Ideas about reptation, tube renewal and tube relaxation applied to the description of the non-linear viscoelastic behaviour in shear of some polydisperse polyethylene melts

Journui of Non-Newtonian Fluid Mechunks, 23 (1987) 189-214 Elsevier Science Publishers B.V.. Amsterdam - Printed in The Netherlands

189

IDEAS ABOUT REf’TATION, TUBE RENEWAL AND TUBE RELAXATION APPLIED TO THE DESCRIPTION OF THE NON-LINEAR VISCOELASTIC BEHAVIOUR IN SHEAR OF SOME POLYDISPERSE POLYETHYLENE MELTS H.C. BOOIJ and J.H.M. PALMEN DSM Reseurch, P.O. Box 18, 6160 MD Geleen (The Netherlunds) (Accepted June 5, 1986)

Summary In melts of polydisperse linear polymers like high-density polyethylene (HDPE) the dynamics of a given macromolecule are affected by the renewal of the confining tube owing to movements of surrounding molecules, the more so according as the matrix contains more shorter molecules. At large strains, the relaxation of the tube due to the retraction of matrix molecules also contributes to stress relaxation. For long-chain branched molecules in a polydisperse melt like that of low-density polyethylene (LDPE), the dynamics are dominated by tube renewal and tube relaxation effects. It is argued that for melts of HDPE as well as LDPE the combination of these ideas leads to a rheological constitutive equation consisting of the sum of two integrals. Both integrands are factored into the product of a strain-independent time function and a time-independent nonlinear strain tensor. Using the relaxation spectrum from oscillation experiments and the nonlinear strain measure from first normal stress growth data, this equation satisfactorily describes the decay of the stresses after step strain or after cessation of steady shear flow for melts of two commercial HDPEs and of two LDPEs. For the HDPE samples one integral suffices, but for the LDPE melts the additional time and strain dependences of the integral relating to tube relaxation are indispensable for a good fit of the experimental curves.

1. Introduction The complex dynamics of polymer melts in processing operations calls for good manageable mathematical models of the time-dependent nonlinear rheological behaviour of these viscoelastic materials. The most sensitive methods for testing recommended constitutive equations describing the 0377-0257/87/$03.50

@ 1987 Elsevier Science Publishers B.V.

190 relationship between stress and strain histories, require the use of more complicated time-dependent flow patterns. However, controlled transients are experimentally much more difficult to achieve than steady-state conditions and give rise to very time-consuming theoretical calculations. Fortunately, more advanced experimental instrumentation and the availability of high-speed computers have now made it possible also to evaluate models and constitutive equations for transient deformations. In a number of recent papers, the authors have examined the usefulness of single-integral constitutive equations in which the time and strain effects are governed by factorable functions for the description of the nonlinear viscoelastic properties of commercial polymeric fluids. A rather empirical and pragmatic approach has been followed, focusing in particular on the form of the nonlinear strain measure or, equivalently, of the nonlinear stress-strain relationship utilized in this type of theory. Experimental data on transient rheological properties have been analysed directly to deduce this measure [1,2,3], leading, for example, to the conclusion that the Seth strain measure is not the proper quantity to use for polymeric fluids [3]. Another way to test constitutive equations is to utilize more or less well-established interrelationships between linear viscoelastic functions of frequency and time and nonlinear transient rheological functions of shear rate and time. For instance, in order to satisfy the Cox-Merz relations, very special well-defined oscillatory forms of the stress as a function of strain are needed [4], quite different from those required for the mirror relations of Gleissle [5]. On the other hand, empirical relations between the steady-state first normal stress difference and the shear viscosity curve lead to different and quite peculiar forms for the nonlinear strain measure, as demonstrated [5] for that of Gleissle. In all the cases mentioned above, the condition of separability of time and strain effects into the product of a strain-independent memory function and a time-independent strain tensor is satisfied. In this paper, the authors try to describe the transient rheological properties of commercial polymer melts for which this prerequisite does not hold. For this purpose, use is made of recent views on the dynamics of macromolecules in the molten state, on the basis of which a type of generalized constitutive equation will be formulated in the next section. In the following sections, the predictions of this equation in certain transient flow fields will be calculated and compared with experimental data obtained for some special polyethylene melts. 2. Form of the constitutive equation 2.1 Chain retraction and reptation In 1967 Edwards [6] argued ment constraints act effectively

that in a dense polymeric system, entangleas an open-ended confining tube enclosing

191

each macromolecular chain along its average contour. The diffusion of the chain is thus restricted to one-dimensional reptational motion. Using this concept in 1971 de Gennes [7] analysed the dynamics of the reptation of a single polymer molecule inside a three-dimensional network of fixed obstacles. In 1978 Doi and Edwards [8] introduced reptational motion into a dynamic model of the nonlinear viscoelasticity of concentrated systems of flexible macromolecules under macroscopic deformation. They derived a rheological constitutive equation by considering the motion of a single polymer molecule in the mean field imposed by the other chains. Those interested in a recent survey of the backgrounds, the relation to the rubberlike liquid model, the practical usefulness and a comparison with experimental evidence of this reptation model are referred to, for example, the monograph of Janeschitz-Kriegl [9] and the evaluation of Osaki and Doi

WI. The Doi-Edwards (DE) model was extended in 1980 by Doi [ll] to include short time-scale relaxation processes. According to this approach, in a step-strain experiment the tube deforms affinely and the stress in a polymer molecule relaxes through three consecutive processes. The fastest one is a local re-equilibration of the contour length of a primitive chain segment between entanglements, consisting of N, monomers, with a relaxation time rA which is the (longest) Rouse relaxation time of this segment. Then a retraction of the chain follows inside the tube to regain its equilibrium length, with a characteristic time rB which is the Rouse relaxation time of a chain consisting of N primitive chain segments ( ~a =’ 2N2r,). The slowest relaxation is a complete disengagement of the chain by reptation through which the chain recovers an isotropic configuration with a disengagement time 7c = 3Nr, = 6N37,. The authors now introduce the following functions: G, = 3vkTN = 3RTp/M,, where v = number of molecules per unit of volume, p = density of the melt, and M, = molecular mass of a primitive chain segment; E = relative deformation gradient tensor; CX( E) = ( 1E. u I),, where u is an isotropic unit vector; FaB( E) = ((E . u),( E - u)~),, which is the Finger strain tensor (divided by 3) and

Q,,(E) = In the first stage of the relaxation process after a- step strain E at t = 0, the components of the stress tensor a+( t, E) relax from G,N,(Y~( E)Q& E) to G,a2( E)Qap( E). In the second step the relaxations proceed to GOQaB( E) and in the last stage the stresses vanish as they ought to do in a fluid with fading memory. The whole relaxation process can be described with the

192 following e+(f,

equation

E) =

for the stress tensor:

G,Q,,(E)g(t/7,)g2(t/7,,

(1)

with

(2) 2

1 + [a(E)

&

c

podd P =

i

- l] exp * i

P + ME) - 11exp(-f/7B)}2, 8 2

p odd P =

( 1 -tp2

exp -

7C

= exp(

-t/q.).

Ii (3) (4)

Various linear and nonlinear rheological properties of a polystyrene sample melt of narrow molecular mass distribution can be described quantitatively by Lin [12] using the original DE theory without the functions E), provided that the distribution was taken into &A,) and g(&, account and 7c was taken proportional to M3.4. However, Menezes and Graessley [13] showed that the nonlinear retraction step is necessary to explain the results of stress growth experiments on polybutadiene solutions with narrow molecular mass distributions. Schausberger et al. [14] and Pfandl and Schwarzl [15] have found it necessary to use the mole-fraction distribution instead of the theoretical weight-fraction distribution and 7c of M3.’ instead of T - A43.4 to describe the linear viscoelastic properties polydisperse polyityrene melts. Both modifications indicate that the long molecules contribute less to the dynamic moduli than is expected theoretically. In the authors’ opinion, this is due to the fact that the surroundings of a given molecule are subject to the same type of motions as the molecule itself. 2.2 Tube renewal In the models mentioned, it is assumed that during the various relaxation stages the environment does not participate in the relaxation of the chain considered. However, ‘constraint release’ through ‘tube renewal’ due to diffusive reptation of the neighbouring chains has already been examined by Klein [16], Daoud and de Gennes [17], Graessley [18], Montfort et al. [19], Marrucci [20], and others. Klein [16] proposed a self-consistent model for the renewal of the tube configuration, considering that in a monodisperse

193 of linear molecules the neighbouring chains constituting the tube reptate likewise inside their own tubes. He arrived at the conclusion that each tube constraint is renewed with a characteristic time T,,,(N) - N2~cN - N5. This means that for all entangled monodisperse melts, the tube renewal time is much longer than the reptation time, so that reptation should be the dominant relaxation mechanism. For a tube containing an N-chain immersed in a melt of chemically identical P-chains, it was calculated [16,17] that T,,( N, P) - N2P3, so that the constraint release will contribute significantly to the diffusion and relaxation of N-chains in a melt of much shorter P-chains. This only holds if the matrix is entangled itself. Otherwise, this material will act as a solvent, giving Rouse-like instead of reptile behaviour [21] and a level of the rubbery plateau modulus equal to G,(p* [20,22], where ‘p denotes the volume (and weight) fraction of the N-chains. The viscoelastic behaviour of binary blends of nearly monodisperse polymers has been investigated experimentally mainly on polystyrene resins [19,23-261. The most extensive experimental research on the effects of constraint release has been performed by Montfort et al. [19] on blends of N-chains and shorter P-chains. At short times, all P-molecules reptate in a matrix of essentially immobile obstacles N with a disengagement time OCR= 2.3 x 10-l’ JV;.~ s at 160°C, and the modulus relaxes from the rubbery plateau value G, to a secondary plateau at a level G,cp. Thereafter, the stress on the remaining N-molecules relaxes faster than would be, expected on the basis of pure reptation of the N-chains in an N-matrix. This is due to the tube renewal process, which is more important the lower are the values of P and cp. For the effects of N, P and cp on the tube renewal time, they found on the basis of a theory proposed by Graessley [18] the formula melt

%*(N,

PY ‘p) =

1.4 x lo-%4;9[ +9tp(1

&4$3

+ 9&l

- (p)*M~~3A4;.3(2M;.3

- &V;%4;.3( + M;.3)-l

+

M/$3 + 2A4;.3)-*

(1 - q)3J4;-ll.

(5)

Their experimental data over three decades can be described by this formula within a factor of 2. In the DE theory, each chain contributes additively to the stress relaxation function of a blend of monodisperse samples, yielding [8]: gblend(t)= &#)gN(t), N

(6)

where cp( N) are the volume fractions of the constituents N and gN( t) are the relaxation functions in the form of eqns. (2-4). We now assume that eqn. (6) also holds when tube renewal occurs. For the binary blends

194 mentioned, g(t)

this leads to the relaxation

= (1 - ‘P)8(f/%J)

function

+ cpg(~/%N)g(%“(~~

P, (P)).

(7)

The right-hand product term for the relaxation of the N-chains can be replaced by one function g(f/T) with an effective disengagement time r given by 7-l = ~6.j + ~~;d, if simple exponential functions like (4) are used. Application of eqn. (5) to the binary 1 : 1 mixtures of M = 128 000 and by SchausM = 70 000, resp. M = 770000 and M = 128 000, investigated berger et al. [14], yields 7 = 0.58 rcN, resp. +r= 0.65 7cN. Applied to the fictional binary mixtures with M = 500000 and M = 100000 as treated by Pfandl and Schwarzl [15], ranging from 5 to 99.8 wt%, values of the effective relaxation time between 0.18 7cN and 0.85 7oN are calculated. Hence, tube renewal contributes essentially in all these cases. 2.3 Tube relaxation Tube relaxation, i.e. constraint release during the chain retraction stage at large strains due to the retraction and reptation of the matrix molecules, is also examined [17,18,33]. This effect can only be of practical importance if N > P. Generally, the behaviour of a small minority of labelled chains embedded in a strained melt of chemically homogeneous material has been studied. The approach used by Viovy et al. [33] for describing the relaxation behaviour in regime 3, i.e. in the case where the reptation time rep of the matrix molecules lies between the retraction time 7sN and the reptation time rcN of the N-molecules considered (rs,, < 7BN < rep < T,--), leads to the result that the stresses decay according to the following function: g”(t)

= s(t/%)g(t/%>

E)h2(t/r,,,

~)g(f/%J)g(f/%,)~

(8)

with h ( t/TI3N 9 E) = 1+ [c&‘~(E)-~)]

exp(-t/TBN).

(9)

The stresses in the N-chains first partially relax by the retraction of the neighbouring molecules, then by the retraction of the N-chains, followed by the reptation of the N-chains, modified by tube renewal. In regime 2, which arises when rrn, c rep c rs& c rcN, Viovy et al. [33] found that g”(t)

=g(t/%)g(t/%p,

E)h20/%,

~)g(t/rcN)g(f/LJ~

(10)

When time rBN has passed, the N-chains have their equilibrium curvilinear lengths, not because of self-retraction but owing to retraction and reptation of the shorter chains in the melt. The next relaxation stage of the N-chains, i.e. the approach to an isotropic chain configuration, is also strongly affected by the motions of the surrounding P-molecules.

195 N,a*.

10-c

-time

----

10~~ TA _ _ TA ~-~-_-----~--_--TA

-

_

‘BP

_,

(log

scale)

\

. _, TBN_ r ,TBP TBN TCP T ,---,_r-------_---_---_regime* ‘CP

_ TCN-

regime

4

--regime

3

‘BNT

Fig. 1. Relaxation functions gN(t) after a step-shear strain of about 10 in regime 4 for N=lOO,inregime3forl%of N=lOOin P=25andinregime2forl%of N=lOOin P=8, calculated using eqns. (l), (8), (10) and (5).

The main difference between regimes 3 and 2 is that in regime 3 the time scales of the. retraction and reptation of the N-molecules are mainly determined by rBN and rcN, whereas in regime 2 the relaxation of the N-molecules is effected through the movements of the environment of which ri3r and rcr are representative. By way of example, the substantial effects of tube relaxation and tube renewal on the relaxation functions after a step-shear strain of about 10 are illustrated in Fig. 1 for 1% solutions of N = 100 molecules in a matrix of P > 100, P = 25 and P = 8 molecules, respectively. 2.4 Stress relaxation

in a melt of polydisperse

linear molecules

Up to this point, the effects on relaxation of a given molecule embedded in a matrix of lower molecular mass have been considered. If P a N (regime 4) the retraction of the matrix molecules may be faster than the reptation of N, but is always slower than the retraction of N, and that is not expected to influence the relaxation behaviour of N. At even larger values of P, undisturbed retractional and reptational motion of N is predicted. The considerations show that in a polydisperse melt only the fraction of the material with lower molecular mass will significantly affect the behaviour of an N-chain. Then two possibilities are left: either rBP < rBN < rep < 7eN

196 (regime 3) or rBp < rep < rBN -C 7cN (regime 2). The boundary is formed at 3P3 = N2, indicating that at N = 10 only molecules with P -e 3 contribute to a regime 2 behaviour and that at N = 100 only molecules with P -c 15 reptate faster than the N-chains retract. For example, assuming MC = 1250 [34] a polydisperse commercial polyethylene sample HDPE 2, with it4, = 10’ g/mol and k&/M, = 6, contains 4 wt% shorter than P = 3, 30% shorter than P = 15 and 80% shorter than P = 100. This means that the greater part of the surroundings of every molecule N in this polydisperse material consists of molecules that are either larger than N or shorter, but reptating slower than N retracts. Hence, we approximate the real situation by stating that all molecules shorter than the considered N-chain cause a regime 3-type relaxation of the N-chain. This fraction introduces a relaxation from a2Q to CXQwith time constants rBp running over the whole time interval below 7aN. The end of this retraction of the lower molecular mass part of the matrix will be that the relaxation of the stress in the N-chain from a2Q to aQ does not proceed steeply around rBN, but much more gradually at shorter times. This time span will be shorter for short N than for long N (because a broader distribution of rBp will be active), but for most N the stress will fall to CYQfar before 7aN. After that the relaxation to Q advances with time constant 7BN, giving rise to the equation

for the N-chains. The stress relaxation proceeds according to eqn. (6) by

of the whole polydisperse

At very small strains, in the linear viscoelastic h2( f/rBN, E) = 1, so that

melt then

range, Qap( E) = Fap( E) and

(13) where q(N) is the differential volume-fraction molar mass distribution function. This expression by definition equals G(t)&(E), where G(r) represents the linear viscoelastic relaxation modulus. For commercial polymeric melts, it is common use to define a hypothetical continuous spectrum of relaxation times H(T) in such a way that G(t)

=/

+mH( T) exp( --f/r) --m

d In 7,

(14)

so that l/7 has to correspond with 1/7cN + l/r,,. In the ideal case of monodisperse N in P 2 N, theory says that 7aN = case, we still assume 2r,N2 = 0.61 r;‘3 &’ . For the present polydisperse

197 that rBN is proportional to e,,&

E) =

to T’. Then eqn. (12) becomes

Qap@)/+%d xexp(-ktTTi)j

approximately

exd-t/d{1 + b(E) - 11 d In 7,

(15)

utilizing eqn. (9), with I= 2/3. This form shows some similarities to the one employed by Menezes Graessley [13]. Because 7RN cz 7, this formula can also be approximated u+(t,

equal

E) = QaB(E)/+PH(7)

exp( --f/T)

+ QaP(B)[I(E)

- I]/+WH(r) --oo

and by

d In 7 exp( --t/r’)

d In r

(16)

or G&,

E) =

Q,,@)G(d

2.5 Stress relaxation

+ Qc&)b(E)

in a melt of polydisperse

- 11G&h

(17)

long-chain branched molecules

Long-chain branching very strongly increases the reptation time of the whole molecule [27-331. Doi and Kuzuu [28] have shown for melts of monodisperse star molecules that the longest relaxation time of the reptation process is I-: = rANa exp( v’Na) with v’ = 0.6, where N, is the number of primitive chain segments of the arm. This indicates that the range of Tc-values is much broader for melts of branched than of linear molecules, at the same level of polydispersity. Therefore tube relaxation and tube renewal effects will be much more dominant in melts of polydisperse branched polymers than in those of polydisperse linear polymers. Especially regime 2 behaviour will prevail, giving rise to relaxation functions of the form of eqn. (10). By a reasoning analogous to the one leading to eqn. (ll), it can be made plausible that the relaxation after some initial time will proceed as g”(t)

= h2(t/rcW

(18)

~)gW,J

assuming that T,,, -=c rcN. Here OCRis some average or effective reptation time, determined mainly by the lower molecular part of the environment of the N-chain. As r,,, is also governed by this part of the material, we again to T,/,,, taking into account that at long rcP suppose that 7cP is proportional the renewal is much more difficult than at short rcP. Then eqn. (18) becomes g”(t)=exp(-t/T){l+[a(E)-l]

exp(-kt/T’)},

and the stress relaxation of the whole polydisperse melt can be described again by eqn. (16), probably with different values of k and 1.

198 It is anticipated that in this case G2( t)/G( t) is much larger than in the case of polydisperse linear polymer melts, as r,, is expected to be much closer to rep than rcN to 7aN. Very recently, Roovers [30] found experimentally that at high molar mass the enhancement in viscosity of H-shaped polymers in which the five subchains are approximately of equal length is even larger than of three-arm or four-arm stars with the same number of entanglements per branch because the interior sections of the chains between two branch points are strongly restricted in their retraction and reptation motions. Moreover, the relaxation spectra are found to be very broad, similar to those of comb polymers. Then the surrounding molecules have to bring about the relaxation of the stresses in these chains, leading to regime 2 behaviour, described by eqns. (18) and (16). Hence, we conclude that polydisperse melts of linear, star, comb as well as randomly branched molecules, all relax according to eqn. (16). Also blends of these various architectures are expected to follow the same line, so that eqn. (16) will be used in the subsequent analyses for HDPE melts as well as for LDPE melts. If one also takes into account the experimental evidence that the various polymers need different strain measures [3,35,36] and for the fact that other theories of viscoelasticity of polymer chain systems, like those of Marrucci and coworkers [20,37,38] or Curtiss and Bird [39], yield strain functions different from Qap( E), it is obvious to replace QJE) by a generalized nonlinear strain tensor SJE), unspecified for the time being, but with the property that it reduces to the Finger strain tensor F,,(E) in the limit of small strains. In order to be consistent, the definition of cr(E) also has to be adjusted so that a2(E),S,,(E) = F&(E). Thus the result is obtained that after a step strain E the stress tensor relaxes according to a,&

E) = G@)&(E)

+

G20&(@

-

09)

ds,,(E)~

indicating that at short times the stress is proportional to a(E) and at long times to S,+(E). Introduction of this generalized measure of strain into the quasi-linear Boltzmann superposition integral yields the following expression for the total stress tensor at time t, after an unidirectional deformation gradient history E(t, t’) describing the strain at past times t’ with respect to the reference state at t:

u,,&, J?) =

/“i G(s) ds;f’ +G2(~) db(E) ~;lL’“‘}~

ds,

0

P-0) where the symbol

E under the integral sign stands for E( r, t’) and s = t - t’.

199

In the limit of small strains this equation reduces to the quasi or finite linear viscoelastic (FLV, [13]) response for second-order fluids:

In the next section, eqn. (20) will be employed some special shear flow histories y( t, t - s).

to predict

the stress fields for

3. Prediction of stresses for various shear flow histories 3. I Small-strain

oscillation experiment

In the limit of small strain, eqn. (20) yields for the complex dynamic modulus G*(o) = G’(o) + iG”(w) = Gd(o) exp is(w) the well-known +O”

G*(w’=/,

H( 7)io7 1+iw7

shear form

(24

dln7.

Between the functions G”(w) and G’(w) the Kramers-Kronig relations exist-and analogous relations have been derived recently by Booij and Thoone [40] between 6(w) and In Gd ( w )-which are real rheological relations in the terminology introduced by Bernstein [41], namely relations between measurable rheological quantities, valid in a particular type of motion, which do not require a knowledge of specific material properties such as the functions H( 7) or S(y). However, these relations ask for a practically impossible integration over the entire frequency range from zero to infinity. Published approximations of these interrelations are normally quite crude or complicated, but one simple approximation has been shown to be very useful [40,42], namely

S(w) = ;

d In G,(U) d In u

(23)

U=O’

Oscillation experiments over broad frequency ranges can be used to calculate H(T) from eqn. (22), e.g. by means of the iteration procedure described by Scholtens and Boo?) [43]. 3.2 Step-strain

stress relaxation

test

If a shear strain of magnitude y is imposed upon a relaxed fluid instantaneously at time t’ = 0, the induced stress tensor will relax at t > 0, according to eqn. (20), with E,, = y. Knowing the relaxation spectrum H(T), the relaxation moduli G(t) and G2( t) can be calculated with eqn. (16) for every value of k and 1. The

200

relaxation of the shear stress a,,(t, y) as well as of the first normal stress difference a,,(t, y) - u,*(t, y)-which for the sake of conciseness will be denoted u( t, y) and Ni( t, y), respectively-can be measured in step-strain stress relaxation tests with varying step height. These data provide the following functions ~(1, Y) = G(~)&(Y) K(t,

+ G,(t)[a(u)

Y) = G(~)[%(Y)

- l]&,(y),

- S,,(Y)]

(24)

+ G,(t)[a(v)

- 11 [S,,(Y)

- %(Y)]. (25)

The identity G(Y)

-

c~*(y)S,~(y)

S**(Y)

= FJy)

implies that the relation (26)

= Y&*(Y)

should hold between the components the rheological relation W>

of the strain tensor Stia. This leads to

Y) = Ye09 Y),

(27)

often referred to as the Lodge-Meissner relation. Lodge [44] proposed classifying constitutive equations on the basis of the value of the stress ratio Ni( t, ~)/a( t, y) in this kind of single-jump shear strain experiment. The limited amount of available experimental data on polymeric liquids seems to favour a time-independent value for this ratio equal to y. 3.3 Stress growth after instantaneously

applied simple shear flow

Suppose an initially stress-free fluid is sheared from t’ = 0 with a constant shear rate y. The build-up of the stresses governed by eqn. (20) is then

u,fp(t, +) =

/‘( G(s) dsfs’ 0 +

dbb) - 1&&s) G2b)

The rates of increase of the shear stress and first normal then proceed according to du+(t,

?>

= Gttl

d&2(@)

dt dN,+(t, dt

?) =G(t)

d&, dt

In principle, relation

the nonlinear

1 d&+(yt) T

_ tde+(yt)

Y

+ G

dt

dt

cl)

db(?t)

2

dt s22

+ G

strain

-

stress difference

11 s,,(+d

7

dt 2

(28)

ds

ttj

db(?t)

measure

-

11 h

-

&,I

dt

S,*(y)

can be obtained

(29) (30) by the

= G(t)S,,(+t) +G,(t)[(~f)“*S:j*(~t)

-

s,,(N]

(31)

201 which has been derived using eqn. (26). In the case where G2( t) = 0, the relation for S,, due to Bernstein [41] is obtained, which has also been acquired in various ways for factorable and strain dependent models with strain independent memory functions [45-471. Equation (29) implies that in stress growth experiments the shear stress exhibits a maximum at a shear strain and a time for which the relation d%*(Y)

0.5G,(t)&/~)“*

=

dy

(32)

- G(t) - G*(f) + O.~G(~)(Y/%)“*

holds, with y = yt. This relation shows that at low shear rates the shear stress is maximum at about the same strain at which S,,(y) is maximum, but that at higher shear rates the peak in a+( t, Jo) occurs at larger strains, in agreement with experimental data obtained on, for example, solutions of polystyrene and polybutadiene [13,48]. Equation (32) has the advantage over the corresponding relation of Menezes and Graessley [13] that at very high shear rates the slope dS,,(y)/dy at the maximum of a’(t, q) does not tend to the unrealistic value of - cc, but to - S,,( y)/y, which condition can normally be reached only at very high y values. According to eqn. (30), in stress growth experiments the first normal stress difference will show a maximum when d%

-

dy

G2(0&

s22

=-

G(t)

- G,(t)

-

s22j1’*

+ 0.5G2(t)y(S,,

- SZZ)-“*

*

(33)

If the function N:(t, 7) has a maximum it occurs at the value of y where S,, - S,, is maximum at low shear rates and where d( S,, - S,,)/dy = - 2S,, at very high shear rates, but always at strains larger than the peak strain of e’(t, ?). If eqn. (26) holds, the following rheological relation, previously derived by Bernstein [41], can easily be deduced from eqns. (29) and (30): 1 a&+(&

$)

1 1 = f a +*Y

. au+(t, at

Integrated forms of eqn. (34), meant shear stress data, have been elaborated

y)

I *

to calculate by Kearsley

(34) the normal stress from and Zapas [45].

3.4 Steady shear flow In the case of steady simple shear flow at a constant shear rate +, the For the stress field is given by eqn. (28) with t tending to infinity. computation of these stresses, G(s) and Sas( +.s) have to be known over a

202 very broad range of s. Inversion of the integral in order to calculate these functions from experimental values of u,~(+) over a range of shear rates yields only very inaccurate results. 3.5 Stress decay after cessation of steady shear flow If the sudden stoppage of a stationary shear flow at a shear rate q is taken to occur at time zero, the decay of stresses in the fluid satisfying eqn. (20) proceeds as a-(t,

9)

=

I”(

(qs)

t

ds12(Yd(ss - G)

(S)d[OL(~(s-t)-llS,2(~(s-I))

+G

ds

2

NJt,

&

3

(35)

i)=/w(G(s)ds1~;s22 t +G

~S~d[a(i(s-t)) 2

-

l&%,

-

s223

ds

For this type of experiment, it is impossible to derive equations analogous to eqns. (24) and (25) or (29) and (30) for the functions S,,(y) and S,,(y) S,,(y). If condition (26) applies, eqns. (35) and (36) yield the rheological relation obtained by Yamamoto [49] for a relative deformation dependent model with strain independent relaxation times, viz.: (37) which has also been mentioned by, for example, Kearsley and Zapas [45] and Osaki [46]. 4. Experimental 4.1 Equipment and materials A Rheometrics mechanical spectrometer, type RMS 7200, is utilized with a parallel platen system for dynamic measurements and a cone-and-plate system for transient experiments. In order to account for compliance problems (see e.g. Menezes [50], Vrentas and Graessley [51]) the radius and the cone angle should be tuned in to the viscosity of the samples and the stiffness of the apparatus. A large cone angle is preferable, and we chose an angle of 0.2 rad and a radius of 1.25 cm except at shear rates below 0.2 s- ‘, where an angle of 0.15 rad and a radius of 2.5 cm is used.

203 In transient experiments, accurate registration of all displacements and forces is indispensable [48]. Therefore, the strain is controlled by a velocity servo drive system, including a DC torque motor, a tachometer, a rotary voltage differential transformer, a position control loop and an HP function generator. The stresses are measured by the standard transducers with signal conditioners. All strain and stress signals are fed through a multiprogrammer into an HP 9836 calculator, which also controls the course of the experiments, analyses the signals, calculates the desired functions, stores the data and steers the printer and plotter. The materials used in this investigation have been selected for melt viscosity and elasticity in such a way that the various types of experiment can be carried out reasonably accurately with the equipment lay-out described. Further, in order to extend the viscosity range of low-density branched polyethylenes (LDPE) used in our previous work [2] (melt flow index 0.3 dg/min) and in the well-known transient investigations of the Meissner, Wagner and Laun group [52-541 (melt flow index 1.4 dg/min), two samples with a higher melt flow index are chosen which, moreover, do not show a simple factorization of stress and strain effects. The other two samples are high-density polyethylenes (HDPE) with melt flow indices of the same order, one of which has a relatively narrow molecular mass distribution. Commercial grades of DSM were preferred, of which some characteristic properties are shown in Table 1. The HDPE samples are considered to consist of linear molecules, but the LDPE samples are polymerized in a high-pressure process and definitely contain long-chain branched molecules. The molecular mass distribution is broad for all samples, but the molecular mass between entanglements is small for polyethylenes, which makes it easy to penetrate far into the entanglement region of the majority of all pieces of the distribution. In this TABLE

1

Various

characteristics

of the materials

Material

HDPE

Grade Density ’ Melt flow index ’ Intrinsic viscosity ’

Stamylan 963 8.0 1.29 19 75 200

M, d M,v d W d a ’ ’ d

1

HDPE 2 9089U

In kg/m’ at 296 K. In dg/min at 463 K. In dl/g at 408 K in decalin. In kg/mol, with gelpermeation polyethylene fractions.

Stamylan 962 2.0 1.66 17 100 500

chromatography

LDPE 2

LDPE 1 9020

Stamylan 918 7.5 1.00 15 180 900

1300C

Stamylan 923 4.4 1.03 18 150 700

on the basis of calibration

34OOC

with linear

region the factorability can be tested most conveniently. On the other hand, in melts with long-chain branched molecules, constraint release effects due to retraction may play a more dominant role. Therefore, in these melts deviations from factorization are most likely to be observed. 4.2 Experimental

procedures

Stationary oscillation experiments are performed automatically in the range of the angular frequency w between 10-l and 10’ rad s-l with steps of 0.2 in log w and at top-top strains of about 0.25 maximum. The calculator gives the ratio of the amplitudes of stress and strain and the phase shift between them, hence Gd( w) and 6(o), together with some related functions. The measurements are performed isothermally at temperatures between 403 K and 523 K, generally every 20 K. In step-strain relaxation experiments, the time dependence of a( t, y) and N1( t, y) at several values of y are recorded. Stress growth experiments give a’(t, +) and &+(t, +) up to steady-state values (or fracture of the sample) after the inception of flow at a series of shear rates +. The course of a-( t, q) and of Nr-( t, f) is determined in stress decay measurements after stoppage of steady flow at different P.S. Obviously, no experimental apparatus can start up or stop along a Heaviside step function, neither in strain nor in strain rate. Also, inertial effects and coupling of stresses in the melt and the measuring system (apparatus compliance and geometry effects) have to be taken into account. Various studies treating these problems are referred to by Attane et al. [55], who themselves are among the few who discussed the problems of noninstantaneous stoppage. A very instructive and complete survey of the experimental problems and results in single-step shear strain experiments has been published recently by Lodge [44], who also referred to the relevant literature. Michele [56] and Pedersen and Chapoy [57] have thoroughly investigated this kind of problems encountered in the use of the mechanical spectrometer. We have tried to avoid these experimental problems and to approach controlled transients to a large extent by replacing the step functions in Desired strains up to 30 are imposed on the strain by ramp functions. samples by shearing with a rate p equal to 31.5 s-l, controlled by the velocity servo system in a time interval between t’ = -At and t’ = 0. If eqn. (20) is confronted with this history, the expression for the stress tensor becomes ua&,

+ At)

=

/““‘i

f

+

G(s)

G,(s)

dsapc;~ - ‘))

db(+b - t)) - 1ls,p(Y(s - t)> ds

ds

1

.

(38)

205 Using the mean-value theorem of calculus and approximating the kernel of the integral on the internal s = t to s = t + At by its value at the midpoint of the interval, hence at s = t + : At, one obtains, analogous to Zapas and Phillips [58,59], the approximate relation

q& - : At,

Y)

= G(t)&,(v) + Gdt){b(v)

-

$%,+(Y)}.

(39)

With a known spectrum H( 7) and a best guess of the parameters k and I in eqn. (16) the nonlinear strain tensor S,,(y) can be obtained from the experimental relaxation curves. In stress growth experiments the strain rate is not instantaneously constant\either. The real 9(t) is measured continuously and deviates substantially from the prescribed shear rate at times smaller than about 0.3 s. In the linear vi coelastic range a’( t, q)/+ and N,+( t, +)/jl’ are independent of -+, and theri fore these quantities are calculated correctly by using the actual y(t). When a steady shear flow is stopped, the actual cessation time can be read from the position control unit. This time is taken as time zero in the subsequent stress decay measurement. 4.3 Experimental

results

Oscillatory measurements For viscoelastic fluids, the most sensitive and least arbitrary method for checking the appropriateness of time-temperature superposition for the results of isothermal oscillation experiments is based on the use of curves of the phase shift S vs log w, because S is not affected by variables such as sample dimensions, thermal expansion, vapour inclusions, etc., which irifluence the values of mod&h. If temperature changes shift these curves along the log w axis without altering their shape or height, the horizontal shifts with respect to the curve at a reference temperature T, are denoted by log aT. According to the Kramers-Kronig relations and eqn. (23), the curves of log G,, vs. log o do not change in shape with temperature either, and have to be shifted horizontally just as much, leaving the possibility for an additional vertical shift log b,. If the log Gd vs log w curves can be superimposed on the curve at T, by the horizontal shifts log a,. and vertical shifts log b,, the time-temperature superposition principle is said to hold. In this way, the master curves at a chosen reference temperature have been obtained. From these data, the relaxation spectra H( 7) shown in Fig. 2 are calculated by means of an iteration procedure [43] and the recalculated curves of 6( oa,) and Gd( oaT) fit the original points excellently. The horizontal shift factors aT can be fitted by Arrhenius equations with thermal activation energies of 62 and 42 kJ/mol for the LDPE and HDPE samples, respectively. Indeed, the spectra for the two LDPE samples are very similar,

206

106

’ ‘. ---.-.

106

HDPE 1 HDPES LOPE1 LDPE 2 463 K

104 103 102 10'

100

10-6

1o-5

Fig. 2. Relaxation

10-4

10-3

time spectra

10-2

10-l

100

10'

102

1 103

of the melts investigated.

but we prefer the examination duplication of one of them.

of these two samples

to, for instance,

the

Stress growth experiments at constant shear rate fiow The build-up of the shear stress a+(t, +) and of the first normal stress difference N:(t, q) has been measured after the inception of various constant shear rates. As mentioned already, the data on Ni+( t, +) are more sensitive to variations in the nonlinear strain measure than those on o+( t, +). Therefore the quantity G(t)-’ d N:( t, y)/dy will be analysed primarily. At numerous points in time, values of N:( t, -j) have been measured. Parts of the double logarithmic plots of N:(t, y) vs time at constant y around a time ti are fitted with a polynomial of the third degree, taking ti as the origin, and this procedure is repeated in the next step around a time 1.585ti, etc. In this way, called ‘floating fitting’, the slope dN:(t, +)/dt is calculated and, using eqn. (14), the quantity G(t)-’ d Ni+( t, +)/dy as function of y at constant + is obtained. This quantity exhibits very specific differences in behaviour in HDPE compared with LDPE melts. In Fig. 3, it is shown at a fixed value of y, say y = 5, for two melts. The HDPE 2 sample shows a time-independent value, whereas for the LDPE 1 sample strongly time-dependent values of this quantity are found. This means that for an HDPE melt, eqn. (20) with G, = 0 can be used, but that for the LDPE melt a G, term will unavoidably be necessary. The nonlinear strain measure S,, - S,, for the HDPE melts obtained from this evaluation of the stress growth can be very satisfactorily described

207

X HDPE 2 + LDPE 1

Fig. 3. Slope of the growth curve of the first normal stress difference at various shear rates at y = 5, divided by the modulus, as a function of time. Symbols: experimental data; curves: calculated with eqn. (30), see text.

by an empirical equation proposed by Soskey and Winter [36]: s,,-s*2= 1

+y2 ayh ’

which covers a number of equations previously suggested. Table 2 offers a survey of the values of the parameters a and b which have been derived by

TABLE

2

Parameters

for the strain measure

S,, = y(l+

ay’)-’

u

b

Fluid

Data/analysis

References

0.222 0.111 0.0066 0.070 0.095 0.199 0.172 0.302 0.145 0.128 0.071 0.145 0.083 0.071

2 2 3 2 2 2 1.39 1.57 1.9 1.8 1.7 1.9 1.7 1.7

PIB solutions PS solutions PIB solutions LDPE melt PS melt PDMS melt LDPE melt PS melt PP melt HDPE 2 melt LDPE melt HDPE 1 melt LDPE 1 melt LDPE 2 melt

Zapas Adams and Bogue Phillips Wagner/Papanastasiou van Aken/Papanastasiou Papanastasiou Soskey and Winter Soskey and Winter Booij et al/this work Booij et al/this work Booij et al/this work this work this work this work

[601 [611 1471

W/631 W/631 [631 [361 1361 PI PI PI

208

analysis of literature data. The Doi-Edwards measure 5 (Q,, - Q.“,) is very closely approximated by eqn. (40) with u = 2/9 and b = 2. For polyolefins, a value of b less then 2 is most suitable. This function implies that a(y) = (1 + .yh)“2. The analysis of data on NC ( t, y) for the LDPE melts by means of eqn. (30) is much more complicated. In view of the number of parameters to be fitted, some extrapolations of experimental data and auxiliary graphs are employed, on the basis of which the following sets of values is estimated: a = 0.083 and b = 1.7 for LDPE 1 and a = 0.071, b = 1.7 for LDPE 2. Then the quotient G,(t)/G(t) is calculated and by trial and error, using a HP 9836 calculator, suitable values of the parameters k and 1 in eqn. (16) are found quite quickly, i.e. k = 2.8 and 1= 0.78 for both LDPEs. Stress decay experiments after cessation of steady shear flow It has been stated earlier [2] that the decay of the first normal stress difference after cessation of steady flow at a constant shear rate 3, hence the function Nlp(t, y), is the transient rheological quantity to hand that is the most sensitive to the nonlinear strain tensor S+(y). This can be most clearly indicated by the values of the first normal stress coefficient +r = NJ?* at a shear rate y of say 8 s-l for the HDPE 1 sample. In the stress growth experiment, +!J:( t, +) is equal to the linear value at t = 0.1 s; at 1 s it is about a factor of 3 below the FLV response and this factor amounts to about 50 in the steady state, say at t > 10 s. However, in a stress decay experiment 1 s after stoppage of this flow the value of $;( t, +) is already a factor of 400 below the FLV limit. (For the shear stress coefficient 77= a/q, these factors are only about 1; 1.2; 1.5 and 20 respectively). On the other hand, a number of trial calculations have demonstrated that for the calculation of $J;( t, y) the exact shape of the relaxation spectrum in the long-time region is also crucial. This part is difficult to obtain experimentally and can only be approximated by trial and error. As mentioned already, the stress decay data cannot be used to calculate the nonlinear strain measure. Nevertheless, they are very well suited for checking the rheological equation (37), which is equivalent to -

This relation appears to hold for all four melts within the experimental inaccuracy, which is quite large, namely in the order of + 25%. Systematic deviations are not found, and this is taken as an indication that the rheological constitutive equation (20) may be employed for the melts investigated.

209 4.4 Theory compared with experimental

data

All parameters needed for the description of the stresses in unidirectional transient shear flow, i.e. H(7), a, b, k and I, are determined for the four polyethylene melts and the stresses can now be calculated by means of eqns. (24), (25), (28), (35) and (36). In Fig. 4, the experimental data (symbols) of the build-up and of the decay of the normal stress coefficient at some of the investigated values of the constant shear rate are shown for the HDPE 1 sample at 463 K. They agree almost perfectly with the curves calculated with the relaxation spectrum of Fig. 2 and the values a = 0.145 and b = 1.9. The upper curves are the FLV limits, showing clearly the nonlinear viscoelastic nature of this rather low molecular and only slightly non-Newtonian fluid. The corresponding data for the HDPE 2 melt are presented in Fig. 5, together with the curves calculated with a = 0.128 and b = 1.8. At higher shear rates, some necking occurs even before reaching a steady state, giving rise to stress decay data which are too low. For the rest, the description is good. This higher molecular, more strongly non-Newtonian fluid shows much stronger nonlinear viscoelasticity than HDPE 1, particularly in K(t, ?). The growth and decay of the viscosity of the HDPE 2 melt is displayed in Fig. 6. As a whole, the agreement is agian satisfactory, only ihe experimental data of q- at small shear rates (not all shown) are significantly lower than the calculated curves. For q+( t, y) both the experimental and calculated

104

104 FLV 103

103

0.667 1.067

102

g6

102

113

4

10’

9 in 5-l

ri.“l 10-2

10’

t in s 10-l

100

10’

102

10-2

10-l

100

10’

102

Fig. 4. Growth and decay of first normal stress coefficient at various shear rates for HDPE 1 melt. The curves are calculated and the upper ones represent the finite linear viscoelastic limits.

8::;;

HDPE

1 2,,463 K

UJi(t,f) in Pa s2

_

0.8 1.25 2.0

in s-‘-lo2

lo22Jq 10-I

100

10'

102

103

10-Z

10-l

100

10'

102

Fig. 5. First normal stress growth and decay coefficients at various shear rates for HDPE 2 melt. tHJPE 2,,463 K

q-(&p in Pa ;klo4

(

103 -1 \

“\

-tin 10-2

10-l

ix\ 100

\\

f 10'

\

:l IO2 --T 102

Fig. 6. Growth and decay of shear stress coefficients at various shear rates for HDPE 2 melt. FLV

104

?!z 1:687 Z*Y7

103

6:667 12.5 t

102

C in s-l 10'

Fig. 7. Growth and decay of first normal stress coefficient at various shear rates for LDPE 1 melt, including the calculated finite linear viscoelastic response.

211

104

LDPE

2,443 K

103

102

IO' 10-2

10-l

Fig. 8. Experimental for LDPE 2 melt.

100

and calculated

10'

stress relaxation

curves after various

step-shear

strains

curves show a slight overshoot at all shear rates, even at values as low as 0.042 s-l. Comparison of Figs. 5 and 6 demonstrates the well-known facts that the shear stress decays faster than the normal stress and that the decay rates increase according as the preceding shear rate was higher. In the present theory this has nothing to do with a disruption of the longest relaxation time mechanisms or other modifications of H(r), but is only caused by the nonlinearity of S,,(y). For the LDPE melts, we need both integrals for a reasonable prediction of transient rheological behaviour. Some of the results are shown in Fig. 7 for LDPE 1. The upper curves again represent the FLV response and at medium shear rates the data can very nicely be described by theory, with a = 0.083, b = 1.7, k = 2.8 and I = 0.78. The same is true for the LDPE 2 melt, but then with a = 0.071. At higher rates, the starting values of $;(t, ?) are too low owing to necking; moreover, the decay is experimentally slower than predicted theoretically. Whether these two effects are linked or not is obscure at present, but the possibility of healing of the neck and a relative increase in the stresses cannot be excluded. Finally, the stress relaxation after step strain is also calculated. As an example, Fig. 8 depicts the results for LDPE 2 at 443 K. The calculated curves are not parallel, owing to the second term in eqns. (24) and (25): for y = 15.53 the value of the modulus a/y is a factor of 4 below the FLV limit at t = 10-l s and a factor of 8 at 10’ s. Experimentally, the Lodge-Meissner

212

relation (27) measurements ideal, but in experiment it correction of instead of the

is very well fulfilled and the averaged values of several are introduced in Fig. 8. The agreement is certainly far from view of the insensitivity and inaccuracy of this type of is impossible to conclude that the theory is inapplicable. The the experimental results for the ramp function excitation step strain with the help of eqn. (39) comes up to expectations.

5. Conclusions

Tube renewal effects have to be taken into account to predict the transient viscoelastic properties of commercial highly polydisperse HDPE melts. A single integral constitutive equation with a factorizable integrand containing an empirical relaxation spectrum H(r) and a strain measure with two material parameters can provide a good description for all measured transient rheological features. Tube renewal effects are supposed to be dominant in LDPE melts and tube relaxation effects are of paramount importance in the nonlinear range. Both effects are due to the reptation of parts of the matrix molecules and therefore give rise to identical intensity functions H( 7) in the time dependent parts of the two integrals. No direct method has been found for the determination of the nonlinear strain measure for these materials, but after some trial and error, values of the parameters were obtained such that the constitutive equation with two integrals very satisfactorily described all transient properties measured on the two (rather similar) LDPE melts. These intermolecular interactions strongly hamper the search for relations between viscoelastic properties and molar mass distribution or other molecular structure characteristics such as long-chain branching. In our opinion, tube renewal is the main reason that even the linear viscoelastic properties of simple blends of anionic polystyrenes cannot be described [14,15] by the original Doi-Edwards theory. For linear polydisperse materials, eqns. (13) and (14) in principle couple H( 7) to cp( N), utilizing the formulas mentioned for rcN and 7ren, but tractable relations are not very evident and call at least for further research. For LDPEs the relations between r, N’, ‘p and branching structure are so complicated that there is little hope that linear viscoelastic characterisation will provide reliable molecular mass distribution data. References 1 H.C. Booij and J.H.M. Palmen, in G. Astarita, G. Marrucci and L. Nicolais (Eds.), Rheology, Vol. 2: Fluids, Plenum Press, New York, 1980, p. 501. 2 H.C. Booij, J.H.M. Palmen and P.J.R. Leblans, *in B. Mena, A. Garcia-Rejon and C. Rangel-Nafaile (Eds.), Advances in Rheology, Vol. 3: Polymers, Mexico, 1984, p. 367. 3 H.C. Booij and J.H.M. Palmen, Rheol. Acta. 21 (1982) 376.

213 4 H.C. Booij, P. Leblans, J. Palmen and G. Tiemersma-Thoone. J. Polym. Sci., Polym. Phys. Ed., 21 (1983) 1703. 5 P.J.R. Leblans, J. Sampers and H.C. Booij, Rheol. Acta, 24 (1985) 152. 6 SF. Edwards, Proc. Phys. Sot., 92 (1967) 9. 7 P.G. de Gennes, J. Chem. Phys., 55 (1971) 572. 8 M. Doi and SF. Edwards, J. Chem. Sot., Faraday Trans. II, 74 (1978) 1789, 1802, 1818; 75 (1979) 38. 9 H. Janeschitz-Kriegl, Polymer Melt Rheology and Flow Birefringence, Springer Verlag, Berlin, 1983. 10 K. Osaki and M. Doi, Polym. Eng. Rev., 4 (1984) 35. 11 M. Doi, J. Polym. Sci., Polym. Phys. Ed., 18 (1980) 1005. 12 Y.-H. Lin, J. Rheol., 28 (1984) 1. 13 E.V. Menezes and W.W. Graessley, J. Polym. Sci., Polym. Phys. Ed., 20 (1982) 1817. 14 A. Schausberger, G. Schindlauer and H. Janeschitz-Kriegl, Rheol. Acta, 22 (1983) 550; Chem. Eng. Commun., 32 (1985) 110. 15 W. Pfandl and F.R. Schwarzl, Colloid Polym. Sci., 263 (1985) 328. 16 J. Klein, Macromolecules, 11 (1978) 852. 17 M. Daoud and P.G. de Gennes, J. Polym. Sci.. Polym. Phys. Ed., 17 (1979) 1971. 18 W.W. Graessley, Adv. Polym. Sci., 47 (1982) 67. 19 J.P. Montfort, G. Marin and P. Monge, Macromolecules, 17 (1984) 1551. 20 G. Marrucci, J. Polym. Sci., Polym. Phys. Ed., 23 (1985) 159. 21 B.A. Smith, E.T. Samulski, L.P. Yu and M.A. Winnik, Phys. Rev. Lett., 52 (1984) 45. 22 W.W. Graessley and S.F. Edwards, Polymer, 22 (1981) 1329. 23 T. Masuda, K. Kitagawa, T. Inoue and S. Onogi, Macromolecules, 3 (1970) 116. 24 N.J. Mills and A. Nevin, J. Polym. Sci., A-2,9 (1971) 267. 25 K. Murakami and K. Ono, J. Polym. Sci., Polym. Lett. Ed., 10 (1972) 593. 26 W.M. Prest and R.S. Porter, Polym. J., 4 (1973) 154. 27 J.P. de Gennes, J. Phys. (Paris), 36 (1975) 1199. 28 M. Doi and N.Y. Kuzuu, J. Polym. Sci., Polym. Lett. Ed., 18 (1980) 775. 29 D.S. Pearson, S.J. Mueller, L.J. Fetters and N. Hadjichristidis. J. Polym. Sci., Polym. Phys. Ed., 21 (1983) 2287. 30 J. Roovers, Macromolecules, 17 (1984) 1196. 31 J. Klein, D. Fletcher and L.J. Fetters, Nature, 304 (1983) 526. 32 J. Klein, D. Fletcher and L.J. Fetters, Faraday Symp. Chem. Sot., 18 (1983) 159. 33 J.L. Viovy, L. Monnerie and J.F. Tassin, J. Polym. Sci., Polym. Phys. Ed., 21 (1983) 2427. 34 V.R. Raju, H. Rachapudy and W.W. Graessley. J. Polym. Sci., Polym. Phys. Ed., 17 (1979) 1223. 35 C.J.S. Petrie, J. Non-Newtonian Fluid Mech., 5 (1979) 147. 36 P.R. Soskey and H.H. Winter, J. Rheol., 28 (1984) 625. 37 G. Marrucci and J.J. Hermans, Macromolecules, 13 (1980) 380. 38 B. de Cindio, Polymer, 25 (1984) 1049. 39 C.F. Curtiss and R.B. Bird, J. Chem. Phys., 74 (1981) 2016, 2026. 40 H.C. Booij and G.P.J.M. Thoone, Rheol. Acta, 21 (1982) 15. 41 B. Bernstein, Int. J. Non-Linear Mech., 4 (1969) 183. 42 H.C. Booij, J.H.M. PaImen and B.J.R. Scholtens, Preprints 27th IUPAC Symp. on Macromolecules, Strasbourg, 1981, p. 791. 43 B.J.R. Scholtens and H.C. Booij, in J.E. Mark and J. La1 (Eds.), Elastomers and Rubber Elasticity, ACS Symp. Series No. 193, Amer. Chem. Sot., Washington, DC., 1982, p. 517. 44 A.S. Lodge, J. Non-Newtonian Fluid Mech., 14 (1984) 67.

214 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64

E.A. Kearsley and L.J. Zapas, Trans. Sot. Rheol., 20 (1976) 623. K. Osaki, Proc. VIIth Int. Congr. on Rheology, Gothenburg, 1976, p. 104. M.C. Phillips, J. Non-Newtonian Fluid Mech., 2 (1977) 109. E.V. Menezes and V.W. Graessley, Rheol. Acta, 19 (1980) 38. M. Yamamoto, Appl. Polym. Symp., 20 (1973) 3. E.V. Menezes, J. Non-Newtonian Fluid Mech., 7 (1980) 45. C.M. Vrentas and W.W. Graessley, J. Rheol., 26 (1982) 359. J. Meissner, Rheol. Acta, 10 (1971) 230. M.H. Wagner, Rheol. Acta, 15 (1976) 136. H.M. Laun, Rheol. Acta, 17 (1978) 1. P. Attane, P. Le Roy and G. Turrel, J. Non-Newtonian Fluid Mech., 6 (1980) 269 J. Michele, Rheol. Acta, 17 (1978) 59. S. Pedersen and L.L. Chapoy, J. Non-Newtonian Fluid Mech., 3 (1978) 379. L.J. Zapas and J.C. Phillips, J. Res. Nat. Bur. Stand., A75A (1971) 33. K.S.C. Lin and J.J. Aklonis, J. Appl. Phys., 51 (1980) 5125. L.J. Zapas. J. Res. Nat. Bur. Stand., 70A (1966) 525. E.B. Adams and D.C. Bogue, AIChE J., 16 (1970) 53. M.H. Wagner and J. Meissner, Makromol. Chem., 181 (1980) 1533. A.C. Papanastasiou, L.E. Striven and C.W. Macosko, J. Rheol., 27 (1983) 387. J.A. van Aken and H. Janeschitz-Kriegl, Rheol. Acta, 20 (1981) 419.