pom-pom'' constitutive equation

Predicting low density polyethylene melt rheology in elongational and shear flows with ‘‘pom-pom’’ constitutive equations N. J. Inkson and T. C. B. McLeisha) Interdisciplinary Research Centre in Polymer Science and Technology, Department of Physics and Astronomy, The University of Leeds, Leeds LS2 9JT, United Kingdom

O. G. Harlen Department of Applied Mathematics, The University of Leeds, Leeds LS2 9JT, United Kingdom

D. J. Groves Interdisciplinary Research Centre in Polymer Science and Technology, Department of Physics and Astronomy, The University of Leeds, Leeds LS2 9JT, United Kingdom (Received 11 November 1998; final revision received 2 February 1999)

Synopsis A recent constitutive equation derived from molecular considerations on a model architecture containing two branch points a ‘‘pom-pom’’ captures the qualitative rheological behavior of low density polyethylene ~LDPE! in shear and extension for the first time @T. C. B. McLeish and R. C. Larson, J. Rheol. 42, 82 ~1998!#. We use a hypothetical melt of pom-poms with different numbers of arms to model the behavior of LDPE. The linear relaxation spectra for various LDPE samples are mapped to the backbone relaxation times of the pom-pom modes. Data from start-up flow in uniaxial extension fixes the nonlinear parameters of each mode giving predictions for shear and planar extension with no free parameters. This process was carried out for data in the literature and for our own measurements. We find that multimode versions of the pom-pom equation, with physically reasonable distributions of branching, are able to account quantitatively for LDPE rheology over four decades in the deformation rate in three different geometries of flows. The method suggests a concise and functional method of characterizing long chain branching in polymer melts. © 1999 The Society of Rheology. @S0148-6055~99!01103-7#

I. INTRODUCTION The branching topology of polymer molecules has a marked effect on the flow properties of the melt, and in general branched polymers have very different rheological behavior from that of linear polymers, especially in extensional flows. The ‘‘pom-pom model’’ of McLeish and Larson ~1998!, presents a theoretical approach to the behavior of the macromolecular segments that lie between the multiple branch points of large a!

Author to whom all correspondence should be addressed. Electronic mail: [email protected]

© 1999 by The Society of Rheology, Inc. J. Rheol. 43~4!, July/August 1999

0148-6055/99/43~4!/873/24/$20.00

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INKSON ET AL.

branched molecules. In this article this model is applied to industrial grade polymeric materials such as low density polyethylene ~LDPE!. The molecular structure of LDPE is known to contain multiple irregularly spaced long chain branches, although detailed information on its molecular topology is not available. Its rheological behavior has posed a long-term challenge for modeling. The most accurate modeling to date has employed integral Kaye-Bernstein-KearsleyZapas ~K-BKZ! @Kaye ~1962!, Bernstein, Kearsley, and Zapas, 1963# constitutive equations @see Ahmed et al. 1995#. These contain sufficiently flexible kernel functions to permit the response in uniaxial extension and shear to be fitted closely. However, once this is done they fail even qualitatively to predict the rheology in planar extension. ~Alternatively, they may be arranged to fit both extensional geometries @Wagner et al. ~1998!# but in this case they fail in shear.! In experiments LDPE strain hardens under planar extension, as in uniaxial extension; by contrast the integral models predict a softening response similar to that of simple shear flow. The failure of single integral constitutive equations such as K-BKZ or Wagner equations @Wagner and Laun 1978# to predict hardening in planar extension is discussed by Samurkas et al. ~1989!. The term hardening ~or thinning! refers to the rise ~or drop! in transient viscosity at strain rates in the nonlinear regime. The reason for the failure of this class of equations must be that they do not take into account the molecular dynamics of entangled branched polymer melts under flow. The molecular issues are greatly clarified by considering model architectures. For example, the pom-pom molecule is an idealized branched polymer consisting of a backbone ~or crossbar! that ends in two branch points consisting of a number, q, of arms @see Fig. 1~a!#. A tube model for the effect of entanglements around a monodisperse melt of this structure gave rise to a set of integral and differential equations for internal structural variables such as segment stretch and orientation @McLeish and Larson ~1998!, Bishko et al. ~1997!#. Together these provide a constitutive relation for the stress on deformation of the melt. It displays the correct qualitative behavior by simultaneously predicting strain hardening in both planar and uniaxial extensional deformation and shear thinning. A key feature of molecular theories of entangled polymer melts, which is essential in the case of branched polymers, is the natural separation of relaxation times for stretch and for orientation @McLeish and Larson ~1998!#. This is not typically an ingredient in phenomenological constitutive equations, such as Phan-Thien-Tanner and other Oldroyd type models where the same relaxation time applies to stretch and orientation. A rigorous quantitative calculation of the stress relaxation in polydisperse LDPE molecules with many layers of branching within each molecule would be too complicated to form a useful constitutive equation. However a tube model would predict, quite generally, that different parts of the molecule have widely seperated timescales for orientational relaxation @McLeish ~1988a!, Rubinstein et al. ~1990!#. These parts would in turn be expected to possess different relaxation times for stretch and orientation. This suggests that we can ‘‘decouple’’ the structure into an equivalent set of pom-pom molecules with a range of relaxation times and arm numbers. The orientational relaxation time distribution would be set by the linear relaxation spectra, with nonlinear measurements fixing the remaining parameters. In this article we report the results of applying this procedure for three intensively studied batches of LDPE taken from the literature: melt 1, IUPAC A and IUPAC X. We also include a commercial sample with a different degree of long chain branching, called LDPE B, which was characterized in our own laboratory ~see Fig. 14 for its complex shear modulus!. This polymer is a high Mw and broad molecular weight distribution ~MWD! LDPE with Mw ; 250 000 and polydispersity of ~Mw/Mn! ; 15.

LDPE RHEOLOGY WITH ‘‘POM-POM’’ EQUATIONS

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FIG. 1. ~a! A pom-pom with three arms at each branch point (q 5 3). At small times the polymer chains are confined to the Doi-Edwards tube. s c is the dimensionless length of branch point retraction into the tube. l is the stretch ratio where L is the curvilinear length of the crossbar and L 0 is the curvilinear equilibrium length. ~b! At longer timescales the pom-poms arms are relaxed and fluctuate on timescales such that the crossbar cannot entangle with them. The crossbar can now reptate out of its widened tube.

II. RHEOLOGICAL DATA AND EXPERIMENTS A. Data taken from the literature Melt 1 is the LDPE sample on which Meissner performed early rheological measurements @extensional viscosity is presented in Meissner ~1971! and the shear viscosity data are in Meissner ~1972!#. IUPAC A is characterized in Meissner ~1975!, with uniaxial extension data given in Laun and Mu¨nstedt ~1979!; and IUPAC X data are taken from Laun and Schuch ~1989!. Note that IUPAC X is a batch of the same material as IUPAC A @which is discussed in Laun and Schuch ~1989!#. The relaxation spectra for IUPAC A is given in Laun ~1986!, and Khan and Larson ~1987!; IUPAC X spectra are in Samurkas, Larson and Dealy ~1989! and melt 1 in Laun

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INKSON ET AL.

~1986!. The spectra are given as sets of relaxation times t i and associated moduli g i . Note that the spectra are taken at different temperatures: 125 °C for IUPAC X and 150 °C for IUPAC A. We obtained a relaxation spectrum for LDPE B from oscillatory shear stress growth of the equivalent shear viscosity using a Rheometrics RDA II rotational rheometer in the steady shear mode. Measurements for strain rates from 0.001 to 0.3 s21 were made at 140 °C in nitrogen using a 25 mm diam 5° cone and plate geometry. The LDPE B was melt pressed into disc specimens again at 180 °C. This provides both a convenient sample form to load into the rheometer and the same melt history as that for the elongational measurements. B. New extensional data The transient uniaxial viscosity measurements for LDPE B were obtained from a Rheometrics RME elongational rheometer. This is essentially the instrument described by Meissner and Hostettler ~1994!. The sample, in the form of a rectangular strip, is gripped by horizontal metal caterpillar belt clamps and pulled symmetrically over a support cushion of nitrogen gas, which passes though a sintered metal frit that forms the surface of the sample table. The force require to extend the sample is measured by a transducer unit attached to the right-hand belt grips, together with the drive motor. The transducer unit itself is, in effect, a horizontal moving beam monitored by a low voltage displacement transducer ~LVDT! and suspended from a fixed beam by twin vertical leaf springs. The sample table, caterpillar grips and transducer are all mounted in a small oven with a nitrogen atmosphere controlled at the melt extension temperature. The strain measure used is the Hencky strain « H 5 ln(Lt /Lo) with a maximum of 7. The stress measurement assumes that the sample remains uniform in cross section, and so it is essential to ensure that the polymer specimens are of high quality and extend uniformly. Therefore, great care is needed in preparing the samples to avoid inhomogeneities. The polymer was cryogenically ground to make a coarse powder feedstock in order to remove effects of granule memory. This was then pressed at 180 °C into rectangular bars 60 mm37 mm31.5 mm and allowed to cool naturally between the press platens at full moulding pressure. Satisfactory uniform extension was obtained at 140 °C for a range of strain rates from 0.001 to 0.1 s21. All measurements were made ‘‘undamped,’’ that is, without the oil dashpot provided to reduce oscillation during the initial tensioning of the polymer sample. III. MOLECULAR RHEOLOGICAL MODEL A. Stress relaxation in branched polymers A linear polymer chain in a melt has the freedom to move but is constrained by its neighbors, which act to keep it in a tube centered on the chain’s primitive path. In order to diffuse the chain ‘‘reptates’’ ~slides! in a curvilinear motion along the primitive path @Doi and Edwards ~1986!#. The presence of a branch point affects the ability of the chain to move in its tube. A star polymer cannot reptate like a linear polymer chain and must rely on star arm fluctuations in order to relax its stress @de Gennes ~1975!, Pearson and Helfand ~1984!, Ball and McLeish ~1989!#. In addition, the presence of branch points affects the stress relaxation of polymer chains after it has undergone a deforming strain. For a linear chain, the tube will deform with the bulk strain and experience a change in contour length followed by the chain retracting to its original length @Doi and Edwards 1986#. For a branched polymer only the free ends can retract back into its stretched tube


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after deformation. Deeper segments will only retract when it is entropically favorable to withdraw the branch points connecting them to the outer segments into the tube @Bick and McLeish ~1996!#. This occurs only at high strains when the tension in the deeper segments exceeds the total force at the branch point from the dangling arms. The entropic force arises from the ends ability to explore more of their surroundings ~by Brownian motion! than more confined inner segments @Doi and Edwards ~1986!#. Thus, in the pom-pom molecule, a stretched segment connected to q outer ~unstretched! ones by a branch point can support a tension of q times the entropic force of the free ends just before arm withdrawal. Consequently the maximum stretch of this segment is equal to q ~providing this is still within its Hookian regime!. These general consequences of the tube theory of polymer melts provide the basic physics underlying the pom-pom model.

B. Pom-pom model Here we briefly review the details of the molecular constitutive equation derived for a monodisperse melt of pom-pom molecules. A pom-pom molecule has a ‘‘crossbar’’ of s b dimensionless entanglement ‘‘lengths’’ (s b 5 M b /M e , where M b is the molecular mass of the crossbar and M e is the entanglement molecular mass!. The crossbar has a branch point at each end connecting it to q arms, each of s a entanglement lengths ~likewise, s a 5 M a /M e ); see Fig. 1. The dominant contribution to the stress is assumed to arise from the crossbar segment, since the arms relax on much faster timescales ~since they behave like star polymer arms!. There are three variables that quantify the state of a pom-pom crossbar at any time, t: l(t), the dimensionless stretch ratio of the crossbar; S(t), the orientation tensor, which measures the distribution of unit vectors of the crossbar tube segments (S 5 ^ uu& , averaged over all orientations!; and s c (t), the dimensionless distance of arm withdrawal into the backbone’s tube ~see Fig. 1!. Additionally there are five structure parameters: the relaxation time for orientation of the crossbar, t b , the relaxation time for stretch (l) of the crossbar, t s , the number of dangling arms ~or priority q! and the dimensionless molecular masses of the crossbar s b and arms s a . We noted above that the stretch of a segment is limited by the balance of entropic tensions to l(t) < q. In situations that would generate larger stretches, the dangling arms are withdrawn into the crossbar tube so that s c (t) becomes nonzero. The crossbar stretch is then locked at q until s c becomes zero again. In this way only one of l or s c is changing in time @McLeish and Larson ~1998!#. The mathematical constitutive equations of the pom-pom model are comprised of three dynamical integro-differential equations ~previously constitutive equations have always taken the form of one integral or differential equation! for each of the dynamical structure variables S(t), l(t) and s c (t) together with an expression for bulk stress. These dynamical equations are supplemented by expressions for the relaxation times for orientation, t b , and stretch, t s . Earlier work @McLeish and Larson ~1998!# has shown that the rigorous integral equation for S(t) may be approximated by a differential equation which shares the same asymptotic structure and greatly reduces computational time. In this article we use these simplified pom-pom equations and ignore the small contribution to the stress arising from oriented arm segments ~the dimensionless arm withdrawal, s c 5 0). In order to prevent l from increasing beyond q, dl/dt is set to zero when l 5 q and the deformation rate would tend to stretch it further. In the differential approximation the orientation tensor S(t) is defined by

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878

A~ t !

S~ t ! 5

trace~ A~ t !!

~1!

,

where A(t) evolves as

] ]t

A~ t ! 1u–¹A 5 K–A1A–KT 2

1

tb

~ A2I! ,

~2!

where K is the deformation rate tensor ~velocity gradient, K 5 ¹u). This differential approximation ensures the correct asymptotic form of the orientation tensor in shear flow, and gives the same qualitative behavior in shear and extension as the full integral expression. In flow, the dissipative drag on the molecule, which is entirely due to the effective friction at the branch point, is in balance with the elastic recovery of the backbone. The elastic recovery of a Gaussian chain of s b steps of size a on relaxation of the arm material has a spring constant equal to 3kT/s b a 2. Equating frictional drag forces on the branch points from the drag of their surroundings to the elastic Brownian force leads to

zb 2

S

K:S2

]L ]t

D

5

3kT sba2

~ L2s b a ! ,

~3!

where L 5 s b al is the curvilinear distance of separation of the branch points along the tube, and z b is the friction coefficient z b @see McLeish and Larson ~1998!#, giving

] ]t

l 5 l~K:S! 2

1

ts

~ l21 ! ,

for l , q.

~4!

In this simplified model, the stress of the melt is entirely due to the crossbar segments and is given by

s5

15 4

G 0 f 2b l 2 ~ t ! S~ t ! ,

~5!

where G 0 is the plateau modulus, and f b is the fraction of molecular weight contained in the crossbar. The stress is quadratic in the backbone extension l. One factor of l arises because the stretched backbone occupies a greater length of the tube equal to ls b a. The other comes from the tension within the backbone, which is proportional to l. The prefactor of 15/4 arises from the tube model calculation of the plateau modulus in terms of the molecular parameter a @Doi and Edwards ~1986!#. McLeish and Larson ~1998! discuss the behavior of this model in start up of shear and extension. The viscosities in different flow geometries are given by

h5

sxy

g˙

,

in shear flow,

~6!

and

hu 5 hp 5

sxx2syy 2«˙

,

in uniaxial and planar extension.

~7!

The behavior in uniaxial and planar extension is found to be very similar and quite different from that in shear. The viscosity in the start up of shear is shown in Fig. 2~a!. In


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FIG. 2. ~a! Transient shear viscosity of a pom-pom melt (q 5 5, t b / t s 5 30) in start-up shear flow at four different shear rates. ~b! Crossbar stretch ratio l~t! in start-up of shear flow against time at four different shear rates.

all these single mode figures showing calculations for a single pom-pom mode t b / t s 5 3, t s 5 1, G 0f 2b 5 1 and q 5 5, with varying dimensionless strain rates of t s g˙ 5 1, 3, 10, and 30. In shear, increasing the shear rate lowers the maximum viscosity. This is due to the behavior of l shown in Fig. 2~b!. At high strain rates l has a large transient overshoot before decaying to a steady state value of l;

1 12

ts

,

for g˙ @ t21 s .

2tb

Thus when t s ! 2 t b , the crossbars are not permanently stretched as one might have expected at shear rates g˙ > t 21 s because the orientational alignment reduces their component along the shear gradient. As a consequence the pom-pom model shares the strong shear-thinning behavior of linear polymer melts.

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INKSON ET AL.

FIG. 3. ~a! Transient uniaxial extensional viscosity of a pom-pom melt (q 5 5,t b / t s 5 3) in start-up at four different extension rates. ~b! Crossbar stretch ratio l(t) in start-up of uniaxial extension flow against time, at four different extension rates.

The behavior in transient extension is very different from that of shear; see Fig. 3~a!. The transient viscosity increases with time ~extension hardens! until it reaches its steady state value. For strain rates greater than the critical strain rate, «˙ crit 5 (121/q)/ t s the steady value of l will achieve its maximum value of q; see Fig. 3~b!. At these high strain rates the transient viscosity separates from the low strain rate viscosity curves and increases rapidly, only to level out suddenly once l reaches q. This rapid increase in viscosity is seen in the extensional viscosity data of LDPE. We now consider the effect of changing the value of q in this model. In shear flow there is no difference in the behavior of the viscosity except at very high shear rates when l may briefly attain a value of q. However, in extensional flows the value of q limits the maximum value of l and hence the plateau viscosity. Figure 4 shows that for «˙ 5 1.0 s21 there is a clear increase in the steady state viscosity with q (q 5 1, 2, 5, 10! due to the pom-pom backbones becoming fully stretched. In fact, since h¯ plateau } q 2 /«˙ high q pom-poms contribute much more stress and hence have much higher viscosities than a


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FIG. 4. Transient uniaxial extensional viscosity of a pom-pom melt ( t b / t s 5 3) in startup at a constant extension rate of «˙ 5 1.0 as a function of arm number (q 5 1, 2, 5, 10!.

melt of low q pom-poms at «˙ > t 21 s . Another consequence of this relation is that the value of the plateau in the viscosity decreases with the extension rate. Thus beyond the critical strain rate, «˙ crit , the viscosity decreases with the extension rate as the stress saturates. Consequently the steady state extensional viscosity has a sharply peaked maximum at «˙ crit ~see Fig. 5!. C. Multiple levels of branching In a large molecule, with multiple branch points, the segmental tension can be traced from the free ends inwards by summing the tensions at each branch point @Bick and McLeish ~1996!#. This value of summed multiples of the entropic force on a given segment is named its priority. It is defined as the ratio of the maximum tension of a segment to its equilibrium tension. Thus in the pom-pom model the cross bar has a

FIG. 5. Steady state uniaxial extensional viscosity, h («˙ ), plotted against rate of extension, «˙ , for a pom-pom melt (q 5 5, t b / t s 5 3).

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INKSON ET AL.

FIG. 6. Relaxation process of a long chain branched molecule such as LDPE. At a given flow rate «˙ the molecule contains an unrelaxed core of relaxation times t . «˙ 21 connected to an outer ‘‘fuzz’’ of relaxed material of relaxation times t , «˙ 21 behaving as solvent.

priority q. The maximum tension of a segment depends on the number of free ends that can be traced back to it. The immediate response of a branched molecule to a large strain depends on the priority distribution of the segments within it. A related picture shows how a large molecule containing multiple branch points relaxes after deformation. The free chain ends will relax rapidly, behaving in the same way as the arms of the pom-pom up to the branch point connecting them to the rest of the molecule. This branch point is able to move one diffusive step after a deep retraction of the chain ends connected to it. This allows the molecular segment up to the next branch point to relax as a star arm, but on the much longer timescale set by diffusion of the outer branch point. In turn, deep retractions of this segment allow the inner branch point to move, and so on. Such a hierarchical process continues until the deepest ~innermost! segments of the molecule relax at the longest timescales ~see Fig. 6!. Therefore, the relaxation time of a segment is determined by the path distance to the nearest free end that is able to release it from its tube constraint by retraction. This statistic, called seniority @Rubinstein et al. ~1990!#, also increases, like its priority, towards the middle of a complex branched molecule. Both statistics are required to calculate the full molecular rheology of a branched polymer. Although real LDPE molecules are highly asymmetric, every segment within them does possess a well-defined seniority and priority. This is because both the relaxation time ~seniority! and the critical stress for branch-point withdrawal ~priority! are both determined by relaxations from one of the two tree structures connected to it. In either process the relaxation arrives first from one of the segment’s ends; it is this that determines its priority or seniority. Since relaxation times increase exponentially with seniority of segments, an appropriate physical picture of a molecule at


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TABLE I. IUPAC A.

t i ~s! 0.001 0.005 0.028 0.14 0.7 3.8 20 100 500

g i ~Pa!

q

tb /ts

1.523105

1 1 1 1 2 6 6 9 22

2.0 2.0 2.0 2.0 2.0 1.7 2.15 1.25 1.1

4.0053104 3.3263104 1.6593104 8.693103 3.1513103 8.5963102 1.2833102 1.8495

a given flow rate «˙ contains an unrelaxed core of relaxation times t . «˙ 21 connected to an outer ‘‘fuzz’’ of relaxed material of relaxation times t , «˙ 21 behaving as solvent ~Fig. 6!. IV. A MULTIMODE POM-POM MODEL LDPE has a random, polydisperse, branched structure as a result of the high-pressure free-radical polymerization process in which it is produced. This also results in LDPE’s high proportion of long chain branches ~LCBs!. The precise structure and degree of branching are still unknown and there do not exist accurate experimental techniques to measure them @see Axelson et al. ~1979! for a discussion of nuclear magnetic resonance ~NMR! techniques#. It is therefore tempting to ask whether the structure of LDPE can be inferred from its rheological behavior. Even if the exact molecular structure were known, a rigorous calculation of the full priority and seniority distributions for a randomly branched polymer would be prohibitively expensive computationally. Instead, we chose to approximate it by taking a theoretical blend of the relatively simple pom-pom molecules with differing number of arms, assigning orientational relaxation times from the linear relaxation spectra. The high priority segments of LDPE are thus represented by pom-poms with larger numbers of arms and longer relaxation times than the outer segments. We expect high priority ~maximum stretch, q! segments to share a high seniority ~relaxation time for orientation t b ). The

TABLE II. IUPAC X.

t i ~s! 0.001 0.003 0.01 0.03 0.1 0.3 1.0 3.0 10.0 30.0 100.0

g i ~Pa!

q

tb /ts

9.23104 6.553104 3.83104 2.573104 1.773104 1.103104 7.313103 3.763103 2.13103 9.43102 3.53102

1 1 1 1 1 2 2 3 4 5 5

2.0 2.0 2.0 2.0 2.0 2.0 2.0 1.9 1.8 1.03 1.0

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TABLE III. LDPE B ~12 mode relaxation spectrum!.

t i ~s!

g i ~Pa!

q

tb /ts

0.01 0.030 305 0.091 841 0.278 33 0.843 48 2.5562 7.7466 23.467 71.146 215.61 653.42 1980.2

7.20623104

1 1 1 1 2 3 4 6 8 10 21 75

5.0 5.0 4.0 3.5 3.3 3.2 3.0 2.9 2.0 1.92 1.52 1.01

3.30983104 2.79423104 1.73163104 1.18383104 6.6293103 3.90583103 1.58353103 7.13693102 2.26483102 87.078 15.915

central approximation is that of ‘‘decoupling’’ the different levels of branching on the same molecules — interactions between segments of different priorities on the same molecule are neglected. Our justification for this approximation is that for a given flow rate «˙ , the flow behavior is dominated by those segments whose relaxation times t b and t s are of order «˙ 21. As noted earlier, modes with relaxation times much smaller than «˙ 21 remain relaxed, whereas longer relaxation time modes become ‘‘saturated’’ in strain. Since the relaxation times increase exponentially with seniority, and both seniority and priority increase in a similar fashion from the outside of a molecule, segments of different priorities have widely separated relaxation times. Consequently, the effects of couplings of stretch and orientation between them are expected to be small. In the multimode model, each ~ith! mode is characterized by four parameters: the stretch and orientation relaxation times t si and t bi , respectively; the number of arms q i ; and the fraction of the stress relaxing, g i . There are now two evolution equations per mode equivalent to Eqs. ~2! and ~4! for the tube orientation Si (t)and stretch l i (t). The stress contribution of all the ~n! modes is additive: n

s5

(

i51

n

si 5

(

i51

g i l 2i ~ t ! Si ~ t ! .

~8!

The plateau moduli g i and the orientation relaxation time may be found from the linear relaxation spectrum. Published relaxation spectra are available for both IUPAC A and IUPAC X and are shown in Tables I and II. To test the robustness of our approach we obtained three relaxation spectra for the LDPE B sample from its complex shear modulus, and they are shown in Tables III–V. A. Relaxation spectra and pom-pom model parameters In Tables I–VI we present the linear spectra of three materials decorated with nonlinear pom-pom parameters for each mode that best fit the data in uniaxial extension. The values of t bi in these spectra are essentially arbitrary. We have chosen to use pom-pom modes that directly correspond to these linear spectra. Thus each pom-pom mode corresponds to a mode in the linear spectrum that determines both the value of g i and t bi of this mode. The two remaining parameters, t si and q i , must be found from the nonlinear response. However, to be consistent with the physical model they must satisfy the following constraints. First, the ratio t bi / t si is 4/p 2 times the average number of


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TABLE IV. LDPE B ~10 mode relaxation spectrum!.

t i ~s! 0.0136 0.145 1.03 4.86 6.72 17.53 100.0 500.0 1000.0 5000.0

g i ~Pa! 1.23105 4.533104 1.5033104 5.1963103 3.03103 1.7443103 9.553102 80.0 50.0 4.0

q

tb /ts

1 1 1 1 2 3 5 10 23 150

4.0 4.0 3.5 3.3 3.2 3.1 2.9 2.5 1.9 1.4

entanglements of backbone section. Although the number of entanglements is unknown, it should lie within a range of roughly 2–10. The physical limit for a completely unentangled segment is t bi 5 t si . Moreover, since t b(i21) sets the fundamental diffusion time for the branch point controlling the relaxation of segment i, we must also satisfy t b(i21) , t si . Thus the t si are physically constrained to lie in the interval t b(i21) , t si , t bi . The number of arms, q i , is determined by the priority. Again this is unknown, but as the priority and seniority increase towards the inner segments, q i should increase with t bi . The values for q i and t si were found by fitting the transient uniaxial extensional viscosity. As noted earlier the shear viscosity is insensitive to q i . Although transient full extension can occur in the single mode model at sufficiently high shear rates, this occurs rarely in the multimode model as increasing the shear rate merely excites faster relaxing modes. For each mode, the evolution of Si (t) may be calculated analytically, while the stretch l(t) was found by integrating Eq. ~4! using a fourth-order Runge-Kutta scheme. The values of q i and t si were then adjusted by ‘‘trial and error.’’ Although with 10 or more modes this optimization might appear to be a complex task, the nonlinear behavior is dominated by modes whose stretch Weissenberg number is of order one ( t s «˙ ' 1). Therefore at each particular strain rate only those modes for which t s «˙ is close to unity behave nonlinearly. Recall that the single mode extensional viscosity has a sharp maximum at t s «˙ ' 1. Thus at each specific strain rate we can isolate the one or two modes that are active. For each strain rate, the level of the plateau in the extensional viscosity is

TABLE V. LDPE B ~8 mode relaxation spectrum!.

t i ~s!

g i ~Pa!

q

tb /ts

0.01 0.0571 0.3261 1.8622 10.634 60.724 346.77 1980.2

8.88423104 4.81093104 2.64833104 1.26663104 4.74533103 1.1923103 251.45 24.445

1 1 2 3 4 9 15 60

3.5 3.3 3.2 3.0 2.9 2.5 2.3 1.15

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886 TABLE VI. Melt 1.

t i ~s!

g i ~Pa!

q

tb /ts

1024

1.293105

1 1 2 2 3 7 8 9

4.5 3.9 3.7 3.6 2.9 2.0 2.1 1.3

1023 1022 1021 100 101 102 103

9.483104 5.863104 2.673104 9.803103 1.893103 1.803102 1.03100

determined by the value of q i , whereas the form of the growth of the viscosity towards the plateau determines the ratio t bi / t si . The results of matching the uniaxial extension viscosity with measurements of IUPAC A, by Laun and Mu¨nstedt ~1979!, are shown in Fig. 7 and the shear viscosity fit is shown in Fig. 8. The corresponding values of q i and t bi / t si are shown in Table I. Note the successful fitting of turn ups and thinning ~end points!. The level of agreement of the shear viscosity is very pleasing as the parameters in the model were chosen only to fit the extensional data and the shear fit follows with no further adjustment of the parameters. In fact the shear response is rather weakly dependent on the nonlinear parameters: adjusting q i makes little difference since maximum extension is often not achieved in shear, and adjusting t s only varies the position of the stress overshoot peak slightly. The shear behavior of the multimodal model matches the start-up viscosity well and the presence of multiple modes smoothes out the large overshoot that is apparent in the single mode pompom model at high shear rates @Fig. 2~a!#. The fit of the extensional viscosity data for melt 1 @Meissner ~1972!# is shown in Fig. 9. The extensional viscosity peaks do not match all the curves precisely, since the data set covers many strain rates where only a few modes are dominating the response. The IUPAC X data of Laun and Schuch ~1989! is significant in that it provides the only measurements of both uniaxial extension and planar extension @which is comprised

FIG. 7. Transient uniaxial extensional viscosity of a 9 mode pom-pom melt in start up plotted against data for IUPAC A LDPE. The extension rates range from 0.01 to 1.0 s21.


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FIG. 8. Transient shear viscosity of a 9 mode multimodal pom-pom melt in start up plotted against data for IUPAC A LDPE. The shear rates range from 0.001 to 20.0 s21.

of ‘‘stressing viscosities’’ in two directions, m 1 1 (t), in the direction of greatest extension (t) in the direction which has a constant length; see Meissner et al. 1982!# as well and m 1 2 as simple shear. Now both of the fits to the latter two data sets are entirely free of parameter choices. Data are only available for two different strain rates of 0.01 s21 ~Fig. 10! and 0.05 s21 ~Figure 11!. By choosing suitable values of q i and t bi / t si we are able to fit all three experiments simultaneously. Note that there is quite a lot of scatter in the data for m 1 2 (t). The values for the parameters are shown in Table II. Both the IUPAC A and IUPAC X parameter sets are in accord with the constraints imposed on the values of q i and t bi / t si . The value of q increases with mode number and hence t b . The values of t b / t s correspond to a range of between two and six entanglements. These are physically reasonable values, and correspond to long chain branch frequencies from 1 per 4000 carbon atoms to 1 per 10 000 carbon atoms.

FIG. 9. Transient uniaxial extensional viscosity of an 8 mode pom-pom melt in startup plotted against data for melt 1 LDPE. Extension rates range from 0.01 to 1.0 s21.

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FIG. 10. Transient uniaxial extensional, planar extensional and shear viscosity of an 11 mode pom-pom melt in start up plotted against data for IUPAC X LDPE. The shear/elongation rate is 0.01 s21.

The values of q and t s were obtained through isolation of the most active modes at a given strain rate. Since extensional data are not available at strain rates which correspond to the action of the shortest time modes, varying the values of q for these modes makes virtually no difference to the quality of the fit. We have assigned the value of q 5 1 to these modes on the grounds that they represent the outermost segments of the molecule and are expected to retract as linear polymers. The values of q for the long time modes are important only at very low rates of strain. The parameter values for the LDPE B sample are shown in Tables III–V. We used three different relaxation spectra in order to establish a connection between the choice of

FIG. 11. Transient uniaxial extensional, planar extensional and shear viscosity of an 11 mode pom-pom melt in start up plotted against data for IUPAC X LDPE. The shear/elongation rate is 0.05 s21.


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FIG. 12. Diagram of the transient uniaxial extensional viscosity of a 12 mode pompom melt in start-up plotted against data for LDPE B. The extension rates range from 0.001 to 0.1 s21 .

spectra and the values of the parameters used in the model to fit the extensional rheology. This is discussed in Sec. IV B. The LDPE B extension and shear viscosity data and theory curves for the 12 mode spectrum are given in Figs. 12 and 13, respectively ~the fits using the other relaxation spectra look much the same!. Note that in extension, the theory curves at the higher strain rates of 0.03 and 0.1 s21 rise significantly above the experimental data. It is not possible to obtain a better fit here because only the slowest ~12th! mode contributes significant stress at all the strain rates. It is possible, however, that the samples broke at these two strain rates before reaching their maximum strain value ~which is set by an extension-thinning instability to be discussed below!.

FIG. 13. Transient shear viscosity of a 12 mode pom-pom melt in start up plotted against data for LDPE B. Shear rates range from 0.001 to 0.3 s21 .

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B. Discussion The most remarkable feature of our model is the consistent coincidence of the rapid changes in gradient of h¯ 1 (t,«˙ ) with the filament failure that ends the extensional experiments. The model continues to make predictions for the stress growth after the severe ‘‘knees’’ in the curves, the data cease at those points because the filament breaks via a necking instability. This may also be the cause of the slight overshoots in some of the experiments. This constitutes another remarkable feature of this approach to long chain branched polymers — a suitable decomposition into pom-pom modes seems able to predict both the upturn in extensional viscosity and the point at which the material will fail due to a necking instability. We conjecture that the strain and strain rates at which a long chain branched melt fails by necking is an inherent property of the material. This failure can be predicted by analogy with the Conside`re criterion @Nadai ~1950!, Vincent ~1960!, and Cogswell and Moore ~1974!# for elastic solids. This predicts that necking will occur at strain when the tangent to the stress-strain curve passes through the origin. This procedure would be expected to hold, approximately, for viscoelastic materials where the stress is primarily elastic, and would predict failure precisely at point where h¯ 1 (t,«˙ ) has a rapid change in gradient. In the preceding models, the number of modes was determined by the number of modes in the linear relaxation spectra of each sample. Therefore the frequency range of the complex viscosity data sets the range of relaxation times. This can lead to problems if, as is the case with the IUPAC X data, there is insufficient low frequency data to probe the longest relaxation time modes. These modes correspond to the innermost sections of the molecules that are the most highly branched. Although these modes have comparatively little effect on the linear rheology ~and hence are often neglected!, they dominate the nonlinear rheology at low extension rates. A lack of long-time modes may be compensated for by allowing the value of t s of the longest mode to approach t b . This produces a greater stress contribution for that mode ~see the choice of parameters for IUPAC X in Table II! at low extension rates. Although, in this case t s has to be set equal to t b to compensate for the lack of low frequency modes, it is expected on physical grounds that t b / t s should decrease for such higher modes. At the longer timescales associated with the higher modes, entanglements are only produced by other unrelaxed molecular sections. Thus the effective entanglement concentration decreases with mode number because segments with faster relaxation times do not act as constraints. The effect of this dynamic dilution is to decrease the number of entanglements s b and hence t b / t s @since t b / t s 5 (4/p 2 ) s b , McLeish and Larson ~1998!#. In fact the innermost segments may not be entangled at all at the longest relaxation times. In addition to the range of relaxation times, the density of relaxation times is also critical. Too high a density results in the solution for the moduli being highly ill posed @see, e.g., Friedrich and Hofmann ~1983!, Winter et al. ~1993!#, whereas too low a density can lead to a single mode being active over a large range of strain rates. This produces problems in fitting the nonlinear response ~an example is the overprediction of the stress maximum of melt 1 at the particular rate of 0.01 s21, see Fig. 9!. However, if our model is truly to reflect the underlying molecular physics, it must be possible to express the model parameters in a form that is independent of the choice of the t bi . This leads to the idea of a ‘‘molecular fingerprint’’ of a long chain branched melt based upon the parameters of the multimode pom-pom model. We obtained three different relaxation spectra from LDPE B data with 8, 10, and 12


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FIG. 14. Storage and loss modulii, G 8 ( v ),G 9 ( v ) of the LDPE B sample obtained from oscillatory shear stress growth of the equivalent shear viscosity using a Rheometrics RDA II rotational rheometer in the steady shear mode.

modes, respectively. The t bi and g i of the 8 and 12 mode spectra were chosen using a numerical error-minimizing routine to fit of the G 8 ( v ) curve of the LDPE B sample ~see Fig. 14!. The 10 mode spectrum was ‘‘hand picked’’ to ensure a very different selection of time parameters t b . In order to visualize how the individual modes in each of the spectra contribute to the viscosity, we plot g t 2/D t against t in Fig. 15. Provided that modes are chosen so that the spectra remain stable, g t 2/D t should follow a smooth curve, characteristic of the material, without much fluctuation of individual g i . Included in Fig. 15 is a 15 mode spectrum of the same LDPE B sample to illustrate how the ill-posed nature of the discretization leads to an instability if the number of modes is too large for the available data.

FIG. 15. Plot of different choices of linear relaxation spectra for LDPE B in the form of g i t 2bi /D t b plotted against t b .

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FIG. 16. 8, 10 and 12 mode pom-pom parameters g i , q i , r i of LDPE B plotted against t b . The parameters were obtained from fitting of the extensional viscosity data.

Since the parameter values required to fit the multimode pom-pom model should be directly related to the branching structure of the material, we wish to identify a material characteristic or ‘‘fingerprint’’ from these parameter values. However, it is clearly essential that this should be independent of the discretization of the linear spectrum. In Fig. 16 we plot the values of g i , q i and r i (r i 5 t bi / t si ) for each fit of the extensional viscosity data for the three different relaxation spectra. It can be seen that the q i distributions are indeed very similar, irrespective of the number of modes. Although the values of q i for the lower modes can vary slightly without affecting the fit to the extensional viscosity it is the higher modes which reflect the degree of branching deep within the LDPE mol-

FIG. 17. Multi-modal pom-pom parameters g i , q i , r i of LDPE IUPAC A and the time-temperature shifted parameters of IUPAC X plotted against t b .


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FIG. 18. Multimodal pom-pom parameters g i , q i , r i of melt 1 plotted against t b . The parameters were obtained from fitting of the extensional viscosity data.

ecule that are most important. It can be seen that these are very similar for the three spectra in these plots. The trend of the r i parameters is less clear and seems to depend more on the choice of the number of very slow modes in the spectrum and their influence on the behavior of the extensional viscosity at very low strain rates. However the shape of the r i envelope is still consistent among the models for LDPE B, and so is also a candidate as a distinguishing feature, although not as strong as q. There is an effective power law for the g i distribution before the cut off. With a power of about 0.3. Rubinstein, Zurek, McLeish, and Ball ~1987! pointed out that the stress-relaxation function for entangled dynamics at or near the percolation threshold for branched polymers is not in fact a power law. However, the predicted function can mimic an apparent power law that depends on the average number of entanglements between branch points (M x /M e ). The apparent power index is of the order of (M x /M e ) 21 so it qualitatively ties in with the material parameters. We can also compare the choice of parameters for the IUPAC A and IUPAC X since they are batches of the same material. However, we must first perform an approximate time-temperature shift of the IUPAC X data because its spectrum was taken at 125 °C, whereas that of IUPAC A was measured at 150 °C. This shift is not exact due to the thermorheological complexity of LDPE, but it does gives us a rough comparison of the parameter choice ~see Fig. 17!. We used a value of the flow activation energy, E a of 13.6 kcal mol21 @Meissner ~1975!#. This gave us a calculated shift in frequency of the IUPAC X data of a T 5 0.363. Again the q i distributions are very similar supporting a claim that they are a property of the material. Comparing these q i distributions to the LDPE B q i distributions it is apparent that they are lower, which is to be expected for a sample with less long chain branching. The parameters for melt 1 are shown in Fig. 18. We plotted the multimodal flow curve of extensional viscosity against extensional strain rate for the IUPAC A model to compare to them experimental data ~Fig. 19!. The large peaks look rather unphysical but demonstrate the action of individual modes at differing rates of strain ~the single mode flow curve is shown in Fig. 5!. The prominent peaks result from the limited number of modes in the relaxation spectrum, each of which corresponds to single species of pom-pom molecules. The discrete nature of the model is therefore much more critical in extension than in shear. In practice, however, there is a

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FIG. 19. Steady state uniaxial extensional viscosity, h («˙ ), plotted against rate of extension, «˙ , for a 9 mode pom-pom melt plotted against data for the LDPE IUPAC A.

continuous distribution of molecular weights that would smooth out the peaks. Nevertheless with only nine modes of the IUPAC A model we do obtain a curve that follows the overall shape of the LDPE data. V. CONCLUSION The remarkable result is that a small set of pom-pom modes with physically reasonable parameters is able to account simultaneously for the three functions ~each of two ˙ variables! h (t, g˙ ), h¯ 1 (t,«˙ ) and h¯ 1 planar(t,« ) over four decades in time and rate. This is quite unprecedented in modeling nonlinear LDPE rheology. This suggests strongly that the physics of the simple pom-pom molecule captures a generic aspect of branched polymer dynamics. It seems possible to approximate very complex branched structures ~which are illunderstood themselves! using the relatively simple behavior of a dual-branched molecule whose properties can be understood via the polymer physics derived from the tube model and branched polymer theory. It has shown that the key aspect in the simultaneous extension hardening and shear thinning properties of LDPE can be put down to the segments connected by branch points and their ability to sustain a large amount of tension. The model’s success in achieving hardening in planar extension while retaining strong shear thinning demonstrates its superiority over current integral constitutive equations and demonstrates the importance of including the relevant physics in the rheological models. Several further applications suggest themselves. The use of such multimode pom-pom equations in non-Newtonian fluid dynamics simulations would enable quantitative prediction of complex flows of LDPE. The relatively simple form of the approximate pom-pom model constitutive equations @involving the differential approximation for S(t)] could be used in such numerical simulations. Such calculations have already been performed for the single mode pom-pom model @Bishko, McLeish, Harlen, and Larson ~1998!#. Furthermore, the nonlinear spectral parameters themselves represent a method for characterizing branched polymer melts that could provide a discriminator for different branched molecular structures. They relate extensional data on branched melts directly to averaged molecular structural quantities, and suggest that strain hardening curves and


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even the strains at break in careful experiments may be characteristics of the materials. Carefully chosen sets of $ t bi , t si , q i , g i % account for all these features as well as for the shear data, and are relatively robust to the choice of mode density. ACKNOWLEDGMENTS This work was supported by a CASE sponsorship from BICC Cables. The authors wish to thank A. R. Blythe and M. Cassidy for useful discussions.

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