Flory approach for polymers in the stiff limit

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Flory approach for polymers in the stiff limit

H. Nakanishi

To cite this version:

H. Nakanishi. Flory approach for polymers in the stiff limit. Journal de Physique, 1987, 48 (6),

pp.979-984. �10.1051/jphys:01987004806097900�. �jpa-00210527�

Flory approach ^for polymers in the stiff limit

H. Nakanishi (~)

Collège ^de France, Physique de la Matière Condensée (+), ^11, place Marcelin-Berthelot,

75731 Paris Cedex 05, ^France

(Reçu ^{le 1 er} juillet 1986, révisé le 30 janvier ^1987, accepté ^{le 24} f6vrier 1987)

Résumé.

On étudie, ^dans l’approximation ^de Flory, la conformation de chaînes semi-rigides, ^linéaires

branchées dans la limite de forte rigidité. ^{On montre} que la dimension marginale dc ^associée

^{effets de}

volume exclu est en général réduite dans cette limite. En particulier, pour

chaîne linéaire

bon solvant

on

trouve dc

^{2 en accord avec} des résultats récents obtenus numériquement par Lee ^et Nakanishi.

Abstract.

We present

Flory approximation for the conformation of semi-flexible linear and branched

polymers ^{in the} extremely stiff limit. We show that the upper marginal dimension dc for the excluded volume effect is in general reduced in the stiff limit. In particular, ^for

linear chain in

good ^solvent

^find dc

^{2 in} agreement ^{with the recent} Monte Carlo result of Lee and Nakanishi.

Classification

Physics ^Abstracts

05.40

05.30

61.40

1. Introduction.

The conformation of semi-flexible long ^chain poly-

mers has been of considerable interest from theoreti- cal [1-4] ^and experimental [3-4] points ^{of view.}

Recently, ^Lee and Nakanishi [5] studied the cros- sover behaviour of such a chain from its extremely

stiff limit using the biased self-avoiding ^{walk model} (BSAW) [6], ^and pointed ^{out a} dramatic difference in the effects of excluded volume in d

2 dimensions and d

3 dimensions. They concluded, ^{based on}

their Monte Carlo simulations, ^that the excluded volume effect is irrelevant in the stiff limit in d = 3.

In this paper, ^we present ^an analysis ^{based on} Flory approximations [7-9] for the conformation of stiff linear and branched polymers. ^This analysis

results from an extension of the form of excluded volume parameter ^used by Odijk [10] ^for polyelec- trolytes ^and also follows the work of Schaefer, Joanny and Pincus [3] who, however, ^{did not discuss} the present ^{limit of extreme} stiffness. Although Flory approximations ^{are known to} rely ^on

«

accidental

cancellations of neglected ^{terms in the}

free energy, their predictions ^{have been} surprisingly

(t) ^{Present address :} Department ^of Physics, ^Purdue University, ^W. Lafayette, IN 47907 U.S.A.

(+ ) Unité Associ6e

CNRS (UA ^N° 792).

accurate for quantities ^{such as the} Flory exponent v and for the value of the _upper marginal ^dimension dc ^for simple polymeric systems [9]. ^In fact, ^{the use} of Flory approximations ^{for the} evaluation of

dc is much like the Ginzburg ^{criterion in critical} phenomena ^{and is} quite reasonable from the theor- etical standpoint ^also.

The BSAW model is simply ^{that of an isolated} self-avoiding random walk on a lattice with a variable parameter characterizing the stiffness. The statistical

weights of BSAW are described by ^the probability p

of taking ^a gauche step (a bend in the walk) ^{and the} probability 1- p ^{of a trans} step (continuing straight), ^{at each} stage of Monte Carlo simulation.

Taking ^the step length (monomer dimension) ^{as a,}

the persistence length ^is proportional ^to lp

^{alp. In}

d dimensions, ^the generalization ^of Odijk’s ^excluded

volume (per persistence length) ^{is then}

This result will be used for the discussion of stiff chains in good solvents and an analogous ^{one will be}

used for 0 solvents. While these results make sense

physically ^{only for d , 2 because the BSAW model}

itself makes sense only ^for integer dimensions of two or greater, ^{it is} possible ^to give ^{a formal} analysis ^for

an arbitrary ^{value of d}

0. Indeed, e.g. for the 0 case, such an analysis may have ^a valid basis for d>, 3/2 as will be seen below. Of _course, direct

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphys:01987004806097900

980

physical conclusions are to be drawn only ^for integer

values of d>, 2. The following analyses always

assume the limit of Np > ^{1 where N is the number of} steps ^{in the chain} (thus proportional ^{to the molecular} weight) ^{so that there are a} large ^{number of}

^{blobs »}

of linear size of the order of the persistence length.

2. Linear chain in good ^solvent.

We first discuss the case of a linear chain in a good

solvent. In this _case, the Flory ^free energy ^can be written as

where the elastic contribution f el ^is given approxi- mately by using ^the leading ^{term in the free, stiff}

chain result [1] ^for Np > 1, ^as

where R - (R2) ^{1/2 is the} root-mean-square ^end-to- end distance. The repulsive interaction contribution

fint ^is given using (1) ^as

to the leading ^{order in monomer} concentration.

According ^to the usual Flory approximation, ^one simply minimizes the free energy (2)-(4), ^{which will} yield

For d

3, ^this expression ^was already given by

Schaefer et al. [3].

Aside from its inherently approximate nature, the intended validity ^{of such an} expression ^{is limited}

from above by the upper marginal ^dimension de and from below by ^the neglect ^{of the} higher ^order

terms in fi.t. ^{In our} problem, it is further constrained because of the assumption (1). ^{Thus, in this} problem,

for the usual

Flory » ^{limit of} fixed p ^and large N, these constraints would restrict the validity ^of (5) ^to

2 , d , 4 (with possible logarithmic corrections at either end).

It is useful for later discussions to go through ^this argument ^{in some detail. Let x} be the first term in

fint (righthand ^{side of} (4)) ^and y be the ratio of ^two successive terms in fi,,t. The value of _y can be obtained from the following assumed form of the

expansion ^of /;nt ^{in terms} of the concentration of

«

rods _», c

Np/Rd :

where

This is consistent with the expected ^form of the third virial coefficient [3, ^11, 12]. ^{From this, we obtain}

To see the validity ^of (5), ^we substitute it into the

general expressions (7) ^{for x and y.} Requiring ^the

results to satisfy XF >> 1 ^and YF >> 1, ^{we obtain} 1 :5: d :5: 4, which then leads to the stated result.

On the other hand, if p is ^{fixed and we ask at what}

size N * the chain conformation crosses over from Gaussian to the excluded volume type (5), ^{then we}

must substitute the leading Gaussian form R oc (Nip a )112 ^{into (7). In this case,}

For the Gaussian form to be valid for N N *, ^we

must have _{XG -} 1 and YG -- 1 ^at N *, ^which gives

This results in N* - 1 /p ^{for d}

^{2 and N* ~} 11p3

for d

3, the latter of which was already given by

Schaefer et al. [3].

We _may, however, be interested in the

stiff » limit of N -> 00, p --. 0 with Np

^Const. (> 1 ).

This interest stems from the objective ^{to write a} scaling ^form [6] ^for (R2) ^{about the} singular point ^at

p

N - 1 = 0. The most obvious and natural variable is Np (the ^{number of}

^rod-like

segments ^{of the}

persistence length), ^{but the} arguments leading ^to equation (9) suggest ^{a second} important ^variable N/N *. Therefore, ^we propose ^{a «} scaling

^relation,

Unfortunately, equation (10) suggests ^that (R2) ^is

in general dependent ^on N and p separately ^{and that}

there is no reduction in the number of variables as true scaling relations would require. ^{However, for}

d

2, ^{the two} variables in f (a, f3 ) ^are identical and

(R2) ^{does have a} simple scaling ^{form. For d}

^{3, we}

have

0 in the stiff limit, ^{and once} again

(R2) ^{will have a} simple (now Gaussian) scaling

form. We show in figure ¹ qualitative diagrams illustrating ^{the crossover} between Gaussian and excluded volume behaviours for two and three dimensions.

We can also see that the upper marginal ^dimension de ^{for the} stiff limit is reduced to 2 by rewriting ^the

Gaussian values _xG and YG in equation (8) ^as

The requirement ^{for the} consistency of the Gaussian

approximation, XG ^{1 and} YG -- 1 ^{for fixed} Np, ^then

yields de ^{= 2 in} agreement ^{with the Monte Carlo}

result of reference [5].

Fig. ^1.

^Crossover phase diagrams for stiff chains in (a) d

^{2 and} (b) d

3 dimensions. The symbols ¹ d, ^{2 d and 3 d}

refer to the effective dimensionality ^{and RW and SAW refer to} the absence

the presence of the excluded volume behaviour.

3. Linear chain in 0 solvent.

For a linear chain in a 0 solvent, ^the leading ^{term in} tint ^{has the form}

and thus the Flory approximation gives

Note that an analog ^to the excluded volume f2 for this case is

which replaces a2 d for a flexible chain. Thus analyses

based on equation (12) ^cannot be valid for d 3/2 for small p ; for d > 3/2, however, ^{such an} analysis

may still make sense formally ^{as an} analytic ^continu-

ation in d into non-integral values in a sense

somewhat similar to the so-called E-expansions [9].

For the usual Flory limit, ^an analysis analogous ^to

that leading ^to equations (8) ^and (9) ^{then shows that} equation (13) ^{should be} limited in validity ^to

3/2 d 3 at best and that de

^{3 as usual for a 19} regime. ^{For the}

^{stiff »} limit, however, ^the only place ^where (13) might apply ^{is the} lower bound of this range, d

3/2, similarly ^{to the} good ^solvent

case.

In the Flory limit of fixed _p, the crossover size N * can be calculated similarly ^to equation (9) :

For d = 2, ^{we have} N*-p- 2 ^and thus, using equation (10), ^{we see that} a -+ 0 in f(a, (3 ) in ^the

stiff limit. This leads to a Gaussian scaling ^{form with} vanishing excluded volume even in two dimensions.

In fact, in the stiff limit de ^{is reduced to 3/2.}

4. Linear chain in poor solvent.

For a linear chain in poor solvent, ^we may suppose that the sign ^{of the} leading interaction term is

reversed,

This form will result if, e.g., ^one supposes ^an

impenetrable ^{rod with a} concentric sheath of attrac- tive region ^as representing ^{a stiff} segment ^which likes to stick to other stiff segments. Normally, ^such

a free energy is considered unphysical since it would be unbounded from below as R -> 0. To rectify ^this problem, ^one usually ^{includes a} positive higher

order term in fint. ^An alternative remedy ^is simply ^to impose ^a hard-core excluded volume as an external constraint for R. The actual minimum thus depends

in general ^on both this constraint and the higher

order terms of fint-

To see this, ^we first obtain the unconstrained minimum of f using ^{the sum of} equations (12) ^and (16) :

and (3) ^for fel. Following ^{de Gennes} [13],,we ^write

an expansion parameter,

and the minimization with respect ^{to this} parameter yields

For d 4, ^the righthand ^{side of} (19) ^is large ^and

negative. Thus, ^{we must have a 1} and conclude

that

982

with no dependence ^{on the stiffness} p. However,

since the hard-core excluded volume _per persistence length ^is f2, assuming isotropy (i. e. , ^as long ^{as there}

is no transition to a nematic state), ^{we must have}

Thus we surmise that the latter constraint actually

determines the allowed minimum f and ^as long ^{as the} polymer ^remains isotropic, equation (21) ^{is the} appropriate Flory expression ^{for a} stiff linear chain in poor ^solvents (with Np > 1). ^{In this case,} higher

order terms of lint ^have larger powers of p ^{and thus}

the term (16) ^is leading ⁱⁿ magnitude ^as required ^for consistency.

5. Branched polymer.

Let us now consider an isolated branched polymer

with fixed branching fugacity. ^To obtain the entropy

term lei

R/Ro in ^{the Flory} approximation, ^we

estimate the Gaussian radius Ro ^{in the} following

way. The number of monomers between two succes-

sive branching ^{units, n, is} proportional ^{to the} reciprocal ^{of the} (fixed) branching fugacity ^{A. Since}

the linear chain between such branching ^{units has a}

linear size of 6 where ^for sufficiently large n (i.e., ^for large np),

we have

For fixed, ^small A, then, ^{we have}

where N is essentially ^{the nuniber} of bifunctional

monomers. In a good solvent, ^{we also have}

as before, ^{to the} leading ^order assuming ^{that the}

interactions arise from stiff segments ^of any branch.

The minimization of F gives ^a Flory expression, ,

whose validity ^for fixed p ^and large ^{N, n} is restricted to 2 d 8 with d,

^{8 as} usual. Since we are

considering ^{fixed A, we} don’t need to worry about the fact that the linear parts of the chain become Gaussian above d

4 [14].

The crossover size N * is obtained by considering

The requirement ^is XG - 1 and YG > 1 ^at N *, ^which

for all d 8. In particular, ^{for d}

3, ^{N *} = p-6/5.

For d = 2, ^{we have} N * = p - 2/3, which should be taken to suggest that the chain shows excluded volume behaviour whenever the conditions for the present approximations ^{are met} (i.e., Np > np > 1).

The stiff limit for fixed ^A implies np - 0 and thus each linear segments ^must be considered to be a

rigid ^{rod of} length ^na instead of (22). ^{In this case the} problem simply ^{reduces to that of a branched} polymer ^with only branching units each of size

§

^{na. In} particular, Flory expressions ^{for such}

chains are well known [14] ^{and we have} d,

^{8 as for}

the Flory limit discussed above. More general ^situa-

tion of N, n and p being ^{all variable} parameters ^is left for a future study.

In a 0 solvent, ^{we take} f el ^{as in} (24) ^but fint ^{as in} (12). Upon minimization, ^we get [14]

which should be applicable only ^{for 3/2 d 6 for}

the Flory ^{limit with fixed small A. The crossover size}

is given by

for all d 6 ; ⁱⁿ particular, ^{N *} = p - 2 for d

^{3 and}

N

p -1 ^{for d}

^2. The situation is analogous ^to

the linear chain in a good ^solvent in that in d

2 there is a confluence of the two variables Np

and N/N *. ^{However, the} stiff limit with fixed A is

again ^{reduced to a branched} polymer ^with only branching ^units and dc, ^{remains to be} 6. The results for _poor solvents are similar to those for linear chains under the present conditions.

6. Concentrated solution of linear chains in good

solvent.

For a concentrated solution of polymers, ^{we must in} general ^consider screening ^{effects even} within the

Flory approximation. Extremely polydispersed ^solu-

tions of flexible, linear chains in good ^{solvents have} already ^been considered by ^{Daoud and} Family [15].

In this case, ^one considers a polydispersity ^{of the}

form of the one-dimensional percolation problem ^so

that the number of N-mers is proportional ^to

qN -1 (1 _ q )2 where q ^{is the} independent probability

that a bifunctional unit has reacted. Here, ^the screening coefficient NW 1 ^{of the} leading ^{term of} tint ^{can be} replaced by ^{N -1 1 when} considering ^a typical chain of size N since all relevant averages

(IV, NW, Nz) scale in the same way. (NW ^{refers to the}

so-called weight average and NZ ^{is the} z-average.) Moreover, ^the screening coefficients of higher ^order

terms in fint should also scale as N-1 ¹ [16].

The stiff chain analog ^{of the same} problem ^{is as} follows. ^{We have} fe, ^{as in} (3), ^but f;nt ^is given ^{to the} leading ^order, by

Thus the Flory expression ^{for the radius is} simply

The N-dependence ⁱⁿ (32) ^{is the same as for the}

flexible case [15]. ^{However, we also have}

For the Flory ^{limit of} fixed p, ^we have x oc y -1 ^{and it}

is impossible ^to have x > 1 and y > ^{1 at the same} time. This problem ^{was first} pointed ^{out to us} by

Daoud [16] ^{in the context of branched, flexible}

chains. This suggests ^{that the} expression (32) ^{has no} region ^of validity ^{in the} Flory limit. Since small p helps ^{to achieve} y > 1, ^one might suppose ^{that the} stiff limit would allow this for a range of d ; however,

we note that,

in the stiff limit, which would suggest ^{the window to} be 2/3 d 2. Since the excluded volume of the form (1) ^{cannot be valid for d 2,} this scheme cannot turn (32) ^{into a} consistent approximation [17].

The consideration of the Gaussian limit reveals that d,, ^{= 2 for both the} Flory ^{and stiff} limits ; ^thus

there is no reduction in d,.

7. Concentrated solution of branched polymers ⁱⁿ good ^solvent.

Lastly, ^{it is} possible ^{to discuss a} polydispersed ^melt

of branched polymers ^{as well.} Again ^{we treat the} good ^{solvent case} only, ^{and assume} that the branch-

ing fugacity ^is fixed and small so that the size N

essentially ^counts only the bifunctional monomers

(which ^are stiffi. ^Since N z =#= NW ⁱⁿ general, ^{when we}

consider a typical polymer ^{of size N -} Nz, ^{we must}

know the particular ^{form of} polydispersity ^{for the} problem ^{at hand. Let us thus assume a} particular

relation NW ^oc Ns ^{for our} problem. Then, using (24)

for tel ^{and, to the} leading order,

we obtain a Flory expression ^{for the stiff} analog :

Assuming ^the screening coefficient of the k-th order term to be N k - 2IN k - ^{[16], we} would then have

Thus, ^{for the} Flory limit, ^once again (36) ^{appears to}

have no region ^of validity ^as pointed ^out by ^Daoud

and Coniglio for the flexible case [16]. The stiff limit with constant branching fugacity ^{does not} help ^this

situation because, ^{as in} 5, ^this limit reduces to branched polymers ^with only branching ^{units. More} general situation with A also being ^{a variable} may,

however, resolve this problem ; this is left as a future

study. ^{We also have}

and thus N * = p - 4/3 ^{for d}

^{3 and N *} =p- 1/2 ^for

d

2 if the mean field percolation value of s = 1/2

were used. The latter should be interpreted ^as suggesting ^{that the} typical chain with Np > np >> ^{1 is} always ^of excluded volume type.

8. Summary.

In summary, by ^a Flory approximation applied ^to

non flexible polymers ^we find that in general ^the

upper marginal ^dimension dc for the excluded vol-

ume effect is reduced in the stiff ^{limit of N .... oo,}

p -> 0 with Np

Const. > 1. For isolated linear chains, ^{we find} simple scaling forms for (R2) ⁱⁿ

d

2 and d

3 in both good ^{and 0 solvents in this} limit, ^{with the} scaling form for d

3 being ^that appropriate ^for Gaussian chains. For isolated branched polymers, ^with fixed, ^small branching probability ( p ), ^{two relevant variables} entering

into (R2) ^{coincide giving a} simple scaling ^{form for}

d

2 in 0 solvents (but ^{not in} good solvents). ^For

both linear and branched polymers, ^{the effect of}

stiffness seems stronger ^{in a} good ^{solvent than in a 0}

solvent

a result somewhat surprising because the chains are more swollen in the former. In a highly polydispersed ^melt of linear polymers ^{in a} good solvent, ^{for a} typical chain, ^the marginal ^dimension dc ^{does not seem to be} reduced in the stiff limit.

Acknowledgments.

I am indebted to P. G. de Gennes for basic ideas that evolved into this work and to J. F. Joanny ^{and M.}

Daoud for important ^comments correcting my mis-

conceptions during ^{the course of} this work.

984

References

[1] ^{KRATKY, O. and POROD,} G., Rec. Trav. Chim. Pays-

Bas 68 (1949) ^1106.

[2] ^{FLORY, P.} J., Statistical Mechanics of ^{Chain Mole-}

cules (Wiley, ^New York) ^1969.

[3] SCHAEFER, D. W., JOANNY, J. F. and PINCUS, P., Macromolecules 13 (1980) ^1280.

[4] AUSSERRE, D., HERVET, ^{H. and} RONDELEZ, F., ^J.

Physique ^{Lett. 46} (1985) ^L-929.

[5] LEE, S. B. and NAKANISHI, H., Phys. ^{Rev. B 33} (1986) ^1953.

[6] HALLEY, J. W., NAKANISHI, H. and SUNDARARA- JAN, R., Phys. Rev. B 31 (1985) ^293.

[7] FLORY, P. J., Principles of Polymer Chemistry, Chap. ^XII (Cornell University Press, Ithaca,

New York) ^1971.

[8] FISHER, ^M. E., ^J. Phys. ^Soc. Japan ²⁶ (1968) ^44.

[9] ^DE GENNES, P. G., Scaling Concepts in Polymer Physics, Chap. I (Cornell University Press, Ithaca, ^New York) ^1979.

[10] ODIJK, ^{T. and} HOUWAART, ^A. C., ^J. Polym. ^Sci.

(Poly. Phys. Ed.) ¹⁶ (1978) ^627.

[11] STRALEY, ^J. P., ^Mol. Cryst. Liq. Cryst. ²⁴ (1973) ^7.

[12] Writing fint/Rd ^{in terms of the} monomer concen-

tration 03A6 :

fint/Rd = 03A6 (B203A6 + B303A62 + ...),

and assuming that the series within the parenth-

eses be of the form g(03A6 / 03A60) ^where 03A60 = B-12 = a-d ^{is the} close-packing ^concen-

tration (independent of p), ^{we obtain}

Ai+1/Ai

Bi+1/pBi

(03A60p)-1 ⁼ ad/p.

We thank J. F. JOANNY and M. DAOUD for

illuminating discussions on this point.

[13] ^{DE GENNES, P.} G., ^J. Physique ^{Lett. 36} (1975) L-55 ;

see also WILLIAMS, C., BROCHARD, ^{F. and} FRISCH, H. L., ^{Ann. Rev.} Phys. ^{Chem. 32} (1981) ^433.

[14] ^The N-dependence ^{in the} expression ^for

good

solvent was already given by J. ISAACSON and T. C. LUBENSKY, ^J. Physique ^{Lett. 41} (1980) ^L- 469;

for a 0398 solvent, by DAOUD, ^{M. and} JOANNY, ^J. F., ^J.

Physique ⁴² (1981) ^1359.

[15] FAMILY, ^{F. and} DAOUD, M., Phys. ^{Rev. B 29} (1984) 1506;

see also DAOUD, M. and FAMILY, F., ^J. Physique ⁴⁵ (1984) ^151.

[16] DAOUD, ^{M. and} CONIGLIO, A. (private ^com- munication). ^The general expression ^{should be}

Nk-2z /Nk-1w ^{for the term} proportional ^to 03A6k

long ^{as the} polydispersity exponent

^{3. For} the definition of this exponent, see, e.g., STAUF- FER, D., Phys. Rep. ⁵⁴ (1979) ^1.

[17] ^{We thank}

referee for pointing ^out the constraint of d ~ 2 for the expression ^{for 03A9 to make sense. In}

this _case, this constraint actually eliminates a

«

window » of validity ^for

Flory expression.