Molecular Rheology and linear viscoelasticity

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

95

Molecular Rheology and Linear Viscoelasticity G. Marin and J.P. Monffort Laboratoire de Physique des Matdriaux Industriels, URA-CNRS 1494, Universit6 de Pau et des Pays de r Adour, 64000 Pau (France) 1. I N T R O D U C T I O N M e a s u r e m e n t of the linear viscoelastic properties is the basic rheological characterization of polymer melts. These properties may be evaluated in the time domain (mainly creep and relaxation experiments) or in the frequency domain: in this case we will talk about mechanical spectroscopy, where the sample experiences a harmonic stimulus (either stress or strain). One may use the linear viscoelastic data as a pure rheological characterization, and relate the viscoelastic parameters to some processing or final properties of the m a t e r i a l u n d e r study. F u r t h e r m o r e , linear viscoelasticity a n d n o n l i n e a r viscoelasticity are not different fields that would be disconnected: in most cases, a linear viscoelastic function (relaxation fimction, memory function or distribution of relaxation times) is used as the kernel of non linear constitutive equations, either of the differential or integral form. That means that if we could define a general nonlinear constitutive equation t h a t would work for all flexible chains, the knowledge of a single linear viscoelastic function would lead to all rheological properties. On the other side, the linear viscoelastic functions of polymer melts are directly related to their molecular structure, and these relations are now defined well enough to derive, with a reasonable accuracy, what would be the linear viscoelastic behaviour of a given species from its molecular weight distribution, at least in the case of linear polymers. Moreover, there is a definite trend to use rheology as a molecular characterization tool, in the same way as other popular spectroscopic methods such as N.M.R spectroscopy or Size Exclusion C h r o m a t o g r a p h y . However, the inverse problem, i.e., getting molecular weight distribution from rheological measurements, is a difficult (and ill-defined) problem, and this is a very up-to-date area of research due to its obvious practical implications in polymer characterization. The relationship between chemical structure and viscoelastic behaviour is established through molecular models considering that polymers relax or diffuse in the same way: they are considered as flexible statistical chains trapped between the topological constraints created by the surrounding chains.

96 2. L I N E A R VISCOELASTIC BEHAVIOUR OF L I N E A R AND FLEXIBLE CHAINS . BASICS AND P H E N O M E N O L O G Y We will begin with a brief survey of linear viscoelasticity (section 2.1) 9we will define the various material functions and the mathematical theory of linear viscoelasticity will give us the mathematical bridges which relate these functions. We will then describe the main features of the linear viscoelastic behaviour of polymer melts a n d c o n c e n t r a t e d solutions in a p u r e l y r a t i o n a l a n d phenomenological way (section 2.2)" the simple and important conclusions drawn from this analysis will give us the support for the molecular models described below (sections 3 to 6). 2.1. L i n e a r viscoelastic functions The measurable linear viscoelastic functions are defined either in the time domain or in the frequency domain. The interrelations between functions in the frequency domain are purely algebraic. The interrelations between functions in the time domain are convolution integrals. The interrelations between functions in the time and frequency domain are Carson-Laplace or inverse Carson-Laplace transforms. Some of these interrelations will be given below, and a general scheme of these interrelations may be found in [1]. These interrelations derive directly from the mathematical theory of linear viscoelasticity and do not imply any molecular or continuum mechanics modelling. In t h e various f o r m u l a t i o n s of the m a t h e m a t i c a l t h e o r y of l i n e a r viscoelasticity, one should differentiate clearly the m e a s u r a b l e and nonmeasurable functions, especially when it comes to modelling: a p a r t from the material functions quoted above, one may also define non-measurable viscoelastic functions which are pure mathematical objects, such as the distribution of relaxation times, the distribution of retardation times, and the memory function. These mathematical tools may prove to be useful in some situations: for example, a discrete distribution of relaxation times is easy to handle numerically when working with constitutive equations of the differential type, but one has to keep in mind that the relaxation times derived numerically by optimization methods have no direct physical meaning. Furthermore, the use of the distribution of relaxation times is useless and costs precision when one wishes simply to go back and forth from the time domain to the frequency domain. This warning is important, given the large use (and sometimes overuse) of these distribution functions. 2.1.1. Functions in the time domain The relaxation modulus G(t) is the value of the transient stress per unit strain in a step-strain experiment. This type of experiment may be achieved with modern rotary rheometers with a limited resolution in time (roughly 10 -2 s). If one wishes to evaluate G(t) at shorter times, it is necessary to derive G(t) from the high frequency G*(co) data by an inverse Carson-Laplace transform. The creep function J(t) is the transient strain per unit stress in a step-stress experiment. The resolution at short times is also limited from instr~ment response and sensitivity. J(t) at short times may also be derived from the high frequency complex compliance data.

97 One m a y also use a memory function m(t) within the integral formulation of l i n e a r a n d nonlinear viscoelasticity; this memory function m(t), which is t h e derivative of the relaxation function G(t), is not a measurable function.

2.1.2. Functions in the frequency domain Three complex functions m a y be used to characterize the linear viscoelastic behaviour in the frequency domain" - The complex shear modulus, which is the complex stress to complex strain ratio" G * (co) = complex stress c*(r - ------- = G'(co)+ j G"(co) y*(~) complex strain

(2-1)

- The complex viscosity, which is the complex stress to complex rate of strain ratio" 11"(o)) =

complex stress = (co-----)) ~* = Tl' (co)- j 11"(o)) = ~ _*(co) _G complex rate of strain ~/* (co) jco

(2-2)

- The complex compliance, which is the complex strain to complex stress ratio" j .(co) = complex strain = 7*(co) = j,(co)_ j j,,(co) = 1 (2-3) complex stress ~* (co) G* (co) These three functions are related via simple algebraic relations" G * (co) = ~

1

J*(r

= jco T1* (co)

(2-4)

Although the complex shear modulus is not the most appropriate function to use in all cases, we will describe the linear viscoelastic behaviour in terms of this last function, which is the most referred to experimentally; furthermore, molecular models are mostly linked to the relaxation modulus, which is the inverse Fourier transform of the complex shear modulus.

2.1.3. The distribution of relaxation times By analogy with a generalized Maxwell model, it is possible to write the relaxation modulus or the complex modulus as a sum of the contributions of n individual Maxwell models" 12

G(t)=~G k=l

12

k(t)=~G ke

----

t

~k

(2-5)

k=l 4oo

and G * (co)= jco ~ G(t) e -j~t dt 0

(2-6)

98 n

n

hence G*(co)= ~ G k *(co)= ~ jOTlk k:l k=ll+f~k

(2-7)

with Tlk=Gk~k. In terms of molecular models (section 3), the set of individual times ~k and weighting factors Gk is imposed by the model. It is also possible to derive a set of relaxation times and weighting factors numerically by optimization or approximation methods from the experimental data. In that case, the relaxation times have no real physical meaning and are simply numerical/empirical parameters which allows one to represent the viscoelastic behaviour as a sum of decaying exponentials which are handy to use for numerical analysis. It is also possible to give integral forms of these sums" oo

G * (co) = jr

H(z) 0 ~+jcoz dz +oo

(2-8)

t

and G(t)= ~ H(z)e ~ d ln(z)

(2-9)

--oo

H(z) is a continuous distribution of the logarithms of relaxation times and is called the "relaxation spectrum" by rheologists, whilst the "true" distribution of relaxation times is zH(z). We have r e p o ~ on Figure I the normalized distribution of relaxation times for 4 polystyrene samples with polydispersity indices ranging from 1.05 to 4.2 [2]. It is clear that the distribution of relaxation times broadens with the distribution of molecular weights; these features will be analyzed in terms of molecular models in sections 3 to 6.

A

v

o

I

-3

-2

-I

0 Log z /Zo

+1

*2

Figure 1 9Reduced distribution of relaxation times for atactic polystyrene samples having different values of polydispersity index P=l.05 (A), 1.25 (B), 2.45 (C), 4.2 (D)[ref. 2]

99 2.2. T h e m a i n f e a t u r e s of the l i n e a r viscoelastic b e h a v i o u r of p o l y m e r melts: We will discuss in this section the variations of the viscoelastic parameters derived from linear viscoelastic measurements; all these parameters may be derived from any type of m e a s u r e m e n t (relaxation or creep experiment, mechanical spectroscopy) performed in the relevant time or frequency domain. The discussion will be focused however on the complex shear modulus which is the basic function derived from isothermal frequency sweep measurements performed with modern rotary rheometers.

log G* (W) -

G,.-...-..~ ~ - . - - . ~

G' ~ G

fMu

-//,,'M2 f

~~

G o'

G

t er min al

e

log UJ plat eau

POLYMERIC

"BEHAVIOUR

transition

glass y

SMALL SCALE MOLECULAR MOTIONS

Figure 2 9 Schematic of the variations of the complex shear modulus of linear polymers (dotted lines: molecular weight M2>M1). The main features of the linear viscoelastic behaviour of polymeric melts in the frequency domain are reported on Figure 2 : - the lowest frequency range describes the slowest relaxation motions of the macromolecules. The double logarithmic plot of G' and G" exhibits slopes of -respectively- 2 and 1, leading to two characteristic parameters. The zero-shear viscosity" ~o = lim G"(co____~) co-,0

CO

(2-10)

100

The limiting compliance: j0=lim G'(r ~-~o [G"(co)] 2

=1

limG'(c~

Tioz ~-,o r ~

(2-11)

which is the elastic parameter governing the main features of the elasticity of the melt (first normal stress difference, extrudate swell, etc...). The zero-shear viscosity is the norm of the relaxation spectr~m :

Tlo= ~~H(z)d In z

(2-12)

The product TIo jo is the characteristic relaxation time zo of the terminal region. In terms of molecular models, this time scales as the longest relaxation time. In terms of the distribution of relaxation times H(z), zo is the "weight-average relaxation time" which is the average relaxation time related to the second order moment of the relaxation spectm~m : ~o = no jo =< ~w >= ~ z2H(z)d In

(2-13)

At intermediate frequencies, monodisperse polymers exhibit a well-defined "plateau region" where G'= constant G~ (Figure 2). For a given macromolecular species, the value of the plateau modulus is a characteristic feature t h a t does not depend on molecular weight. The only way to lower the plateau modulus is to add small compatible molecules, either of the same species or not : this is, for example, what is done for Hot-Melt Adhesives (HMAs) when adding a "tackifying resin" which softens the polymer and improves the "tack". In terms of the distribution of relaxation times, the ratio TIo/G~ is an average relaxation time (we may call it "n-tuber-average relaxation time"), which is the first order moment of the normalized relaxation spectm, m : < ZN >= no/ G~ = i ~H(z)d In H(~)d In z

(2-14)

The ratio: < Zw >

o co = Je

(2-15)

<~N >

is a "polydispersity index of relaxation times" which characterizes the broadness of the distribution of relaxation times. For monodisperse species, the experimental value of this ratio lies between 2 and 2.5, whatever the polymer nature. This value increases largely as polydispersity increases. One of the direct practical

lO1 applications of molecular modelling will be to relate the distribution of molecular weights to the distribution of relaxation times.

2.2.1. The effects of chain length 2.2.1.1. The zero-shear viscosity Tl0 The variations of the zero-shear viscosity of monodisperse polymeric melts and concentrated solutions exhibit two domains, each being characterized as a first approximation by a power law exponent" at low molecular weights, corresponding to less than 200 monomeric units, the exponent lies between 1 and 1.5. This domain will be analyzed in section 5. above a critical molecular weight Mc, the power law exponent is 3.4-3.5. As far as viscosity is concerned, Mc defines the begining of a regime where the macromolecular chains are viewed as "entangled", which explains the large molecular weight dependence of viscosity. The entangled regime will be modelled later using the "reptation" concept (see section 3). -

-

f

!

i _

i 0 K) /

o/+

13

vi r I0 e n

/,

I'--I

(J

c;

/

#o,+

IO s -

~_

g

o

o

.

/

o/

_

0

10 4 -

10 2 10 3

I

I

I

10 4

10 5

10 6

log M

Figure 3 : Molecular weight dependence of the structure factor of nearly monodisperse polystyrene samples [3]. Figure 3 illustrates this type of behaviour for various series of anionic polystyrene samples with a fairly narrow distribution of molecular weights. In the case of entangled polydisperse materials (Mw>>Mc), the zero-shear viscosity follows approximately the same molecular weight dependence as for monodisperse species when the viscosity data is plotted as a function of the weight-average molecular weight Mw; i.e. : 110 = A(T) M~ 4

(2-16)

I02 2.2.1.2. The plateau modulus G~ The plateau region begins to be developed at molecular weights somewhat above Mc; it is however a well-defined elastic parameter for a given chemical species of high molecular weight. Comparing different polymer species, its value increases with the flexibility of the chain (i.e: GNPolystyrene< o o GNPolyethylene). The use of the theory of rubber elasticity may give an order of magnitude of the average molecular weight Me between entanglements which create a temporary network: G~ = pRT

(2-17)

Me '

p being the polymer density and T the absolute temperature. The critical molecular weight Mc is roughly two times the molecular weight between entanglements Me.

2.2.1.3. The limiting compliance jo In the entangled regime, the limiting compliance jo of monodisperse samples is also an elasticconstant characterizing a given polymer chemical species: contrary to the plateau modulus, its value depends on the distribution of molecular weights, i.e.,on the polydispersity index as a firstapproximation. This is a very important point, as m a n y elasticeffects(firstnormal stress difference,extrudate swell, ...)of the melt are governed by the limitingcompliance. For purely monodisperse samples, the product jo G ~ has a value close to 2 for all flexiblepolymers. Below the entangled regime, the limiting compliance follows approximately a linear dependence with molecular weight according to Rouse's theory (see section 6)jo = 0.4 M . pRT

(2-18)

The molecular weight value M'c where the compliance becomes independent of molecular weight is larger than M c (M'c--3Mc), which indicates that the "polymeric" regime seems to appear at higher molecular weights for elastic properties compared with viscous properties. So one has to keep in mind that the chain length (or molecular weight) at which "entanglements" effects begin to appear depends strongly on the physical property measured (melt viscosity, melt elasticity,self-diffusion,etc...)(see in particular chapter L1). In the case of polydisperse polymers, the limiting compliance increases strongly with the broadness of the distribution of molecular weights. The limiting compliance is not, however, a simple function of the polydispersity index, because its value depends on the shape of the distributionitself.There is indeed no simple correlation with any molecular weight moments (averages), and molecular models will be really helpful to describe the elasticity of the melt.

103

2.2.2. The effect of temperature: For a given polymer, the viscoelastic curves (either moduli or compliances) obtained at different temperatures in the plateau and terminal regions are simply afflne in the frequency (or time) scale, in a double logarithmic plot. The use of this time-temperature equivalence allows one to obtain "master curves" at a reference temperature, which enlarges considerably the experimental window. For glass-forming materials such as polystyrene, polymethylmetacrylate, polycarbonate, polymerists describe the shift factor aT in terms of the WLF equation: -c~ -To) In aT = (co + T - T o ) '

(2-19)

T being the experimental temperature and To the reference temperature to which the data is shifted. The WLF may be reduced to the Vogel equation which describes the viscosity of molten glasses and supercooled liquids"

B/af

In a T = _T0 ---~-

B/af W-'-~_ '

(2-20)

where the limiting temperature Too may be related to the glass transition temperature Tg by the approximate rule: Tg-Too = c2g -- 60 ~ C. The entropic nature of the elasticity of the melt implies also a slight vertical sbJR in the plateau and terminal regions. This shi~" b T = P~176 pT'

(2-21)

may be neglected when using the time-temperature equivalence in a limited range of temperatures. The time-temperature equivalence implies that the viscosity (or relaxation times) of polymers may be written as the product of two functions : no = $(P(M)). M (T)

(2-22)

The mobility factor M (T) describes the segmental mobility of the chain : it depends mostly on temperature and pressure, but may be affected by the presence of small chains (such as solvent molecules or small chains of the same chemical species as the polymer). For concentrated polymer solutions, the addition of small molecules affects mostly the glass transition temperature (hence Too), and the value of B (eq.2-20) is essentially the same as for the bulk polymer. A plastifyer will decrease the value of Too, and hence increase the segmental mobility. On the contrary, the addition of a tackifying resin which has a higher Tg than the polymer will increase the segmental mobility of the polymer in the case of formulations of Hot-Melt adhesives.

104 The structure factor $(P(M)) describes the topological relaxation of the macromolecular chains: t ~ s is the function which will be described by molecular models, P(M) being the distribution of molecular weights. Here lies a very impo~t point: if one wishes to "isolate" the topological effects in order to test molecular models, one has to use rheological functions defined at the same segmental mobility, and hence the same value of the mobility factor: as far as viscosity is concerned, the reduced function Tlo/M (T) will be used instead of the viscosity itself.

2.2.3 The effects of concentration (concentrated solutions): In the case of concentrated entangled solutions, the "elastic" parameters follow power law dependences as a function of polymer volume fraction r : ( 0

0 GN)sol=(GN)bulk

{~{~

( jO)sol=(jO)bulk ~ - a

(2-23) (2-24)

with an exponent a - 2-2.3 for entangled chains, so the product jo G~(which reflects the polydispersity of relaxation times) remains the same whatever the concentration. That means that the effects of the addition of small compatible species on the elastic parameters are mainly topological, i.e., the nature itself of the solvent molecules has a very small effect on the melt elasticity and the shape of the distribution of relaxation times. On the contrary, the effects of dilution on the polymer viscosity will be twofold : - a topological effect on the structure factor $ that will be described by molecular models; a change of the mobility factor M, that may either increase or decrease, depending on the plastifying -or antiplasfif3dng- effect of the molecules added to the polymer. -

2.2.4 The self-similarity of the viscoelastic behaviour of flexible chains The above phenomenological description of the viscoelastic behaviour of polymer melts and concentrated solutions leads to the following i m p o r t a n t conclusions 9if one focuses on the behaviour in the terminal region of relaxation, what is usually done for temperature (time-temperature equivalence) may also be done for the concentration effects and the effects of chain length; one may define a "time-chain length equivalence" and "time-concentration equivalence"[4]. For monodisperse species, the various shifts along the vertical (modulus) axis and horizontal (time or frequency axis) are contained in two reducing parameters: the plateau modulus G~ and a characteristic relaxation time, either Zw = qo jo or ZN = rio/G~. A plot of G*(cOZo)/G~ - where zo is either T~Vo r 1; s - in the terminal relaxation region is a universal function independent of temperature, concentration, chain length, and independent also of the chemical nature of the polymer (Figure 4). This self-similarity of the viscoelastic behaviour of monodisperse linear chains, whatever their chemical structure, may be extended to polydisperse species having the same shape of the molecular weight distribution (i.e., the same

105 polydispersity index as a first approximation). This implies some universality in the large-times relaxation processes of entangled polymers. As a consequence, the general features of the mechanical relaxation of long and flexible polymeric chains will be described by molecular models that do not "see" the local structure but describe the overall diffusion and relaxation of these chains in a universal way. Hence the power of the models described below lie in their universality:, it is easy to shift from one polymer to another, changing only a few parameters linked to the local scale structure of the polymer under study.

0-

o

@

-2

l

-I

I

0 log (~qo G~

I

I

,_,

I

2

Figure 4 : Master curve for the linear viscoelastic behaviour of entangled polymers in the terminal region of relaxation : V Polystyrene, bulk (M=860000, T=190~ Q Polyethylene, bulk (M=340000, T=130~ A Polybutadiene solution (M=350000, polymer=43%, T=20~ [from ref.4]. 3. THE CASE OF E N T A N G L E D M O N O D I S P E R S E L I N E A R S P E C I E S : PURE REPTATION

3.1. The basic reptation model The reptation concept was introduced by de Gennes [5] in 1971: it is based on the idea that long and flexible entangled chains rearrange their conformations by reptation, i. e., curvilinear diffusion along their own contour. De Gennes considered the reptation of a linear chain among the strands of a crosslinked network which create p e r m a n e n t topological obstacles. First, the dynamics of the wriggling motion of the chain along its own contour (what Doi and Edwards called later the primitive path) was described by de Gennes in terms of a diffusion equation of a "defect gas": he showed that this motion is fairly rapid: its longest relaxation time Teq is proportional to M 2, where M is the molecular weight of the chain (that time Teq would be equivalent to the ~B relaxation time in the slip-link model; see text below and Fig.8).

106

9

9

9

9

9

9

9

9

Q

9

Figure 5 9The basic concept of P.G. De Gennes 9 reptation of a chain trapped in a tube-like region by migration of "defects" along the chain. At times t >Teq, the wriggling motion results merely in a fluctuation around the primitive path, so the chain moves coherently in a one-dimension diffusion process, k e e p i n g its arc length constant. The macroscopic diffusion coefficient of a reptating chain scales with chain length (molecular weight) as" D o, M-2,

(3-1)

and the time for complete rearrangement of conformation" 1; o~ M 3.

(3-2)

This time is the longest relaxation time of a linear chain. We will refer to it as the "reptation time".

e

,

~

-

,

.

e

/

.-

Figure 6 9The tube concept: The real chain is trapped between entanglements and is wriggling aroud the "primitive chain" (full line).

Doi and Edwards used this concept to derive the viscoelasticproperties of polymer liquids from the dynamics of reptating chains [6]. They ass1~med that

107 reptation would be the dominant relaxation process for polymer melts, even in the absence of a permanent network. This is a very strong assumption, as the topological constraints are made by surrounding chains that also diffuse by reptation. This assumption was justified later when analyzing the constraints release (or tube renewal) process (section 4).

~/

I "- \ ?

J

f /

~\

\ __kJ

Figure 7 9Reptation: the chain disengages from its initial tube by back-and-forth motions; the time necessary for a complete renewal of its initial configuration is the "reptation time" which is the longest relaxation time of the polymer. When analyzing the overall diffusion of a single chain due to Brownian motion, the topological constraints made by the surrounding chains confine the chain in a tube-like region. The centreline of the tube is called the "primitive path" and can be regarded as the curve which has the same topology as the real chain relative to the other polymer molecules; the real chain is then wriggling around the primitive path. The Doi-Edwards calculation is based on the theory of rubber elasticity. In order to calculate the time-dependent properties, the contribution of individual chains to the stress following a step strain is evaluated; then the relaxation of the stress is related to the conformational rearrangement of the chains by a reptation process. For this calculation, the topological constraints along the chain are represented by frictionless rings around the chain (Fig. 8), which is another way of describing entanglements. The succession of segments ("primitive segments") joining these "slip-links" along the chain is called the "primitive chain". As the sliplinks are now local constraints, the continuous nature of the constraints is accounted for by assuming a natural curvilinear monomer density along the chain. So all three models (reptation among fixed obstacles/tube model/slip-links model : Figs. 5 to 8) are equivalent in essence. When submitting the polymer sample to a sudden deformation (step strain), the primitive path is distorted by affine deformation (Fig. 8b), and the curvilinear monomer density is perturbed from its equilibrium value.

108

B

D

A

C

a) t < O : e q u i l i b r i u m

A

C

F

E

c) t--_ 1;a ( A p r o c e s s )

state

E

b) t = O : s t e p s t r a i n C

E

d) t _=_ ~s (B p r o c e s s )

C~

B'

E~

D'

e) t =_ 1:c (C p r o c e s s : reptation)

Figure 8 " Relaxation of a polymeric chain after a step-strain deformation[6]: process A (8c): reequilibration of chain segments; process B (8d): reequilibration across slip links; process C (8e): reptation. Then the chain will relax: The f i ~ t relaxation process (called the A relaxation process; Fig. 8c) which occurs at the shortest times will be a local reequilibration of monomers without slippage through the slip-links. In other words, it is basically a Rouse relaxation process between entanglement points which are assumed to be fixed in that time scale. The characteristic relaxation time of this process is rather short and is independent of the overall chain length (see below). The second relaxation process (B process; Fig. 8d) is a reequilibration of segments along the overall chain, i.e., across slip-links. It is basically a retraction of the chain to recover its natural curvilinear monomer density, which may be depicted as a Rouse relaxation process along the entire chain. In the last relaxation process (C process; Figs. 7 and 8e), the chain renews its entire configuration by reptation. The viscosity, p l a t e a u modulus, limiting compliance and m a x i m u m (terminal) relaxation time derived from the basic D-E model are power laws of the molecular weight M: Tlo or M 3, G~ or M o, jo or M 0, to = ~o jo o~ M 3,

(3-3)

109

and

jOG~=6. 5

All viscoelastic functions may be expressed in terms of a single reptation parameter (for example the plateau modulus or tube diameter) and the monomeric friction coefficient (or mobility factor in our terminology), in agreement with the above phenomenological presentation. If the general features of the viscoelastic behaviour are well depicted, the experimental molecular weight dependence of these parameters is : 110or M3.4, G~ o~ M o, jo o~ M o,

(3-4)

~0 = T10jo or M3.4, and j O G ~ = 2 . In particular, the fact that the experimental "polydispersity of relaxation times" jo G~ is larger than the theoretical value indicates the presence of other relaxation processes. In the following section we will describe a somewhat different analytical derivation of the Doi-Edwards model taking into account additional relaxation processes (in particular the Ta (glass transition) relaxation). Our derivation, which gives a good quantitative agreement with the observed linear viscoelastic behaviour for a large number of linear polymers, keeps however the basic physical concepts of the reptation model. Hence this model is dedicated only to the case of entangled linear monodisperse species. The case of branched polymers, polydispersity and short chains effects will be presented in, respectively, sections 4,5 and 6. The same basic models are also used in chapter 1.1 and 1.3. In chapter 1.1, the mutual diffusion in polymer melts is related to the reptation and constraint release processes, whereas in chapter 1.3 the relaxation of chain segments along the polymer chain is investigated in terms of the some relaxation mechanisms as described below. 3.2. Detailed d e r i v a t i o n of linear viscoelastic p r o p e r t i e s of l i n e a r s p e c i e s [7] 3.2.1. The (A) relaxation process That relaxation process may be defined as a Rouse diffusion between entanglement points. The characteristic relaxation time of the (A) process is : ~ob2 N 2, 1;A = 6~2ksT

(3-5)

where Ne is the number of monomers between e n t a n g l e m e n t points, ks Boltzmann's constant, T the absolute temperature, b the effective monomer length

110 (b=C~o 1,1 being the monomer length) and ~o the monomeric friction coefficient. The relaxation function associated to the (A) process is :

FA(t) =

expF-L P~At],J

(3-6)

p=l

where Ne is the number of Kuhn segments between entanglements. In order to define completely the relaxation function, we have to determine the initial modulus

G'N SO: O(t) = GN FA(t),

(3-7)

G'N is a function of strain and may be written as : GN = pRT

1

Me (~Ul>o ,

(3-8)

w h e r e p is the polymer density, E is the strain tensor, u a unit vector corresponding to the vector linking two entanglement points (slip-link segment) [6]; 1 the < ~ / o term expresses the fact that the molecule is outside its equilibrium configuration" from Doi-Edwards theory G N = 4 pRT.

5 Me

We may then write the relaxation modulus corresponding to the (A) relaxation process (short times) as :

N~e x p [ _ p t21 GA(t) = 4 pRT ~ 5 M e p=l L ZA J"

(3-9)

3.2.2. The (B) relaxation process Doi and Edwards postulate that the chain recovers its equilibrium monomer density along its contour by a retraction motion of the chain within the tube. That motion, induced by the chain ends, leads to a relaxation function : 8

FB(t) =

poddZp2/t;2

exp[-p2t] L -~-BJ'

(3-10)

with: ~0 b2N2 ~B =

3g2kBT

where N is the number of segments along the chain.

(3-11)

111 Viovy [8] describes that re-equilibration process as an exchange of monomers between neighbouring segments : he calls that process "reequilibration across sliplinks", and the corresponding relaxation function may be written as : N/Ne Ne [ p2t] FB(t)= ~ - - ~ - e x p - - - - - . p=l

(3-12)

~;B J

This function corresponds to a Rouse spacing of relaxation times and gives a better fit of the experimental data than Eq. 3-10. Hence the relaxation modulus of the (B) process may be written as a hmction of the entanglement density N/Ne:

G B(t)= 4 pRT N~~ Ne

5 Me

[ p2t]

-'NeXPL-Bj

(3-13)

3.2.3. Reptation: relaxation process C In that final relaxation process the molecule recovers its final isotropic configuration by a reptation motion. The characteristic time for the reptation process is (see also chapters L1 and L3):

1 ~0a2Ne(N) 3 ~:c =--g kBT "~e '

(3-14)

where a is the tube diameter (a 2 = Ne b2). The relaxation function is given in the original Doi-Edwards picture [5-8] by the equation" Fc(t)= E p28~2exp[-p2t]

podd

(3-15)

k "~'CJ

The plateau modulus of that relaxation domain is :

=RT

(3-16)

Me ' and hence the relaxation modulus is: Go(t)= pRT

8 exp[-pat] podd

(3-17)

k -~-cJ

The refinement introduced by Doi [9] who considers tube length fluctuations is more relevant to experimental scaling laws as far as the viscosity/molecular weight dependence is concerned. Following that concept, Gc(t) can be cast into the integral form :

112

Oct':0RT[ Me JO4N/N~

e

'/ '

/

- ~ ( i i d~ + ~2v/s, exp - ~(2i

~,'

(3-18)

where N ~4v4 ~(1) = Ne 16 Zc

2v ~ < ~]N/Ne,

v

~(2) =

~ - 4N / Se

(3-19)

2v

~c

1 > ~ > ~~/'N/N~,

(3-20)

The v parameter may be approximated to 1 for highly entangled chains.

3.2.4. The Ta high-frequency relaxation domain: the transition region between the rubbery and glassy regions In order to give an analytical representation of the mechanical properties in the high frequency range characterizing local motions in the molecular chain, we used analytical forms derived from studies on dielectric relaxation : a Cole-Cole or Davidson-Cole equation generally gives a good fit in the transition region; we used in the present case a Davidson-Cole equation, that presents the advantages of being truncated at large times and to give analytical forms both in the time and frequency domains" G *HF = Goo -

Go. (1+ j ~ H F ) 1/2'

(3-21)

with"

1;HF -

~o 12 X2kBT

(3-22)

The inverse Fourier transform of Equation 3-21 gives the relaxation modulus"

]

(3-23)

Other m a t h e m a t i c a l forms may be used to describe the high frequency relaxation. These various equations, either phenomenological or based on diffusion defect models lead to a characteristic relaxation time ~;HFof the glass transition (Ta) domain of the same order of magnitude.

113 As a s u m m a r y , the characteristic relaxation times of the various relaxation mechanisms presented here above are linked to each other by (see also Chapter

1.3):

I:.~ = ~1 I:i Ne2 SB = 21;A( ~ ) 2 = ~l 1;i N2 N

(3-24)

N3

1;c = 3ZB ~ee = I;i Ne with ~i = ~0 b2 ~2kB T " 3.2.5. Viscoelastic function in the whole time/frequency domain Thus the relaxation modulus may be calculated from a very limited number of physical parameters (G ~ Goo and ~i), with no "ad-hoc" parameters, in a time range covering the initial glassy behaviour down to the terminal relaxation region. For a typical polymer, this range exceeds ten decades of times. The complete expression of the relaxation modulus is :

NINe exp - zr

G(t) = Me

+-

2v e exp 4N/N

~ 2)

[ p2t 1 N~eNe [ p2tll + G~ [ 1 - e f t exp--+ --~-exp5 Me L p=l I:AJ p=l ~BJJ

4pRTIN~e

(#--~) ] .

(3-25)

The complex shear modulus is the Fourier transform of Equation 2-25 : G * (co) = G~

2,,

f04N/Ne

jcozr 1+ jco~(1)

d~ +

~1

jco~r ] 2v d~ ~]N/ Ue 1+ jC0Z~(2)

NNe

+ -G~ e Ne jc0(1:B/p2) + jc0(~A/p2) 5 k p=l N l+jco(~ B/p2) p=l 1+ jco(xA/p2)

] I

+ Goo 1-

1 1

1 .(3-26) (1+ jO~HF)2

We have reported on Figs 9 through 12 a comparison between the experimental complex shear modulus and its theoretical calculation (full line) for two polymer species. The model fits reasonably well the linear viscoelastic properties of a large number of linear polymers ranging from polyolefins to glass-forming polymers. This calculation gives us the basic "long-chains monodisperse behaviour" which feeds the more complete derivation taking into account the effects of constraints release (section 5).

114

,oj.

1

f 8.-

- 8I Eo

~-6

~,6 = o

4

4

i

-5

l

0

5

I0 log ~

log t~ (sec -I]

(sec-al

Figures 9 and 10 9 storage (G') and loss (G") moduli of nearly monodisperse polystyrene samples at 25~ 9 (A)M=900000; (Q)M=400000; (O)M=200000; (0) M=90000; full line ( )" theory (eq.3-25) [from ref. 7].

I~ f

:

~'

I

1

i

"

'l

1

1

/ 8~-

Io

l

~

;

~

~

;

J

!i 1

8

4 i

loq ~

(sec J]

log ~J (sec -~)

Figures 11 and 12: Storage (G') and loss (G") moduli of nearly monodisperse polybutadiene samples at 160~ (A)M=361000; (~)M=130000; (O)M=39400; full line ( ): theory (eq.3-25) [from ref.7]. 4. E N T A N G L E D M O D E L . B R A N C H E D POLYMERS Branched polymers may be classified into two categories from the point of view of rheology : - polymers with short branches (Marm<
115

lower plateau modulus, so the formalism used for linear flexible polymers may still be used with a good approximation; - polymers with long entangled branches. In spite of the fact that the mlmber of branches per molecule is generally small, experimental data show a tremendous viscosity increase with branch length and a much wider distribution of relaxation times compared with linear polymers. On the other side, the plateau modulus values as well as the temperature dependence of viscosity and relaxation times are close to what is obtained for linear chains of the same species. Another difference with linear chains is the regular increase of the limiting compliance with molecular weight. We will deal in this review article with monodisperse, model-branched polymers in order to describe the basic relaxation modes of branched polymers. The concepts described below are the source of current attempts to describe the viscoelastic properties of complex tree-like structures which are close to those found in low density polyethylene, for example. One may found interesting approaches of that problem in recent papers presented by Mac Leish et al [10]. If one considers that the reptation process is dominant for linear chains, one has to imagine additional processes of diffusion for polymers with long branches. The experimental data suggest strongly, however, that the basic kinetic unit of the chain (whatever it is) is the same as for linear chains : the Rouse-like A and B processes are still there, which are still strong imprints of the "tube".

......

:

:

-.:

(a)

(b)

(c]

(d)

Figure 13 : T h e picture of de Gennes: "reptation of stars": the branch has to go back to its attachment point to renew its configuration. The basic models consider well-defined star-branched polymers. De Gennes [11] imagined in 1975 a simple relaxation mechanism of a branch based on the Brownian motion of an arm of a star-branched molecule in a network of fixed obstacles (Figure 13). From statistical considerations, the time necessary for a branch to renew its configuration is : I;m or f ( N a r m ) e x p ( k N arm)

where Narm stands for the number of chain segments per arm.

(4-1)

116 The e x ~ n e n t i a l dependence of the relaxation time with arm length is a constant feature for all models describing the renewal of configuration of long branches, and the debate has focused on the non-exponential term f(Narm). Doi and Kuzuu [6] have proposed a somewhat different approach based on the tube concept. They start with three basic assumptions : - the segments of a branch are confined within a tube; - the tube deforms amnely with the macroscopic deformation; the centre of the star is assumed to be fixed during the relaxation of the branch. The relaxation of a branch occurs by a retraction process within the tube and the branch end is a s s u m e d to be in a potential barrier. The m a x i m u m relaxation time of a branch is analogous to De Gennes' result (eq. 4-1) with a predicted value of v=8/15. Doi and Kuzuu [6] subsequently derived the the relaxation modulus of a star-branched polymer as : -

G(t)

4---G~ 15

ex

-

d~

with

z~ = Zrn ~2exp(a(~2-1))

(4-2)

(4-3)

8 Narm 15 N e "

with a=~.

"

i

" ~-A

9

9

9 ~e

9

9

9

9

"]'X2

9

i

: (a]

[b]

Figure 14 9Another picture of the disengagement of a branch from its initial p a t h is a '"oreathing" of the branch by fluctuations in path length. Pearson and Helfand [12] used a somewhat similar approach to determine the characteristic relaxation time for a branch to disentangle; their calculation leads to a similar form, with a different exponent for the front factor :

~rn~

Ne )

exp v Ne ).

(4-4)

I17

Ball and Mac Leish [13] used the same concept of the free end of a branch in a potential well, creating a process of"dynamic dilution" (Figure 15) which results in a v a r i a t i o n of the molecular weight between e n t a n g l e m e n t s during the disentanglement process of a branch. The mad'mum relaxation time of a branch is the same as that given in equation (4-4), but the relaxation modulus which takes into account the gradual disentanglement of the branch is given by" G(t) = ~

Jo

1

exp -

(4-5)

ds

Mann being the molecular weight of a branch.

(a)

(b)

(c)

Figure 15 : The various models of Mac Leish lead rather to disengagement of a branch by fluctuations from its ends. The tree-like cloud is the trace of the agitation of chain segments over a period of time of the order of Zm. Some general important remarks may be formulated regarding the characteristic viscoelastic parameters of these polymers with long chain branching : the maximum relaxation time does not depend on the total molecular weight, but depends essentially on the molecular weight of the branch, and more precisely on the entanglement density of the branch Narm/Ne. - all models predict an exponential dependence of the zero-shear viscosity and terminal relaxation time as a function of the entanglement density on the branch. This is confirmed by the experimental data obtained on model-branched monodisperse samples. The v factor which appears in the exponential term is fairly close for all models, ranging from 0.5 to 0.625; this parameter is also related to the polydispersity of relaxation times jo G O(see section 2.2) as 9 -

jo G~ = 5 v Narm 4 Ne jo G~ = v -Sarrn --Ne

(Doi-Kuzuu)

(Pearson-Helfand)

(4-6)

(4-7)

118

1 Narm Je~ G~ = ~ v Ne . (Ball-Mac Leish)

(4-8)

which explains the broadening of the distribution of relaxation times as well as the linear dependence of the limiting compliance with respect to molecular weight (as opposed to linear polymers). It is possible to recalculate the entire relaxation function, including the A, B, and glass transition processes in the same way as we did for linear polymers, with yet another remark: it can be shown by theoretical arguments that the equilibrium relaxation time of the entire branch along its own contour, which is the equivalent of the B process of a linear chain, is 4 times the value of ZB of a linear chain of same length: 4 ~o b2 Teq = ~/t2kB T N2arm"

(4-9)

We have reported on Fig. 16 the complex shear modulus of two star-branched polybutadiene samples at 25~ The full lines have been calculated using the Ball and Mac Leish model for the terminal relaxation region, whereas the same relaxation functions as for the linear polymers have been used regarding the A, B and glass transition domains. Hence: M +

5

G~

SO 1 e Ne

exp-z-~

jco(Teq/p2)

+

L p=l N 1+ j~(Teq/p2)

109/

~IO s "6

ds

p=l

e J~(~:A/p2) 1+ j - ~ 2 7~-2)

i

i

i

10-2

I

i I0 2

+G.. 1 -

i

I

I

i I0 4

I I0 6

1 I0 e

1 -1.(4-10) 1 (I+jo HF)

-

i0 3

"~10

I0 -'~

r

ioIO

-~)

Figure 16 : Complex shear modulus of two nearly monodisperse star-shaped polybutadiene samples at 25~ ( - ) 3 branches, total M=164 000; (O) 4 branches M= 45 000.

119

The essential physics of the mechanical relaxation of star-branched polymers seems to be well-understood when the branches are highly entangled. However, the transition domain where the molecular weight of the branches goes down to Mc is not yet well-described : as molecular weight decreases, one observes a strong decrease of the plateau modulus as well as important effects of constraints release, along with maybe additional relaxation processes. Also, when one deals with polydisperse branched polymers, or blends of linear and branched polymers, constraints release (section 5) becomes rapidly the dominant relaxation process, and it is difficult at the present time to give a clear picture of the effects of polydispersity of molecular weights or polydispersity of branches as simply as it is done for linear polymers. 5. E N T A N G L E D P O L Y D I S P E R S E LINEAR CHAINS : DOUBLE REPTATION 5.1. T u b e r e n e w a l The simple reptation concept proposed by de Gennes and developed by Doi and Edwards deals with permanent entanglements creating a fixed tube around each chain. However, as in a melt, the tube is made of similar chains diffusing also by reptation, a self-consistent model should consider the motion of the surrounding chains. Then, each topological constraint or entanglement should be assigned a finite lifetime and the attached segment of the tube will be lost when it is visited by one end of the passing chain (Fig. 17). Therefore, the constraint release mechanism leads to the vanishing of internal segments of the initial tube due to simple reptation of passing chains whereas by pure reptation the initial tube disappears by losing its end segments. Both mechanisms could be combined by saying that a tube segment is lost when one or the other chain involved in the corresponding entanglement has one end in the close vicinity of the entanglement. In t h a t sense, we are not thinking in terms of individual chains (simple or pure reptation) but in terms of coupled chains (double reptation [14]).

Figure 17 : The tube renewal concept : an internal tube segment is lost when the attached entanglement disappears. Various authors [15,16,17] postulated that for a tube made of N/Ne segments the longest relaxation time ~ren accounting for contraint release is the same as the

120 Rouse time of an N-chain with an elementary time ~cr directly connected to the reptation time Zc of each passing chain.

~ren = (N / N e)2 Zcr -= ( S / N e)2 ZC"

(5-~)

The so-called tube renewal time 1:ren can be compared to the reptation time zc if the prefactor is known in the above relation. Zren/ ZC = (N / N e)2 / 18~2 for Klein [15] and Zren / ZC = 4~2(z-1)(N / N e ) 2 / z-12z with z = 2 to 4 for Graessley [17]. z is a coordination number accounting for the hypothesis that an internal tube segment is lost when the first constraint among z "active" constraints has been removed. Both expressions predict that Sren is higher than zc for monodisperse samples with N higher than about 12 [15] or 4 [17]. This means t h a t a pure reptation description is correct for highly entangled polymers as shown in section 3.1. For weakly entangled polymers or long N-chains surrounded by shorter Pchains, a tube renewal time Zren shorter t h a n Zc can be expected. If the two mechanisms are assumed to be independent of each other [17], the overall relaxation time z can be put as the harmonic average of the two times : -1

z-1 = z51 + Z~n.

(5-2)

This combination is equivalent to saying that the overall diffusion coefficient is the s:lm of the coefficients of the two processes. Watanabe and Tirrell [18] suggested that the rate constant of constraint release depends on the tube configuration, but the comparison between the two models [19] does not allow us to conclude in favor of either one of the models. Thus we will use the simplest one given by relation

(5-2). 5.2. Effects of p o l y d i s p e r s i t y for e n t a n g l e d c b - l n s In a polydisperse sample, each N-chain is surrounded by chains of different lengths. Therefore, the constraint release time Zcr varies according to the reptation time of the passing chain. Some entanglements can be considered as p e r m a n e n t (P>>N), while others will disappear quickly (P
121

.4l

N=50

P=5

.3

o

0

= .06 .I

O, -4

-3

-2

-I log

0

i

2

Figure 18 : Variation of the terminal relaxation domains of the two components of a binary mixture of polystyrene [19] samples ( ~ : concentration of N-chains). Their tube is made of N and P-chains (P<
~ren(N,p) _=_(N/Ne)2~c(p) or N2p 3

(5-3)

and a comparison with the reptation time of the N-chain can be made through

122

Graessleys' expression with the coordination number z---3" ~ren(N,p) / ~ 4 p3 %(N) = ~g) NN,2 For N / N ~ >

(5-4)

(P/Ne) 3, the tube renewal meeh~_ism is dominant and can be

directly measured. O t h e r ~ s e , it can be derived from relation (5-2) and the experimental value of the overall relaxation time. Experimentally, only average times are determined and the most commonly used are either (i) the weight-average relaxation time Zw = TloJ~ or (ii) the reciprocal of the frequency corn1 at the maximum of TI" as shown in Fig. 18. Both times depend on the relaxation time distribution function H(z) and can be compared to the longest time zmWe showed [19] t h a t theoretically C~m1 is quite insensitive to the width of the distribution and depends somewhat on its shape" corn1 = 0.73zm for a Doi-Edwards spectr,,m and corn1 = 0.63~ m for a Rouse distribution. The weight average time Zw ~2 is more sensitive to the shape of the distribution: Zw = ]-~ zm (Doi-Edwards) and ~2 ~:w - - ~ ~:m (Rouse). For diluted chains, a Cole-Cole plot of the complex viscosity (Fig. 19) exhibits a relaxation domain well-separated from the matrix allowing one to measure the same average relaxation times as above. However, the weight average time Zw has to be corrected by the matrix contribution. Watanabe [20] cast it into the form : Zw = lim

G"

(5-5

-4)p Gp

where the subscript P stands for the matrix.

._.1.6 (b)

d o_ o~ 0

%

/

\

/

~'"

+%. ~ + + , . b +q~ + +

//

. . . .

I 1.6

,

\+

/

,\

rl'(lO3Po,

,

, s.)

l 3.2

\+

\++~+++++

+ 9

,

,

i

/ 3

L

|

~'(I02pO.

\ - t .

1 6

S.)

Figure 19" The terminal relaxation domain of diluted polymethylmethacrylate (a) and diluted polyisoprene (b) can easily be distinguished from the matrix one (dashed lines) in a Cole-Cole representation of complex viscosities [19]-

123 As the data of com are straightforward and experimentally connected to the longest time by O~m1 = 0.7z m [21], they will be used along with the reptation time ~c in order to calculate the tube renewal time Zren of the diluted chains: -1 1;ren (N,P)

= zml(N,P) - ~cl(N)

(5-6)

We have conducted experiments on different polymers [21, 22] (polystyrene, polyisoprene, polymethylmethacrylate) and used data in the literature on polystyrene [20] and polybutadiene [23] in order to check the scaling laws predicted by relation (5-3). All the data confirm that the tube can be viewed as a Rouse chain (1;re n or N 2) but with an elementary time Zcr which is not the reptation time of the passing chain. The experimental P dependence of the tube renewal time is ~ren or p2.5+o.1 and has been interpreted in terms of multiple contacts between the N-chain and each given P-chain [21, 24, 25]. For Klein [24], the N-chain can be divided in blobs containing p monomers with a volume scaling as p3/2. In t h a t volume , the number of passing chains is roughly proportional to p3/2 / p = p1/2. Therefore, the n u m b e r of contacts between the N-chain and an identified P-chain is p / p 1 / 2 = p1/2. Assuming that the constraint release time Zcr is modified in the same way, we can write" 1:ren (N,P) o, N2~c(p)/p1/2 or N2p 5/2,

(5-7)

which is observed experimentally. Furthermore, the chemistry of the polymer can be taken into account by means of the number N e of monomers between entanglements or the molecular weight Me. -G.O

I

PB

1

I.

I

. . . .

I

'

'

I

"

-6.5 r'-i r Z -70-

_.a.

.J

-75

-80

85 3O

!

i

3.2

3.4

......

I

3.6

J,.

3.8

l

4.0

,,, l.,

4.2

4.4

log Me

Figure 20 9 Experimental relation between the tube renewal time 1;ren and the reptation time z c of the matrix for different molecular weights of various polymers [ data from ref 19].

124 1

Fig. 20 shows that the ratio ~ren(N,P).M~p/Zc(P).M 2 scales as Me 2 and all the data can be east into the form" 4(MN ~2 zc(Mp) 1:ren(N,P) = ~,'~e J M - - - ~ '

(5-8)

which confirms that the tube is made of N/Ne segments and the constraint release 1

time 1:cr= 4z c / M 2 differsfrom the theory. For monodisperse polymers, the tube renewal time scales as"

M3/2 1;ren (N) = 41:c(N). ~ ~e

(5-9)

which implies that Zren = Zc for M = (Me/2)}. This number of segments varies from 4.9 for polybutadiene to 10.4 for polystyrene and is consistent with the theoretical predictions[15,27]. Consequently, for highly entangled polymers, the pure reptation model holds as described in part 3.1. but it has to be corrected for weakly entangled samples according to the double reptation model. The m a i n modification is to use the overall time z instead of the pure reptation time z c, defined by relations (5-9, 3-14 and 5-2). Furthermore, in Doi's formula (3-18) the numerical constant v accounting for the contribution of contour length fluctuations should vary as a function of N as v = 1 - 0 . 5 / ~ ] N / N e in order to recover the well-known scaling law for the zeroshear viscosity 11o o, M 3-4. Nevertheless, the expression for Zren does not hold until N=Ne where we expect the overall relaxation time to merge with the Rouse time zB (relation 3-11). The entangled- unentangled transition remains to be clarified even though a recent approach by des Cloizeaux [26] looks very promising.

5.2.2. Polydisperse samples Let P(M) be the normalized molecular weight d i s t r i b u t i o n function ( ] S P(M)dlnM = 1 ) g i v e n , for example, by the S.E.C. technique; the weight x,

J

average molecular weight Mw is defined as: M w = ~? P(M)dM

(5-10 )

5.2.2.1. Relaxation times Each molecule is surrounded by the above distribution of chain lengths and the tube renewal time has to take into account the distribution of the attached constraint release times. For a monodisperse sample, Graessley [17] defines the constraint release time ~cr from the Doi-Edwards relaxation ftmction F(t) such as"

125

z

%r = ~[[F(t)] dt = f(z).~c

(5-11)

where z is the coordination n u m b e r . As the experimental connection between Zcr and zc is not so simple, we suggest using the simplest relaxation function : F(t) = exp(-t/ZZcr) with Zcr = 4zc/~t-M as deduced from relations (5-1) and (5-9). For a polydisperse entangled polymer, the distribution of chain lengths malting every tube is given by P(M) and an average relaxation function can be defined by:

(5-12)

< F(t) >= ~.'_ F(t,M).P(M)dlnM The subsequent average constraint release time < %r > is given by" <%r >= ~o~< F(t)>"dt

=

(5-13)

~Z[P(M)exp(-~/z~r

which yields the following tube renewal time for each N-chain in the sample" zt(M, P(M)) =

(5-14)

< Zcr >

The expression of < ~c, > has been checked for binary blends of monodisperse polystyrenes (Fig. 21). The tube renewal time of the high N-component is measured at different volume fractions CN and the molecular weight distribution is defined by two step functions : 1-CN at Mp and r at MN. The experimental data fit well the model with z=3.

f0

_

//

--

//

~

,~ i[ ."/( /'.," / II I

0

0.5 q>N

Figure 21: Variations of the tube renewal time of N-chains (MN = 2 700 000 g.mo1-1) in a matrix of shorter chains (Mp = 100 000 g.mo1-1) of polystyrene, as a function of concentration ~N [21].

126 Therefore, the longest relaxation time of a chain in a polydisperse somple is modified by a shift factor ~. defined by:

~'(M) =

'r(M, P(M)) 'rtl(M:) + zc(M) -1 z(M) = -~ = 'rren(M'P(M))+'rc(M)-I

3/-----~ 4M §1 2 .... 9 're ( - - ~ ) + 1 <'rcr >

(5-15)

For chains such that 'rot(M)=< 'rcr> the relaxation time is unchanged. For longer chains, for which < 'rcr> is lower than Zcr(M), the relaxation time is decreased (~.< 1) whereas for shorter chains the relaxation time is higher in the blend than in a monodisperse environment. 5.2.2.2. Relaxation functions The double reptation approach allows us to visualize the blend of n different species of the s~me polymer (molecular weights : M1, M2, ..., Mi, ... Mn ; volume fractions r as a network of (i, j) knots accounting for the entanglements between an i-chain and a j-chain [14, 19, 22, 27]. Therefore, the time-dependent density Fij (t) of initial knots in the blend is proportional to the relaxation function of each species involved in the knot, t h a t is to say : Fi5(t) a Fi(t) Fj(t).

(5-16)

From t h a t relation, it can be shown t h a t the density of (i, j) knots is equal to the geometric average of the density of knots between similar chains : (5-17)

Fij(t) = [Fii (t).Fjj(t)] 1/2

As the volume fraction of (i, i) knots is ~i2, and that of (i, j) knots is given by 2 {~i.{~j, the overall average number density of initial knots can be written as : ~

F(t) = Z Z r

(t)= [Z r

1/2

2

(t)].

(5-18)

On the other hand, the relaxation function of (i~i) knots is directly connected to that of i-chains by a mere shift, e.g. : Fii (t) = F i (at). In other words, in a monodisperse polymer the knots are renewed at a rate proportional to t h a t of the chain segments. The shift coefficient a is assumed to be length-independent, allowing the same shift factor to be applied to the overall relaxation function F(t) = F(at). Then, we recast relation (5-18) into :

F(t)=

r

"

(t)

.

(5-19)

127

The individual relaxation function Fi (t) is defined from Doi's expression (relation 3-18) where Fi (t) = Gc(t) / G~ and % is replaced by: x(M,,) = [zj1 + (Me/M)2(%r)-l] -1 accounting for the molecular weight distribution. An experimental check of such a quadratic blending law is given by the storage modulus G'(m) of binary blends which exhibits a plateau G'N at intermediate -I -i, frequencies, 2;(M1) < (DO < 1~(M2) corresponding t o $(M2) < 1;0 < 2;(M1) for the relaxation function. Therefore, for blends such as M 1 >> M~, the blend relaxation function is given by F(to)_--r 2 leading to G'N = ~12 G~, which is observed experimentally [28]. For a polydisperse polymer defined by its MWD function P(M), the relaxation function is given by"

F(t) =

P(M)

2(t,M)d In M

(5-20)

if only the reptation process is taken into account. But, for large polydispersities, the Rouse process (B) of the long chains overlaps the reptation process of the short chains. Consequently, the most general expression of the relaxation function (or relaxation modulus) must include all the relaxation processes described in part 3.2. As the Rouse dynamics is assumed to be linear with respect to the MWD and that the A and HF processes are mass independent, we define the relaxation modulus of a polydisperse linear polymer by :

[f

G(t)= +~P(M) G~I2 (t,M)dlnM

+

(5-21)

P(M)GB(t,M)dlnM+GA(t)+G~(t),

which is consistent with rel. (3-24) for monodisperse samples. 5.2.2.3. Viscoelastic behaviour The relaxation modulus is the core of most of the viscoelastic descriptions and the above expression can be checked from experimental viscoelastic functions such as the complex shear modulus G*((D) for instance. In addition to the molecular weight distribution function P(M), one has to know a few additional parameters related to the chemical species :the monomeric relaxation time xo, the rubbery plateau modulus G~ and the glassy plateau modulus G~. The temperature dependence is included in the relaxation time Xo and more precisely in the friction coefficient ~o- Expressed in terms of free volume fraction f which increases linearly with temperature and expansion coefficient af, the WLF equation gives the temperature dependence from two parameters C1 and C2 o

o

128 at a reference temperature To. The product C~ C2 = B / af is constant as long as the free volume expansion factor af can be considered as temperature and mass independent. An alternative description such as the Vogel equation introduces a temperature T~ = TO- C 2 which is a constant for a given high polymer species. Therefore, the friction coefficient can be written as : ln~o = lnA +

B

af(T - T.)

.

(5-22)

The values of B / o~f and To. are tabulated for different polymers [29] and the value of A can be derived from the elementary relaxation time zi measured in the transition zone. The high-frequency domain does not depend on molecular weight value and distribution, and thus the tabulated values of ~o at a given temperature are applicable to commercial samples. Figure 22 gives two examples of the description of the viscoelastic data of commercial polypropylene and high-density polyethylene samples by the expression for the complex shear modulus derived from expression (5-21).The first term is dominant for highly entangled systems.

~3

2_

I i

+.§

IoQw

Figure 22 : Experimental data and theoretical curves (expression 4-21) of the complex shear modulus of commercial polypropylene (M w = 348 500, Mw/M N = 6.1) and high density polyethylene (Mw = 210 000, Mw/M s = 11.7) [19] The agreement is satisfactory but it is worth noting that the fit will be poorer if the high molecular tail is not described properly or more generally if the relaxation time shi~ function ~(M) is not correct. For example, we showed [19] that failure to take into account the shift; factor k leads to a large discrepancy between the model and the experimental data.

129 Another important point is that, when approaching Me, the tube consistency becomes weaker or in other words, the constraint release scaling law is modified and the rubbery plateau disappears whereas the steady-state compliance jo decreases. A self-consistent approach should predict that around Me, the reptation modes would be gradually replaced by Rouse modes in order to describe the non entangled- entangled transition. 6. E F F E C T S OF NON-ENTANGLED CHAINS

6.1. The unentangled r e g i m e It is commonly admitted that a linear flexible polymer melt behaves as a dilute solution as long as the molecular weight is sufficiently low so that entanglement effects do not occur. The Rouse formulation of the bead-spring model with no hydrodynamic interactions holds for such undiluted polymers because of the presence of segments belonging to other chains within the coil of a given molecule. The Rouse description predicts a relaxation modulus given by GB(t), Equation (313), where the product MeN/Ne is replaced by the molecular weight M, so the longest relaxation time is : ~oR2N

6M

(6-1)

"~Rouse - 6~2kT - 11o p~2NART,

Rg being the radius of gyration and NA the Avogadro's number; it follows that : ~oR2N A 0 0.4M. no = P 3-6~oo and Je = pRT ~

(6-2)

The temperature dependence of Tlo is mainly included in the friction coefficient ~o (relation 5-22). Therefore, 11o can be expressed by : In 11o= ln($(M)) +

B o~f[ T - Too(M ) ]

,

(6-3)

where the structure factor $(M) describes the variations of the radius of gyration. Furthermore, the temperature Too is no longer a constant. In the free volume models, T~ accounts for the variations of the free volume fraction f (f = a f ( T - Too)) which is assumed to be mainly due to the concentration of chain ends. As the chains become shorter, the free volume fraction f increases, hence Too decreases.

130

2O

I

1

i

I

$

15-

_~I0

I

-

I/) 0 (J

._. > 5-

+ 0

1

0

I

50

I

I

1

I00 150 2 0 0 2 5 0 3 0 0 temperoture (~

F i g u r e 23 9 D a t a of zero-shear viscosities of polystyrene fractions ranging from 900 g.mol -z to 30 000 g.mo1-1 as a function of t e m p e r a t u r e [29-37]. The m a s t e r curve is obtained by experimental shifts from the data of a reference mass of 110 000 g. mole -z. It includes more t h a n one hundred experiments lying within the experimental b a r error. A least squares analysis gives the p a r a m e t e r s of the reference mass and the other ones are deduced from the shiit factors. The plot of a m a s t e r curve of the thermal variations of 11o for various molecular weights and temperatures (Fig. 23) shows that the expansion coefficient af can be considered as a constant in a wide range of t e m p e r a t u r e s . The vertical and horizontal shift factors respectively describe the mass dependence of the radius of gyration and temperature T.. Polystyrene is a good example for analyzing the non-entangled regime because the molecular weights available are as low as Me/20. Consequently, the experimental data are significant in a range of molecular weights exceeding one decade.

/ v

,

,

,

v

4O

z:: 01--I0~

0

No I

I

0.2

0.4

! "~

0.6

103/Mw

I

0.8

,

J

i. i.0

F i g u r e 24" The horizontal shift factors of the master curve of Fig. 23 give the t e m p e r a t u r e Too as a function of molecular weight (reference Too = 49.4 ~ C).

131 The variations of T=. are derived from the horizontal shift factor and can be expressed by (Fig. 24) : D T.. = (T..)= - - M

(6-4)

For polystyrene, D = 83 500g.mol-t, which means t h a t beyond a mass of approximately D, the temperature T=. is fixed - (T=)=. = 49.7 _+0.3~ in the entangled region, the free volume fraction is constant at a given temperature and the iso-free volume state merges into the isothermal state. From the vertical shift factor of the master curve, we are able to describe the mass dependence of the zero-shear viscosity in the iso-free volume state which is directly connected to the radius of gyration of the chains. In the molten state, it is generally assumed that the chains exhibit a Gaussian conformation and therefore the viscosity should be proportional to the molecular weight. Unexpectedly, we observed (Figure 25) an unambigous different scaling law (% a M 6/5) which confirms previous results [31] and should be explained by the variations of the radius of gyration. This hypothesis is consistent with direct measurements of Rg by SAXS experiments on solutions of polystyrene in O conditions [38]; the same scaling law is found for molecular weights lower than about Me (Fig. 26). 104 _

I0 ~

6/5

~r

Io' . "" 4 t

sO

,o~

i/

I01 103

I

104 Mw

IO s

Figure 25 9 the vertical shift factor of the master curve of Figure 23 gives the structure factor A' as a function of molecular weight. In addition, Pearson [39] made numerical simulations of the mean-square radius of gyration of polyethylene by using a rotational isomeric state method. For nalkanes and low molecular weight polyethylenes below M c, he also found a stronger increase of the calculated radius of gyration with molecular weight than expected from Gaussian statistics. Therefore, Gaussian statistics does not seem to apply to short chains as shown by numerical simulations [40] but the observed

132 scaling law, which is the same as for solutions in a good solvent has no connection with an expansion of the chains due to excbaded volume effects as the absolute value of Rg is lower than the extrapolated gaussian value (Fig. 26). I

i

a./-

10 4 -

r

A

..,~ IOs

_

/a

.y

N~

/O

-61' I0 2/

/ ~l/._j 615 I~ E! I

I 10 4

10 3

IO s

Nw

Figure 26 : Molecular weight dependence of the mean-square radius of gyration of solutions of PS in cyclohexane at 34.5 ~ C [38]. The steady-state compliance Je~ follows the Rouse expression until M'c --5 M e. Then, the longest relaxation time is expressed by : 15 Tlo j0 ZRouse -

~2

(6-5) "

From dynamic experiments and applying the time temperature superposition principle, the complex shear modulus is measured over about five decades and the Rouse model can be checked extensively [37]. I0 ~

I0 e ,,,,..,

iO s

na 10 4

b I0 2

IOOi I0 ~

/i

1 I0 2

I

i I0 4

1

1 I0 e

I

1 I0 e

~

J I0 I~

c~(s -4)

Figure 27 9Experimental complex shear modulus of unentangled polystyrene (M = 8 500 g.mol-1) compared to the Rouse model [37].

133 The agreement is good over the whole range of molecular weights below M e (Figure

27) provided that one adds the high frequency term (equation 3-21) accounting for the very local relaxation modes of the chains. The overlap between the two domains becomes significant at frequencies corresponding to 1 000 zi and 100 ziaccording to the molecular weight because of the very different orders of magnitude of the rubbery and glassy plateau moduli. 6.2. Effect o f s h o r t c h a i n s i n t h e e n t a n g l e d r e g i m e When short M S chains are introduced into a sample of entangled M L - chains with a volume fraction r of long chains such as r M L > M e, the blend can be viewed as a concentrated solution of the long chains, or in other words, the M s component is acting as a solvent at least in the terminal zone of relaxation of the long chains. According to Doi-Edwards theory, the reptation of the long chains will occur in a tube whose diameter a varies as r Thus the number of m o n o m e r s between entanglements will scale as r Accordingly, the reptation time zc (relation 3-14) should be proportional to r as a first approximation, the zero-shear viscosity Tlo and the steady-state compliance j0 should respectively scale as r and r E x p e r i m e n t s conducted on concentrated solutions of high molecular weight hydrogenated polybutadiene in a commercial oil [41] showed t h a t the expected law for 11o and the average terminal relaxation time corn1 is satisfied (Fig. 28).

I

"1

I '

I0 ~

107

!

I'

I

I0 e

,E

0

10 5

iO-I

10 4 10 3 _l,~m IO'J

[

j

j (JD

!

l

lnLn

I

I0 "j

I

n

n

n

~

t

J

I

Figure 28 9 Zero-shear viscosity and average relaxation time of concentrated solutions of entangled hydrogenated polybutadiene (M/M c = 300) [from ref 28]. The e x p e r i m e n t a l scaling laws are T10 a r and corn1 a r in a g r e e m e n t w i t h DoiEdwards' theory.

134 For melts of long chains containing short chains, the contribution of the unentangled component is in some cases non-negligible and has to be removed from the data of the blend in order to isolate the contribution of the entangled component:

TIoL = ~lO - (1- r

(6-6)

and JeOL= J ~ TI_x_O)2, 1]OL

(6-7)

as defined for concentrated solutions. Moreover, as far as relaxation times are concerned, the reptation mechanism in an enlarged tube should lead us to favour the tube renewal process. The expression of the tube renewal time Zren (relation 5-8) shows a ~3 scaling which implies that ~ren(ML,~)=zc(ML,~) for M=(Me/2dp) 4/3. Therefore, the double reptation approach applies in an extended range of molecular weights because the concentration is low. The overall relaxationtime z(M L, ~) can be cast into the form

,~(ML ' ~)-1 = [lii ,I;c (M)]-I + [{il3Zren (M)] -1

or 1;(ML,~))= ~ zc(ML) [1+ 4~2ML 3/2M2 ]-1,

(6-8)

whereas the zero-shear viscosity TIoLwill vary as r z (ML, 4)). The experimental evidence of the importance of the tube renewal mechanism when short molecules are added to a high polymer is provided by blends of narrow polystyrene (M s = 8 500 and M L = 900 000) [28]. In a large range of concentrations (4)> 0.05) where the high component is assumed to behave as an entangled melt, the variations of the terminal relaxation time ~m-: in the iso-free volume state (Fig. 29) confirms relation (6-8). As the steady-state compliance J~L scales as ~-2 (Fig. 30), the zero-shear viscosity T10L varies as expected and the plateau modulus G~ which reveals the entanglement network is proportional to r The description of the relaxation modes including the behaviour of the entangled M L - chains and unentangled M S - chains remains to be done. At high concentrations, the long chains will dominate the viscoelastic behaviour in the terminal zone but, when approaching 4) M L = Me, the tridimensional diffusion of the short chains impeded by the surrounding long molecules will bring a noticeable contribution which could depart from the Rouse description.

135

10 3 s s

._E i0 z

==-

s

!

I0 i I0 o .i.,,

, i

~i

10-2

l.,lll

i i

I

I0 ~

I

I

@

Figure 29 : Average relaxation time of a high molecular weight polystyrene (M = 900 000) in the presence of short chains (M = 8 500). The dotted line represents pure reptation and the full line stands for the contribution of tube renewal according to relation (6-8).[from ref. 28]

10-3

I

I

10-4 -3

lO-S

I0"6

lit,,

t

IO"~

~

i

i

i

l

I

Figure 30 9 Steady state compliance as a function of concentration (same blends as in Fig. 29). 7. P R O B L E M S STILL P E N D I N G Along the same lines as described above, several questions arise as far as molecular weight distribution is concerned. The main question is : in the near future, will we be able to predict the viscoelastic behavior of linear polymers w h a t e v e r the molecular weight distribution, encompassing the broadest dispersion from oligomers to very long molecules ? The

136 answer could be positive provided t h a t we know more about the environmental impact on the relaxation modes of each individual chain, whatever its molecular weight. Remembering t h a t three characteristic molecular weights are defined from viscoelastic parameters : Me, the molecular weight between entanglements, M c _=2 - 3 Me, the cross-over mass for viscosity and M'c = 5 - 7 M e, the cross-over mass for compliance, intermediate situations should be explored which could be, in the last stage, merged into a "universal" law : All the masses lie beyond M'c : t h a t situation has been described in this chapter and it could easily be extended down to Mc by taking into account the decrease of the compliance, All the masses lie below M e : in the literature, the Rouse description of the relaxation modes of non-entangled melts or solutions is also used for polydisperse samples by means of a linear blending law. In order to consider the free volume variations of each mass in the blend, the relaxation times have to be shii~ed by a factor ~. which is the ratio of the monomeric friction coefficients of the blend ~Ob and of each species (~o)For a polydisperse sample [42], the relaxation modulus can be cast into : G(t) = I Z P ( M ) G ( k t , M ) d l n M ,

with

(7-1)

k = ~o ~Ob

As a consequence, the zero-shear parameters are : Rg2(M) d i n M tlo b = P 36Na m o ~0 b l'~p(M) -~

(7-2)

0.4 and jo = pRT rl--'-----~K M

(7-3)

P(M)R 2 (M)d In M.

Therefore, for Gaussian molecules, the above parameters are functions of moments of the molecular weight distribution 9 tl0 a M w and j0 a Mz.Mz+l /Mw. Otherwise, the mass dependence should be slightly different for tl0 and a large deviation from a combination of various average molecular weights is expected for the steady-state compliance. For mass distributions extending over the cross-over region between the nonentangled and entangled regimes, the situation is more complicated as anticipated in section 6. When the average molecular weight Mw - or other m o m e n t of the distribution - is lower than Me, a Rouse diffusion could be expected with relaxation times shifted according to the free volume variations, even for the high part of the distribution. Reciprocally, when the average molecular weight is higher than M c,

137 the reptative motion of the long chains will be made easier by the short chains according to the description given in section 6. For weakly entangled monodisperse and polydisperse polymer melts, J. des Cloizeaux [26] proposed a theory based on time-dependent diffusion and double reptation. He combines reptation and Rouse modes in an expression of the relaxation modulus where a fraction of the relaxation spectrum is transferred from the Rouse to the reptation modes. Furthermore, he introduces an intermediate time ~i, proportional to M 2, which can be considered as the Rouse time of an entangled polymer moving in its tube. But, in the cross-over region, the best fit of the experimental data is obtained by replaced ~i by an empirical combination of 1;i , ~c a n d

1;Rous e .

In a more empirical way, Lin [43] establishes an expression of the stress relaxation modulus of monodisperse polymers including the Rouse motions between entanglements and a relaxation process related to the slippage of the polymer chains through entanglement links. The associated relaxation function is a single exponential with an empirical relaxation time ~x- Moreover, the reptation process is assumed to be accompanied by a fluctuation one relevant to a Rouse description and which is the only mechanism remaining at M = Me. Consequently, he qualitatively describes the entire range of molecular weights including the entangled and unentangled regimes. Therefore, more work remains to be done in order to interpret the viscoelastic behaviour of very broad polymers with average molecular weights ranging from low values, lower t h a n Me, to strongly entangled situations. A comprehensive MWD - viscoelastic properties relationship will be of great help for designing materials for specific applications but a new challenge has been around for the last few years. It consists in developing numerical and analytical methods to invert a linear viscoelastic material function to determine the molecular weight distribution. There are several reasons to pursue such an objective - m a n y commercial polymers are slightly or not at all soluble in usual solvents, thus techniques like gel permeation chromatography or light scattering are inapplicable. Rheological characterization can be performed on-line and in real time and it is also a less cots-effective technique. So far, the important issue on how to determine MWDs from rheological data has been addressed with limited success, mainly for three reasons. The monodisperse relaxation function F(M,t) must be described precisely in the terminal and plateau regions, one has to provide a correct blending law yielding the polydisperse relaxation modulus G(t) ; and even if these two obstacles are overcome, specific mathematical procedures are needed in order to solve the illposed problem of numerical inversion of integrals. Many different sets of solution parameters are not physically m e a n i n g ~ l and appropriate constraints have to be imposed in order to determine an acceptable MWD. For entangled systems, the two first conditions are fulfilled in the framework of reptation theories : a comprehensive expression of the monodisperse relaxation modulus G(M,t) is given by expression 3-24 and the double reptation model generalized to a continous molecular weight distribution provides the integral relation between the MWD function P(M) and the polydisperse experimental

138 complex shear modulus G*(co) derived from the polydisperse relaxation modulus G(t) (equation 5-21). A recent publication by D.W. Mead [44] investigates numerical and analytical methods involving the double reptation mixing rule used with specific relaxation functions. A single exponential or a step-function monodisperse relaxation function are relevant to analytical methods whereas more general multiple time constant monodisperse relaxation functions require numerical treatments. The first results sound encouraging, making t h a t rheological measurements which imply bulk, macroscopic techniques are able to catch important molecular features such as chain length distribution. This is further evidence of the sensitivity of rheological techniques to nanoscopic changes, moving them into analytical tools.

REFERENCES 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13 14. 15. 16. 17. 18. 19. 20. 21. 22. 23. 24. 25. 26. 27. 28.

B. Gross "Mathematical structure of the theories of viscoelasticity", Hermann. Paris (1953). G. Marin, J.J. Labaig, Ph. Monge. Polymer, 16 (1975) 223. J.P. Montfort, Doctoral Thesis, Universitd de Pau (1984) G. Marin, J.P. Montfort and Ph. Monge. Rheol. Acta, 21 (1982) 449. P.G. de Gennes, J. Chem. Phys., 55 (1971) 572; see also "Scaling concepts in Polymer Physics", cornell U. Press, Ithaca (1979). M. Doi and S.F. Edwards ' ~ e theory of polymer dynamics" Oxford U. Press, Oxford, (1986). A. BenaUal, G. Matin, J.P. Montfort and C. Derail. Macromolecules, 26 (1993) 7229. J.L. Viovy J. Polym. Sci, Physics, 23 (1985) 2423. M. Doi J. Polym. Sci. Lett., 19 (1981) 265. T.C.B. Mac Leish, Europhys. Lett., 6 (1988) 511. P.G. de Gennes, J. Phys. (Paris), 36 (1975) 1199. D.S. Pearson and E. Helfand, Macromolecules ,17 (1984) 888. R.C. Ball and T.C.B. Mac Leish, Macromolecules, 22 (1989) 1911 J. des Cloiseaux, Europhys. Lett., 5 (1988) 437. J. Klein, Macromolecules, 11, (1978) 852, ASC Polymer Prep., 22 (1981) 105. M. Daoud, P.G. de Gennes, J. Polym. Sci., Phys-Ed., 17 (1979) 1971. W.W. Graessley, Adv. Polym. Sci, 47 (1982) 67. H. Watanabe, M. TirreU, Macromolecules, 22 (1989) 927. P. Cassagnau, J.P. Montfort, G. Marin, Ph. Monge, Rheologica Acta, 32 (1993) 156. H. Watanabe, T. Sakomoto, T. Kotaka, Macromolecules, 18 (1985) 1436. J.P. Montfort, G. Marin, Ph. Monge, Macromolecules, 17 (1984) 1551. P. Cassagnau, Doctoral Thesis, Pau (1988). J. Roovers, Macromolecules, 20 (1987) 184. J. Klein, Macromolecules, 19 (1986) 1. G. Marin, J.P. Montfort, Ph. Monge, J - N. Newt. Fluids Mechan., 23 (1987) 215. J. des Cloiseaux, Macromolecules, 25 (1992) 835. C. Tsenoglou, Macromolecules, 24 (1991) 176. J.P. Montfort, G. Marin, Ph. Monge, Macromolecules, 19 (1986) 393 and 1979.

139 29 30. 31. 32. 33. 34. 35. 36. 37. 38. 39. 40. 41. 42. 43. 44.

J.D. Ferry Viscoelastic properties of polymers, Wiley, New York 3rd ed. (1980). G.C. Berry, T.G. Fox, Adv. Polym. Sci., 5, (1968) 261. R. Suzuki, Doctoral Thesis, Strasbourg (1970) unpublished results. S. Onogi, T. Masuda, K. Kitagawa, Macromolecules, 3 (1970) 1098. D.J. Plazek, V.M. O'Rourke, J. Polym. Sci., A9 (1971) 209. R.W. Gray, G. Harrisson, J. Lamb, Proc. Roy. Soc., London, A 356 (1977) 77. T.G. Fox, V.R. Allen, J. of Chem. Phys., 41 (1964) 344. J.P. Montfort, Doctoral Thesis, Pau (1984) unpublished results. J.C. Majestd, A. Benallal, J.P. Montfort, submitted to Macromolecules. T. Konishi, T. Yoshizaki, T. Santo, Y. Einaga, H. Ysmakawa, Macromolecules, 23 (1990) 290. D.S. Pearson, G. Ver Strate, E. Von Meerwall, F.C. Schilling, Macromolecules, 20 (1987) 1133. C. Degoulet, J.P. Busnel, J.F. Tassin, Macromolecules, 35 (1994) 1957. V.R. Raju, E.V. Menezes, G. Matin, W.W. Graessley, L.J. Fetters, Macromolecules, 14 (1981) 1668. J.C. Majestd, J.P. Montfort, submitted to Macromolecules Y.H. Lin, Macromolecules, 17 (1984) 2846. D.W. Mead, J. of Rheology, 38, 6 (1994) 1797.