Theory of the Single ChainBY S. F. EDWARDSThe Schuster Laboratory, Manchester University, lManchester, M 13 9PLReceived 19th February 1970The theory of a polymer chain is discussed with special emphasis on the role of entanglements inthe conformation and dynamics of a very long polymer.The single chain has always received much attention from polymer scientistssince it is not only the first thing to consider in any polymer problem, but also initself it can give rise to many of the problems which occur in polymerized materialsand solutions. The material has been reviewed many times, so I shall not attempta general survey here, but concentrate on aspects of the single chain problem whichhave not received so much attention as yet, mainly because the mathematicaltechniques required to solve them are unusual, and because experiments which canisolate the properties I wish to discuss are difficult, and also novel.The first pointto raise is one of mathematical technique. When discussing theoretically how apolymer can be constructed out of monomers one evidently will obtain differenceequations in many variables, i.e., be led to matrix equations. These equations tellone what configurations of bond angles and lengths are permitted, their probabilitiesand energies etc., and clearly all this detail is needed to understand local behaviour.However, over large distances, and possibly over large times, this detail may not benecessary, and in that case one may describe polymers by differential or integralequations and replace matrix algebra by the calculus.This leads to an enormoussimplification in that it becomes much easier to see the difficulties and tackle them,but it is not clear how valid this transition is. My belief is that until the continuummodels of polymers have been fully understood one will not obtain mastery over theproblems of real polymers and I shall concentrate on these models, and since acomprehensive treatment of what we know of real chain configuration has just beenpublished by Flory,l will refer to those problems only incidentally.The first question that arises is that of knowing the simplest possible descriptionof a chain. If the chain is made by adding monomers with a certain probabilityprescription p depending solely on the position of the last monomer (i.e., a Markovprocess), the probability of end points Ro, RN; P is given byso that, in Fourier transform,Defining an effective step length I byP(RN,RO) = Snd%p(R,,RN-l)p(R,-,,RN-,) p(R1Ro), (1)P(k) = pN&) = exp [ -N log p(k)J = exp [ - Nk2((a2p/(ak2)k2 = 01 - NO(k4).(a2p/ak2)kz = 0 = 12/6P(RN,Ro) = (3/21n)* exp [ - (3/22)(RN -Ro)2J.Here p need not be a smooth function.T h e result is true for S functions, i.e., R,-Ra-l having only a choice of fixed values. This model gives (RN-RJ2 = NZ2 = sl,444 TIJEORY OF rrrw S I N G L E CHAINs the arc length, and correctly characterizes the long-distance behaviour of a non-interacting chain, and thus a first sight would appear to give a complete theory nearthe Flory temperature.However, this is only true for conformation problems andno others. So it will be necessary to generalize the problem to that in which theprobability of RN depends not only on RN-1 but also on RN+1. In continuum terms,the curvature then enters. Now the random flight model is well known to satisfythe differential equationin the limit of large distances. This equation describes a system in which the tangentat each successive point is completely random, and is similar to the Schrodingerequation of quantum mechanics where the velocity at each successive interval of aparticle, whose position is known, is completely uncertain. When a probability isattached to the curvature of our polymer, it is similar to the transition from quantumto classical mechanics, and e.g., if a Boltzmann factor exp (-fK2/kT) is attached tothe polymer (IC the curvature, = (a2r/as2)' or in discrete terms I C ~ = @,+I-2R, + R,- ,)2/422, the result is a differential equation of the typea a 3- + v- + -v2 + as ar 21 6a2av2G(r,rr,v,v',s) = 6(r - r')6(v - v')6(s)where a is a convenient representation of the constants kT, f (and in general neednot depend on kT in any simple way) ; as before r is the position, v is the derivative.The end-to-end distance is now(RN - R,J2 = l{s- (a/2)[ 1 - exp (- 2s/a)]),which is as before for large s, but for small s is Zs2/a.Such seems to be the minimumcomplexity needed for chains to be treated in differential form.SELF INTERACTION OF A CHAIN ( I )We can learn more of principle by considering a simple chain cross-linked toitself, than from the effect of forces; so we consider first that one is at the Florytemperature, and that cross-links are introduced into a chain.We can arrangechemically that the cross-links are entirely randomly spaced along the chain (e.g.,direct substitution of reactive groups into an inert chain, at the low degree limit,then intramolecular linkage of the groups. Members of the research group arecurrently attempting to illustrate this point) and consider this case for simplicity.There will then be, say, M cross-links distributed at random along the chain.To discuss the size and shape of such a chain, we suppose the cross-links restrict it asif it were subjected to a harmonic well of radius w;', i.e., as if the diffusion equationwere 1 a i a 2 1 ( as 6ar2 2- + - - + - w $ r 2 P = O .The entropy of this system is readily determined to be bounded bywhere -woL/2 is the configuration entropy of the chain in the well and pwz loss ofentropy due to the cross-links, where p is the chemical potential of the N linksS/k = - W O L / ~ + ~ W ~M = -as/apS.F. EDWARDSso that since s is a bound,and wo = 3M/L.Now we consider the much more complicatedlinking agents which, it will be supposed, act fairlydS/dw, = 0,45problem of introducing cross-quickly and are then removed.Then the number of cross-links will be proportional to number of times that particularchain was within a certain distance of itself. When this probability is to be estimatedit will be seen that the basic trouble lies in entanglements of the chain.Manyconfigurations are rapidly interchangeable, but the presence of, say, a knot, willtake a long time to remove. Thus, it can be suggested that as a first approximationone should decompose the chain into those classes which for closed or infinite chainsare permanent, i.e., will not transform into one another as time goes by. We labelthese classes c, so thatP = CP,.C(3)In each c there will be probability of segments of the chain meeting one another, sothe number of links formed will be M,. The final ensemble probability densitywill then beP - , ~ ~ c . m , ,where P,,,, is the probability distribution of a chain in class c with M, cross-links.We now try a rough calculation of this effect.The label c will represent the values ofthe set of invariants characterizing the chain. The simplest of these is the self-angle,i.e., the angle swept out by the chain along itselfc = ~f(drlxdr2) . V(l/r12) (= 4n x +integer for closed curve)and we shall confine ourselves to this quantity for simplicity. (Higher invariantswhich treat the ordering of knots and so on do not alter things in principle.) Nowthe quantity c can be positive or negative corresponding to the existence of positiveand negative knots and to the fact that the equation r(s) is vectorial, i.e., it mattersto c which end of the chain is s = 0. If we calculate the mean value of c2 it divergesfor the random-flight diffusion equation, which is not surprising as the differentialequation permits total changes in direction over infinitesimal arc length and suchloci are infinitely entangled.But when the next order of accuracy is used, ie., eqn(2), a finite answer is obtained andwhere y is a numerical constant. Though not exact, it is reasonable to approximatethe distribution of c as(c2> = ul(L/al (4)since the c take on the values of 471 x +integer (for closed curves) and so can beapproximated uniformly by the continuum (- co < c < GO). The value of c buildsup rather like a one-dimensional random walk as one goes along the chain,positive and negative entanglements appearing at random. Entanglements can spanthe whole chain, but nevertheless the fact that the mean square exists and has thevalue (4) for large c suggests that the picture is a sensible one.One can thereforethink of decomposing the end-to-end probability,a,P(r,L) = ('>i.xp( 271Ll - 2Ll "> = zp,(r,L) , 2: -03 dcp(c,r,L)46 THEORY OF THE SINGLE CHAINwherep(c,r,L)dc = ( L y e x p ( - 2LZ "> s", 2nL2andThe simplest way to developp(c,r,L) is to use the Hermite form,p z (3/2Ll~)3(alqL~)* exp (- 3r2/2L1 - c2a/2qL)[1 - A(r2 - LZ)(c2 - qL/u)].(As with q, A is a number which can be obtained from fitting with the invariant, butspace precludes a derivation.) At this point, by analogy with the simplest case, theprobability of two points of the polymer being a distance I apart in spacewhere we have used the distribution derived for end-to-end probabilities also forintermediate probabilities of two points arc length s apart.Thus, we have M,and the calculation can proceed as before, with a chemical potential pc, and then afinal summation in eqn (3). Space does not permit the final calculation, but I wishto emphasize not so much the detail of the calculation, but the fact that this kind ofquestion can be raised, and answered at least approximately.This discussion leads to the question of whether a long polymer is localized inspace. If a polymer is long enough over a corresponding time scale, the variousinvariants are indeed invariant, and the entanglements can be considered permanent.This leads to the question : if at one time the polymer has a form rl(s), then will itsform r2(s) at a subsequent time have any relationship to rl(s) ? Or will rl(s) be essentiallyunrelated to r2(s) except in the sense that on working out the invariants they will bethe same? In other words, given a segment of the polymer cannot pass throughneighbouring segments, is the effect of neighbouring segments enough to box in thefirst segment, or is the system, in spite of the constraints virtually free to take up anyconfiguration.If the polymer were confined to a box of side much less than the" natural radius " (LZ)*, a single polymer will cross and recross the box many timesand the resulting system will not essentially different from a rubber or glass. Clearlythen a segment will be trapped by its neighbours. Will it also for a free polymer?One can try to answer in this way.If the probability distribution of the chain rl,isp([r,]), of r2 p([r2]), then the joint probability (before normalization) will bewhere 11, I, are the invariants of rl, r, and r the (Kronecker) delta function which isunity when 1, = I,, and zero otherwise. One can try to model SIIz2 which has a" mirco canonical " structure, by a canonical formand this in its turn by a constraint sufficiently simple to integrate, say,exp ( - w1-exp (- wgI(ry - ri)2ds).As before one can now evaluate the entropy of the system as a function of wo, andfind the minimum value. Again, I do not have the space to detail the calculationbut can only state the result : which is that for small a the system is indeed alwayS . F. EDWARDS 47localized but as a increases a critical value is reached after which there is no localiza-tion and the effects of the invariants can be obtained by treating them as a smallperturbation. We recall that the size of the polymer (R2) = LZ (L large) does notdepend on a, but the entanglement does.Thus, one can have two polymers of thesame mean square radius, one of which is fully entangled with itself like the manypolymers in a glass, whereas the other has very little constraint on its configurations.So far the discussion has been restricted to chains at the Flory temperature. Theeffect of constraints can be discussed above and below that temperature. In parti-cular, below the 6 point the self-attracting chain will tend to contract, being heldfrom collapse by the repulsive forces at small distances, and also, over a time scale,by the entanglements which will always work against the configurations of minimumvolume.These different mechanisms can be put together after the style of the van derWaals’ theory of a dense gas, and allowing for polymer solvent interaction, a theorydeveloped of the natural size and distribution of a polymer below the Flory tempera-ture. Once one goes any way below this temperature, polymers collapse and one isthen really dealing with a small piece of solid. We quote only the simplest form for thefree energy, We suppose the self-attraction confines the polymer to a box of side 9,and the van der Waals’ self attraction is v(T), excluded volume of step I is h2.ThenThe terms of F/kT are due respectively, to (i) the change in entropy due to confine-ment ; (ii) the effect of the excluded volume (which unlike the free chain can be assessedsince there is a reference volume g3); (iii) the net attractive potential effect for thepolymer below its &point ; (iv), (v) are polymer solvent effects.When u is large, Fhas a minimum for a definite 92 and the polymer is self confined. But as the tempera-ture increases one reaches a point where 9+co and there is no confinement. (Thederived expression is not accurate as B-+co, but serves as do van der Waal’s typetheories in general, to illustrate the point.)One cannot do justice to the case of the high temperature, T> 0 polymer in a briefdiscussion like this, so I shall leave that topic and discuss the motion of the chain.DYNAMICSTo discuss dynamics one needs to understand the behaviour at arbitrary timeintervals, so the entanglements of the polymers can never be neglected in principle ;they may not matter in special cases, but that has to be proved.To start this problemone needs to think of the probability distribution of the entire chain, not just the endpoints. Suppose that in (1) the probability from R, to R,,, can be approximatedby a gaussian. Strictly this cannot be true but it simplifies the mathematics withoutloss of generality. ThenP([R]) = N exp [-(3/2Z2)~(Rn-R,-J2].nN is the normalization factor. It is convenientcomponents,R,, = Z exp (2nimn/N)r(m),orR(s) = gdw exp (iws)r(w)in the continuum.to think in terms of Fourie48 THEORY OF THE SINGLE CHAINP([R]) is then N exp (( - 3/22)Z[ 1 - cos (2nmn/N)] I R, I 2 > ,a (diagonal) gaussian in the Fourier components.We now consider the motion of aparticle in a potential well V(r). In free space a particle will diffuse according toFick's equation- + - -2 p(r,t) = 0 (:z 2" as:>andp = constant is the equilibrium distribution. In the well one expects the equili-brium distribution in r to beE exp (- V(r)/kT)With the usual kind of assumptions of kinetic theory, the modification of Fick'sequation having the solution exp (- V/kT) can be shown to beand if V is an harmonic potential $q2r2(-+--(--+-+r))p=o. a I c a a 1at 2 ar ar kTThis equation has the Hermite functions for its solutions, and if at t = 0COP = C pnHen(r)n=Othenso that as t-m, all distributions tend to the equilibrium distribution n = 0.Thissuggests that the dynamics of the polymer may have a simple form in which eachr(w) acts like a particle in a potential well with equilibrium distributionexp (- (3/21)w2R(w)dw),i.e.,The symbol 6/6R means functional derivative to get the right number of dw, butthe reader unacquainted with this can treat it as a partial derivative and always thinkof (8) rather than (10). This equation can indeed be derived under certain circum-stances. For example,2 we suppose that the force acting on the polymer is that derivedfrom differentiating the free energy, and that this overcomes the friction of a solvent.ThuswhereF = TS = Tk log P so that (6F/JR) = (3/2Z)w2R(w).and 4 is the random fluctuating forces due to the solvent, and due to the fact that thethermodynamics only gives the mean force (about which there is a fluctuation whichin this case, unlike normal statistical physics, has a mean square larger than its ownuR(w) = [6F/6R(w)J + 4S.F. EDWARDS 49square mean). If it is considered to fluctuate instantaneously, this leads by standardarguments to a Rouse type equationIn this form the diffusion constant is independent of w. There are big assumptions inderiving this form and its range of validity is not clear since it would be surprising toget a complete solution without invoking the mechanism of molecular attachment.Perhaps a more realistic attempt is to consider the exchange of configurations acrosspotential barriers of the C-C type.Suppose we have a configurational idealized toovercoming a potential barrier Q. Then the rate at which this occurs in a thermalsystem according to Kramers isWAWC P(R,--+Rk+AR) = - exp (-Q I kT) = E ,2nPsay, where wA,wc are the frequencies associated with configurations (A) and (B)and p is the relaxation time of excitations of the bonds, i.e., characterizes the time inwhich energy is transported without the (A) +(B) transition.Butso that going into the continuumand we are led to the same form withAR = Rn+1-2Rn+R,-1k(w) = E W ~ R ( W ) + ~but E reflects the local structure and not the overall thermodynamics. We note thecurious law that whereas(R(s,t) -R(s',~))~ ~ ( S - S ' ) ~ ,(R(s,t)-R(s,t'))2-E(t - t')',or for FIory law excluded volume,(R(s,t)-R(s',t))2 N ( s - s ' ) ~ J ' ~(R(s,t)-R(s,t'))2 N ( t - t')6'11.So far no account has been taken of entanglements.I can, however, give a simplemodel to show how they can appear. We suppose our polymer is in two dimensionsheld at fixed points R(O), R(L) and suppose the " entanglement " simply means thatthe angle swept out around an obstacle (perpendicular to the plane) at the origin, isconstant, e.g., in fig. 1,FIG. 150 THEORY OF THE SINGLE CHAINWe now ask what is the dynamics of a chain which moves (as in the analysis above)but cannot get into the configuration, say, of fig. 2,FIG. 2.This means the dynamics has a Lagrange multiplier to ensure that Sd6(s) the integratedangle is that of fig. 1 and not that of fig. 2. One can show that the dynamical equationisE - +-w’R(w)+iAqR)])G, 1 = 0, (&+j%&+w) {&.) IwhereZ = - iAaA/as + i;ldy/ds x curl AA= (y(s)llR(s)l, -W/lR(dl)and if’ the angle swept out is 0, the final solution is 1 I m p (i;lO)G,dd3, playing the role of Lagrange multiplier. Evidently to solve such equations andextend them to three dimensions is a formidable task, but my point is that it ispossible to write these problems down, to define them, and from then on progress canbe made.CONCLUSIONI have deliberately presented a very unconventional emphasis on the problems ofa single chain, but this is a discussion paper so T think these really difficult problemsshould be aired. The work of this paper has mostly been generated by discussionwithin the polymer group at Manchester and I should like to thank the group fortheir stimulus, particularly to Prof. Allen and Prof. Gee.P. J. Flory, Statistical Mechanics of Chain Molecules, (Interscience, New York, 1969).P. G. de Gennes, Rep. Prog. Phys., 1969,32 ; Physics, 1967, 1,37