statistics of the entanglement of polymers: unentangled loops and primitive paths

Statistics of the entanglement of polymers: Unentangled loops and primitive pathsEugene Helfand and Dale S. Pearson

Citation: The Journal of Chemical Physics 79, 2054 (1983); doi: 10.1063/1.445989 View online: http://dx.doi.org/10.1063/1.445989 View Table of Contents: http://scitation.aip.org/content/aip/journal/jcp/79/4?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Theory of nanoparticle diffusion in unentangled and entangled polymer melts J. Chem. Phys. 135, 224902 (2011); 10.1063/1.3664863 Viscoelasticity of Entangled Semiflexible Polymers via Primitive Path Analysis AIP Conf. Proc. 982, 536 (2008); 10.1063/1.2897854 Viscoelasticity and primitive path analysis of entangled polymer liquids: From F-actin to polyethylene J. Chem. Phys. 128, 044902 (2008); 10.1063/1.2825597 Primitive Chain Network Model for Entangled Polymer Blends AIP Conf. Proc. 708, 261 (2004); 10.1063/1.1764134 Stress relaxation in unentangled and entangled polymer liquids J. Chem. Phys. 104, 5284 (1996); 10.1063/1.471257

This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP:

150.135.239.97 On: Mon, 07 Jul 2014 06:28:37

http://scitation.aip.org/content/aip/journal/jcp?ver=pdfcovhttp://oasc12039.247realmedia.com/RealMedia/ads/click_lx.ads/test.int.aip.org/adtest/L23/1691523420/x01/AIP/JAP_HA_JAPCovAd_1640banner_07_01_2014/AIP-2161_JAP_Editor_1640x440r2.jpg/4f6b43656e314e392f6534414369774f?xhttp://scitation.aip.org/search?value1=Eugene+Helfand&option1=authorhttp://scitation.aip.org/search?value1=Dale+S.+Pearson&option1=authorhttp://scitation.aip.org/content/aip/journal/jcp?ver=pdfcovhttp://dx.doi.org/10.1063/1.445989http://scitation.aip.org/content/aip/journal/jcp/79/4?ver=pdfcovhttp://scitation.aip.org/content/aip?ver=pdfcovhttp://scitation.aip.org/content/aip/journal/jcp/135/22/10.1063/1.3664863?ver=pdfcovhttp://scitation.aip.org/content/aip/proceeding/aipcp/10.1063/1.2897854?ver=pdfcovhttp://scitation.aip.org/content/aip/journal/jcp/128/4/10.1063/1.2825597?ver=pdfcovhttp://scitation.aip.org/content/aip/proceeding/aipcp/10.1063/1.1764134?ver=pdfcovhttp://scitation.aip.org/content/aip/journal/jcp/104/13/10.1063/1.471257?ver=pdfcov

Statistics of the entanglement of polymers: Unentangled loops and primitive paths

Eugene Helfand and Dale S. Pearson

Bell Laboratories. Murray Hill, New Jersey 07974 (Received 10 March 1983; accepted 10 May 1983)

Attempts to apply the reptation and tube model to describe polymer disentanglement and associated relaxations have had wide success. However, questions have arisen in various applications as to the statistics of entanglement of a random walk with an obstacle net. Calculations are presented of a number of probabilities which characterize the degree to which a polymer, represented by a lattice walk, entangles with an array of obstacles. In particular, we have calculated the number of ways that such a walk can form unentangled closed loops of various types. If one reels in a general random walk from its ends, pulling out unentangled loops, one is left with the so-called primitive path, which is taken to represent the path of the tube. The probability that an N step random walk has a K step primitive path has been determined. Asymptotic formulas for this probability are presented.

I. INTRODUCTION

The fact that a system of polymer molecules forms an entangled mass is responsible for many of the unique properties of these materials. The necessity for disen-tanglement as the final stage of the relaxation process accounts for the slowness of diffusion and flow, for long memory effects, and for the prominent nonlinearities in rheological behavior. Fortunately, in the past decade much progress has been made in describing the disen-gagement process. Use is made of de Gennes' concept of a single molecule seeing the entanglements as a tube-like constraint from which it escapes by a reptative mo-tion. 12 Doi and Edwards have employed these concepts to create a rheological theory. 3 As efforts have pro-ceeded to apply these ideas with further refinement and to more complex systems a need has developed for a fuller description of the polymer molecule in the entangle-ment net. One can cite a number of works in which this is clear. Doi4 and Graessley5 have shown that common fluctuations of the polymer in the tube (those within a standard deviation) make important contributions to tube disengagement. De Gennes, 6 and Doi and Kuzuu, 7 in con-sidering the relaxation of star-shaped macromolecules, and Curro and Pincus, 8 in studying the relaxation of dangling ends in a rubber network, have to look at less probable fluctuations. Other papers have raised ques-tions about the response of the tube to stress, the relax-ation of the tube, and the need for effective fields to keep the polymer stretched out in the tube. A number of ef-forts have been made to address these questions and test the existing hypotheses. One approach has been to con-duct computer simulation studies. 9-12 Another is by ana-lytic solutions for the properties of model systems. 2.5.6

In this paper we shall present some exact results on the statistics of the entanglement of polymer molecules with an obstacle net. In the next section we define the model. The polymer states will be represented as ran-dom walks on a lattice. The lattice is permeated with a net of lines (points in two dimensions) with which the walks can entangle. In Sec. III we look at the probabil-ities of following certain types of loops which do not en-tangle with the net. Following Doi and Edwards3 we re-gard the "primitive path" as representing the axis of the

tube. It is taken as the path of the walk which remains after the chain has been reeled in as much as possible from its ends10 (cf. Fig. 1). In Sec. IV a calculation is presented of the probability that an N step random walk has a K step primitive path. This probability is a strong-ly peaked function; i. e., the root-mean-squared fluctua-tions of the length of the primitive path about the average value is O(NI/2 ). As indicated above, the less likely fluc-tuations are also physically important, so a formula for the full distribution is presented. The paper concludes with discussion of these results in relation to earlier works. In the body of this report equations for the re-quired counts of random walks are derived and solved for in terms of generating functions. Inversion of the gener-ating functions to obtain the counts is done, in most cases, by contour integration, details of which are pre-sented in the Appendix.

II. THE MODEL

The states of the polymer molecule will be represented by unrestricted (i. e., no excluded volume) random walks on a lattice. The steps of such a walk are meant to

o

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 en-tangle. 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.

2054 J. Chern. Phys. 79(4), 15 Aug. 1983 0021-9606/83/162054-06$02.10 1983 American Institute of Physics


150.135.239.97 On: Mon, 07 Jul 2014 06:28:37

E. Helfand and D. S. Pearson: Entanglement of polymers 2055

represent sections of the chain and not the real polymer bonds. The lattice will be specified to have a coordina-tion number q, meaning that after each step the next step can go in any of q directions (including the direction from which the walk came). No more detailed speCification of the lattice need be given, not even its dimensionality.

We will further specify that there is a net of fixed ob-stacles with which the walk can, potentially, entangle. This is illustrated in Fig. 1 for the case of a square lattice. The circles represent the lattice vertices and the crosses (at half integer lattice points) form the en-tanglement net. The entanglement net for a simple cubic lattice is also easily visualized. The vertices of such a lattice can be specified by triplets of integers (t, m, n), with all integer values of 1, m, and n allowed. Consider a lattice of points located at the integer plus (t, t, t) triplets. The bonds of the latter lattice form the en-tanglement net.

A simple generalization for the construction of the en-tanglement net corresponding to any lattice is as follows: Consider the space as filled with a solid material. For the given lattice drill channels along the bonds, i. e. , the allowed steps. The remaining solid serves as the obstaele net. Topologically it can be shrunk down to a set of lines, but for the present considerations it is not necessary to do so. Research is in progress on the gen-eral speCification of lattices corresponding to obstacle nets. 13

The case of more widely spaced entanglements is of interest in determining the effects of concentration on the tube and on relaxation. The statistics for this case will be discussed in a subsequent report.

III. STATISTICS OF UNENTANGLED LOOPS

It is important to define and count certain types of unen-tangled loops. Such loops can be pulled out, and thus are not part of the primitive path. From a physical point of view, unentangled loops quickly form and disappear, and so are not as slow to relax as the constraints of the prim-itive path.

A loop is a walk which begins and ends at the same pOint. In general we shall say that a loop is unentangled if it retraces its own path (perhaps several times) in re-tUrning to the origin. If an unentangled loop is reeled in from its ends it can be pulled completely down to a pOint. In counting numbers of loops or other types of walks we

B(N-2)

A (N) =

FIG. 2. An A loop begins at the origin po. goes off in any of q directions to PI> forms an N - 2 step B loop off Pl. and returns to Po on the last step. This figure is a schematic of Eq. (3.3).

B (N) =

FIG. 3. An N step B loop begins with a smaller. 1 step. Bloop (0 " 1 " N - 2). and ends with an N -1 step A loop. The first step of the final A loop can go in only q -1 directions. This figure is a schematic of Eq. (3.4).

will conSider that a walk has a direction associated with it, to avoid having to divide all counts by a symmetry factor of 2. This does not effect probabilities. Define the following quantities and types of loops:

(i) A(N} is the number of N step walks which return unentangled to the origin for the first time on step N. We will call these A loops.

(ii) B(N} is the number of walks which return unen-tangled to the origin Po on step N, not necessarily for the first time. Furthermore, on the first step from Po, and after every subsequent unentangled return to the origin, one of the q possible directions for leaving Po is prohib-ited. These are termed B loops, and are of central im-port in the subsequent discussion.

(iii) C(O, N} is the number of walks which return un-entangled to the origin on step N, not necessarily for the first time. These will be called C loops.

For all unentangled loops N must be even. For N = it will be convenient to define

A(O}= , (3.1) B(O)=C(O,O}=1. (3.2)

An A loop of N steps may be described as follows. It takes its first step from Po in any of q directions to Pl. On step N -1 it returns unentangled to Pl. If it was on P 1 unentangled on any earlier step it could not go from P 1 to Po, or this would have represented an earlier un-entangled return to the origin. Thus the loop off P j is an N - 2 step B loop. Finally on step N the walk goes from P 1 back to Po to complete the loop. It follows that A(N} is given by

A(N} = qB(N - 2} , (3.3) as is illustrated in Fig. 2.

A B loop of N steps may be described in the following general terms, as illustrated in Fig. 3. It is composed of a B loop of 1 steps, where 1 is even and O:S 1:S N - 2. On step 1 + 1 the walk leaves the origin, not to return again unentangled until step N; i. e., the final portion is an N - 1 step A loop. However, this A loop starts in only q -1 of the possible q directions since one direction is prohibited for step 1 + 1. Thus an equation for B(N}, for N"22, is obtained by summing over the allowed 1 the

J. Chern. Phys., Vol. 79, No.4, 15 August 1983


150.135.239.97 On: Mon, 07 Jul 2014 06:28:37

2056 E. Helfand and D. S. Pearson: Entanglement of polymers

product of the numbers of ways of constructing the A-loop and B-loop pieces:

-1 N , B(N)=-q -L: A(N-l)B(l) ,

q zo (3.4)

where the prime on the summation indicates a sum over even I only. The term I =N may be included because A(O)=O.

To solve Eqs. (3.3) and (3.4) it is convenient to define a generating function

00

A(x) = L:' A(N)0 , (3.5) NcO

and likewise for B(x). Equations (3.1)-(3.4) are easily converted to

A(x)=qx2B(x) ,

B(X) = 1 + q - 1 A(x)B(x) . q

The solution is

1 q {" [ J 112} A(x)="2 q-1 1- 1_4(q_1)x2j "'

A 1 {[ ~11/2} B(X)=2X2(q_1) 1- 1-4(q-1)x2J .

(3.6)

(3.7)

(3.8)

(3.9)

A(N) and B(N) may be obtained as the coefficient of 0 in an expansion. Before stating the result, note that there are qN distinct N step random walks, so that the probability of an N step walk forming an A or B loop is obtained by dividing by this factor:

PA(N) = A(N)j qN (3.10)

_ (q _1)


B~(nK_ll PK

B(nKl (q-Il

PK-I

FIG. 4. The general decomposition (described in the text) of a random walk into unentangled loops and steps of the primitive path.

termed the primitive path (cL Fig. 1). From the discus-sion below will emerge an alternative to the "Edwards-Evans reel" as a means of identifying, for any walk, the primitive path.

The object of this section is to count the number of N step random walks which have a K step primitive path, C(K, N). Such a walk may be generally described as follows (cr. Fig. 4). It begins at point Po and forms an no step unentangled loop. The number of ways of doing this is C(O, no), previously determined. The range of no is Os no s N -K; i. e., there may not be such a loop, or the loop may be as large as to include all steps ex-cept the minimum number needed to follow with a K step primitive path. On step no + 1 the walk goes in one of the q possible directions to a point Pj' There is a feature that distinguishes this step from any of the previous steps that may have been taken off Po after completing an unentangled loop of size smaller than no' It is that never again will the pOint Po be visited unentangled. In particular, this means that the step after no + 1 from Ph and all subsequent steps from PI after completing an unentangled loop, cannot go in direction P 1P O' Starting after step no + 1 the walk may form an nl step unen-tangled loop returning to Pj' This must be a B loop be-cause of the restriction on steps in direction, PjPo It can be formed in B(nj) ways. The range of nt is Os nj s N -K - no; i. e., there may be no B loop, or it may use up to all the steps except the minimum number needed for a K step primitive path and the no steps used in the C loop off Po. On step no + nj + 2 the walk goes from PI to P 2, i. e., in any of q - 1 directions (not PjPo), never again to visit PI unentangled. It then forms a B loop off P 2 of n2 steps (OSn2SN -K -no -nl)' This continues until the walk arrives at PKon step no + ... +nK-l +K; and forms, finally an nKstep(nK=N-K-no- -nK_j) B loop, returning to P K on step N. On the basis of this argument one can write as a formula for C(K, N) and K?:.1,

x C(O, nO)B(nl)" B(nK_t)B(N -K - no _ ... - nK_l) . (4.1)

We could proceed in terms of this equation, but it is somewhat easier to work with an equivalent difference equation, derived as follows. Consider that after an 1 +K -1 step walk one is at point P K-t of a K -1 step primitive path. On step 1 + K one can go in any of q - 1

directions (not P K-tP K-2) to point P K, never to return to P K-1 unentangled. Finally, the walk forms a N -K -l step B loop off P K The number of N step walks with K step primitive paths is given by (K?:. 2, N?:.K)

C(K,N)=(q-1) ~'B(N-K-l)C(K-1,l+K-1). (4.2) 1=0

For K = 1 the prefactor on the right should be q rather than q - 1, because the walk can leave the origin Po in q directions. This equation is easily solved if one uses M = N -K and defines the generating function

... C(K,x)=L' C(K,K+M)xM. (4.3)

M=O

The solution for the generating function is (K?:. 1)

C(K, x) =q(q _l)K-! [B(x)]K C(O, x) , (4.4)

q {1_[1_4(q_1)X2)1/2}K

=2 K-1X2K (q_2)+q[1_4(q_l)x2)112 (4.5)

[Equations (4.4) and (4.5) are correct for K == 0 if the right-hand side is multiplied by a factor (q -l)/q.) C(K, N) is recoverable by means of the complex x-plane integral

1 f C(K,x) C (K, N) = -2. dx --:::r-rr+r 1Tt x

(4.6)

with contour counterclockwise about the origin. The in-tegral is of a form known to be given in terms of hyper-geometric functions, 14 but we have not worked this out in detail. For small N -K it is easiest to expand the gen-erating function in powers of x2 For large N it is sim-plest to derive directly asymptotic forms for C(K, N) from the integral, details of which are presented in the Appendix.

The major asymptotic results for large N are as fol-lows. Define K = K/ N. For K = 0(1), although possibly quite small,

4qK [ 1 J1I2 1 Pe(K,N)-q_2+qK 21T(1_K2) ]7T'l

{[1 q2 1 2

xexp -N 2" log 4(q -1) +2" log (l-K )

K K 1 + K]1 - - log (q - 1) + - log -- . 2 . 2 1-K

(4.7)

This function is strongly peaked at K = [(q - 2)/ q )N, with fluctuations inK of 0(N!/2). In this regionPC

2058 E. Helfand and D. S. Pearson: Entanglement of polymers

z ~,.., ........

W 0 a. Ol o

o

~

\ GAUSS IAN

\ \ \

\ \

EXACT ~ ~

/ 7

o~~~~ __ ~ __ ~ __ ~~~ __ ~~~~~~

0.0 0.2 0.4 0.6 0.8 1.0

KIN

FIG. 5. A comparison of the exact form (shown as the solid curve) of Pc(K, IV) , Eq. (4.7), and the Gaussian approximation, Eq. (4.8). The limit of N-ao is assumed, or, equivalently, the plot is of the coefficient of N in the exponent.

{ [1 q2 1 2

X exp - N "2 log 4 (q _ 1) +"2 log (1 - K )

-i log (q - 1) + i log ~ ~ : J}. (4.9) As K goes from O(N) to 0(1) the pre factor of the expo-nential goes from 0(}T1/2) to 0(}T3/2).

Earlier in this section a description was presented of the general walk in terms of unentangled loops and other steps. It is clear that the primitive path is composed of these steps remaining after all unentangled loops are excised, which is the same as pulling loops out by reel-ing in the ends. It also becomes clear that what remains for the primitive path is a nonreversal random walk; i. e., a random walk with the one constraint that no step instantly reverses onto the previous step. The statis-tics of such walks have been studied. 15 The root-mean-squared end-to-end distance of a [(q -2)/q]N step non-reversal random walk is the same as that of an N step unrestricted walk. This is in accord with Eq. (4. 8), since every walk and its primitive path must have the identical end-to-end vector.

V. DISCUSSION

It was mentioned earlier that there have been previous studies of the same model employed in this paper. de Gennes discussed a calculation of what we term C(O, N) in an Appendix of Ref. 6. In his book2 he pOints out that the earlier calculation was incorrect, but that the asymp-totic form

C(O, N) - f3(N) exp( - aN) , (5. 1)

where f3(N) has a power law dependence on N, still holds. Reference 6 gives a as

1 a = 2" log (q log 2) , (5.2)

which is a DG = O. 712 62 for q = 6. (A corrected value of a is not presented in Ref. 2.)

Evans and Edwards10 have determined the probability distribution of the number n of steps of the chain per primitive path step, which should be p B(n - 1) for walks with N n. From this they report that for q = 6, a EE ::::0.7.

Our formula for a is

(5.3)

which is aHP= 0.29389 for q = 6. The discrepancy is considerable, and it is worthwhile investigating whether it is due to differences in definitions, whether the simu-lations need refinement, or whether there is a defect in the present theory.

Because unentangled walks cannot form loops the problem of counting them is equivalent to counting ran-dom walks on Cayley trees. After we submitted this article, Needs and Edwards pointed out to the authors that the latter problem has been addressed by Economou16

in the context of electronic motion on a Cayley tree, treated in the tight binding approximation. The elec-tronic Green's function which Economou derives is re-lated to our generating function C(K, x).

APPENDIX: CONTOUR INTEGRALS

In this Appendix we shall perform some of the contour integrals necessary to invert the various generating functions.

1. C-Ioop probability Pc (O,N)

According to Eqs. (3.18)-(3.20) Pc(O, N) is given by

2(q-l)j,dx 1 pdO,N)= 2rriqN yxmr(q_2)+q[I_4(q_l)x2]1I2'

Make the change of variables (AI)

(A2)

to obtain

2(q -1) [4(q _1)]


2 q -1 [4(q -1)]

statistics of the entanglement of polymers: unentangled loops and primitive paths

Documents