinning or necking, leading to rupture. Since for

other purposes related to the properties of the final product long-chain branching is

sometimes undesirable, optimal tailoring of branch content has become a major goal of

the polyolefin industry. Such tailoring may now be possible at the commercial scale,

because of the development of metallocene catalysts Vega et al. 1996 . It is thus

crucial that the effects of long-chain branching on polymer melt rheology, especiallyextensional rheology, be understood and hopefully even predicted from molecular theory.

One would ideally like to develop a constitutive equation relating stress to deformation

history that somehow accounts for the effects of long-chain branching.

Existing constitutive theories for polymer melt rheology have not really come to grips

with the problem of long-chain branching, however. The most robust constitutive theory

is the single integral model of K-BKZ type Kaye 1962 ; Bernstein et al. 1963 ; Wag-

ner and Laun 1978 , given by

t

dtm1 tt, I1 ,I2 B t,t m 2 tt,I1 ,I2 C t, t . 1

Here B and C are the Finger tensor and the Cauchy tensor, with B C1 for anincompressible melt. The tensor B is related to the deformation gradient tensor E by B

ETE see standard treatments of nonlinear rheology, such as Larson 1988 or Tan-

ner 1985 . The invariants of the tensors B and C are given by I1 trace( B) and

I2 trace(C). In the limit of small strains, for which I1 I2 3, the functions m1and m2 reduce to functions of time only which must match the linear viscoelastic behav-

ior; i.e., m1( tt,3,3)m2( tt,3,3) (d/dt)G( tt), where G(s) is the linear

viscoelastic modulus.

At higher strains, outside of the linear viscoelastic regime, the dependences of m1 and

m2 on the invariants I1 and I2 become important and determine the strain-hardening or

strain-softening character of the predicted response. The functions m1(tt,I1 ,I2) and

m 2(tt,I1 ,I2) can, in principle, be measured in step-strain experiments, or can be

calculated from start-up experiments such as those mentioned above. Because of thedependences on two invariants, the degree of strain hardening predicted by the model can

change from one type of deformation to another, for example from shear to uniaxial

extension. For a shearing deformation, I1 I2 identically, while for uniaxial extension,

I1 becomes much greater than I2 for large deformations. Thus, by judiciously choosing

dependences of the K-BKZ strain measure on I1 and I2 it is possible to fit simultaneously

highly strain-softening behavior in shear, and highly strain-hardening behavior in

uniaxial extension. See, for example, Wagner et al. 1979 . However, for planar exten-

sion, as for simple shear, I1 I2 . This implies that while the strain softening or hard-

ening character of uniaxial extension can in principle diverge arbitrarily far from that in

82 McLEISH AND LARSON


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simple shear, the behavior in planar extension is of necessity linked to that of simple

shear, for the K-BKZ model Samurkas et al. 1989 .

Thus, although the K-BKZ equation is very flexible in its ability to fit simultaneouslydiffering flow characteristics as one changes from one type of flow to another for a given

material, it is not infinitely flexible. In particular, when the K-BKZ kernel functions are

obtained by fitting the highly strain softening behavior in shear, they fail to predict the

highly strain-hardening response seen in planar extension for LDPEs.

The planar extensional viscosities have been measured for two different long-chain-

branched polyethylenes, in one case by Laun and Schuch 1989 and the other by Meiss-

ner 1996 . For both materials, the dominant planar extensional viscosity was found to be

highly strain hardening, nearly as much so as the viscosity in uniaxial extension. As

discussed by Samurkas et al. 1989 , the K-BKZ equation fails to predict the observed

degree of strain hardening in planar extension when the kernel functions are adjusted to

fit the observed degree of strain softening in shear.

This failure of even so robust a model as the K-BKZ equation to describe the rheology

of long-chain branched polymers suggests that some new molecularinsight is needed intothe nonlinear relaxation processes that occur in such melts under flow.

The development of a realistic molecular theory for commercial LDPE melts is, how-

ever, a daunting task. The branching structures in typical LDPEs are topologically and

geometrically irregular, and molecular weights are very polydisperse. Thus, a quantita-

tive molecular theory for commercial branched polymers is at present out of reach.

The best way forward would therefore seem to lie in the development of a molecular

constitutive theory for the nonlinear rheological properties of an idealizedbranched melt,

one that is simple enough to consider theoretically, but still having the generic properties

of LDPEin particular the strain hardening in extension and strain softening in shear.

The most important aspect of the branching structure of such molecules is the presence of

multiple branch points on the same molecule. Because of the multiple branch points,

there are molecular strands that lie between two branch points and hence have no free

ends. As described in the following, we believe that such molecular strands produce thestrain hardening properties of multiply branched melts such as LDPE in extensional

flows. Such molecules differ in a significant way from molecules with a star architec-

ture; stars have only a single branch point, and each arm of the star has a free end. Under

flow, the free end allows the branch to retract quickly, and hence large stresses cannot

readily build up. Star architectures are therefore too simple to produce the strain harden-

ing seen in LDPE.

The simplest architecture that has a molecular strand with no free end is one with just

two branch points. This architecture is therefore the simplest one that might produce the

nonlinear rheological characteristics of commercial long-chain branched polymers such

as LDPE.

II. MOLECULAR RHEOLOGY OF THE POM-POM POLYMER

The class of branched polymers chosen for this study can be thought of as a gener-

alization of the H-polymer structure Roovers 1984 , McLeish 1988 . The molecules

contain just two branch points of chosen functionalitya backbone links two pom-

poms of q arms each see Fig. 1 . The attraction of this family of polymers is that the

parameter space describing molecular structure is sufficiently large to be interesting:

variables are the molecular weight of the backbone Mb , molecular weight of the arms

Ma , and the number of arms on each branch point q . The entanglement molecular

weight Me is another important molecular parameter, but will serve only as a scale for

83BRANCHED POLYMERS: THE POM-POM MODEL


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polymer molecular weight so that we define the dimensionless partial molecular weights

sa Ma /Me and s b Mb /Me . The quantities sa and s b are also measures of the

entangled path lengths of the arms and backbone, respectively. On the other hand, the

symmetry of the structure, and the separation of timescales between the backbone and the

arms see below will permit a tractable constitutive formulation, valid for an interesting

range of shear rates. In this section we introduce the physical features of the model withreference to existing theoretical and experimental work on well-characterized branched

polymers.

We will see that the structure is indeed complex enough to exhibit the special nonlin-

ear rheology of branched polymers. This is suggested by the application of the tube

model Doi and Edwards 1986 to branched polymers: we recall that in this theoretical

model of the dynamics of entangled polymers the topological constraints of any given

chains neighbors are assumed to be represented by a confining tube along the chain

contour. Perpendicular to the tube axis, a chain segment is not permitted to move more

than the tube diameter, a, but parallel to the tube a linear polymer chain may freely

diffuse and thereby move its center of mass reptation . It may also contract by virtue

of its entropic elasticity following an imposed stretch retraction . However in the

case of a branched polymer such as the pom-pom, the strand between the two branch

points is not free to retract within its tube after a step strain as freely as a chain with a freeend. This leads to less strain-softening than in linear polymers, and in particular was

shown for the H-architecture to lead to extension-hardening McLeish 1988 . In con-

firmation of the prediction that it is material between branch points that gives rise to the

special rheological properties of branched polymer melts, the nonlinear rheology of

simple star polymers follows closely that of linear melts Fetters et al. 1993 .

A. Branch point withdrawal

Of course the resistance to retraction in a flow cannot be unlimited even in a highly

branched polymerotherwise the response would be that of a network. Fortunately the

FIG. 1. The structure of a pom-pom polymer with a backbone and 2 q dangling arms, with q 3, under

various degrees of stretch. The stretch of the backbone is denoted by the variable (t), the path length of arm

withdrawn into the backbone tube by the variable s c(t).



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tube model provides a mechanism which naturally limits the curvilinear stretch any

segment can support. Once the constraints on every chain by its neighbors are modeled

by a tube along its contour of diameter a , the requirement that Gaussian chain statisticsare maintained in equilibrium implies a net Brownian force on every free chain end of

magnitude kT/a, tending to maintain the curvilinear path of the chain segment within its

tube. This means that, for the q 3 pom-pom polymer of Fig. 1, the backbone may be

stretched by the three free ends until it sustains a curvilinear tension of 3 kT/a. Since all

segments share the same equilibrium tension, this amounts to a maximum allowed stretch

of three the primitive path occupied by the backbone segment may be no longer than

three times its equilibrium length . Beyond this, the tension in the backbone is sufficient

to withdraw the free ends into the tube originally occupied by the backbone see Fig.

1 c . This is not prohibited sterically since tube diameters are always much larger than

monomer dimensions. Neither are there density nonuniformities penalizing the process,

as conjectured to occur in reptation for very high molecular weight polymers Semenov

1997 . This is because in a given tube volume there are many of order Ne3/2

/N pom-

pom molecules. Any mismatch between the number of dangling ends leaving and enter-ing the volume may be accommodated by local elastic strain of the entanglement network

at an energy cost kT. An alternative way of explaining the effect is to employ an

entropy balance: eventually it becomes less costly to lose entropy in the free ends by

withdrawing them all into the same contour tube than to lose it by continuing to stretch

the backbone and thereby to restrict its own configuration space. This new effect of

branch point withdrawal has been used recently to calculate damping functions for a

range of molecular topologies Bick and McLeish 1996 .

The mechanism of branch point withdrawal introduces in a natural way two important

dynamical variables which will be necessary to include in any molecular constitutive

formulation. The first of these is the dimensionless stretch ratio, , of the path length of

the backbone to its equilibrium length. The second is the path length of arm which is

forced to adopt a configuration set by the backbone orientation along with a similar

section of all other attached arms due to withdrawal of the branch point along the tube.

The latter we measure in units of the tube diameter and denote by sc so that 0 sc Ma /Me . Partially withdrawn configurations such as that of Fig. 1 may occur during

continuous deformation as well as after a step strain, so both and s c are functions of

time. The use of a single stretch parameter for the entire backbone is justified because any

local fluctuations in stretch very rapidly equilibrate on timescales on the order of the

Rouse time of the backbone. As we will see below, these timescales are much shorter

than those of rheological interest. The stretch and withdrawal play complementary roles:

the configurational entropy balance means that sc 0 whenever q but s c assumes

its own dynamics when the flow history is such that q, its maximum value. Con-

versely we will find that whenever sc is nonzero the stretch of the backbone is fixed at q.

The conclusion is that in a general dynamical history at any one time either or s c is

changing with time, but not both. We will see that the role of the configurational changesset by sc is to change the effective relaxation times of stretch and orientation of the

backbone. In this way the molecules avoid in a natural way the infinite extension in

elongational flows predicted for Gaussian chains or dumbbells with fixed friction. We

now turn to the linear stress relaxation of the pom-pom model to see how the various

relaxation times arise.

B. Linear stress relaxation

Before proceeding to general nonlinear deformations, we need to derive the conse-

quences of the tube model for the linear stress relaxation of a pom-pom melt and deduce



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the configurational relaxation times of the various parts of the molecule. The tube model

implies no extra terms in the expression for the stress of a polymeric fluid than those

arising from the rubber-elastic orientational entropy of the chain segments Doi andEdwards 1986 . So each segment of a polymer melt contributes to the stress with a time

dependence that reflects exactly its orientational relaxation. Because the orientation of a

segment is in turn constrained by the orientation of the tube segment it occupies, the

orientational relaxation times are just the characteristic times for chain ends to visit the

corresponding tube segments and thereby destroy all memory of their constraints.

We consider how an ensemble of mutually entangled pom-pom polymers reconfigure

the orientations of their segments at t 0 by escaping from their original tube con-

straints. The first consequence of the tube model for these branched polymers is that, at

early times, the backbone segments do not relax at all beyond the confined motion within

the tubesthe branch points are effective pinning points under infinitesimal deformation

as observed above and therefore no free ends have access to backbone segments. So at

first all relaxation is confined to the arms. These behave in a similar manner to a melt of

entangled star polymers, in which it is now well established that tube segments are lost bydeep retractions or breathing modes of the arms. These occur by the random formation

of unentangled loops Helfand and Pearson 1983 which become exponentially more

rare as their size increases. A consequence of this is that the very rapidly relaxing

segments at the extremities of the arms behave as unentangled solvent at the longer

timescales characterizing the relaxation of segments nearer the branch point Ball and

McLeish 1989 , Milner and McLeish 1997 . This has a very strong accelerating effect

on the distribution of timescales for relaxation along the star arms, modifying the result

for star arms in a fixed tube:

x 0 exp15

4s a

1x2

2 2

into the result for a melt of pure stars:

x 0 exp15

4s a

1x2

2

1x3

3 . 3

In both these cases (x) is the relaxation time of a tube segment a fraction x from the

branch point to the free end. The constant 0 is an attempt time for deep retractions of the

entangled dangling arm. In fact 0 carries an additional weak x dependence which has

recently been calculated Milner and McLeish 1997 but this is not relevant to the

backbone relaxations central to this work.

The enormous effect of the cooperative relaxation of the starlike arms can be seen in

the different predictions for the terminal times of the modified and unmodifiedexpressionsthe former is faster by a factor of exp (15/12)s a . In the melt of pom-pom

polymers, the free arms relax as if they were in a melt of star polymers with the same

molecular weight of arms except for one aspectthe backbones remain part of an effec-

tive network throughout the relaxation process of the arms, and so give a spectrum of

relaxation times for the star arms intermediate to the two extreme forms above. The

calculation is equivalent to the case of a blend of star polymers with very long linear

polymers studied by McLeish and OConnor 1993 . In this case the fraction of effec-

tively fixed backbone material b enters into the expression for the pom-pom arm relax-

ation time spectrum, a(x), in a natural way:



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a x 0 exp15

4sa

1x 2

2 1b

1x 3

3 . 4

This prediction has been checked recently both in the case of a star-linear blend and for

a monodisperse H-polymer Allgaier et al. 1997 . The relaxation of occupied tube

segments for timescales less than and including a(0) can now be written as a simple

integral over the arm segments, with a fixed backbone. Following the quadratic depen-

dence of the modulus on concentration of oriented polymer segments, we find in this

regime of timescales:

G t G0 b 1b0

1et/a x dx

2

. 5

Before proceeding to the slower viscoelastic modes originating from the backbones, it is

worth noting in anticipation that the exact form of 4 of the distribution of arm relaxation

times is not at all essential to the nonlinear rheological properties that are our mainconcern. The important property of this fast regime of the relaxation spectrum is that the

segmental relaxation times should decay rapidly from the branch point towards the free

ends of the arms. This requirement is certainly met by the particular exponential form of

4 predicted by the tube model.

The physical picture of the pom-pom polymer at timescales longer than the maximum

orientational relaxation time of an arm is quite different from the star-arm dynamics

outlined above, although it is consistent with it. Now the backbones themselves are the

dynamical objects, and all arm material behaves as solvent because of its much more

rapid dynamics. The backbones therefore diffuse by reptationthe polymers are now

effectively linear objects. However, the tube diameter is larger than it is at high fre-

quency: the only entanglements that are effective in constraining the backbone are those

with other backbones, so that the effective number of entanglements along the backbone

becomes sbb . This argument was advanced some time ago to explain early rheologicaldata on H-shaped polystyrene melts McLeish 1988 and has been confirmed in recent

experiments Allgaier et al. 1997 .

There is one further difference between the reptation of the pom-pom backbones and

that of ordinary linear polymers: in the case of the branched polymers at long times all of

the effective friction to curvilinear motion is located at the branch points rather than

distributed along the chain. We can see this by calculating the diffusion constant of one

of these branch pointsafter a time a(0) the branch point has moved within the con-

straining tube typically the distance of the tube diameter itself it might be guessed that

the diffusive step would explore typically the dilated tube, but here we take a model for

the branch point diffusion which relies on the residence time of the completely retracted

state, which is much shorter than the terminal time for the arm at which dilation is valid ,

so a2 2Dca(0). Using an Einstein argument and requiring that the drag on a branch

point increases linearly with the number of arms q gives a branch-point friction constantb of:

b kT

Dc 2kT

a 0

a2 q. 6

This is exponentially dependent on the arm molecular weight, whereas the contribution to

the total friction of the backbone from the backbone monomers themselves is linear in the

backbone molecular weight. The characteristic time for orientational relaxation of the

backbone material, b , is therefore the diffusion time of a one-dimensional walker of



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diffusion constant Dc/2 there are two branch points on the backbone to traverse a mean

square distance of the dilated primitive path between the branch points, L2. Assuming the

inverse proportionality of the entanglement molecular weight on the concentration ofentangled strands which is b in our case , we have effective values for the dilated tube

diameter and primitive path of aeff ab1/2

and sbeff s bb so that L2 sbeff

2aeff

2

a2bsb2

. The final result for the backbone relaxation time, b , is:

b 4

2 sb

2ba 0 q. 7

The prefactor, as in the standard reptation theory, comes from the solution of the one-

dimensional diffusion equation with absorbing boundaries at either end of the linear

domain see Appendix A .

We could carry the theory of the linear relaxation spectrum to greater refinement by

including the full reptation spectrum of modes, and incorporating the effect of orienta-

tional relaxation by path length fluctuation of the backbone. In fact this is done elsewherein a discussion of experimental results on the H-polymer McLeish 1988 , Allgaier et al.

1997 . For the purpose of this exercise, which is to develop a workable constitutive

equation for this class of polymers in nonlinear flows, it will be sufficient to assign a

single relaxation time to the backbone material, so that the full relaxation modulus be-

comes:

G t G0 bet/b 1b

0

1et/a x dx

2

. 8

C. Slow-mode restriction and expression for the stress

We will assume that the stress may be calculated as a function of molecular orienta-

tion and, as is standard in the tube model, that this in turn may be calculated from theorientation distribution of occupied tube segments Doi and Edwards 1986 . For ordi-

nary linear polymers whose distribution of tube-segment orientations u is known, and

which are not stretched beyond the equilibrium contour length of chain per tube segment,

the stress tensor may be written:

154 G0 uu , 9

where the angular brackets denote an average over the orientation distribution. As usual

it is the second moment of the orientation distribution function that governs the stress. At

this point we observe that the wide separation of timescales between the spectrum of arm

relaxations and the backbone relaxation means that a wide range of deformation rates,

between b1

and a(0 )1, give rise to nonlinear response and therefore have Weissen-

berg numbers greater than one , while only marginally perturbing the configuration of thearm material away from equilibrium. Within this wide window of flow rates, the impor-

tant contribution to the stress comes from the backbone segments of the moleculesthe

arms merely contribute to an effectively Newtonian background viscosity. Therefore we

will henceforth work with variables pertaining to the backbone alone orientation distri-

bution and stretch , noting that when s c 0, some arm material essentially becomes part

of the backbone. The stress tensor expression will then be valid at timescales longer than

all arm relaxations, when the relaxed arm material is acting as solvent. It has long been

established as a useful and accurate approximation that the concentration dependence of

the shear modulus in theta-solvents or melts is approximately quadratic Ball and



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McLeish 1989 , Colby and Rubinstein 1990 . Hence, in terms of the high-frequency

plateau modulus G0 , we expect a modified expression for the stress, arising from the

backbone alone, given by:

15

4G0b

2 uu with b

s b

s b2qs a. 10

One factor ofb is simply accounting for the number of elastically active segments; the

other accounts for the dependence of the tube diameter on concentration.

Further modification of the stress expression is required in the case of flows that

produce either backbone stretching ( 1) or branch-point withdrawal ( s c 0). If the

backbone is stretched, then it may occupy more than the equilibrium number of tube

segments each of length a . Since each occupied and oriented tube segment contributes

to the stress, this effect produces a linear dependence of stress on . Moreover, as

discussed above, when the backbones are stretched, the thermodynamic tension which

they carry grows proportionately with the stretch. The final result is that the stress con-tribution from the backbone is quadratic in .

Still higher values of the molecular deformation lead to partially withdrawn configu-

rations as illustrated in Fig. 1 , when some of the arm material near the branch point is

also oriented. This material must also contribute to the stress because it shares the same

orientation distribution as the end segments of the linear backbone. At this point we make

an approximation by assuming that the orientation distribution is a weak function of

position along the backbone, and that in consequence the tensor S uu is the uniquemeasure of orientation also imposed upon the withdrawn arm material. S( t) becomes the

third dynamical variable which we shall require for the molecular rheological model,

along with ( t) and s c(t) S is, of course, the variable least surprising to a rheologist .

The final expression for the stress in terms of the dynamic variables becomes

15

4G0b b2 t 2qs c t2qs as b S t with b

s b

2qsasb. 11

The contribution from the withdrawn arm segments carries just one power ofb from the

diluted entanglement environment arising from the backbones alone. It might be objected

that the inclusion of contributions to the stress from arm material is inconsistent with our

restriction to deformation rates much slower than any of the arm segments natural

relaxation times. However, there is no inconsistency since the local deformation rate

within a tube segment surrounding a withdrawn arm segment is actually much larger than

the bulk deformation rate. This will be demonstrated in the next section. On the other

hand, the consequence of the rate restriction is indeed reflected in the stress formula in

that the withdrawn segments do not contribute to the entanglement network which is still

controlled by b even though they are oriented. The rapid star-arm retractions are still

fast on the timescale of our deformations, so the arms cannot act as topological obstacles

for backbone material, even though they may be oriented by it.

D. Dynamical equations

It is already apparent that the constitutive structure suggested by our molecular model

is not a closed equation relating stress to strain, but will instead be cast into a dynamical

system of evolution equations for the auxiliary variables S( t), ( t), and s c(t) which in

turn are used to construct the stress. At this stage we make the further assumption that

each backbone spans a sufficient number of tube segments to self-average over both the



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orientation and the stretch. This allows us to carry unique values for the dynamical

variables within an element of the melt over which the deformation history is constant,

rather than work with distribution functions for them in which state the problem was leftby McLeish in 1988 . In the limit of large s b this approximation will become increasingly

accurate. This in contrast with the dilute case, where the serious deficiency of pre-

averaging approximations in dumbbell models has recently motivated highly sophis-

ticated treatments of the full orientational ensemble Feigl et al. 1995 . The highly

entangled backbone of pom-pom polymers seem to provide a molecular justification of a

desirable approximation in the phenomenology which the physics has elsewhere refused

to legalize! We take each variable in turn in the following discussion of their dynamical

equations.

1. Dynamics of orientation tensor St

The orientation tensor S measures the distribution of unit vectors describing the ori-

entation of tube segments in the deforming melt. We have made the approximations of a

single relaxation time and of a unique orientation distribution for backbone segments.

However, as we noted above, an essential aspect of the flow-induced molecular configu-

rations at high strains is that they reduce the effective friction constants of the branch

points and in consequence the relaxation time for the tube orientation. In particular, the

expression for the backbone relaxation time has a generalization to a flow for which sc 0:

b 4

2 sb

2ba xc t q, 12

where x c sc /sa . The essential feature is that the relaxation time is itself in general

time-dependent and takes its equilibrium value as an upper bound. It is actually quite

straightforward to write an integral expression for the time evolution of S under these

conditions, and was done recently in the case of wormlike surfactant micelles for which

the approximations are exact under certain circumstances Cates 1990 . Tube segments

are created at a rate b1

(t) and are thereafter convected and extended by the flow. A

segment created with orientation u at time t has an orientation E( t,t)u/ E(t, t)u at

time t, where E(t, t) is the local deformation gradient tensor between those times

providing it survives. The segment will also increase in length, and therefore in the

amount of chain it carries, by E(t, t)u . Segments created with orientation u at ttherefore carry a relative weight of E( t,t)u / E(t,t)u in the distribution at time t again the angular brackets denote an average over orientations u . The survival prob-

ability itself is the exponential of the time integral of destruction rates b1

( t) in the

interval t t t. The resulting expression for S(t) is therefore:

S t

t dt

b texp

t

t dt

b t 1 E t,t u

E t, t uE t,t u

E t, t u . 13In common with tube theories for ordinary linear polymers Doi and Edwards 1986

and entangled wormlike micelle solutions Cates 1990 , this has interesting asymptotic

properties in extensional and shear flow. In uniaxial extension with x the direction of

stretch, the difference in normal components SxxSy y approaches a constant in the limit

of infinite extension rate, whereas in simple shear with x the flow direction and y the

gradient direction Sxy goes as the inverse of the shear rate for large shear rates.



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2. Dynamics of stretcht

The molecular dynamics of the backbone stretch is controlled by the local force

balance of dissipative drag and elastic recovery. In a flow, the extension of the tubearound the backbone tends to drag the backbone segment with itthe coupling is just the

effective friction constant of the branch points calculated above. In opposition is the

Gaussian elasticity of the backbone whose effective spring constant is kT/sba2.

As above, we denote the curvilinear distance of separation of the two branch points

along the tube by L(t); since the equilibrium separation is just s ba , we have

L/sba. Now the relative velocity of separation of the branch points along their con-

necting tube is just L/ t. However, the velocity of separation of the tube relative to the

branch points against which they drag depends on the average rate of extension of the

tube segments linking the branch points as they are deformed by the flow. If we define

the deformation rate tensor K such that E(t, t)/ t K.E(t, t) then the average

increase of length per unit length of tube is just K:S, so that the relative curvilinear tube

velocity is LK.S. Equating the frictional drag force from the relative velocity to the

elastic force restoring L to its equilibrium length, we write:

b K.SLL

t

kT

sba2 Lsba . 14

Substituting for the branch point friction constant b from Eq. 6 and re-expressing the

force-balance in terms of the dimensionless stretch parameter gives the second evolu-

tion equation:

t K:S

1

s 1 strictly for q . 15

The stretch relaxation timescale s is given by

s sba 0 q. 16

This expression is of the form /2 for the relaxation time of a spring of spring

constant working against friction at each end of. The effective friction constant of the

branch point clearly scales with both the terminal time of the arm, a and the number of

arms q. The spring is the Gaussian chain of the backbone, whose elastic constant

sb1

. This time is shorter than the terminal time for orientation relaxation by one

power of s b but it is relevant to our declared range of deformation rates because it is

longer than the longest arm relaxation time a(0). Relaxations with this timescale only

appear in nonlinear deformations.

3. Dynamics of branch-point withdrawal sct

As discussed above, if ever the melt achieves a local state in which q , the model

insists that dynamical evolution of stops in favor of the variable s c describing branch-

point withdrawal. Evolution of recommences if and when sc returns to zero. We may

think of the evolution of sc in a similar way to that of, in terms of a balance between

an elastic restoring force and a frictional drag force. In this case, however, the drag comes

from the parts of the arms that are not withdrawn into the backbone tube the blob of

randomly oriented material from the q arms at the extremity of the molecule in Fig. 1

because by its rapid retraction dynamics this material creates dissipation by sliding

against the continually extending tube. Using similar arguments to those for b we find



12/30

blob 2kTa x c

a2 q . 17

The relative velocity of the extending tube and the monomer at sc , assuming that the

mid-point of the backbone is at rest, is (qsb/2sc)K:S sc / t . The factor qs b/2

enters because the backbone must be stretched by a factor max q before branch-point

withdrawal occurs. The drag on this relative velocity is balanced by the equilibrium

tensions of the arms qkT/a , since the arms are never stretched at the deformation rates

we are interested in. It might be objected that a higher deformation rate must always lead

to greater tension in a frictional polymer chain embedded in the deforming medium. But

as Ajdari et al. 1994 have shown in the context of chains pulled out of a network, the

tube model gives rise to a self-adjusting friction constant by growing or shrinking the

blob of disoriented material near the chain end. For all chain velocities for which the

chain moves less than a tube diameter in the Rouse time of an entanglement segment, the

tension is kept at the equilibrium value. Our theory for s c in a deforming melt containsthe same physics that Ajdari et al. applied to chains dragged out of a network. Balancing

the drag and elastic tension, and cancelling all dimensional factors of the tube diameter

gives our evolution equation for sc :

sc

t q sb

2sc K:S

1

2a s c, when q . 18

We now see how it is that arm material may be aligned by bulk flows at deformation rates

far slower than the inverse relaxation times of the armthe bulk deformation rate pro-

jected onto the tube orientation distribution K:S is amplified by the typically large

factor of the path length, ( qs b/2s c), from the center of the molecule to the blob of

nearly relaxed material in the driving term for sc(t).

The three equations for S(t), ( t) and s c(t), Eqs. 13 , 15 , and 18 together with

the expressions for the self-consistent variable timescales within them Eqs. 12 and

16 and the expression for the stress Eq. 11 , constitute the simplified constitutive

formulation for a melt of pom-pom polymers. It does not fit neatly into any phenomeno-

logical category of constitutive equation such as the K-BKZ equation , since it shares

both integral in the case of S and differential in the cases of and s c character.

Moreover its structure has the flow coupled to auxiliary variables from which the stress is

constructed, rather than evolving the stress directly. In the next section we will explore

the consequences of the model for a range of viscometric flows. However, we may

remark that we anticipate both extension hardening and shear-thinning from the model:

before any reduction of relaxation times sets in, the model is essentially an entangledversion of a simple Gaussian dumbbell, or Oldroyd model. Such models always exhibit

stretches which grow without limit in extension above a critical extension rate. In mod-

eling dilute solution rheology, this problem is usually circumvented by imposing a maxi-

mum strain on the molecule at which the force required for further extension diverges.

The entangled pom-pom polymers avoid the divergence in a gentler way: by renormal-

izing their relaxation times, they postpone the divergence to ever higher deformation

rates, remaining always in a Gaussian regime. However, we expect to see some growth of

the stress in extensional flows just before this renormalization takes place. This is not

expected in shear, since for simple dumbbells the driving divergence does not arise.



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To summarize this section, we gather the effective constitutive equation set together:

Pom-Pom Equation Set

Expression for the stress:

15

4G0b b

2 t 2qs c t

2qsas bS t with b

s b

2qsasb,

Evolution of orientation:

S t

t dt

b texp

t

t dt

b t 1

E t,t u E t,t uE t,t u E t,t u ,

Evolution of backbone stretch:

t K:S

1

s 1 strictly for q ,

Evolution of arm-withdrawal measure s c :

sc

t q sb

2sc K:S

1

2a x cstrictly for q .

Timescales:Backbone orientation:

b 4

2 sb

2ba x c t q with xc

sc

s a this timescales changes

Arm spectrum:

a x 0 exp15

4s a

1x 2

2 1b

1x 3

3 this timescale changes

Backbone stretch:

s sba 0 q this timescale is fixed .

III. BEHAVIOR IN STARTUP AND STEADY FLOWS

We now explore some consequences of the set of equations derived in the previous

section by computing the stress-growth on startup of extensional and shear flows. A

representative molecular structure is chosen so that neither the arms nor the backbone

dominate the total amount of material. It is of interest to begin with a molecule that has

all parts well-entangled, and for which the backbone is permitted to stretch by a signifi-

cant amount before the arms begin to withdraw. Finally, we observe that is also of

interest to explore the case in which b and s are reasonably well-separated as times-

cales. When this is not so, the orientation and stretch contributions to the stress meet their

respective nonlinearities at the same deformation rates, and the pom-pom polymer would



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be expected to mimic an ensemble of simple elastic dumbbells, albeit with a novel

nonlinear softening. Such models do not show the extreme differences in shear and

extensional behavior seen in branched melts. On the other hand, if the nonlinearity instretch is postponed to deformation rates already in the highly nonlinear regime, the

aligning effect of the shear flow may suppress the tendency to stretch the K:S term in the

stretch equation is small as strong orientation by shear has already begun at shear rates

near the inverse orientational relaxation time . Such a separation cannot, however, pre-

vent an extensional flow from exploring the novel features of this model under a strong

stretch because in that case the stretch-coupling to the flow K:S remains large. We

observe that it is the entanglednature of the effective dumbbell of the pom-pom polymer

that generates this separation in timescales:

b

s sbb . 19

The structure specified by the molecular parameters q 5; s a 3; sb 30 was

chosen to begin with, in the light of these considerations. In all that follows, we make

deformation rates dimensionless in terms of the arm relaxation time a(0 ). This is not, of

course, a terminal time, but stands apart from the two important long times, the orienta-

tional relaxation time b and the stretch relaxation time s . For our chosen model s 150 and b 912 in these units. To estimate the significance of a nonlinearity and

b molecular stretch in a flow Weissenberg numbers may be defined in each case by

multiplying the deformation rate by the appropriate dimensionless time constant. We also

make all stresses dimensionless in terms of the plateau modulus G0 . The pom-pom

equations are solved by a first-order time integration, using small timesteps. To simplify

the calculations, the strain tensors are approximated using formulas derived by Currie

1984 , as described in the next section.

Figure 2 a indicates the growth of extensional stress xxy y , normalized by the

extension rate, in start-up of uniaxial extensional flows over a range of deformation rates.

In terms of the arm relaxation time, these represent dimensionless deformation rates from

0.0003 to 0.04. As expected, the low rates 0.0006, corresponding to orientational

Weissenberg numbers less than 1, exhibit simple stress growth to a steady-state plateau.

As the extension rate is increased, a very small amount of extension thinning is observed

before the backbones begin to stretch at 0.0025 which is where the stretch Weis-

senberg number begins to approach values of order one . This can be seen Figs. 2 b and

2 c , which show on the same time-axis the evolution of the molecular dynamical vari-

ables and sc . Throughout the range of deformation rates over which the plateau

viscosity is growing, the equilibrium value of is also rising. At still higher rates, a

marked change of behavior sets in: reaches its maximum value in finite time, and

thereafter branch-point withdrawal occurss c rapidly rises and finds its equilibriumvalue, typically after a small overshoot Fig. 2 c . The stress-growth curves Fig. 2 a

now show a rapid hardening behavior which is cut off, again with a small overshoot, as

the maximum sustainable backbone stretch is reached. For still greater extension rates,

the extensional viscosity thins again as the steady-state extensional stress is not much

altered by further increases in extensional rate. The qualitative similarity with extensional

data by Laun for LDPE Laun 1984 is remarkable.

The computations for shear flow show very different behavior. Figure 3 shows the

growth of shear viscosity and first normal stress coefficient for the same range of defor-

mation rates as in the extensional flows Fig. 3 a , together with the molecular dynami-



15/30

cal parameters Fig. 3 b . Overshoots in both the shear stress xy and the normal stress

difference xxy y are evidentin fact the overshoots in the shear stress occur at lower

shear rates than in the normal stress. The shear behavior is similar to the results of

previous models incorporating molecular stretch in entangled linear polymers Pearson

FIG. 2. a Time-dependent extensional viscosity (xxyy )/ on startup of uniaxial extension for the pom-

pom model with q 5, sa 3, and sb 30. Extension rates are 0.0003, 0.0006, 0.00125, 0.0025, 0.005,

0.01, 0.02, and 0.04 in terms of the arm relaxation time a(0). Weissenberg numbers for orientation and stretch

may be obtained by multiplying these rates by 912 and 150, respectively. In this and all figures, times and rates

are made dimensionless using a(0 ) and viscosities are made dimensionless using G0a(0). b , c The time

evolution of the molecular variables ( t) and s c(t) for the same flows. High stretch correlates with extension

hardening; rapid changes in sc with equilibration of stretch.



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et al. 1991 . Although the backbone may stretch transiently, and even reach its maxi-

mum value, no hardening effect occurs, and the shear response is uniformly thinning.

The steady-state flow curves for the pom-pom model are shown in Fig. 4 solid lines ,

where the simultaneous shear-thinning and extension-hardening is manifest. Our suspi-

cion that the shear-flow dynamics never sample the stretching behavior of the mol-

ecules is confirmed for this set of structural parametersthe steady-state stretch only

occurs in the extensional case, giving rise to the peak in extensional viscosity. The

magnitude of the peak and the extension rate at which it occurs may be controlled by

varying the molecular structure. Crudely, the maximum stretch sets the height of the

peak, and this is in turn determined by q , whereas the extension rate at the peak will be

of the order ofs1

. Since the low rate limiting viscosity is approximately G0b , and the

peak viscosity, from the preceding arguments, is G0q2s , the ratio of these two is

proportional to q2s /b . Since q 1 and s /b 1, the existence of an extensional

viscosity peak higher than the low rate viscosity is delicatethe separation of the two

FIG. 3. a Time-dependent shear stress and first normal stress difference for shear startup flows using the same

model parameters and deformation rates as in the extentional calculations of Fig. 2. b The time evolution of

(t) and sc( t) in shear.



17/30

fundamental timescales s

and b

must not be so great as to negate the hardening due to

stretch. On the other hand, when the timescales are widely separated, a regime of exten-

sion thinning will exist before the hardening regime. This constitutes an interesting pre-

diction to be tested by future experiments.

A very important prediction of the model is that the response in planar extension falls

into the same class as uniaxial extension, not that of shear. This is clear in Fig. 5 a where

we compare the steady-state uniaxial and planar extensional viscosities as functions of

extension rate. At low extension rates the planar and uniaxial viscosities differ by a

modest factor of 4/3, but at higher strain rates this difference disappears entirely. Thus,

qualitatively, the difference between shear and extension shown in Fig. 4 applies to both

uniaxial and planar extension. In Fig. 5 b , we plot the transient shear and normalized

uniaxial and planar viscosities for a deformation rate of 0.01. Again, the planar and

uniaxial viscosities are similar to each other and both greatly differ from the shear

viscosity in the nonlinear region at long times. The corresponding experimental data fromLaun and Schuch 1989 are represented in Fig. 5 c . Again the qualitative agreement

with the pom-pom model is striking. The differentiation of flow types arises because both

uniaxial and planar extensional flows allow the molecular orientation to couple strongly

to stretch, in contrast to the special, softer, behavior in shear. Thus, there is experimental

evidence that in branched polymers the two extensional flows fall into the same class, in

agreement with the pom-pom model. Generalizations of the molecular model to more

randomly branched material are expected to retain this behavior because of the connec-

tion between molecular stretch and strain hardening. The ability of integral schemes to

mimic the behavior of this model will be further investigated in the next section.

FIG. 4. Steady-state shear and extensional viscosities as functions of deformation rate. Note the strong exten-sion hardening and shear thinning. The dashed lines give the prediction of the K-BKZ-type integral equation

which fits the step-strain behavior of the pom-pom model exactly.



18/30

We next explore the effect of varying molecular structure: keeping s a and s b fixed as

well as a(0),q is varied from 2 to 8 in extensional flows of low and high rates. The

results are shown in Fig. 6 in the same format as the previous calculations. At early times,and uniformly for low deformation rates, the viscosity decreases with increasing q . Of

the two competing factors of increasing relaxation times on the one hand and reducing

effective modulus because of the diluting effect the arms have on the backbone on the

other, the modulus dominates at low extension rates:

0 3G0b2b

12

2 G0sb

2a 0 b

3q. 20

On the other hand, at high extension rates the stretch introduces a factor of q2 in the

stress so that at any fixed high extension rate,

plateau 15

4G0b2q2

1. 21

When q is not too large, this plateau viscosity increases with increasing q , leading to the

cross-over in the ordering of response from early to late times in Fig. 6.

Although the pom-pom model is not meant to represent LDPE quantitatively, since it

was derived specifically for the pom-pom architecture, it does have features in common

with LDPE. The structure of LDPE is both random and polydisperse, but it contains the

essential feature of significant material in segments linking two branch points. In Fig. 7

we reproduce data from Meissner 1972 on LDPE in startup of uniaxial extension and

shear, together with the behavior of the pom-pom model for the set of structural param-

eters explored in this article. Many features are common to both. In particular we draw

attention to the gentle overshoot in shear lying below the linear response and extension

FIG. 5. a Steady-state uniaxial and planar extensional viscosities (xxyy )/ as functions of extension rate

, where x is the direction of stretch and y the direction of contraction. For planar extension the samples

dimension does not change in the z direction. b Transient uniaxial, planar and shear viscosities for deforma-

tion rate 0.01. The uniaxial and planar viscosities have been normalised by dividing by the Trouton

ratios of 3 and 4, respectively, so that all three viscosities match at short times. c Data on transient viscosities

in these geometries from Laun and Schuch 1989 on LDPE at 0.05 s1 replotted in the same formatas b . The line shows the linear viscoelastic response.



19/30

hardening lying above. Also we note the sigmoidal shape of the extensional stress growth

curves and the marked difference in the time at which extension and shear responses

depart from the linear curves.

Data on well-characterized melts of pom-pom polymers are still awaited. However a

further qualitative comparison with LDPE is possible by treating the output of the pom-

pom model as data for which a fit to a K-BKZ-type integral model. The shortcomings or

FIG. 5. Continued.



20/30

success of such a fit may be compared with the performance of integral models in fitting

LDPE behavior itself. This is done in the following section.

IV. COMPARISON WITH K-BKZ TYPE MODELS

In the previous section, we showed that our constitutive model for entangled pom-

poms captures qualitatively a distinctive attribute of the rheology of long-chain-branched

polymer melts, namely strain hardening in extension both uniaxial and planar combined

with extreme strain softening in shear. Section I discussed the inability of the K-BKZ

equation to describe this behavior. To further illustrate this point, in this section we

derive a K-BKZ analog of the pom-pom model: one that reproduces the time-

dependent stresses of the pom-pom model exactly for all step deformations, both

shear and extension. Since a K-BKZ equation is completely specified by its predictions inthese deformations, this procedure uniquely determines the appropriate K-BKZ analog of

the pom-pom model. We then show that this K-BKZ analog, while matching the step-

strain behavior of the pom-pom model by construction , predicts behavior in start-up of

steady shear that is not as softening as that of the pom-pom model, while in start-up of

extension is not as strain hardening as the pom-pom model.

We derive the K-BKZ analog from a version of the pom-pom model that is simplified,

but still qualitatively and quantitatively almost identical to the model described in Sec. II.

First, as alluded to in Sec. III, we replace the strain measure of the K-BKZ equation by

an analytic form proposed by Currie 1984 . That is, in Eq. 13 , we replace

FIG. 6. Start-up flow viscosities in shear and extension for the pom-pom model with structural parameter as for

Figs. 2 and 3 but varying the arm number q to 2, 4, 6, and 8. Note the non-monotonic dependence on q in the

extensional response.



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FIG. 7. a Start-up transient viscosities in uniaxial extension and shear for the LDPE melt IUPAC A studied

by Meissner 1972 at 150 C; b Start-up viscosities for the q 5, s a 3, sb 30 pom-pom model in

shear and extension Deformation rates are given in the caption to Fig. 2.



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15

4 E t, t u E t,t uE t, t u E t,t u 22

by

Q 5J 1

B5

J 1 I213/41/2 C 23

with

J I12 I213/41/2. 24

This analytic expression by Currie has been found to be a very accurate representation of

the tensor given in Eq. 22 for both shear and extensional deformations. In fact we usedthis representation already in all calculations of Sec. III.

Second, we neglect all the consequences of the withdrawal of the branch point into the

central tube except the limit on the stretch, . This approximation may seem drastic, but

in fact makes little difference in the predicted stresses, at least for the parameter values

considered here. Figure 8, for example, shows the steady-state shear and extensional

viscosities versus deformation rate for sb 30, s a 3, and q 5 predicted by the

pom-pom equations with the Currie tensor Q in Eq. 23 , with and without this approxi-

mation. In the former case the second contribution to the stress tensor in Eq. 11 is zero

since s c 0 , and the relaxation time b is a constant. Nevertheless, it is evident from

FIG. 8. Comparison of the flow curves in extension and shear predicted by the full equation set solid line and

by the approximation of no branch-point withdrawal except for the consequence of the crucial upper limit to

the backbone stretch shown as a dashed line. In shear, no branch-point withdrawal occurs even in the full

pom-pom model for the parameter values considered here.



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Fig. 8 that the neglect of this consequence of branch point withdrawal makes little

difference in the predictions of the theory. This is because withdrawal only occurs when

q, when the factor of

2

in the first term of Eq. 11 makes it then the dominantcontribution for q 3. In addition, when branch-point withdrawal is allowed, the back-

bone relaxation time is reduced, which allows the backbone to disorient somewhat, re-

ducing its contribution to the stress, and so off-setting the small stress contributions from

the retracted arms. In fact the amount of arm material required to stabilize the backbone

stretch at q is typically rather small even over a wide range of deformation rates. This is

a consequence of the well-attested exponential dependence of entangled arm relaxation

times. So the exact form of this relaxation-time spectrum Eq. 4 is not essential to the

nonlinear response of the model, which merely requires a very strong dependence of

a(x) on x.

When the effect of branch point withdrawal is approximated in this manner, the

KayeBKZ analog of the pom-pom model is easily constructed. At a time t after a step

strain, the stress tensor predicted by the pom-pom model is

t G t, I1 , I2 Q, 25

where

G t,I1 ,I2 G0et/b I1 ,I2 1 e

t/s1 2. 26

Here is a function of the strain invariants I1 and I2 . This function can be computed for

a shear or a uniaxial deformation by solving the pom-pom equation set in a step strain

i.e., a very fast strain , and the results stored in a table to be read in by the computer

program that solves the K-BKZ equation. The K-BKZ constitutive equation that yields

the pom-pom results of the tabulated equations in section II in a step strain is just

t G0

t

dtm tt, I1 , I2 Q t,t , 27

where

m t, I1 , I2 d

dtG t, I1 , I2 G0e

t/b 1 et/s1

1

b 1 e

t/s1 2

s 1 e

t/s . 28With this KBKZ analog of the pom-pom model, we can compute the stresses in steady

shear and extensional flows. Figure 4 shows the viscosities in these flows versus strain

rate, compared to those for the pom-pom model with branch-point withdrawal approxi-

mated . Note that at high strain rates, there are large divergences of the Kaye-BKZ analog

model from the behavior of the true pom-pom model. The direction of these devia-tions is highly significant. In uniaxial extensional flow, the K-BKZ model shows hardly

any strain hardening; it fails nearly completely to predict the maximum in extensional

viscosity. The deviation in shearing flow is almost as large, but in the opposite direction!

The K-BKZ equation fails to predict the level of shear thinning shown by the pom-pom

model. Thus, as expected, the K-BKZ model is unable to show simultaneously severe

strain hardening in extension and extreme shear strain softening in shear.

Hence, the mathematical structure of the pom-pom equations permits a difference

between shear and extensional-flow rheology that is much greater than can be supported

by the K-BKZ single integral. Note that this distinction is not restricted to uniaxial



24/30

extension versus shear; in planar extension the strain hardening is almost the same as in

uniaxial extension see Fig. 5 . The large distinction between shearing and extensional

flows emerges from the equation for backbone stretch, Eq. 15 . In an extensional flow planar or uniaxial , the velocity gradient tensor K is diagonal and the term K:S in Eq.

15 remains large as the molecules are oriented, and the tensor S asymptotically ap-

proaches a limit in which only Sxx does not approach zero. In shear, however, K is skew;

as the molecules are oriented and all components of S approach zero except Sxx , the term

K:S becomes small, since K does not couple to Sxx . Hence, it is the vorticity in shearing

that leads to its extreme strain softening in the pom-pom model. The K-BKZ model

contains only strain tensors and the dominating effect of vorticity is absent.

An important difference between the predictions of the pom-pom model and the ob-

served rheological behavior of many commercial long-chain branched polymers is in the

approximate time-scale factorability of the latter. Factorability means that after imposi-

tion of a step-strain, the stress can be represented as a product of separate functions of

time and strain. Such behavior has been observed for branched LDPE Laun 1978 . In

the pom-pom model, however, the stress after a step-strain is not factorable into strain-and time-dependence functions. Instead, there are two characteristic timescales, the re-

traction time and the backbone reorientation time. Factorability is only achieved on time

scales longer then the retraction time. Even monodisperse entangled melts of linear

molecules show nonfactorability at time scales shorter than the retraction time Einaga

et al. 1971 . Thus we believe that the factorability observed in the much more complex

polydisperse, branched polymers must be the result of an averaging process that smears

out the nonfactorability that would be observed in monodisperse branched or linear

melts, making it less noticeable. Indeed, very recent experiments on the nonlinear re-

sponse of a monodisperse H-shaped polymer melt do show nonfactorability Allgaier

et al. 1997 . An example that shows how smearing due to a broad spectrum of relax-

ation times resulting from polydispersity can transform an intrinsically nonfactorable

response at the level of individual components into an apparently factorable response is

the Leonov equation, which is nonfactorable when only one relaxation time is included,but is nearly factorable when a spectrum of relaxation times, characteristic of polymer

melts, is used Larson 1987 . Indeed, it can be shown that if a power-law distribution of

relaxation times is incorporated into a nonfactorable constitutive equation, the step-strain

response will be factorable Larson 1987 . Thus, we believe that if the pom-pom model

were generalized to allow for molecular polydispersity, its step-strain response would be

nearly factorable.

V. APPROXIMATE DIFFERENTIAL MODEL

We have already seen in the previous section that some of the more subtle features of

the molecular model may be discarded with impunity, providing one keeps the essential

features of two distinct relaxation times for orientation and stretch as well as the maxi-mum limiting stretch of the backbone. The smaller equation set still reflects quantitative

features of molecular structure, while retaining great simplicity the dynamical equation

for sc(t) and the arm relaxation spectrum are not needed .

However, there remains considerable computational complexity in the integral equa-

tion for the orientation tensor S(t) Eq. 13 . Carrying out the full double integral at

every timestep requires a computational time which grows quadratically with the flow

time. This is feasible for the calculation of viscometric homogenous flows, but becomes

prohibitive when spatially inhomogeneous flows are considered. It would, however, be

very desirable to explore the consequences of the pom-pom model in spatially varying



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flows such as contraction flows, because of the special behavior of LDPE observed in

flow visualizations. This would be possible if a differential dynamical equation could be

found for S(t) which would approximate the integral form.Since the important aspects of the model all touch on the way the stretch couples to

the flow via the orientation, we expect that the important aspects of S(t) to get right are

the asymptotic forms of its components in extension and shear. This rules out the sim-

plest candidate for the evolution of a tensor of unit trace in a flow:

tS t K.SS.KT 2 S:K S

1

b S 1

3I . 30

Although the shear-response Sx y of Eq. 30 does have a maximum as a function of shear

rate, it does not exhibit the right asymptotic form; it decreases as 2/3

rather than as1 for the integral equation. The latter shares this highly shear thinning behavior with

that of the strain measure in standard DoiEdwards theory for linear polymers. The less

steep shear-rate dependence of Eq. 30 does not ensure the shear-thinning response of

the model, since the important coupling term K:S then grows as 1/3 rather than

reaching a plateau value, as it does when the integral equation is used for S. So with the

differential form of 30 stretch is observed in steady-state shear as well as in the startup

transient.

More complex forms of differential equations akin to Eq. 30 exist Hinch et al.

1976 , but may have multiple solutions which threaten instabilities when used in nu-

merical flow-solvers. Instead, we shall retain the simplicity and stability of Eq. 30 while

ensuring the correct asymptotics by using the convection equation for line elements

embedded in the flow, and then dividing by the trace of their second moment tensor at the

each time the tensor S is required, rather than insisting on working with a single tensor ofunit trace. We are grateful to Dr. Oliver Harlen for this suggestion. The recipe for

calculating S(t) is therefore:

tA t KAAKT

1

b A 1

3I ,

S t A t

trace A t. 31

In Fig. 9 we show a comparison of startup flows in extension and shear of the differential

version of the pom-pom equations. In this case all other variables including s c are kept

i.e., all consequences of branch-point withdrawal are kept . Comparing Fig. 9 to Fig.

7 b , it can be seen that, at least in these flows, the behavior of the differential approxi-

mation is qualitatively similar to that of the integral form. For readers who would prefer

to avoid the mouthfulls of integral-version pom-pom and differential-version

pom-pom the abbreviations Ron-Pom and Tom-Pom, respectively have been

suggested, in accordance with the respective computational competance of the authors.



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For ease of reference, we summarize the drastic but unproblematic simplifications of the

full pom-pom model discussed in the last two sections in the following table:

Simplified Pom-Pom Equation Set

Expression for the stress:

15

4G0b

22 t S t with b

sb

2qsas bEvolution of orientation:

tA t KA AKT

1

b A 1

3I ; S t

A t

trace A t

Evolution of backbone stretch:

t K:S

1

s 1 strictly for q

Timescales:

Backbone orientation: b 4

2 sb

2ba 0 q

Backbone stretch: s sba 0 q

FIG. 9. Start-up transient viscosities in uniaxial extension and shear computed using the differential equation

for S given in Sec. VI. Deformation rates are given in the caption to Fig. 2. The results should be compared with

those of the full model in Fig. 7 b .



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We defer further discussion of the pom-pom equations in complex flow geometries to

another paper Bishko et al. 1997 .

VI. CONCLUSION

We have derived a molecular constitutive equation for the dominant backbone stresses

for an ideal multi-branched polymer, the pom-pom. The pom-pom was contrived as

the simplest branching architecture whose nonlinear rheological properties are qualita-

tively like those of long-chain branched commercial melts, such as low density polyeth-

ylene LDPE . By neglecting rapidly relaxing branch motions and concentrating on the

slow backbone relaxation, we derived a tractable integro-differential constitutive equation

that predicts rheological properties in shear and extensional flow remarkably similar to

those of typical LDPEs. In particular, the pom-pom constitutive equation shows strain

hardening in extension, but extreme strain softening in shear.

Furthermore, the predicted behavior in extensional flow is not sensitive to whether the

extension is uniaxial or planar, in agreement with recent data and with intuitive expec-

tations. This last feature is unattainable even in the versatile K-BKZ equation, which can

only predict strong strain hardening in uniaxial extension and strong strain softening in

shear if the kernel function is chosen in such a way that the behavior in uniaxial extension

differs markedly from that in planar extension. Thus, the pom-pom model not only

provides a constitutive equation for an unusual branched architecture, but also suggests

that generalized integro-differential equations of this type might be good candidates for

phenomenological constitutive equations for commercial branched melts.

The key physics of the model survives, even quantitatively, in a number of approxi-

mations to the full equation set, including a fully differential set which holds promise as

an algorithm for the calculation of the non-Newtonian stress in numerical flow-solvers.

ACKNOWLEDGMENTS

The authors would like to thank D. Bick, G. Bishko, R. Keunings, O. Harlen, S.Milner, and M. Wagner for stimulating discussions, and the Isaac Newton Institute for the

Mathematical Sciences, where much of this work was done.

APPENDIX A: DERIVATION OF THE BACKBONE RELAXATION TIME

The stress-relaxation of the backbone is assumed to be dominated by reptationat

long times the pom-pom polymers renormalize to linear chains in tubes whose diameters

are given be entanglements between the backbones only. Stress from occupied tube

segments is lost when a free end in this case a branch point diffuses to the segment. The

total stress is then just the sum of the survival probabilities p(s ,t) at time t of a tube

segment originally at a curvilinear distance s from one of the free ends. This obeys a

linear diffusion equation with absorbing boundaries and has the initial condition

p(s ,0) 1 Doi and Edwards 1986 .

tp s,t Dc

2

s2 p s,t ; p 0,t p L, t 0. A1

By an Einstein argument, the curvilinear diffusion constant Dc is just given by

Dc a

2

4a 0 q A2



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from the arguments preceding Eq. 6 and from the recognition that two branch points

hinder the diffusion of the backbone. We recall from Sec. II that L2 s b2

a2b .

The partial differential equation is solved in terms of the eigenfunctions sin(ns/L),labeled by the integer value n . The lowest eigenfunction carries the dominant part of the

total probability p(s ,t) over most of the chain, so we need only its time-constant for the

level of approximation of our treatment. Substituting the n 1 eigenfunction into A1

yields for its amplitude a1( t) the ordinary differential equation

d

dta1 t

2Dc

L2 a1 t . A3

This gives exponential decay with the time constant b of

b L

2

2Dc

4

2 sb

2b 0 q, A4

which is Eq. 7 .

APPENDIX B: NOMENCLATURE

viscoelastic stress tensor

B Finger tensor

C Cauchy tensor

E deformation gradient tensor

I1 ,I2 first and second invariants of E

G(t) time-dependent linear viscoelastic modulus

a tube diameter

q number of dangling arms on each branch point

sa molecular weight of dangling arms in terms of the entanglement molecular

weightsb molecular weight of backbone segment in terms of the entanglement molecu-

lar weight

k Boltzmanns constant

T absolute temperature

( t) stretch ratio of the backbone

sc(t) number of entanglement segments of arms drawn into the backbone tube

a(x) relaxation time of an arm-segment a fraction x from the branch point to the free

end

0 attempt time for deep retractions of the entangled dangling arm

b volume fraction of backbone material

G0 plateau modulus

Dc

curvilinear diffusion constant of a branch point

b effective friction constant of a branch point

L entangled primitive path length of the backbone segments

S( t) second moment of the orientation distribution of backbone segments

xc fraction of arm withdrawn into a backbone tube

K deformation rate tensor

blob renormalized friction constant of a branch point when some arm is withdrawn

extension rate

0 zero-extension rate limit of extensional viscosity

shear rate



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