Faraday Symp. Chem. SOC.,1983 18 159-171 Dynamics of Entangled Star-branched Polymers BY JACOB KLEIN* The Cavendish Laboratory Madingley Road Cambridge CB3 OHE and Polymer Department Weizmann Institute of Science Rehovot Israel AND DIANNE FLETCHER The Cavendish Laboratory Madingley Road Cambridge CB3 OHE AND LEWIS J. FETTERS? Akron University Akron Ohio 44325 U.S.A. Received 26th August 1983 The diffusion and longest-relaxation process of star molecules entangled in a fixed-obstacle matrix and in melts of a linear polymer are considered. The dynamics in a fixed matrix are considered in terms of a general diffusive motion in an ‘entropic’ potential field. The resulting diffusion coefficient D and longest relaxation time z are calculated and scale as D,cc (l/Nb)exp( -aN,) z cc Nbexp(aNb) (for a 3-arm star with Nb monomers per arm where a is a constant).For the case of a linear melt matrix it is argued that ‘tube-renewal’ effects will dominate the dynamic behaviour above a certain N,,. We report the first experimental study of the diffusion coefficient D(N) of 3-arm deuterated polybutadiene N-mer stars diffusing in a highly entangled melt of linear polyethylene. Our results provide strong support for the calculated form of the diffusion coefficient at low values of N and suggest that at high N values ‘tube’ renewal effects become important. Considerable progress has been made in recent years in our understanding of the dynamics of entangled polymer chains. The central idea is the approximation of the entanglement condition -that chains may not cross through each other -by the confinement of such chains to ‘tubes’.I9 For the case of linear molecules the resulting diffusion of molecules (and thus their longest relaxation) takes place by reptation;2 this has formed the basis of molecular theories of viscoelasti~ity~~ which provide quite satisfactory agreement with e~periment.~ Diffusion studies provide a more direct test of the reptation model ; several independent studies using i.r.microdensitometry,6 forced Rayleigh ~cattering,~ n.m.r. techniques8 and computer-simulation,9 have shown that D cc NP2 (1) for the translational diffusion coefficient D of entangled linear N-mers in agreement with the reptation prediction.2 It was realised some time ago that the ‘tube’ confining any entangled molecules consisting as it does of entanglement constraints due to intersections with its neighbours would itself relax with time as these neighbours diffused away.lo.l1 It was argued however that for the case of linear molecules the ~~, tube relaxation time Ttube $ T~ the time for a molecule to renew its configuration by reptation so that tube-renewal effects would be essentially negligible in entangled linear systems.More direct support for the weakness of tube-renewal effects in such t Present address Exxon Research Corporation Clinton N.J. U.S.A. 159 DYNAMICS OF ENTANGLED STAR POLYMERS Fig. 1. Illustrating the basic diffusion step where the centre monomer C is essentially fixed over an arm-retraction process.is the size of the ‘step’ from the original (+) to the new position of C. (a)Starting configuration. (b)An arm has fully retracted within its original tube and can take a step into a new topological configuration (c). systems has come from recent studies7* l2 showing that D for a given linear diffusant was independent of the matrix molecular weight M (for Mm > Me the critical entanglement molecular weight). The situation both theoretical and experimental is considerably less well resolved for the case of entangled non-linear molecules such as branched or star-like polymers. In an early paper de Gennes13 considered the motion of anf-arm star (Nb monomers per arm) entangled in a fixed non-crossable obstacle lattice. He proposed that translational centre-of-mass diffusion of the star could take place only when an arm retracted as shown in fig.1 without crossing any of the obstacles to the position of the centre monomer and then moved out again. This would move the centre-of-mass by a lattice spacing or a tube-diameter 5 in the representation of fig. 1. De Gennes calculated the probability P,(Nb) of configurations such as in fig. 1(b)as &(Nb) exp Nb) (2) with a a constant and proposed that the rate of steps such as in fig. 1 would be proportional to Pl(Nb). Finally he conjectured a characteristic time zi,(Nb) for the arm relaxation zkl(Nb) zrep(Nb)/P1(Nb) X Z1 N3b eXp (a,Nb) (3) where Trep(Nb) =-Nb32 is the corresponding reptation time for a linear Nb-mer and z is a microscopic jump time.3 The corresponding translational diffusion coefficient is (forf= 3) rv -Drep(Nb) exp (-a Nb) Nb where Drep(Nb) is the reptation diffusion coefficient for a linear Nbmer.Using a different approach Doi and Kuzuu14 also obtained a relation similar to eqn (3) for an arm relaxation time while very recently Graessley in a further modification of the essential idea of fig. 1 suggested15 %2 %2 X +exp(-alNb) (f= 3) Nb Je KLEIN D. FLETCHER AND L. J. FETTERS 161 where the characteristic time for a step < as in fig. 1 is now taken to be ze(Nb) z z1Nb2 the equilibration time of a linear Nb chain within a ‘tube’ (equivalent to its Rouse relaxation time).3 In all these approaches it was assumed (i) that the centre monomer is essentially fixed over an arm-retraction time and (ii) that the entangling matrix is topologically invariant over this time (i.e.no ‘ tube ’-renewal effects). Recently Evansls has reported computer simulations of star polymers diffusing in a fixed-obstacle matrix and found that the variations of the arm relaxation time 7; and the diffusion coefficient DL were better fitted by power laws rather than exponentials. Edwards and Needs,17 however extending the Evans simulation to obtain better statistics found a variation of 7; and Di with Nb significantly more rapid than a power relation and better fitted by an exponential law. Experimentally the main indications to date have come from zero-shear-rate viscosity (qo)data on entangled star melts,l* One expects3 qo to be proportional to a relaxation time z (such as & above) but while q,,does appear to vary approximately exponentially with Nb,the same data have also been adduced16 in support of a power law such as relation (6).In this paper we treat briefly the diffusion of an entangled star-branched molecule with a centre monomer which need not be fixed during the time for a topological step and also the dynamics of star molecules diffusing in an entangled linear melt. In the latter case one might expect ‘tube-renewal’ to be important as qube may be comparable with or less than z,. Finally we describe the first experimental study of the diffusion coefficient of entangled 3-arm star polymers diffusing in a linear melt and consider the implications of our results for current theoretical understanding.DIFFUSION OF ENTANGLED STARS IN FIXED NETWORKS AND IN MELTS In addition to the mechanism illustrated in fig. 1 there is another possibility by which the centre of mass of an entangled star in a fixed topology may make a genuine topological step. This is illustrated in fig. 2 and involves motion of the centre monomer C a distance n monomers down one of thef-arm ‘tubes’ dragging the other df-1) arms behind it [fig. 2(b)].The equilibrium probability of such a configuration is expected to be* P2(n) cc exp [a2ndf-2)1 (012 > (8) Motion of C in an increasing n direction will be opposed by an entropic force,13 which will lead to an effective additional potential (arising from configurational entropy loss) U(n)= a2(nlkT(f-2) (9) (n < 0 being taken for our purposes as motion of C down a different arm to that shown in fig.2). Wherever In[= Nbthe other (f- 1) arms can completely escape their original ‘ tubes ’ [fig. 2 (c)],and a real stochasticaly independent step < can be taken by the centre monomer C [fig. 2(d)].To evaluate the overall expectation time (z) (for In1 + Nb) * Helfand and Pearson have very recently recalculated P,(N) and &(A’) and find a pre-exponential (power) N dependence as well as the dominant exponential term. In this paper we use the exponential form for P(N). 6 FAR DYNAMICS OF ENTANGLED STAR POLYMERS Fig. 2. An alternative mechanism for centre-of-mass diffusion. The centre monomer C of a star starting with configuration (a)moves down one of the arm tubes ‘dragging’ the other arms with it (6)until (c) they are fully retracted past the original position of C( +).The arms can now return to new topological configurations (d),allowing C to make a diffusive step 5. for this process it is necessary to solve for the diffusive motion of the centre monomer in the centrosymmetric field U. The resulting diffusion equation is20 where z(n) is the expectation time for the centre monomer starting at position n to reach In1 = Nb (down any of the arms) D,is the curvilinear diffusion coefficient for a reptating linear N,-mer (cc l/Nb) and is taken as the monomer size. (This corresponds to a rescaling such that n and Nb are the number of entanglement lengths rather than monomers but has no effect on the final result.) Eqn (10) has the form of the familiar diffusion equation of a particle in a potential field slightly modified by recasting it in terms of an ‘escape time’ z(n).The boundary conditions for eqn (10) are z = 0 whenever In1 = N,.Solving eqn (lo) one finds an overall expectation time for the process n -+ Nb whose dominant term has the form (omitting a weak pre-exponential f-dependence). The corresponding diffusion coeffi-cient assuming a step size 5 as before is then 42 = r2 - Nb Drep(Nb) exp( -Nb) (f= 3). (12) J. KLEIN D. FLETCHER AND L. J. FETTERS co. 5CI Fig. 3. A diffusion step taking place by a combination of the processes illustrated in fig. 1 and 2 the centre monomer C moves partly down an arm (nmonomers) and the remaining.(l\',-n) monomers of one of the other arms now retract (b)to allow C to make a step of size <(c).The forms of zs2and D, differ from the earlier proposed forms zil and DLl in eqn (3)-(5) (by factors of Nb2and Nb respectively); this suggests that in calculating the longest times for a process such as in fig. 1 one needs to solve for the motion using an equation similar to eqn (10). The resulting values have the form and More generally one might expect the basic diffusion step in a fixed topology to take place by a combination of the configurations shown in fig. 1 and 2 as indicated in fig. 3 (f=3 for clarity) the centre monomer moves n units down one arm dragging 6-2 DYNAMICS OF ENTANGLED STAR POLYMERS the other (f-1) arms with it and (f-2) of the remaining arms then ‘retract’ (Nb -n) monomers each as in fig.3(b). The static probability of such a configuration is P(n,Nb -n)OC &(Nb -n)P2(n). (13) From eqn (12)’ (4’) and (13) one expects a diffusion coefficient associated with this ‘nth mode’ of the form D(n) % a2N~Dre,(N~)eXp{-[al(N~-n)+~,n]> (f=3,a N a,). (14) Since the processes for different n values must be independent the overall diffusion coefficient for the star entangled in a fixed topology is N Ds = 2 D(n). n-o Thus allowing the centre monomer to move (in contrast to the earlier approaches) does lead to some enhancement in the diffusion coefficient. This enhancement would be greatest if a = a = a leading to a form Di % a2N2,Drep(Nb)eXp(-UNb) (f=3) (16) i.e.by a pre-exponential factor Nb at most. In fact one expects a = a2/2(= a) so that (especially for large Nb) the enhancement relative to a ‘fixed-centre’ approach is less than the factor Nb and is of order l/a or recalling DreD(Nb) K 1/N& The computer-simulation l7 of 3-arm stars entangled in a fixed-obstacle matrix do in fact show a considerable displacement (% < the obstacle spacing or equivalent tube diameter) of the centre monomer before a stochastic diffusion step is achieved. This suggests that the composite process shown in fig. 3 is the one followed in practice for a star in fixed surroundings. TUBE RENEWAL In concentrated polymer solutions and polymer melts one can no longer consider the surroundings of any given molecule as fixed and tube-renewal effects may become important.While such effects are expected to be generally weak and have not to date been directly observed for linear polymer systems,12 they may be important for the case of a star molecule entangled in a melt of linear molecules. It is argued elsewhere that whenever ‘tube’renewal -by release of tube constraints due to the reptating away of entangling neighbours-is dominant it is a hydro- dynamically unscreened or Zimm-like process.20721 For a linear N-mer in a chemically identical linear P-mer melt one then expects the diffusion coefficient due to ‘tube’-renewal effects to have the form Pc Dtube(N [g][TIDrep(N) where Pc = N is the number of monomers per ‘entanglement’ length.Consider a 3-arm star [for simplicity -the extension to f-arms is reasonably straightforward its main effect being to introduce a factor (f-2) in the exponential in the expression for D,] Nb monomers per arm diffusing in a chemically identical linear P-melt. We expect J. KLEIN D. FLETCHER AND L. J. FETTERS 1 10 100 Nb Fig. 4. Schematic plot of D,(N) according to eqn (19). The broken line shows Docc 1/Nb2 (i.e. the reptation relation for a linear N,-mer) on the same scale. the overall diffusion coefficient D to be a sum of the intrinsic coefficient D of the star in fixed surroundings and a ‘tube’-renewal contribution Dtube,as the two processes are independent giving Ds(Nb) -k Dtube(Nb where we have used D from eqn (17).We note that when cast in this form i.e. in terms of the number of entanglement lengths a z 0.5 both as calculated in a lattice mode119 and from empirical fits with viscoelastic data.14 From eqn (18) we seen that the diffusion is dominated by tube-renewal processes wherever Nb $ N* where For Pin the usual range (3-100) Pc we find N* 1:(5-20) N,. Although we have omitted numerical prefactors in our discussion the crossover value N* is insensitive to these due to the exponential on the left-hand side of eqn (20). The form of D,(Nb) is shown in fig. 4. Similar considerations apply to the longest overall relaxation times z,(Nb) of the star arms (except that these should be essentially independent off) and one would expect the overall relaxation rate to be a sum of the intrinsic or fixed surroundings (1/T,) and tube-renewal (1 /qube) values although a detailed discussion will not be presented here.We note in passing that for the case of a melt of stars tube-renewal effects are not straightforward to calculate. In the limit of infinitely high Nb (in a melt DYNAMICS OF ENTANGLED STAR POLYMERS Table1. Molecular characteristics oflinear and 3-am star-branched polybutadiene (PBD) samples 10-3~4~ sample linear (MW/Mn) N CDS-B-3 2.6 1.04 I90 LF- 1 10.8 1.02 770 LF-2 22 1-04 1570 LF-3 53 1.03 3750 LF-4 96 1.06 6830 3-arm star-branched JK-8A 3.1 I .07 220 JK-5A 6.8 1.07 490 JK-6A 9.3 I .04 660 WG-2A 16.8 I .04 1200 WG-1A 21.9 1.06 1560 JK-4A 27.6 1.04 1960 JK-3A 31.2 1.04 2220 JK-2A 37.2 1.05 2650 JK-IA 66.3 1.05 4720 linear polyethylene HDPEI of star-branched molecules) one could argue20 that tube-renewal effects become self-consistently negligible but this may not be the case for intermediate (and practically important) molecular sizes.DIFFUSION OF 3-ARM STAR POLYMERS IN LINEAR POLYMER MELTS EXPERIMENTAL We have measured the diffusion coefficient D(N)of linear and 3-arm star-branched deuterated polybutadienes (d-PBD) of total degree of poplymerisation N diffusing in a high-molecular-weight linear polyethylene melt (designated HDPE 1) and in melts of hydrogenated PBD (p-PBD) samples. The experimental technique is based on infrared microdensitometry and has been described in detail elsewhere.6 Essentially it involves the setting up of a step function in concentration of deuterium-labelled diffusant molecules within a matrix of unlabelled (protonated) polymer and monitoring (by i.r.microdensitometry) the diffusion broadening with time of the original diffusant concentration. The extent of broadening over a known time permits evaluation of D. MATERIALS The molecular characteristics of the polymers used are shown in table 1. The precursor PBD polymers had a microstructure consisting of 92% 174-isomer and 8 % 1,2-isomer7 and were saturated under high D (or H for the hydrogenated samples) pressure in the presence of a palladium catalyst as described by Rachapudy et aL2 The reaction is D CH,-CH=CH-CH 2dCH,-CHD-CHD-CH atm J.KLEIN D. FLETCHER AND L. J. FETTERS and results in a polyethylene-like backbone microstructure with nominal 25 % deuteration as indicated. In practice the extent of deuteration as revealed by independent i.r.-absorption control studies was ca. 33% indicating some H/D exchange in addition to the saturation reaction. The backbone structure was essentially linear with one ethyl (-C,H,) branch every 60 or so backbone C atoms on average owing to the 1,2-component in the precursor polymer. The PBD molecules were anionically p~lymerised~~ and their characteristics determined via osmometry and size-exclusion chromatography. Infrared measurements following deuteration/hydrogenation indicated essentially complete saturation (dis- appearance of the 960 cm-l peak) of the precursor polymers while a redetermination of the molecular characteristics via size-exclusion chromatography showed negligible degradation had occurred.Independent measurements on the stars indicated a degree of branchingf = 3.0 f0.1. The N values given in table 1 are the numbers of main-chain CHJCHD units in the diffusants while M and polydispersity values for the stars are obtained from the corresponding values for the linear precursors. 3% w/w solid solutions of labelled diffusants in HDPE1 and the hydrogenated PBD samples were made by precipitation from a common solvent by adding the hot solution to an excess of cold methanol. Step functions were then set up as described earlier.6 The diffusion runs were carried out in an oxygen-free N atmosphere and the resulting broadened profiles analysed in the usual way.The broadened profiles fitted the theoretical Fickian curves closely across the entire concentration range; this is in contrast to the previous study6 of the diffusion-broadening of deuterated polyethylene samples (d-PE) where polydispersity effects resulted in deviation from the Fickian profile other than at the interface between the high- and low-concentration regions.6 This is due to the essentially monodispersed nature of the d-PBD samples used in the present experiments (table 1 M,/Mn = 1.05) compared with the moderate polydispersity (Mw/MnII 2) in the earlier investigation. This close fit was important in the context of measuring the diffusion of the star diffusants as the form of 13 against N plots could be obtained directly without analysis for polydispersity effects; in addition the closer fit of the experimental profiles with the Fickian curves implied lower D values could be more reliably measured.RESULTS AND DISCUSSION Fig. 5 and 6 show D plotted against N on a double-logarithmic scale for the linear and star d-PBD diffusants in HDPE1 at 176 f0.5 OC. Each point represents a separate experiment run over different lengths of time (varying by a factor of up to ca. 4 within sets of points for a given N value) from a few days to several months and is the mean of ca. 10 independent concentration-profile measurements. Samples were analysed for thermal degradation in a separate control study under the same conditions as the diffusion experiments no detectable degradation was observed after a period of two months at 176 OC.The solid straight line fig. 5 is a regression analysis best fit (correlation coefficient r2 = 0.99) to a power law for D(N)for the linear diffusants (15 experiments) and is D = DLN-1.95k0.1 (21) with D = 2.8 x cm2s-l. This is in accord with the inverse square law expected for pure reptation and with previous experimental studies. The absolute value of D (for a given N) is greater (by a factor of ca. 2) than that measured for d-PE diffusing under the same conditions,6 although the functional dependence of D on N is the same. (This may be partly due to the slightly different microstructure of the two polymers and also -to a greater extent -to the way in which the reported N values of the d-PE DYNAMICS OF ENTANGLED STAR POLYMERS lo2 103 lo4 N Fig.5. Diffusion coefficient D(N) for linear deuterated PBD N-mers diffusing in an HDPE1 melt at 176OC. The solid line is a least-squares best fit to the data of the form D = DLN-1.95f0.1. diffusants were originally evaluated. These are currently being systematically redetermined using the same procedures as for the d-PBD samples.) We note that the range of N in the present study is considerably larger than in the previous investigation. Fig. 6 shows D(N)for the 3-arm star-branched diffusants (Nis the total main-chain degree of polymerisation i.e. N = 3Nb). The variation of D with N is initially appreciably more rapid than a power law. A very good least-squares fit (r2= 0.99) to the first five points (17 experiments) is obtained by a power-exponential relation of the form calculated in eqn (17) D D(N)=2exp (-aN) N withD = 8.2x lops cm2 s-l a = 2.82 x loh3(solid curve fig.6). A good fit (r2= 0.99) is also obtained for the first five points using a pure exponential relation D(N)=D,exp (-aN) with D = 3.9 x cm2s-l a =4.2 x However if one tries to fit a simple power-law dependence (Dcc Na) to the initial data the best (least-squares) fit is considerably less good (r2N 0.94) while a power exponential of the’ form D K (1/N2)exp (-aN)is an even poorer fit (r2=0.85). For N 21500 the experimental data in fig. 6 show a progressively weaker dependence of D on N. This may be attributed to the increasingly dominant role of ‘tube-renewal’ effects due to the mobility of the linear (HDPEl) matrix molecules as discussed in the previous section.The general form of the D(N) data in fig. 6 compares well with the expression of eqn (19) and the corresponding plot in fig. 4. J. KLEIN D. FLETCHER AND L. J. FETTERS N Fig. 6. Diffusion coefficient D(N) for 3-arm star-branched deuterated PBD N-mers diffusing in an HDPE1 melt at 176 OC. The solid curve is a least-squares best fit to the data of the form D D = ‘exp(-aN) N (see text for values). The dotted line shows D(N) for the linear N-mers from fig. 5. This provides the first direct indication of ‘tube-renewal’ effects in an entangled polymer system. We are currently investigating the effects of tube-renewal more systematically by carrying out Dmeasurements for given star molecules in a series of linear hydrogenated polybutadiene (p-PBD) matrices of different molecular weights M,.[We recall’? l2 that for the corresponding case of a given linear diffusant in entangled matrices D does not vary (within error) with M,.] Preliminary indications are that for a star-branched diffusant of N N 500 [star JK-SA point (1) on fig. 61 D is independent of M for M > lo5.For a star of N N 2000 [star JK4 point (2) on fig. 61 D increases sharply with decreasing M for M < 1.5 x lo5 (the highest M value in our study). These indications are consistent with the notion that tube renewal is important for the longer star (whose D value is significantly higher than the predicted exponentially decaying ‘fixed-tube’ variation shown in fig.6) but not in the case of the shorter one. This study of the effects of M is not yet complete however and we shall not consider it in more detail here. We have also measured D as a function of temperature for star JK6 in HDPEl over the range 136-176 “C. The effective activation energy for diffusion Q = 38 & 3 kJ mol-l is significantly higher than for a corresponding linear d-PE molecule diffusing in the same conditions for which Q = 24f2 kJ mol-l. This DYNAMICS OF ENTANGLED STAR POLYMERS difference is consistent with the differences observed between activation energies for viscous flow in linear and branched polyethylene and hydrogenated polyisoprene me1 t s.23 SUMMARY We have considered the diffusion and longest relaxation process of (f-arm Nb monomers per arm) stars entangled in a fixed obstacle matrix in terms of a general diffusive motion (within ‘tubes’) in an ‘entropic’ potential field.The resulting diffusion coefficient D,and longest relaxation time z are found to vary as (omitting a weak pre-exponential f-dependence) where D,,,(N) and z,,,(N) are the corresponding values for a reptating linear N-mer. For diffusion of stars in a linear polymer melt it is proposed that ‘tube-renewal’ effects will dominate the dynamics of the star beyond a certain Nb. We have also measured D(N) for linear and for 3-arm star N-mers diffusing in a melt matrix of a chemically similar linear polymer. The results for the linear diffusants are consistent with pure reptative motion D = 1/N2 in accord with earlier studies.For the star diffusants our results show a rapid initial variation of D with N closely resembling the power exponential variation of D,calculated above. For the longest stars D(N) varies considerably more weakly with N suggesting that in this regime (N & 1500) ‘tube-renewal’ effects dominate the dynamics of the star in the linear matrix. J.K. is particularly grateful to Dr R. Ball and Prof. S. F. Edwards for their help in solving the diffusion problem [eqn (lo)] and for very fruitful discussions and to Dr J. Deutsch for useful comments. We thank Prof. W. Graessley and coworkers for useful suggestions and advice on the high-pressure deuteration procedure.We thank the S.E.R.C. for financial support (D.F.) and the N.S.F. (U.S.A.) for support of the work done at Akron University (grant no. DMR-79-008299). S. F. Edwards Proc. Phys. Soc. 1967 91 513. P. G. De Gennes J. 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