Furuday Symp. Chem. SOC.,1983 18 83-102 Dynamics of Molten Polymers on the Sub-molecular Scale Application of Small-angle Neutron Scattering to Transient Relaxation BY F. BouEi,* M. NIERLICH AND K. OSAKI S.R.M. C.E.N. Saclay 91 191 Gif-sur-Yvette France Received 2nd September 1983 The application of small-angle neutron scattering (SANS) to transient viscoelastic relaxation is advantageous in that it can for example give the form factor of a single chain in a polystyrene melt undergoing transient relaxation namely restoration (stress relaxation) at constant shape after a sudden deformation. The deformation is typically a uniaxial extension of ratio = 3 and the form factor is measured instantaneously on samples quenched from deformation temperature at different times t after the extension.Complete data are given here for one partly reported experiment [F. Boue et al. J. Phys. (Paris) 1982 43 137; J. Phys. (Paris) Lett. 1982 43 L585/L591 and L593/L600] covering an increased range of scattering vector q and time t. These data permit comparison of calculated form factors for the Rouse model and the tube model (reptation) confirming the following. (a) At small times the data agree both with the Rouse model and with the tube model which assumes a three-dimensional Rouse motion over lengths smaller than the tube diameter. At larger times the data disagree with the Rouse model. (b)A previously reported discrepancy with the tube model concerning the process of contraction of the deformed chain in its ‘tube’ persists [F.Boue et al. J. Phys. (Paris) 1982 43 137 M. Doi and S. F. Edwards J. Chem. Soc. Faraday Trans. 2 1978 74 1789 1802 and 18181. (c) At larger times data are not in disagreement with the tube model (disengagement process) if one accepts that contraction does not occur but that the chain has already relaxed by this stage up to the scale of tube diameter. Supportive results from other experiments which we have carried out are briefly presented. 1. PRINCIPLE OF THE EXPERIMENT In this work one measures the elastic neutron scattering at small angles from polystyrene melt samples containing some perdeuterated chains (PSD) in a matrix of non-deuterated chains (PSH). Scattering is elastic (which however gives information on chain dynamics) because the samples examined correspond to successive states depending on time of a melt undergoing transient relaxation.In practice the sample is made from a strip of PSH +PSD mixture which has been carefully moulded and annealed. This strip is then clamped in a stretching machine and heated at a constant temperature T above the glass-transition temperature. Then it is extended during a relatively short time interval t and thereafter maintained at constant deformed length and temperature. These are the conditions required for transient states approaching equilibrium at infinite time. Since the samples are not cross-linked the equilibrium value of the tensile strength is zero. For experimental purposes the sample is quenched at time t measured from the end of the deformation period t,.Samples are prepared in this way for each value of t and the whole set is subsequently examined in the quenched condition by neutron scattering. From each sample at a given t one obtains the static form factor of one embedded chain:? t The small-angle neutron scattering (SANS) technique is explained by Prof. Monnerie in his paper at this Symposium. Also see ref. (1) for pioneer work and ref. (2) for the use of high PSD concentrations as employed here. 83 DYNAMICS OF MOLTEN POLYMERS where i and j index the N monomers of the chain ri(t)and ri(t) are their positions at the same time t and q is the scattering vector. This approach has the advantages that it introduces the wavevector q which permits exploration in space and the establishment of a (q t)relation isotopically labelled chains (unlike chemically labelled species) do not display a propensity to demix in the bulk the available q-range easily covers submolecular lengths from monomers (5 A; q FZ 2 x 10-l A-l) to chains of M = los (300 A; q x 7 x A-l) and finally the techniques can be applied over a large t-range.The upper time limit is related to defects in the sample but can be 1/10 of the terminal time for stress relaxation in contrast to the case of inelastic scattering from melts at rest where smaller ratios are obtained for the most rapidly relaxing melts (PDMS at 100 0C).3[Values for characteristic times such as the terminal time are given in ref. (4).] POTENTIAL AND LIMITATIONS It is now necessary to discuss critically the potential and limitations of the technique in terms of the practical ranges of the parameters (q and t) and of the uncertainties in temperature deformation and derived form factor.STATE OF DEFORMATION Extension and relaxation can either be performed successively in the same heating medium (air or oil) or separately should it be necessary to examine the stretched sample before conducting the relaxation in another device. Both the first5-’ and second procedures have been employed.8 If the relaxation is very long inhomogeneity will ultimately appear but as for the extension the essential conditions are the use of a well annealed isotropic strip which is initially constant in thickness and width effective clamps but most of all an oven temperature constant throughout the heating medium.Good homogeneity of the final strip is the best guarantee of processing and indeed is required for proper characterisation of the deformation. A geometrical test is to measure the initial thickness eo(r)at a point r and the distance between dots painted on the surface in the two directions parallel zo(r),and perpendicular yo(r),to the extension. The same quantities are then measured in the final state ef,zf,yf,at the same point in the material (whose position vector is now r’). We write The scatter of the three quantities &,A and A, which ideally should be equal gives the uncertainty AA. We obtained AA < 0.05 for all samples which had relaxed for a period t less than one hundredth of the terminal time.We believe that AA < 0.02 is achievable by careful work. For larger values of t only AA < 0.15 was obtained but it should be possible to reduce this spread by better annealing and an improved temperature distribution down to perhaps 0.05 for t < T,,,/5. It is also possible if e is both sufficiently constant and less than 0.3 mm to use an optical birefringence test using white light where one seeks uniformity of colour. DEFORMATION HISTORY If the final state appears homogeneous by the above and other tests there is good reason to believe that the deformation history was identical at all points. It is measured by the variation with time of the total length [(t)(A[/[ < 1%) the tensile strength (AF/F < 5% ) and the light transmission between crossed nichols [cos2 e(t)(n,-n,) (t)] F.BOUE M. NIERLICH AND K. OSAKI n (n,) being the refractive indices in directions z 01).Repeated extension under the same conditions showed these quantities to be reproducible to within 5% .7 TEMPERATURE One has to distinguish spatial and time variations of temperature for each sample and also reproducibility between each sample for perturbations due to all variations of the external temperature ;this can modify space distributions before each extension. The spatial variation arises mainly from vertical temperature gradients in the equipment but is also affected by impulsive draughts during and following the extension because the relative positions of some metallic components are changed. Tests with a sensor in place of the sample7 indicated that spatial and time variations were bounded within & 0.5 "C.However by careful adjustment of a@ air oven or by using a thermostatted room it proved possible to obtain variations < 0.1 "C.Such a value is easily attainable with an oil bath if one removes the vertical gradient; a practical limit would be 0.05 "C since an uncertainty of f0.05 "C is caused from heat produced or absorbed on stretching7 TIMES The first two relevant times concern the act of extension.This was conducted at a constant speed gradient s and thus lasted a time t = (1/s) (In A). As s was varied appropriately to the chosen temperature between 0.06 and 0.18 s-l t ranged for A= 3 between 7 and 20 s; the inverse of s spans 5-17 s.These numbers illustrate the rapidity of the deformation. The duration of the quenching must be added but was found7 to be equal to 15 s. These times are to be compared with the characteristic times of the polystyrene melt variation of which led to the time-temperature superposition4 given previ~usly.~ Quenching times could roughly be reduced to < 5 s. t and l/s are already the lower limits for the corresponding temperatures at which they were used (see table 1). Were faster stretching employed plastic deformation or breaking would occur. We have compared the relaxation of two strips one stretched at a given rate so the second at a rate 2s,. For relaxation of duration t > 2tso,no significant difference appears either in recorded curves of stress relaxation or in the form factors of samples quenched at t.It is thus possible for t > 2t, to disregard the exact elongation history and thus to compare the present results with models which assume a sudden deformation. Even for t "N 2t, it remains true that the elongational histories were very similar for different samples (scatter in stress was believed to be < 5%). In practice the minimum value of t must be set at the minimum of t + t, equal to 10+ 15 = 25 s at 117 "C. t cannot be reduced to < 5 s by reducing the temperature because the sample would break so one can only achieve a reduction in t, the quenching time. The maximum value of t is easy to obtain by increasing the temperature which can rise to 140 "C. However here the essential limiting factor is the deterioration of the strip; it is difficult for t to exceed one tenth of the terminal stress relaxation time of the matrix.SCATTERING VECTOR RANGE The range of 141 presently available is 7 x 10-3-2 x 10-1 A-l. The lower limit is larger than the lowest value of q handled by the most powerful small-angle machine the spectrometer D11 at the I.L.L. which is ca. 1 x A-l. However at this q-vector counting times become very long because of the large sample-detector distance. In addition scattering by microvoids becomes important for 141 < 5 x A-1. Since these voids are difficult to suppress the lower limit of the spectrometer is not a serious DYNAMICS OF MOLTEN POLYMERS problem. The upper limit is imposed by the problem of subtraction of the incoherent background2.(see also Prof. Monnerie's paper at this Symposium). Thus for a high concentration (C = 15%) of labelled chains the ratio of coherent to incoherent scattering will be for the most oriented sample (no. 71) 1280/840 in the perpendicular direction and 834/772 in the parallel direction for 141 = 2 x 10-1 A-l. If the thickness and the neutron transmission of the sample necessary to subtract the substantial incoherent background are known within lo% the uncertainty of the form factor AS,(q)/S,(q)will be > 30% whereasit is < 10% at 141 = 1 x 10-1 A-l. Thisessentially determines the maximum value of 141. TREATMENT OF NEUTRON MEASUREMENTS Assuming that the classical treatment we used2 is correct the essential uncertainties arise as follows.First is the subtraction of the incoherent background at large 141 the error decreases very rapidly as (41 decreases. For 1q1 = 3 x A-l it is < 2 x Second is the normalisation of the form factor. This quantity has to be compared for different samples of different volumes (thicknesses) and different neutron transmissions so it is necessary to divide the coherent scattering by these quantities. This leads to a normalisation uncertainty of ca. lo% although actual experiments appeared to be better than this. By using homogeneous samples of the same thickness this uncertainty could be reduced to less than a few percent. SCIENTIFIC BACKGROUND The present experiment operates at the microscopic scale (5-300 A) yet in the same field as various macroscopic techniques which have been compared with or even inspired by modern the~ries.~? lo Thus in the regime of large times near the terminal time for stress relaxation there have been measurements of the stress terminal time (qercc lW4), the self-diffusion coefficientll and fracture welding,l2? l3all of which were to be in good agreement with the de Gennes theory of reptation (disengagement).In the regime of smaller times macroscopic experiments have involved measurements of overstress in the regime of non-linear deformation and of its relaxation; these were found to be in agreement with the Doi-EdwardslO prediction. In an endeavour to link the macroscopic and microscopic approaches an attempt is made in this paper to discuss the results of one experiment in terms of the Rouse and reptation models9~ lo at the microscopic level in the two regimes of long and short times.Finally other experiments will be introduced into this comparison. 2. EXPERIMENTAL Some of the data presented below have already been reported namely variation of the transverse radius of gyration R,,(t) extracted at small q,5and observation of S,(q)in the parallel direction for large t.5Following those papers and comments above the experimental description will be brief. There then follows an improved comparison with models using all the data and calculated functions S,(4). Samples were made from a mixture of PSD and PSH of molecular weights given in table 1. They were moulded in vacuum (2 x Torr) and annealed; the stretching conditions are given in table 2.The speed gradient was held constant and the oven was heated by air.* Five different temperatures were used with the uncertainty explained above allowing coverage of a large range of the ratio t/T,,,(T) (for polystyrene Tgz 100 "C).The tensile force and the birefringence {in fact cos2[e(t) An( t)])were measured during the stretching as well as during the * We used a machine in Prof. Monnerie's laboratory [a description is given in ref. (6)]. F. Bod M. NIERLICH AND K. OSAKI Table 1. Preparation characteristics of the samples (isotropic) rate of duration of sample number stretching /s-l stretching /s duration of relaxation T/"C 71 0.189 7 30 s 113 10 and 7 0.06 20 10 s 117 19 0.06 20 20 s 117 18 0.06 20 1 min 117 6 0.06 20 4 min 117 9 0.06 20 20 min 117 24 0.115 10 10 s 122 23 0.1 15 10 1 min 122 25 0.115 10 20 min 122 42 0.189 7 30 s 128 44 0.189 7 1 min 128 43 0.189 7 8 min 128 46 0.189 7 30 min 128 49 0.07 18 1 min 134 48 0.189 7 1 min 134 50 0.189 7 16 min 134 relaxation [see ref.(5) and (7)]. Birefringence An,and stress appeared to vary in the same way during the relaxation for t > 22,; for I < 2t and during stretching that was the case only for the highest temperatures 128 and 134 "C. For lower Tthe stretching was indeed much faster at a given s and the stress curves were vertically shifted by a quantity A(s T).The relaxation of this part A took < 1 s following the end of stretching. The concentration C, of labelled species was set high (1 5 % ) allowing more precise neutron data and the material handled as described earlier in this The two spectrometers D11 and D17 (1,L.L.) gave data on three partly overlapping ranges.Data were corrected in order to superimpose in the overlapping domains.6 The total q range was then 7 x 10-3-2 x 10-l A-l. Comparison of t-values with the characteristic times of the sample employs the W.L.F. superp~sition,~ for which we obtained5 T,,,, a 5 x lo3 s and q,,z 5 x lo5s but with large uncertainty5 while the temperature-reduced values of I P7,lie between 20 and lo5 s. 3. RESULTS The data for S,(q) are presented in fig. 1 in the representation q2St(q)as a function of log, (qRgiso),Rg being the radius of gyration measured for the isotropic sample.The q2St(q)data in this figure are presented in two sections. Thus in the lower part of the figure are the curves for different values of t in the parallel direction. For each time t as q increases the curve rises so becoming closer to the isotropic curve. This means that the chain is more isotropic and thus more relaxed at small distances. An increase in t also moves the curves systematically closer to the isotropic limit. For large t an inflexion point appears. In the upper part of the figure are plotted the curves for the perpendicular direction. They all display a maximum which comes from the quantity q2St(q).For small q this quantity which can be regarded as the linear density of the chain increases just as the isotropic case but lies above the isotropic curve because the chain contour is more dense and more compressed than the isotropic at this scale.For large q the chain DYNAMICS OF MOLTEN POLYMERS ....... OO 0.5 1 1.5 2 lOg,o qRg is0 Fig. 1. Data for the single-chain form factor S,(q) in the representation q2St(q)plotted against log, qRgisofor the following samples e,71 (reduced time at 1 17 "C = 10 s) ; + 18 (reduced time at 117 "C= 60 s); A,49 (9000 s) and 0, 50 (130000 s). The dotted line represents affine deformation and the dash-dotted line represents the isotropic Gaussian conformation. as with the parallel direction becomes more isotropic. The form factor returns to the isotropic case the linear density decreases and a maximum is produced.The lines in fig. 1 denote the theoretical value for the isotropic chain [i.e.the Debye function for a Brownian chain of the same R, D(q2),multiplied by q2]and for a chain deformed in a totally affine way i.e. S(q) = D(qBq),where B is the Finger tensor A 0 B= (0 l/z/A 0 0 0 l/dA (the curves are given for the two directions). F. Bod M. NIERLICH AND K. OSAKI 89 One can see that in the parallel direction all the experimental data except at the smallest t,have departed from the affine curve. In the perpendicular direction for small t and at small q the data coincide with the affine curve before departing from it at a higher q. The small-q region corresponds for the affine curve to the upper boundary of its Guinier range (qRgI< l) where the behaviour of the form factor can be characterized entirely by the value of the radius of gyration.Thus the coincidence of the data and the affine curve in this range is equivalent to the fact that the transverse radius R,,(t) is the affine one at small t this was reported previously in ref. (5) for L = 3 as well as for L = 2 and 1.5. In contrast for the parallel direction one can see that the Guinier range of the affine curve is well below the experimental q-range; it is then impossible to extract the value of RgII.However we have already remarked that at the smallest t the data are close to the affine parallel curve we can deduce that they would coincide at smaller q supporting an affine value of R,,, (t)for this very small value of t.TIME-TEMPERATURE SUPERPOSITION One new result of this experiment is to show within experimental uncertainty that the time-temperature superposition works for S,(q) on a submolecular scale. Two samples prepared at two different temperatures with the same value of reduced time tTo= (aT0/aTl) t,(T,) lead to the same form factor. This has already t,(T,) = (aTO/aT2) been reported for R,(t); it has now been checked for the whole form factor for several values of reduced time as in fig. 2. 4. DISCUSSION COMPARISON WITH MODELS THE ROUSE MODEL In this classical mode114 the chain behaves as if were free in a viscous medium. This gives a set of modes zp = (p/W2TRouse corresponding to the motion of sequences of p sub-units. The rate of relaxation from a deformed shape will thus depend on the length scale at which it is observed the characteristic time at scale r will be z(r)cc r4 [as r K d(zp)for Brownian chains].Consequently at time t the form factor will vary with q in the following way. (i) At high values of q corresponding to small values of r the chain conformation appears to be relaxed i.e. the form factor will be close to the one for the isotropic Brownian chain. (ii) At small values of q the chain conformation will remain close to the initial deformed shape at t =0. As t increases the onset of relaxed behaviour (return to the isotropic form factor) will be obtained at a smaller value of q proportional to t-lI4. We have calculated Spouse(q), taking as the initial condition an affine deformation of the chain at all scales.This calculation is given elsewhere;15 the results are the dotted and solid lines of fig. 3 again in the q2S,(q)against log,,q representation for different t/TROuSe ratios. The relaxation of the form factor is as qualitatively described above in both directions the curves move from the affine case back to the curve for the isotropic material rising for the parallel direction and falling for the perpendicular direction which in that case produces a maximum in the curves. The abscissa of the onset of this departure depends on time. Let us first compare predictions and experiment for the smallest value of t e.g. t/ TRouse= 5 x lop5compared with data for t = 10 s (a)and t = 60 s (+).If we make a very qualitative comparison we can say that the shape of the curves is similar in both directions.If we increase the accuracy of the comparison we observe some DYNAMICS OF MOLTEN POLYMERS 0.23 h v) c. O+ .d E +a 1 0 + e v 0 h CIt b v O+ h” + n c7. 0 + 0 + 0 + 0.115 0 + 0 + 0 + I 1 1 0 0.5 1 1.5 2 log10 qR,, Fig. 2. Agreement with the time-temperature superposition showing a comparison of two samples stretched and relaxed at two different temperatures in order to have the same temperature-reduced time f1170C = 80 s + 18 T = 117 “C;0,24 T = 122 “C. systematic differences. The filled circles (0,t = 10 s) lie on the theoretical line in the parallel direction only at low q; they depart from it by staying closer to the affine line (i.e.more oriented than predicted).In the perpendicular direction they always lie above the theoretical line. Points marked by a cross (+ t = 60 s) lie above the theoretical line in the parallel direction; in the perpendicular they lie slightly below the theoretical line at low q and slightly above it at large q (i.e.again more oriented at high q than predicted). If we now take the theoretical line for t/ TRouse= 10-4 we see an agreement in the parallel direction with data for t = 60 s (+).In contrast the data for the same volume of t in the perpendicular direction lie well above the line except at low q. In summary the difference in shape between experimental and theoretical curves permits F.BOU& M.NIERLICH AND K. OSAKI ....*-.....I. ........ I 1 3 I 1 2.5 2 h *v, .-e e 2 1.5 n b v 4-n b 1 0.5 Fig. 3. Calculated form factor S,(q)for a Rouse model in the [q2St(q),log, qR,,,,] representation for t/TRouse= 5 x lop3 5 x lop2 lop1 2 x 10-l and 4x 10-l (full lines). Extrapolations following eqn (2) in the text are plotted for 5 x and as the dotted lines. The line made up of large dots represents affine deformation and the dash-dotted line represents the isotropic Gaussian conformation. The symbols (which are the same as those in fig. 1) show a comparison with the data. DYNAMICS OF MOLTEN POLYMERS 0.23 h U .d E e W h a vp. hi a+x 0 0 N cl. a+' +* 0 0.115 -8 + x 0 0 + x 0 0 0 I log, (4f;) Fig.4. (a) For legend see opposite. only a cruae owervation me experimental aara ror IU <t/s <ou overlap wirn theoretical lines for 5 x <t/TROuSe < For larger values of t/TROuSeno overlapping is possible. From this an experimental value of TRouse can be obtained as lo4 <Tg&se/s<5xlo5 at T =117 "C.This can be compared to (i) the value of the terminal time that we obtained from the literature4 and from our stress measurements:5 5 xlo5 <Ter/s <5 xlo6 at 117 "C and (ii) the value of the Rouse time of a free chain extrapolated from data for low M,* assuming TRousecc M2:5xlo3 <TRouse/s<5 xlo4. Tkx&, lies between the maximum for (i) and the minimum for (ii).For larger values of t/TRousewe have then to retain this estimate of T"R"opuse. The t/TRouseline must be compared to data for 50 <t/s <2500. However even data for t =9000 s appear much less close to the isotropic line than the Rouse line! The disagreement is still larger at larger q. F. BOUE M. NIERLICH AND K. OSAKI I I I (b ... 0 0 0 0 0 0 0.5 1 1.5 2 25 log, (4& Fig. 4. (a) Test of a qtl/*superposition for different values of t1170C using a horizontal shift of OC curves in the [q2St(q),log,,qRgi,,] representation for small times t117 = 10 (O), 20 (+) 60 (x) and 240 s (0).(b) Test of a qf1/4 superposition for different values of f1170C using a OC horizontal shift of curves in the [q2St(q),log, qRgi,,] representation for large times t117 = 9000 (@) and 130000 s (0).Thus the evolution of S+(q)with t appears qualitatively to be equally fast at small t but slows down at large t and the Rouse model appears much too fast. The comparison that we have just made has some disadvantages which arise from the difference in shape between the experimental and theoretical curves and from the need for a value for T,,,,,. This can be avoided as follows. THE qf114SUPERPOSITION LAW This alternative test of the Rouse model is based on the following theoretical prediction the fact (see above) that a return to the isotropic form factor begins at a value q* cc t-l14 leads more precisely to the asymptotic law15 at high q the behaviour Sis,(q)cc l/q2 corresponding to the plateau of q2Sis,(q)at high q (fig.1). Eqn (2) is a (q,t) superposition law. By plotting q2St(q)against log, qt114it should be possible to superimpose the curves; fig. 4(a) shows that this is indeed the case in the perpendicular and parallel directions for small ratios t/TRouse, but not for higher DYNAMICS OF MOLTEN POLYMERS 0 t (117°C) Fig. 5. Values of the horizontal shift log, q obtained by superimposition of plots as in fig. 4(a) and (b) plotted against log, t. Solid lines have the slope 1/4. ratios confirming earlier findings. (In the perpendicular direction the superposition can only occur after the maximum in order for it to be in the asymptotic regime.) However a second apparent superposition appears [fig. 4(b)] for the largest times reached it has already been reported for the parallel dire~tion,~.but is not found in the perpendicular direction. This superposition can also be achieved by measuring the horizontal shift necessary to overlap two plots of q2St(q)when plotted against log, t. Fig. 5 shows (i) a slope 1/4 at small t in agreement with the Rouse model (ii) a departure from that slope at higher t with a weaker slope corresponding to the slowing down of the real process and (iii) a second part at high t also of slope 1/4 although not on the same straight line. This is the one previously rep~rted,~? and it will be discussed below. To summarise the Rouse model starting from an affine deformation agrees with present data at small t but predicts too rapid a relaxation as t increases.However a different qt1/4superposition seems to appear at large times. THE TUBE MODEL This model considers the chain to be embedded in the matrix as confined in a tube made by the surrounding chains. Three motional processes are included (i) a wriggling three-dimensional motion inside the tube at scales smaller than its diameter D; (ii) a one-dimensional Rouse motion along the tube of maximum time Tes= T~,,,,cc N2 which is the time fluctuation of the length of the tube; (iii) a consecutive back-and-forth motion of the whole chain along the tube axis leading to a progressive disengagement of the chain from the original tube creating by the two ends a new tube within a time Tdiscc N3. These concepts were first developed by de Gennes9 and then extended by Doi and Edwards, especially for deformed systems.This work was the basis for the calculation of S,(q).16 The three processes are regarded as strictly consecutive because the chain is long enough. Recall that in the first process the form factor accords with the Rouse model up to a time t = T, which is the Rouse time of a chain of N monomers of size D (D cc Z/N, T cc NE). This process then stops. The second process which for a melt at rest leads to a fluctuation of tube length L leads for a deformed melt systematically to a contraction of L because the value after affine deformation Laffis larger than the equilibrium value Lo.During that contraction the form factor at small q related to the density of monomers in the chain will increase whatever the direction of q F.BO~, M. NIERLICH AND K. OSAKI because the contorted tube axis encompasses three dimensions; at the end of the second process the form factor increased for a given q by a factor Laff/L,,. The third process will start only once this situation has been attained and is desirable as disengagement from the deformed tube thereby creating two new isotropic ends of the tube the centre-deformed part decreasing in size with time. A detailed calculation by Dr Ball shows that the result is close to the form factor of a three-block copolymer isotropic-still deformed-i~otropic.~* l5 This is equivalent at high q to simply adding the contribution from the Np(t)monomers of the centre part and the M1 -p(t)] monomers of the end parts99 l5 l6 St(q) = ~(t) Saniso(q) + 1 -~(t)Siso(q)* (3) In eqn (3) the rate of relaxation is the same for different q values given the condition that q is high enough this is very different from the Rouse model and gives a much slower relaxation.At small angles eqn (3) is not valid; at the length scales of the radius of gyration the relaxation is again independent of q but with a different function. These questions are developed in ref. (1 5) with numerical evaluations. COMPARISON FOR THE FIRST PROCESS There is in this case no difference from the three-dimensional Rouse process which agrees with experimental data at small t. However as t increases this Rouse behaviour should stop and must be replaced with the form factor ofa chain relaxed on this length scale of a monomer sub-chain i.e.D.The corresponding time T, can be estimated if one knows N, it is classically the number of monomers between entanglements and is extracted from the plateau modulus value.3 For PS an acceptable value is N z 200 which gives Te= (Ne/N)2TRouse= (200/7000) x 5 x lo3 = 5 s at 117 "C. In fig.6are compared form factors for this timescale with an equivalent representation of the result of a Rouse relaxation. This avoids the difficulty of estimating T,explained above. The concept is of a Brownian chain simply obliged to pass affinely displaced points which are separated by N monomers along the chain:15 below the length scale of the distance between those points (ca.0)the chain is relaxed. Fig. 6 shows that best agreement is achieved for N/N 90 thus N = lo2 for samples 10 and 19.However for sample 18 (?17 = 70 s > q)relaxation does not seem to have stopped. One reason could be that another process has already started at this time. Let us now test if it is the second process of the Doi-Edwards model. SECOND PROCESS 'CONTRACTION ' This process should occur between Te and Tdis.According to our estimates4 the experimental t range covers all times when it could occur. One should observe an increase in the form factor in the parallel direction as well as in the perpendicular direction in the region q c 1/D. D can be estimated as the radius of gyration of a Brownian chain of N monomers (molecular weight Me):D = 0.275 2/M z40 A (from neutron-scattering data). For q < 1/40 = 2.5 x A-l the increase in &(q) should move the curve q2St(q)further from the isotropic one for that process.For the experimental data for the perpendicular direction only a continuous return down towards the isotropic curve and never up is observed as t increases. Thus in fig. 7 experimental data are contrasted with the calculated form factor after the end of contraction q c l/D (dotted line) the data lie below the graph. This result has been noted before using instead of S,(q) an extracted value of Rg(t).Note however that the variation with t is the opposite if St(q)increases R,(t) decreases. The same discrepancy with theory appears. It is apparent that the contraction could be masked I I 1 n Y ." E d W n P W t4- U b 3 2 1 .-r 1 I -1 -0.5 0 0.5 1 1.5 2 log, qR Fig.6. Comparison of data for the factor of a chain relaxed between entanglements S(q) = EC,Zjexp(-q2r:j/6) with i-j > N,,affine values for r$i i-j < N, isotropic value for r& (dashed line) and r&= (i-j)2/NZ,(R2-1) Nea2+(i-j) a2(full line) for N = N/30 N/60 N/90 N/120 (N = 7000). a,10 + 20 and x ,60 s [see ref. (7) and (16)]. F. BOUE M.NIERLICH AND K. OSAKI ">-' /' / / / I I I I 1 I Ii' /--1 0 1 2 log10 qRgiS0 Fig. 7.Comparison of data (same symbols as in fig. 1) with a form factor calculated by assuming the disengagement of a chain contracted in a tube of diameter zero. Chosen times are such that t/qer=0.1 0.2 0.4 and 0.8.4 FAR DYNAMICS OF MOLTEN POLYMERS by the beginning of disengagement this question will be discussed in section 5 with reference to newly available data. THIRD PROCESS DISENGAGEMENT The solid lines in fig. 7 show the results of the theoretical calculation16 assuming that contraction has occurred and is completely achieved and that the tube diameter Dis sufficiently small that qD 61. The figure allows a comparison with experimental data which shows that the effect of the disengagement on the form factor does not permit us to keep both assumptions and to account for the data. The disengagement induces an over-slow relaxation at low q in the perpendicular direction; at large q in both directions it always leads to a plateau as it averages [eqn (3)] between Saniso and Siso,which both give a plateau.Accordingly S,(q) was calculated taking account of our observations about the first and second processes we include D# 0 and we exclude the contraction. The initial state for disengagement then corresponds to the theoretical form factor plotted in fig. 6. The same calculation as was done for an initial contracted state in fig. 7 is then applied to the form factors of fig. 6. A corresponding comparison with the data is illustrated in fig. 8(a)and (b).Here it was necessary to choose the same value of N (here N,”,) for all the calculated curves however at least fig. 8(a)-(c) show that a common value of N/Ne = 120 does lead to a satisfactory agreement with t/Tter t/Tdis-DISENGAGEMENT AND FACTORISATION LAW We now provide a direct test of the disengagement removing the preceding modifications needed to calculate the form factor.As remarked above a law of this type expressed in eqn (3) is implied by the theoretical calculation of the disengagement. Agreement between the calculation and experimental data means that eqn (3) is obeyed within experimental uncertainty. Eqn (3) can be termed a factorization law since it can be written as St(q)-siso(q) = [saniso(q) -siso(q)l dt) (4) where the q-and t-dependences can be factorised. Thus for different t,the curves of log,,[S,(q) -Siso(q)] against qshould superimpose at high qby a q-independent vertical shift. The variation with tof this shift has been compared with two other quantities log,,[Ri,(t) -and log,,a(t) where CT is the stress.At large tthe three functional dependences coincide and are close to a linear variation as reported in ref. (6). However higher values of t/qer> 1 are necessary to check this variation. To avoid sample destruction at such values an experiment is planned in which long labelled chains are dissolved in a matrix of molecular weight three times larger than the already large molecular weight of the labelled chain. 5. COMPARISON WITH OTHER EXPERIMENTS The results discussed above show discrepancies with both the Rouse and the tube models. However a comparison with the Rouse model leads to the same conclusions as do macroscopic observations. At small times (or for small molecular weights) agreement is observed which vanishes at long times (see for example the variation with z of the retardation spectrum H(1n T)~ which follows the Rouse prediction (ccIn c at small z).On the other hand comparison with the tube model leads to discrepancies which are not observed at the macroscopic level. These discrepancies are discussed here. For this purpose supplementary SANS experiments were conducted on samples I Y " -1 -0.5 0 0.5 1 1.5 2 -1 -0.5 0 0.5 1 1.5 log qR 1% qR N e Fig. 8. Comparison of data with a form factor calculated by assuming the disengagement of non-contracted chain with a non-zero tube diameter such as N = N/30 N/60 N/90 N/120 as in fig. 6. (a) Data from sample 50 compared with model for t/T = 0.4. (6) Data from sample 46 compared with model for t/ T = 0.1.DYNAMICS OF MOLTEN POLYMERS Table 2. Molecular-weight distribution of polymer samples PSD 784000 1.41 PSH 760000 1.51 PSD 95 500 1.17 PSH 117000 1.11 PSD 2 600 000 1.15 PSH 2 640 000 1.28 where the molecular weight of the chains is varied. Data for one mixture of very long PSD and PSH chains of about the same weight (M z 5 x los) and for one mixture of short chains (M z lo5) dissolved in a matrix either of the same molecular weight or one much higher (M > lo6) are categorised in table 2. HIGHER-MOLECULAR-WEIGHT CHAINS AND THE CONTRACTION PROCESS This experiment was planned to check the lack of contraction. The chosen values for t lie from 0 to above the estimated value of TRouse.Two extension ratios were used A= 3 and a higher one A = 4.6 which is judged to increase the contraction effect. The results are reported in detail in a forthcoming paper but the present conclusion is that while all the experimental orders of magnitude would in accord with theory lead to a contraction effect the latter is not observed. In particular the difference between TRouse and GiSis for this molecular weight large enough to prevent a complete masking of contraction by the commencement of disengagement. SMALL-MOLECULAR-WEIGHT CHAINS TUBE DIAMETER AND DISENGAGEMENT This experiment covers the usual q range and a time range of 10 s at 1 17 "Cto 20 min at 117 "C (qerx 5 x lo3 s at 117 "C for M = 95000). The essential results reported in ref.(6) are as follows. At small times agreement with the curve calculated from the Rouse model is found as for the mixture 780000 (PSD)+ 760000 (PSH) described earlier. At time t z Te,satisfactory agreement is found with the form factor of a chain relaxed at the length scale of the tube diameter with N = 2 x lo4,which is the same value of N as in the preceding experiment. From this given state one can calculate the effect of a disengagement process and it is found to agree with the data at the largest time (t117 = 20 min i.e. t/qerz 2/10). Let us recall that the divergences from Rouse behaviour would in any case be weak because the chains are short. EFFECT OF THE MATRIX For each value of t at different temperatures one sample has been made for the two matrices M x lo5 and M x 1.5 x log to enable a comparison of the two form factors to be made.For t/qerx (t/TRousez 5 x no difference appears; this is in agreement with the Doi and Edwards model. For the largest times (t/cerz 2/ 10 t/ TRousez l) one can see a difference of the order of the experimental uncertainty in the parallel direction where the form factor for the matrix of smaller molecular weight is more relaxed. However this smaller molecular weight could provoke sample deterioration as the ratio of t to the terminal time of the matrix is ca. 1/10 while it is lo3 times more for the other. A more suitable experiment has to be done for large t. F. BOUE M.NIERLICH AND K. OSAKI 6. GENERAL REMARKS AND CONCLUSIONS Our observation of the return of S,(q) to the isotropic form factor with time owing to a time-temperature superposition leads to systematic behaviour each stage of which is characterized by a different variation of the function S,(q) the rate at a given 4.When t is small a strong dependence upon q is observed such that only the small distances relax; the variation is close to that of the Rouse model [S,(q)x q4]. Then a second stage appears where the relaxation slows down very much while large distances now relax as well as short ones. This evolves to a third stage where it is possible to observe a q-independent rate [S,(q) = c,,]. This succession of stages almost exactly corresponds to the relaxation of stress with time (in linear regime) for a similar stepstrain experiment.The first stage corresponds to the transition zone at the end of this zone the slowing down of the stress relaxation ~. ~ starts for a time z~The time of~onset of the slowing down of S,(q) is of same order as T,~.Consecutively the Rouse model works for t,, for SANS as for stress. Evidently this is the same small-t region of the inelastic neutron-scattering experiment3 which also leads to the suggestion of Rouse motion. For t,, the Rouse model fails for S,(q) it also fails for stress relaxation for long chains. We are then led to test the tube theories which explain this slowing down. The disengagement process which is proposed leads to a very large terminal time. The q-independent rate (factorisation law) observed in the third stage of evolution of S,(q) (t/?& < 0.I) is characteristic of this disengagement.These conclusions in favour of the tube model must be modified by two remarks. First the test of a q-independent rate must be extended to larger t/qerratios. This may solve the paradox6 that S,(q) in the parallel direction seems to accord both to a superposition law and to a factorisation law.? The second remark concerns the second stage ofevolution of S,(q).For the non-linear regime the Doi and Edwards prediction agrees with the stress relaxation experiment the stress is higher at t 2Ttr than the linear value [i.e.0= (A2-l/A) G; G; being the plateau modulus4] and returns to the linear value for t x M2.In our SANS experiment the corresponding effect (contraction of the dimensions of the chain) is not observed.It then remains to be explained how both the stress behaviour and the lack of contraction can be consistent. We therefore propose to observe what motions occur when the tube model motions cannot be present i.e.when reptation is hindered this is the case for cross-linked chains. The same experiment would be performed. A strip of cross-linked polystyrene would be suddenly stretched and observed instantaneously by quenching using SANS during transient relaxation. This experiment has been started. Preliminary results are plotted in fig. 9. As the time t after the sudden deformation increases the form factor goes t The following explanation could be advanced starting from low q the onset of the behaviour exemplified by eqn (3) is encountered when it is possible simply to add the new and old tube contributions the calculation shows that this is possible when I/q is higher than the sizes of the different parts of the ‘triblock copolymer’.The relevant size is the smaller giving the smaller q; here it is the size of the new part rnew(t)zc J(:[I -p(f)]). From the expression of p(f)given in ref. (9) and (lo) we can develop it for f/Tdls < 1/10 81 ~(t) = X 7~XP(-p2t/GiJ z 1 -t (t/Tdls). PL This gves rnewa t1I3 an onset of eqn (3) at q a leading to an apparent superposition in the range of the onset and factorisation at higher q. DYNAMICS OF MOLTEN POLYMERS 1 oL I 1 1 I I 0 0.02 0.04 0.06 0.01 0.1 q1A-l Fig. 9. Data for preliminary experiment of transient relaxation upon cross-linked melt from a mixture of PSD M = 600000 in PSH MH = 600000.Approximate mass between cross-links M x 50000. Stretching temperature 117 "C.0, t = 10 s; x ,t = 100 s; + isotropic sample and 0,isotropic melt. back to the one at t infinite. This shows that it is possible to cover the time range where rearrangements occurs in a deformed crosslinked rubber before reaching the plateau behaviour. J. P. Cotton D. Decker M. Benoit B. Farnoux J. Higgins G. Jannink R. Ober C. Picot and J. des Cloizeaux Macromolecules 1974 7 863. F. Boue M. Nierlich and L. Leibler Polymer 1982 23 29. D. Richter A. Baumgartner K. Binder B. Ewen and J. B. Hayter Phys. Rev. Lett. 1981 47 109. J. D. Ferry Viscoelastic Properties of Polymers (Wiley New York 3rd edn 1982).F. Boue M. Nierlich G. Jannink and R.C. Ball J. Phys. (Paris) 1982 43 137. F. Boue M. Nierlich G. Jannink and R. C. Ball J.Phys. (Paris) Lett. 1982,43 L-585; L-591; L-593; L-600. ' F. Boue Thtse d'Etat (University of Orsay 1982). F. Boue M. Nierlich J. Hertz and K. Osaki to be published. This paper will report tests of the tube model especially the Doi and Edwards contraction for very long chains. @ P. G. de Gennes J. Chem. Phys. 1971,55 572. lo M. Doi and S. F. Edwards J. Chem. SOC., Faraday Trans. 2 1978 74 1789; 1802; 18 18. l1 J. Klein Macromolecules 1981 4 460. l2 R. P. Wool and K. M. O'Connor J. Appl. Phys. 1981,52 5953. l3 K. Judd H. H. Kaush and J. C. Williams J. Muter. Sci.,1981 16 204. l4 P. E. Rouse Jr J. Chem. Phys.1953,21 1273. See also ref. (3) or P. G. de Gennes Scaling Concepts in Polymer Physics (Cornell University Press Ithaca 1979 USA). l5 F. Boue K. Osaki and R. C. Ball accepted for publication in J. Polym. Sci.,Polym. Phys. Ed. It is shown in ref. (6) that in spite of the apparent intention of the author to use a reptation model the calculation is as given in S. Daoud J. Phys. (Paris) 1977 38 751. lo P. G. de Gennes and L. LCger Dynamics of Entangled Chains to be published. l7 K. Osaki and K. Kurata Macromolecules 1980 13 671. l8 L. Leger H. Hervet and F. Rondelez Macromolecules 1981 14 1732.