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.