Faraday Symp. Chem. SOC.,1983 18 57-81 An Experimental Approach to the Molecular Viscoelasticity of Bulk Polymers by Spectroscopic Techniques Neutron Scattering Infrared Dichroism and Fluorescence Polarization BY LUCIEN MONNERIE Laboratoire de Physic0 Chimie Structurale et Macromoleculaire Ecole Superieure de Physique et de Chimie Industrielles de Paris 10 rue Vauquelin 75231 Paris Cedex 05 France Received 29th July 1983 Small-angle neutron scattering (s.a.n.s.) infrared dichroism (i.r.d.) and fluorescence polariz- ation (f.p.) are techniques which provide information on the molecular orientation of stretched polymers; they can thus be used to study the viscoelastic behaviour of bulk polymers on a molecular scale. The basic principles the experimental requirements and the molecular parameters available from experiments are described for each technique.Thus s.a.n.s. leads to overall dimensions of deuterated chains in a mixture of deuterated and hydrogenated polymers. In contrast i.r.d. yields the chain segment orientation averaged over all the chains of the sample whereas orientation determined from f.p. only deals with labelled sequences (central or end sequences) of labelled chains mixed with normal chains. Experimental results obtained by i.r.d. and f.p. on polystyrene samples stretched at a constant strain rate above are discussed. The influence of the molecular weight of either labelled chains or polymer matrices leads us to consider first a topological coupling between the relaxation processes of the labelled chain and those of the surrounding chains and secondly an orientation effect arising from the anisotropy of the strained polymer medium.An i.r.d. study of the orientation of compatible blends of polystyrene and poly(2,6- dimethyl-l,6phenylene oxide) is reported. Each polymer component exhibits different orien- tation behaviour. A recent renewal of interest in the viscoelasticity of entangled bulk polymers has arisen from the theoretical approach of chain relaxation developed by de Gennes Edwards and Doi. These theories are based on a molecular description of chain motions which involves reptation of the chain along the 'tube' formed by neighbouring chains. Although such theories can be tested through their predictions of macroscopic viscoelastic properties of bulk materials such as viscosity and elastic and loss moduli it is very tempting to obtain direct information on chain dynamics on a molecular scale.For this purpose spectroscopic techniques such as small-angle neutron scattering (s.a.n.s.) infrared dichroism (i.r.d.) and fluorescence polarization (f.p.) are particularly suitable. S.a.n.s. yields information either on the molecular dimensions of the whole chain or on the structure of part of the chain whereas i.r.d. and f.p. applied to stretched samples lead to information about the chain segment orientation. In this paper each technique will be presented with a particular emphasis on its application to studies of the molecular viscoelasticity of bulk polymers. Two examples are discussed the first dealing with polystyrene the second concerning 57 MOLECULAR TECHNIQUES AND VISCOELASTICITY the compatible blends of polystyrene and poly(2,6-dimethyl- 1,4-phenylene oxide) and these are compared with predictions of molecular viscoelasticity theories of polymer me1 t s.SMALL-ANGLE NEUTRON SCATTERING Small-angle neutron scattering is an experimental technique which has been used during the last ten years to investigate polymer conformation over the entire concentration range from dilute solution to the melt. Indeed owing to the wavelength of the thermal neutrons used (2-20 A) and to the low-angle spectrometers which are available it is possible to investigate sizes in the range from 1 to 5 x lo3A corresponding to the characteristic dimensions of polymers.A number of excellent reviews are available which deal with neutron-scattering theory1** and its application to ELASTIC NEUTRON SCATTERING THEORY In a scattering experiment involving any type of radiation (light X-rays y-rays or neutrons) an incident beam with a well defined wavelength A and wavevector k (k is a vector in the direction of travel of magnitude Ik I = 271/1) reaches the sample under investigation. The radiation scattered in a direction 8 is characterized by a wavevector k. If no energy transfer occurs between the incident radiation and the system i.e. the system does not undergo any motion on the timescale of the experiment the scattering is purely elastic and A = A,. Then the momentum transfer hq defined as hiq = k-k is simply expressed by 471 .8 q = -sin-. A0 2 Thus an elastic-scattering experiment consists of simply measuring the scattered intensity at various values of the scattering angle 8. A quantity commonly used to describe radiation scattering is the differential of the scattering cross-section da(q)/dR which gives the ratio of the flux scattered in the q direction to incident flux per unit solid angle i2. The method applied to calculate da/dR is the same for the various types of radiation including neutrons. If one considers the interaction potential between the radiation and the sample under investigation V(r),where r is the position vector of the interacting species da/dR is written as the spatial Fourier transform of the pair correlation function of the interaction potential = (V(q) V(-4)) dR with V(q)= d r exp (iq r) V(r) the angular brackets indicating a thermodynamic average.However the particular characteristics of neutrons lead to important differences. A neutron has a mass m = 1 a.u. which is much larger than that of an electron (m= 1836 me).Furthermore it is uncharged and has a spin of 1/2. The main interaction of a neutron with the sample under investigation and the only one considered hereafter is that involving the nuclei. Such an interaction is very different from that involved in light or X-ray scattering which concerns electrons. The interaction characteristic distance between neutron and nucleus is exceedingly L. MONNERIE short range ( A) and therefore is very small compared with the wavelength of the neutron.Thus the scattered wave is isotropic whereas it is anisotropic for X-rays and is characterized by its amplitude of -b where b is termed the scattering length for dimensional reasons. b is complex; its real part can be positive or negative and its imaginary part is a measure of the neutron absorption which can occur. Owing to this nuclear interaction the scattering length is different for each isotope as well as each element unlike the X-ray scattering length which depends only on the atomic number of the element and not on the isotope. Instead of the scattering length b another commonly used quantity is the cross-section ratio CT of the scattered neutron flux to the incident flux which is given by CT = 4nb2.First let us consider an array of N nuclei corresponding to the same isotope and without any spin; the nuclei are denoted by subscripts i and j. The only interaction of a neutron with such nuclei occurs via purely nuclear forces and there is a scattering length b which is identical for all nuclei. The interaction potential is V(Ri) = bd(R-Ri). Thus the scattering cross-section is expressed by = b2 C (exp [iq(R -Itj)]) = b2IX exp (iq Ri)I2 i i i where Ri and Riare the position vectors of nuclei i and j. Thus da/dR is a coherent scattering equal to the square of the sum of the scattering amplitudes of each atom. Let us now look at what happens for an array of N nuclei corresponding to the same isotope but with a spin I.Owing to the spin of 1/2 of the neutron in addition to the previous interaction there is a magnetic interaction between the spins of the neutron and the nucleus which will be different whether the spins are parallel or antiparallel i.e. whether the total spin state is I+ $ or I-4. This leads to two different scattering lengths b+ and b- respectively and the mean scattering length (b) can be derived taking into account the degeneracy of total spin states I+ 1 (b) =---b++-I b-21+1 2r+1 and in the same way (b2) = I+ 1 -(b+)2+-(b-)2. I 2r+ 1 2I+ 1 The above expression of do/dR can be separated into two terms corresponding to i =j and i # j leading to = N(b2)+ 2 (bibjexp[iq(Ri-Rj)]). dR i #j As there is no correlation between the position of the nuclei and their spin one obtains MOLECULAR TECHNIQUES AND VISCOELASTICITY which can be rewritten as Thus two fundamentally different cross-sections are obtained where the coherent contribution is = (b)2Cexp [iq(Ri-Iti)]= b& I Cexp (iq Ri) l2 (g)coh i i i and the incoherent part is = ((b2)-(b)2)N =(b-(b))2N = b:ncohNn (&) incoh At this stage it is worthwhile to indicate several points (1) The coherent scattering depends on the mean value of the scattering interaction and has the same form as for other types of radiation.It is phase dependent and leads to information on the structure of the investigated system. For this reason it will be the only contribution considered hereafter.(2) The incoherent scattering arises from fluctuations in the scattering length away from the mean value (b). It is isotropic therefore no information can be obtained about the relative positions of the nuclei. It leads only to a background over which the coherent scattering is superposed. (3) The incoherent scattering is unique to neutrons. It arises from the fact that the neutron has a spin which interacts with the spin of the nucleus. Of course incoherent scattering does not exist for atoms without spin such as 12Cor lSO. (4) Table 1 gives the values of coherent incoherent and absorption cross-sections for various elements. Note that the values of bcoh are different for H and D in such a way that it is possible to obtain a contrast between hydrogenated and deuterated molecules.Furthermore the value of oincoh for H is almost an order of magnitude greater than that for any other nucleus and it is also considerably larger than all the other coherent cross-sections. Therefore the scattering from any sample containing hydrogen will be dominated by the latter's high incoherent scattering. We now consider the case of systems containing different types of atoms. As each type of atom has a different scattering length it is worth considering a local scattering-length density defined by b(R)= I:b,6(R-Ri) or b(q)= C biexp (iq Ri). i i Thus the coherent-scattering cross-section is simply expressed by When dealing with a binary mixture of particles a and p if one assumes that the local fluctuations of the number of atoms are negligible (i.e.that the system is incompressible) it can be shown that b(q)is given by where pa is the local density of particles of type a and Vaand Vpare the partial molar L. MONNERIE Table 1. Scattering lengths incoherent cross-sections and absorption cross-sections for some elements H D 1/2 1 -0.374 0.667 79.7 2 0.19 -0 'ZC 0 0.665 0 -0 14N 1 0.94 0.3 1.1 lS0 0 0.58 0 -0 28Si 0 0.42 0 0.06 c1 3/2 0.96 3.4 19.5 volumes of particles a and /I,respectively. Therefore b(q) is the Fourier transform of the density of particles a only weighted by an apparent scattering length K termed the contrast factor Thus one obtains (g) = ~2 (Pa(q)Pa(-q))* coh This result is very important for it shows that for a binary mixture one can extract information about the relative positions of atoms of one species.This is the basis of the deuterium-labelling technique which allows one to study deuterated species in a matrix of hydrogenated species. This approach has been extended to the case of a solution of labelled and normal polymers leading to interesting and unique contrast possibilities. POLYMER STRUCTURE STUDIES OF ISOTROPIC SAMPLES PRINCIPLES As pointed out previously only the q(or angular) dependence of coherent scattering contains information on the structure of the sample. Thus in a study of mixtures of deuterated and protonated polymers it is necessary to subtract the large incoherent scattering arising from the hydrogen atoms. In the previous section all the systems considered were arrays of atoms in which each atom was a scattering particle.In the case of polymers which consist of long chains of chemically bound monomer units the whole macromolecule is too large compared with the neutron wavelength to be considered as a unique scattering particle. However on the scale of a monomer unit or of a statistical segment provided that their dimensions are smaller than the q-l values investigated interference between the waves scattered by the various constituent atoms is negligible and the centres of mass of these units can be considered as scattering points. Then the coherent-scattering length associated with the centre of mass of a unit is the sum of the scattering lengths of the individual atoms.For example this leads to bcoh= -0.166 x 10-l2and 3.998 x 10-l2 cm for C2H and C2D, respectively and to bcoh= 2.328 x 10-l2 and 10.656 x 10-l2 cm for hydrogenated and deuterated styrene. The contrast factor K is obtained from the expression given previously and the coherent scattering can be expressed as a function of the deuterated species only (denoted by D): (g) = K~Z (exp[iq(~,~-~?)]). coh i,j MOLECULAR TECHNIQUES AND VISCOELASTICITY In this expression the sum involves all deuterated monomers both those in the same chain and those belonging to different chains. Low concentrations of labelled polymers If the concentration is low enough to neglect the interchain correlation the single-chain correlation function of the labelled polymer can be obtained.Then the treatment is identical to that developed for light scattering by polymer solutions. Thus (&) = K2sE coh (4) coh with sy coh (4) exp (k* R)P(R)dR where P(R)is the probability of having two scattering centres at a distance R. For an ideal Gaussian chain of N scattering centres P(R)is well known and leads to 2 sycoh(q) = N:fD = N:,--"exp(-q2(R~))-1+q2(R~)1 4 (R&) wheref, is the classical Debye function and (I?& ) represents the mean-square radius of gyration of the chain. For a concentration c (with the dimensions of weight per unit volume) of a labelled polymer of molecular weight M the intensity of scattering per unit volume Z(q) is where N is Avogadro's number. Depending on the q values investigated different types of information can be obtained on the structure of the labelled chain.The characteristic molecular parameters of a chain are the radius of gyration RG,the persistent length a and the statistical step length 1. The values of q compared with the reciprocal of these molecular parameters define four q regions. (A) q c RG~, called the Guinier range corresponds to a long-range correlation between monomers and gives information on the overall chain dimensions. In this range the exponential term infD can be developed leading to Therefore from measurements performed at various q values the molecular weight and the value of RG for the labelled polymer are obtainable. (B) RG~ < q < a-l. In this range termed intermediate correlations at short distances between monomers inside a chain are observed.However unlike the Guinier range no characteristic distance can be derived. Only the type of q dependence can give information about the chain structure on this scale. Thus for a random Gaussian chain Z(q) varies as q-2 whereas for more or less rigid filaments Z(g) varies as q-l. (C) a-l c q c I-l. In this range the chain behaves as a rod and thus Z(g) presents a q-l dependence. (D) I-' < q. In this high-q range the scattering is governed by the structure of the chain segments. L. MONNERIE High Concentrations of labelled polymers The above requirements of low concentrations of labelled polymers to obtain information on single-chain structure implies in practice that experiments should be performed at several concentrations and then extrapolated to c+O using the well known Zimm plot.Such a procedure needs the use of the neutron beam for a rather long time. Recently it has been shown theoreti~ally~~ that the single-chain scattering function can be obtained from measurements at concentrations as high as 50% (wt/vol). The calculations are based on the assumption of incompressibility and for the case of hy- drogenated and deuterated chains with identical molecular-weight distributions they lead to where b, is the volume fraction of labelled polymer and S,(q) the single-chain scattering function of the labelled chain. Thus information on the deuterated chain structure can be derived from only one scattering experiment at any concentration.If the normal and labelled polymers have different molecular-weight distributions5 both S,(q) and S,(q) are involved and one obtains where x is the Flory interaction coefficient between H and D polymers. Thus two or three measurements are required to determine SD(q) and S,(q) depending on whether x is neglected or not. EXPERIMENTAL RESULTS A description of the neutron-scattering apparatus is beyond the purpose of the present paper. Relevant information can be found in ref. (2) and (6). The main problem in coherent neutron scattering studies on hydrogenated and deuterated polymer mixtures arises from the subtraction of the hydrogen incoherent scattering. This implies that one should make similar measurements on a blank which contains the same relative proportions of D and H atoms now randomly distributed over all the molecules.In the case of low concentrations of deuterated polymers the results obtained in the Guinier range on several polymers have proved that in the bulk state the chains are Gaussian and their dimensions are identical to those under 8 conditions. In the intermediate q range satisfactory results have been obtained but it appears that in this range it is necessary to take into account the conformational structure of the polymer.' Furthermore for high q values (qR > 5) corresponding to correlation distances comparable to the size of a few monomers the cross-section of the chain should be considered as shown experimentally.8 Thus the test of a given q dependence is delicate.For high concentrations of deuterated polymers the theoretical previsions have been successfully checked5. in the Guinier and intermediate q ranges when dealing with normal and labelled polymers with close molecular-weight distributions. In the case of very different molecular weights the problem of extracting the information on single-chain structure remains. For high-molecular-weight polymers (M > 109 it is more and more difficult to reach the Guinier range with the low q values experimentally achievable. MOLECULAR TECHNIQUES AND VISCOELASTICITY APPLICATION OF S.A.N.S. TO MOLECULAR VISCOELASTICITY STUDIES As previously shown s.a.n.s. yields information on polymer structure in various ranges. It may therefore be used for studying deformed (e.g.stretched) samples in order to gain a better understanding of the molecular mechanisms involved in many processing techniques as well as to test molecular theories of either rubber elasticity or chain relaxation.The time required to perform an s.a.n.s. experiment is too long to allow direct measurements during stretching or relaxation when dealing with uncrosslinked polymers. In the case of crosslinked rubbers for which the equilibrium state can be achieved at each elongation ratio (A) the problem is different and studies can be carried out directly. For uncrosslinked polymers only materials with a glass-rubber transition temperature (r,>higher than room temperature can be studied. The method consists of stretching the sample at the chosen temperature until a given A value quenching it at room temperature and subsequently performing the s.a.n.s.measure- ments. The same procedure is used for relaxation studies after stretching at a given A value; here the samples are simply quenched after various relaxation times. st retching direction Fig. 1. Schematic view of the small-angle scattering of a strained sample. The q vector lies in the detector plane perpendicular to the incident neutron beam of wavevector k,. The characteristic directions 11 and Iare defined with respect to the sample stretching direction. Curves of equal intensity are represented as ellipses in the q plane. It is obvious that to get sufficiently large changes in polymer structure it is necessary to use A values larger than 2.Thus the s.a.n.s. measurements are performed on significantly deformed polymer chains. Using the whole plane of q shown in fig. 1 one can consider curves joining all the points (detector cells) which correspond to the same given scattered intensity. Of course an isotropic sample gives circles whereas a uniaxially strained sample leads to curves which should be ellipses if the polymer chains are affinely deformed relative to the overall sample dimensions. For uniaxial stretching one can study the scattered intensity along directions either parallel or perpendicular to stretching direction (qI1and ql respectively).T\us both the Guinier range and the intermediate q range can be studied for ql,and ql. In the same way as for isotropic samples measurements in the Guinier range lead to the mean-square radius of gyration of the chain but now there are two values (R&) and (I?&,) corresponding to qtland ql respectively.For the intermediate q range it has been ShownlO that the q dependence of S(q) provides information on the mechanism of deformation. Thus for an affine deformation L. MONNERIE the q dependence is changed compared with the isotropic case whereas for a mechanism in which chain deformation is achieved by end-to-end pulling the q dependence is identical to isotropic chains but with a lower value of S(q,,).Experiments performed on stretching polystyrenell show that the actual behaviour lies between these two extreme cases. The affinity is more pronounced for deformations carried out at temperatures closer to q.A more elaborate deformation model has been developed recently.12 Thus the s.a.n.s.technique offers two different ways of studying the viscoelastic behaviour of uncrosslinked polymer chains. First one can look at the evolution of the radius of gyration as a function of either the elongation ratio or the relaxation time at constant strain and compare the resulting curves with the predictions of molecular viscoelasticity models. The second way is to consider the intermediate-q range and to study the behaviour under various experimental conditions (temperature strain rate relaxation time etc.). The s.a.n.s. measurements provide a unique means of studying the overall chain behaviour. However note that owing to the q values that are available practically in low-q region and owing to the fact that (RGII) increases nearly as A these molecular characteristics can be determined as a function of R only for polymers with molecular weight ca.lo5Dal. For higher molecular weights only (RGI)can be derived. Another interesting feature of neutron scattering deals with the technique of labelling using deuterium. Indeed in amorphous polymers there is no segregation between hydrogenated and deuterated chains. Thus it is possible to obtain information on the structure of a labelled chain which can be directly related to normal polymer behaviour. Furthermore it is possible to use a labelled chain with molecular characteristics (uw, chain branching etc.) very different from those of the matrix.At present such studies are only valid for low concentrations of labelled chains. Finally polymers can be labelled either all along the chain or in particular sequences (internal or end sequences) yielding the possibility of examining the behaviour of particular chain sequences. Although these studies require further theoretical develop- ment to derive structural information from s.a.n.s. they constitute for the future the most powerful experimental test of molecular-viscoelasticity theories. In contrast to these advantages an important problem arises from the lack of flexibility in the experiments owing to the use of high-flux neutron reactors. From a more technical point of view difficulties appear in thermoplastic samples of avoiding any microvoids both in isotropically moulded samples and in stretched samples.These requirements are greater for samples with a high content of deuterated polymer awing to the higher contrast between air and deuterium than with hydrogen. These microvoids give additional scattering at low qvalues which can affect the determination of R,. ORIENTATION DISTRIBUTION FUNCTION Before considering the spectroscopic techniques which lead to a measurement of the orientation distribution of characteristic vectors we introduce some convenient quantities to describe the orientation of uniaxially symmetric systems. Each vector is characterized by the angle 6 that its direction makes with the symmetry axis. The orientation distribution is represented by a function f(6) which can be expanded in terms of Legendre polynomials in cos 8 as follows with b = (1/2 n)(21+ 1)/2 (P,(cos 8)) 3 FAR MOLECULAR TECHNIQUES AND VISCOELASTICITY where (~(COS 8) averaged over the distribution 8)) is the value of ~(COS (p2(C0s8)) = (112)(3 cos2 8-1) 8)) = (1 /8) (35 COS~8-30 COS~ (~~(cos a+3) (cosn8) = s"f(9)cosn 8sin 8dO.0 It has recently been shown13 that for an uniaxial distribution the determination of (4)and (p4) is sufficient in most cases. INFRARED DICHROISM PRINCIPLE OF ORIENTATION MEASUREMENTS Adsorption in the infrared region of the spectrum deals with vibrational motions of the various atoms of a molecule and corresponds to a change in the vibrational energy level of the system.For the harmonic-oscillator approximation this energy change can be correlated with the structure and force constants of the molecule by performing a classical analysis of a vibrating system. This leads to a set of normal vibration frequencies associated with the normal modes of oscillation of the molecule. In order to be active in the infrared a vibrational mode must imply a change in the dipole moment of the molecule. Although a normal mode involves to some extent all the atoms in a molecule a large number of these vibrations are localized to a high degree of approximation in small groups of atoms. Thus some absorption bands can be assigned to the stretching of specific bonds the bending of bond angles or the wagging twisting and rocking motions of a given group of atoms.Owing to the change in the dipole moment of the molecule during the infrared-active normal vibration each mode will have a transition moment M with a definite orientation in the molecule. The intensity of an infrared absorption band depends upon the angle the electric vector E in the incident radiation makes with the transition moment M. The in- dividual absorbance a is given by a = log, I,/I = IMI2(M E)2 where I and I are the incident and transmitted intensities. This equation forms the basis for the use of polarized infrared radiation as a tool to study the orientation of a molecule in an anisotropic material. Let us consider a single molecule in an oriented sample with a preferential orientation axis A. The transition moment M of the considered mode makes an angle a with an axis of the molecule the direction of which is at an angle 8 from the orientation axis A (fig.2). For incident radiation polarized along A the related absorbance all will be all = I MI2(cos2acos28+4 sin2a sin28). In the same way for incident radiation perpendicular to A we get al = IMI2[+cos2asin28++ sin2a(1+cos28)]. If we deal with a set of identical molecules characterized by an orientation distribution function f(6) relative to the axis A the parallel and perpendicular overall absorbances will correspond to All = J,rn ailf(8)sin 8d8 L. MONNERIE preferential orientation axis M lar Fig. 2. Positions of molecular axis and transition moment with respect to the reference axis A.A convenient quantity is the dichroic ratio R defined as R=AII/Al. The value of R can range from zero (no absorbance in the parallel direction) to infinity (no absorbance in the perpendicular direction). For an isotropic sample no dichroism occurs and R = 1. For a set of molecules perfectly aligned along A the resulting dichroism is R = 2cot2a. Using this value leads to the following expression for R jon[1+(R,-1) cos20]f(0)sin 0 d0 R= i," [1+;(R~-1) sin2 elj-(e)sin 0 de 1 +(R,-1) (COS~0) or R= 1 +;(R,-1)(1 -(COS20))' Thus the measurement of the dichroic ratio of an infrared absorption band for which R is known yields the second moment of the orientation distribution function (cos2 e). The orientation function (P2(c0s0)) is obtained by APPLICATION TO THE ORIENTATION OF POLYMERS Infrared dichroism has been applied for many years to orientation measurements involving polymers.These studies have been reviewed re~ent1y.l~ Dichroism measurements are easily performed either on dispersive instruments or more conveniently on a Fourier-transform apparatus. Measurements can be carried MOLECULAR TECHNIQUES AND VISCOELASTICITY out either on stretched samples or during stretching. The main practical problem in the case of Fourier-transform measurements arises from the requirement of band absorbance lower than ca. 0.7 absorbance units in order to permit use of the Beer-Lambert law. This means one must obtain sufficiently thin films. Depending on the extinction coefficient of the considered band the required thickness can range from 1 to 200 pm.From this point of view polymers with strong absorption bands (e.g. polycarbonate) are difficult to study. The main difficulty in using infrared dichroism to study orientation is the necessity to find infrared absorption bands of the polymer which are sufficiently well assigned to normal vibrations of specified atomic groups. Such an assignment can be achieved by making a normal-coordinate analysis and experimentally by looking at deuteration effects and dichroic behaviour. Furthermore it is necessary that these well defined vibrational bands do not overlap with other bands resulting from another normal mode a harmonic or a combination of other modes. 9 0 0 / , / 0 0 0 0 \ I I 0 I I I /'-0 $3,.' (b) Fig.3. Local chain axis (a) in polystyrene and (b)in poly(2,6-dirnethyl-l,Cphenyleneoxide). Another problem arises from the choice of a local chain axis which is convenient to describe the chain orientation. When dealing with vibration modes which are independent of the local conformation of the main chain (trans or gauche) the in- frared dichroism will lead to the average orientation of chain segments and the local chain axis must be chosen in such a way that the same value of the orientation function is obtained when various absorption bands of the same type corresponding to different a values are used to calculate Pz.Examples of such local chain axes are given in fig. 3.Some vibration modes and absorption bands are characteristic of specified chain conformations (ttt for instance) and thus infrared dichroism will lead to the L. MONNERIE orientation value of these particular chain conformations. This can be done in the same way for absorption bands characteristic of the crystalline structure. Finally note that infrared dichroism studies do not require any labelling; however consequently the derived orientation refers to an average over different chains. Some experiments using deuterium-labelled polystyrene chains blended with hydrogenated polystyrene have been performed recently by J. F. Tassin in our laboratory. Some specific bands do not overlap but allow a measurement of the orientation of each species. Unfortunately this method requires concentrations of deuterated chains in the blend > lo% which makes studying the effect of the molecular weight of the labelled chains or of the polymer matrix difficult.In the case of blends if specific bands which do not overlap can be found for each polymer infrared dichroism is a good tool to study the orientation of each component in the blend. An example of this type is given later. FLUORESCENCE POLARIZATION PRINCIPLES Fluorescent molecules have the property of re-emitting in the form of visible light part of the energy acquired by the absorption of luminous radiation. After illumination by a very short pulse at time to the fluorescent light emitted at time t,+u is proportional to exp (-u/z) where z is the mean lifetime of the excited state (usually called the fluorescence lifetime).The most frequent z values range from 1 to 100 ns. When absorbing light of a suitable wavelength a molecule behaves as an electric dipole oscillator with a fixed orientation with respect to the geometry of the molecule. Such an equivalent oscillator is termed an absorption transition moment M,. In the same way for the fluorescence emission we have an emission transition moment M. When such a molecule receives an incident beam polarized along the P direction (fig. 4) the absorption probability is proportional to cos2a,. In the same way the fluorescence intensity measured through an analyser A is proportional to cos2p. Thus for the P and A directions of polarizer and analyser the observed luminescence intensity is proportional to cos2a0cos2~.Owing to the lack of phase correlation between t observation Fig. 4. Polarized absorption and fluorescence emission P,polarizer; A analyser. MOLECULAR TECHNIQUES AND VISCOELASTICITY excitation and emission lights fluorescence emission can be described as resulting from three independent radiations respectively polarized along the X,Y and 2axes with intensities Ix Iy and I,. The Curie symmetry principle applied to excitation light polarized along 2,leads to Ix =Iy. The fluorescence polarization is characterized by the emission anisotropy r =(111-Id/(lIl +w where Illand Zl correspond to the fluorescence intensity obtained with an analyser direction parallel and perpendicular respectively to that of the polarizer.When dealing with an isotropic set of fluorescence molecules in such conditions that the relaxation times of the molecular motions are in the range of the fluorescence lifetime f.p. yields information on the mobility of the molecules; such an application has been reviewed re~ent1y.l~ ORIENTATION OF UNIAXIALLY SYMMETRIC SYSTEMS For our present purpose we are mainly interested in the use of f.p. to look at the orientation distribution of fluorescence molecules. The main results which can be derived are presented below; more details can be found in a recent review15 and in the original paper.l6 7 Fig. 5. Illustration of the angles which define the orientation of molecular axis Moat time to and M at time to+u with respect to the fixed frame OXYZ.In the following we will assume that the transition moments in both absorption and emission coincide with a molecular axis A4of the molecule whose direction is specified by the spherical polar angle R =(a,8) in the reference frame (fig. 5).Let us introduce the angular functions N(Ro,to) the orientation distribution of M at time to (Moin fig. 5),and P(R tIno,to),the conditional probability density of finding at position R at time t a vector M which was at position R at time to. After illuminating the sample by a linearly polarized short pulse of light at to,the intensity emitted at time (to+u) for the P and A directions of polarizer and analyser is given by i(P,A,t,+u) =~~~~(~o,to)~(~,t,+~l~o,to) x cos2(P, M,)cos2(A,M)exp (-u/z)dRo dR L.MONNERIE where K is an instrumental constant. In this expression to corresponds to the macroscopic evolution of the sample for example in a rheological experiment whereas u corresponds to a microscopic reorientational motion in the scale of the fluorescence lifetime z. In most cases the todependence of N and P can be ignored within the time z (ca. s) and the fluorescence intensity emitted under continuous excitation is given by i(P,A toz) = i(P,A,to+U) du. Jo* In the case of a uniaxial symmetric distribution of the molecular axes M the intensities corresponding to the P and A directions lying along the fixed-frame axes (2 corresponds to the symmetry axis) can be conveniently expressed through the following quantities Gg) = $(3cos28,-1) Gg) = i (3 COS~8-1) Gg)= ~((3C0S260- 1)(3cos28- 1)) G& = (sin 80 cos 8 sin 8cos 8cos (/3 -Po)) G$i) = s9a (sin2 80 sin2 8cos 2(p-/30)).Thus for example I(Z,2)= (K/9)(1 +2Gg) +2Gg) +4G#) i(2,X)= (K/9)(1 +2Gh;) -G# -2Gi:)). UNIAXIAL FROZEN SYSTEMS In such cases no molecular motion occurs during the fluorescence lifetime z and the quantities G# (= Gh;)) and GLF) can be rewritten with only two independent quantities cos2 8and COS~8. All the information on the fluorescence intensities may be displayed in a 3 x 3 tensor I ’ # (sin48) (sin48) 4 (sin28cos28) I = K b (sin4 6) (sin4 8) 4 (sin2 t?cos2 8) . (sin28cos28) i (sin28cos28) (COS~8) It is easy to see that the second and fourth moments of the orientation distribution (cos20) and (COS~S) respectively can be derived from fluorescence intensity measurements Ixx I, and Izx.UNIAXIAL MOBILE SYSTEMS In this case both orientation and mobility contribute to the fluorescence polariza- tion and the two effects have to be separated from the measured intensities. This is possible16if we assume that during the fluorescence lifetime (ca. s) the orientation distribution does not change. Nevertheless special optical equipment” is required and only (cos26) can be obtained from the measurement of five intensities i(P,A) corresponding to P and A directions which are not contained in the same plane. On the other hand the mean amplitude of the motion performed during the fluorescence lifetime is available from the data.MOLECULAR TECHNIQUES AND VISCOELASTICITY The same information on frozen and mobile systems can be obtained even if the absorption and emission transition moments are no longer parallel. APPLICATION OF FLUORESCENCE POLARIZATION TO ORIENTATION MEASUREMENTS IN POLYMERS Since f.p. is an optical technique only samples that are sufficiently transparent can be studied; i.e. amorphous polymers of any thickness or thin films (< 100pm) of semi-crystalline polymers. The use of f.p. to derive the orientation functions [Pz(cos 8) and p4 (cos e)]of a set of fluorescence molecules requires that there is no energy transfer between the fluorescent molecules implying a concentration of fluorescent species in the sample below 100 ppm. Thus the intrinsic fluorescence of a monomer unit cannot be used e.g.phenyl groups in polystyrene. Q Fig. 6. Labelled chain containing an anthracene fluorescence group (the arrow represents the transition moment). In order to correlate unambiguously the orientation of the transition moment of the fluorescence molecule with that of the polymer chain it is necessary to carry out covalent labelling. This can be achieved by performing an anionic polymerization and deactivating the living chains by 9,1O-bis(bromomethyl)anthracene.In this way the resultant polymer contains a centrally located fluorescence group in which the transition moment lies along the local chain axis (fig. 6).The labelled polymer is then incorporated into normal polymer at a concentration of 0.5-1 %.End-chain labelling can be obtained by using a monofunctional anthracene derivative. Such a method has been successfully applied in our laboratory to polystyrene and various polydienes. The main interest of f.p. is the great sensitivity of the fluorescence intensity measurements which allows the use of a very small amount (< 1%) of labelled polymer. In this way it is easy to look at the orientation of a labelled chain surrounded by normal chains and to vary the molecular weight of either the labelled chain in a given matrix or the matrix with a given labelled chain. In tmcase of polymers mobility studies performed by f.p. on isotropic bulk polymerslg have shown that a significant motion during the fluorescence lifetime only occurs at temperatures more than 5OoC above Tg.Thus polymers below this temperature range can be considered as frozen systems. F.p. measurements can be performed either on samples stretched above and then quenched or during stretching. For this last case special equipment has been developed in our laboratoryl7? l9 which allows measurements on both frozen and mobile systems. L. MONNERIE MOLECULAR-VISCOELASTICITY STUDIES OF BULK POLYSTYRENE . The techniques described above were first applied to the molecular behaviour of polystyrene chains during either stretching or relaxation at constant strain. As the s.a.n.s. results are presented and discussed in the paper by Boue we will focus on the experiments performed by i.r. dichroism and f.p.; most of these have been obtained recently in our laboratory.They deal with the influence of the experimental conditions of stretching (temperature strain rate and molecular weight) on the orientation of uniaxially stretched atactic polystyrenes (PS) the characteristics of which are reported in table 2. Table 2. Number-average molecular weight and polydispersity of poly-styrene samples polymer PS 1 149 800 1.70 PS 100 105 000 1.12 PS 160 160 000 1.16 PS 200 190 000 1.17 PS 400 420 000 1.24 PS 600 660 000 1.15 PS 900 855 000 1.19 PS 1300 1 300 000 1.28 INFRARED DICHROISM (I.R.D.) STUDIES2' Thin films suitable for i.r. spectroscopy have been obtained by solution casting. The stretching was performed on a hydraulic machine developed in our laboratory,19 operating at a constant strain rate (i) in the range 0.008-0.115 s-l up to 600% deformation of a sample of 6 cm length between its jaws.It is equipped with a special oven to obtain good temperature stability ( 0.02 "C)and a temperature homogeneity along the stretching axis of at least 0.03 OC. The PS films were stretched at a given II and then suddenly quenched at room temperature. 1.r.d. measurements were later performed using a Nicolet 7 199 Fourier-transform spectrometer. The resulting orientation function (P2(c0s 8)) refers to the local chain axis of PS shown in fig. 3. The 1028 and 906 cm-l absorption bands used are not sensitive to chain conformation and thus lead to a mean orientation of PS chains. Measurements in either the temperature range (1 10-128.5 "C) or the strain-rate range (0.008-0.115 s-l) yield a variation of (P2(cos8)) with draw ratio (1 = l/l,,,where 1 is the initial length of the sample and 1 is its length after drawing) which is nearly linear up to = 4 within the accuracy of our measurements.In order to illustrate the orientation behaviour the value of (Pz(cos 8)) at 1 = 4 is plotted as a function of T and log k-l in fig. 7 for PS 100 and PS 900 samples. For a stretching temperature close to the two polymers behave in the same way. An increase in Tor a decrease in i results in a greater orientation for PS 900 than for PS 100. These results suggest that a relaxation of orientation occurs during stretching. Therefore it is of interest to treat the orientation data in a more quantitative way in order to derive an orientation relaxation function 8(t)in a similar manner to that MOLECULAR TECHNIQUES AND VISCOELASTICITY Fig.7. Orientation function (Pz(cos 6)) at 1= 4 as a function of temperature and strain rate for monodisperse polystyrene samples. (-) PS 900 from i.r.d. ;(- -) PS 100 from i.r.d. ;(- -) PS 400-PAP 370 from f.p. -1.5-Fig. 8. Log 6(t) plotted against log t for PS 100. (1) 110 (2) 113 (3) 116.5 (4) 122 and (5) 128.5OC. commonly carried out in viscoelasticity studies for the relaxation modulus E(t). Thus the data have been analysed according to the method developed by Lodge2' for birefringence using the constitutive equation proposed by this author to describe the orientation during deformation.This leads for PS 100 to the variation of log O(t) as a function of log t shown in fig. 8 for the five temperatures studied. From these curves L. MONNERIE Fig. 9. Master curves of log O(t) plotted against log t for monodisperse samples between 110 and 128.5 "C. Reference temperature T,= 115 OC. (1) PS 900; (2) PS 100. Fig. 10. Comparison between log O(t) and log E(t) plotted as a function of log t for PS 1. Reference temperature T = 115 OC. (1) 1.r.d. orientation relaxation (2) mechanical relaxation. it is possible to obtain a master curve using the W.L.F. shift factor with the coefficients obtained by Plazeck22 for polystyrene. Master curves corresponding to PS 100 and PS 900 at a reference temperature of 115 "C are shown on fig.9. In order to compare our results with the relaxation modulus E(t) viscoelasticity measurements have been performed using dynamic shear experiments. In fig. 10 E(t) and O(t) curves are compared for sample PS 1. The two curves are very similar indicating that the relaxation of the orientation is closely related to the plateau region and the beginning of the terminal zone. The molecular-weight dependence of the orientation master curve (fig. 9) is clearly explained by the extent of the plateau region at long time with the increase of molecular weight as found from the mechanical viscoelasticity studies. MOLECULAR TECHNIQUEs AND VISCOELASTICITY FLUORESCENCE POLARIZATION STUDIES Unlike the i.r. dichroism experiments which yield only the chain segment orientation averaged over all the polymer chains in the sample the labelling required for f.p.measurements allows us to look at the orientation behaviour of only labelled chains. In f.p. studies on PS performed in our laboratory the anthracene fluorescent group was located in the middle of the polymer chain (fig. 6)and < 1 % (wt/wt) of labelled chains were mixed with normal PS chains. The molecular weights of the labelled chains used denoted PAP are reported in table 3. Stretching was carried out on the same machine as for i.r.d. studies but now f.p. measurements are performed during the stretching. The samples moulded under pressure and annealed were 8 cm long 2 cm wide and 0.2 cm thick. Table 3. Molecular weight of anthra-cene-labelled polystyrene chains labelled polymer M PAP 17 17 000 PAP 33 33 000 PAP 77 77 000 PAP 140 140 000 PAP 287 287 000 PAP 370 370 000 PAP 500 500 000 PAP 930 930 000 COMPARISON OF F.P.AND I.R.D. ORIENTATIONS The first point is to compare the orientation evolution with T and 6 given by f.p. and i.r.d. measurements. Results derived from f.p. for PAP 370 in a PS 400 matrix are presented in fig. 7 with the data for PS 100 and PS 900 obtained from i.r.d. At temperatures far from there is a satisfactory agreement the orientation of PS 400 lying between those of PS 100 and PS 900 as expected. On the other hand at 113 OC f.p. leads to a much lower degree of orientation than i.r.d. Such behaviour has been confirmed by the fact that the orientation observed by f.p.does not seem to depend on the glassy part of the stress but is related to the rubber-like comp~nent.~~ A possible origin of this is discussed later. INFLUENCE OF THE MOLECULAR WEIGHT OF LABELLED CHAINS Another interesting feature is the influence of the molecular weight of the labelled chain when that of the polymer matrix is kept constant. Thus experiments have been recently performed by J. F. Tassin and C. Ayrault at 128.5 OC with a PS 160 matrix. Results are presented in fig. 1 1 at a strain-rate value i = 0.115 s-l; similar behaviour is observed at other strain rates. First there is an increase in orientation with the molecular weight of the labelled chain proving that the orientation determined by f.p. is sensitive to overall chain relaxation.However a more surprising result is the rather high orientation obtained for PAP 17 chains. Indeed as the molecular weight of this labelled polymer is around the mean molecular weight between entanglements for PS (Me 1 16000) one would have expected a rather low orientation or even no orientation if only topological constraints L. MONNERIE / A Fig. 11. Orientation function (P2(cos 0)) plotted against draw ratio for various molecular weights of the PAP-labelled chains in a PS 160 matrix. Stretching temperature T= 128.5OC. Strain rate d = 0.115 s-l. (1) PAP 17 (2) PAP 33 (3) PAP 77 (4) PAP 140 (5) PAP 370 (6) PAP 500 (7) PAP 930. were considered. Such chains are too short to be oriented efficiently by the deformation of the physical network.On the other hand it seems that their orientation comes from the anisotropy of the surrounding medium. Note that similar orientation effects have been observed for pendant polyisoprene chains in a chemically crosslinked The pendant chains labelled inside the chain or at the end of the chain exhibit an orientation which increases with A although always remaining lower than the orientation of the labelled chain involved in the permanent network. Furthermore free fluorescent probes made up of 9,lO-dialkylanthracene with 16 CH groups are oriented at > 3.23Thus from these results it appears that in addition to orientation arising from topological effects (a physical entanglement network) and which could be described by molecular-viscoelasticity theories based on the tube concept there is another contribution arising from the interactions with the surrounding anisotropic medium.Although it could be argued that this effect is specific to f.p. and originates from the perturbation introduced by the fluorescent label the fact that the f.p. orientation behaviour observed at 128.5 *Cis similar to that found from i.r.d. measurements indicates that the contribution from the anisotropic medium is also involved to some extent in i.r.d. Indeed the statistical unit of a polymer chain corresponds to an anisotropic object which can also be oriented by interaction with the strained surroundings. It is now possible to account for the fact that f.p. orientation unlike that of i.r.d. MOLECULAR TECHNIQUES AND VISCOELASTICITY does not depend on the glassy part of the stress which means that f.p.orientation only appears over some values of A (A > 1.2-1.3). Indeed the label inside the chain should induce a longer anisotropic object than the statistical segment of the polymer chain and it has been shown e~perimentally~~ that the longer an anisotropic flu- orescent probe is the larger is the value of A required to orient it. INFLUENCE OF THE MOLECULAR WEIGHT OF THE POLYMER MATRIX The last study which has been performed concerns the influence of the molecular weight of the polymer matrix on the orientation of PAP 287.25The stretching was performed at 128.5 OC for various constant strain rates. PAP orientations obtained at A =4 are plotted in fig.12 as a function of the ratio of molecular weights of the polymer matrix and the labelled chain M,,/MpAp.From this figure three main features can be pointed out (1) there is a limiting value of (P2(c0s8)) which is independent of ,4 (P2,max 0.04); (2) the molecular weight of the matrix required to = reach this limit depends on d and increases as d decreases; (3) for matrices smaller than PS 900 an influence of d is observed P2 increases as d increases up to the limiting value P2,max. 0.051 Fig. 12. Orientation function (P2(cos 0)) measured at A. =4 at various strain rates plotted against the ratio Mn(matrix)/MPAP.t =0.115 s-l; A,d =0.059 s-l; **, 0 d =0.029 s-l. These results are very interesting for they raise a question concerning the validity of the theory of Doi and Edwards.26 Indeed from the latter’s slip-link model at a given d the same orientation of the labelled chain should be obtained independent of the molecular weight of the matrix as far as Mmatrix>MPAP,i.e.if the environ- ment is fixed relative to the motion of the labelled chain. Such behaviour is qualitatively observed at the higher t (0.1 15 s-l) but for lower values the orientation increases with increasing molecular weight of the matrix. Among the three relaxation processes considered by DO^,^' it appears that the disengagement of the chain from its original tube is not involved under the experimental conditions used since it has been shown19 that deformation of the sample is completely recoverable by annealing treatment above q.Thus it seems that the orientation is mostly affected by the shrinking of the chain into its deformed tube characterized by a relaxation time zBwhich should scale as M2,where M is the chain molecular weight.Indeed if the data of fig. 12 are plotted against the quantity all the points lie close to a single curve shown in fig. 13. At first sight such qualitative agreement would *be surprising since it has been shown above that the f.p. orientation contains two contributions one related to topological effects and tentatively described by the slip-link model the other reflecting the anisotropy of the surrounding medium. However it is clear that this medium L. MONNERIE I I 0 9 18 10-10 EM2 n Fig. 13. Orientation function (Pz(cos8)) measured at 2 = 4 plotted against the parameter -Data obtained from .i = 0.115 s-l; A,.i = 0.059 s-1; * i = 0.029 s-1; 0, i:M:(matrix).0 data obtained for M(matrix) = MpAp.anisotropy is governed by the same topological relaxation effects. Thus in the experiment concerned in which the shrinking of the chain is the dominant relaxation process a Mi(matrix) dependence should be expected. Although these results qualitatively support the M2 dependence of zB they nevertheless prove unambiguously that an improvement of the Doi-Edwards theory is required to take into account the effect of the polymer matrix molecular weight. This has been recently done28 by taking into account that in strained polymer melts unlike free chains in a deformed permanent network there is a coupling between the relaxation of a labelled chain and the relaxation of the matrix chains.Thus in the slip-link model a self-consistent treatment has to be made which leads to a new concept of tube relaxation i.e. partial disappearance of the topological constraints on the labelled chain resulting from the relaxation of the surrounding chains. Of course the tube relaxation effect depends on the relative molecular weights of the labelled and matrix chains; the theoretical predictions are in qualitative agreement with the experimental results. NEW INSIGHT INTO POLYMER MELT RELAXATION The studies reported above on the orientation of uniaxially stretched polymer melts show that there is a good agreement between the relaxation curve of the average chain orientation determined from i.r.d.and the modulus relaxation curve. However the f.p. results show that although the M2 dependence of the chain- shrinkage relaxation time predicted by Doi2’ seems to be confirmed qualitatively it is necessary to improve the slip-link model first by considering a topological coupling between the chains in the melt and secondly by taking into account the anisotropy of the strained medium on the orientation of chain segments and its consequence with regard to chain relaxation. At the present time only the topological coupling has been considered.28 ORIENTATION- AND CHAIN-RELAXATION BEHAVIOUR OF COMPATIBLE POLYMER BLENDS As mentioned above i.r. dichroism can be a powerful technique for examining the chain orientation of each partner in a polymer blend.Such studies have been carried out in our laboratory29 on mixtures of atactic polystyrene PS 1 and poly(2,6- MOLECULAR TECHNIQUES AND VISCOELASTICITY dimethyl-l,4-phenylene oxide) (PDMPO M = 23000 M = SOOOO) and they lead to compatible blends over the entire concentration range. The local chain axis used in PDMPO is shown in fig. 3 and the convenient vibration bands are 906 and 865 cm-l for PS and PDMPO respectively. Films from various blends with PDMPO concentrations in the range 0-35 % (wt/wt) have been obtained from solution and then stretched at constant strain rate at a temperature T = Tg = 11.5 OC where Tg refers to the glass-rubber transition temperature of the blend under investigation.-3 I I I The first unexpected result concerning orientation is that the two polymers do not behave in the same way when the composition of the blend is varied. The PDMPO orientation is high and does not depend on its concentration in the blend unlike the PS orientation which is affected by the amount of PDMPO present. Orientation data for each polymer have been treated according to the Lodge method to obtain orientation-relaxation functions O(t) in a similar way as indicated earlier for pure PS. Only one master curve is obtained for PDMPO orientation independent of the composition of the blend. In contrast each composition leads to a different master curve for PS orientation as shown on fig. 14. At short times the PS orientation increases wth PDMPO content up to a concentration of lo% beyond which it remains constant.For long times different behaviour is observed the PS orientation relaxation becoming increasingly slow as the DPMPO concentration is raised. It is reasonable to assign the orientation behaviour observed in PS/PDMPO blends to the high stiffness of the PDMPO chains. Because of this stiffness PDMPO chain relaxation occurs on a timescale longer than that which is involved during stretching in the experimental conditions used. Owing to entanglements one can consider that after deformation PDMPO chains participate in a highly oriented physical network. Interaction between PS and PDMPO chains hinders PS relaxation over the entire timescale studied. For short relaxation times the interaction factor certainly prevails in hindering local motions of the PS chains.For longer relaxation times besides the interaction term topological constraints arising from the highly oriented and slowly relaxing PDMPO network hinder large-scale motions of the PS chains (such an L. MONNERIE 81 interpretation is supported by the influence of the molecular weight of the PDMPO chains). In some ways the effects of PDMPO chains in the matrix are similar to those reported above for high-molecular-weight polystyrene matrices. G. Allen and J. S. Higgins Rep. Prog. Phys. 1973,36 1073. C. Picot in Static and Dynamic Properties of the Polymeric Solid State ed. R. A. Pethrick and R. W. Richards (N.A.S.I. Series D. Reidel Dordrecht 1982) pp.127-172. A. Maconnachie and R. W. Richards Polymer 1978 19 739. A. Z. Akcasu G. C. Summerfield S. N. Jahshan C. C. Han C. Y. Kim and H. Yu J. Polym. Sci. Polym. Phys Ed. 1980 18 863. F. Boue M. Nierlich and L. Leibler Polymer 1982 23 29. J. P. Cotton D. Decker H. Benoit B. Farnoux J. Higgins G. Jannink R. Ober C. Picot and J. des Cloizeaux Macromolecules 1974 7 863. ' D. J. Yoon and P. J. Flory Polymer 1975 16 645. M. Rawiso and C. Picot to be published. M. Beltzung J. Herz C. Picot J. Bastide and R. Duplessix ZUPAC Znf. Bull. 1981 275. lo H. Benoit R. Duplessix R. Ober M. Daoud J. P. Cotton B. Farnoux and G. Jannink Macro-molecules 1975 8 451. l1 F. BouC and G. Jannink J. Phys. (Paris) C2 1978 39 183. l2 F. Bod J. Bastide M. Nierlich and K.Osaki to be published. l3 D. I.Bower J. Polym. Sci. Polym. Phys. Ed. 1981 19 93. l4 B. Jasse and J. L. Koenig J. Macromol. Sci.,Part C,1979 17 61. l5 L. Monnerie Static and Dynamic Properties of the Polymeric Solid State ed. R. A. Pethrick and R. W. Richards (N.A.S.I.Series D. Reidel Dordrecht 1982) pp. 383414. l6 J. P. Jarry and L. Monnerie J. Polym. Sci. Polym. Phys. Ed. 1978 16 443. J. P. Jarry P. Sergot C. Pambrun and L. Monnerie J. Phys. E 1978 11 702. J. P. Jarry and L. Monnerie Macromolecules 1979 12 927. l9 R. Fajolle J. F. Tassin P. Sergot C. Pambrun and L. Monnerie Polymer 1983,24 379. 2o D. Lefebvre B. Jasse and L. Monnerie Polymer in press. 21 A. S. Lodge Trans. Faraday SOC. 1956 52 120. 22 D. J. Plazeck J. Phys. Chem. 1965 69 3480. 23 J.P. Queslel Thbe Docteur-Znginieur (Paris 1982). 24 J. P. Jarry ThPse Docteur-6s Sciences (Paris 1978). 25 J. F. Tassin and L. Monnerie J. Polym. Sci. Polym. Phys. Ed. in press. 26 M. Doi and S. F. Edwards J. Chem. SOC. Faraday Trans. 2 1978 74 1789; 1802; 1818. 27 M. Doi J. Polym. Sci. Polym. Phys. Ed. 1980 18 1005. 28 J. L. Viovy L. Monnerie and J. F. Tassin J. Polym. Sci. Polym. Phys. Ed. in press. 29 D. Lefebvre B. Jasse and L. Monnerie Polymer in press.