Reptation of a Polymer Chain in the Presence of Fixed ObstaclesP. G. de Gennes

Citation: The Journal of Chemical Physics 55, 572 (1971); doi: 10.1063/1.1675789 View online: http://dx.doi.org/10.1063/1.1675789 View Table of Contents: http://scitation.aip.org/content/aip/journal/jcp/55/2?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Rheology and reptation of linear polymers. Ultrahigh molecular weight chain dynamics in the melt J. Rheol. 48, 663 (2004); 10.1122/1.1718367 Entropic trapping of a flexible polymer in a fixed network of random obstacles J. Chem. Phys. 111, 1778 (1999); 10.1063/1.479439 The size of a polymer chain in an imperfect array of obstacles: Monte Carlo results J. Chem. Phys. 108, 3310 (1998); 10.1063/1.475728 A random walk chain reptating in a network of obstacles: Monte Carlo study of diffusion and decay ofcorrelations and a comparison with the Rouse and reptation models J. Chem. Phys. 94, 3222 (1991); 10.1063/1.459791 On reptation in polymer melts J. Chem. Phys. 84, 5922 (1986); 10.1063/1.449905

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572 J. C. TULLY AND R. K. PRESTON

32 See M. J. Romanelli in Ref. 31, Chap. 9. 33 For distances greater than 11 a.u. the probability function

P(R, R) is essentially unity except for very small values of V.L

A typical four-branched trajectory (ants method) required about 9 sec of computer time, using the Yale Computer Center's IBM 7094. This is to be compared with about 6 sec for an ordinary unbranched classical trajectory.

34 Reference 10 and private communication with J. Krenos and R. Wolfgang. The D+ /D2 + ratio of 1.4 observed bv Krenos and Wolfgang is not in very good agreement with our theoretical value of 0.9. This is probably due to the fact that, because of the particular experimental mode of operation used to obtain the results of Ref. 10, ions scattered with high laboratory velocities in the forward direction were detected more efficiently than those formed at large angles and/or low velocities. The present calculations predict similar angular (mostly backward) and

THE JOURNAL OF CHEMICAL PHYSICS

velocity distributions for HD+ and D,+, but not for D+ (mostly forward and higher velocity).

35 Since it is possible to calculate very accurate ab initio potential surfaces for this system, it would be valuable to perform the surface hopping calculations on such surfaces. This would elimi­nate the possibility of disagreement with experiment being due partially to inaccurate surfaces, and would restrict the error to the assumptions of the model.

36 We estimate that in the present calculation, if the coupling terms were changed by a factor of 2, the resulting change in the cross sections would be of the order of 10%.

37 For discussions of the applicability and extensions of the LZ approximation, see Ref. 18. Also, D. R. Bates, Proc. Roy. Soc. (London) A2S7, 22 (1960). C. A. Coulson and K. Zalewski, ibid. A268, 437 (1962). E. E. Nikitin, Advan. Quantum Chem. 5,135 (1970).

VOLUME 55, NUMBER 2 15 JULY 1971

Reptation of a Polymer Chain in the Presence of Fixed Obstacles

P. G. DE GENNES

Laboratoire de Pilysique des Solides, Fac1tlte des Sciences, 91------Orsay, France

(Received 18 January 1971)

We discuss possible motions for one polymer molecule P (of mass M) performing wormlike displace­ments inside a strongly cross-linked polymeric gel G. The topological requirement that P cannot intersect any of the chains of G is taken into account by a rigorous procedure: The only motions allowed for the chain are associated with the displacement of certain "defects" along the chain. The main conclusions derived from this model are the following:

(a) There are two characteristic times for the chain motion: One of them (Td) is the equilibration time for the defect concentration, and is proportional to M2. The other time (T,) is the time required for complete renewal of the chain conformation, and is proportional to M3.

(b) The over-all mobility and diffusion coefficients of the chain P are proportional to M-2. (c) At times t< T, the mean square displacement of one monomer of P increases only like «r,-ro)2) =

const t1/4•

These results may also turn out to be useful for the (more difficult) problem of entanglement effects in unlinked molten polymers.

I. INTRODUCTION (2) In the liquidlike region J(t) is linear in time:

The stochastic motions of a single polymeric chain dissolved in a solvent of low molecular weight are reasonabl y well understood in terms of a model set up by Rouse l and improved by Zimm.2 On the other hand, the dynamical properties of concentrated (or molten) polymers are still poorly understood. An excellent review of the experimental situation has been given by Ferry.3 The main features appear to be the following:

(1) The "creep compliance" J(t) (i.e., the delayed response in strain induced by a step function increase in stress) displays a glasslike behavior at short times (t«T') , then an approximate plateau (rubberlike behavior T' <t< T"), and finally a liquidlike behavior (t»T"). The molecular mass dependence of the "terminal time" T" has been measured on a few typical systems, and appears to be3

(Ll)

J(t)~·.feO+ (t/'Y/).

The zero-frequency viscosity 'Y/ depends strongly on M for large M, and follows the celebrated law4

(I.2)

(3) The self-diffusion coefficient D for one polymeric chain moving among the others has been measured only in a few cases, and the dependence of D on M is not very well established. For polyethylene, McCall et at. have found;;

(1.3)

Qualitative arguments leading to Eq. (1.2) for the viscosity have been given by Bueche6 and Graessley7;

but they lead to rather unsatisfactory exponents when applied to Eq. (Ll) (T",-...,M4.a) or to Eq. (1.3) (D,-...,M-3.a). Another attractive approach to describe entanglement effects is based on the idea of a transient networkS; the relaxational modes of a set of chains

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REPTATION OF A POLYMER CHAIN 573

which are intercoupled by friction at some points have been worked out.9 However, some of the statistical assumptions on which the model is based (e.g., the fact that entanglement takes place on the same monomer at all times) are still open to some doubt.

These observations have prompted us to try another approach: instead of a molten polymer, we focus our attention on a different system, where entanglements are also dominant, but where their effects can be under­stood in terms of a more precise statistical analysis. The system that we have in mind is made of a single, ideal, polymeric chain P, trapped inside a three­dimensional network G, such as a polymeric gel. A two-dimensional representation of this situation is shown on Fig. 1, where G reduces to a set of fixed obstacles 0 10 2 " ·On···. The chain P is not allowed to cross any of the obstacles; however, it may still move in a wormlike fashion between them. The details of the model are given in Sec. II. In Sec. III we present an elementary calculation of the over-all chain mobility (or of the self-diffusion coefficient, which is related to the mobility by an Einstein relation). We find that both coefficients behave as M-2. In Secs. IV and V we carry out a more detailed investigation of some microscopic properties: Brownian motion of one monomer, fluc­tuations of the end-to-end vector, etc. The two different times which emerge from these resul ts are discussed in Sec. VI.

II. A MODEL FOR REPTATION

We consider a freely jointed chain of N "monomers." The positions of successive monomers are labeled rl, r2, .. " rn, "', rN. The vector intervals

an = rn+l- rn~ar/ an are assumed statistically independent

(an(t) ·am (!) )=onma2.

(ILl )

(II.2)

We assume that the length of the chain is very large when compared to the distance between neighborinO' links in the fixed network: then the possible chai~ motions are strongly restricted. We postulate that the

FIG. 1. The chain P is free to move between the fixed obstacles 0 but it cannot cross any of them.

o

p

stored length b

c

(0)

(b)

FIG .. 2 .. \ "defect" moves from A towards C along the chain. When It crosses monomer E, this monomer is displaced by an amount b.

only allowed motions correspond to the migration of certain "defects" along the chain. A qualitative picture for such a defect is shown on Fig. 2. The curvilinear interval between two monomers (n) and (m) would be equal to (n- m) a in the absence of defects. But if we have II defects in this interval, each of them storing an amount b of length, the curvilinear interval becomes

(II.3)

For simplicity we take all defects with the same "stored length" b. (A distribution of b values can easilv be included, but leads to no interesting change in the-final results.) The most important consequence of Eq. (II.3) is that there is a conservation law on defect number: II changes only when a defect crosses one of the end poin ts (n or m) of the interval. This conserva­tion law, plus an assumption of short-range forces (no backflow effects of the Zimm2 type), leads to a one­dimensional diffusion equation for the gas of defects along the chain. To write down this equation, let us introduce the number p of defects per unit length of the extended chain. [p=II/(n-m)a in the example of Eq. (II.3).J We shall constantly assume that p is small, i.e., that we have a dilute gas of defects. Let us also define a defect curren t J n (= number of defects passing through point n per unit time). Then the conservation law is

(llA)

and I n has the explicit form

(II.S)

.<:l is the diffusion coefficient of the defects along the chain: It is a microscopic constant, characteristic of local jump processes, and is independent of the molecu­lar mass M (for large M). The second term in Eq. (II.S) represents the drift of defects due to external

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P. G. DE GEN~ES

forces fn applied on the monomers. 'Pn is the force per defect; to relate 'Pn to fn we write down the work per­formed when one defect moves by an amount (aon) along the chain: this means that on monomers are dis­placed, each of them by an amount b, where b is a vector of magnitude b, tangential to the chain. Thus

onfn • b = aon'Pn,

(ll.6)

Equations (ll.4) and (ILS) for the gas of defects must be supplemented by boundary conditions at both ends of the chain n=O and n=N.lO We assume that, at the ends, the density p relaxes very rapidly towards a constant equilibrium value p,

PO=PN=p.

In the absence of external forces, the general solution of the diffusion equations (II.4) and (I1.S) then takes the form

Pn-P= L Cp sin (p7f"nl.Y) exp( -tITp) , (II.7) p

where p is a positive integer, labeling the various relaxation modes, Cp is a constant, and the relaxation times T p are given by

larger than the equilibration time of the defects Td •

This inequality will be confirmed in Sec. V. When it holds, we can derive the chain mobility (defined as the ratio of the drift velocity to the applied force) by a very simple argument.

Consider the chain P in reptation under a constant (weak) force F, applied along a certain direction (Z). The force per monomer is fn = FIN and the force on one defect is, according to Eq. (11.6),

(llI.1 )

The vectors b are time dependent, but they change only on the scale T r • On the other hand, the gas of defects reaches a steady state under external forces in a much shorter time Td • Thus, we can treat the 'Pn' as constants, and write down the response of the gas of defects as a steady state conduction current}:

J = pJ.l~y-1 iN dn'Pn, (1II.2) o

where J.l= t:,.lkJlT is the defect mobility. From Eq. (llI.1) we get

J = (pJ.L!JV2a) fF·bdn

= [pJ.lbl (:Ya)2JfF· dl, (III.3)

(II.S) where dl is the line element of the chain. The integral

The longest among these relaxation times is

Td=T(p=!) = 7f"-2(N a)21 t:,. (IL9)

and it is proportional to the square of the molecular mass. We may say that Td is the time for equilibration of the gas of defects.

When the defects move, the chain progresses, as shown on Fig. 4. The velocity of the nth monomer is related to the defect currentJn by

drnldt= bJn. (IUO)

Equation (ILlO) must also be supplemented by boundary conditions. For instance, when a defect leaves the chain at the extremity (N), a new terminal arc (of length b) is set vp and may take various orienta­tions: in zoological terms, the head of the snake must decide which gate it will choose in the cross-linked network. We assume that this choice is at random, i.e., that the last vector b is completely uncorrelated to the preceding ones.

The reptation equations (II.4) , (11.6), and (II.10) are significantly more complicated than the same equations for a free chain.! However, as we shall see in the following sections, it is possible to extract from them some definite predictions on the stochastic behavior.

III. ELEMENTARY CALCULATION OF THE OVER-ALL MOBILITY

Let us assume for the moment that the time Tr for complete renewal of the chain conformation is much

fdl=P (III.4)

is the end-to-end vector, and we have for the average defect current

J = [pJ.lbFI (.Ya)2JPz. (IlLS)

The center of gravity g of the chain moves with a velocity

N

g= .\,-1 i dnrn •

o

We can transform rn by Eq. (ILlO) and insert (IlLS) as the current, obtaining

. J iN Jb g= - bdn= -Po .\' 0 iY a

(IIL6 )

Averaging over the values of P we are left with one nonvanishing velocity component:

where the over-all mobility J.ltot is explicitly given by

J.ltot = J.l[pb2(PZ2)1 ("Ya) 3J (III.7)

Equation (III.7) may also be written in terms of the self-diffusion coefficient Dtot=kBTJ.ltot. Both coefficients are seen to decrease like .V-2 (or M-2). This is to be compared with the case of a free Rouse chain, where Dtot,,-,M-l.

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REPTATION OF A POLYMER CHAIN 575

IV. DISPLACEMENTS OF ONE MONOMER

The quantity

(IV.l)

provides a good description of the motions of one monomer. It turns out that ret) may be computed rather simply when t is smaller than the equilibration time of the defects Td [provided that (n) is not too close from the ends (0) or (N)]. Then, end effects may be omitted and we may consider that the relevant portion of the chain (around 12) worms its way inside a thin, permanent, undeformable tube (Fig. 3).

The position of the monomer (12) may then be defined (instead of using rn) by its curvilinear abscissa along the tube, Sn (t). Let us discuss first the quantity

(IV.2)

e(t) is the number of defects which have passed from the "left" to the "right" of monomer n during time t, minus the number of those which went from right to left. The average (e(t) is zero for symmetry reasons. On the other hand, (e2

) is that average number of defects which commuted either from right to left or from left to right, during the time t. Since in a time t diffusion takes place over a length (tlt) 112, this number will be of order p(tlt)112. The calculation given in .-\ppendix A confirms this, and gives

(IV.3)

It is also possible to prove, by a standard Holtsmark­:Vlarkoff analysis" that the distribution pt(e) of e at a given time t is Gaussian, provided that the "number of migrating defects" p(tlt) 112 is much larger than one:

pt(e) = [1/(21l'(e2 ») 112J exp[ -e2/2(e2(t)J

[p(tlt)112»1]. (IV.4)

\Ve can now proceed and describe the spread in three­dimensional space of rn(t) -rn(O). Two points separated by a curvilinear distance s= be have a mean square distance in space given by

(IV.S)

because the tube has the conformation of a random

T

FIG. 3. An infinitelv long chain P cannot move sideways: It is 'trapped in a thin tube T.

coil. Thus we may write

([rn(t) -rn(0)J2)=bafdept(e) I e I = ba (2/ 1l') 112[ (e2 (t) ) JI12

([rn(t) -rn(O) J2)= (2/-il14) bapl 12 (tlt) 114

(Td»t» 1/ tlp2). (IV.6)

Equation (IV.6) is one of our central resul ts: It describes a very special type of Brownian motion. For comparison we shall write down the laws for systems which are more familiar:

(a) diffusion of one small molecule

(r2)"'t.

(b) motion of one monomer in a free Rouse chainl2

(r2 )"'tI12

(t«N2TV-" where TV is a fundamental jump frequency).

Equation (IV.6) shows that for a chain in reptation (r2)"'tI14: The motion is extremely slow. For this reason Eq. (IV.6) cannot be tested by inelastic neutron scattering (at least in the present state of the art). On the other hand, this very slow creep might lead to observable effects in nuclear relaxation. This is dis­cussed for one limiting case in Appendix C.

We end up this section with a word of caution: although the distribution pt(e) is Gaussian [when p(tlt)112»lJ the distribution of r=rn(t) -rn(O) is not Gaussian: It decays roughly like exp( _r413 ) at large r.

V. THE RENEWAL TIME Tr

Our aim in this section is to discuss the correlation for the end-to-end vector P= Ln an:

(P(O) ·P(t) )=cf>(t). (V.l)

After deriving cf>(t) we shall define a renewal time Tr as the time above which cf> (t) becomes negligibly small.

It will turn out that Tr is much larger than Ta: This observation simplifies the analysis considerably. For times of order Tr we may forget completel y the details of the reptation process inside the chain, and neglect in particular the fluctuations of its length L=X a(l-pb)"-'~Ya.

A. Relation between Vector Correlations and Migration along a Tube

In terms of the vectors an, the function cf>(t) [Eq. (V.I) J becomes

cf>(t) = L (an(O)·am(t» n,m

(V.2) n,rn

Our task is to derive the "vector correlation functions" Gnm(t) in the limit t»Td •

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576 P. G. Dr: GENNES

A

/

" /

, , I

I

FIG. 4. Successive steps for the long time (t~T,) motions of a chain inside a gel. (a) Initial position: The chain is restricted to a certain tube T. (b) First stage: The chain has moved to the right by reptation. (c) Second stage: The chain has moved to the left; the extremity I has chosen another path (IIf,). Note that, for this example, a certain fraction of the cbain (II J 2)

is still trapped in the initial tube at stage (e).

Let us consider first the chain P at the initial time t = O. It extends (on a length "-' iYa) on a curvilinear path (1010) which we call the "initial tube" [Fig. 4(a)]. Let us call A the position of monomer (n) and B the position of (m), both taken at t= O. The initial tube has the conformation of an ideal coil: thus the orientations of an and am (both measured at the same time t = 0) are uncorrelated for n~m. This may be written as

(V.3)

Now let us consider a later time t: The chain has moved back and forth and some portions of it are now out of the initial tube [Figs. 4(b), 4(c)]. We shall have non­vanishing correlations Gnm(t) only if some part of the chain is still in the tube. For monomers (r) belonging to this part, we may still define a curvilinear abscissa ST(t) , measured along the initial tube. Since, for the time scale ("-'TT) of interest, changes in the chain length are negligible, we may write that all the monomers r have been displaced along the tube by the same amount !T(t) :

STet) = S,(O) +!T(t). (V.4)

We get a nonvanishing contribution to Gnm(t) if, and only if, the following conditions are satisfied:

(a) The monomer (m) must have moved to point A

(because different line elements along the tube are uncorrelated) .

(b) The vector am(t) must be in the direction imposed by the tube: That is, (m) must belong to the family of monomers (r) defined above. This demands that, at all times t' in the interval (Ot), the extremities (1 and J) of the chain have not passed through point A. (Figure 4 shows how the correlation is lost at a point F if I has passed F.)

Thus we must have

Sm(t) =sn(O), (V.Sa)

sn(O) -SI(O) >!T(t') >Sn(O) -SJ(O). (V.Sb)

Finally, we may interpret Gnm(t) as the probability for the random function !T(t') to go from 0 (at time 0) to sn(O) -sm(O) = (n-m)a (at time t), without ever exceeding the I imi ts (V .Sb) .12.

B. Migration Along One Infinite Tube

The stochastic properties of !T(t') depend only on the density fluctuations of the gas of defects, and are independent of the three-dimensional structure of the chain: To study !T(t') , it is enough to solve a restricted problem, where the chain is constantly confined to one unbranched tube of infinite length (Fig. 4).

The restricted problem is one dimensional, and is thus easily solved. The results, for the curvilinear abscissa Sn- of one monomer, are the following:

(a) At times t«Td (and for n not too close to the end points) Eq. (IV.3) holds and we have

([Sn(t) -s,,(O) J2)= (2/7r1/2 )pb2 (/lt) 112.

(b) At times t"-'Td or t> Td a more detailed analysis, taking into account end effects, is required: It is described in Appendix B. The results become simple in the limit t»Td :

(t»Td). (V.6)

Equation (V.6) describes the over-all diffusion of the chain along its fixed curvilinear path. The curvilinear diffusion coefficient Dc is given explicitly by

(V.7)

Dc (or the corresponding curvilinear mobility Ilc= Del kJlT) is inversely proportional to the molecular mass: This is natural since the friction exerted by the chain (in uniform curvilinear motion) is proportional to its length.

The result (V.6) is independent of the index n: on the scale of times t(»Td ) and of curvilinear distances ("-'Na) which is of interest here, the inner constrictions of the chain can be neglected, as was announced below Eq. (V.l): Eq. (V.6) is an equation for the over-all displacement !T:

(V.6')

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REPTATION OF A POLYMER CHAIN S77

It can also be shown that, when t»Ta, l1(t) becomes a stochastic function with independent increments. This observation is crucial, because it is only for functions of their class that restrictions such as (V.Sb) are tractable.

C. The Second Spectrum of Relaxation Times

Let us now restrict our attention to times t»Ta, and introduce a function Wet, (1) giving the statistical weight for finding 11(0) = 0 and l1(t) =11. The function W is ruled by a standard diffusion equation

(V.8)

We want to count in W only the random functions l1(t') which remain inside the interval (of length Na) defined by Eq. (V.Sb). This is obtained if we impose the boundary conditions

Wet', Sn(O) -sr]= Wet', Sn(O) -SJ]=O. (V.9)

Then W differs from Gnm only by a proportionality factorl3:

Gnm(t) =aW[t, (n-m)a]. (V.lO)

To find the explicit form of Gnm [satisfying the initial condition (V.3)] we then expand the solution of Eqs. (V.8) and (V.9) in a series of sine waves vanishing at both ends of the interval, and obtain

Gnm(t) = (2N)-1 L sin(1rpm/N) sin(1rpn/N) p

where p is again a positive integer, and the Op's represent a new set of relaxation times

1/Op=1r2p2(Dc/N2a2) = p2(1/Tr) ,

Tr= (N a) 2/1r2Dc = (N a) 3/1r2pb2t::.. (V.12)

Two remarks concerning Tr should be made at this point:

(a) Tr is proportional to N3 or M3, while Td , as given by Eq. (II.9) , is proportional only to N2. Thus, for large N, Tr»Td as announced.

(b) Td is independent of the density of defects (p) and of the stored length per defect b. On the other hand, Tr is strongly dependent on both.

Finally, we insert Eq. (V.11) into (V.2) and derive explicitly the correlation function t/>(t) for the end­to-end vector

<p(t) = (P(O) ·P(t»

= (a2/LV) L L exp( -t/Op) p nm

Xsin(p1rn/N) sin(p1rm/N). (V.13)

Replacing the sums over nand m by integrals, we get

t/>(t) as a rapidly converging series

t/>(t)/Na2= (8/1r2) L (1/p2) exp( -p2t/Tr). (V.14) (p odd)

For t< Tr t/>(t)--+N a2. For i»Tr t/>(t) decays exponentially with time constant Tr: This shows that Tr is indeed the renewal time for chain conformations.

VI. DISCUSSION

A. Reptation in a Gel

We found two sets of relaxation times (Tp and Op) for a chain in reptation. Each set has a structure reminiscent of a Rouse chain; however, the physical processes associated with both sets are very different:

(a) The first set (Tp) corresponds to relaxation of the density of defects along the chain. The basic time for this set is Td"-'M2.

(b) After a time i"-'Td the chain is still essentially confined in the "tube" which held it at time t= O. Each monomer has then been displaced only by a small amoun t along the tube [8s"-'b (paN) 1/2].

(c) To disengage the chain completely from its initial tube we require a much longer time Tr"-'M3. Tr may be interpreted as a dielectric relaxation time, if the mono­mers have a finite permanent dipole along the chain axis an.

Of course, at present, there are essentially no experi­mental data on the dynamics of free chains trapped inside a gel; but the self-diffusion coefficient D should be comparatively easy to measure. Our prediction is D"-'M-2. The extra mechanical dissipation in the gel due to the presence of the chains P would also be of interest, but it would be extremely hard to measure it at the very low concentrations which are required to have independent chains. In any case the theoretical analysis of the viscosity 'Y} appears difficult, and the molecular mass dependence of'Y} remains to be found.

What are the weaknesses of the model?

(1) At first sight our introduction of "defects" appears rather arbitrary. However, all our results are insensitive to the details of the reptation process: what is important is to take into account the sequestration in "tubes" and the fact that the total chain length is conserved.

(2) Back-flow effects have not been included. For a chain inside a conventional solvent, they are in fact very important2: However, in a gel, we expect that back flow around a moving object decays exponentially within a finite distance t-, as first shown by Debyel4 : thus our results are probably not too seriously affected by backflows, provided that the mean radius of the chain is much larger than t-.

(3) Excluded volume effects are not taken into account: they might modify slightly the exponents of the various laws. Also no special consideration has been

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578 P. G. DE GE)l"NES

given to the knots which the chain may perform on itself. This is probably not too serious since these knots can also be unwound by our reptation process.

B. Conjectures on Polymer Melts

Obviously the time Tr is closely related to the "terminal time" derived from mechanical studies on concentrated or molten polymers. The M3 dependence of Eq. (V.12) is in fact not very far from the experi­mentaJ3 (M3.3) exponent.

It is also tempting to interpret Td as the char­acteristic time for the glass-rubber transition. However, all these interpretations are merely guesses at the present stage: The transposition of our results (for one chain in a fixed gel) to a system of many mobile chains is extremely delicate and will require separate theo­retical studies. If it turned out that the exponents are the same for both problems, we would come out with laws which are very different from the predictions by Bueche6 and Graessley,7 for both D and T r•

ACKNOWLEDGMENT

The present author has greatly benefited from dis­cussions with G. Agren, who is currently attempting to test some of these ideas by computer simulation.

APPENDIX A: MEAN SQUARE CURVILINEAR DISPLACEMENT OF ONE MONOMER

We consider a gas of defects with positions ml' , 'mi' , , along an infinite chain P. The curvilinear abscissa of monomer (n) may be written as

where

Sn= const+b L lJ(mi- n ), i

IJ ( x) =! sign ( x) . We then have

(c2(t»= (1/b2) ([Sn(t) -sn(O) J2)

= L «(lJmi(t)-lJmi(O) (IJmj(t)-lJmj(O)).

'if

Different defects being uncorrelated, only the terms i=j remain. The defects which contribute are those for which IJm(t) is of opposite sign to IJm(O). This leads to

(c2(t) )=ap L [p(mO I m't)+p(m'O I mt)J, m>n;m'<n

where p(mO I m't) is the probability for one defect initially at m to move to m' in time t. To compute (c2 )

it is convenient to differentiate it with respect to time and to make use of the diffusion equation

ap/at= (ll/a2) (a2P/am2)

= (ll/ a2) (a2p / am'2)

= - (ll/ a2) (a2p / amam') .

Then the integrations on m and m' are easily done,

and give (a/ at) (c2) = 2p(ll/ a) p (nO lilt).

Using the standard diffusion result

p(mO I m't) = (27l'U)-1/2 exp[ - (m-m')2/2u2J,

(u2 = 2D.t/ a2)

and integrating, one obtains Eq. (IV.3).

APPENDIX B: CURVILINEAR MOTIONS OF ONE MONOMER FOR A FINITE CHAIN IN

AN INFINITE TUBE

Although the physical content is very different, the calculation of ([Sn(t) -Sn(O) J2) is formally identical to the appendix of Ref. 15; thus we shall give here only the main lines of the argument. The equation for curvilinear motions, under applied forces in (we take in colinear to the tube at each point) is derived from Eqs. (II.S) and (I1.6), and reads

. as" ( pb2

1 a2Sn )

8n=;;t=bln=D. al?liT in+-;Zan2 •

The responses (.~n) to a time dependent set of forces in may then be obtained by an expansion in Fourier series. The result has the causal form

'~n(t) = L [ dt'{3nm(t-t')im(t') , tn -Xi

where the response functions (3nm are given explicitly by

where 0+(1) is a delta function with its peak at t= +0; P is a positive integer. From the {3's one derives correla­tion functions by the Kubo theorem

(1)0)

and by two time integrations one obtains, for the diagonal term (n = m) ,

([Sn(O) -Sn(t) J2)= 2p~~ a

p

The second term in this equation is dominant at times t< Td , and is then equivalent to Eq. (IV.3). The first term is the main contribution at large times (t> Td )

and is the only term kept in Eq. (\'.6).

APPENDIX C: NUCLEAR RELAXATION BY REPTATION

To be specific, let us assume that the chain P carries some protons: their spin-lattice relaxation includes contributions from the modulation of their dipolar interactions by the motions of the chain.16 Among these

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REPTATION OF A POLYMER CHAIN 579

motions we shall consider only the slow reptations, omitting all the more rapid (and comparatively trivial) components such as methyl group rotation, etc. The reptation might indeed become dominant if the nuclear frequency Wn is sufficiently low.

The spin-spin couplings which are modulated may involve either two protons of P [case (a) ] or one proton of P and one fixed spin belonging to the obstaclesl7

[case (b)]. In both cases we have to follow a moving object linked to a given monomer (n) : a pair of protons (i and j) for case (a), or a single proton for case (b).

\Ve shall now restrict our attention to a medium with densely distributed obstacles (i.e., a chain P moving inside a rubber), for the following reason: Consider a monomer (n) which, at time t=O, has a certain position and orientation. After a rather large time t, it has moved by reptation. The dipolar interaction of interest JeD (t) is then changed strongly, and becomes uncorrelated with JeD (0), except if by chance (11) has returned to its ini tial position (or wi thin a small in terval OC from it): The corresponding probability is Pt(c=O)Oc. If the surrounding medium is rigid and dense, when monomer (n) returns to its initial position, it also recuperates its initial orientation: thus, the correlation function for the dipolar interaction JeD may be taken of the form:

(JeD (O)JeD(t) )=constpf(c=O).

Using Eq. (IV.4) (for large t) this gives

(Jen (0) JeD (t) )=constt-1/4.

The spin-lattice relaxation rate liT! is proportional to the Fourier transform of correlation functions of this type, taken at the nuclear frequency Wn or at the double frequency 2wn • Since

this leads to

f'" dtt-1/4 COSW/= constw-3/4, o

l/T! = constwn- 3/4 (for wn---tO) .

This exponent may also be derived, in case (b), bv a different argument: One writes the relaxation rate liT! in terms of a probability distribution '

P(r,O; r/l; rj)

for finding spin (i) at ri at time 0 and at r/ at time t, in the presence of the (fixed) spin U) with which it interacts. One then makes the approximation

P(riO; r/t; rj)---tw(riO; r/t) ,

where w(riO; r/t) is the self-correlation function for which a second moment is given by Eq. (IV.6). What is needed is the function w for space intervals

r=r/-r,:

which are comparable to an interatomic distance. An explicit calculation of 7V for such intervals (and large t) givesl8

w(riO 1 r/t) = (3/27rrab)pt(c=0).

Again this leads to a 1 t 1-!/4 dependence for the correla­tion function at large times.

To conclude this Appendix, we should emphasize that the wn -

3/4Iaw for the relaxation rate refers only to

one highly idealized limit: it may be unobservable in practice for various reasons:

(1) Effects related to the detailed monomer struc­ture, to the chain rigidity, etc., have not been con­sidered.

(2) In time intervals t"-'wn- 1 the reptation process must involve many monomer units [p(~t)!/2»1] for the asymptotic laws to be meaningful. This may be hard to obtain, even with high temperatures and low fre­quencies.

It is also not clear whether these results remain meaningful for the related, but different, problem of polymer melts. Here a recent experiment!9 indicates 1/T!"-'wn- o.5 in the low-frequency limit.

1 P. E. Rouse, J. Chern. Phys. 21, 1272 (1953). 2 B. H. Zimm, J. Chern. Phys. 24, 269 (1956). 3 J. D. Ferry, Viscoelastic Properties of Polymers (Wiley, New

York, 1970), 2nd ed. 4 A recent review on this field is G. C. Berry and T. G. Fox,

.\dvan. Poly. Sci. 5, 261 (1968). 5 D. ,"V. McCall, D. C. Douglass, E. W. Anderson, J. Chern.

Phys. 30,771 (1959). 6 F. Bueche, J. Chern. Phys. 20, 1959 (1952). F. Bueche,

The Physical Properties of Pol3'mers (Interscience, New York 1962) .

7 W. W. Graessley, J. Chern. Phys. 43, 2696 (1965) and 47, 1942 (1967).

8 A. S. Lodge, RheoL Acta 7, 379 (1968). 9 A. J. Chompff, W. Prins, J. Chern. Phys. 48, 235 (1968). 10 Note that, in the limit of large N which is of interest here,

we make no difference between n = 0 or n = 1. For the statement of boundary conditions leading to the modes of Eq. (IT.7) it is more convenient to put the origin at n=O.

11 For a review on the Holstein 2'l1arkoff method, seefor instance S. Chandrasckhar, Rev. Mod. Phys. 15,2 (1943).

12 P. G. de Gennes, Physics 3,37 (1967). 12. '"'" e assumed here that, if monomer (111) returns to its

initial position in the tuhe, it also recuperates the exact initial orientation. Tn fact, the arguments of Secs. v.n and V.C require only that (III) recuperates a finite fraction of its initial orientation. This fraction will appear as a time-independent scaling factor in Eq. (V.10).

13 Gn", is a statistical weight per monomer while W is a weight per unit length.

14 P. Debye, J. Chern. Phys. 14,636 (1946). 15 E. Dubois Violette, P. G. de Gennes, Physics 3, 181 (1967). 16 See, for instance, A. Ahragam, Principles of N1Iclear Mag-

netis1ll (Oxford U. P., London, 1961), Chap. 8. 17 The fixed species might he a nuclear spin, or a paramag­

netic impurity of spin S (acting through terms such as S zT + in the dipolar interaction).

18 Again it must be emphasized that w is strongly non-Gaussian: imposing a Gaussian form, one would come out with a very different correlation function (~. I t I -3/8).

19 R. Lenk, J. P. Cohen .\ddad, Solid State Commun. 8, 1864 (1970) .

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