Experimental validation of non linear network models

Rheology for Polymer Melt Processing J-M. Piau and J-F. Agassant (editors) 9 1996 Elsevier Science B.V. All rights reserved.

141

Experimental validation of non linear network models

Christian CARROT, Jacques GUILLET, Pascale REVENU, Alain ARSAC Laboratoire de Rhdologie des Mati~res Plastiques. Universitd Jean Monnet. Facultd des Sciences et Techniques. 23 Rue Dr Paul Michelon. 42023- Saint-Etienne. Cedex 2. FRANCE

1.INTRODUCTION 1.1.Constitutive and model equations for computer simulation of complex flows. Nowadays, industrial and research applications of computers are widely used in the field of polymeric materials. Computer aided rheology and simulation of processing operations involving the flow of polymeric liquids have received growing attention for the understanding of the physics of the processes, for the design of the equipments and for control purposes. The theoretical foundation of this software is provided by principles of continuum mechanics, together with improved numerical methods for the solution of the mathematical equations and by the use of pertinent constitutive equations for the description of the theological behaviour of molten polymers. Most flows encountered in processing or experimental problems are characterized by complex kinematics involving different geometries combining shear and elongation, different time dependences and amplitudes of the deformations. Moreover, the behaviour of polymeric materials under such conditions exhibits a large variety of specific features such as the shear rate dependence of the viscosity, appearance of norma| stresses, memory effects and high resistance in elongation. Among the large m~nber of existing models, only a relatively few equations can predict the variety of phenomena encountered in the flow of polymer melts and can give a fair description of the physics involved in the theological

142 behaviour of these materials, both from a microscopic and a macroscopic point of view. From the physicist's and chemist's points of view, the ideal constitutive equation should be able to describe the behaviour of polymers in any flow situation without any adjustable parsmeter. Linear and non linear viscoelastic characteristics of the material should be theoretically described from molecular dynamics, knowing the basic properties of the polymer molecules, from the monomeric unit to the molecular weight distribution as obtained from the polymerization process. Many advances have been performed in this sense in the past few years (see Section 1.5), however the complexity of the equations that might be obtained to take into account these various features may lead to numerical difficulties. Indeed, the computation of complex flows, such as those encountered in typical processing conditions, sets its own practical requirements on the constitutive equation, especially when reasonable computation times and memory requirements are expected. The numerical properties of the equation generally do not match those of the physicist. In this sense, for example, the use of a large number of relaxation modes or of a continuous spectrum to describe the memory function is time and memory consuming and the economic spectrum using only a few contributions, though unsatisfactory from the molecular dynomics point of view, remains the rule. Thus, this constitutive equation is bound to be replaced by an unsatisfactory but easy to handle model equation which involves a minimum of violation of basic principles of material physics. This equation will necessarily contain a few adjustable material parameters, which have to be easy to determine in a limited number of well defined flow experiments. Obviously, the minimum requirement that could reconcile the two stand points should be the ability of the model equation to describe properly the response of the polymer to simple viscometric flows (simple shear, uniaxial, biaxial and planar extension) in which quite perfectly controlled conditions can be obtained. From this, it can be hoped that the situation might be at its best in the combined complex flow. The aim of this section is to perform comparisons between the predictions of some constitutive equations and experimental results in simple shear and uniaxial elongation on three polyethylenes. In addition, this is expected to provide well-defined sets of material parameters to be used in the model equations for the computation of complex flows.

143 L2.Network theories for polymer melts and related models. Among the various approaches in use for the depiction of the interactions of the polymer molecules in the melt, these being known to be at the origin of the observed theological behaviour, the network theories enable the building of reasonable models t h a t fulfill the previous requirements for the sake of simplicity. These theories are based on the classical theories of rubber elasticity of maeromoleeular solids, wherein permanent chemical crosslinks connect segments of molecules, forcing them to move together. This central idea can be applied to polymeric liquids. However in this case, the interactions between molecules are assumed to be localized at junctions and are supposed to be temporary. Whatever their nature, physical or topological, these crosslinks are continually created and destroyed but, at any time, they ensure sufficient connectivity between the molecules to give rise to a certain level of cooperative motion. The stress is considered to be the sum of the contributions of segments between junctions that are still existing at the present time. By the way, these segments, t h a t were created in the past, may be of different ages and complexities. This time dependence is generally described by a relaxation spectrum t h a t gives rise to the linear rheologieal behaviour. Strain dependence, at the origin of nonlinearity, can be described either by changing the motion of the network relative to the continuum or by special rates of creation and loss of junctions. Owing to their relatively fair tractability and because they retain some physical consistency, network models are widely used in computer simulation of the flow of polymer melts. Thus, the attention of the present article is focused on constitutive equations of this type. l~.Integral and differential forms of the models. At this point, the question of the use of either an integral or a differential equation arises. Integral forms are closer to those obtained by the recent molecular dynamics concepts for entangled polymer melts. Unfortunately, their use requires the knowledge of the Finger strain tensor in complex kinematics, which together with the fluid memory, involve the description of the material history. This, in turn, sets the difficult task of particle tracking. Though it has been difficult to cope with, alternative descriptions (Protean

144 coordinates) and new ideas (such as those of the streAm-tube, see Section III-2) enlighten the subject and bring new hope for this kind of equation. In this sense, differential equations appear more tractable since they do not require particle tracking. Indeed, the solution of the coupled equations of mass, momentum and energy balance including the material equation, properly described on a suitable finite element mesh, theoretically provides the material lines. Nevertheless, the correct description of the basic experiments ot~en requires the use of strong nonlinear terms. Such improvements may be unsatisfying from the numerical point of view since they can lead to stiff systems of nonlinear equations and to many convergence related problems. Considering these previous remarks, two network models, thought to be representative of each class of equation, have been investigated, namely the Wagner model and the Phan-Thien Tanner model, 1.4.Experimental validation of network model~ P a r t 2 presents a summary of the theoretical considerations and basic assumptions t h a t lead to the model equations. Part 3 discusses some experimental aspects and focuses on the measurements in various shear and uniaxial elongational flow situations. Part 4 and 5 are devoted to the comparisons between experimental and predicted rheological functions. Problems encountered in the choice of correct sets of parameters or related to the use of each type of equation will be discussed in view of discrepancies between model and data. 2~THEORETICAL ASPECTS 2.1.The basic integral and differential non-linear constitutive equation~ 2.1.1.The linear M~xwell model and its limits.

Constitutive equations of the Maxwell-Wiechert type have received a lot of attention as far as their ability to describe the linear viscoelastic behaviour of polymer melts is concerned. From a phenomenological point of view [1-4], these equations can be easily understood and derived using the multiple springdashpot mechanical analogy leading to the linear equation :

145

~i(t) + ~ ~

= Th~ and ~=(t) = .~ ~(t)

(1)

1

where

~i(t) is the contribution of the ith Maxwellian assembly (spring and

dashpot in series) to the extra-stress tensor ~(t), d ~-~is the tensor time derivative for small displacements, ~ = Vu + (Vu) t is the rate of strain tensor (u being the fluid velocity), is the relaxation time of the ith element, Tli = gi ~ is the viscosity contribution of the ith element, gi is the modulus contribution of the ith element. This differential form can be integrated to give the integral form of the model which can also be derived from the Boltzman superposition principle using the concept of fading memory of viscoelastic liquids: t

~(t) =- j re(t- t') dt(t') dt'

(2a)

.-OO

or

t ~=(t)= j G ( t - t ' ) ~ dt'

(2b)

-00

where

Th t m(t) = ~ ~.2 exp{-~.} is the memory function, Th t din(t) G(t )= ~ ~. exp{-~.} =- dt is the relaxation modulus, dt(t') is the infinitesimal strain tensor (t being the reference of the

deformation, so that the strain at time t' is relative to that at time t ). Satisfactory agreement is achieved from these equations for depiction of the main features of linear viscoelastic properties that can be obtained with the experimental tools, either in transient or in oscillatory rheometry. These equations are all the more attractive in that similar mathematical forms can also be obtained from molecular considerations for the description of

146 the flow behaviour of non-entangled [5] and entangled [6-10] melts at least in the case of narrow molecular fractions, so that any parameter in equations (1) or (2) becomes physically meaningful. However, the latter approach leads to complex relaxation spectra with a large number of Maxwellian modes (and maybe even more complex if polydispersity and branching are taken into account). Recognizing that only one or two modes per decade are generally sufficient to get a proper description of the linear viscoelastic behaviour [11], such an economic spectrum can be numerically adjusted [11-17] and is often used, while keeping in mind that, in this case, one loses the physical meaning of such modes. Though a simple Maxwell model in the form of equations (1) and (2) is powerful to describe the linear viscoelastic behaviour of polymer melts, it can do nothing more than what it is made for, that is to describe mechanical deformations involving only infinitesimal deformations or small perturbations of molecules towards their equilibrium state. But, as soon as finite deformations are concerned, which are typically those encountered in processing operations on polymers, these equations fail. For example, the steady state shear and elongational viscosities remain constant throughout the entire rate of strain range, normal stresses are not predicted. 2.1.?~The ~ e

and UCM models.

One major discrepancy of the previous model can be attributed to the use of the infinitesimal strain tensor and to derivatives restricted to time changes. Indeed, in the case of large deformations, one has to refer to finite strain tensors, such as the Finger c t ' l ( t ') or Cauchy Ct(t') strain tensors (t being the present time as a reference of the deformation), and to derivatives involving time and space changes known as convected derivatives. Making these modifications, the integral form of the equation of state becomes: t ~(t) =-p'~ + ~ r e ( t - t ' ) C t l ( t ') d t ' = - p ' ~ + ~=(t) -OO

where

~ is the total stress tensor, 5=is the unit tensor,

(3)

147

p' is an arbitrary isotropic term. For an incompressible liquid, because of the arbitrary term, normal stresses are known only up to a constant using the constitutive equation. This equation was proposed by Lodge [18, 19] in the case of entangled polymer molecules, considering that the response of polymers to flow is that of a temporary network of junctions. The strands of the network can be of different topology (i) but they are considered as Gaussian springs that deform like the macroscopic continuum. This latter a s s u m p t i o n of affine motion m e a n s t h a t any s t r a n d end-to-end vector, initially coincident with a macroscopic vector embedded in the continuum, remains parallel to and of equal length with it during deformation. The s t r a n d s are continuously destroyed and rebuilt by thermal effects only. Assuming t h a t the rates of creation and destruction of the ith segment are constant, an equation for the memory function similar to that of equation (2) can be written, with viscosity contributions Tli connected to the stiffness of any ith spring and to its rate of creation, and relaxation time ~i connected to the survival probability of the ith junction. This formulation provides the way to cope with the economic spectrum discussed previously as a crude physical description of the entangled melt. The Lodge equation can also be obtained in a differential form known as the Upper Convected Maxwell equation (UCM): ~i(t) + ~ ~i(t) = Tli ~

and ~=(t) = .~ ~(t)

(4)

1

where ~i(t) is the upper convected derivative of the contribution of the ith mode to the extra stress tensor ~=(t). It should be noted that this is the exact derivation of equation (3). These models are usually referred to quasi-linear models and display qualitatively correct predictions of typical phenomena of elongational flows such as the occurrence of the strain-hardening effect in transient extension. Nevertheless the predicted elongational viscosity is never bounded in the long time range and a steady state value can only be expected for small elongation rates. Moreover, the shear behaviour r e m a i n s unrealistic as compared to the experiment, especially because of constant predicted viscosity and first normal

148 stress coefficient. The cure of these discrepancies either requires much more complexity in the formulation of the constitutive equation or an additional attention to some characteristics of the viscoelastic material that might only be displayed in nonlinear flows. 2.2.An i n t e g r a l constitutive equation: the Wagner modeL

2.2.1.The general K-BKZ model. Additional complexity can be brought to the constitutive equation in its integral form. Indeed, the idea of rubber elasticity that is inherent to the Lodge model has been generalized by Kayes, Bernstein, Kearsley and Zapas [20-23] in a large class of constitutive equations. In a perfect body, the strain energy W may be linked to strain and stress by:

-

8W be

(5)

1 where e= is a diagonal strain tensor (Hencky strain tensor) such as ~== ~ In C "1.

This can also be rewritten in terms of the Cauchy and Finger strain tensors as:

1: =2~IW1 C -t 8W =

= "~I2 C=}

(6)

Since W is a scalar value, it depends on scalar characteristics of the strain tensors, namely on the invariants I1 and I2 defined as: I1 = tr[C 1]

1

I2 = ~ {(tr[C

(7a)

-1])2

- tr[C2]} = tr[C]

(7b)

In r u b b e r and viscoelastic fluids, these two quantities are sufficient since, when incompressibility is taken into account, I3 = 1. For viscoelastic fluids, both strain energy and stress can be assumed to depend on the strain history through the strain invariants:

149

t W = ~ u ( t - t', I1, I2) dt'

(8)

-OO

and thus:

~(t) = ~ 2 {

C t l ( t ' ) " ~22 Ct(t')} dt'

(9)

-00

or

t ~(t) = ~ {Ml(t- t', I1, I2) c t l ( t ' ) - M2(t- t', I1, I2) Ct(t')} dt'

(10)

--OO

where M l ( t - t', I1, I2) and M2(t- t', I1, I2) are m a t e r i a l functions. The K-BKZ has the interesting property being t h e r m o d y n a m i c a l l y consistent because one can theoretically derive two material functions which depend on a single potential function u in the general form: Mi(t- t', I1 I2) = 2 8u(t- t', I1, I2) ~Ii

(11)

'

The Lodge model is a special case of this class of models, where the material functions are selected as: M l ( t - t', I1, I2) = m ( t - t')

(12a)

M2(t- t', I1, I2) = 0

(12b)

1 u ( t - t', I1, I2) = ~ m ( t - t') (I1- 3)

(13)

However, although it has some t h e r m o d y n a m i c consistency, the l a t t e r model failsto describe the non linear viscoelastic b e h a v i o u r properties, especially in shear, wherein the shear-thinning behaviour of the viscosity and of the normal s t r e s s coefficients are not predicted. As a c o n s e q u e n c e , m o r e complex

150 n o n l i n e a r functions have to be introduced to get a proper description of the observed behaviour. 22..2.Time-strain separability. Some experimental features might simplify the problem. Considering t h a t in step shear strain of constant ,mplitude 7 starting from the material at rest, the K-BKZ model leads to the following equation for the shear stress: t 9 (t, 7) = ~ {Mi(t - t', Ii, I2) - M2(t- t', Ii, I2)} 7 dt' 0

(14)

and since, in this case: Ii=I2=3+

T2

(15)

one can write" t 9 (t, 7) = ] M(t- t', Y)7 dt' 0

(16)

The nonlinear relaxation modulus is then obtained as:

G(t, T) = Z(t, 7) 7

(17)

As a l r e a d y mentioned by various authors [24, 25], it is found experimentally t h a t at various shear strains, for polydisperse polymers, the logarithmic plots of G ( t , 7) v e r s u s time are only shifted vertically. This indicates t h a t the nonlinear relaxation modulus might be factorizable: G(t, 7) = G(t ) h(7)

(18)

where h is a strain-dependent function, between zero a n d unity, called the d a m p i n g function. This t i m e - s t r a i n s e p a r a b i l i t y seems to hold in the experimental range of shear strains. From a theoretical point of view, this can only be achieved if u ( t - t', Ii, I2) is also factorizable:

151

u(t - t', I1, I2) = m ( t - t') U(I1, I2)

(19)

As a consequence, the K-BKZ model is now written in the less general form of the factorizable K-BKZ model: t 8U 8U ~(t) = ~ 2 m ( t - t') {~-~1ct'l(t') " ~ 2 Ct(t')} dt'

(20)

-00

or

t ~(t) = ~ re(t- t') {hl(I1, I2) c t l ( t ' ) - h2(I1, I2) Ct(t')} dt'

(21)

-OO

It is worth mentioning that the strain function is not t e m p e r a t u r e dependent and t h a t the influence of temperature is only applied on the memory function or relaxation modulus through the shortening of the relaxation times with increasing t e m p e r a t u r e s . 2.2.3.The Wagner m o d e l Since second normal stresses are generally difficult to obtain from the experimental point of view, it may seem attractive to cancel the Cauchy term of the K-BKZ equation setting h2(I1, I2) = 0 and to find a suitable material function

hl(I1, I2). Wagner [26] wrote such an equation in the form: t 9=(t) = ~ m ( t - t') hl(I1, I2) C t l ( t ')

(22)

-OO

Cancellation of the Cauchy term may bring some discrepancies, the more evident one being that, whatever h is, it leads to a zero second normal stress difference. A more subtle one concerns the loss of the t h e r m o d y n a m i c consistency of the model. Indeed, it is not possible to find any potential function in the form U(I1, I2) with h2(I1, I2) = 0 unless hi only depends on I1. As mentioned by Larson [27, 28], this can induce violation of the second principle in complex flows such as those encountered in processing conditions.

152 22~4.The g e n e r a l i z e d i n v a r i a n t The form of the function h can be described from shear experiments. Indeed, since in these experiments" I1 = I2 = 3 + 7t2(t')

(23)

The mathematical form of the function can be derived simply from a fit of the experimental h(T) as obtained in step shear strain for example. However, the problem is further complicated if one now takes into account flows where the two invariants differ from each other as, for example, in uniaxial elongational flows where: I1 = exp{2et(t')} + 2 exp{-et(t')}

(24a)

I2 = exp{-2et(t')} + 2 exp{et(t')}

(24b)

In order to conciliate these features, Wagner [29, 30] proposes the use of a generalized invariant: I = ~ I1 + (1-~) I2

(25)

which enables the derivation of a single form of the damping function, since in s h e a r flows I = I1 = I2. It should be pointed out t h a t one m a y find other generalized invariants which satisfy this condition. These have been proposed by various authors but are less frequently used, for example: I = (I1)a(I2)1~

(26)

In any case, the new parometer of the generalized invariant has now to be obtained from additional experiments in elongational flows. But the Wagner equation can now be written in the unified form:

~=(t) =

t ] m ( t - t') h(I) c t l ( t ') -OO

(27)

153 In terms of network description, Wagner considers t h a t the d a m p i n g function may reflect an additional process of destruction of network junctions by strain effects, as described by the generalized invariant, thus involving some peculiarities of the flow such as its geometry and the s t r a i n intensity. As regards the rate of creation of network junctions, it is a s s u m e d to remain constant as in the Lodge model. 2~2.&Interpretation a n d mathemotical forms of the d a m p i n g functiom In its form, the model of equation (27) is very useful since using a proper damping function enables a correct description of various shear or uniaxial elongational basic experiments. Going back to the Lodge model, Wagner keeps the assumption of constant creation rate (connected with Tli for the ith segment in the memory function) but assumes t h a t the loss probability of junctions is now a combination of two independent mechanisms. One is related to Brownian motion as in the case of the Lodge model and depends on the time elapsed between the creation of an entanglement and the present time (t- t'), the survival probability being related to 1/ki. The second reflects the network breaking by deformation and depends on the kinematics of the flow and not on the segment configuration, it is expressed as a survival probability between times t' and t. Since the processes are assumed to be independent, the total loss probability is j u s t the sum of the former and this leads to a separable nonlinear memory function : M(t- t') = m ( t - t') h(I).

(28)

Wagner proposes a single exponential form of the damping function: h(I) = exp(-n~] I- 3).

(29)

The exponential form is interesting because, in shear, the response of the model can be analytically derived. However because of the exponential, it decays very rapidly, even at low deformation and therefore it cannot take into account the linear viscoelastic domain, which is sometimes found to extend to relatively high values of the strain (typically 0.5). Another interesting form was used by Papanastasiou and al. [31]:

154

1 h(I)=l+a(i.3)

"

(30)

Using a factorizable K-BKZ equation (21), Wagner and Demarmels [32, 33] showed that an equation of the damping function such as 9 h(I1, I2)=

1 1 + a~](I1 : 3)(I2 - 3)

(31)

m a y be suitable for s h e a r and uniaxial extensional flows. Though it is not written in term of a generalized invariant, it degenerates to equation (30) in shear. The i n t e r e s t of equation (30) also lies in its s i m i l a r i t y with a p p r o x i m a t i o n s of n o n l i n e a r functions obtained from the Doi-Edwards constitutive equation [8] for the reptation theory, at least in shear. Indeed, the later authors have developed a simplified equation in the form of the K-BKZ model considering the "independent alignment" ass,lmption, which states that the strands contract back after the strain is imposed and before relaxation occurs, so that they are only oriented and not deformed. Currie [34] found an accurate approximation for the related potential function U(I1, I2) which, in shear, leads to:

h(7) =

5 N~4~+ 25 + 10 (~2+ 2)~J4r~+ 25 + (4r~+ 25)

(32)

Equation (30) is an approximation of equation (32) in shear. Larson [35] has further simplified the Currie potential to obtain U(I1) and he derived h(7) in the form of equation (30) ~ i t h a = 0.2. In terms of network analogy, the damping function may be viewed as the expression of the retraction of the strands as compared to the continuum. The Lodge model thus corresponds to no retraction (affine deformation, a=0 in equation (30)), the Doi-Edwards equation corresponds to complete retraction (a=0.2), whereas incomplete retraction makes the damping function more softly decreasing (0 < a < 0.2). In the later cases, the deformation is non-affine since there is a difference between t h a t of the continuum and t h a t of the network strands. Wagner [33] showed that the Doi Edwards strain function

155

exaggerates the strain dependence and that obviously complete retraction is not consistent with the data. Another form which, due to a third parameter, enables a slightly better approximation of equation (32) and of the experimental data was used by Soskey and Winter [36] in the form: 1

h(I) = 1 + a (I- 3)b~2

(33)

Parameters a (and b) can be associated with the completion of the retraction process together with the strain amplitude. 2,2.6.Integral t e m p o r a r y n e t w o r k models and molecular theories. Wagner and Schaeffer [37-39] made an interesting attempt to reconcile the different aspects of the temporary junction network model together with the Doi-Edwards model. They proposed a simple picture of the effects of large deformation on the stress in terms of a slip links model. Assuming t h a t the entanglements can be thought as small rings through which the chain may reptate freely, in addition to equilibration and reptation, two deformation processes are assumed to give rise to the observed nonlinear behaviour after a step strain. The first process is connected to equilibration of the monomeric units between the entanglements by a slippage of the chains and is described by a normalized slip function S, giving the number of monomers in a deformed s t r a n d relative to equilibrium. The second process is related to a loss of junctions at the chain ends or along the chains by constraint release described by a normalized disentanglement function D giving the mlmber of strands for a deformed chain relative to equilibrium. These functions are connected to the tube dimensions (relative length of the strands or tube segments u' and tube diameter a). Since the number of monomers on a chain is balanced, at equilibrium, the average over the configuration space of their product is unity : = 1. Calculating the stress with various assumptions on the functions leads to different types of equations with different strain measures : -No slip and no disentanglement (D = 1 and S = 1) leads to the Lodge model. -Isotropic slip and disentanglement (D = , S = , D . S = = 1) leads to the Wagner model with h(I1, I2) = D 2. -Slip related to relative strand extension but constant tube diameter (S = u', the molecular tension in a deformed chain is equal to its equilil~rium value)

156 and anisotropic or isotropic disentanglement (D. S = 1 or D . = 1) provides the Doi-Edwards model with or without the " i n d e p e n d e n t alignment" assumption. -Slip related to relative strand extension but constant tube volume (S = u 'y2, the molecular tension in a deformed chain depends on the individual segmental stretch) and anisotropic d i s e n t a n g l e m e n t (D . S = 1) slightly improves the predictions [40]. -Wagner and Schaeffer ass~lmed t h a t the tube diameter is a function of the average stretch through a molecular stress function (S = u' f'l[], the molecular tension in a deformed chain depends on the average segmental stretch) and anisotropic disentanglement (D . S = 1). The function f can be theoretically derived from the experimental damping function. 2 ~ 7 J r r e v e r s i b i l i t y assumptiom As mentioned by several authors, there is experimental evidence that the process of loss of junctions, whatever its nature and whatever the domping function, may be irreversible. This led Wagner and Stephenson [41, 42] to consider t h a t their original equation is only valid in experiments wherein the d e f o r m a t i o n is monotonicallly increasing. To t a k e into a c c o u n t the irreversibility of the loss of junctions, which means t h a t these are never rebuilt in a decreasing deformation following an increasing one, they suggested the use of a functional r a t h e r than a damping function. So that, in the original model, the damping function should be replaced by: H(II(t,t'), I 2(t,t ' )) = ~ f m tC. _= t t, h(Ii(t',t'), I 2(t, " t ' ))

(34)

In a monotonically increasing deformation, H(I1, I2) = h(I1, I2).

2.3.A differential constitutive equation: the Phan-Thien T a n n e r modeL 2~3.12VIodifications of the UCM m o d e l Many improvements or modifications to the UCM model can be found in the literature. These can lead to various classes of constitutive equations keeping the differential nature of the equation [2, 3, 35]. As pointed out by Larson [43], a systematic classification of these can be performed by rewritting the UCM model as:

157

~(t)- ~

1

(35)

_

The various changes that may be carried out can be either on the convected derivative or in the right term of equation (35) or both; these imply the removal of some assumptions of the initial model. Such a possible modification, that was claimed to give a correct description of the essential phenomena of the nonlinear viscoelastic behaviour of polymer melts, is that proposed by Phan Thien and Tanner [44-46] involving the use of a special convected derivative and special kinetics of the junction. ~2~on ~ m o t i o n . The G o r d o n - S c h o w a l ~ derivative. The first kind of modification to the UCM model that may be conceivable is t h a t of the convected derivative. This leads one to consider that the motion of the network junctions is no more t h a t of the continuum and thus, the affine assumption of the Lodge model is removed. Among the various possibilities, P h a n Thien and Tanner suggested the use of the Gordon-Schowalter derivative [47], which is a linear combination of the upper- and lower-convected derivatives, instead of the upper-convected derivative: ~(t) = (1 - ~) a ~i(t) + ~a

(t)

0
(36)

This is equivalent to the assumption that there is some slippage of network junctions with respect to the continuum during the deformation. The local velocity field is thus different from the macroscopic one. The adjustable p a r a m e t e r a materializes the slippage and is called the slip parameter. To avoid confusion, it should be noticed that equation (36) is often written as:

~i(t) = -1- 2~ ~(t)+ ~

~(t)

-1 < ~ < +1

(37)

Using this derivative in the former UCM model, Johnson and Segalman proposed a model [48] that improves the predictions especially in shear, leading to normal stress differences and shear viscosity which are now shear rate dependent. Unfortunately, although it appears to be attractive in shear, the use of such a derivative can lead to some physical paradoxes that will be discussed

158 later. Moreover, although it brings some improvement in shear flows towards the UCM equation, it can be shown easily t h a t such a modification does not basically change the elongational behaviour. For example, the elongational viscosity remains unbounded at large times at high elongation rates (and a steady state value cannot be obtained). This could be expected since the GordonSchowalter derivative mainly takes into account rotational effects which are absent in elongation flows. ~Time-dependent segment kinetic~ This led P h a n Thien and Tanner to consider an additional change of the constitutive equation, removing the ass,lmption of constant creation and loss rates for the junctions. This could be done in m a n y ways, for example by m a k i n g these quantifies functions of the invariants of the stress or strain tensors. Phan Thien and Tanner assumed that, for each type of junction, these rates might be dependent on the average internal extension of the surrounding strands towards their equilibrium length. They wrote this in the form of viscosity contributions ~'i and relaxation times ~'i t h a t are now dependent on the trace of the extra-stress tensor (tr[~=i]), together with the equilibrium characteristics of the segment (Tli, ~-i) in the form:

~.'i = y(1]i, X~,Xt4r . L~J) _~. ,

and

Tl'i = V(l~i, ~i, tr[T_i])

(38)

where Y is some increasing function of the trace. The complete Phan Thien Tanner model is: 9

9

Y(rli, ~, tr[l:~(t)]) 1;i(t) + ki ~i(t) = Tli ]~,

and

~(t)= = ~. ~i(t).=

(39)

Two suggested forms of Y are"

Y(Tli, ki, tr(1;i))= 1 + e

tr(1;i),

Y(Tli, ~, tr(~i)) = exp{ e ~i tr(Ti)} ,

(40a)

(40b)

159 where e is a positive adjustable parameter. Provided that a suitable Y function is chosen, this is claimed to give a correct p i c t u r e of m a n y p h e n o m e n a displayed in simple s h e a r and uniaxial elongational flows. One should note that the original model of Phan-Thien and T a n n e r uses the non-affine derivative together with nonlinear stress term.

3.EXPERIMENTAL ASPECTS 3.12Vlaterials. Three different polyethylenes referred to as HD, LD, LLD have been investigated. They mainly differ in their molecular weight distribution and in the structure of the chains which can be either very linear as in the case of HD (high density polyethylene) or short-branched in the case of LLD (linear low d e n s i t y polyethylene) or long-branched in the case of LD (low density polyethylene). The weight average molecular weight and the polydispersity index of the samples are summarized in Table la. 3.2.IAnear viscoelastic b e h a v i o u r a n d m e m o r y function. 3.2.1J)ynamic measurements. In order to get a good description of the linear relaxation modulus, which remains the basis of any nonlinear model, experiments have been carried out in the linear viscoelastic domain in oscillatory shear flow. Dynamic moduli have been measured in a parallel plate rheometer (Rheometrics RDA700) for frequencies r a n g i n g from 0.01 to 500 rad/s. The frequency window at a reference temperature of 160~ has been slightly expanded by the use of timet e m p e r a t u r e superposition from experiments performed between 130 and 200~ Though in the case of polyethylene, the activation energy (Ea) of flow is often so low t h a t no appreciable gain can be expected (see Table lb), the achievable frequency range at 160~ covers about five decades in the terminal zone from 0.005 to 1000 rad/s. Some terminal viscoelastic parameters have also been evaluated at 160~ using a Cole-Cole expression for the dynamic viscosity. Table lc shows the zero shear viscosity (T10), the characteristic relaxation time (~.o, corresponding to the

160 m a x i m u m of the imaginary part of the complex viscosity) and the relaxation time distribution parameter (h, obtained from the eccentricity of the Cole-Cole plot towards the real axis). ~

t e relaxation SlmCWUnL From the dynamic moduli, discrete relaxation time spectra have been calculated by a nonlinear minimization procedure [17], which lets the number of relaxation modes and the time value of these modes freely adjustable. The spectra of Table ld have been obtained and their use in a Maxwell model enables a very good recovery of the experimental values of the storage and loss moduli in the explored frequency window (within 4% error) as shown on Fig.1. It is worth noticing that the spectrum of LLD for the terminal zone seems to be r a t h e r complete whereas t h a t of the two other polymers are more or less sharply cut. This occurs respectively in the long time range for LD and in the short time range for HD. This peculiarity should be kept in mind when dealing with n o n l i n e a r models since it can lead to some discrepancies for the predictions in nonlinear situations. These problems m i g h t be expected especially when the spectrum is truncated in the long time range, wherein these modes are known to be very important for the viscoelastic behaviour of polymers.

Table la: Molecular weight characteristicsof the s,,amples. Material .

.

.

.

.

HI) LD LLD

Mw [g/mol]

Mw / Mn

90000 ~ 120000

4.6 4.5 6.3

,

,

Table lb" T h e r m a l characteristics of the samples (WLF coefficients and activation energy for flow at 160~ Material

=,,

HD LD LLD

,,,,

_~6o

C1

2.38 4.81 2.49 9 ||

,.

.

.

.

_16o

C2

[o]

311 308 311

.

.

. _'~6o

Ea

[kJ/mol] 27.5 56.0 28.9

161

Table !c: P a r - m e t e r s of the terminal zone, at 160~ from a Cole-Cole plot. Material

110 [Pa.s]

HD

3500

0.088

0.47

LD LLD

426(D 154O0

8.82 0.38

0.41 0.49

.

..,

,,,

,

.

,,

.

.,.,

.

.

.

.

.

.

.

~o [s]

.

.

,,

.

,

h

,,

.

=

,.,

,,

9

Table 1.d:,Discrete relaxation spectra at 160~ HD

[s]

LD ki [s]

Tli [Pa.s]

234

0.000645

98.1

0.000128

236

514

0.(D535

285

0.00612

1350

0.0258

842

0.0285

768

0.041

3360

0.113 0.485 2.30

765 576 312

0.155 0.891 4.58

2620 6460 ~

0.277 2.01 15.7

4{~ 3730 2010

11.2

164 85.5

23.4 118

13200 7~)

135

956

0.00104 0.00573

71i [Pa.s]

LLD

.

80.3

,

.

..,

,

,

,,

,,,

~ [s]

,

.,

Tli [Pa.s]

,

,...I 106 ~.~ 105 -,=1

~

104

" 0 o

~o

103 [

J

lO2 ~

c-

m

101 c

~_ lO~ ,~

~ I 0

10-1

10~

10z

102

103

Frequency [rad/s]

Figure la: Storage (o) and loss (Q) moduli of HD at 160~ (full line indicating a fit obtained from the spectrum of Table ld).

162

10 6 ~ n

''

[

lO s

"

-,=4 ,---4

-u 10 4 0

tn

j

o

10 3

10 2 ~1

13 era

~

101

0o

L 0

.

L

o~ 10 -1 10 - 2

1 0 -I

i0 o

Frequency

101

102

103

[rad/s]

Figure lb: Storage (o) and loss (Q) moduli of LD at 160~

(full line indicating a

fit obtained from t h e spectrum ofTable ld).

106 F n ~-~ 105r -,-4

104 r 0

I

103 o "

~c

102

01 0 I

,~ lO~

~

L 0 4-J

G~

I

m I0 -1 10-2

10- I

i0 o

101

102

103

Frequency [ r a d / s ]

Figure lc: Storage (o) and loss (o) moduli of LLD at 160~ fit obtained from the spectmlm of Table ld).

(full line indicating a

163

3~3.Experimental data in ~mple shear. 3~.l~teady state shear flow. Steady state shear flow experiments have been performed using a capillary rheometer (Instron 3211) at 160~ with different dies (diameter range: 0.251.25mm; length range: 2.25-25mm) so that Bagley corrections were taken into account. Other experiments have been performed using a cone and plate geometry (Rheometrics RMS800 rheometer) at different temperatures. The time temperature superposition has been applied on both the shear stress (1:(~)) and the first normal stress difference (NI(~)). Combining the two sets of results for each material, the viscosity data TI(~)extends to nearly five decades from 0.01 to 1000 s -1. The first normal stress results only cover a limited window from 0.1 to 10 s -1 for HD and about three decades for LD and LLD. The main measurement problem is linked to the low value of the normal force at low shear rate and to instabilities at high rates. The "extended" Cox-Merz rule [49] can be successfully applied for HD and LLD. This rule states that the viscosity and elasticity coefficients for oscillatory and steady state shear flows are related, according to: (41)

Tl*(co) = Tl(~) and 2G'(co)/a~2 = VI(~) when co = T,

where

~1(~) is the first normal stress coefficient, is the shear rate, co is the frequency.

Whenever it is applicable, such a comparison can lead to a considerable widening of the shear rate range, this is especially interesting in the case of normal stresses which are generally difficult to measure on a broad window of rates. However, significant differences were noted in the case of LD preventing the use of the previous rule and any enlargement of the data set. 3~3~Stress g r o w ~ Using the cone and plate geometry, stress growth experiments have also been performed using different temperatures and different shear rates. 9

9

§

9

§

9

Correct tangential (1:+(t,7);Tl+(t,7))and normal stress (N i(t,7);~i(t,7))data were

154 obtained from 0.1 s until the steady state flow is obtained.The s h e a r r a t e r a n g e which enables satisfying m e a s u r e m e n t s covers about one decade typically from 0.2 to 2 s-1. ~tress

r e l a x a t i o n a f t e r a s t e p st~_inMeasurements of the nonlinear relaxation modulus G(t,7) have also been

carried out using the plate-plate geometry. Various step strains were applied on the sample and the stress relaxations were recorded. Since the shear strain is known to be inhomogeneous in such a geometry, a correction of the apparent relaxation Ga(t,T) modulus has to be taken into account to get the real relaxation modulus for the m a x i m u m strain in the disk sample. This procedure is very similar to that proposed by Rabinowitch in Poiseuille flow, wherein the shear rate is also non-homogeneous, and has already been described by Soskey and Winter [36].The correctionfactor is:

G(t,7) = 4 + n Ga(t,T) with n

n =

dlnGa(t,T) . dln7

(42)

In this way, the nonlinear relaxation modulus was measured from the linear domain to strains up to 10.

3.4.Experimental data in ~

elongation.

3A.1.Stress growtl~ M e a s u r e m e n t s of the elongational d a t a during stress growths have been obtained on a constant strain rate, exponentially increasing length, tensile rheometer (Ballman type) at 160~

The elongation rates achievable in this type

of r h e o m e t e r for the three materials range between 0.05 and 2.6 ~-1. The largest window was obtained for the more viscous m a t e r i a l (LD) a n a in this case correct m e a s u r e m e n t s were possible in the shorter times (0.01 s) w h e r e a s this limit is slightly g r e a t e r for the other materials. In any case, the m a x i m u m Hencky strain does not exceed 2.5. This range could not be extended because of the force transducer sensitivity, either because of the low viscosity (short times) or because of the v a n i s h i n g sample cross area (long times) or b e c a u s e of filament breaking and inhomogeneous deformation of the tested sample.

165

3.42~to~dy state viscosity. Since any elongational stationary viscosity can hardly be obtained in the t r a n s i e n t experiments unless in the Troutonian regime (low strain rates), determination of a steady state viscosity has been performed on LD and LLD by indirect methods using the Cogswell analysis of convergent flows and entrance pressure losses [50, 51], as derived from the Bagley plots of the capillary experiments. According to Cogswell, the polymer, during its flow from the reservoir to the die, will form its own streamlines, leading to the lowest pressure loss. In this situation, for an axisymmetric geometry, the entrance pressure loss is mainly due to elongational stresses and thus one can calculate an average elongational stress as: 3

~E = ~ (n + 1) APE ,

(43)

and an average elongational strain rate on the flow axis as: 9

e =

4~/1;

3 ( n + 1)APE

where

'

(44)

n is the flow index, APE is the entrance pressure loss, is the shear stress (using the Bagley correction), is the apparent shear rate.

Though the Cogswell analysis was derived for 180 ~ entrance angle, the pressure losses were found to be independent of the angle between 90 and 180 ~ provided that the die length is short enough to avoid material compressibility. It should be noticed that in such cases, exit pressure was also claimed to be the origin of important errors, nevertheless the entrance pressure loss was found to be insensitive to the die diameter, and conversely to the contraction ratio, in the range of 4.76 to 12.7. A question arises concerning the residence time of particles in the convergent part of the die. Indeed, it is important to evaluate and check this residence time in order to assess whether it is sufficiently long to get a stationary value of the viscosity or not. At low shear rates (high residence times), for LLD, a good agreement was found between the calulated values of the converging flow analysis and those of the transient experiments.

166 But at high shear rates, this might be questionable in some cases since in the case of L L D for example, at elongation rates greater than 17s -z, the residence time was found to fall in the order of the m a x i m u m relaxation time of the spectrum.

3 , 5 ~ t a l rauges of data for n o - H m ~ r viscoelasticity experiment~ The experimental ranges of strain rates (or strains) are summarized in Table 2 for the various types of experiments. Time-temperature superposition was successfully applied on the various steady shear flow and transient shear flow data. The shift factors were found to be exactly the s_~me as those obtained for the dynamic data in the linear viscoelastic domain. Moreover, these were found to be also applicable in the case of entrance pressure losses leading to an implicit application to elongational values. Table 2a: Experimental ranges for shear data (shear rates or shear strains, (*) indicates that the Cox-Merz rule was applied). N1 (*)

Material and

G', G"

11

TI(*)

Nz

Function

co [rad/s]

~ [s-l]

~ [s-l]

; [s-l]

Tl+ ,Nz +

G

0.5-5 0.2-0.8 0.2-2

0-10

,

HD LD LLD

0.01-1000 0.15-2000 0.01-2000 0.15-7 0.01-1000 0.005-1000 0.01-300 Non valid 0.01-7 Non valid 0.005-150 0.01-600 0.005-600 0.01-3 0.005-150 ,,

,

,.,

=

|.

.,

,

. . . .

0-10 0-6

Table 2b" Experimental ranges for elongational data (elongation rates or Hencky str~ns).

Material and Function

TIE ~ [s-Z]

HD LD LLD

0.09-9 0.2-100

TIE+ [s-q (c)

0.125-1 (2.0) 0.05-2.6 (2.5) 0.05-1.4 (2.5)

167

4.EXPERIMENTAL V~ATION

OF THE WAGNER MODEL

4.1.Time-strahl separability. Figure 2 shows some results obtained in step shear strain experiments, in terms of nonlinear relaxation modulus, in the case of LD. On logarithmic scales, the curves are only shifted vertically, i n d i c a t i n g t h a t they are superimposable in the experimental time range a n d t h a t the separability, expressed by equation (18), seems to hold.

10 5 n

",,,..,. 10 4 o ~E

tO

",,,,~.

"~ 10 3 "X'.'.. \

x t~ e~

10 2

i0-2

. . . . . . . . .

lO-i

I 0~

10 i

10 2

Time [s]

Figure 2: Linear and nonlinear relaxation modulus of LD obtained from step shear strain experiments at 160~ (-----): linear, (..... ): T = 2, ( - - - ) : T = 3. 4,2.The d a m p i n g function. 4.2.1~rimental determination. Damping fimctions in shear can be obtained experimentally by two different methods. Firstly, by direct e x p e r i m e n t s in step s h e a r as indicated in paragraph 2, and secondly by a derivation from the tangential stresses (l:+(t,~)) in t r a n s i e n t experiments as proposed by Wagner in elongation [52] and Fulchiron et al. in shear [53]:

158

9+(t,~)

t fl+(t',~) m(t')

dt'}.

(45a)

7t

This latter method can also be used in uniaxial elongation using the transient §

9

+

9

+

9

elongational stress (oE(t,e)= ~zz(t,e)-Ve2(t,e)): §

1 h(e)

=

+.

9

aE(t,s

expl2~t} - exp{-~t} { G(t)

~@.

t oE(t,e) m(t') " { G2(t') dt'} 0~

(45b)

Figure 3 shows a comparison between damping functions obtained from step s h e a r and transient data for LD.

1 . 0

=08

:..k

o .r-i

9

O

u

(-0.6

LL

('0.4

-w-4 (:1. E

A

r' 0. 2 00

&

'

0

~p

2

'

'

4 6 Shear Strain

A ~

'

8

I0

F i g u r e 3" E x p e r i m e n t a l d a m p i n g function of LD obtained from t r a n s i e n t experiments either in step shear (filled symbols) or in step shear rate at 0.2 and 0.5 s -1 (open symbols). The shear damping function obtained from transient experiments remains unchanged whatever the shear rate is in the experimental window, so t h a t one m a y assume t h a t it is not shear-rate dependent. More surprising is the large discrepancy between the results between the two methods, wherein the step

169 shear experiment leads to smoothly decreasing functions. One could argue t h a t the delay time for application of strain in a classical rheometer is a possible origin of this discrepancy, however this is in contradiction with the higher value of h. More probably, this might contradict the former conclusions concerning the shear rate independence for very high initial shear rates such as those encountered at the start of step strains. 4 . 2 ~ t h e m a t i c a l form of tho. d~mping functionIn this study, the mathematical form of h, equation (33), is chosen to fit the experimental function obtained from transient data. In the case of LD, it is found to be very close to equation (30) since the value of the parameter b is nearly equal to 2. As far as the elongational damping function is concerned, it was found to be independent of the elongational rate for the experimental range. Figure 4 shows a plot of the damping function in shear and in elongation for LD together with the fit using equation (33) with invariant (25). Indeed, from the elongational data, the last parameter ~ describing the generalized invariant can be theoretically obtained. However, it should be noted that this might be sometimes difficult, especially when strain hardening is not very marked as it is the case for LLD and HD for example. Indeed, in this case, equation (45), which implies simultaneously the difference of two terms and numerical integration, can lead to large errors mainly due to uncertainties in the experimental data. In the short time range, this can lead to physically unrealistic values of the domping functions (greater than unity), because of the slight difference between the Lodge (h = 1) and the Wagner model and because of the finite starting time of rheometers. On the other hand, in the long time range, there may be large uncertainties on the calculated functions and thus on the fitted parameters because of the difficulty to obtain the steady state in constant elongation rate experiments. As a consequence, when strain hardening is not marked, one has to refer to the transient function itself to fit the ~ parameter and even more, doing that, one may be faced with multiple choices unless the steady state viscosity data can be obtained from indirect experimental methods (converging flow analysis for example).

170

1.0

c

0.8

o

-J-I

o

c

0.6

c

0.4

.,~

Oo

E

rl$

o

0.2

0.0 L 0

, 2

, ~ J 4

_. 6

i 8

.... 10

Strain

Figure 4: Damping functions of LD in shear (o) and in elongation (o), fitting is done according to equation (33) with invariant (25) (a = 0.084, b = 2.06, ~ = 0.019). In spite of uncertainties in ~ in the case of LLD and HD, the values of Table 3 were obtained for the three polyethylenes. The corresponding shear damping functions are plotted on Figure 5. It can be seen t h a t the decrease of the function is smoother in the case of LD and this might indicate some resistance of junctions to strain as described by the retraction assumption. This is consistent with the branched nature of LD where long side branches might hinder complete retraction of the polymer chains and lead to a conservation of network junctions. Though the value of ~ is r a t h e r uncertain, its observed constancy was already mentioned by Papanastasiou et al. [31]. However, this might be questionable since one may think that this parameter should reflect an additional influence of the geometry towards the retraction process. Table 3" Fitte d Ear .ameters of the damping function for the three polyet_hylenes Material

a

b

HI) LD LLD

0.106 0.084 0.086

2.32 2.06 2.56

0.020 0.019 0.020

171

1 01 c

08

o .~1 U

c 06 c 04

-.=4

E

.

~

~"~--.z..z.... --z.z_-_L:_-:::.~.=

O0

|

0

2

I

=

I

4 6 Shear Strain

I

8

,,

I

10

Figure 5: Damping functions of the three polyethylenes calculated with equation (33) and invariant (25). (----): LD; (..... ): HI); ( - - - ) : LLD ~rimental validation of the Wagner model in simple flow~ This section presents different results obtained using the Wagner equation with the form of the damping function of equation (33) together with the generalized invariant of equation (25). It is primarily devoted to comparisons b e t w e e n predictions and experimental results in s h e a r and uniaxial elongational flows described in paragraph 3 for LD. 4 ~ l . S t e a d y state f l o w ~

Figure 6 shows the predictions of the Wagner model compared to experimental data for elongational viscosity, shear viscosity and first normal stress difference of LD. These have been calculated according to: oo

Tl(~) = J re(s) h(~/s) s ds ,

(46a)

172

oo

~1(~)= J re(s)h(~ s) s 2 (:Is,

(46b)

oo

~]E(~) = 1 g m(s) h(~ s) {exp(2~ s)- exp(-~ s)} ds

(46c)

L _ j m--J

~-~o "~.

10 5

o u)

> O0

10 4

2~

' . ' oL 10 3 f_Z o ~

~_1 "-

CO

02

10

10-2

10-1

10 o

101

10 2

10 3

Shear or Elong. Rate Is -1]

Figure 6: Steady state functions for L D at 160~ (experimental and calculated). (o): elongational viscosity, (o)- shear viscosity, (A): first normal stress difference.

4.3~.Transient s h e a r flow~ Figures 7 and 8 show the predictions of the Wagner model compared to experimental data for t r a n s i e n t shear viscosity and first normal stress coefficient of LD. These have been calculated according to: t Ti(t,~) = ~ re(s) h(~ s) s ds + t G(t) h(~ t), 0

(47a)

t vl(t,~) = j" re(s) h(~ s) s 2 ds + t 2 G(t) h(~ t). 0

(47b)

173 ~Transient elongational flows F i g u r e 9 shows the predictions of the Wagner model compared to experimental data for transient elongational viscosity of LD. These have been calculated according to:

qz(t,~) _.1 -dt =

re(s) h(~ s) {exp(2~ s) - exp(-~ s)} ds

8

1 + ; G(t) h(~ t) (exp(2~ t)- exp(-~: t)}.

(48)

E

10 5 r

_ n.o~o

ooo

ooo

o o

>, ~

m 0 U

10 4

-0-,l

L

e"

10 3

10 -1

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

10 o

101 Time

10 2

Is]

Figure 7: Transient shear viscosity for LD at 160~ calculated). (o): 0.2 s -z, (o): 0.5 s -1, (A): 1 s o1.

(experimental and

174

106

-

0 CJ

O oOOO U'; U~ q,; L. r--1

O

10 5 ,-jO~n

,,,%

nn

nnnnm

nn

mn L. 0 Z

10 4 &&A^

e

m f,_ ~ b_

I0 ~

101 Time

102

Is]

F i g u r e 8: T r a n s i e n t first n o r m a l s t r e s s c o e f f i c i e n t for L D (experimental and calculated). (o): 0.2 s -1, (o): 0.5 s -1, (A): 1 s -1.

10 6

at

160~

-

>, lO s -~..4

-r.t

>

10 4

0

Ld

10 3

10-2

9

,

,

. . . . .

I

,

. . . .

, , , i '

10 -1

10 o

'

. . . . . .

!

,

101

. . . . . .

iI

10 e

Time [s]

Figure 9: Transient elongational viscosity for LD at 160~ calculated). (o): 0.05 s -1, (~): 0.5 s -1, (A): 1 s -z, (0): 2 s -1.

(experimental and

175

4.4.Conclusion. The previous set of comparisons shows that the Wagner model enables a good description of the experimental data in various experiments in simple shear and uniaxial elongation. However, it is worth pointing out that in most cases, there is a major difficulty concerning the determination of the parameters of the damping function. Firstly, it has been shown that there may be many experimental problems in a direct determination of the experimental function. In shear, damping functions obtained from step strain and from step strain rate experiments do not match each other. This poses an important question on the validity of the separability assumption in the short time range. Significant departures from this factorization have already been observed in the case of narrow polystyrene fractions by Takahashi et al. [54]. These authors found that time-strain superposition of the linear and nonlinear relaxation moduli was only possible above a certain characteristic time. It is interesting to note t h a t this is predicted by the Doi-Edwards theory [10] and according to this theory, this phenomena is attributed to an additional decrease of the modulus connected to a tube contraction process and time-strain separability may hold after this equilibration process has been completed. Other examples of non-separability were also reported by Einaga et al. [55] and more recently by Venerus et al. [56] for solutions. Another explanation of the observed discrepancy can be found in the numerical treatment of the stress growth experiments, either in shear or in elongation, which sometimes gives rise to physically unrealistic values of the damping function because of uncertainties in the measured data. This is especially the case for transient elongational data on low viscosity materials or on polymers having nearly Troutonian behaviour in the experimental range of elongation rates (LLD or HD). To avoid these problems, one can prescribe a mathematical form of the equation of the damping function and of the generalized invariant. Then, minimization of the experimental rheological functions is expected to provide the values of the required parameters. Unfortunately, the adjustable parameter of the generalized invariant, which can only be obtained from elongational data, cannot be found accurately in many cases. Indeed, accurate determination of this value requires both a pronounced strain hardening behaviour and a steady state viscosity in the long time range, which is seldomly

176 achieved in current experiments. The latter problem can be avoided by resorting to indirect methods for the determination of the elongational steady state viscosity such as the convergent flow analysis, while keeping in mind that this technique has not yet received sufficient theoretical justification. Combined flows, involving both simple shear and uniaxial elongation could be of great help in providing additional tools to select the parameters of the damping function with better accuracy. Other strain histories may also be used for this purpose such as that proposed by Giacomin et al. [57], who proved that large amplitude oscillatory s h e a r flows are a useful tool to obtain the parameters of the damping function reliably. Unfortunately, the authors also showed that the concept of irreversibility described by equation (34) is not sufficient to get an appropriate depiction of the rheological phenomena in reversing flows. Prediction of the second n o r m a l stress difference in s h e a r and thermodynamic consistency obviously requires the use of a different strain measure including of the Cauchy strain tensor in the f o r e of the K-BKZ model. With the ratio of second to first normal stress difference as a new parometer, Wagner and Demarmels [32] have shown that this is also necessary for accurate prediction of other flow situations such as equibiaxial extension, for example.

5 J ~ P E R I M E N T A L VALIDATION OF THE PHAN THIEN TANNER MODEL 5.1s o ~ n ~ f i n g from the use of the Gordon-Schowalter derivative. The use of the Gordon-Schowalter derivative, equation (36), brings some discrepancies that can be easily pointed out considering the limiting case of a = 0 [58]. This case is known as the Johnson-Segalman equation: ~(t) + ki ~(t) = qi ~ ,

and

~=(t) = .~ ~i(t).

(49)

1

The slip parameter can be easily determined from various experiments in shear situations by some fit of the steady state shear viscosity and primary normal stress coefficient. Analytic expressions are easily derived in steady state and transient flows in the form:

177

(50a) i 1 + a(2- a)(X i })2 2Thki

~(~)

(50b)

= 2

i 1 + a(2- a)(X i ~)2

Tl+(t,~/) =v.

i 1+

a(2-a)(~)2

{1- e-t&i (cosP4a(2 - a ) ~ t ] - s ~4a(2 - a) sin[~/a(2 - a)~t])} ,

+

(50c)

9

~l(t,7)= Z

i 1 + a(2- a ) ( ~ 1 2 {1- e-t/~ (cos ['4a(2 - a)~t] +

sin [~/a(2 - a)~t])} .

(50d)

~]a(2 ' a)~s Second normal stress difference is predicted with the following dependence: N__~2_

(51)

_a

N1-'2

"

Alternatively, in t r a n s i e n t flows the slip p a r a m e t e r can also be determined using the time position of the m a x i m u m of the experimental functions (tw for t a n g e n t i a l stress and tN for normal stress). However, this can only be p e r f o r m e d a c c u r a t e l y if the stress overshoot is large enough to avoid uncertainties in these values and this can only be achieved at high shear rates. In this case, according to equations (50c) and (50d) : /t

trr =

, 2 "4a(2- a)

(52a)

178

(52b)

tN'-

~]a(2- a ) ~ The values of the calculated slip factors are listed in Table 4. Table 4: Slip p a r a m e t e r s of the various m a t e r i a l s Material

aT

aN

HD

0.25

0.8O

LD

0.08

0.24

LLD

0.24

0.77

As can be seen from t h e s e r e s u l t s a n d from Fig.10 (for HD), s a t i s f a c t o r y a g r e e m e n t cannot be achieved by the use of a single value of the slip p a r a m e t e r for b o t h t a n g e n t i a l and normal stresses.

~-~ lOs 1:1.

r

~1,~ ,El

L_.J ID

~--~ 104 -,"4

10

o

u~

o

~

c L o 102 ~z l-

ffl L

~.

I01 10 . 3

......... 10-2

10 -1 i0 o 101 Shear Rate Is -1]

102

103

Figure 10: Viscosity (o) and first n o r m a l stress coefficient (~) of HD a t 160~ (experimental data and fit using ( ~ ) :

a = 0.25 or ( ..... ): a = 0.80)

Using the corresponding value of a obtained by a fit on either the tangential or n o r m a l stresses (named aT a n d aN), e q u a t i o n s (50, a a n d b) give a r a t h e r good fit of the experimental curves on a large r a n g e of s h e a r rates. The same

179 conclusion is also valid in t r a n s i e n t shear flows. Two different values are obtained from the maxima of either the tangential or the normal stresses and a single value of the slip parameter cannot describe simultaneously the maxima of both the viscometric functions. Nevertheless, the values aT and a s , as determined from each function, remain independent of the shear rate for a p a r t i c u l a r material. Moreover, similar values of the slip p a r a m e t e r are obtained for a particular material whatever the time-regime of the shear flow experiment is. The fact that a single value of the p a r a m e t e r is not able to describe both tangential and the normal stresses can be considered as a manifestation of the violation of the Lodge-Meissner rule [59] by the JohnsonS e g a l m a n model, as previously described by m a n y a u t h o r s [43, 60]. This relationship states that, after a step strain, the ratio of the first normal stress difference to the shear stress should be equal to the strain magnitude 9 T l i - T22

= 7.

(53)

It requires that the principal stress axes should coincide with the principal strain axes. This rule has been experimentally checked by m a n y authors [24, 56] Actually, the use of the Gordon-Schowalter derivative involves the violation of the Lodge - Meissner rule, indeed when a equals 0 or 2, either the upper or the lower convected derivatives implies that the relationship is respected. In the general case, the double value of the slip parameter is a natural way to accommodate this rule. The connection between the double value of the slip parameter obtained from the viscometric functions and the violation of the Lodge-Meissner rule becomes more evident when the time-strain separability of the model is considered. For this purpose, the Johnson-Segalman model can be rewritten under the form of a single integral equation, cancelling the Cauchy term, which gives the following form in simple shear flows: t =~(t) = / m ( t - t ' ) h ( y ) Ctl(t ') dt'. -00

(54)

180 From this equation, derivation of equations (50) cannot be obtained with a unique form of the damping function. Derivation of equations (50a) and (50c) requires:

sin[~T] h(T)=

qa(2'a)Yy

"

(55a)

Derivation of equations (50b) or (50d) requires: h(~,) = 2 (1, c o s [ ~ ~ / ] ) a(2- a) T2

(55b)

Because of the sine form of equations (55), physically unrealistic values of h(7) 2x occur at strain higher than . ~]a(2 - a)

Nevertheless, a rather good fit of the experimental damping function (as determined by equation (45) of the preceeding section) can be obtained until T = 4 for all the materials with equations (50), provided that the corresponding slip parameters are used (aT for equation (50a) and aN for equation (50b)). The significance of the double value of a is clearly shown in equations (50) since respect of the Lodge-Meissner rule or of material objectivity requires the identity of equations (50a) and (50b) which can only be achieved by two different values for a:

sint~]aw(2 aT)T] 2(1-cos[~]aN(2-aN)T]) ~]aT (2- aT)T

-

aN (2- aN) T2

(56)

Solving equation (56) numerically gives the following relation between aT and aN : aT = 0.3 aN,

(57)

which is valid whatever the material is and for strains lower than 4. This equation is consistent with the experimental values derived previously from the steady and transient experiments.

181 However, it is now worth pointing out that the difference in aT and aN imply t h a t the initial model should be replaced in shear by a pair of two independent correlations for shear stress (eq. 50a and 50c) or for first normal stress coefficient (eq. 50b and 50d). But at this point some questions arise concerning the choice of the proper value (aT or aN) to be used in any other flow situation. Though it is possible to imagine equation (49) including some variation of a with flow history or invariants, it could hardly be different in two equations for the same flow kinematics. The sine form of the "damping function ~ leads to another major problem, which lays in the occurrence of undesirable oscillations in t r a n s i e n t shear flows (Figure 11). This phenomenon may be misleading for example when modelling instabilities in complex flows, since it is then hardly possible to distinguish between real phenomena and those generated by the model itself.

3500

~aooo ~

-~

>

2500

2000

0

'~

'

2

4

''

6

'-

'

8

10

Time [s]

Figure 11: Transient viscosity of HD at 160~ (experimental data (o): 0.5 s "1, (~): 5 s -1 and fit with a = 0.25) Keeping in mind the previous remarks, it must be recognized that the lower the value of the slip parameter, the smaller the deviation to the Lodge-Meissner relation. This may be especially interesting, since in such a case, one can expect a single value of a t h a t enables some kind of compromise for an acceptable depiction of the rheological shear functions. This can be expected,

182 for example in the case of LD (see Table 4) for which r a t h e r low values of the slip p a r a m e t e r are obtained. At least, even if the Gordon-Schowalter derivative is obviously an improper tool for the description of non-affine deformation, its basic m e a n i n g retains some consistency with experimental observations, especially concerning the loss of junctions in materials with very different molecular structures. Indeed, going back to the significance of a, its low value in the case of LD is in agreement with the assumption of some kind of resistance to slip of junctions in branched materials, whereas the opposite trend is observed in the case of linear polymers (LLD and HI)), for which higher and r a t h e r similar values are found (Table 4). As a final comment, it is worth mentioning t h a t the J o h n s o n - S e g a l m a n model predicts an elongational behaviour that is very close to t h a t of the UCM equation. Indeed, the t r a n s i e n t elongational viscosity remains unbounded in the long time range at high elongation rates and no steady state value can be obtained in the general case. In this type of flow, the slip p a r a m e t e r only has an influence on the occurrence of the strain hardening effect, whose time position changes towards the UCM equation (Figure 12). Divergence of the transient elongational viscosity occurs in the Johnson-Segalman model when : 1

t>

if0 < a < 1,

(58a)

2 ~ (1- a) (~)max or

1

t>

if1 < a < 2,

(58b)

e (a- 1) (ki)max instead of (UCM model): t>

1 2 ~ (~)max

.

(58c)

183

106

// /

o

Q_ i__1

>, lOs ~ tn 0 t~ 01 -e-i

>

1 0 4 ..

0 i,i

1

10 3 -10-2

10 -1

10 o

101

102

Time I s ]

Figure 12: T r a n s i e n t elongational viscosity of LD at 160~

(experimental data

(o):0.5 s -1, (Q):2 s -1 and fit using the Lodge model (--): a = 0 and the JohnsonSegalman equation (- -): a = 0.24). 5.2.Choice of t h e s e g m e n t idnetic~ Since the slip p a r a m e t e r does not basically change the divergence behaviour of the elongational viscosity of the UCM model, the limiting case of the P h a n Thien T a n n e r model with an u p p e r convected derivative (a = 0) m a y give an indication of t h e influence of the function Y t h a t governs the segment kinetics and the choice of its mathematical form. The linear equation for Y is obviously improper since, as can be seen from Fig.13, an unrealistic behaviour of the steady elongational viscosity is predicted (constant value a t high elongation rates). In t h i s sense, the e x p o n e n t i a l form is more realistic. Using the exponential form, the description of the pronounced m a x i m u m of elongational viscosity requires r a t h e r low values of p a r a m e t e r e, of the order of 0.01 when the upper convected derivative is used (a = 0, Table 5). As a general rule in this case, it is obvious that the higher the value, the less pronounced the maximum, the extreme case being t h a t of s = 0 (Lodge model and divergence of the viscosity).

184

This is unfortunately contradictory with the value that is needed to describe the shear functions. Indeed, although the Lodge-Meisner rule is valid w h e n the upper-convected derivative is used, the shear thinning effect on shear viscosity and primary normal stress coefficientrequires rather high values of e (Table 5 and Fig.14) which are, in any case, one order of magnitude greater than in elongation, the extreme case being again that of e = 0 (Lodge model and no shear rate dependence of the shear functions). Table 5: Parameter e for the various materials . . . . . .

(elongational data)

e (shear data)

HD LD

O.085 0.030

0.7

L[~

0.105

Material

e

_

.

,=.

0.15 0.7

106 f---i e@

EL

>, 105 tn o u

.r.4

>

10 4

r0

t~ ..j

.._.

1010 3 - 3 . . . . .10 . . . -2 ...

......

|

,

.

. 1,..,a

. . . . . . . .

i

10'- 1 10 o 10 1 10 e Elongational Rate [ s - t ]

. . . . . .

!

10 3

Figure 13: Steady state elongational viscosity for LD at 160~ (experimental d a t a and fit using (.... ) linear form and (-----) exponential form of Y).

185

r---i

t'-Jt--=J

~

K,~4~a"

4-J -,-41 s >'"me-'~

~ 8 ~~ . g x ~'"

~

o

2.1

m -F,4

>~.~mL 104 I

""'""-.

~u=L~-u~

o~

L

4/

~ L

~r "~

u~

10 2

[

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

1O- 3

1 O- 2

'

1 O- 1

................................... 10 ~

Shear" or Elong.

101

Rate

10 2

10 3

[s-l]

Figure 14: Steady state functions for LLD at 160~ (experimental d a t a (o): elongational viscosity, (Q): shear viscosity, (A): first normal stress difference and fit (--) e = 0.105, (- -): e = 0.7)

It is thus impossible to find a single value that enables correct description of both shear and elongational data. This m a y be understood considering the efficiency of the Y function in describing the shortening of the junction lifetimes. In the model, this shortening is all the more important since the stress magnitude is higher. Since it is generally observed that materials which exhibit the highest stress in elongation (elongation thickening) also show the opposite trend in shear (shear thinning), the weighting by the function can hardly be achieved in any coherent way with a single value of e in different flow geometries. Once more, as in the Johnson-Segalman equation, this sets an important limitation for the easy handling of such an equation. 5 ~ C o m b i n a t i o n of the two modifications-Experimental vah'dation of the P b a n Thien T a n n e r model The

original

Phan

Thien

Tanner

equation

was

written

using

simultaneously both modifications: Gordon Schowalter derivative and segment kinetics term. The segment kinetics term (exponential form) enables a more

186

realistic description of the steady elongational behaviour, giving rise to a bounded viscosity in the long time range. The Gordon Schowalter derivative has its major influence on the shear properties and additionally predicts a second normal stress difference as in the case of the Johnson-Segalman model, equation (53). Unfortunately it also introduces, conversely, the infringement of the Lodge Meissner rule. Considering the previous remarks, one must keep in mind the following important points. The smaller the value of a, the lower the deviation to the Lodge-Meissner rule. However, in this case the flow behaviour of the model is primarily described by e and the simultaneous description of shear and elongational data using a single value of this p a r a m e t e r is impossible. The smaller the value of e, the higher the value of the steady elongational viscosity can be. However, in this case the shear flow behaviour of the model is described by a so that the violation of the Lodge Meissner rule may become important. The predictions are then very close to those of the JohnsonS e g a l m a n model with the associated discrepancies such as spurious oscillations in transient shear. The determination of a couple of values (a,e) is then bound to be a compromise obtained from a simultaneous fit of the elongational and shear functions (Table 6). Table 6: Parameter a and e for the various materials Material

a

E ,

,

. . ,

.

.

HD LD LLD

,

.

9

.

,

0.50 0.15 0.35

,

,.

,

,

,,==

0.050 0.025 0.060

Figures 15 to 18 show the predictions of the model for LD at 160~ in steady state and some transient flows in shear and uniaxial elongation.

187

~ ~ I06 ~'--'

~

o

o

i0 s

U

>

~

10 4

E

o~

~Zo~b103. "~

10 2

I0 -3

///:" . . . . . . .

.

I

~

.

I0 -2

.

.

.

I

I0 -I

. I-ill

. . . . . . . .

I

. . . . . . . .

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

I0 ~

103

I(0 2

I0 x

Shear or Elong. Rate [s - I ]

Figure 15: Steady state functions for LD at 160~ (experimental and calculated). (o): elongational viscosity, (u): shear viscosity, (A): first normal stress difference.

10 5

-

U~

.

EIO

130

131"100rl

o~

10 3

I0 - I

. . . . . . . .

j

. . . . . . . .

10 ~

,

. . . . . . . .

101

,

10 2

Time [ s ]

Figure 16: Transient shear viscosity for LD at 160~ calculated). (o): 0.2 s -1, (u): 0.5 s -1, (A): 1 s -1.

(experimental and

188

10 6 o

O0

105

0

L

p-g m E (. 0 Z

L o,-g LL

10 4

103

10-1

10~

101

102

Time [s]

F i g u r e 17: T r a n s i e n t p r i m a r y stress coetticient for LD a t 160~ a n d calculated). (o): 0.2 s -1, (~): 0.5 s -1, (A): 1 s -1.

10 6

(experimental

-

105 -,-4

-a=4

>

10 4

o W

103 10-2

.

.

.

.

.

.

,,I

10 -1

9

,

. . . . . .

i

. . . . . . . .

10 o

,i

i

i

l

101

.....

i

10 2

Time [ s ]

F i g u r e 18: T r a n s i e n t elongational viscosity for LD at 160~ calculated). (o): 0.05 s -1, (Q): 0.5 s -1, (A): 1 s -1, (0): 2 s -1.

(experimental and

189

5.4.Conclusion. The ability of the Phan Thien Tanner equation and related models for the prediction of data in shear and elongation has been investigated. Attention has been focused on special simplified cases of the original equation which enable

the understanding of the influence of each parameter. The use of a single parameter equation, removing the affine assumption of the U C M model by replacement of the upper-convected derivative by the Gordon-Schowalter derivative, is the case of the Johnson-Segalman model. This kind of modification does not significantly improve the elongational prediction towards the U C M equation. Moreover, in shear, though the improvement is obvious, discrepancies remain, especially concerning the nonuniqueness of the slip parameter for tangential and normal stresses. The change of kinetics of the junctions also leads to a single parameter equation in the form of the original PTT equation but using the upper-convected derivative. Only the exponential form of the stress term gives a realistic description of the steady elongational behaviour in the long time range. Though this is shown to improve the behaviour in elongation, this was conversely found to be contradictory with a better description of the trend in simple shear because of opposite requirements on the value of the parameters. Indeed, the parameter that controls the kinetics promotes both an elongation thickening behaviour together with a shear-thinning trend which is in contradiction with experimental data on LD for example. At least, using the complete Phan Thien Tanner equation, with non-affine motion and modified kinetics enables a correct description of the data in shear and in elongation. However, the parameters t h a t can be determined for this model are bound to be some compromise. This is n e c e s s a r y in order to minimize the deviation to the Lodge-Meissner rule, due to the use of the Gordon-Schowalter derivative. This is also r e q u i r e d to give a d e q u a t e description of both the shear and uniaxial elongational behaviour. Additional undesirable phenomena in some flows have also been pointed out such as oscillations in transient flows. At least, it is worth noticing that the Phan Thien Tanner model is, in its m a t h e m a t i c a l form, a non-separable equation. However, it has been pointed out t h a t , for some special forms of the r e l a x a t i o n spectrum, a p p a r e n t separability may be displayed [61].

190 6.CONCLUSION Two different constitutive equations, namely the Wagner model and the P h a n Thien Tanner model, both based on network theories, have been investigated as far as their response to simple shear flow and uniaxial elongational flow is concerned. This work was primarily devoted to the determination of representative sets of parameters, that enable a correct description of the experimental data for three polyethylenes, to be used in n u m e r i c a l calculation in complex flows. Additionally, a d v a n t a g e s and problems related to the use of these equations have been reviewed. Both these models find their basis in network theories. The stress, as a response to flow, is assumed to find its origin in the existence of a temporary network of junctions that may be destroyed by both time and strain effects. Though the physics of time effects might be complex, it is supposed to be correctly described by a generalized Maxwell model. This enables the recovery of a representative discrete time spectrum which can be easily calculated from experiments in linear viscoelasticity. Though the Wagner and Phan Thien Tanner equations seem to give adequate description of the observed behaviour either in shear or in uniaxial elongation, it is worth mentioning some peculiarities and key points that should keep the attention of the user to avoid misleading conclusions. These constitutive equations differ in their mathematical form: the Wagner equation is an integral equation whereas the Phan Thien Tanner model is a differential one. This induces important differences in the way they might be treated for calculations in complex flows, since integrals will require particle tracking whereas differential equations will not. These numerical t r e a t m e n t s are generally mutually exclusive since, in the general case, the problem of correspondence between integral and differential forms is not solved. Attempts at finding such correspondences may be found in various papers by Larson [62, 63] especially concerning the Wagner model and the Phan Thien Tanner equation with upper-convected derivative. On the other hand, integral forms are closer to the results of our knowledge of molecular dynamics in entangled polymers and hybrid theories combining

191 simplified molecular models and temporary network equations are worth thinking over. The Wagner equation finds its theoretical basis in the derivation of the more general K-BKZ equation. Unfortunately, it loses p a r t of its original thermodynamic consistency since, for simplification purposes, only the Finger strain measure is taken into account. Doing so, it is no more derivable from any potential function and additionally it does not predict second normal stress differences any more. The equation leads to the definition of a time and strain-dependent memory function which can be further factorized into a time-dependent part (the linear memory function) and a strain-dependent damping function. Though on one hand, there is some experimental evidence for this in limited time ranges, on the other hand, a few experiments might question this strong hypothesis since, for example, the damping function obtained from step shear rate data is found to be different from that in step shear strain. The construction of a single mathematical form of the damping function in shear and uniaxial elongational flows requires the use of a generalized invariant, which includes the effect of both strain invariants I1 and I2 through a proper combination of them. The simplest one being a linear combination can be used in various equations for the damping function, including a limited number of adjustable parameters. Including these features, the Wagner model can give a proper description of experiments in shear and in uniaxial elongation for increasing deformations. When deformation is non-increasing, since the damping function reflects the loss of junctions under the influence of strain, and since it should obviously be an irreversible process, a functional damping term has to be introduced. Nevertheless, this key point for any use in complex flow calculations has to be improved. In its general form, the Phan Thien Tanner equation includes two different contributions of strain to the loss of network junctions, through the use of a particular convected derivative which materializes some slip of the junctions and through the use of stress-dependent rates of creation and destruction of junctions. The use of the Gordon-Schowalter derivative brings some improvement in shear and a second normal stress is predicted, whereas the

192 influence of the kinetics through the trace of the stress tensor is much more important in elongation. Unfortunately, the use of the Gordon-Schowalter derivative brings large discrepancies, especially as far as material objectivity is concerned. Indeed, using it, the principal axes of strain and stress do not rotate together during shear flows and this induces the violation of the Lodge Meissner rule. Consequently, the slip parameter of the derivative is found to be different for tangential and normal stress functions. This becomes evident in the limiting case of the Johnson-Segalman model which, for representative parameters, is found to be a good approximation of the Phan Thien Tanner model in shear. Moreover, this kind of derivative induces spurious oscillations for transient rheological functions. One may try to avoid the problem by the use of the upper-convected derivative, which ensures the coincidence of the principal axes of stress and strain. But doing that, it appears that any kinetics based on the stress amplitude is improper, since materials which exhibits thickening behaviour in elongation are, to the contrary, shear-thinning. Consequently no unique dependence can be expected for these two kinematics. The determination of a single set of parameters in various flows in then bound to be a compromise. Table 7 gives a summary of qualitative performances and problems encountered for simple shear and uniaxial elongational flows, using the Wagner and the Phan Thien Tanner equations or more simple models as special cases of the former. Additional information may also be found in papers by Tanner [46, 64]. All equations presented hereafter can be cast in the form of a linear Maxwell model in the small strain limit and therefore are suitable for the description of results of the linear viscoelasticity in the terminal zone of polymer melts.

193 Table 7: Wagner and Phan Thien Tanner equations- Problems in simple shear and uniaxial extension. Model Comments cD

~D

UCM-Lodge Eqs.(3) or (4) o

Constant viscosity and first normal stress difference. Second normal stress difference is zero. No overshoot and linear limits in transient stress growth. Linear relaxation modulus in step shear strain. Unbounded transient viscosity at high rates. Strain hardening at short time. Linear results or infinite value for steady state viscosity. Second normal stress difference is zero.

Wagner Eqs.(33 ) and

(25)

JohnsonSegalman Eq.(49)

Inaccuracy on the generalized invariant parameter. o p==r

Exaggerates shear-thinning. Lodge Meissner rule unsatisfied (2 slip parameters). Oscillations in transient stress growth. Negative relaxation modulus in large step shear strain. bb o ~=~

Phan Thien Tanner with UCD

c~ cD

o

c~

Phan Thien Tanner Eq.(39) o

Unbounded transient viscosity at high rates. Linear results or infinite value for steady state viscosity. Non-separable equation.

Linear form of the junction kinetics unsuitable. Parameter of the junction kinetics differs from shear. Non-separable equation. Lodge Meissner rule unsatisfied. Oscillations in transient stress growth. Linear form of the junction kinetics unsuitable. Compromise is necessary for the parameters.

194 R~'~C~.

1. 2. 3.

4. 5. 6.

7.

8.

9. 10. 11.

12.

13.

J.D.Ferry,~Viscoelastic Properties of Polymers ~, 3rd edition, JohnWiley &Sons, 1980. R.B.Bird, R.C.Armstrong, O.Hassager,~Dynamics of Polymeric Liquids" Vol.1, Fluid Mechanics, 2nd edition, John Wiley & Sons, 1987. R.B.Bird, C.F.Curtiss, R.C.Armstrong, O.Hassager,~Dynamics of Polymeric Liquids ~, Vol.2, Kinetic Theory, 2nd edition, John Wiley & Sons, 1987. N.W.Tschoegl, ~The Phenomenological Theory of Linear Viscoelastic Behavior- An Introduction ~, Springer Verlag, 1989. P.E.Rouse, A theory of linear viscoelastic properties of dilute solutions of coiling polymers, J. Chem. Phys. 21 (1953), 1272-1280. M.Doi, S.F.Edwards, Dynamics of concentrated polymer systems- Part 1: Brownian motion in the equilibrium state, J. Chem. Soc, Faraday Trans / / 74 (1978), 1789-1801. M.Doi, S.F.Edwards, Dynamics of concentrated polymer systems- Part 2: Molecular motion under flow, J. Chem. Soc, Faraday Trans H 7..44(1978), 1802-1817. M.Doi, S.F.Edwards, Dynamics of concentrated polymer systems- Part 3: The constitutive equation, J. Chem. Soc, Faraday Trans H 74 (1978), 18181832. M.Doi, S.F.Edwards, Dynamics of concentrated polymer systems- Part 4: Rheological properties, J. Chem. Soc, Faraday Trans H 75 (1979), 38-54. M.Doi, S.F.Edwards, ~The Theory of Polymer Dynamics", Clarendon Press, 1986. M.Baumgaertel, H.H.Winter, Determination of discrete relaxation and retardation time spectra from dynamic mechanical data, Rheol. Acta 28 (1989), 511-519. M.Baumgaertel, H.H.Winter, Interrelation between continuous and discrete relaxation time spectra, J. Non-Newt. Fluid Mech. 4.~4(1992), 1536. M.Baumgaertel, A.Schausberger, H.H.Winter, The relaxation of polymers with linear flexible chains of uniform length, Rheol. Acta 29 (1990), 4(D-408.

195

14.

15. 16.

17.

18.

19. 20. 21.

22. 23. 40

25. 26.

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