摘要:

Faraday Symp. Chem. SOC.,1983 18 145-158 The Shape and Stiffness Dependence of Transport Properties of Macromolecules BY S. F. EDWARDS The Cavendish Laboratory Madingley Road Cambridge CB3 OHE Received 1st September 1983 It is argued that the impenetrability of macromolecules or of bodies in suspensions is better treated by honouring topological invariance than by using infinitely repulsive forces. The topological invariance is best treated by accurate analysis of the coupled equations of the polymers and their surroundings. Soft forces between polymers can only be treated by the use of collective coordinates so a comprehensive set of equations of motion needs to involve fluid velocities and appropriate collective coordinates for the polymer. The appropriate collective coordinate is defined and shown to lead to closed equations.These equations now contain the coupled motion normal (soft) forces the property that polymers cannot pass through each other etc. Some discussion is given of solutions for these formidable equations in various limiting cases. 1. INTRODUCTION The equations governing the motion of suspensions of molecules or particles are of growing interest with much progress having been made but with many unsolved problems. A central difficulty is that situations that are intuitively obvious for example the interaction of two hard but non-spherical objects turn out to be very awkward to put into a mathematical form. Consider the interaction of the two bodies shown below which are made of some hard material.Suppose there are no forces unless the bodies touch when there is an infinite repulsion. If one tries to write a symbol C,V(R,-RR,)for the potential energy where R and R are any points on surfaces 1 and 2 then Vis infinite when R,= R,and when R is inside body 2 or R,inside body 1. However this means that it cannot be V(R,-R2)alone but V([R,][R,]), where V puts the earlier part of this sentence into mathematics. How does one put it into 145 TRANSPORT PROPERTIES OF MACROMOLECULES mathematics? First one has to describe the surface. Neither surface permits any simple coordinates such as polars. Surface 2 is roughly toroidal but it is easy to see distortions which wreck toroidal coordinates. One has to use some intrinsic coordinates r = R(s,,s,) where s and s represent labels on a geodesic system built into the surface not an easy thing in general and leading to more complications for things like Suppose nevertheless that we have equations R1(sP),sp)) R,(si2),sp)) and want to say that no point of R is inside 2 or vice versa.One has to have a third coordinate dl) say which labels a nest of surfaces of which R,is an example c(l) = a say. In section we have . . .-The dotted lines are c > a and c < a. One is therefore seeking a generating function t(r) such that t(r)= a (1.1) is the surface r = R,(s,,s,). The interior of the surface is c < a the outside c > a. A point of R,is inside if 4R2)< a and outside if 4R2)> a. A criterion for being outside is then s,” G(c[R,(s,,s,)] -x>dx = 0 for all s, s,.Thus the infinite repulsive force can be replaced by a subsidiary condition (1.4). Of course this leaves the key problem which is the invention of tgiven R(sl,s2) and which we will not pursue at this point. Similar but simple problems arise with polymers which can be considered as limiting cases of R(s,,s,) when s1 has a large range of values buts denotes the perimeter of a small section; in fact for flexible polymers the problem is basically specified by R(s,)alone although for rod polymers the familiar divergence of the potential theory of rods makes it worth carrying a finite section. S. F. EDWARDS 147 The ‘interaction’ problem for polymers is the entanglement problem that one polymer cannot pass through another but must go via an end.This problem persists even if there is no interaction in the usual sense present i.e. if we shrink a potential between two rods to have a smaller and smaller interaction radius the effect on the free energy correspondingly reduces right down to zero in the limit but the dynamical effect remains right down to the mathematical line molecule. For example for two long rods the statement that the two rods are in configuration (i) as against (ii) is expressed by giving a sense of direction n,,n to the rods and noting that if m is the vector from rod 1 to rod 2 along the line of minimum distance m*(n,x fiz)l(i) = -m*(ni x n&l(ii)* (13) To keep rod 1 on the same side of rod 2 one can therefore ask that m-(n,xn,) does not change sign.This result can be put more formally in terms of the Gaussian invariant rr 1 is constant for two infinite or closed curves which are moved in such a way that they do not cut. However the converse is not true. (One can move curves in such a way that after several intersections leading to a new topological class the Gaussian integral nevertheless remains the same.) Several powerful new papers1* have recently been written on this subject but it does appear that although some progress can be made by topological arguments on the whole this is too cumbersome a method. A way around these difficulties is to recognise that the interesting problem lies in the dynamics of the systems and that when the bodies are immersed in a medium satisfying some reasonable viscoelastic equations with a reasonable boundary con- dition the constraints imposed on the dynamics by the boundary conditions do keep the topological constraints satisfied.Or to put it another way one may not know how to write down the topological invariants of motion in which one body does not penetrate another but one can write down equations of motion in which they behave in such a way as to conserve these invariants. These equations will be considered in the next section. They lead to coupling the motion of the macromolecule or shaped bodies etc.,with the equations of the medium in which they are embedded. This still does not give the final equations of motion for the macroscopic behaviour of the suspension or solution.It is not at all obvious what coordinate system ought to be used to make such a description. Take the simple example of a suspension of rods much studied because of the importance of liquid crystals. Each rod has only two vector degrees of freedom its centre of mass and the direction in which it is pointing (or rather the angle in which it is pointing if it is symmetrical under inversion through the centre of mass). The centres of mass are say R and the density coordinate p(r) or pk = X exp (ik R,) in Fourier transform. TRANSPORT PROPERTIES OF MACROMOLECULES If the angle held by rod a is Oa da then one can invent Pk;1,m = I= &mc0a da)exp (ik Rg) (1-7) where Y are the usual spherical harmonics. For symmetrical rods one expects only even I to appear.The isotropic case is I = 0 and the orientation and its transport is contained in the ~~;~,~(t). This of course is a problem which has been much studied for rods but it exists also for flexible polymers where it is more subtle. The difficulty lies in the fact that whereas a rod has two internal degrees of freedom the flexible polymer has an infinite set. The problem arises as to what kind of collective coordinates will describe the position of the molecules and the internal degrees of freedom. There is a considerable literat~re~-~ on for example dumbell models where the polymer is modelled essentially by a spring but this method is inadequate in the sense of completeness. I.e. one wishes to use collective coordinates of a type which will describe the system with the same level of completeness as the direct use of polymer coordinates but in addition will close.It is not easy to make this point clear without going into detail and this is done anyway in section 3 where equations are deduced which do indeed close. The great challenge which then emerges is that as the density of polymers or inclusions increases there comes a density (or molecular weight) at which apparently a phase change or something close to it takes place and a much slower form of motion prevails. Although the actual transition is hard to describe the direct coordinates allow one to give a reasonable description of flexible and of rod polymers at high densities (other problems such as the flow of closely packed solid shapes also show the transition to high viscosity but are not so easy to describe).The question is can motions like reptation be deduced directly from the collective equations? The problem is studied in this paper but not solved. 2. DYNAMICS OF EMBEDDED OBJECTS The argument goes like this. If there is a fluid (of unit density) surrounding the object the Stokes boundary condition on R [which can be R(s,) or R(s,,s,)] is R = u(R) or if R is called U U = u(R). This can be built into equations of motion by a Lagrange multiplier of the structure ;r:14s){ U(s)-u[R(s)l)ds (2.3) or Then if the equation of motion of the fluid without the suspension is 9u=f (2.5) (where 9 is in general non-linear forfexternal forces) the effect of the suspension is to modify eqn (2.5) to 9~+ ZJ a($) 6[~-R(s)]ds =f.(2.6) If the suspended object has a free energy F(R)the equation of motion neglecting its inertia and any internal friction is -= tr. aR S. F. EDWARDS (It is not difficult to add dynamics to this by replacing F by a Lagrangian or a Rayleighan if internal friction is present.) Thus aF LBu+Zd(r-R)-aR =f H = u(R) are the coupled equations of motion. The problem of completely rigid bodies will come out either by a limiting case of the free energy or better by using the rigidity in the Lagrange multiplier. These comments are of course completely well known in the literature. What will now be done is to specialise to flexible polymers and see what the formalism gives. The analysis follows the paper by Edwards6? to which the reader is referred for details.The basic model is the Rouse model in which the coupling of the polymer with the fluid is ignored and replaced by a friction coefficient. The free energy comes from the number of steps in a random walk where JV the total number of possible steps 1 is the step length and R(s) is now in Weiner measure or in Rouse’s language R’(s)ds is the distance between balls in a ball-and-spring model. Thus the entropy S = -gJR2(s)ds (2.10) and 3k~ F = -1 T R’2(~) ds. (2.11) 21 Finally in Fourier (ie.Rouse) modes (2.12) The Rouse equation for the Brownian motion of the polymer is then vR,--3k~Tq~ R =fq(t) (2.13) 1 2n where fqis the random driving force so that the probability distribution of the RF) is (2.14) or in s-space (2.15) with the Einstein relation D=-kB (2.16) V abeing the polymer label.TRANSPORT PROPERTIES OF MACROMOLECULES This can be extended to the case with hydrodynamic interaction taking the place of the friction which gives where G is the Oseen tensor. A simpler and illuminating version of this equation arises if all polymers except one say are kept fixed when it can be shown6 that that one satisfies where H vanishes when any point of R(l)meets any point of any of the other R(@. This vanishing of the diffusivity implies that the polymer 1 is unable to cross the others so the topology is preserved except in as much as the ends can move past obstacles. This behaviour is implicit if not so obvious in eqn (2.17) which means that the diffusion equation (2.17) does contain all the topological constraints correctly.If one argues they are not important one can average the Oseen tensor retaining only the a=B terms to recover the Zimm approximation (G) -4-i (2.19) (2.20) However this approximation will not give reptation and it is clear that any obvious approximation of eqn (2.17) will lose the restriction to reptation expected at high density or with very long chains. The 'hard' nature of the 'topological' obstacles is represented by the zero of H and however small it is a non-zero value implies a failure of an approximation to represent the constraint properly. The same kind of analysis will hold for rods say when R(s)is replaced by the centre of mass R and an angle n and the Oseen tensor contains these variables and an integration along the length of the rod.For deformable bodies one has a similar formulation also with R(sl,s,) but now the Oseen tensor has to respect the condition that the fluid lies outside the surface which brings extra complications. Returning to flexible polymers if one now looks at the effective hydrodynamics of the system of fluid and polymers one can write this down in the preaveraged form at non-dilute concentrations. Clearly since the preaveraged form destroys the topological invariants this can be an inconsistent approach but of course the topological invariants are there in dilute solution also because they affect the internal motion of a single molecule.The hope is that below the point at which the much higher dependence of viscosity on molecular weight sets in it may be valid to preaverage. (2.21) iwR +J(q)R = u[R(s)]/exp (iqs) ds (2.22) where' (2.23) (2.24) S. F. EDWARDS 151 Here C(k,q) is the two-point correlation function which for a polymer is k21/6 (2.25) and these equations represent the self-consistent-field approach to polymer solutions. However there has been no mention here of forces between the polymers. When these are added it is clearly impossible to consider the problem as reducing to that of a single effective polymer in an average fluid. What one wants is to describe the polymers by some collective coordinates and this we do in the next section.3. COLLECTIVE COORDINATES The simplest problem to start the discussion is that of the free energy of a solution of polymers fin exp (-F/kB T)= n n dR(")(sa) exp -X ([R]) J-J a s where the simplest model of a random-walk polymer of step length 1 with an interaction W has A collective coordinate replaces the set R(a)(sa) by a function defined over space p(rl,Y, r3. . .) = E f(Yl Y, Y3 . . . R(a)R@). ..) (3.3) @Y the simplest being the density If we introduce the definition of p(rl,r,...) into the integral for F rr This is of course purely formal and although there are many degrees of freedom (in fact an infinite number of degrees of freedom) what has been done is merely to write a Jacobian just the same as Z = Jp(x)dx (3.7) = JP(4J 4Y -gWl dY (3.8) = J P(Y)dY (3.9) since Hence J S[Y-g(x)] dx = F(y) the Jacobian.(3.10) [In textbooks there is usually a 1 1 relationship between variables in a Jacobian but TRANSPORT PROPERTIES OF MACROMOLECULES this is not at all necessary provided one can stomach 6functions i.e. a density p(r) Y continuous which can be related to a set of N points Rca)etc.] If these ideas are applied to the density as the collective coordinate one has rr $j J ds,dsgW(Rca)-R@))=z "W^kPkPz (3.11) k where wk is the Fourier transform of W(r).The random-walk weight exp (-Rt2 ds) where A(k) is the Debye function <I exp [ik R(s,) -R(s2)]ds ds2)iree chains and .. . . . . refers to higher terms cubics quarks etc. which turn out to be important unless the mean density is very high or very low; it is supposed that the mean density conforms to one of these limits.Then \ (3.12) which is a very simple result. One can now expect that a Brownian motion equation for pk might have the structure (3.13) with exp (-2) as the equilibrium solution. This does indeed emerge in the Rouse model (3.14) (3.15) a = ik exp [ik R(a)(s)]-(3.16) k aPk giving (3.17) for the diffusion term. The random phase approximation writes pod(&+j1 for &+j and the 'dynamical friction' term is inferred from the fact that exp (-X)is equilibrium. Thus the Brownian fluctuations of the Rouse model satisfy [-,k..-(-+-)]Pa a a at k aPk ap-k a2 ap-k = 0. (3.18) 6 = In the limit of A(k)= -and wk being independent of k wk w eqn (3.18) becomes k21 (3.19) S.F. EDWARDS 153 where the screening length 5-2 =-6WPo (3.20) 1 valid for large p,,. For semi-dilute solutions a more complex formula is needed for 5 but it is available.* (Notice an inconsistency here if the density is high one cannot expect the Rozse model to be correct; there will be reptation. The static theory embodied by X is correct; this Brownian equation is only valid if chains can pass through each other -'phantoms'.) Now try this method with the hydrodynamic interaction. The Oseen tensor has the form G = (Sij -kikj/k2)>(k) (3.21) where g(k) is a scalar function (qok2)-l. Improved versions screen this into qo-1(k2+c~2)-1 in an appropriate limit it being the solution q-l(k) of the effective equations (2.23) which is like qk2 for small k and like q0(k2+tfk2) for large k in the higher-density regions.However for the present it is the tensor structure that matters. This now gives (3.22) or (3.23) The random-phase approximation now gives k =j (3.24) If the integral is approximating by ignoring all kinds of screening one has (3.25) So the Brownian fluctuations using Zimm level dynamics give (3.26) or more consistently dropping 4 also in the friction term (3.27) So far Pk seem good coordinates but of course the 'R'answer has simply been transferred into 'p'. If one tries to base the whole development on the pk one finds that one cannot couple the pk scalars to the uk vectors.Put another way one can have much movement in a system with constant density. The key omission physically lies in the fact that the polymers are stretched under motion. Crudely speaking Pk is satisfactory for a set of statistically spherical rigid objects but not for extensible objects. This has been extensively studied in the literature using dumbell model^.^-^ However these models do not satisfy our ambition of gaining closed equations of motion for the collective coordinates and to use them then to gain closed coupled equations for ff uid and polymers. TRANSPORT PROPERTIES OF MACROMOLECULES The system which seems satisfactory is to introduce P&,j= ds,exp ([ik*R(a)(s,)]+~*[R(a)(sa)-R(a)(0)]).(3.28) aJ The boundary condition is p&j = (i(k +~9 u[R(@(s,)]-g u[Rca)(0)]} eXp (ik Rca'(sa)+9-[R(''(s,)-Rca'(0)]) (3.29) = i(k +J?'UIPk+l j-g* UIPk+I,j+l (3.30) = rkjlmnUIPm,n say (3.31) or more briefly p = rup (3.32) which is used in the Lagrange multiplier constraint akl.@kj-ruP)-Suppose there is a 2 for these variables call it X(p).As with 2in terms of the p it is an unpleasant series.It can be written in closed form in Fourier transform i.e. Jexp [i @&j p&j- &@)I n dpkj = an explicit function of d&,j kj [as can $&)I but the simplest version will be rr (3.33) where C is [(pp) -(p)(p)]-l for free polymers. Proceeding now as before a few lines of algebra yield an equation of the type (3.24) where u is the mean u when the local fluctuations are smoothed out.Henceforward replace u by u to get the coupled equations (3.35) (3.36) where A = l-GT (3.37) or in full (3.38) With JP in the Gaussian approximation one gets (3.39) A=AC (3.40) S. F. EDWARDS 155 likewise f in au vp -+-fP(P -<P>) =fext . at Pf luid (3.41) If now the pre-averaging process is applied it replaces &v@-<P>) by MPP> (P -(PN (3.42) which can now be inverted to give (3.43) and so finally q(k)= <rp((iw +Y)-'} rp)+qok2. (3.44) The author has succeeded in evaluating Y,doing the final integrals and recovering the usual forms of the viscosity v(k). The structures are the same as they must be but the coefficients differ since some of the integrals appearing although pure numbers are not familiar.The result of all this effort so far is only to reach the point which has been reached already by just coupling u and the Ha). However the unaveraged equations contain all the topological invariance possessed by the original equations which is not for example possessed by dumbell theories and as the equations have not been linearized the process of reptation must still be in there. Let us now bring together what one can expect and what is already available. 4. PERSPECTIVE Consider the hierarchy of suspension/solution problems. (a) Rods. / / / / \ The key to the effects of entanglement here is to imagine the centres of the rods fixed and consider the sphere swept out by the ends of any particular rod.Project the intersection of other rods within the sphere from the centre onto the surface. Draw TRANSPORT PROPERTIES OF MACROMOLECULES the result in Mercator’s projection. Then the chosen rod appears as two points its ends and any other rod which offers a constraint appears as a curve on the projection. If all rods but one chosen rod are frozen our chosen rod will have the freedom of the pair of points. This sequence presents itself as the ratio rod length to mean spacing increases (a. 1) No constraint surface of the sphere is accessible. (a. 2) Some closed regions; but if the other rods regain their freedom one can expect the whole surface to become available. (a. 3) Very little space. Even when all rods can rotate the system is locked.Although this is just illustrative it suggests that for rods fixed at their centres a glass transition takes place for a critical ratio of rod length to mean spacing. This suggests that there should be a marked change in the viscosity of suspensions of rods from the dilute value to the very high values predictedg in the ‘log jam’ situation. (b) What is the analogue of this for flexible polymers? The fact that rods can always move along their length has an analogue in the permanent availability of reptation for unbranched chains. However the glass transition expected if a point of each rod is fixed does not have an obvious analogue. If the number of other chains here S. F. EDWARDS 157 represented by rings in a two-dimensional model increases clearly the time taken for one chosen polymer to get from some configuration slows up 0 0 0 V 0 0 It may be that cooperative effects result in a sudden decrease in this mobility and mark the sudden increase in the viscosity from an M to an M3+regime but it is not easy to be sure that the change is sudden or that there is a simple explanation of the experimental values found for ‘Mc’.Another way to describe this is to say that the competition between motion along the reptation tube and across it clearly starts at low concentration with the dominance of the ‘across’ mode and as the concentration increases only the reptative mode survives.c tube .-/ Cooperation could make the elimination of the ‘across’ mode sharp.I have explored the mathematics of this change which is all there in the equations of sections 2 and 3 but the results are not so far convincing. (c) Fatten the polymers into volume possessing shapes with surfaces R(s,,s2). There are two obvious limits and in these the left-hand one amounts to independent trajectories whilst the right-hand one involves a kind of extension of reptation where motion involves the squeezing of one shape between the others. Whereas polymer motion is dominated by entropy the situation here is dominated by internal energy; TRANSPORT PROPERTIES OF MACROMOLECULES it is a problem inelasticity and hydrodynamics with no real effect from thermodynamics. Even so there clearly are different regimes of motion and the effective viscoelasticity will show a cross-over as the shapes move into juxtaposition.This problem is a real one in colloidal suspensions and in the kind of flow encountered with micelles and the sorts of molecules involved in food processing and it presents a splendid challenge to the theorist. J. Des Cloizeaux and R. C. Ball Commun. Math. Phys. 198 1 80 543. B. Duplantier Commun. Math. Phys. 1982 85 221. R. B. Bird 0.Hassager R. C. Armstrong and D. F. Curtiss Dynamics of Polymeric Fluids (Wiley New York 1977). M. Grumela and P. J. Carreau J. Chem. Phys. 1983 78 5164. This paper contains a general attack on the problem. J. M. Aubert and M. Tirrell J. Chem. Phys. 1982 77 553. This paper contains further references. S. F. Edwards Proc. R.Soc. London Ser. A 1982 385 267. S. F. Edwards and M. Muthukumar Macromolecules submitted for publication. M. Muthukumar and S. F. Edwards J. Chem. Phys. 1982,76 2720. M. Doi and S.F. Edwards J. Chem. SOC., Faraday Trans. 2 1978 74 918.

ISSN:0301-5696    

DOI:10.1039/FS9831800145

出版商:RSC    

年代:1983

数据来源: RSC

摘要:

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. Chem. Phys. 1971 55 572. M. Doi and S. F. Edwards J. Chem. Soc. Faraday Trans. 2 1978,74 1789; 1802; 1818. G. Marrucci and G. de Cindio Rheol. Acta 1980 19 68. J. D. Ferry Viscoelastic Properties of Polymers (John Wiley New York 3rd edn 1980). J. Klein and B. J. Briscoe Proc. R. Soc. London Ser. A 1979 365 53. L. Leger H. Hervet and F. Rondelez Macromolecules 1981 14 1732. * R. Kimmich and R. Bacchus Colloid Polym. Sci. 1982 260 91 1. K. E. Evans and S. F. Edwards J. Chem. Soc. Faraday Trans. 2 1981,77 1891. lo J. Klein Macromolecules 1978 11 852.l1 M. Daoud and P. G. de Gennes J. Polym. Sci. Polym. Phys. Ed. 1979 17 1971. l2 J. Klein Philos. Mag. Sect. A 1981 43 771. l3 P. G. De Gennes J. Phys. (Paris) 1975 36 1199. l4 M. Doi and N. Kuzuu J. Polym. Sci. Polym. Lett. 1980 18 775. W. W. Graessley Adv. Polym. Sci. 1982 47 67. l6 K. E. Evans J. Chem. SOC. Faraday Trans. 2 1981,77 2385. l7 R. Needs and S. F. Edwards Macromolecules in press. (a) T. Masuda Y. Ohto and S. Onogi Macromolecules 1971 4 763; (b) W. W. Graessley and J. Roovers Macromolecules 1979 12 959. J. KLEIN D. FLETCHER AND L. J. FETTERS 171 l9 E. Helfand and D. Pearson to be published. 2o J. Klein to be published. *l J. Klein Am. Chem. SOC. Div. Polym. Chem. Prepr. 1981 22 105. 22 H. Rachapudy G.G. Smith V. R. Raju and W. W. Graessley J. Polym. Sci. Polym. Phys. Ed. 1979 17 1211. 23 W. W. Graessley Macromolecules 1982 15 1 164. 24 B. J. Bauer and L. J. Fetters Rubber Rev. 1978 51 406.

ISSN:0301-5696    

DOI:10.1039/FS9831800159

出版商:RSC    

年代:1983

数据来源: RSC

摘要:

Faraday Symp. Chem. SOC.,1983 18 173-188 Studies of the Concentration Dependence of the Conformational Dynamics of Solutions containing Linear Star or Comb Homopolymers BY C. J. T. MARTEL M. G. DIBBS R. L. SAMMLER, T. P. LODGE,? $ T. M. STOKICH C. J. CARRIERE AND J. L. SCHRAG* Department of Chemistry and Rheology Research Center University of Wisconsin Madison Wisconsin 53706 U.S.A. Received 26th August 1983 The concentration dependence of the conformational dynamics of polymer solutions as revealed by measurements of oscillatory flow birefringence (0.f.b.) has been obtained for narrow-distribution linear comb and regular-star molecules for concentrations in the range c[q] 5 11. The data obtained show that the relaxation-time spectrum is affected markedly by concentration.The longest relaxation time is affected most; for the linear comb or 3-armed star polystyrenes studied to date z exhibits an exponential dependence on concentration for c[q] < 3. The Muthukumar theory correctly predicts the dependence of z on concentration for solutions containing linear molecules for which c[q] < 5. There is some indication that z1as a function of concentration exhibits two different concentration regimes one above and one below the onset of significant entanglement effects. The shortest relaxation times are almost unaffected by concentration. The measured frequency dependences of 0.f.b. properties for finite concentrations have also been compared with the predictions of the Muthukumar and Freed and the Muthukumar theories for the concentration dependence of the relaxation times for the bead-spring model.Extensive studies of the oscillatory flow birefringence (0.f. b.)l-I0 and linear visco- elastic (v.e.)11-22 properties of polymer solutions have provided considerable insight into the dynamics of conformational change in both the dilute and the semi-dilute regimes. Both experiments frequently have utilized high viscosity solvents (such as the Aroclors) and time-temperature superposition to extend the experimentally accessible effective frequency range to as many as seven decades thus providing information about polymer dynamics ranging from the slowest overall shape-change modes down to quite local rearrangements for which motions of a small number of monomer units form the basic motional repeat unit.Both techniques are sufficiently sensitive and precise to permit extrapolation of low-concentration data to obtain infinite-dilution properties;8-10v11 14-199 21 however the 0.f.b. experiment is generally sufficiently more sensitive and precise that most of the dynamics information discussed here has been obtained from birefringence measurements only. Both experiments provide the same dynamics information except in the high-frequency regime. Fig. 1 illustrates the excellent agreement normally observed at low and intermediate frequencies ; the solution contained 0.0304 g cmP3of 390 000molecular-weight linear atactic polystyrene in Aroclor 1248. Here the frequency dependence (w is the angular frequency) of phasing of the birefringence with respect to the sinusoidally time-varying shear rate t Present address Department of Chemistry University of Minnesota Minneapolis Minnesota 55455 U.S.A.$ Present address Dow Chemical Company Midland Michigan 48640 U.S.A. 173 100 90 80 70 60 50 X 40 30 20 10 0 0 1 2 3 4 5 6 log (w aT) Fig. 1. Comparison of the frequency dependence of -(0 -0,J (birefringence data solid line) and the equivalent angle x and lq* -qLIR from viscoelastic measurements for 0.0304 g cm-3 solution of 390000 molecular-weight PS in Aroclor 1248 reduced to 25.0OOC. (Os0 is the low-frequency limiting value of 0,.) 0,T = 24.97 “C aT = 1.00; 0, T = 10.00 “C aT = 14.61; a,T = 0.02 “C aT = 233.51.~100= 3.44 poise. is represented by the solid line; the individual data points have been replaced by the smooth curve for clarity. The viscoelastic data are shown as individual data points; the quantities plotted are the magnitude of the complex viscosity coefficient q*(y~*= q’ -ir”) minus the high-frequency frequency-independent value of v’ usually denoted as &,and the angle x defined by x = tan-l [q”/(q’-&)]. Time-temperature superposition has been employed to superpose data obtained at three different temperatures; the subscript R on the quantity Iq* -q’,lR indicates reduction by multiplication by the factor T,p,/Tpa where the T are absolute temperature the p are the solution densities the subscript 0 refers to the reference temperature 25.00 OC and aT is the shift factor.22 (It is necessary to use q’ rather than the bulk solvent viscosity vssuggested by most theoretical treatments since the polymer-chain dynamics contribution to the v.e.properties is being sought; the use of r’ apparently removes additional contributions from various sources including a substantial solvent modi- fication in the immediate neighbourhood of the chain due to polymer-solvent interactions.23) There have been several studies of the initial concentration dependence of 0.f.b. and v.e. properties of linear monodisperse polymers.9* 22 Except for very low molecular weights bead-spring-model prediction^^^-^^ (isolated-molecule theories) have been shown to be in very good agreement with infinite-dilution properties in both good and 8 solvents at low and intermediate frequencies when exact eigenvalues are employed (8 solvent agreement is poorer).For dilute solutions the relaxation times of the bead-spring model are expected to exhibit some concentration dependence while the corresponding relaxation strengths are usually assumed to be independent of c.J. T. MARTEL et al. concentration.22 Experimental work has shown that the intermolecular interactions occurring in dilute solutions affect relaxation times significantly and the longest relaxation time more than the others.g- l1 Recently explicit equations for the initial concentration dependence of the relaxation times of the bead-spring model have been obtained by Muthukumar and Freed2i and by Muthukumar.28 This paper reports observed concentration dependences for extensive 0.f.b.measurements for solutions of polystyrene (PS) and poly(a-methylstyrene) (PMS) for concentrations in the range c[q]5 11 and compares the observed dependences with theoretical predictions. All comparisons assume that relaxation strengths are essentially independent of concen-tration for the solutions studied. EXPERIMENTAL MATERIALS The studies reported here have been carried out with narrow-distribution (Mw/Mn< l.lO) atactic polymer samples linear polystyrenes 6a (M,= 860000) 3b (Mw= 390000) 4b (Ew 111000) 60817 (E,= 53700) and 8b (M,= 10000) (manufacturer's data; Pressure = Chemical Co.); a linear poly(a-methylstyrene) PMS no. 5 (Mw=400000) generously provided and characterized by L.J. Fetters Exxon Corp.; regular (equal-arm molecular weight) star polystyrenes E-3 (three arms total M = 192600) F-3 (3 arms total a,= 224300) 5-12 (12 arms total M = 805000) and 10-12 (12 arms total Mw= 2090000) also provided and characterized by Dr Fetters ;and a regular 25-branch (equal-length branches) comb polystyrene C632 (total M =913000; Mwof backbone = 275000; M of branch = 25700) generously provided and ~haracterized~l by J. E. L. Roovers of the National Research Council of Canada. All solutions were prepared in the chlorinated biphenyl solvent Aroclor 1248 lot KM 502 (Monsanto Chemical Co.). This solvent was selected for its large dependence of viscosity on temperature22 and the close match of its index of refraction with that of PS and PMS to eliminate form birefringence effect^.^ i* All solutions were prepared by weight; concentrations were converted to g cmP3 assuming additivity of volumes (assumed densities are 1.445 1.060 and 1.080 at 25 "C for Aroclor 1248 PS and PMS respectively).In general the initial (highest concentration) solution was prepared by direct addition of polymer to solvent. Solvation was assisted by moderate heating (<60 "C) and occasional gentle stirring; total solvation time ranged from 6 to 9 weeks depending on the sample molecular weight. The initial poly(a- methylstyrene) solution was prepared by first dissolving the polymer in analytical-reagent-grade benzene which was subsequently stripped out in uacuo after the addition of Aroclor 1248. All other solutions were prepared by direct dilution and were subjected to low heating (< 40 "C) and occasional gentle stirring for at least 2 weeks prior to use.METHOD The second-generation thin-fluid-layer 0.f.b. apparatus and the measurement technique have been described el~ewhere.~. The transducers have been interfaced to second- and third- 32 generation computerized data-acquisition and processing systems which increase the effective frequency range of the sets of apparatus and improve their sensitivity; these systems are also described el~ewhere.~~-~~ All data reported here were obtained at an optical wavelength (in air) of 5770 A. Solution temperatures were controlled to within kO.01 "C and were determined by means of thermistors calibrated against an N.B.S.calibrated standard platinum resistance thermometer. The Aroclors are themselves weakly birefringent when subjected to a shearing deformation ;thus the total birefringence observed has contributions from both the polymer and the solvent. The polymer contribution is obtained by correcting the measured values for the solvent contribution based on the assumption of simple additivity of the polarizability tensors for the various constituents (procedure of Sadron).i,8 Thus the tensor sum of the polarizability contributions of the volume fractions of pure solvent and polymer is assumed to correspond to the solution properties. For the low-shear-rate conditions employed in the thin-fluid-layer 0.f.b. instrument the principal polarizability directions remain at &45 "with CONFORMATIONAL DYNAMICS OF POLYMER SOLUTIONS respect to the streamline direction throughout the sinusoidally time-varying cycle of deformation which leads to a particularly simple vector-subtraction correction procedure which is discussed el~ewhere.~.As mentioned previously in connection with the viscoelastic properties apparently there is a substantial solvent modification in the immediate neighbourhood of a PS or PMS chain which also causes the above birefringence correction procedure to be detectably in error for molecular weights below ca. lo5. RESULTS AND DISCUSSION The 0.f.b. data are reported in terms of the frequency dependence of the magnitude SMand the phase angle 8 of the complex mechano-optic coefficient S* defined as (phasor notation) -An* S* = S exp (i8,) = S’+ is” = -(1) 3* where An the real part of An* is the difference between indices of refraction n and n corresponding to the principal polarizability directions and j the real part of ?* is the sinusoidally time-varying shear rate as defined The quantities SM and 8 are utilized rather than S’ and S” since 0 is by far the most sensitive function of the four and the 0.f.b.experiment is sufficiently precise to be able to utilize this sensitivity. Measurements obtained at various temperatures for each solution were reduced to the 25.00 OC reference temperature resulting in reduced-variable plots of bg(&/aT) and 8 against logfa, where f is the frequency in Hz and a is the time-temperature superposition factor., Scatter in measured SMand 0 before superposition are estimated to be 0.3% and f0.3’ respectively throughout the working frequency range except for the lowest molecular weight the lowest concentrations and the highest frequencies where the solvent contribution becomes large.Infinite-dilution properties are reported as [SM] = limSM/c and [O,]= limo,. c-0 c+o The general character of the concentration dependence of the 0.f.b. properties is illustrated in fig. 2. The finite-concentration results for five solutions of PS 3b are displayed; the individual data points have been replaced by smooth curves for clarity and these have been overlayed directly. The location on the reduced-frequency .axis of the initial departure of 8 from the -180’ low-frequency limit is governed by the longest relaxation time z,.Thus for PS 3b z increases by a factor of ca. 40 as the Concentration is increased by a factor of ca. 10. At high reduced frequencies (log fa 25) however 8 depends only weakly on concentration. In terms of the concentration dependence of the bead-spring-model relaxation times {zp)these results suggest that the effect of concentration is strongly mode-dependent ;with increasing mode number p (pincreasing corresponds to zp decreasing) the effect of concentration is decreased. Thus z has a strong concentration dependence whereas zN (the shortest relaxation time for the model) is almost independent of concentration. For frequencies above log fa x 5,8 exceeds the -270’ limit given by simple chain-dynamics theories for all PS and PMS samples studied to date.This anomalous behaviour is the principal subject of another article.3s Comparisons of the frequency dependence of infinite-dilution v.e. and 0.f. b. properties for linear molecules with the theoretical (isolated-molecule) bead-spring- model predictions indicate that in this concentration regime the observed properties tend to correspond more closely to the theoretical non-free-draining (dominant hydrodynamic interaction) predictions although the behaviour is generally inter- c. J. T. MARTEL et al. I'l'l'l"'!'~'! l 1 -6 -300 -i -7 -280 3 \ --8 -260 h -b -? 5 -9 -240 M 0 -2 ---220 -10 --200 -11 --180 111111111,!,111 -1 0 1 2 3 4 5 6 log UQ,) Fig.2. Plots of log(S,/a,) and 8 against logfu at various concentrations for 390000 molecular weight PS in Aroclor 1248 reduced to 25.00 OC. A = 5770 A. Values of c/g as follows (1) 0.1115 (2) 0.0713 (3) 0.0356 (4) 0.0215 (5) 0.0109. mediate between the free-draining and non-free-draining limits ;the appropriate value of the theoretical hydrodynamic interaction parameter h* required to fit experimental results is a function of the goodness of the solvent.8~9~22 In general excellent quantitative agreement has been found between experiment and bead-spring-model predictions8* 22 in the low- and intermediate-frequency regimes. The values of the two g9 theoretical parameters N and h* (N is the number of modes or gaussian subchains representing the polymer molecule and is proportional to the molecular weight) used to fit the 0.f.b.infinite-dilution properties of linear PS for all molecular weights studied correspond to a subchain molecular weight of ca. 5200 and h* ranging from 0.125 at high molecular weight to 0.175 at low molecular weight while for the linear-PMS data the subchain molecular weight is ca. 8000 and h* is 0.15. Analysis of the results for the PS comb and stars with small numbers of arms produces Nand h* values that agree with the linear-PS values. Fig. 3 illustrates the excellent agreement usually obtained; for the 390000 molecular-weight PS solution N = 75 and h* = 0.15. However for stars with large numbers of arms the agreement between model predictions and infinite-dilution properties is substantially poorer in the frequency regime where branching effects are most visible; fig.4 illustrates the trends generally observed. In general the 8 data show a 'crossover' behaviour. At low frequency ($a < 2.1) the observed behaviour is like that of linear polymers while for higher frequencies (fa > 3) the 8 data agree with the predicted values; the frequency regime 2.1 <faT < 3 shows a depression of the predicted 8 branching peak. The same trends are seen in v.e. data. Model calculations too extensive to be described here suggest that the crossover behaviour results from a suppression of the contribution from the centre of such stars; the central regions of any molecular geometry contribute by far 178 CONFORMATIONAL DYNAMICS OF POLYMER SOLUTIONS -6 -7 I-8 O.2 - u) L M -0 -9 -10 -11 0 1 2 3 4 5 6 log Cfar1 Fig. 3. Plots of log ([&&]/a,)and [e,] against logfa for 390000 molecular-weight Ps in Aroclor 1248 reduced to 25.00 OC. Curves Zimm theory for N = 75 h* = 0.15. Lo = 5770 A. Values of T/OC as follows @ 25.00; 0 15.88; 0, 2.81; 0 -1.42. -5 - l ' l ' l l ' T l I 1 d-300 i --280 -6- - I -260 2 - -7 - -240 0, -.-c.-P m -2 &a-00 -220 ---9 ---200 --10 ---180 -1l.III1I1~I1~'' 2 3 4 5 6 C. J. T. MARTEL et al. the major part of low-frequency v.e. or 0.f.b. properties. The reasons for such a suppression are not clear yet but excluded-volume effects caused by the high segment density in the centre of such stars would produce changes in the right direction.No quantitative evaluations of the effect on v.e. and 0.f.b. properties are available to date but Miyake and Freed have recently evaluated the effect of excluded volume on intersegment distances for linear and star molecules;37 their predictions appear to be consistent with this picture. Because of this discrepancy between predictions and infinite-dilution properties for stars no comparisons between finite-concentration predictions and measured frequency-dependent 0.f. b. properties will be presented here. The theoretical bead-spring-model predictions employed here (linear star or comb geometries) are calculated from1? 35 N (3) [Sol P-1 and (4) where 6kTllp NkT C T;/T~ P-1 is the infinite-dilution value of the relaxation time for the pth mode N is the number of modes (or gaussian subchains) for the model q' is an optical factor N is Avogadro's number k is Boltzmann's constant M is the polymer molecular weight o is the radian driving frequency [q]is the steady-flow intrinsic viscosity and A are the exact eigenvalues of the Zimm H-A or Lodge-Wu B matrix.38 For regular (equal-molecular-weight arms) stars eqn (3)-(5) may be written in a particularly useful form owing to the degeneracies introduced in the H*A or B matrices by geometrical symmetry.P-1 P-2 odd modes even modes Odd even t wherefis the number of arms and Nb is the number of gaussian subchains in one arm of the star.Thus for regular stars with several arms the contributions of the odd-numbered modes are enhanced markedly and steady-flow properties such as [So] or [q] will be dominated almost totally by the slowest mode. Also from the degenerate H-A matrix it is clear that T? will be controlled by 2Nb or twice the arm molecular weight hereafter designated as the 'span moleciilar weight ' rather than total molecular weight. (For a linear molecule the two are equal.) CONFORMATIONAL DYNAMICS OF POLYMER SOLUTIONS Fig. 5 shows the general character of the concentration dependence (at 25.00 "C) of z for the 400000 molecular-weight PMS and the 390000 molecular-weight PS for which the most extensive concentration range has been studied.The z1 values are obtained by superposing theoretical curves and experimental data. Assuming that all mode strengths are essentially concentration independent there is a unique relaxation- time distribution that will produce a given curve shape. The error bars shown for the higher concentrations are not a result of experimental uncertainty but are a reflection 0 0.05 0.10 c/g crn+ Fig. 5. Plot oft against c for 390000 molecular-weight PS (0) and 400000 molecular-weight PMS (0)in Aroclor 1248 reduced to 25.00 "C.(--) z = zy[1 +cA -~'2(cA)~+2(cA)~], M; (-) z1= zyexp(cA); (--) z = zy(l+cA) M.F.zY = 2.02ms A = 41. of inadequate theoretical fits. The highest concentration corresponds to c[q]x 11 for which zl/zy x 36. For c[q]c 3 the dependence of z on c is nearly exponential; how- ever for higher c an interesting change is observed that is readily seen in a semi- logarithmic plot for the same 390000 molecular-weight PS sample in fig.6 (which also shows z1values for four other molecular weights of linear PS). For this sample the low-c region (c[q]< 3) clearly shows an exponential concentration dependence while at intermediate concentrations (4 c c[q]c 9) there is a sigmoidal curvature apparently followed by a second regime showing a nearly exponential concentration dependence (c[q]from 9 to 11). This change in character in the 4 < c[q]< 9 region suggests the existence of two different concentration regimes centred about c[q]= 6 or 7. Interestingly from v.e. data one would expect entanglement effects to become dominant for this molecular weight at a concentration of 0.085 g ~m-~, suggesting that this transition may be caused by entanglement^.^^ None of the curves for the other molecular weights shown in fig.5 extends to sufficiently high concentrations to show c. J. T. MARTEL et al. this transition clearly although the data for the 860000 molecular-weight PS may show its onset; for this sample v.e. data suggest that entanglement effects should become dominant at a concentration of 0.048 g ~m-~. The z1for all other molecular weights (1 11 000 53 700 and 10000) show an exponential concentration dependence for the entire range of concentrations studied to date. The initial slope of these curves is a strong function of molecular weight particularly at higher molecular weights as I I1 10-61 I I1 I1 1 I I I I I I 1 1 1 1 11 11 I 0 0.0 5 0.10 0.1 5 0.20 clg cm-' Fig.6. Concentration dependence of z for several molecular weights of linear polystyrene in Aroclor 1248 reduced to 25.00 "C. (a) M = 860000 initial slope = 32.5 A = 75; (b) M = 390000 initial slope = 17.8 A = 41 ;(c) M = 11 1000 initial slope = 9.7 A = 22; (d) M = 53700 initial slope = 7.4 A = 17; (e)M = 10000 initial slope = 5.6 A = 13. noted in the fig. 6. Fig. 7 and 8 present values of log(z,/zy) plotted against c for linear and star molecules; each figure shows results for molecules having nearly the same span molecular weight. Fig. 7 illustrates that linear and 3-armed stars show the same exponential character with the slope of the curves increasing with span molecular weight as expected.For PS 5-12 a 12-armed star the dependence is not strictly exponential and the initial slope is larger than expected. Fig. 8 presents the results for molecules with a larger span molecular weight; here the span molecular weight CONFORMATIONAL DYNAMICS OF POLYMER SOLUTIONS 10 0-.-c- 1 0 0.01 0.02 0.02 0.03 0.05 0.06 0.07 c/gcme3 Fig. 7. Concentration dependence of (T,/z:) for several polystyrene stars and a linear polystyrene all of similar span molecular weights in Aroclor 1248 reduced to 25.00 OC.(-) Linear span M = 111000 T = 0.27 ms; (8,PS 5-12 (12-arm star) span M = 140000 T = 0.35 ms; 0 PS F-3 (3-arm star) span M = 135200 T = 0.31 ms; 0,PS E-3 (3-arm star) span M = 116400 z = 0.29 ms.clg ~rn-~ Fig. 8. Concentration dependence of (T/z:) for linear comb and 12-armed star polystyrenes of similar span molecular weights in Aroclor 1248 reduced to 25.00 OC. 0,Linear span M = 390000 T; = 2.0 ms; 0,PS 10-12 (12-arm star) M = 336000 T = 0.83 Ins; 0,PS C632 (comb) span M = 326400 T = 3.1 ms. estimate for PS C632 the 25-armed comb is probably too high since it is assumed to be the sum of the backbone molecular weight and twice the arm molecular weight. The results for the linear and comb samples seem reasonable but the 12-armed star shows a marked deviation. If the deviation is due to normal interchain interactions this may suggest that theoretical evaluations of the concentration dependence of relaxation times for star molecules will be more difficult than for linear polymers; the c.J.T. MARTEL et al. observed dependences are clearly different. However the infinite-dilution 0.f. b. properties for the 12-armed stars also show anomalous behaviour as noted earlier and whatever is responsible for this may also cause the different concentration behaviour shown in fig. 7 and 8. Fig. 9 presents semilogarithmic plots of measured values of the low-frequency (‘steady-flow’) limit of S, denoted as SMo divided by c plotted against c for the linear and star samples with span molecular weights of 1 1 1 000-140000. Again as for the z plots an exponential concentration dependence is observed for (S,,/c) for the linear and 3-armed stars.The initial slope of the curves is consistent with the span 10-4 I M m 5 v1 . n 0 -. 0 m 2 10-7 0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 cJg~rn-~ Fig. 9. Concentration dependence of (S,,/c) for a linear polystyrene and several polystyrene stars all of similar span molecular weights in Aroclor 1248 reduced to 25.00 OC. (-@-) linear span M = I1 1000; 0,PS 5-12 (12-arm star) span M = 140000; 0,PS F-3 (3-arm star) span M = 135200; 0, PS E-3 (3-arm star) span M = 116400. molecular weights. The 12-armed star results show curvature as did the z values plotted for this molecule in fig. 7. Note that [SMO] is nearly the same for all samples although the total molecular weights differ by up to a factor of 8; the span molecular weight is the major controlling factor for [SMO].Fig. 10 is a semilogarithmic plot of values of z and K(S,,/c) as a function of c where K is an arbitrary constant introduced so that the shape and slope of the concentration dependence of these quantities can be compared for the same linear and star molecules (span molecular weights of 111000-140000). Clearly the same dependence is observed for both z and S, for this concentration regime for these molecules even for the lower-molecular- weight 12-armed star. This is also observed for all other molecules studied which have lower span molecular weights as well as for the comb sample but not for the 390000 and 860000 molecular-weight linear PS or the PS 10-12 star except at the highest concentrations for the 390000 molecular-weight linear PS.If one assumes that the concentration dependence of relaxation times is mode-dependent as the observed 0.f.b. frequency dependence indicates then from eqn (3) and (4) [and (6) and (7)] one would predict a difference in concentration dependence for z and S,,/c of the type exhibited by the high-span-molecular-weight samples except at the highest concentrations. The lower-molecular-weight (S,,/C) results may be misleading owing CONFORMATIONAL DYNAMICS OF POLYMER SOLUTIONS o-2 c L ---,OR - - - - - 1 ti4 I l l 1 l l 1 I I 1 0 0.05 0.10 c/g ~rn-~ Fig. 10. Comparisonofthe concentration dependences of z (0) polystyrene and several polystyrene stars all of similar span molecular weight in Aroclor 1248 and K(S,,/c) (:) for a linear reduced to 25.00 "C.(a)linear M = 111 000; (b)PS E-3 (3-arm star) span M = 116400; (c) PS F-3 (3-arm star) span M = 135200; (d)PS 5-12 (12-arm star) span M = 140000. to concentration-dependent contributions caused by modifications of solvent in the neighbourhood of the polymer chains as was noted previously. Recently Muthukumar and Freed2' and Muthukumar28 have obtained explicit expressions for the initial concentration dependence of the bead-spring-model relaxation times by considering the effect of intermolecular hydrodynamic interaction (quasi-static limit). The Muthukumar-Freed (M.F.) result for the pth relaxation time is zp = zO,(l +Acp-"+ ...) (8) and the Muthukumar (M) expression is zP = T;[ 1 +AcP-"-~'2(Ac/l-~)$+~(AcP-~)~-...] (9) where A is a positive constant obtained from the initial slope of the lnz against c curve and K is a positive exponent with a value of 0.5 in 8 solvents and 0.65-0.80 in good solvents.The M.F. expression was expected to be restricted to concentrations for which c[q] 5 1 while the M result would be expected to apply for a larger range of concentration. In order to compare predictions based on eqn (8) and (9) with experimental results values of N,h*,A and K are required. Nand h* are determined by fits to infinite-dilution properties as noted previously. IC has been arbitrarily selected to be 0.65 for 185 c. J. T.MARTEL et al. PS/Aroclor solutions (moderately good solvent conditions) ;the resulting predictions are insensitive to small changes in IC so its value is relatively unimportant.Fig. 5 includes curves corresponding to the predicted 2 concentration dependences from the M.F. and M theories and also shows an empirical exponential dependence (solid line) zp = zpexp (AcpK). (10) The M.F. theory clearly does not describe the observed dependence for the linear 390000 molecular-weight PS and the 400000 molecular-weight PMS for c[q] L 1 while the M result works fairly well for c[q] up to nearly 5. However when one looks at the shapes of predicted and experimental 0.f.b. frequency dependences for these molecules it is clear that although the M.F. result predicts quite accurately the observed relaxation-time spacings for concentrations such that c[q] s 2,9 the M expression and the arbitrary exponential form work equally well in this regime.However as one goes to higher concentrations the predicted 0.f.b. frequency dependences no longer match the measured properties. For c[q] > 3 for the 390000 molecular weight PS the M.F. result slightly underestimates the 8 peak and the breadth of the relaxation-time spectrum while the M equation leads to an overesti- mation of the 8 peak and predicts a correct relaxation-time spectrum breadth. The empirical exponential form also overestimates both the 8 peak and the breadth of the relaxation time spectrum at high concentrations. For c[q] = 1.9 for the 53700 molecular weight PS the M.F. result gives a good match in 8 peak shape but an incorrect spectral breadth.The M result and the exponential form overestimate the Bs peak but predict the correct spectral breadth. Fig. 11 and 12 illustrate the character of the disagreement between the M.F. and M predictions and the observed properties at even higher concentrations that are well -5 m-300 -6 ' 1-280 -7 -2 60 -<b g -8 -240 M --9 -220 -10 -200 -1 1 -1 80 -1 0 1 2 3 4 5 6 lo€! cf4,) Fig. 11. Plots of log(S,/a,) and 8 against logfa for 0.1000 g cmP3 solution of 53700 molecular weight PS in Aroclor 1248 reduced to 25.00 OC. & = 5770 A. T/OC as follows 8 25.00; 0, 15.88; (D 2.81 0,-1.42. Theoretical curves (N =9 h* = 0.175 A = 17 K = 0.65) 1 solid line M.F.; 2 dashed line M exponential. CONFORMATIONAL DYNAMICS OF POLYMER SOLUTIONS -4I j -320 t -5 -300 -6 -280 3 2 -0" -7 -260 -8 -240 -10 -200 -11 -180 ~111l11l1,1,1l1l ~ -2 -1 0 1 2 3 4 5 6 log Va,) Fig.12. Plots of log(S,/a,) and Os against logfa for 0.1115 gcmP3 solution of 390000 molecular-weight PS in Aroclor 1248 reduced to 25.00 OC Lo = 5770 A. T/OC as follows 0 45.00; 0,25.00; a,15.88; @ 2.81; 0 -1.42. Theoretical curves (N= 75 h* = 0.15 A = 41 K = 0.65) (--) rpaexp (AcP-~),(1,-) M.F. (2,--) M. A Fig. 13. Molecular-weight dependence of A for linear polystyrene in Aroclor 1245 reduced to 25.00 OC. beyond the expected range of applicability. Interestingly the M.F. form appears to produce a better relaxation-time spacing for lower-molecular-weight polymers in this regime (c[q]z3) and the M result works surprisingly well for the 390000 molecular-weight PS (c[q]= 11) while the empirical exponential form does not.It is clear that the experimental establishment of ranges of applicability for such theories must await completion of extensive molecular-weight-dependence and concentration-dependence studies. However fig. 13 presents a log-log plot of A against molecular weight for the linear polystyrenes studied to date. c. J. T. MARTEL et al. 187 CONCLUSIONS 0.f.b. and v.e. data clearly demonstrate that the relaxation times for slow modes have a strong concentration dependence while the fast motions are nearly independent of concentration. For the linear comb and 3-armed regular-star molecules studied to date z exhibits an exponential dependence on concentration for c[q]c 3.For linear molecules the Muthukumar theory correctly predicts the dependence of z on concentration for c[q]< 5. For the 390000 molecular-weight linear PS solutions for which the most extensive studies are available z as a function of concentration appears to exhibit two regimes the second appearing at concentrations just above that at which entanglement effects are predicted to become dominant from v.e. studies. 12-armed star PS does not exhibit the same exponential concentration dependence. Contrary to expectations the low-frequency frequency-independent values of SM denoted as SM0,show the same concentration dependence as z for linear and 3-armed stars for which the span molecular weight is below 150000.However the higher- molecular-weight linear PS samples show different concentration dependences for z (stronger dependence) and S, in the low-concentration regime and the same dependence at higher concentrations; this is predicted by any theory in which concentration effects are mode dependent. Regular stars with several arms would be expected to show more nearly the same concentration dependence for both z1and S, due to enhancement of the contributions from odd-numbered relative to even-numbered modes; this appears to be the case for the 12-armed stars studied to date. G. B. Thurston and J. L. Schrag J. Chem. Phys. 1966 45 3373. G. B. Thurston J. Chem. Phys. 1967 47 3582. G. B. Thurston and J. L. Schrag J. Polym.Sci. Part A-2 1968 6 1331. G. B. Thurston and J. L. Schrag Trans. SOC. Rheol. 1962,6 325. J. W. Miller and J. L. Schrag Macromolecules 1975 8 361. A. L. Soli and J. L. Schrag Macromolecules 1979 12 1159. M. G. Minnick and J. L. Schrag Macromolecules 1980 13 1690. T. P. Lodge J. W. Miller and J. L. Schrag J. Polym. Sci. Polym. Phys. Ed. 1982 20 1409. T. P. Lodge and J. L. Schrag Macromolecules 1982 15 1376. lU T. P. Lodge and J. L. Schrag Macromolecules in press. l1 R. M. Johnson J. L. Schrag and J. D. Ferry Polym. J. 1970 1 742. l2 D. J. Massa J. L. Schrag and J. D. Ferry Macromolecules 1971 4 210. l3 K. Osaki and J. L. Schrag Polym. J. 1971 2 541. l4 Y. Mitsuda K. Osaki J. L. Schrag and J. D. Ferry Polym. J. 1973 4 354. l5 K. Osaki Y. Mitsuda R.M. Johnson J. L. Schrag and J. D. Ferry Macromolecules 1972 5 17. l6 K. Osaki J. L. Schrag and J. D. Ferry Macromolecules 1972 5 144. T. C. Warren J. L. Schrag and J. D. Ferry Macromolecules 1973 6 467. N. Nemoto Y. Mitsuda J. L. Schrag and J. D. Ferry Macromolecules 1974 7 253. l9 R. W. Rosser N. Nemoto J. L. Schrag and J. D. Ferry J. Polym. Sci. Polym. Phys. Ed. 1978 16 1031. 2o B. G. Brueggeman M. G. Minnick and J. L. Schrag Macromolecules 1978 11 119. 21 K. Osaki Adu. Poly. Sci. 1973 12 1. 22 J. D. Ferry Viscoelastic Properties of Polymers (Wiley-Interscience New York 1980). 23 T. M. Stokich and J. L. Schrag to be published. 24 P. E. Rouse Jr. J. Chem. Phys. 1953 21 1272. 25 B. H. Zimm J. Chem. Phys. 1956 24 269. z6 G. B. Thurston and A.Peterlin J. Chem. Phys. 1967 46,4881. 27 M. Muthukumar and K. F. Freed Macromolecules 1978. 11. 843. 28 M. Muthukumar personal communication ; Macromolecules submitted for publication. N. Hadjichristidis A. Guyot and L. J. Fetters Macromolecules 1978 11 668. 3o N. Hadjichristidis and L. J. Fetters Macromolecules 1980 13 191. 31 J. E. L. Roovers Polymer 1979 20 843. 3z J. W. Miller Ph.D. Thesis (University of Wisconsin 1979). 188 CONFORMATIONAL DYNAMICS OF POLYMER SOLUTIONS 33 A. L. Soli and J. L. Schrag Macromolecules 1979 12 1159. 34 T. M. Stockich M. G. Dibbs T. P. Lodge and J. L. Schrag to be published. 35 M. G. Dibbs Ph.D. Thesis (University of Wisconsin 1983). 36 T. P. Lodge and J. L. Schrag Macromolecules in press. 37 A. Miyake and K.F. Freed Macromolecules submitted for publication; personal communication. 38 R. L. Sammler J. L. Schrag and A. S. Lodge Rheology Research Center Report no. 82 (University of Wisconsin Madison Wi 1982). 39 K. Osaki K. Nishizawa and M. Kurata Macromolecules 1982 15 1068.

ISSN:0301-5696    

DOI:10.1039/FS9831800173

出版商:RSC    

年代:1983

数据来源: RSC

摘要:

Faraday Symp. Chem. SOC.,1983 18 189-197 Rheological Behaviour of Branched Polymer Molecules BY DALE S. PEARSON* Exxon Research and Engineering Company Corporate Research-Science Laboratories Annandale New Jersey 08801 U.S.A. AND EUGENE HELFAND Bell Laboratories Murray Hill New Jersey 07974 USA. Received 18th August 1983 A theory is presented for describing the viscoelastic properties of star-shaped branched polymers in concentrated solutions or melts. We describe the dominant mode of relaxation as diffusion of the chain in a potential field as others have done. Some indication of the form of the potential function is obtained by studying the statistics of random lattice walks entangled with a net of obstacles. The theory is compared with experimental data on well characterized stars covering a wide range of branch-point functionality.We show that it correctly describes the molecular-weight dependence of the zero shear-rate viscosity qo and the steady-state compliance Jt as well as the frequency dependence of the dynamic shear moduli G(o)and G(u>).Some comments are made about the behaviour of branched polymers with a structure more complex than stars. When a sudden strain is applied to a viscoelastic liquid a resisting stress appears which decays away with time. If the sample is a high-molecular-weight polymer the rate of stress relaxation is a measure of the time required for the molecules to disentangle from their oriented configurations and return to an isotropic state. The essential constraint which controls this disengagement is that polymer chains cannot pass through the contours of the neighbouring chains which surround them.In 1971 de Gennesl introduced the concept that the primary mode of relaxation was a simple snake-like diffusion of the polymer along a path approximating its own contour a motion he aptly termed reptation. In 1978 Doi and Edwards2 extended these ideas by developing a theory that predicted the viscoelastic response of entangled liquids. These theories as originally presented apply to Zinear polymer molecules. If the polymer contains long branches it is expected that this simple curvilinear translation will be suppressed. This paper addresses the issue of how branched polymer molecules disentangle. We first provide a description of a model for the entanglement of random lattice walks with a net of fixed obstacles.Certain statistical properties of these walks are obtained and then used to develop a theory for the time-dependent behaviour of star-shaped branched polymers. It is shown that this theory can quite successfully describe the linear viscoelastic properties of model star molecules. We conclude with a review of what is known about the behaviour of more complex branched polymers such as those with a comb-like or H-shaped structure. STATISTICS OF ENTANGLED CHAINS After the initial success of the reptation theory it became clear that further progress would require a deeper understanding of how molecules are entangled with each other. This was evident from work on the early stages of the relaxation p~ocess,~ on the 189 RHEOLOGICAL BEHAVIOUR OF BRANCHED POLYMERS Ilx X X Fig.1. An unrestricted random walk shown by solid lines on a square lattice represented by the open circles. The crosses form a net of obstacles with which the walk can entangle. The walk has several unentangled loops such as the one beginning one down from the upper left corner. Reeling in the walk from its ends produces the primitive path which is traced by the dashed lines. anomalous ‘3.4 law’ of viscosity (r,,xM3.4),4on the behaviour of branched polymer^^-^ and even on the equilibrium properties of rubber network^.^ In the model of Doi and Edwards2 a polymer is assumed to be confined to a tube-like region by the other polymers which surround it.Edwards and Evans studied some of the properties of this tube by using computer simulations of random walks entangled with fixed obstacles.8 In particular they determined the length of a walk which remains after it is reeled in from its ends without intersecting any of the obstacles. In this section we discuss some analytical results we have obtained on a model very similar to that of Edwards and Evans. Polymer chains are depicted by unrestricted random walks on a lattice with a coordination number q. The steps of the walk are meant to represent sections of the molecule and not the real polymer bonds. Furthermore the lattice is permeated by a net of fixed obstacles. This is illustrated in fig. 1 for a planar lattice (q =4) where the obstacles are placed at the centre of each lattice square.The position of the entanglement net for more complicated lattices can be ~pecified.~ A walk which begins and ends at the same point is called a loop. A loop will be termed unentangled if it retraces its own path (perhaps more than once) in returning to its origin. In the Edwards and Evans reel such unentangled loops will be completely pulled out (cf. fig. 1). The walk which remains is defined as the axis of the tube or the so-called primitive path. We have calculated the probability pc(K,N) that a random lattice walk of N steps has a K step primitive path or equivalently that N-K of the steps are part of unentangled loops. For N large and K =O(N) the function pc(K,N) is given by9 log(q-where IC =K/N.This function is strongly peaked at a value of K given by D. S. PEARSON AND E. HELFAND 191 0'5m 0.1t I .';...- ,/' \. 1 KIN Fig. 2. A comparison of the exact form of pc(K,N) eqn (1) (-) and the Gaussian approximation eqn (3) (--). The value of q is 6 and the limit N + co is assumed or equivalently the plot is of the coefficient of N in the exponent. In this region pc(K,N) can be approximated by the Gaussian function9 A comparison of the exact form of pc(K,N) and eqn (3) is shown in fig. 2. It is of interest to point out that the configuration of the primitive path is the same as that of a no-reversal random walk. A walk of this type containing [(q -2)/q]N steps has the same mean-squared end-to-end distance as an unrestricted random walk of N steps.ll The probability of a chain's end returning unentangled to the origin arises in ionnection with the diffusion of star-shaped polymers.The result we find is The form of eqn (4) was anticipated by de Gennes5 and a very similar expression has been recently obtained by Needs and Edwards.lO They have also provided an experimental test of this result by generating walks on a simple cubic lattice (q = 6) and counting unentangled returns. Essentially quantitative agreement with eqn (4) is found. From a physical point of view unentangled loops along a polymer chain form and disappear. Small-scale fluctuations in their number and size are related to the 'defect diffusion' which transport the chain1 and to the rapid changes in the length of the primitive path which effect viscoelastic proper tie^.^ Large fluctuations provide a mechanism for the relaxation of dangling ends in a rubber network7* l2and the arms of branched polymers.RHEOLOGICAL BEHAVIOUR OF BRANCHED POLYMERS VISCOELASTIC PROPERTIES OF STARS The simplest type of branched polymer is a star-shaped molecule consisting offarms connected to a central branch point. The viscoelastic properties of stars have been studied by a number of research gro~ps.~~-~~ The different time-dependent behaviour of stars when compared with linear polymers was attributed to changes in the mechanism by which they di~entang1e.l~ X 0 w II III 1 1 IIIIIIIIIIIIJ Fig. 3. An arm of a star in two separate positions of escape.The probability of retracting a distance y is proportional to exp [-U(y)/kT]. In the reptation model of de Gennes' a linear molecule is assumed to diffuse in free Brownian motion along the axis of the tube. If the chain is oriented by an applied strain it will return to an isotropic state because the ends of the chain adopt random configurations as they emerge from the tube. Long branches attached to a polymer molecule should obstruct this simple translational motion. However an arm of a star can still renew its configuration by retracting along its axis pushing out unentangled loops into the surrounding matrix. The low probability of adopting such contracted states can be thought of as creating a potential in which the end of the arm is diffusing.The lifetime of the tube is then related to the time required for the end of the arm to diffuse back to the branch point. Let an arm of the star consist of N segments each containing N freely jointed monomers. N is the number of monomers between entanglements and there are N =NN monomers per arm. Following Doi and KUZUU~ we assume that motion takes place in a potential U. U is taken to be a quadratic function of the position of the chain end x which varies from x =0at its equilibrium position to x =L, at the branch point (see fig. 3) kT U(x)=a--x2. L& The quantity a =v'(N,/N,) proportional to the length of the arm is normally a large number. Experimental data to be presented below will be used to estimate the constant D.S. PEARSON AND E. HELFAND v’. The use of a quadratic form for U can be motivated by the results presented earlier [see eqn (2) and (3) and fig. 21. The problem of determining the time required for an arm to retract a distance y in its tube can be obtained by solving the Smoluchowski diffusion equation.la However an alternate derivation suitable for our needs follows from a Kramers-type analysis for motion of a particle in a well with an infinite sink at y.17 Let x be the root-mean-square fluctuation in the position of the chain end. Its value determined from U is x = ~,,/(2a)k (6) The time z = CLtq/2akT cc NZ (7) is a measure of the time required to diffuse to x,; i.e. z = xt/D where D = kT/[and [is the friction factor for the chain. Then for y % x and t % z the Kramers method gives the probability F(y,t) that the end of the chain has not reached y (y < Lea) When eqn (6) is substituted for U we obtain The fraction of undestroyed tube i.e.the part which has not been visited by the chain end is 1 f(t) = -I”’“dy F(y,t). (10) Le xe Note that eqn (8) and (10) provide a way to generalize our calculation for the case where the potential is not quadratic. A more detailed derivation of eqn (9) is found in ref. (18). If o(t)is the stress at time t after applying a small strain y at t = 0 then the stress relaxation modulus is given by G(t) = a(t>/r. (1 1) The fraction of the initial stress which remains at t should be proportional to the amount of tube which has not been vacated i.e.where Gois the initial modulus usually referred to as the plateau modulus. Eqn (10) and (1 2) provide a means to calculate the zero shear-rate viscosity q019 Because [ L, and a are all proportional to molecular weight of the arm Ma eqn (1 3 b) predicts that (vf2) Me qocc (‘)‘exp 1 (14) * Kramers did not explicitly give this result but it is easily derived. 7 FAR I94 RHEOLOGICAL BEHAVIOUR OF BRANCHED POLYMERS i.e.the viscosity increases exponentially with molecular In a similar manner the steady-state shear compliance is calculated by19 =a/Go a% 1 or equivalently J:G0 = v'-. Ma Me 20.0 I I I I I I I I -I -r-I-1 18.0 16.0 14.0 12.0 8.0 6.0 4.0 2.0 O"b.0 210 4t0 610 8;O 10.0 lh.0 li.0 16.0 li.0 2d.0 2i.O 24.0 .o MdMe Fig.4. Product of the steady-state compliance J and the plateau modulus GoN for the series of star-shaped polymers. The symbols indicate polybutadiene 4-arm (0); polystyrene melts,14 4-arm (A) and 6-arm (V); and polyisoprene 4-arm (0) and 6-arm (0). Eqn (15c) suggests a method to determine the constant v'. In fig. 4 we have plotted the product of JE and the plateau modulus against Ma/Mefor polyisoprene stars,12 polystyrene starsL3 and polybutadiene stars.14 Both 4-arm and 6-arm stars are included. A linear relationship is obtained which by least-squares analysis gives a value v' M 0.6. The connection between the viscosity of stars and the arm molecular weight also involves the constant v'. In fig. 5 we show the logarithm of qo/(Ma/Me)iagainst MJM for polyisoprene polystyrene rneltsl4 and polyisoprene melts.18-20 These data include stars with functionalities from 4 to 12.In agreement with eqn (14) the relationship is linear at Ma/Me% 1. The full curves were calculated with eqn (13 a) using v' = 0.6. D. S. PEARSON AND E. HELFAND eqn (12) and (13). I I I I - 100 02 u -.-n K hE lo-’ 3 W u 02 u 1 n --10-1 100 10’ 102 103 104 OT,, Fig. 6. Dynamic moduli G(o)(A) and G(o)(O) normalized by the plateau modulus GoN for an 8-arm polyisoprene melt.’* The value ofz is 1050 s. The solid curves were calculated with eqn (1 2) and (1 6). RHEOLOGICAL BEHAVIOUR OF BRANCHED POLYMERS The frequency dependent dynamic moduli of linear viscoelasticity can be obtained from G(t) as G*(w) = iw dt exp (-iot) G(t) (164 JOrn = G’(w)+iG”(w).(16b) An additional test of our theory is provided by comparing it with experimental values of G’(w) and G”(w)obtained on an 8-arm star polyisoprene.20 This is done in fig. 6 where the solid curves were calculated using a v’ value of 0.6. A discussion of the functions G(t)and G*(w) including useful approximations for correlating experimental data can be found in ref. (18). REMARKS Star-shaped polymers have a structure such that all sections of the molecule are connected at only one end to a branch point. Although these sections cannot translate freely they can renew their configuration by fluctuations in the manner discussed above.Branched polymer molecules that have the shape of an H,21 a comb22 or a tree contain internal sections that are connected at both ends to a branch point. Even though the mechanism by which these sections relax is not clear it seems certain that they will have longer relaxation times than the arm of a star.’ An indication that this is the case can be found in the study of Graessley and Roovers22 on comb-shaped polystyrene. The frequency-dependent dynamic moduli contain two sets of relaxation times in the terminal zone. The first is associated with the arms of the comb and the other at still lower frequencies with the backbone. Realizing that JZ is dominated by the longest relaxation times they demonstrate that JE OC wbb/&b (17) where n/fobb and (bbb are the molecular weight and the volume fraction of the backbone.Eqn (17) indicates that in the determination of Jt the relaxation times of the arms can be ignored. The structure of a branched polymer may also effect its non-linear (strain- dependent) properties. Non-linear properties of star polymers have been measured and they appear to be similar to those of linear However experiments in the non-linear range on well-characterized branched polymers with internal sections have not been done. We thank L. J. Fetters for providing materials used in this study and for pointing out to us that viscosity data on stars of different functionality could be reduced to a common curve by plotting viscosity against arm molecular weight. ’ P-G. de Gennes J.Chem. Phys. 1971 92 572. * M. Doi and S. F. Edwards J. Chem. Soc. Faraday Trans. 2 1978,74 1789 1802 1818. M. Doi J. Polym. Sci. Polym. Phys. Ed. 1980 18 1005. M. Doi J. Polym. Sci. Polym. Lett. Ed. 1981 19 265. P-G. de Gennes J. Phys. (Paris) 1975 36 1199. M. Doi and N. Y. Kuzuu J. Polym. Sci. Polym. Lett. 1980 18 775. W. W. Graessley Adv. Poiym. Sci. 1982 47 67. K. E. Evans and S. F. Edwards J. Chem. SOC.,Faraday Trans. 2 1981,77 1891 1913 1929. E. Helfand and D. S. Pearson J. Chem. Phys. to appear. lo R. J. Needs and S. F. Edwards to be published; R. J. Needs personal communication. l1 C. Domb and M. E. Fischer Proc. Cambridge Philos. Soc. 1958 54 48. D. S. PEARSON AND E. HELFAND 197 l2 J. G. Curro and P. Pincus Macromolecules 1983 16 559.l3 W. W. Graessley T. Masuda and J. E. L. Roovers Macromolecules 1976 9 127. l4 W. W. Graessley and J. Roovers Macromolecules 1979 12 959. l5 V. R. Raju E. V. Menezes G. Marin W. W. Graessley and L. J. Fetters Macromolecules 1981 14 1668. l6 S. Chandrasekhar Rev. Mod. Phys. 1943 15 1. H. A. Kramers Physica 1940 7 284. l8 D. S. Pearson and E. Helfand Macromolecules submitted for publication. l9 J. D. Ferry Viscoelastic Properties of Polymers (John Wiley New York 3rd edn 1980) chap. 3. 2o F. Jerome Vitus M.S. Thesis (University of Akron 1979). 21 J. Roovers and P. M. Toporowski Macromolecules 1981 14 1174. 22 J. Roovers and W. W. Graessley Macromolecules 1981 14 766. 23 D. S. Pearson unpublished results.

ISSN:0301-5696    

DOI:10.1039/FS9831800189

出版商:RSC    

年代:1983

数据来源: RSC

摘要:

GENERAL DISCUSSION Prof. L. Monnerie (E.S.P.C.I.,Paris France) said In his Introductory Paper Dr Graessley mentioned that the reptation theory and the reptation theory plus the tube constraint release both lead to q cc M3 whereas the reptation theory plus the tube constraint release and chain breathing results in a 3.4 exponent. Could he tell us what exponent is obtained by considering the reptation theory plus the chain breathing only? Dr W. W. Graessley (Exxon Annandale N.J. U.S.A.)replied The problem of reptation and chain breathing (fluctuations in path length) was first considered by Prof. Doi. He showed that the result for viscosity could be then fitted by q cc M3.4 over a wide range of chain lengths. We did little more than substitute his result for relaxation time into the equation that governs the contribution from constraint release.Adding this third process does not change the apparent exponent from Doi’s result but it provides closer agreement with the observed magnitude of the viscosity. Dr G. B. McKenna (N.B.S. Washington D.C. U.S.A.) said Although as Dr Graessley indicated in his talk the reptation and tube models of polymers have been useful in studying their viscoelastic behaviour it seems to me that because of the deficiencies in these models (e.g. the GkG product qo cc etc.) consideration should be given to other models as well. One such model was recently proposed by Dr K. L. Ngai,l which he has found to be applicable to a large range of relaxation phenomena in condensed matter including polymers.The model predicts a universal mechanism for coupling of a ‘primitive’ relaxation mode with its surroundings. As described below the constant n,which gives the amount of coupling also scales the viscoelastic functions with for example molecular weight. As we shall see then the expression q cc Mk4 falls out naturally from the model as do other observed relationships. Two related predictions which result from the model are that a relaxing quantity obeys G(t)= Go exp (-t/~,)l-~ (1) for sufficiently long times and that the effective relaxation time z is related to the ‘primitive’ relaxation time z, by z = [( 1 -n)co z, exp (AS /R)]l’(l-n). (2) Here nis an effective measure of the coupling of the mode to its surroundings co is a cutoff frequency and AS is an entropy change.It is immediately observed that eqn (1) is the Williams-Watts empirical relaxation function which dates from much earlier work by Pierce for mechanical relaxations. The second prediction given by eqn (2) is entirely new. It is a statement of the basic physics of the Ngail model; i.e.not only is the primitive relaxation mode modified from an exponential to a fractional exponential but also the primitive zo is no longer directly observed as it is shifted to a longer time z, according to eqn (2). 199 GENERAL DISCUSSION Eqn (1) and (2) are related because if eqn (1)is used to determine the coupling parameter n and the effective relaxation time z, then the same n and z must be used in eqn (2). Despite an additional unknown constant cot exp (AS,/@ eqn (2) offers a powerful prediction of the relations between the dependences of z and ze on temperature T,molecular weight M, etc.For example if zo is thermally activated with activation enthalpy EA so that zo =z exp (EA/RT),then z = zz exp (EA/RT) has an activation enthalpy of Ez = EA/(1 -n). This provides a stringent test since only n appears and it is predetermined by the experimental data. Such a relationship between the apparent and primitive activation enthalpies has been verified in many instances for polymer glasses.2 In the case of polymer melts and concentrated solutions eqn (1) and (2) are found to be relevant in the entanglement regime. Through the entanglements each polymer chain couples to its surroundings.The Rouse modes of relaxation of each chain indexed by i have primitive relaxation times zoi given by the familiar expression The relaxation mechanism given by eqn (3) now works through the entanglement coupling by shifting each of the primitive zoi to its effective zei. If we examine the molecular-weight dependence then according to eqn (2) 7 . a MW(n-1). (4) ea Since the degree of cooperativity of Rouse modes decreases as i increases the n is maximum for i = 1 and decreases rapidly to zero with increasing i. Then owing to the non-linearity of eqn (4) zel %-zeZ,ze3 etc. The terminal relaxation and viscosity are then determined primarily by the i = 1 mode characterized by n and z,,. Its contribution to stress relaxation is G(t)= G exp (-z/z,,)'-~~ (5) where r denotes the gamma function.The product of the recoverable compliance and plateau modulus is given by Literature data for G(t),G'(w),G(w)and J(t)for monodisperse linear polymer melts were analysed by NgaL2? * Without exception the data were well described by eqn (1) and its associated functions. The n values determined from these analyses lie between 0.40 and 0.47. Then the relationships for the scaling of viscosity with M and JZ GON were for n = 0.4 qoa Mk3 J",; = 2.05 and for n = 0.47 qoa Mk8 J",Gk = 2.63 which is in good agreement with experimental data. In this treatment the reptation hypothesis has not been invoked. The primitive relaxation species is the terminal Rouse mode and the mechanism of coupling exemplified by eqn (1)and (2) is able to describe rheological behaviour in entangled linear melts and solutions in the linear viscoelastic range.GENERAL DISCUSSION 20 1 Ngai3+* has also found that the model appears to describe the viscoelastic behaviour of branched polymers. At high molecular weights larger n values are found. For example star-branched polystyrenes and polyethylenes have n = 0.7 which predicts that ‘locc Mk7and GG = 15.2. These predictions also agreed with the experimental observations. K. L. Ngai Comments Solid State Phys. 1979 9 127; 1980 9 141. K. L. Ngai Polym. Prepr. Am. Chem. SOC.,Diu. Polym. Chem. 1981 22 289. K. L. Ngai and R. W. Rendell Polymer Prepr. Am. Chem. SOC.,Div. Polym. Chem. 1982 23,46. K. L. Ngai Naval Research Laboratory Washington D.C.personal communication (1983). Dr W. W. Graessley (Exxon Annandale N.J. U.S.A.)replied It is certainly true that my remarks about molecular theories have dealt almost exclusively with reptation and tube models. The Doi-Edwards theory embodies those ideas and it produces defi- nite and testable predictions about a wide range of properties both mechanical and non-mechanical that depend on the slow dynamics of the chains. Although starting from a very simple picture it gives many results correctly. Its failures seem at least at present to arise from oversimplifications rather than some fundamental flaw in the basic ideas. It represents the current state in the long search for a physically understandable unified and quantitative explanation of entanglement effects.That is why I emphasized it. The Ngai formulation deals only with linear mechanical properties. As far as I know it provides no prediction about the diffusion coefficient or the large deformation response. It makes no testable statement about the connection between chain dynamics and dynamic response indeed the assertion of complete generality seems to preclude any such statement. It says in effect that linear viscoelastic behaviour of entangled polymers is something apart from the other dynamic properties to be understood in detail only as part of some universal law governing coupled systems. Nevertheless if that viewpoint works as a general principle even if only for linear viscoelastic properties then it certainly deserves some attention.Agreement with ‘locc M3.*for linear chains becomes interesting however only if the consequences for other chain architectures are also correct. The Ngai coupling constant n is larger for entangled star polymers reflecting a broader relaxation spectrum but this requires an increase in the temperature coefficient of viscosity Ez = EA/(1 -n,). Experiment-ally the temperature coefficients of linear and star polystyrenes’ and polybutadienes2 are practically the same as those of their linear counterparts. Moreover the proposed general form seems to give a rather poor representation of the low-frequency response for star polymer^.^ Thus there are real questions about whether Ngai’s formulation applies broadly even to linear viscoelasticity in entangled systems.Other attempts to deal with entanglement effects were also omitted from the discussion but for different reasons. The early suggestion that entanglement simply increases the frictional coefficient for Rouse modes beyond some critical wavelength4 produces a plateau region and can be made to fit ‘locc M3.4,but it inevitably gives the erroneous JZ cc M and D K cc MP3.*.Later efforts asserting modifications in the Rouse spectrum for various reasons (incoherence of the velocity field5 or intramolecular coupling6 in entangled liquids) succeed in giving JZ G& = constant but they still lead to D cc ‘lo1.Even aside from these differences from observations however the physical basis for the modifications is unclear and they supply no connection with the non-linear response.W. W. Graessley and J. Roovers Macromolecules 1979 12 959. W. E. Rochefort G. G. Smith H. Rachapudy V. R. Raju and W. W. Graessley J. Polym. Sci. Polym. Phys. Ed. 1979 17 1197. GENERAL DISCUSSION K. L. Ngai and R. W. Rendell Polym. Prepr. Am. Chem. Soc. Div. Polym. Chem. 1982,23,46 (see fig. 2). J. D. Ferry R. F. Landel and M. L. Williams J. Appl. Phys. 1955,26 359. W. W. Graessley J. Chem. Phys. 1971 54 5143. D. R. Hansen M. C. Williams and M. Shen Macromolecules 1976 9 345. Dr M. Adam (S.R.M.,Gif-sur-Yvette,France) said The Doi calculation taking into account the fluctuating tube leads to a non-identical dependence on molecular weight of reptation time and zero-shear viscosity and thus to a molecular-weight dependence of the shear elastic modulus.This is in contradiction to experimental results. Thus one wonders what experimental facts support the idea of a fluctuating tube or in other words the idea that the 3.4 molecular-weight exponent value is due to a crossover regime between an entangled and non-entangled chain? Dr W. W. Graessley (Exxon Annundale N.J. U.S.A.)said The fluctuating-tube idea gives a reasonable account of the main facts for star polymers. Assuming that idea is correct fluctuations in path length must also be considered for linear poly- mers. Their contribution should become negligible relative to reptation for sufficiently long chains; qo cc M3must be recovered eventually but how far beyond M,? The main conclusion of Doil was that the crossover associated with a reptation-fluctuation competition is gradual enough to accommodate an apparent power law for the viscosity qo/M3cc M0a4,in the experimental range.The other predictions that Joe and G&would vary slightly with chain length through the crossover may be artifacts of the particular model he used but in any case would be difficult to confirm or deny experimentally. Data for Joe and G; are fewer and much more subject to error than those for viscosity. Also Joe is especially sensitive to slight variations in chain-length distribution and the values of M/Me needed to obtain unambiguous estimates of G; are already quite large. At the moment I do not know of other ways to test the idea so it remains plausible (I think) but unproven.M. Doi J. Polym. Sci. Polym. Phys. Ed. 1981 19 265; 1983 21 667 Dr B. Ewen (University of Mainz West Germany)said Although there are various theories1 on the screening of hydrodynamic interactions in semidilute solutions no experimental data have been available on this subject. As a consequence it was in fact unknown until very recently at what point the screening of these interactions becomes complete. However this gap has now been filled by the results of quasielastic neutron-scattering experiments which examine the dynamics of single polymer chains over the whole range of c~ncentration.~ From these investigations it can be shown unambiguously that the screening of hydrodynamic interactions is incomplete and that the values of the corresponding screening length tH(c) (where c is the polymer concentration) are comparable with those related to the complete screening of excluded-volume interactions (see fig.1). In order to verify these results one may start from the fact that the Zimm which takes account of completely unscreened hydrodynamic interactions and the Rouse model5 have been found to be an adequate description of the scattering behaviour of the dilute solution and the melt respectively.6* Under the assumption that the hydrodynamic interactions at a fixed concentration c (c* < c 6 1 where c* is the overlap concentration) are completely screened for distances larger than tH(c) a crossover from Zimm to Rouse behaviour is expected to take place. This crossover should appear first at small q values and be shifted more and more to larger q values with increasing concentration.However in contrast to GENERAL DISCUSSION these predictions it is quite evident from the scattering data that the transition from a dilute polymer solution to the corresponding melt is accompanied by two different successive crossover effects. The first crossover exhibits the features outlined above Rouse behaviour at small q and Zimm behaviour at larger q whereas the situation is inverted for the second crossover. Both crossover effects can easily be detected from the q-dependence of the lineshape parameter n [see fig. 9 of ref. (S)] provided one is aware of the implicit dependence of n on q within the scatteringalaws of the Rouse and Zimm model.The existence of two successive crossover effects is understood in terms of a new theoretical concept based on the assumption of incomplete screening of hydrodynamic interactions. The corresponding scattering law where tH(c)enters as one parameter predicts three different regimes to be characterized by unscreened Zimm enhanced Rouse and screened Zimm behaviour and allows one to determine tH(c)quantitatively. In fig. 1 tH(c)and t(c) the screening length of the excluded-volume interaction are 3.0 -2.5 5 n 0 W X z E: -2.0 '. * 1-5 I I I I I 0.15 0.18 0.23 0.38 0.45 C Fig. 1. Hydrodynamic screening length tH(c) plotted as a function of concentration 0 semidilute regime and 0, extrapolation to higher concentrations where the double crossover is observed.For comparison <(c),the screening length of excluded-volume interaction,6 is also shown (--); < = 3.4 x c-O.~'. plotted. At present there are not sufficient data to extract the power law for tK(c). However one can see that the absolute values of both quantities do not exhibit large differences. ' P. G. de Gennes Macromolecules 1976 9 587 594. * M. Muthukumar and S. F. Edwards Polymer 1982 23 345. B. Ewen B. Stuhn K. Binder D. Richter and J. B. Hayter Polym. Commun. in press. B. H. Zimm J. Chem. Phys. 1956 24 269. P. E. Rouse J. Chem. Phys. 1953 21 1272; P. G. de Gennes Physics 1967 3 37. B. Ewen D. Richter J. B. Hayter and B. Lehnen J. Polym. Sci. Polym. Lett. 1982 20 233; D.Richter B. Ewen and J. B. Hayter Phys. Rev. Lett. 1980,45 2121. D. Richter A. Baumgartner K. Binder B. Ewen and J. B. Hayter Phys. Rev. Lett. 1981 47 109; GENERAL DISCUSSION 1982,48 1695. A. Baumgartner K. Kremer and K. Binder Faraday Symp. Chem. Soc. 1983 18 37. Dr M. Muthukumar (University of Massachussets Amherst U.S.A.) said As pointed out by Prof. Freed the simple static approximation leads to poor numerical results1 for the intrinsic viscosity and the Huggins coefficient. We2 have recently generalised the famous Kirkwood-Riseman model for the chain to include the interchain hydrodynamic interactions via the multiple-scattering formalism. Our calculated results of the concentration dependence of the self-friction coefficient and the dependence of the Huggins coefficient on the goodness of the solvent (in the zero-frequency limit) are in excellent agreement with the experimental data.3 Although the zero-frequency transport coefficients can be calculated using this simple method one must resort to the Freed-Perico method to obtain the frequency dependence.For example my expression for the non-linear dependence of the polymer-mode relaxation time on the concentration to explain Prof. Schrag’s data was obtained in the spirit of the original Muthukumar-Freed theory. K. F. Freed and S. F. Edwards J. Chem. Phys. 1975 62 4032; M. Muthukumar and K. F. Freed Macromolecules 1977 10 899. M. Muthukumar J. Phys. A 1981 14 2129. M. Muthukumar J. Chem. Phys. 1983 78 2764; M.Muthukumar and M. T. DeMeuse J. Chem. Phys. 1983 78 2770; M. T. DeMeuse and M. Muthukumar Macromolecules to be submitted. Prof. K. F. Freed (University of Chicago U.S.A.) said Dr Muthukumar has performed two interesting types of calculations of the concentration dependence of polymer viscosity. One approach generalizes the Kirkwood-Riseman model by treating the concentration-dependent dynamics of rigid po1ymers.l The configuration average is introduced after the hydrodynamic calculation. In this model the polymers only have overall translational and rotational motion and this considerably simplifies the complicated mathematics necessary when there is internal chain dynamics. Apart from the necessity of correcting the eigenvalues in this approach by a factor of 2lI2 as in the Kirkwood-Riseman treatment the description of the dependence of the Huggins coefficient on the quality of the solvent comes out very well from this simple model.Muthukumar2 has attempted to use this simple model to describe the concentration dependence of the individual mode relaxation times but this generalized Kirkwood-Riseman type model does not appear to be adequate to treat these dynamical quantities. Muthukumar’s most recent calculation as summarized in the paper at this Symposium by Schrag and co~orkers,~ generalizes our previous treatment4 of the leading concentration dependence to higher concentrations using our effective-medium appr~ach.~?~ I am pleased to see that this method produces results in excellent agreement with the experiments of Schrag and coworkers3 despite some of the strong approximations in the theory.As noted in the Symposium paper of Perico and myself,6 this treatment introduces a static approximation which leads to a qualitatively correct description of the concentration dependence but to quantitative errors such as an error of a factor of two in the calculated intrinsic viscosity. When the various pre- factors are treated as empirical adjustable parameters to account for the quantitative problems of the static approximation the net result is excellent agreement with e~periment.~ The treatment of Perico and myself6 is designed explicitly to include this extra dynamics in order quantitatively to predict the prefactors as well as to consider some of the higher-frequency phenomena which may arise specifically by virtue of this added dynamical coupling in polymers.GENERAL DISCUSSION M. Muthukumar J. Phys. A 1981 14 2129; J. Chem. Phys. 1983 78 2764; M. Muthukumar and M. T. DeMeuse J. Chem. Phys. 1983 78 2770. M. Muthukumar J. Chem. Phys. 1983 79 4048. C. J. T. Martel T. P. Lodge M. G. Dibbs T. M. Stokich R. L. Sammler C. J. Carriere and J. L. Schrag Faraday Symp. Chem. SOC. 1983 18 173. * M. Muthukumar and K. F. Freed Macromolecules 1978 11 843. K. F. Freed and S. F. Edwards J. Chem. Phys. 1974 61 3626; K. F. Freed in Progress in Liquid Physics ed. C. A. Croxton (Wiley New York 1978). K. F. Freed and A. Perico Faraday Symp. Chem. SOC. 1983 18 29. Dr R. A. Pethrick (University of Strathclyde Scotland) (communicated) The theory assumes that the momentum distribution of the solvent is unperturbed by the introduction of the polymer.Recent infrared studies of polymer solutions appear to indicate that the high-frequency momentum distribution of the solvent is perturbed by the solvent. We believe that in part this is the molecular basis of hydrodynamic screening and may account for certain anomalies in the high-frequency viscoelastic response of polymer solutions. R. A. Pethrick M. M. Omar T. G. Parker and A. M. North J. Chem. SOC. Faraday Trans. 2 1983 79 687. Prof. K. F. Freed (University of Chicago U.S.A.) (communicated) Dr Pethrick’s experiment involves very-high-frequency dynamics whereas our theories of hydrody- namic screening1 consider the near-steady-state limit.Our theory explicitly includes all of the perturbations in the momentum distribution of the solvent due to the introduction of the polymer. A single polymer perturbs the velocity field of the fluid and the modifications of this perturbed velocity field by other polymers in solution are in part what leads to hydrodynamic screening. Hence we do not assume an unperturbed-solvent momentum distribution. I have another suggestion concerning the strange behaviour observed by Schrag and coworkers2 in their concentration-dependent viscoelastic measurements and I discuss it separately in a comment on Prof. Schrag’s paper. I believe that the puzzle may be related to the well known presence of molecular-weight-independent contributions to the intrinsic viscosity of low-molecular-weight polystyrene.This empirical observation still remains to be fully explained but it appears to describe adequately the data in Schrag’s paper. K. F. Freed and S. F. Edwards J. Chem. Phys. 1974 61 3626; K. F. Freed in Progress in Liquid Physics ed. C. A. Croxton (Wiley New York 1978); K. F. Freed and H. Metiu J. Chem. Phys. 1978 68,4604. C. J. T. Martel T. P. Lodge M. G. Gibbs T. M. Stokich R. L. Sammler C. J. Carriere and J. L. Schrag Faraday Symp. Chem. SOC.,1983 18 173. Dr M. Adam and Dr M. Delsanti (S.R.M. Gif-sur-Yvette France) said The zero-shear viscosity q and the longest relaxation time TR have been measured1* for semi-dilute polystyrene solutions in benzene. The experiments were performed at a monomer concentration c < 10% but larger than the overlap concentration c* (c* = M/R3 G-620M-0.785)for a large range of molecular weight (1.7 x lo5 < M < 2 x lo7)and reduced concentration c/c* (4 < c/c* < 70).Some results are in agreement with theoretical prediction^,^ namely that the relative viscosity qr (fig. 2) and the ratio &/TI (fig. 3) of the longest relaxation time TRto the first-mode characteristic time of a single chain are functions of the reduced concentration c/c* only and that the shear elastic modulus is dependent on concentration only G = 8.32 x lo5 c2.36 dyn cm-2. GENERAL DISCUSSION 0 2 5 10 20 50 100 c/c* Fig. 2. Relative viscosity plotted as a function of the reduced concentration (on a log-log scale).The slope corresponds to an exponent 4.07 f0.1 (theoretical value 3.9). The following symbols are used for the different molecular weights M, A 20.6 x lo6; 0 3.84 x lo6; 6.77 x lo6; 0 0,2.89 x lo6; + 1.26 x lo6; x 4.22 x lo5; V 1.71 x lo5. I I 1 I 2 5 10 20 50 CIC* Fig. 3. Log-log plot of the reduced relaxation time TR/q,as a function of c/c*. The slope corresponds to the exponent 2.05fO.l (theoretical value 1.6). Symbols are the same as in fig. 2. GENERAL DISCUSSION 3.6 0 c/c*> 8 3.2 X 0 c/c* 312 -0 c/? 34.7 Fig. 4. Variation of the value of the molecular-weight exponent X with concentration. (Experiment performed at 0,35 and x ,60 "C.) However some results cannot be explained using the simple reptation model.Thus at a given concentration the molecular-weight exponents X determined through viscosity or longest relaxation time are identical qr K TR K (M,)x~,but the molecular- weight exponent XMis an increasing function of concentration (fig. 4). Our experimental results do not confirm the fluctuating-tube model4 as being responsible for a molecular-weight exponerd larger than that predicted by the reptation model3 in the semi-dilute regime. M. Adam and M. Delsanti J. Phys. (Paris) 1983 44 1185. M. Adam and M. Delsanti Rev. Phys. Appl. 1984 19 253. See for example P. G. de Gennes Scaling Conceptsin Polymer Physics (Cornell University Press Ithaca N.Y. 1974). M. Doi J. Polym. Sci. Polym. Phys. Ed. 1981 19 265. Prof.K. F. Freed and Mr J. F. Douglas (University of Chicago U.S.A.)said The paper notes that it is now possible fully to describe excluded-volume effects on certain polymer-dynamical properties in the infinite-dilution limit. We use the usual two- parameter model of excluded volume and consider preaveraged dynamical (or equilibrium) quantities Q which scale in the two-parameter theory as1 where (S2),is the mean square radius of gyration in the unperturbed state G,( S2)fi2 is Q in this unperturbed state and where the second line applies for the second virial coefficient. GQand CQmay depend on segment indices number of branches etc. An approximation to the second-order renormalization group predictions for these properties Q,for linear ring and regular star and comb polymers has been shown to be2 (d= 3) Q= {" Q( 2)f/2 (1 +32~/3)*/'(1 +aQA,) for z < 0.15 (6.441~)*(~"-~)(1 GQ(S2)f/2 +aQ) for z 2 0.75 (3) 208 GENERAL DISCUSSION 0.4 0.3 \k 0.2 0.1 0 1 2 3 4 4 Fig.5. Comparison of different theoretical expressions for Y with experimental data. Data points are from Berry5 for polystyrene in toluene at 12 “C (0)and in decalin at various temperatures (0). The curves are for so-called ‘compatible theories’ (capable of predicting both the second virial coefficient and the radius of gyration) (l) (2) (3) are the original Flory-Kurata theory the modified Flory-Kurata theory and the Kurata-Yamakawa-Tanaka the~ries,~ respectively. Similar data are given in Norisuye et ~1.~ Tanaka et al.’ and Matsumoto et a1.* for polychloroprene poly(pmethy1styrene) and polyisobutylene in several solvents.where we use the first line of eqn (2) v = 0.592 A1 = (321/3)/(1 + 32~/3). (4) The scaling variable z is treated as an empirical parameter as is the z-variable in the two-parameter theory. G and p are readily available and only aQneed be determined from either renormalization-group or two-parameter theory calculations or from experimental data. We illustrate the theory by comparison with experimental data for which z has been eliminated between two direct observables and the first line of eqn (3)is used for I in the range (0.15-0.75). Fig. 5 presents the universal plot of the most important infinite-dilution equilibrium properties the penetration function Y against a3,Z.The figure is reproduced from ref. (3) (fig. VII. 11) with the renormalization-group predictions from eqn (3)labelled as RG and with the good-solvent limit denoted as Y*. The other three curves are older theories as described in Yamakawa. Fig. 6 presents data from ref. (3) fig. VII. 17 for A M/[q]against agz -1. Our renormalization-group calculation for [q] is in the non-draining limit and the experimental value of (9 is used to help correct for the preaveraging approximation in the calculations. The renormalization-group predictions are added to the figure and labelled as RG again. BF designates the best empirical fit to the data. The excluded-volume dependence of a)/@,is presented in fig. 7 using data from Noda et aZ.* The solid curve is the renormalization-group non-free-draining-limit GENERAL DISCUSSION I 1 I I 1 .oo -i= Y . A v q3 N T' 0-50 0 0 0.5 1.0 1.5 a;* -1 Fig.6. Experimental data for 7t = A2M/[q]for polychl~roprene~ in carbon tetrachloride at in n-butylacetate at 25 "C (6) for p~ly(p-methylstyrene)~ 25 "C (a) and in t-decalin (0); in toluene at 30 "C (a),in cyclohexane at 30 "C (0-), in methyl ethyl ketone at 30 "C(0)and in diethyl succinate at various temperatures (0). Similar data may be found in ref. (7) and (8). 1.0 1.2 1.4 1.6 1.8 2.0 2.2 2.4 &St Fig. 7. The universal ratio @/(Do plotted as a function of criz = (S2)/(S2)ofor poly(cr- methylstyrene). The data are reproduced from ref.(4) with the RG prediction added. More data are given by Matsumoto et aL8and Fujita et ~1.~ with similar agreement between theory and experiment. GENERAL DISCUSSION prediction. The relative effects of excluded volume are adequately described here in the preaveraging approximation. Calculations have also been made for hydrodynamic and static properties of rings regular stars and the H-comb. Comparisons for the branched polymers have only been made in the good and theta solvent limits. In general the agreement between theory and experiment for linear polymers is comparable to that in fig. 5-7. J. F. Douglas and K. F. Freed Macromolecules 1983 16 1800. * J. F. Douglas and K. F. Freed submitted to Macromolecules. H. Yamakawa Modern Theory of Polymer Solutions (Harper and Row New York 1971).I. Noda K. Mizutani T. Kato T. Fujimoto and H. Nagasuwa Macromolecules 1970 3 787. G. C. Berry J. Chem. Phys. 1967 46 1338. T. Norisuye K. Kawahara A. Teramoto and H. Fujita J. Chem. Phys. 1968 49,4330. ' G. Tanaka S. Imai and H. Yamakawa J. Chem. Phys. 1970,52,2639. T. Matsumoto N. Nishioka and H. Fujita J. Polym. Sci.,1972 10 23. K. Kawahara T. Norisuye and H. Fujita J. Chem. Phys. 1968 49 4339. Dr G. B. McKenna (N.B.S. Washington D.C.,U.S.A.)said The paper of Prof. Binder is of great interest because it suggests that for linear chains in a melt reptation may not be the predominant mode of deformation even above the entanglement molecular weight. Could this be related to our findings? (described below) that there is no great difference between the zero-shear melt viscosities of linear polystyrene fractions and their cyclic non-reptating counterparts? Narrow fractions of polystyrene (PS) molecules in the form of closed uncatenated rings (cycles) were synthesized by reacting bifunctional 'living ' precursors with an appropriate coupling agent at very low c0ncentration.l Recent theories2 of the flow behaviour of molten polymers model the chain motion by reptation along a curvilinear 'tube' formed by the constraints of the surrounding matrix.Since a ring molecule cannot reptate in the conventional sense its behaviour in shear might be expected to differ appreciably from that of its linear homologue. In the present work the molecular-weight dependence of the zero-shear viscosity (qo) of melts of cyclic and linear PS fractions is investigated over an Mw range encompassing the critical molecular weight (M z 40000) above which entanglements occur in linear PS.4 Viscosity measurements were performed using a cone-and-plate viscometer at low shear rates < y/s-l < and over a wide range of temperatures (120 < T/"C < 210).All the tests were carried out under dry nitrogen to minimize possible degradation. The results show that the temperature dependence of qo of the ring molecules is indistinguishable from that displayed by linear chains including data obtained earlier by Pierson5 and Suzuki6 using the same instrument. In fact for Mw> M, all. the (log qo,T) curves can be superimposed within experimental accuracy by simply shifting them along the log qo axis.The same holds even for the lowest Mwring for which obviously chain ends do not contribute to molecular mobility. The dependence of qo on Mwis depicted in fig. 8 for 150 "C. This shows that the difference between the viscosity of the rings and that of their linear precursors is quite small. Note however that the viscosity of the rings is better described by a power law qo cc Mgg,which involves a slightly larger value of the exponent s than the one (ca. 3.4) characterizing linear chain^.^-^ These results raise some important questions about the motion of uncatenated rings in a matrix made of similar rings. The unexpectedly small differences between the t Work performed at the laboratories of the Centre de Recherches sur les Macromolecules Strasbourg France by G.B. McKenna G. Hadziioannou C. Strazielle G. Hild P. Rempp and A. J. Kovacs. GENERAL DISCUSSION 21 1 I I I I 3 .O 3.5 4.0 4.5 5.0 5.5 log, (M,) Fig. 8. Zero-shear viscosity plotted as a function of molecular weight for fractions of linear and cyclic polystyrene molecules at 150 "C:0,cycles; W linear (this study); A,linear (data of Pierson); V,linear (data of Suzuki). zero-shear viscosity of non-reptating cycles and that of their presumably reptating linear homologues is surprising. The close similarity in the behaviour of these systems strongly suggests that the contribution of reptation to shear flow may be less important than assumed by recent theories.2T3 Further work is needed however to assess this conclusion.G. Hild C. Strazielle and P. Rempp Eur. Polym. J. 1983 19 721. P. G. DeGennes J. Chem. Phys. 1971 55 572. M. Doi and S. F. Edwards J. Chem. SOC.,Faraday Trans. 2 1978 74 1789. G. C. Berry and T. G. Fox Adu. Polym. Sci.,1968 6 261. F. Pierson Thesis (Fac. Sci. Univ. Strasbourg 1968). R. Suzuki Thesis (Fac. Sci. Univ. Strasbourg 1970). Dr A. Baumgartner,Dr K. Kremer and Prof. K. Binder (KFA Jiilich West Germany) replied These data on ring-polymer viscosities are most interesting and certainly represent a challenge for a theoretical explanation along the lines of reptation theory. We also think that for the behaviour at intermediate wavevector and on the timescales studied in our work (timescales which are much shorter than needed for a chain to 'creep out' of a tube if tubes are meaningful objects in our case) it probably does not matter too much whether the chain is a closed ring or has free ends and hence the behaviour might well be related as Dr McKenna suggests.GENERAL DISCUSSION Dr R. F. T. Stepto (UMIST Manchester) said The simulation method used by Baumgartner Kremer and Binder is based on Metropolis sampling in which a single segment is randomly moved to create a new configuration. The method which implies individual and sequential segmental movement does not truly reflect segmental dynamics in real time. As far as I am aware Metropolis sampling was first applied to the simulation of polymer chain configurations by La1 and Spencer1? and has subsequently been used3-j in several computations of equilibrium statistical properties of polymer chains.The generation of new configurations is by arbitrarily chosen segmental movements. This is satisfactory for equilibrium statistical properties where it is required merely to generate a population of the correct (Boltzmann) frequency. It is not satisfactory for the simulation of properties which depend on how configurations interchange. The method of perturbation from one configuration to the next will then affect the final results. M. La1 and D. Spencer Mol. Phys. 1971 22 649. M. La1 and D. Spencer in Applications of Computer Techniques in Chemical Research ed. P. Hepple (Applied Science Publishers London 1972).M. La1 and R. F. T. Stepto J. Polym. Sci. Polym. Symp. 1977 61 401. C. J. C. Edwards D. Rigby and R. F. T. Stepto Macromolecules 1981 14 1808. C. J. C. Edwards D. Rigby R. F. T. Stepto and J. A. Semlyen Polymer 1983 24 395. Dr A. Baumgartner Dr K. Kremer and Prof. K. Binder (KFAJiilich West Germany) said The simulation methods used in our work are well known and it was not the purpose of our paper to review technical aspects of these methods;l early attempts to use Metropolis sampling for dynamic properties of chains were made by Verdier and Stockmayer,2 for instance. The remarks that methods of this type do not truly reflect segmental dynamics in real time certainly is correct; for instance oscillatory motions are not included nor any motions related to hydrodynamic flow.However the point is that the simulated model of a single chain in the limit of small wavevectors and long times must still give results which reduce to the Rouse model. This fact has been checked explicitly and is nicely verified.l For a many-chain system the 'entanglement restriction' that chain motion must not lead to chain interaction is also included in the simulation. This model although it may not be realistic for the description of the dynamics of real polymer melts on very short timescales has all the necessary ingredients for a check on the validity of reptation theory (which is general enough to encompass also this model but on the other hand not expected to be valid for very short timescales). This was the point of interest for our work.The comparison with corresponding experimental data which we have presented indicates that the model is in fact satisfactory. For reviews see A. Baumgartner in Applications of the Monte Carlo Method in Statistical Physics ed. K. Binder (Springer Berlin 1984); and A. Baumgartner Simulation of Polymer Motion preprint from Annu. Rev. Phys. Chem. in press. P. H. Verdier and W. H. Stockmayer J. Chem. Phys. 1962 36,227. Dr F. BouC (CEA-CENS, Gif-sur- Yvette France) said Dr Baumgartner presented supporting his results experimental data [ref. (27) of his paper] obtained from the inelastic scattering of neutrons on the spin-echo spectrometer IN11 of I.L.L. (Grenoble). These results have been widely discussed. However to clarify the present discussion it would be useful to give simply the estimated values from the following three viscoelastic quantities of three characteristic times of PDMS chains used in the experiment (molecular weights 3800 30000 and 60000; temperature 100 "C):(i) the GENERAL DISCUSSION relaxation time of the chain between entanglements T, usually equal to the maximum Rouse time of a chain of molecular weight Me (where Me is the molecular weight of that part of chain comprised between two successive entanglements; see our paper at this Symposium); (ii) the maximum Rouse time of the total chain TRouse, for the different molecular weights; (iii) the terminal time of the chains.These times could then be easily compared with the characteristic times of the IN1 1 spectrometer lying between lop9 and 5 x lopRs.Dr A. Baumgartner Dr K. Kremer and Dr K. Binder (KFA Jiilich West Germany) said The maximum Rouse time of the total chain given by1 is calculated (using c = 6 1 = 2.3 A and lo/12 = 5.3 x lo5g cm-2 s) to be zRouseM 5 x lo-* s for M = 30000 and zItouseM 2 x lop7s for M = 60000. The entanglement length for PDMS is N M 340. The characteristic relaxation times z,are estimated accordingly to be z = zRouseNZ/N2z 3 x s. The disengagement time,2 defined as zd = T~~~~~ N/N, is now estimated to be zd M 6 x lops s for M = 30000 and zd M 5 x lop7s for M = 60000. W. W. Graessley Adu. Polym. Sci. 1982 47 68. P. G. de Gennes J. Chem. Phys. 1980 72 4756. Dr J. S. Higgins (Imperial College London) said The interpretation of the Q-dependence of the time power-law exponent in terms of hydrodynamic screening as presented in fig.9 of Dr Baumgartner’s paper raises two objections about the assumptions underlying the investigation of the crossover regime. To extract the time power law one has to assume that (1) The coherent dynamic structure factor S,,h(Q t) is correctly evaluated from the Rouse model which is however only valid for the range Qo + I where o is the step length associated with one bead.l (2) The simplified forms of the structure factor may be used namely in the Zimm limit. These are however the asymptotic limits at long times of the full variations calculated by de Gennes.’ It should be pointed out that the data are obtained using neutron quasielastic scattering techniques for which neither assumption is strictly valid ; the violation of the second probably being much worse than the first.We now consider the possible effects of these experimental limitations in detail. (1) Experimental Q-range. The Rouse step length cannot be easily estimated for real polymers. However Akcasu2 showed that when the range Qa zz 1 is explored the time variation of Scoh(Q,t) changes from the Rouse or Zimm forms towards a Lorentzian (ie. the power to which t is raised increases as Q increases). At the same time the Q-dependence of the characteristic inverse relaxation time Q reduces from Q4(Rouse) or Q3(Zimm) towards Q2.Examination of the Q variation of R can indicate when the range Qa M 1 is reached.We have examined the results for a number of polymers in dilute solution3 and shown that for polydimethylsiloxane (PDMS) o = 16 A. Thus the higher$ data for PDMS as given in fig. 9 of the paper under discussion may well GENERAL DISCUSSION 10 1 c 8 h 4 oi z c -I 0.1 0.01 10 100 1000 4 * BIA Fig. 9. The coherent normalised dynamic structure factor Scoh(Q, t)for Rouse and Zimm models compared with a Lorentzian [exp (-Rt)] and plotted as In [-In S(Q,t)] against In B (where B is proportional to time). In each case R = lo*s-l. The dashed lines indicate the long-time limits R and 2 given in eqn (1). The arrowed line on the abscissa gives the experimental range of B/A = 5.33 x 10" s. explore the range outside the validity of the pure Rouse behaviour and hence show an apparent variation in n.(2) Experimental timescale of the neutron spin-echo techniq~e.~ Because of the limited time range the experiments cannot observe the simple forms of s,,h(Q t) which apply in the asymptotic limit even for the fastest relaxation (highest-Q data). In fig. 9 of this comment we show the Rouse (R) Zimm (Z) and Lorentzian forms of s,,h(Q t) plotted as In [-In Scoh(Q, t)] against In t in order to demonstrate the various time power laws. As can be seen the asymptotic limits (dashed lines R and 2,) are reached well outside the experimental range (shown as the arrowed line on the abscissa) for the value of SZ chosen here los s-l. Now for a PDMS dilute solution the experimental values of SZ range from lo7s-l at Q = 0.026 k1up to 2.6 x lo8 s-l at Q = 0.15 A-1.3 The curves shown in fig.9 are thus typical of a polymer solution. Moreover as R increases with Q the experiment will explore an increasing range of the s,,h(Q t) and will thus show variation in the apparent power law n. * P. G. de Gennes Physics 1967 3 37 and P. G. de Gennes and E. Dubois-Violette Physics 1967 3 181. * Z. Akcasu M. Benmouna and C. C. Han Polymer 1980 21 866. L. K. Nicholson J. S. Higgins and J. B. Hayter Macromolecules 1981 14 836. J. B. Hayter in Neutron Diffraction ed. H. Dachs (Springer-Verlag Berlin 1978). GENERAL DISCUSSION 215 Dr A. Baumgartner Dr K. Kremer and Prof. K. Binder (KFA Jiilich West Germany) said :The points raised by Dr Higgins are of course well known to everybody involved in neutron scattering from polymer systems.Ewen et al.l9 have been fully aware of these problems in their analysis. We have not discussed this point in our paper because the data of our fig. 9 have only been mentioned as a side remark that there is the problem of Zimm-Rouse crossover in addition to the problem of Rouse-reptation crossover when one considers concentrated polymer solutions rather than melts. It was not our intention to squeeze a second paper on neutron-scattering studies of polymers into a paper devoted to computer simulations. Thus at this point we only mention that the analysis of ref. (1) and (2) is not just based solely on a fit to the simplified asymptotic forms of the Rouse-Zimm model.Rather the treatment of de Gennes3 has been generalized to calculate the full scattering function S,,,(q t) for the crossover between Rouse and Zimm behaviour on the basis of the new concept of incomplete screening of hydrodynamic interactions. The analysis1 of the neutron-scattering data with this full scattering law is consistent with the results presented in our fig. 9. Hence the points raised by Dr Higgins cannot serve to criticize our conclusions on hydrodynamic screening. Note that for very concentrated solutions and melts where we observe essentially Rouse behaviour the full Rouse function is not a better fit to the actual neutron data than is its simplified asymptotic form. This indicates that the Rouse model does not describe the relaxation of melts at short timescales exactly which fact is not unexpected it is a model for a single chain in a heat bath rather than for an ensemble of strongly interacting chains.Hence the approximation of the use of the Rouse model in the first place seems to be a more serious approximation than any of the points raised by Dr Higgins in our case. B. Ewen B. Stuhn K. Binder D. Richter and J. B. Hayter to be published. B. Stiihn Dissertation (University of Mainz 1983). P. G. de Gennes Physics 1967,3 37; P. G. de Gennes and E. Dubois-Violette Physics 1967 3 181. Dr F. Ganazzoli (Milan Polytechnic Italy) said I completely agree with Dr Higgins' remarks about the different time ranges observed experimentally and predicted also by the Rouse or Zimm models for the quasielastic scattering laws.In the case of atactic PS in CS solution at T = 30 OC for instance the half-peak time of the coherent scattering curve ti plotted against the wavevector modulus Q gives ti = const where /?is 2.7 0.2 for 0.05 < Q/kl < 0.13 and 2.0 f0.2 for 0.1 1 < Q/E1< 0.27,' quite far from the exponents 4 or 3 predicted by the Rouse or Zimm models respectively in the long-time limit. However I would also like to add a word of caution concerning the spatial ranges. For wavevector values Q 2 0.1 k1one is probing distances of the order d 5 25 A where the local chain structure comes into play. Hence the scattering law has no universal behaviour and no scaling of Q or time gives a unique curve as can be seen by comparison between atactic PS and PDMS in dilute solution.2 We have attributed these differences to the different chain rigidity which has two independent sources.The first one is purely conformational and arises from first- neighbour correlation of the chain torsion angles. The second one is dynamical (internal viscosity) and depends on the energy barriers hindering the rotations around the skeletal bonds3 in a Kramers-like activated pro~ess.~ Thus unlike the conform- ational rigidity the internal viscosity is dependent on temperature. Along these lines we can see that PS and PDMS may be regarded as close to the two extremes of high and low rigidity among the random-coil polymers the former has both high conformational rigidity and high internal viscosity enhanced by the GENERAL DISCUSSION bulkiness of the phenyl rings which must approach each other during a skeletal rotation.The latter is very flexible for both aspects due to the very low intrinsic barrier around a Si-0 bond and the opening of the Si-0-Si angles (143") which keeps the methyl groups on neighbouring Si atoms apart. With these ideas we can also understand qualitatively the data obtained recently on melts5 from polymers ranging from very flexible (PDMS) to very rigid (PIB with two geminal methyl groups on the same skeletal atom) through intermediate ones (PTHF and PPO). Here the p exponent for Am K Qb (Aw K tcl) increases with flexibility when going from PIB to PDMS and for the same polymer increases with T qualitatively in keeping with what predicted for the internal viscosity effect.Only PDMS does not follow this trend its conformational flexibility being very high and its internal viscosity being virtually nil its p is always 4 i.e. the greatest among the polymers reported in ref. (5). G. Allegra J. S. Higgins F. Ganazzoli E. Lucchelli and S. Bruckner Macromolecules in press. * F. Ganazzoli G. Allegra J. S. Higgins J. Roots S. Bruckner and E. Lucchelli paper in preparation. (a)G. Allegra J. Chem. Phys. 1974,61,49 10;(b)G. Allegra and F. Ganazzoli Macromolecules 1981 14 1110. H. A. Kramers Physica 1940 7 284. G. Allen J. S. Higgins A. Maconnachie and R. E. Ghosh J. Chem. SOC. Faraday Trans. 2,1982,78 21 17. Dr A. Baumgartner Dr K. Kremer and Prof.K. Binder (K.F.A. Jiilich West Germany) replied We fully agree with Dr Ganazzoli that the scattering function at intermediate wavevectors reflects the local chain structure and in this regime (in the notation of our paper this is the regime where qZ no longer is very small compared to unity) one has to be very careful with the interpretation of the scattering function and the decay law which clearly in this regime is not universal. However this regime was not of interest for the discussion of our paper which restricted attention to the universal regime ql << 1. Of course with respect to the experimental data on PDMS quoted in our paper one has to make sure that one is sufficiently close to this universal asymptotic regime. We feel that there is a great deal of experimental evidence supporting that this is indeed the case (cf.for instance the neutron-scattering investigations of Ewen et a1.l and Richter et a[. and those quoted in our paper). The fact that this would not hold for atactic polystyrene in CS solution for similar absolute values of q is not counter- evidence of course since PDMS and PS seem to have different values of the effective length I; it is emphasized by Dr Ganazzoli himself that PS and PDMS may be regarded as close to the two extremes of high and low rigidity among the random-coil polymers and hence PS must have a larger value of I than PDMS. The latter material thus seems indeed the most well suited material for neutron scattering purposes since one can study the universal behaviour at rather large values of q.This point is also borne out by explicit calculations of Stuhn,3 who has performed explicit calculations with the method of Allegra et aZ.4 for the case of PDMS thereby taking explicitly into account effects due to the local chain structure short-range forces etc. These calculations demonstrate convincingly that these ' local ' effects are indeed unimportant at the wavevectors studied in ref. (l) (2) and (9,and mentioned in our paper. B. Ewen D. Richter J. B. Hayter and B. Lehnen J. Polym. Sci. Polym. Lett. 1982 20 233. * D. hchter B. Ewen and J. B. Hayter Phys. Reu. Lett. 1980 45 2121. B. Stiihn Dissertation (University of Mainz 1983) and to be published. G. Allegra and F. Ganazzoli Macromolecules 1981,14,1110; G.Allegra J. S. Higgins F. Ganazzoli E. Lucchelli and S. Briickner to be published. B. Ewen B. Stiihn K. Binder D. Richter and J. B. Hayter to be published. GENERAL DISCUSSION Dr J. L. Viovy (E.S.P.C.I. Paris France) said The paper by Prof. Binder and his co-workers describes several attempts to observe reptation directly on a molecular scale. Since most of them reveal no deviation from Rouse behaviour it is tempting to use previous knowledge and try to answer at least qualitatively the question was reptation really expected to be observable in these experiments? Clearly chains with 16 bonds at a concentration of ca. 0.25 are not expected to be entangled and the results of Section 2 are not surprising. In Section 3 the authors use ‘long’ chains with 200 bonds on a tetrahedral lattice.Owing to the lattice constraint and several other simplifications a quantitative comparison between this system and real chains can only be tentative. Nevertheless previous simulations in dilute solution1 showed that the equally weighted lattice chain was a reasonably good approximation for polyethylene. For this compound viscoelastic measurements in the melt lead to a value of M between entanglements of ca. 1400 (W. W. Graessley this Symposium). Thus the number of bonds between entanglements at C = 0.344 in a B solvent should be N M 140. If the extrapolation to lattice systems is allowed one might fear that chains with 500 or 1000 bonds will be necessary to observe the entangled regime at this concentration.On the other hand the authors seem to observe a reptation-like behaviour with the pearl-necklace model (Section 5) but this model is more difficult to compare with real chains. The authors also recall experiments on PDMS performed using the neutron quasielastic scattering device IN 11 [ref. (27) in the paper]. The observation of reptation on this apparatus should depend mainly on the possibility of exploring times greater than the crossover time 7 cc W-l d$ [ref. (1 5) in the paper]. Thus to observe reptation the tube diameter dT is a much more critical parameter than the jump frequency W. PDMS which has a high mobility but also a high molecular weight between entanglements Me (W. W. Graessley this Symposium) is probably not the best system.Reptation should be easier to observe in ‘thin’ polymers with no side groups low Me and low Tg.Indeed Dr Higgins and her colleagues recently observed (poster this Symposium) for such a polymer PTHF an important deviation from Rouse-like behaviour in the entangled regime. ’ F. Geny and L. Monnerie J. Polym. Sci. Polym. Phys. Ed. 1979 17 131. Dr A. Baumgartner Dr K. Kremer and Prof. K. Binder (KFA Julich West Germany) replied Dr Viovy first suggests that chains with 16 bonds are not expected to be entangled. However we would like to remark that this system was entangled and reptation nicely verified when only one chain was allowed to move and the others were held fixed. He then estimates the number of bonds between entanglement points for polyethylene solutions at a concentration c = 0.344.Our simulation of chains on the tetrahedral lattice should not be compared with such a solution but with a solution of much higher concentration or even a melt for the lattice model the nominal concentration c = 1 does not yield a melt but a perfect (and absolutely immobile) crystal and even at c z 0.7 one tends to have a glassy state with no large-scale motion possible for reasonable computing times. A detailed study of static properties of the lattice model at c = 0.344 [ref. (25) of our paper] also reveals that this model has properties rather close to those of melts. Prof. L. Monnerie (E.S.P.C.I. Paris France) remarked The PDMS chains have a very unusual low internal viscosity which could explain the finding that only Rouse behaviour is observed in neutron quasielastic scattering experiments.Concerning the GENERAL DISCUSSION particularly small constraint of chain connectivity on the segmental motion I want to mention the experimental results that we have obtained studying the excimer fluorescence emission of 10,lO’-diphenyl-bis-(9-anthryl)methyloxide (diphant) in PDMS matrices’ ranging from tetramer up to chains of molecular weight as high as 4 x lo5 daltons. In diphant molecules the most stable conformation corresponds to anthracene rings far from each other. If during the lifetime of the singlet excited state of one anthracene group intramolecular motion leads to a conformation in which the two anthracene groups are parallel a specific excimer fluorescence occurs.Surprisingly we have found the same rate of intramolecular motion and the same activation energy independently of the molecular weights of the PDMS. Thus although the bulk viscosity of PDMS matrices considerably varies from the tetramer (very low viscosity liquid) to the high molecular weight (similar to a rubber) the same dynamic flexibility exists locally. E. Pajot-Augy L. Bokobza L. Monnerie A. Castellan H. Bouas-Laurent and C. Millet Polymer 1983 24 117. Dr A. Baumgartner Dr K. Kremer and Prof. K. Binder (KF’ Jiilich West Germany) replied We understand Prof. Monnerie’s comment as suggesting that PDMS may have an unusually large tube diameter (due to unusually low internal viscosity) and therefore it could not be identified by neutron scattering.This suggestion may well be correct. What is lacking is any theory which links the phenomenological concept of the tube diameter to any more microscopic chain properties. This suggestion is also in agreement with our point that the tube diameter should not be linked to geometric properties (topology of entanglement etc.) alone; dynamic aspects must also be very important. Dr J. M. Deutsch (Cavendish Laboratory Cambridge) said :The preceding paper (by Prof. Binder and coworkers) has dealt with the simulation of flexible polymers. The results have been interpreted as being different from reptation and therefore cast some doubt as to the agreement of the reptation model with computer experiments at a microscopic level.I wish to point out that neither the Rouse model nor the new model proposed by Dr Baumgartner to explain his results can serve as alternate explanations to the dynamics of long entangled polymers as both must disagree with the now well established experimental result that the diffusion coefficient D,is proportional to 1/N2.To do so I will now derive the following theorem. Define the time-dependent displacement of the ith monomer and similarly for the centre of mass gcm(t) = <[rcrn(t)-rcrn(0)12)* The following inequality holds rigorously IMt)I’-kcm(t)PI 2~g(i) (3) where Rg(i) <(ri -rcm)2) equilibrium (4) i.e. Rg(i)is the radius of gyration of the polymer about the ith monomer. To prove eqn (3) consider the reduced time-dependent displacement GENERAL DISCUSSION 219 Expanding eqn (5) we have ([ri(t)-rcrn(t)12) +([ri(O)-rcrn(0)l2) -2([ri(t)-rcrn(t)I* [ri(O)-rcrn(O)l)-(6) The first two averages are equal-time averages and hence equilibrium averages.Thus the first two averages are equal and also equal to R:(i). The last average can be bounded by the Schwarz inequality (i.e. (u=u) < du2do2).So gred(t) d 4((ri -rcrn)2)equilibriurn 44(0* (7) To obtain an upper bound on gred(t),one can rearrange the terms in eqn (5) so that gred(t) = ([ri(t)-ri(O)l -[rcrn(t)-rcrn(0)I2)-(8) Expanding this out gives ([ri(t)-ri(0)12) +(rcrn(t> -rcrn(0)12) -2([ri(t)-ri(O)I * [rcrn(t) -rcrn(0)I)-(9) The last term can again be bounded using the Schwarz inequality so that gred(l) < ((([ri(t) -ri(0)12 )1' -{([rcrn( t -rcrn(o)12 -(10) Combining both inequalities (7) and (10) gives ({ <[ri(t)-ri(011 )>e -i([rcrn(t)-rcrn(011 >) d ~RE(I) (1 1) and hence proves eqn (3).This result is physically obvious since gcrn(t)and g(t) cannot differ by more than a few radii of gyration. Note also that for Gaussian chains Rg(i)< 2iRg. (12) We now apply this result to Dr Baumgartner's interpretation of his computer data. In his interpretation there is an intermediate crossover regime from t K N8l3 to t K N24/7,where g(t) K NP. The fact that this is a positive power is what leads him to find two timescales. An N24/7timescale and an N3 timescale. However pick any time between N8/3and N24/6,for example t a N3.Then b k(t)]i-kcrn(t)]$ N = ANitA-B-= Ni(AN&-B) < 2 x 2:R = 2 x (2N):I (13) where A and B are constants of proportionality and I is the step length of the polymer.Eqn (1 3) implies (AN&-B) < 2 x 241 (14) which is violated for large N. There is a further problem with Dr Baumgartner's interpretation of his results. If we superimpose plots for two different chain lengths of g(t) against time we see that for intermediate times monomers on the longer chain have moved further than monomers on the shorter chain. We can make the ratio of the two amounts moved arbitrarily large by increasing both chain lengths. I.e. if we have a system of chains of length Nl and a system of chains of length N with corresponding time displacements gl(t) and g2(t)then the maximum ratio of g,(t)/g,(t) is I now return to eqn (3) and use it to show that the Rouse model is also in disagreement with experimental data concerning the diffusion coefficient.The Rouse model has correlation function g(t) proportional to PI2 and then t. The correlation GENERAL DISCUSSION function for the centre of mass g,,(t) is proportional to t. We also know the prefactor dependence on N i.e. The short-time behaviour of g(t) is clearly independent of chain length at constant density. I.e.if g(t) = 0(l2) then the value of g(t)should be independent of chain length for very long polymers and at constant density. Thus if g(t) cc t1/2 then there can be no prefactor dependence on N for long polymers. This is indeed confirmed by all the simulations in the previous paper and also in my own simulations.The crossover between the t1l2 law for g(t)and the t/N2 law occurs when t cc N4. If we choose a time t cc N3then [g(t)]e- [g,,(t)]; = Ah-B-ti = AM- BN; 6 2 x (2N);l N but this implies AN;-B 6 2 x 241 which is violated for large N. The above remarks do not imply the simulations in the previous paper are invalid but just that there are faults in the interpretation of the results obtained. The simulation of chains of length 200 on a diamond lattice by Kremer do not appear to be in a regime where one would expect to find reptative behaviour (due to low density and high persistence length) and therefore do not invalidate my own simulations on this subject (which finds evidence for reptation and where all chains are allowed to move freely).For a more detailed account of the problems with the interpretation of Kremer’s simulation and a rebuttal against claims that my own work is invalid see ref. (1). It appears to me that Dr Baumgartner’s recent simulation could in fact be interpreted in terms of the reptation model with an entanglement length of order 25 to 50 which is remarkably close to the entanglement length I obtained in my simulations ranging from ca. 15 to 50 at similar density. I would be interested to know if Dr Baumgartner does find it possible to interpret his results in this way. J. M. Deutsch Phys. Rev. Lett. 1983 51 1924. Dr Baumgartner Dr K. Kremer and Prof. K. Binder (KFA Jiilich West Germany) replied Dr Deutsch has made an interesting comment.Of course we have not claimed that the Rouse model can explain the diffusivity law D cc N-2.In addition more recent computer simulations of very long chains (N = 162) do not support the interpretation of the dynamics of shorter chains (see fig. 7 of the presented paper). The data are in fact not inconsistent with reptational behaviourl (cf. fig. 10). However the error for the estimated exponents of disengagement time rd cc Nu (a z 3) and correlation function g(t)cc tb (b= 0.27 f0.04 at intermediate times t < rd) are too large in order to exclude adequacy of any other model except of the reptation model. Still the experimental fact rd cc N3.4 remains an open question. Concerning the validity of previous simulations of polymers confined to a cubic lattice,2 we refer to a detailed recent comment3 and reply4 to this paper.2 A.Baumgartner Annu. Rev. Phys. Chem. 1984 35. J. M. Deutsch Phys. Rev. Lett. 1982 49 926. K. Kremer Phys. Rev. Lett. 1983 51 1923. J. M. Deutsch Phys. Rev. Lett. 1983 51 1924. GENERAL DISCUSSION 221 1o2 --lo-' I I I I IIIII I I Illlll I I I111111 I I I Illlll I I I IIIU. Fig. 10.Log-log plot of Monte Carlo data for the monomer displacement g(t) against time obtained from simulations of the 'pearl-necklace ' model. 32 0.064 f0.007 1.3f0.1 0.50 f0.05 n 50 0.071 f0.008 1.3f0.1 0.55 k0.06 0 72 0.072f0.006 1.3k0.2 0.56 & 0.12 0 162 0.075f0.008 1.2_+ 0.3 0.54f0.16 Prof. K. F. Freed and Dr A. Perico (University of Chicago U.S.A.) said The treatment in our paperf and a more detailed version thereof provides an understanding of the fact that hydrodynamic screening affects the Rouse modes at different rates.Hence as concentration is increased the long-wavelength modes are screened first whereas the screening of the shorter-wavelength modes may in fact be rather small. This implies that short segments along the chain may still retain their full correlations in their perturbations of the fluid flow. This situation has been suggested in the paper by Binder3 and is physically quite reasonable. By simple extrapolation of the results of our paper1? to higher concentrations we can exhibit how this arises. In dilute solutions the eigenvalues A of the matrix HA of Zimm and Rouse may be approximated for a large number of segments n by 1 = nfv (1) GENERAL DISCUSSION where 5 is the bead friction q, is the solvent viscosity l is the step length and a is the mode number.The Rouse eigenvalues are and v is an approximation to the diagonal element of the diagonal matrix N obtained from the hydrodynamic interaction matrix H by the transformation where Q is the matrix which diagonalizes matrix HA. [Note that expression (2) for v is obtained from the Kirkwood-Riseman approximation valid under theta conditions.] To order c in concentration our results'? for Gaussian chains display the relaxation frequency in the form A = AE +A? v:( 1 -c[q] 6,). If we simulate the effect of higher concentrations by renormalizing the concentration- dependent term similar to the form used empirically by Schrag4 and derived by one of us in a separate comment5 we have Aa = @+A," v exp (-6 c[q]).For c -*a eqn (6) goes to the simple limit A = A,". However fig. 1 of our paper2 shows that 6 decreases rather rapidly as a function of the mode index a for small a. Hence the second term in eqn (6) gets screened out at lower concentrations for the first mode a = 1 than for the second mode a = 2 etc. Our paper1 shows 6 approaches a constant for higher a and these modes.perhaps may never be entirely screened out. This analysis indicates how the hydrodynamic interactions are screened more slowly over short distant scales. The transition from a Zimm-like viscosity to a Rouse-like viscosity occurs when the concentration is high enough so that about the first three or so modes are screened.Hence we have the Rouse-like viscosity when 6 c[q]$-1. However at this concentration the other modes are slowly becoming screened out. Eqn (5) is obtained provided c is not too large [see ref. (4)] so we cannot rigorously use it to comment on hydrodynamic screening in the melt. Nevertheless it does present the possibility that the shorter-wavelength modes may not be fully screened even in the melt. K. F. Freed and A. Perico Faraday Symp. Chem. SOC.,1983 18 29. * A. Perico and K. F. Freed J. Chem. Phys. to be published. A. Baumgartner K. Kremer and K. Binder Faraday Symp. Chem. SOC.,1983 18 37. C. J. T. Martel T. P. Lodge M. G.Dibbs T. M. Stokich R. L. Sammler C.J. Carriere and J. L. Schrag Faraday Symp. Chem. SOC.,1983 18 173. K. F. Freed Faraday Symp. Chem. SOC.,1983 18 239. Dr A. Baumgartner,Dr K. Kremer and Prof. K. Binder (KFAJiilich,West Germany) said We are grateful for this very interesting comment. It substantiates our feeling that there is no contradiction between our phenomenological assumptions and theoretical approaches treating hydrodynamic screening on a more fundamental theoretical level. Such approaches are very desirable as our phenomenology cannot say anything about concentration dependences of the parameters involved. It would be intriguing to try to extend the experimental work to lower concentrations in order to reach a regime where one might compare the data more directly to these GENERAL DISCUSSION 223 more powerful theories.Unfortunately this requires longer chain lengths as well as smaller scattering vectors and hence would be rather difficult. Prof. M. Doi (Metropolitan University Tokyo Japan) said In presenting my paper I would like to report our latest result which supports the tube model in the system of rod-like polymers. We carried out a computer simulation for the Brownian dynamics of thin rodlike polymers (of b = 0) in the concentrated solutions well above the entangled region (cL3 is increased up to 150). The result is (i) at high concentration (cL3 > 80) the effective rotational diffusion constant D,shows the concentration dependence predicted by eqn (2.12) of the paper; (ii) the numerical factor B is of the order of 500.This result agrees with the results of real experiments of electric birefringence and dynamic light scattering. Thus the validity of the tube model appears to be well established for rod-like polymers. Prof. A. M. Jamieson (Case Western Reserve University Cleveland U.S.A.) said The theoretical work of Doi and Edwards and later modifications offers the exciting possibility of relating viscoelasticity of concentrated solutions of stiff coils to macromolecular structure and prompts a survey of existing rheological data. Unfortunately such comparisons are rendered difficult because of several factors such as polydispersity intermolecular aggregation partial flexibility and thermodynamic non-ideality. In previous work Doi has noted that as c +c** the concentration where the appearance of a liquid-crystal phase occurs one might anticipate a more drastic slowing of rotational diffusion because of finite-rod-width effects and he suggested an equation of the form1 D,= DE c-~L'F-~(c/c**) (1) where F = (1 -Bc/c**)-' (2) with B 2 1.0.This leads to a Newtonian viscosity of the form Unfortunately for high degrees of asymmetry this equation is relatively insensitive to the structural factors discussed above as evident in the experimental study of Chu et a1.' A more useful equation suggested by the latter work,2 is k Trs(c* *)3 L6 where yI** = D; (5) A survey of systems for which data on c** qo and DE can be evaluated indicates two distinct types of behaviour. Data on systems which exhibit only short-range interactions e.g.dispersive forces or strongly screened electrostatic forces fit eqn (4) but with B < 1. Examples include poly(y-benzylglutamate) in dimethyl f~rmamide,~ poly(p-benzamide) in dimethyla~etarnide,~ polyisocyanates in several organic solvents5 and xanthan in 0.1 mol dmP3 NaCl.6 On the other hand data reported by Chu et a1.2 on highly charged polyamide chains in sulphonic acids of low ionic strength fit eqn (4) but with B z 1. Thus it appears GENERAL DISCUSSION the degree of rotational mobility of rods near c** may be strongly dependent on the range of the intermolecular interaction. M. Doi J. Phys. (Paris) 1975 36 '607. S. G. Chu S. Venkatraman G. C. Berry and Y. Einaga Macromolecules 1981 14 939.G. Kiss and R. S. Porter J. Polym. Sci.,Polym. Symp. 1978 65 193. S. P. Papkov V. G. Kulichikhin V. D. Kalmykova and A. Ya. Malkin J. Polym. Sci.,Polym. Phys. Ed. 1974 12 1753. S. M. Aharoni Ferroelectrics 1980 30,227. M. Milas and M. Rinaudo Polym. Bull. 1983 10 271. Prof. L. Monnerie (E.S.P.C.I.,Paris France) said Prof. Doi reported results on concentration dependence of the viscosity of PBLG in a rod-like form. The overall chain conformation of PBLG can be changed from a rod to a coil by a modification of the solvent. (Such an overall change in form originates from a destabilization of the helical structure.) Thus with the same polymer sample it is possible to go from a set of rigid rods to a set of coils and to probe the concentration dependence of the viscosity in these two extreme situations.Could Prof. Doi tell us what the results are and how they fit the corresponding theories? Prof. M. Doi (Metropolitan University Tokyo Japan) replied It is a difficult question to answer. As far as I know an experiment which particularly aims at examining the effect of chain flexibility has not been done. Theoretically the reptation of semiflexible polymers is difficult to analyse because an additional dimensionless constant i.e. the ratio between the segment length and the chain length appears. In the case that reptation occurs in a fixed network a rigorous prediction can be made.l? However for a real solution of semiflexible polymers I do not know the answer because the tube diameter will perhaps depend on the chain flexibility and I do not know how to estimate that.Th. Odijk Macromolecules in press. M. Doi J. Polym. Sci. in press. Dr W. W. Graessley (Exxon Annandale N.J. U.S.A.)said Could Prof. Monnerie please explain the difference between tube relaxation and constraint release (tube renewal) in accounting for matrix effects on the labelled chains? Prof. L. Monnerie (E.S.P.C.I. Paris France) replied The constraint release proces~~-~ is the effect on one chain of the reptation motion of the surrounding chains. It can occur in polymer melts at rest as well as in strained polymers. In this latter case it involves chains which have yet recovered their equilibrium contour length and it follows the length-preserving assumption. On the other hand the tube relaxation proces? that we have introduced is a constraint release on a chain which has been drawn above its equilibrium contour length by an external strain.In this process the length-preserving assumption no more applies. Depending on the relative molecular weights of the considered chain and of the surrounding chains either the retraction or reptation motions of the surrounding chains relax the contour length of the chain under consideration faster. M. Daoud and P. G. de Gennes J. Polym. Sci. Polym. Phys. Ed, 1979,17 1971. J. Klein Macromolecules 1981 14 460. W. Graessley Adv. Polym. Sci. 1982 42 67. GENERAL DISCUSSION 225 Dr M. P. Dare-Edwards (Thornton Research Centre) said Has Prof. Monnerie considered the use of Raman spectroscopy and in particular Raman microprobe techniques in the study of polymer systems under in situ strain conditions? Although there are few polymers for which full polarization information is available valuable qualitative information could be obtained with quantitative interrelationships within any given polymer system.The constraint on extremely thin samples could of course be removed by use of the Raman technique and use of microprobe techniques can allow for spatial selection to avoid or ‘burn-out ’ the fluorescent impurities contained in many bulk polymers. Furthermore through the use of recently developed multi- channel spectrometers time-dependent studies have become feasible down to the microsecond timescale of particular value in studies of both transient and oscillatory strain conditions.Prof. L. Monnerie (E.S.P.C.I. Paris France) replied It is correct that Raman spectroscopy can be used to look at chain orientation1. and to remove the constraint of very thin samples which is required for i.r.d. Raman spectroscopy provides a measurement of not only (Pz(cos 8)) but also (P4(cos 8)). However it requires that the various components of the polarizability tensor of the considered mode should be known. In fact such information is not directly available; it implies that the exact symmetry of the considered mode has been established. This is a strong limitation to the use of Raman spectroscopy to measure orientation in any polymer. Furthermore many bulk polymers exhibit a spurious fluorescence which overlaps Raman lines.B. Jasse and J. L. Koenig J. Macromol. Sci. Part C 1979 17 61. D. I. Bower in Structure and Properties of Oriented Polymers ed. I. M. Ward (Applied Science Publishers London 1975) pp. 187-21 8. Dr H. Killesreiter (Clausthal-Zellerfeld West Germany) said Fluorescence polar- ization measurements are surely one of the best means of studying the orientation of molecules in a polymer. The question arises however as to how quantitative the method can be. In Prof. Monnerie’s paper energy transfer has been excluded referring to a very low concentration of the fluorescent sample. However resonance transfer of energy with subsequent concentration quenching can range up to intermediate distances of > 100 A. Moreover applied strain causes an approach of the fluorescing groups perpendicular to the strain and might thus reveal a two-dimensional fluorescence quenching effect.Have any attempts been made to consider this very complicated problem? Prof. L. Monnerie (E.S.P.C.I. Paris France) replied Indeed if fluorescent mole- cules are too close to each other energy transfer can occur leading to a change in fluorescence polarization and fluorescence intensity. In any case such an energy transfer can occur between molecules > 50 A apart and in all our experiments the concentration of the fluorescent label is ca. 10 ppm corresponding to much larger distances between the labels even after a stretching at A = 6. This point was checked in the initial experiments. Dr. F. BouC (C.E.A.,Gif-sur-Yvette France) said In the section of his paper dealing with the influence of the molecular weight of the labelled chains Prof.Monnerie reported a surprisingly high orientation for ‘chains too short to be oriented efficiently by the deformation of the physical network’. In other words the time corresponding to the inverse of the strain rate i is expected to be much larger than the characteristic maximum time of relaxation of the conformation of the short chains. So these would 8 FAR GENERAL DISCUSSION appear isotropic which is not observed. Prof. Monnerie also reported similar findings for the deformation of pendant chains in a chemically crosslinked network. While they are fixed to the deformed network only by one point and are smaller than the mesh size Mmesh,these chains appear to be oriented.We reported similar results observed by means of neutron scattering. The orientation appears at high q (small distances) whatever the molecular weight Mlabof the labelled chains embedded in the other chains. The form factor in the direction perpendicular or parallel to the elongation axis is much further from the isotropic value than predicted by the theories.132 This occurs (i) at values of q corresponding to lengths less than the distance between two entanglements (Dcc Mi Me being the molecular weight of a part of chains between two entanglements) the effect persisting down to the size of the monomer; (ii) for all times t elapsed after the strain step edge larger than the time of relaxation between two entanglements T,.This (q,t) range corresponds to Prof. Monnerie’s observation of chains of Mlab<Me,of maximum time T < T,,observed at a strain rate such that 1/k > T,.Simi-lar observations have been made in the case of deformed permanent networks2 The orientation of a labelled path (i.e. a labelled chain crosslinked to the network by a large number of its crosslinks) appears again at high q to be more oriented than predicted by the models. In view of that comparison theoretical curves have been plotted for the ‘junction affine’ and the ‘phantom network’ models. High q implies a scale of length less than the size of the mesh down to the scale of the monomer. For the case of the melt see fig. 6 of my paper and for a more detailed discussion F.Boue Long Thesis (Orsay Paris Sud) no. 2662. * Both cases of melts and of rubbers are discussed by F. Bout and J. Bastide J.Phys. (Paris) submitted for publication. Prof. L. Monnerie (E.S.P.C.I.,Paris France) said I now wish to respond to comments made informally by Dr Fernandez Sir Geoffrey Allen and Prof. Silberberg. First Dr Fernandez. A set-up using a rapid-scanning Fourier-transform infrared spectrometer coupled to a hydraulic stretching device which can be used for harmonic deformation experiments has been reported by Prof. Hsu.~ This equipment can only be used with solid polymers or rubber networks. On the other hand it is possible to perform i.r.d. measurements on a chosen vibration band during oscillatory experiments on polymer melts.In such a set-up a cone-plane attachment with an i.r. transparent window is used and the intensity measurements are obtained through lock-in amplifiers monitored by the harmonic deformation. Equipment of this type is presently under construction in various groups. I turn now to Sir Geoffrey Allen. In the case of PS +PDMPO compatible blends we have previously interpreted our results considering the higher stiffness of PDMPO chains. In order to test this interpretation we have recently performed2 similar orientation studies on PS +PVME [poly(vinylmethylether)] blends. Indeed PVME chains can be considered as very flexible chains owing to the low Tgvalue (-30 “C). Surprisingly we have found that at the same (T-T,)value the PVME chains also lead to an increase in orientation of PS chains and that PVME is as efficient as PDMPO.However PVME chains are not oriented at all whereas PDMPO chains were highly oriented. Such a result rules out the previous interpretation on PS+ PDMPO. We now think that the molecular origin of such an effect could be an increase in the friction coefficient due to the specific interactions which exist between chains of compatible polymers. A change in friction coefficient should affect the behaviour GENERAL DISCUSSION 227 of both polymer species but in our experimental conditions [(T-T,) and i values] we are only able to see the effect on PS chain orientation. Indeed the relaxation times of the other chains are either too long (PDMPO) or too short (PVME) relative to the time range which is explored.If this interpretation is correct in the case of PS +PoClS [poly(o-chlorostyrene)] compatible blends the relaxation times of the two types of chains being close we should observe an increase in orientation for both polymer chains. A study on this system is presently carried out in our laboratory. Finally I address myself to Prof. Silberberg. In fluorescence polarization studies the effect of a labelled species on the conformation of adjacent segments is always questionable. For this reason we do not claim the absolute value of the obtained orientation but we look at the effects of other molecular factors (molecular weight of the labelled chain molecular weight of the matrix chains) in such a way that local effects arising from the labelling remain constant in all the investigated systems.In the case of i.r.d. the choice of a vibration band related to the side chains or the main chain is governed by the requirements of a suitable absorbance and an absence of overlapping with other bands. For atactic polystyrene it has been shown3 that similar behaviour (T-dependence effect of draw ratio) is observed using the sym- metrical stretching vibration of the CH group at 2850 cm-l the 1028 cm-l band which corresponds to a transition moment along the C,-C axis of the phenyl ring or the 906 cm-l band the transition moment of which is perpendicular to the phenyl ring. Of course the angle between the transition moment and the local chain axis has to be taken into account.The similarity of the results proves that stretching above Tgdoes not change the position of the phenyl ring relative to the main chain (eventual rotation of the phenyl ring around the C1-C4 axis or distortion of the angle between the C,-Cp and C,-C bonds). With polymers bearing a longer side chain in which conformational changes can occur as in PMMA it is certain that the use of a vibration band corresponding to a transition moment not rigidly connected to the chain backbone can lead to misleading results. D. J. Burchell J. E. Lasch R. J. Farris and S. L. Hsu Polymer 1982 23 965. J. P. Faivre B. Jasse and L. Monnerie to be published. B. Jasse and J. L. Koenig J. Polym. Sci.,Polym. Phys. Ed. 1979 17 799. Dr J. S. Higgins (Imperial College London) said I address my remarks to Dr Bouk.During the discussion several statements have been heard to the effect that ‘the neutron quasielastic scattering results show only a Rouse-like motion for polymer chains in the melt’. These remarks are based largely on results published recently for poly(dimethylsi1oxane) (PDMS) melts.’ The well known high flexibility of this polymer gives rise to fast motion which can be resolved by the neutron-scattering However if the reptation model has any validity there are two relevant molecular parameters the first is the Rouse relaxation time and the second is the tunnel diameter. Even when fast Rouse motion occurs (which may well be the case for PDMS) a large tunnel diameter could preclude observation of entanglement effects within the distance scale of observation of the neutron experiments.This scale can be roughly defined as Q-l [where Q is the momentum transfer = (47z/L) sin (0/2)]. Graessley4 once calculated the tunnel diameter for PDMS from viscoelastic data as 50 A. In order to investigate these effects we have performed a series of coherent quasi- elastic neutron experiments on poly(tetrahydr0furan) (PTHF) melts and obtained some interesting result^.^ PTHF may be close to polyethylene which has a high plateau modulus and therefore a small tunnel diameter of ca. 30 A. Four samples were investigated a perdeutero and a hydrogenous version of each of two molecular-weight fractions. These were M = 2300 (M << Me) and M = 45000 (M % Me). The 8-2 228 GENERAL DISCUSSION 0 -0.1 -h k 03 -0.2 z c --0.3 Fig.11.Dynamic coherent structure factor S,,,(Q t)plotted logarithmically against time for two poly(tetrahydr0furan) melts at 145 "C:0,high-M chains in a low-M matrix; x ,high-M chains in a high-M matrix. The dashed curve is the Rouse model with QR = 5.6 x lo7s-l. The solid line is the de Gennes local Rouse model with QR = 5.6 x lo7s-l; D = 30 A; Tep= lop3s. A = 5.3 x 10-11 s. observed molecules were the hydrogenous high-molecular-weight samples. They were mixed respectively in the low-molecular-weight (unentangled chains) and the high- molecular-weight (entangled chains) deuterated species. Fig. 1 1 compares the coherent dynamic structure factor Scoh(Q,t) for the two samples at Q = 0.09 A-l.Similar results were obtained at other Q values. While at short times the behaviour is very similar for the two samples the entangled molecules show a much slower long-time decay than the unentangled ones. The dashed curve shows the best fit of the Rouse model to the Sco,(Q t) for the unentangled molecules. It is not possible to fit any Rouse model through the whole set of data ( x ) for the entangled molecules even by adjusting the relaxation time. The solid line represents the best fit of de Gennes' modified Rouse or local reptation6 model. The parameters used are a Rouse relaxation time taken from the unentangled curve R = 5.6 x lo7s-l a tunnel diameter of 30 I$ and a reptation time Tep= s. The major aim of these experiments is not however an over-emphasis on the best fit of a particular model to the results.We wish to indicate that the neutron experiments do observe for this system effects which it seems can only be attributed to chain entanglements. D. Richter A. Baumgartner K. Binder B. Ewen and J. B. Hayter Phys. Rev. Lett. 1981 47 109; 1982 48 1695. J. S. Higgins R. E. Ghosh W. S. Howells and G. Allen J. Chem. SOC.,Faraday Trans. 2 1977 73 40. G. Allen J. S. Higgins A. Maconnachie and R. E. Ghosh J. Chem. SOC.,Faraday Trans. 2,1982,78 21 17. W. Graessley J. Polym. Sci.,Polym. Phys. Ed. 1980 18 27. J. S. Higgins and J. E. Roots J. Chem. SOC.,Faraday Trans. 2 to be published. P. G. de Gennes J. Phys. (Paris) 1980 42 735. GENERAL DISCUSSION I I I I time + Fig.12. Radius of gyration R of PDMS molecules (M = 10000) between cross-link points in a network deformed cyclically at 2.8 Hz with a maximum strain 1.5 (0,parallel; M perpendicular to deformation). Dr A. R. Rennie (University of Mainz West Germany) said The technique of ' real-time ' small-angle neutron scattering from deformed polymers described in the papers by Monneriel and Boue et a1.2has recently been extended to higher strain rates. The method developed in Mainz and Grenoble3 is that of periodic time-resolved scattering. A sample is cyclically deformed and separate spectra are recorded for several different phases of strain. The total neutron counts from many cycles are averaged to achieve sufficiently accurate data. This is a very promising technique for molecular studies on materials that can be cyclically deformed at frequencies up to 10 Hz (possible strain rates in excess of 10 SKI).First experiments have been conducted in collaboration with the CRM Strasbourg on model PDMS networks and as an example of the results of such experiments fig. 12 shows the radius of gyration of molecules between the cross-link points in directions parallel and perpendicular to the strain measured at 2.8 Hz. The significance will be described elsewhere. L. Monnerie Faraday Symp. Chem. Soc. 1983 18 57. F. Boue M. Nierlich and K. Osaki Faraday Symp. Chem. SOC.,1983 18 83. A. R. Rennie and R. C.Oberthur to be published. Dr J. L. Viovy (E.S.P.C.I.,Paris France) said Dr Boue has reported experiments suggesting that after a time comparable to its Rouse time a labelled chain is less relaxed when it is embedded in a melt of longer chains than in an homopolymer melt.We have made similar observations using fluorescence po1arisation.l In that latter case we were able to account for the variation of the orientation of a labelled chain as a function of MW2of the surrounding chains,2 using the recently introduced tube relaxation process. Does Dr Boue think that neutron-scattering experiments are sufficiently precise to check quantitatively some predictions of this model? The case of a long chain in a melt of shorter chains would also be interesting to study by neutron scattering since the tube relaxation model predicts in that case a partial relaxation of the N chain when the surrounding P-chains retract.J. F. Tassin and L. Monnerie J. Polym. Sci. Polym. Phys. Ed. 1983 21 1981. J. L. Viovy L. Monnerie and J. F. Tassin J. Polym. Sci. Polym. Phys. Ed. 1983 21 2427. GENERAL DISCUSSION Dr F. Boue‘(C.E.A.,Gif-sur-Yvette France) said All of the comments made are very interesting and correct. Dr Higgins pointed out an improvement in the inelastic observation of long-chain dynamics this is to use a polymer with a ‘tube diameter’ as small as possible such as PTHF. This seems to allow observation of entanglement effects while still not being in the time range necessary to test terminal behaviour such as disengagement (cer s compared with s). The experiment of Dr Rennie = is as attractive; it is in fact an alternative to our experiment using essentially another deformation history sinusoidal instead of a step strain.One needs to decorrelate the results from that story which is not necessary for transient relaxation (after step strain). It may thus be less simple to test theoretical predictions or to see small effects. Finally the comment of Dr Viovy touches on some of our present developments. Experiments are planned where the same labelled chain will be observed inside matrices of different molecular weights during relaxation. Dr R. A. Pethrick (University of Strathclyde Scotland) said During the course of these discussions it has become clear that the analysis of data does not correctly reflect the molecular dynamics of a polymer melt. In our paper presented later in the Symposium we present data on the viscoelastic relaxation of poly(dimethylsi1oxane) melts.Fig. 9 of our paper describes the relaxation response of the entangled polymer system. The plateau region at ca. lo4 Hz corresponds to the limiting high-frequency condition of the Doi-Edwards model. The response between lo4 and lo8 Hz is associated with motion of chains between entanglements segmented motion occurs above lo9Hz. Neutron-scattering experiments will probe a region between lo8 and lo7Hz dependent upon the q-range and spectrometer. Thus it is not very surprising that dynamic experiments can be analysed in terms of rapid segmental motion and a Rouse-like motion. A detailed analysis of the viscoelastic data indicates that the distribution is not exactly a Rouse spectrum.The interpretation of the deviation is not simple and probably reflects the topography of the melt. In most polymers separation between the Rouse and segmental motions may not occur and hence the scattering function should reflect this fact. Dr H. Killesreiter (Clausthal-Zellerfeld West Germany) said We are concerned with viscosity measurements on oils with mean molecular weights of 302 422 and 602 g mol-l.lv In this case a rather linear relationship holds for the above compounds between their viscosity and molecular weight. On the other hand Kuss3 carried out a series of experiments on pure substances with comparable molecular weights. He found that the viscosity is mainly influenced by mutual steric effects of the molecules.Looking at the numerous and convincing results for high-molecular-weight mole- cules in fig. 1 of Dr Rahalkar’s paper there arises the question as to whether the influence of the molecular structure of a molecule can be screened totally by molecular entanglement thus forming balls and/or tubes? H. Killesreiter Erdol Kohle Erdgas Petrochem. 1982 35 428. H. Killesreiter K-D. Klie H-J. Neumann and D. Severin Erdol Kohle Erdgas Petrochem. 1983,36 71. E. Kuss Ber. Bunsenges. Phys. Chem. 1983 87,33. Dr R. R. Rahalkar (Food Research Institute Norwich) replied The only major comment offered regarding this paper was on the possible effect of molecular structure upon viscosity. An important aspect of structure which can affect the viscosity is branching. The polymer used during the present work polydimethylsiloxane is GENERAL DISCUSSION 23 1 essentially a linear polymer and hence effects due to branching could be neglected.The Doi-Edwards theory in its present form does not take branching into account. Thus branching was not an important factor during the theoretical or experimental aspects of the present work. There is however another structural aspect which influences viscosity and which was very crucial for this work viz. polydispersity. It was observed that for the same value of M a greater value of M,/M results in higher viscosity. Referring to fig. 1 in the paper samples S1-S5 are more polydisperse than samples S5-S9 and are also more viscous. In any attempts theoretically to derive viscosity as a function of molecular weight it is necessary to take into account polydispersity.Also in experimental measurements of the viscosity-molecular weight relationship it is essential that all the samples plotted on the same graph should have the same degree of polydispersity. The same holds true for other reptation parameters in general. Some parameters (e.g. plateau modulus) change very little with polydispersity while others (e.g. the longest relaxation time the recoverable compliance etc.) are strongly dependent upon molecular mass distribution. As an example the quantity G&JZ can be expressed as where G; is the terminal-region plateau modulus is the recoverable compliance and F is the molecular mass distribution dependent factor. For a monodisperse poly- mer F = 1 and the expression reduces to that for the monodisperse case.l However for the most probable molecular mass distribution (M,/M = 2) the factor F turns out to be 20.This illustrates the influence of polydispersity upon the rheological behaviour in general and upon the recoverable compliance in particular. Molecular structure also affects the viscosity to the extent that it may have an effect upon the conformation of the molecule. The Doi-Edwards theory as presented here assumes the polymer to be a flexible random coil. There may be structural factors which render a rod-like or a stiff coil conformation to the molecule e.g. the presence of bulky side groups the presence of polar groups capable of intermolecular interactions etc. If any of these factors are influential the polymer is not a flexible random coil and hence Doi and Edwards theory cannot be applied.The dependence of viscosity upon molecular mass may be totally different depending upon the molecular conformation (for some rodlike systems viscosity increases as the 6th power of molecular mass as compared with the 3.5th power for random-coil polymers). Thus viscosity very much depends upon molecular structure and even following our present work (taking into account the polydispersity) assumptions made in the Doi-Edwards theory place restrictions upon the type of system to which it may be applied. W. W.Graessley J. Polym. Sci.,Polym. Phys. Ed. 1980 18 27. Prof. A. M. Jamieson (Case Western Reserve University Cleveland U.S.A .) said I address my remarks to Dr Ross-Murphy.(1) The recent work on xanthan solutions summarized in this paper beautifully demonstrates the important role played by specific cations in the self-association behaviour of xanthan in aqueous salt solutions. There can be no doubt that such aggregation exists and as recognized in our earlier work this property makes the xanthan+water system a less than ideal model for solutions of rod-like polymers. More definitive results may be expected with the availability of homogeneous xanthan samples. For example the enzyme-hydrolysed xanthan (M = 235000) prepared by GENERAL DISCUSSION Milas and Rinaudol exhibits clearly the classical viscometric properties anticipated for concentrated solutions of thin rods uiz.the shear viscosity q increases as c3.0up to a concentration c** where ordered domains are formed and there is a sudden decrease in shear viscosity. Likewise xanthan subjected to thermal renaturation in 4 mol dm-3 urea as described by Ross-Murphy et al. might be expected to show similar behaviour. Has this been observed for xanthan in 4moldm-3 urea? If not is it possible that the thermal renaturation may produce a molecule more flexible than the original ‘native’ species (e.g. because of point defects) even though the ‘degree of conformational order ’ as measured by optical rotation is indistinguishable? (2) It does not seem to me that the viscoelastic studies reported in this paper definitively establish that entanglements are not responsible for the ‘weak-gel’ behaviour of xanthan in aqueous salt solutions.Clearly the data for G’ and G” (fig. 7 of the paper) indicate that for a 0.5% w/v xanthan in 0.02 mol dmP3 KCl the behaviour is predominantly elastic in the frequency range studied (w > 10-1 rad s-l). However as these authors note (p. 127 paragraph l) for their system c[q]z 15 and thus may be expected to conform approximately to the situation seen in our experiments at c = 0.14% w/v since [q]w lo4cm3 g-l M = 2.2 x lo6 in our sample. Our S(q t) data indicate (fig. 2 of our paper) only a partial relaxation of the xanthan solution ‘structure’ at times of the order 10-l s. Estimating the free-particle rotational diffusion coefficient D;for the ‘native’ sample of xanthan used by Ross-Murphy et al.to be 0,“ w 20 s-l which would seem consistent with the reported [q],M and TEB data and with the proposed rod-like structure L x d w 1100 x 25 nm and assuming D,= /?D;(CL3)-2M D;(O.~/C[~])~ corresponding to p w lo3 we deduce D,w s-l for c[q]z 15. For a rod-like polymer it is not unreasonable that comparatively little energy will be dissipated for w > D,. The ‘weak-gel’ response could reflect the fact that the system is simply more congested at c = 0.5% (w/w) in the absence of urea. (3) It seems well established that for concentrated polymer solutions of high molecular weight,2 a breakdown of the Cox-Merz rule referred to by the authors is observed such that Iq*(u)l,+ S-q(v). This arises from the appearance in q(j) of non-linear viscoelastic effects at low shear rates in these highly entangled systems.Thus the Cox-Merz rule cannot be used as a criterion to distinguish ‘weak-gel’ effects from ‘entanglements ’. ’ M. Mulas and M. Rmaudo Polym. Bull. 1983 10 271. * K. Osaki M. Fukuda S. Ohta B. S. Kim and M. Kurata J. Polym. Sci. Polym. Phys. Ed. 1975 13 1577. Dr S. B. Ross-Murphy (Unileuer Colworth House Bedford) replied The essential difference between the urea-treated samples and the native xanthan illustrated in fig. 7 of our paper is that under these conditions the viscoelastic properties are grossly affected by urea treatment although measurements of intrinsic viscosity show little change. This confirms that intermolecular interactions other than entanglements have a major effect.It is still possible that the urea-treated samples would show the classical maximum in zero-shear viscosity qo just before the transition from isotropic to anisotropic behaviour but we have not investigated this. Such behaviour would of course occur at much higher concentrations than reported here although the viscosity transition is not always observed even when expected because of the tendency to aggregate. Further studies of the urea-treated system would be valuable although light-scattering measurementsare complicated by the number of components present. The rigid-rod equations applied by Jamieson and coworkers have only limited GENERAL DISCUSSION applicability in view of the overall wormlike character of xanthan (see comment by Prof.Burchard). The deviations from the Cox-Merz rule discussed by Osaki et al. are in the opposite direction to those discussed by us and occur with an roplateau. The rheological properties observed for xanthan are much more akin to the 'yield stress' (Bingham fluid) properties of concentrated colloidal dispersions. Similarity with colloidal systems is also apparent in the very pronounced strain dependence of r* (co,~). Despite this all measurements were performed in the linear viscoelastic region say y < 0.1. Prof. W. Burchard (University of Freiburg West Germany) said Xanthan is sometimes considered to be a rigid rod although this cannot still be true for very large degrees of polymerization. Thus the question is how stiff is xanthan? We have two ways of determining the persistence length of the Kuhn statistical segments i.e.(1) from the molecular-weight dependence of the mean square radius of gyration (S2)z and (ii) from the shape of the particle scattering factor. The mean-square radius of gyration of monodisperse semiflexible chains is given by' (s2)= 1i{Nk/6- 1/4+ 1/4Nk-(1/8N;)[l -exp (-?-Nk)]) where the Kuhn length lk (equal to 2q with q the persistence length) and the number of Kuhn segments Nk is related to the contour length L = Nklk = nl where n is the number of repeat units in the chain and I = 0.98 nm the length of these units. Thus only 1 remains as a fitting parameter. Fig. 13 shows measurements of (s2)iby Paradossi and Brant.2 The dotted line corresponds to monodisperse chains with ik = 200 nm; the two full lines represent the behaviour of polydisperse chains (M,/M, = 2) with lk = 200 and 150 nm respectively.A fit with monodisperse chains is not possible since even the fully stretched xanthan chain has a lower radius of gyration than was measured. NK 0 5 10 15 20 M,/ 1Os dalton Fig. 13. Measurements of (S2)!;from ref. (2) for three values of I in nm. GENERAL DISCUSSION 3000 -i 2000 c -Y ‘D 1 h k 1 0 4 s 1000 0 1 2 3 4 5 6 q (S?! Fig. 14. Particle scattering factor measured by Paradossi and Brant2 for (a)native xanthan and (b)pyruvate-free xanthan. The particle scattering factor for semiflexible chains has been calculated by K~yama.~ The shape of the curves as a function of q changes from that of a rigid rod to that of a flexible Gaussian coil as Nk is increased.Fig. 14 shows some measurements by Paradossi and Brant2 for native xanthan and one example of our results from a pyruvic free xanthan. Polydispersity M,/M = 2 has again been assumed in the calculations. The figure by the curves indicates the number of Kuhn segments. Evaluation of these curves reveals two fact^:^^^ (i) the Kuhn length 1 in both types of xanthan is 200 A20 nm but (ii) native xanthan is a double-strand chain whereas the pyruvic free chain consists of only one strand. These results are cross-checked by comparing the radii of gyration; that for the modified xanthan chain (A in fig. 13) falls on the curve for the native xanthans when corrected for double strands (B in fig.13). In summary we find that xanthan is one of the stiffest chains known (the Kuhn length is ca. ten times larger than for cellulose derivatives) but it is not a rod. Chain stiffness is not changed when the pyruvic group is removed but the double strand of the native xanthan dissociates into single chains. H. Benoit and P. Doty J. Phys. Chem. 1953 57 958. G. Paradossi and D. A. Brant Macromolecules 1982 15 874. R. Koyama J. Phys. Soc. Jpn 1973,-84 1029. M. Schmidt G. Paradossi and W. Burchard in preparation. T. Covielio M. Dentini V. Crescenzi K. Kajiwara and W. Burchard in preparation. Dr E. R. Morris (National Institute for Medical Research Mill Hill London) and Mr R. K. Richardson (Unilever Research Colworth Laboratory Bedford) said In discussing the solution properties of xanthan it is important to remember that on heating the molecule ~ndergoes~-~ a cooperative order-disorder transition from the highly persistent structure described by Ross-Murphy et aL4 and Jamieson et al.,5 to a disordered ‘random-coil ’ conformation similar to that of other water-soluble GENERAL DISCUSSION a" A E --a * a c a -1 o2 a a A cellulose derivatives.The mid-point temperature (T,) of this conformational transition increases with ionic strength in accordance with polyelectrolyte theory.6 For solutions of xanthan in deionised water ionic strength is determined by the counterions to the uronate and pyruvate ketal groups of the polymer and thus decreases with decreasing polymer concentration.In sufficiently dilute solution T may fall below ambient temperature so that the disordered form of the polymer is adopted. We suggest that this behaviour explains the apparent discrepancy between the high persistence length rep~rted~.~ in two papers at this Symposium and the typical 'random-coil' properties in very dilute solution mentioned in an unrecorded comment by Prof. G. B. Thurston. On addition of simple electrolyte (KCl) to induce the ordered conformation however we have observed typical 'weak-gel' mechanical spectra at polymer concentrations comparable to those studied'~ by Prof. Thurston with no indication of a Newtonian plateau in the frequency (w) dependence of q* (fig. 15). E. R. Morris D.A. Rees G. Young M. D. Walkinshaw and A. Darke J. Mol. Biol. 1977,110 1. * G.Holzwarth Biochemistry 1976,15 4333. M. Milas and M. Rinaudo Carbohydr. Res. 1979,76 189. S.B. Ross-Murphy V. J. Morris and E. R. Morris Faraday Symp. Chem. SOC. 1983,18 115. A. M. Jamieson J. G. Southwick and J. Blackwell Faraday Symp. Chem. SOC. 1983,18 131. I. T. Norton D. M. Goodall S. A. Frangou E. R. Morris and D. A. Rees J. Mol. Biol. in press. ' G. B. Thurston J. Non-Newtonian Fluid Mech. 1981,9,57. * G.B.Thurston and G. A. Pope J. Non-Newtonian Fluid Mech. 1981,9,69. Dr W. W. Graessley (Exxon Annandale N.J. U.S.A.) (communicated) Model systems have been extremely helpful in understanding flexible chain liquids. Flexible polymers with nearly monodisperse distributions and relatively weak intermolecular forces are available for wide ranges of chain lengths and architecture.As a result the effects of large-scale structure and concentration can be studied in relative isolation from complicating factors such as polydispersity and molecular association. Does Prof. Jamieson see any hope for a similar approach in rod-like polymers? The hard problem seems to be to eliminate association. Is there something inherent in a rod-like structure that makes association so pervasive or could it be eliminated by an appropriate synthesis? GENERAL DISCUSSION Dr S. B.Ross-Murphy (Unilever Colworth House Bedford) said In an unrecorded remark Dr Klein has suggested that some biopolymers are suitable as model rigid-rod polymers for example in studies of lyotropic liquid crystals.As has already been pointed out although for example DNA and xanthan have very large persistence lengths q z 100 nm the native samples are often of such high molecular mass M, that the contour length L is appreciably larger than q. For example the mass per unit length M,/L for double-stranded DNA is ca. 1950 nm-l,I so that for L z q M = 2 x lo5. Since native DNA typically has 2 x lo6 < M < 12 x lo6,only fraction- ated samples of DNA are rod-like and difficult fractionation procedures would be required to prepare significant amounts of sample. Hagerman2 has pointed out the uncertainties inherent in most studies of DNA stiffness viz. the size heterogeneity (especially in sheared samples) the presence of single-stranded DNA and the relative contribution of excluded-volume effects and local-chain stiffness (especially at low ionic strength).He notes however that recombinant DNA technology may now be capable of producing monodisperse samples with M up to ca. 4 x lo6.For the present however this may explain why experimental studies of more concentrated solutions of rigid rod polymers have largely been restricted to synthetic polymer^.^^ Z. Kam N. Borochov and H. Eisenberg Biopolymers 1981 20 2671. P. J. Hagerman Biopolymers 1981 20 1503. S-G.Chu S. Venkatraman G. C. Berry and Y.Einaga Macromolecules 1981 14 939. C.C. Lee S-G.Chu and G. C. Berry J. Polym. Sci. Polym. Phys. Ed. 1983 21 1573. Prof. M. Doi (Metropolitan University Tokyo Japan) said The idea of using the new order parameters Pk;l or Pk,j is very interesting.The scalar order parameter Pk which has been used in the earlier theory for the interpretation of the dynamic light-scattering data is insensitive to reptation while Pk; will reflect reptation directly. I imagine that the characteristic time in the time correlation function of the order parameter of pk;l is the reptation time. Is that correct? One wonders if Sir Sam Edwards could tell us more of this new approach. Dr P. Thirion (E.S.P.C.I.,Paris France) said The stress relaxation kinetics of an entangled star polymer derived by Drs Pearson and Helfand from a quadratic potential field [eqn (9)-( lo)] is quite different from the logarithmic relaxation directly deduced by de Gennes from the probability of the end chain retracing its steps.What prediction can be made in this respect for the mechanisms considered by Drs Klein Fletcher and Fetters to calculate the diffusion coefficient of the star centre of mass? Dr J. Klein (Cavendish Laboratory Cambridge) said The de Gennes picture] for stress relaxation of a long dangling-end in a rubber network for example is based on a mechanism similar to that in fig. 1 in our paper. Since the dangling end is connected at its ‘root’ to aJixedcrosslink point this seems the correct picture to use i.e. there can be no ‘centre-monomer’ motion of the type envisaged in our fig. 2. Then our treatment predicts only a pre-exponential dependence which is different from that of de Gennes.Thus for a length n(t) (entanglement lengths say) of the dangling end to have relaxed over a time t following application of constant strain e (at t = 0) we might expect [see e.g. expression for zS1eqn (3’) in our paper] GENERAL DISCUSSION where z is a microscopic time and a a constant of order unity. 1.e. lnn(t) 1 n(t>+--a = -In(:). a This is only slighily different [in the lnn(t) term] from the de Gennes picture,l but might provide an improved expression at shorter times [at long times n(t) becomes large and the lnn(t) term becomes less important]. If we assume’ the stress at time t as a( t) = e[E,-kn(t)] where E is an elastic modulus and k a constant proportional to dangling-end concentration we have the stress varying slightly differently from logarithmically especially at short times with n(t) given by the solution to eqn (1).P. G. de Gennes Scaling Concepts in Polymer Physics (Cornell University Press Ithaca N.Y. 1978). Dr P. Richmond (A.R.C. Norwich) said Towards the end of his presentation I understood Dr Klein to say that the two curves in fig. 6 of his paper should tend to the same asymptotic limit as N-0. I do not see that this follows from the solution obtained i.e. D cc exp (-aN)/N and would welcome clarification on this point from the authors. As a supplementary if one of the arms of the star is of different molecular weight say N’ then the diffusion coefficient of the star D*= D(N,N’) and as N’ -+ 0 D* +D for a linear chain. Have the authors considered how this limiting transition takes place? Furthermore could one conceive of extending the experiments to more complex branched molecules which are embedded in a linear chain matrix such as those illustrated below where the branches may be varied in a systematic manner so changing the diameter of the cylindrical molecule or radius of the spherical molecule? Dr J.Klein (Cavendish Laboratory Cambridge) said Both the accepted expression for the reptation of linear chains Dlin cc 1/N2 and our solution for entangled star-polymers D,cc 1/N exp (-aN) are obtained under conditions where entangled behaviour is assumed i.e. N > N, the entanglement length. For sufficiently low N (4N,) neither stars nor linear molecules are expected to exhibit entangled behaviour in the high-molecular-weight linear melt matrix (HDPE1) in our experiments; i.e.their size will be less than a ‘tube diameter’ or ‘interentanglement spacing’ in terms of current models.For such low N values the mobility of diffusants will probably be determined by Stokes-Einstein-type friction (typically cc N) rather than by en- tanglement constraints; for a given total N (4N,) therefore both linear and star- branched species should have a similar mobility. It is true however that over the molecular-weight range for the linear molecules (fig. 5 our paper) this low-N extreme does not appear to be attained as D cc 1/N2 for all the data points. Concerning the supplementary point we have extended the treatment in our paper to cover T-shaped molecules where the ‘leg’ of the T is of variable length (from a 238 GENERAL DISCUSSION symmetrical 3-arm star down to zero -a linear molecule).Experiments to measure the effect of a variable third arm are also in preparation. In principle the extension of the experiments to molecules such as you propose can be also carried out using the i.r. microdensitometry technique. The difficulty would be in synthesising such molecules with a narrow molecular-weight distribution. Dr A. R. Rennie (University of Mainz West Germany) said :Is there an explanation for the large difference in activation energies for linear and star polymers observed by Klein et a1.l From the results of von Meerwall et aZ.2 the opposite effect might be expected. In the n.m.r. experiments the activation energy was found to decrease as the concentration of chain-ends increases.There 'free-volume'effects seemed roughly in accord with the temperature coefficient of the activation volume for diffusion in polymers found by Rennie and Tabor.3 Some other more significant process must be playing a role in the motion of the star polymers. There would be considerable elucidation of the discrepancy between experiment and theory shown in fig. 6 of the paper by Klein et aZ. if data were available for diffusion of star molecules in a matrix of star molecules. Are there any results for such a system? J. Klein D. Fletcher and L. J. Fetters Furuday Symp. Chem. SOC. 1983 18 159. E. von Meerwall J. Grigsby D. Tomich and R. van Antwerp J. Polym. Sci.,Polym. Php. Ed. 1982 20 1037.3 A. R. Rennie and D. Tabor Proc. R. SOC. London Ser. A 1982 383 1. Dr J. Klein (Cavendish Laboratory Cambridge) said :The activation energy for diffusion for both stars and linear molecules in our studies are for diffusion in the same high-molecular-weight linear-melt matrix. Since the concentrations of both types of diffusants in our experiments are very low (ca. 3 %),the concentration of chain ends remains essentially constant on going from linear to three-arm stars in our system. As noted in our paper an increase in activation energy on going from linear to star-branched melts was also observed for the case of viscous flow and the explanation proposed by Graessley [ref. (23) our paper] to account for this may also apply to diffusion. The effect of 'tube renewal' on stars diffusing in a star-melt should be very weak and we are currently preparing experiments on such systems to check this directly.Dr W. W. Graessley (Exxon Annandale N.J. U.S.A.)added The observed differ- ences of temperature coefficients for diffusion in linear and star hydrogenated polybutadiene (polyethylene) parallels differences for the viscosity [(Ev)Land in activation-energy terms]. There we know that (~5~)~ is 30 kJ and contrary to any free-volume effects the difference between linear and star grows in proportion to arm molecular weight. For three-arm stars1 (E,)B =30 +(2.2 x10-4 MB). (1) The suggestion has been made that this difference is caused by two factors,2 the contracted conformation of the transition state for stars in general according to the am-retraction mechanism (papers by Drs Klein and Pearson) and the higher energy of such conformations for hydrogenated polybutadiene in particular as inferred from relatively large and negative temperature coefficient of chain dimensions for polyethylene K =d In (R2)/dT x-1.1 x OC-l.Thus the transition state for polyethylene stars has higher energy which adds a term proportional to arm length to the overall activation energy for flow. In species such as polybutadiene itself where GENERAL DISCUSSION 239 K is probably small the difference between (Ev)Land (EV)Bis also small. The same considerations should apply to diffusion. We are now measuring self-diffusion coefficients for hydrogenated p~lybutadiene.~ We find (ED)L= 26 kJ suggesting AED = (ED)B-(ED)Lz 12 kJ for the sample JK-6A (MB= 9300/3 = 3100) of Klein et al.Eqn (1) predicts a smaller increase for the viscosity AEv z 7 kJ perhaps reflecting a larger influence of tube renewal effects on that property. W. W. Graessley and V. R. Raju J. Polym. Sci. Symp. in press; see also J. Polym. Sci. Polym. Phys. Ed. 1979 17 1223. W. W. Graessley Macromolecules 1982 15 1164. C. R. Bartels W. W. Graessley and B. Crist J. Polym. Sci. Polym. Lett. 1983 21 495 and work in progress. Prof. K. F. Freed (University of Chicago U.S.A.)said The empirical eqn (10) of Prof. Schrag's paper,' namely z,(c) = 7; exp (ACP-~) (1) can be derived by combining effective medium hydrodynamic concepts with scaling arguments in the manner that Adler and Freed2 used previously to derive the empirical Martin equation (v -vo)/cvo = [vl exp (kH 4vI) (2) where c is the polymer concentration qo is the solvent viscosity [q]is the intrinsic viscosity and k is the Huggins coefficient.Let us first consider the derivation of eqn (1) for a solution of Gaussian chains. Then scaling arguments similar to eqn (3.1) and (3.3) of Adler and Freed2 indicate that the concentration-dependent relaxation time for the pth mode must scale as where 7 is the Rouse relaxation rate at infinite dilution f is an unknown scaling function to be determined and h is the draining parameter given by (h = [/vol)n1/2 where [ is the bead friction coefficient I is the effective step length and n is the number of effective units.Schrag and coworkers have been quite successful in simulating the concentration- dependent behaviour of polymer solutions by empirically introducing into the single-chain Rouse-Zimm equations a draining parameter h(c) which varies with concentration. A similar type of idea emerges from our hydrodynamic theory. The theory shows that the screening of hydrodynamic interactions produces an effective concentration-dependent Oseen tensor and we can associate it with an effective draining strength h,(c) that differs in general for the various Rouse modesp. Hence we expect that the concentration-dependent relaxation times should be expressible in terms of the single scaling variable h,(c) as z,(c) = 7:f'[h,(c>l (4) withf again to be determined.Comparing eqn (4) with eqn (3) implies that the only possibility is to have h,(c) = hg,(c[vl) (5) where g is a function that is determined below. A specific form for the function ofg may be obtained by invoking effective-medium arguments2 similar in spirit to those previously employed by Bri~~kman.~ We can GENERAL DISCUSSION regard h,(c) as specifying the draining strength for the pth mode of a single chain in the effective produced by a solvent plus all the other chains in the solution. The addition of more polymers with concentration 6c to the solution alters the draining parameter h,(c) to h,(c) g,(dc[q])in order to account for the hydrodynamic screening of the 6c added chains in the 'effective' solvent containing the chains at concentration c and an effective pth-mode draining constant h,(c).However this new solution is just one having a total polymer concentration of C = c+6c with the draining parameter h,(C). Consequently the draining parameter calculated in both these ways must be the same h,(C) = h,(c +64= h,(4 g,(6c[vl). (6) Combining eqn (5) for h,(c) with eqn (6) yields Eqn (7) is well known in statistical mechanics' as relating the equilibrium statistics between two weakly coupled systems. For continuous arguments eqn (7) has the unique solution g,(c[rl) = exp (-a,c[rl) (8) with a to be determined below. Hence eqn (4) reduces to the simple form of qc) = TEflCh exp (-a c[rI>>. (9) The functionf can be deduced with the aid of simple boundary conditions for the c +0 limit.In the long-chain or large-draining-parameter limit the non-draining limit ensues where the individual bead friction coefficient must become irrelevant. Since zp is proportional to [ it implies that when the argument off becomes small it behaves as f,(X) +AX-1 as X+ co (10) with tgA/h the non-draining limit relaxation time T:". Hence so long as the concentration is not too high and we can safely take h exp (-a c[q])9 1 then the boundary condition (10) can be applied to eqn (9) producing the form qc) = 7:" exp (apc[471) h exp (-a,44)9 1 (1 1) whose range of validity is even prescribed. The small-concentration limit enables us to expand the exponential in c and thereby identify the parameter a from the theory of the leading order in contribution to 7,.Note that the power-law dependence of [q]on molecular weight M implies that eqn (1 1) is not just a power-law dependence on c and M as in ordinary scaling treatments. The effective-medium methods provide a highly non-trivial addition. This completes the derivation in theta solvents but we must consider the case of good- and intermediate-quality solvents. Now the second virial coefficient A becomes a relevant parameter describing the interaction between polymers. The scaling function in eqn (3) can depend on the additional scaling variable cA,M or equivalently the scaling variable A M/[q]with M the molecular weight. The function g in eqn (5)can likewise depend on A,M/[q].The application of the effective-medium ideas in eqn (5)-(7) still leads to eqn (8) where the coefficient a can depend on the A M/[r].However the final result is still eqn (1 1).Using the theory for small c to determine a returns us to the empirical eqn (1) with A cc [q]. This type of analysis emphasizes the importance of the effective-medium concept GENERAL DISCUSSION 241 and ideas of hydrodynamic screening. I am gratified that Muthukumar has extended our original leading concentration calculation8 to higher concentrations by using the effective-medium equations that Edwards and I had originally deri~ed.~-~ His formula1 agrees well with the experimental data’ to concentrations even higher than the simple result eqn (1). Our derivation in eqn (1 1) exhibits a high concentration cut-off of the form of eqn (1) after which blob-type scaling arguments may be used.2 A full theory like Muthukumar’s of course is necessary in the end.Both treatments have the same hydrodynamic concepts in common; they are just pursued with different degrees of detail. C. J. T. Martel T. P. Lodge M. G. Dibbs T. M. Stokich R. L. Sammler C.J. Carriere and J. L. Schrag Faraday Symp. Chem. Suc. 1983 18 173. R. S. Adler and K. F. Freed J. Chem. Phys. 1980 72 4186. H. C. Brinkman J. Chem. Phys. 1952. 20 571. K. F. Freed and S. F. Edwards J. Chem. Phys. 1974,61 3626. K. F. Freed in Progress in Liquid Physics ed. C. A. Croxton (Wiley New York 1978). ti K. F. Freed and H. Metiu J. Chem. Phys. 1978 68 4604. ’ F. Reif Fundamentals of Statistical and Thermal Physics (McGraw-Hill New York 1965).M. Muthukumar and K. F. Freed Macromolecules 1978 11 843. K. F. Freed and A. Perico Faraday Symp. Chem. Sue. 1983 18 29. Prof. J. L. Schrag (University of Wisconsin Madison U.S.A.)said This derivation predicting an exponential concentration dependence of z is astonishingly straight- forward. It appears to provide welcome additional insight regarding our parameter A as well. Prof. K. F. Freed (Uniuersity of Chicago U.S.A.)said The paper by Perico and myselfl considers the leading concentration dependent relaxation times for the simplest case of Gaussian chains. The theory exhibits the non-draining-limit concentration-dependen t relaxation times z in the simple scaling form zp = 1 + c[q]6 + 0(c2)) where z; is the relaxation time for the pth mode at infinite dilution c is the concentration of polymers and [q]is the intrinsic viscosity.This implies that the A parameter2 in Prof. Schrag’s paper3 can be written as A = S,[q] for the Gaussian chain limit. In good solvents general scaling arguments imply that 6 could be a function of the ratio A M/[q]where A is the second virial coefficient and M is the molecular weight. If we consider the simple form in eqn (I) it is well known empirically4 that the molecular-weight dependence of the intrinsic viscosity is found for low-molecular- weight polymers to obey the equation [q]= KMa+B where K B and a are constants for a particular polymer solvent and temperature. Since we show’ that Schrag’s parameter A is proportional to [q],then A has a power-law term Ma as well as a molecular-weight-independent portion B which enters for low molecular weights.Some weak dependence of the coefficient 6 on the ratio A M/[q] would not affect this general dependence of A on [q].Hence the anomalous behaviour observed by Schrag and coworkers3 for the concentration-dependent viscoelastic properties of dilute and semidilute solutions is consistent with the known dependence of the steady-state intrinsic viscosity on molecular weight summarized in eqn (2). To my knowledge no one has yet fully explained the molecular origins of the constant B in eqn (2),but Thurston and Schrag4 offer the reasonable suggestion that the stiffness i.e. non-Gaussian character of short chains is partially responsible.The GENERAL DISCUSSION excellent experiments of Schrag and coworkers3 stress the need for understanding this dependence in eqn (2). However given the above analysis their puzzling data reduce to a question about the steady-flow intrinsic viscosity of low-molecular-weight polymers a problem which is much simpler to study than the problem arising in the context of Schrag’s data involving frequency- and concentration-dependent viscoelastic properties. K. F. Freed and A. Perico Faraday Symp. Chem. Soc. 1983 18 29. M. Muthukumar and K. F. Freed Macromolecules 1978 11 843. C. J. T. Martel T. P. Lodge M. G. Dibbs T. M. Stokich R. L. Sammler C. J. Carriere and J. L. Schrag Faraday Symp. Chem. Soc. 1983 18 173. G. B. Thurston and J. L. Schrag J.Polym. Sci. Part A2 1968 6 1331. Prof. J. L. Schrag (University of Wisconsin Madison U.S.A.)said Prof. Freed’s observation that the experimentally observed constant B in the empirical intrinsic viscosity expression may be another manifestation of the anomalous behaviour observed for the molecular- weight dependence of our A parameter at low molecular weights is indeed interesting. At this point the experimental studies have not been completed for a sufficient number of low molecular weights to test this suggestion adequately. It is also interesting that he predicts different ‘break points’ for plots of [v] against M and A against M (our A parameter). We hope to be able to test these predictions experimentally in the near future. Prof. M. Doi (Metropolitan University Tokyo Japan) said The birefringence of dilute polymer solutions has two origins the intrinsic birefringence (the birefringence due to the orientation of the chain segment) and the form birefringence (that due to the deformation of the polymer coil).The concentration dependences of these two factors are different. In Prof. Schrag’s data analysis it appears that only the intrinsic birefringence is taken into account. Is the conclusion affected when the effect of the form birefringence is taken into account? Prof. J. L. Schrag (Universityof Wisconsin Madison U.S.A.)said The importance of ‘form birefringence ’ effects in our oscillatory flow birefringence studies is not entirely clear. However there are several additional pieces of evidence that suggest that for the Aroclor-based solutions of polystyrene or poly(a-methylstyrene) studied here such effects are not detectable.In particular the low-frequency regime dynamics from which z is obtained correspond remarkably closely to the dynamics displayed by viscoelastic measurements on the same solutions; the two values of z are indistinguishable. In addition the close match of solvent and polymer indices of refraction together with the observed independence of higher effective frequency birefringence properties with respect to molecular weight and concentration strongly suggest that the form effect should be insignificant here. Our preliminary attempts to date to study the form effect (non-matching solvent and polymer refractive indices) in the oscillatory-flow birefringence experiment have shown overall frequency- independent birefringence level shifts but apparently no modification of the dynamics information being displayed.Dr M. P. Dare-Edwards ( Thornton Research Centre) said We have performed some 13C n.m.r. Tl relaxation measurements at Thornton Research Centre to study the GENERAL DISCUSSION effect of V.I. improver polymers on the microscopic mobility of model base solvents. These studies allow for entirely species-specific determination of local mobility or viscosity of solvent molecules in the presence of polymer additives. We have observed considerable effects for example on hexadecane solvent mobility with polymer concentrations of 10% (w/w) causing increases in averaged solvent local viscosity of > 10%.An expansion of similar studies should greatly assist in the development and testing of theoretical models which include the solvent modification effects. Prof. J. L. Schrag (University of Wisconsin Madison U.S.A.)said It is indeed gratifying to learn of additional evidence for solvent modification due to polymer additives obtained from an experiment that is more sensitive to local motional effects. I would point out that the solvent modification we sense seems to be much larger than Dr Dare-Edwards quotes for the same concentration (10% w/w) for the poly- styrene/Aroclor systems we studied. We find that r/oo/rsz 2.6; the major contribu- tion to the difference q’ -rs is now thought to be from solvent modification in the neighbourhood of the chain.This large effect may be common when high-viscosity solvents are employed; presumably the solvents used here were of substantially lower viscosity. Dr M. Adam (S.R.M.,Gij-sur-Yvette France) said Prof. Schrag and coworkers measure the first Zimm mode z of a polymer and prove that it is concentration dependent. This z can be considered (i) as the translational diffusion coefficient (concentration dependent) measured at a transfer vector q such that qR = 1 (R being the radius of gyration of one polymer) or (ii) as the rotational diffusion coefficient which using latex spheres, has been shown to be concentration independent. How can these two concepts be reconciled? D. Esteve C. Urbina M. Goldman H. Frisby H. Raynaud and J.P. Strzelecki Phys. Rev. Lett. submitted for publication. Prof. J. L. Schrag (Universityof Wisconsin Madison U.S.A.) said At qR z 1 one measures a ‘mean’ relaxation time z by dynamic light scattering which has contributions from translational diffusion and all internal modes. Only when qR 4 1 does one recover pure translation; when qR % I information related to internal modes is obtained. My understanding is that the relation between this ‘mean’ z and the first normal mode z obtained from our measurements has yet to be quantitatively established. In addition incomplete flexibility would further complicate the question since although z is triply degenerate for the isolated very flexible chain it splits into two distinct relaxation times as chain stiffness increases.These times correspond to an end-to-end stretching mode and two (degenerate) mainly rotational modes [see ref. (l)]. In addition finite solution concentrations will affect the relaxation times in a mode-dependent manner for solutions of flexible polymers since the interaction and interpenetration of neighbouring molecules gives rise to screening caused by a combination of intra- and inter-molecular hydrodynamic interaction. These mode- dependent concentration effects are what are being treated in the approaches of Muthukumar and Freed Muthukumar and Freed and Perico for example. The concentration dependence of the rotational diffusion coefficient for latex spheres would not be expected to show the same concentration dependence since these are large relatively rigid non-interpenetrating nearly spherical particles.However I would still expect to see a concentration dependence of the rotational diffusion coefficient although there may be limited concentration ranges (very dilute solutions) GENERAL DISCUSSION over which the effect of concentration may be minimal. (I believe that Freed has also considered this problem.) Thus I believe the two concepts you mention really deal with quite different systems and thus cannot be reconciled. W. H. Stockmayer in Molecular Fluids,ed. R. Balin and G. Weill (Gordon and Breach New York 1973). Prof. M. Doi (Metropolitan University Tokyo Japan) said As Dr Klein pointed out the disengagement of the arm from the tube depends both on the diffusion of the central monomer and the retraction of the arm.Are these two motions independent of each other? If not how can one incorporate them into a single diffusion equation and what is its effect on the final result? Dr J. Klein (Cavendish Laborutory Cambridge) said It is difficult to envisage a very strong coupling between the two types of mechanism shown in fig. 1 and 2 of our paper although a weak effect is possible. Dr Doi’s observation that the flow viscosity of three-arm star melts is a little lower than for stars with higher functionalities (for a given arm length) is interesting. The enhancement in arm relaxation rate expected from the availability of additional n-modes (fig. 3 of our paper) is most significant when (f-2) = 1 i.e. for a three-arm star.This is because the motion of a centre monomer down a tube (fig. 2) is suppressed exponentially with increasing f [e.g. eqn (1 1) of our paper] and becomes negligible for f 9 3 relative to the single arm retraction (fig. 1). Thus in our model the zero shear-rate viscosity (proportional to longest relaxation time) for a given arm length should be lower at f= 3 than at higher functionalities as observed. The extent of enhancement of the relaxation rate can be deduced in a straightforward manner following a summing procedure for the different mode rates similar to that indicated for D, eqn (15) of our paper. Typically the longest relaxation time for a three-arm star in our model where the centre is allowed to move is shorter by a factor of ca. 1/a [ca.3 according to current models see e.g.ref. (17) and (19) in our paper] relative to the ‘fixed-centre’ calculation for the same arm length. For a four-arm or higher functionality star (fa4) the extent of enhancement of relaxation rate depends on the arm length but is essentially negligible whenever exp [(f-3) > l/a [cf eqn (16) in our paper] i.e. as soon as Nb 2 3 entanglement lengths. Dr R.F. T. Stepto (UMIST Manchester) said The reptation concept has proved valuable in describing the dynamics of linear. chains where one dimension of the molecule (the contour length) is much larger than the other two. I wonder if it will eventually prove so useful for branched molecules. The papers by Klein Fletcher and Fetters and Pearson and Helfand investigate mechanisms based on ‘arm retraction’ and ‘tube renewal’ and appear to require a ‘transition state’ where the molecule has become ‘linear’.This is clearly illustrated in the figures in the paper by Klein et al. Such a transition state is less likely to occur for star molecules with a large number of arms and ‘tube-renewals’ will occur with a higher frequency. Movements of the surrounding chains become more important as the reference molecule is then not required to undergd such large simultaneous movements of several arms. Indeed as an extreme the movements from the first to the last sketches in fig. 1-3 in the paper by Klein et al. would occur with less frictional dissipation if the ‘tubes’ moved and the arms of the reference molecule moved with them (although obviously the segments surrounding an arm cannot always remain the same).GENERAL DISCUSSION 245 Dr D. S. Pearson (Exxon Annandale N.J. U.S.A.) said Our model of the relaxation of the orientation of star-shaped polymer molecules neglects the diffusive motion of the central monomer. We believe that this assumption is justified when the functionality (f) of stars is large. In comparing our theory with experimental data we have used results where f 3 4 and found that the viscosity is a function of arm length but not f. We do not know if the viscosity of three-arm stars differs systematically from the results in our paper. Also we have not tried to calculate the entropy loss when the central monomer moves alongthe tube and hence cannot comment on whether these configurations will have an important effect on the dynamics of disentanglement.Dr J. Klein (Cavendish Laboratory Cambridge) said We usually associate ‘tube renewal’ in an entangled system with the release of constraints about any given chain due to the diffusing away of adjacent constraining molecules. For very slow diffusion of a star in a linear matrix as in fig. 6 (or schematically in fig. 4 of our paper) one might expect ‘tube-renewal’ effects to be important for example eqn (1 8) in our paper. However for a star in a melt of stars tube-renewal effects become self-consistently very weak especially at high-molecular-weight values. For example a characteristic renewal time1 for a tube enclosing an N-mer is ztuhe z (N/N,)2z, where z is a constraint lifetime and N an entanglement size.If z is itself an arm retraction time z~~~~is much larger than the longest-arm relaxation time and tube renewal will provide only a weak perturbation to the relaxation dynamics in a star melt. This will apply to both low and high star-functionalities for sufficiently long arms. The detailed calculation of constraint release effects for intermediate (and practically important) star-sizes is complicated but may be significant for short arms. I J. Klein Macromolecules 1978 11 852 Dr D. S. Pearson (Exxon Annandale N.J. U.S.A.) said Although we have achieved some success in understanding rheological properties of star molecules by just considering ‘arm retraction’ we believe that the process of tube renewal may be important and we are studying that possibility now.Dr J. Roovers (N.R.C. Ottawa Canada) said I would like to point out that the loss-moduli-frequency curves of narrow-molecular-weight-distributionpolymers show a small maximum in the plateau region. The frequency of this maximum is found to be independent of the architecture of the branched polymer for stars comb and H-polymers. The frequency of the maximum depends on the molecular weight of the branch but the exact dependence has not been established. The relaxation mechanisms envisaged for polymers (reptation chain breathing) require that each polymer segment is directly connected to at least one chain end. In H-polymers the central bridge is connected to two branch points however. We have now obtained experimental evidence for very high melt viscosities (long relaxation times) for H-polystyrenes (Macromolecules,1984).When the zero-shear viscosities of the H-polymers are plotted as in fig. 5 of Dr Pearson’s paper all points lie above the line for the polystyrene stars. The high-molecular-weight slope for the H-polymers is approximately twice that for the stars. We have recently obtained zero-shear melt viscosities for ring polystyrenes with molecular weight up to 3.35 x lo5. From the results it appears that the zero-shear viscosities of rings are substantially higher than those of linear and star polymers of the same size. These preliminary data do not refute the reptation model for linear polymers. GENERAL DISCUSSION Dr G.B.McKenna (N.B.S. Washington D.C. U.S.A.)said I want to take this opportunity to respond to Dr Semlyen’s informal comments concerning the poly- styrene ring molecules. At the same time my response will address the comments made by Dr Roovers and Sir Sam Edwards on the same subject. First although Dr Semlyen is correct that fractionation is a problem in the production of samples of high-molecular-weight ring molecules he is incorrect in stating that it is poor fractionation which explains the difference in results obtained by different groups for the molecular-weight dependence of the viscosity of the polystyrene ring molecules (by inference the results obtained by Roovers and our results on the C.R.M. samples). This is incorrect because there is virtually no difference between our results and those of Roovers in the raw data.The difference is one of interpretation of these results. The observation is (see my comments on Prof. Binder’s paper) that the viscosities of the ring molecules are lower than or about the same as those of the linear molecules for the same molecular weight at molecular weights > M,. The greatest difference observed is a factor of ca. 2-3 at the lower molecular weights (M < M, < 2MJ. At the highest M molecular-weights (ca. 270000) the viscosities are the same. We have interpreted this to be a surprisingly small difference given that the ring molecules cannot reptate. Roovers on the other hand upon correcting his results using the theoretical Zimm-Stockmayer contraction factor g = 0.5 finds that for the same molecular structure gM, the cyclic molecules have a higher zero-shear viscosity than do the linear molecules.He interprets this to mean that there is a very large difference in behaviour. Since a change in M (or gM,) of 0.5 means a change in viscosity by a factor of 10.6 we can see why there is a difference between our appreciation of the raw data and that of Roovers for the corrected data. I make one other comment about the results on the ring molecules. Roovers interpreted his viscosity dependence on molecular weight to be an exponential in molecular weight. We have used the more classical approach and drawn two straight lines one with slope 1.0 below M and one with slope 3.9 above M,. Given that the highest molecular weights are limited to (6-8) x M, it is perhaps of interest to obtain data at some higher M in order to establish definitively the M dependence of viscosity.From a practical point of view however this may be impossible due to the fractionation problem alluded to by Dr Semlyen. Finally in addressing Sir Sam Edwards I merely put forth the comment that although he is not surprised at finding a ‘large difference’ in viscosities between linear reptating chains and cyclic non-reptating chains as proposed by Roovers perhaps he should be surprised because there is not a great difference in viscosities when one examines the raw data and interprets them as we have done.

ISSN:0301-5696    

DOI:10.1039/FS9831800199

出版商:RSC    

年代:1983

数据来源: RSC

摘要:

LIST OF POSTERS Tube relaxation and self-consistency in the viscoelastic theory for polymer melts J. L. Viovy J. F. Tassin and L. Monnerie Ecole Supkrieure de Physique et de Chimie de Paris France Constraint-release effects in entangled polymer melts J. P. Montfort G. Marin and Ph. Monge UniversitP de Pau et des Pays de l'Adour France Dilute solutions of macromolecules in a Poiseuille flow W. Stasiak and C. Cohen Cornell University Ithaca U.S.A. Computer simulation of the rheological properties of simple molecular liquids D. Brown M. Whittle and J. H. R. Clarke UMIST On dilute solution properties of polymers M. Muthukumar University of Massachusetts Amherst U.S.A. A determination of molecular relaxation times from macroscopic measurements J. F. Tassin P.Thirion and L. Monnerie Ecole Supkrieure de Physique et de Chimie de Paris France Effect of entanglements on dynamics of polymer molecules in the melt observed by neutron spin echo J. S. Higgins and J. E. Roots Imperial College London Factors affecting the dynamics of polystyrene chains in solution G. Allegra F. Ganazzoli E. Lucchelli and S. Briickner Istituto de Chimica del Politecnico Milan Italy and J. S. Higgins Imperial College London Ultrasonic and viscometric studies of cyclic and linear poly(dimethylsi1oxane) melts K. Dodgson R. A. Pethrick and J. A. Semlyen Shefield City Polytechnic Relation between the rheological properties of PMMA solutions and the molecular structure of the polymer W. Oppermann G. Rehage and D. Wagner Technical University Clausthal West Germany Melt rheology of ring polystyrenes J. Roovers National Research Council of Canada Ottawa Canada The molecular-weight dependence of the zero-shear viscosity of polystyrene ring molecules in the melt G. McKenna G. Hild C. Strazielle P. Rempp and A. J. Kovacs CNRS Strasbourg France Diffusion coefficients of polymer chains in dilute solution C. J. Edwards A. Kaye and R. F. T. Stepto UMIST Critical helix contents in gelatin gels D. Durand J. R. Emery and J. Y.Chatellier UniversitP du Maine France Micellar aggregates in viscoelastic solutions studied by proton n.m.r. spectroscopy U. Olsson 0.Soderman and P. Guering University of Lund Sweden

ISSN:0301-5696    

DOI:10.1039/FS9831800247

出版商:RSC    

年代:1983

数据来源: RSC

摘要:

AUTHOR INDEX* Adam M. 202 205,243 Kremer K. 37 211 212 213 215 216 217 Barlow A. J. 103 218 220 222 Baumgartner A. 37 211 212 213 215 216 Lamb J. 103 217 218 220 222 Lodge T. P. 173 Binder K. 37 211 212 213 215 216 217 McKenna G. B. 199 210 246 Martel C. J. T. 173 218 220 222 Blackwell J. 131 Boue F. 83 212 225 230 Burchard W. 233 Carriere C. J. 173 Dare-Edwards M. P. 225 242 Delsanti M. 205 Deutsch J. M. 218 Dibbs M. G. 173 Doi M. 49 223 224 236 242 244 Douglas J. F. 207 Edwards S. F. 145 Ewen B. 202 Fetters L. J. 159 Fletcher D. 159 Monnerie L. 57 199 217 224 225 226 Morris E. R. 115 234 Morris V. J. 115 Muthukumar M. 204 Nierlich M. 83 North A. M. 103 Osaki K. 83 Pearson D. S. 189 245 Perico A.29 221 Pethrick R. A. 103 205 230 Rahalkar R. R. 103 230 Rennie A. R. 229 238 Richardson R. K. 234 Richmond P. 237 Freed K. F. 29 204 205 207 221 239 241 Roovers J. 245 Ganazzoli F. 215 Ross-Murphy S. B. 115 232 236 Graessley W. W. 7 199 201 202 224 235 Sammler R. L. 173 238 Schrag J. L. 173 241 242 243 Harrison G. 103 Semlyen J. A. 103 Hawthorne W. 103 Southwick J. G. 131 Helfand E. 189 Stepto R. F. T. 212 244 Higgins J. S. 213 227 Stokich T. M. 173 Jamieson A. M. 131 223 231 Thirion P. 236 Killesreiter H. 225 230 Viovy J. L. 217 229 Klein J. 159 236 237 238 244 245 * The page numbers in heavy type indicate papers submitted for discussion.

ISSN:0301-5696    

DOI:10.1039/FS9831800249

出版商:RSC    

年代:1983

数据来源: RSC