Flory approach for polymers in the stiff limit

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1 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 < /jphys: >. <jpa > HAL Id: jpa Submitted on 1 Jan 1987 HAL is a multi-disciplinary open access archive for the deposit and dissemination of scientific research documents, whether they are published or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers. L archive ouverte pluridisciplinaire HAL, est destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

2 05.30 On We J. Physique 48 (1987) JUIN 1987, 979 Classification Physics Abstracts Flory approach for polymers in the stiff limit H. Nakanishi (~) Collège de France, Physique de la Matière Condensée (+), 11, place Marcelin-Berthelot, Paris Cedex 05, France (Reçu le 1 er juillet 1986, révisé le 30 janvier 1987, accepté le 24 f6vrier 1987) 2014 Résumé. étudie, dans l approximation de Flory, la conformation de chaînes semi-rigides, linéaires ou branchées dans la limite de forte rigidité. On montre que la dimension marginale dc associée aux effets de volume exclu est en général réduite dans cette limite. En particulier, pour une chaîne linéaire en bon solvant on trouve dc 2 en accord avec des résultats récents obtenus numériquement par Lee et Nakanishi Abstract. present a 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 a linear chain in a good solvent we find dc 2 in agreement with the recent Monte Carlo result of Lee and Nakanishi. 1. Introduction. The conformation of semi-flexible long chain polymers has been of considerable interest from theoretical [1-4] and experimental [3-4] points of view. Recently, Lee and Nakanishi [5] studied the crossover 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 polyelectrolytes 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 U.S.A. (+ ) Unité Associ6e au 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 theoretical 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

3 980 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 written as can be where the elastic contribution f el is given approximately by using the leading term in the free, stiff chain result [1] for Np > 1, as 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 where R - (R2) 1/2 is the root-mean-square end-toend 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 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 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 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 a -+ 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].

4 Crossover Fig. 1. 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 or the presence of the excluded volume behaviour. 3. Linear chain in 0 solvent. For a linear chain in a 0 solvent, the leading tint has the form term in 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, 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 continuation 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. This form will result if, e.g., one supposes an impenetrable rod with a concentric sheath of attractive 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

5 982 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 the polymer remains isotropic, equation (21) is 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- to the sive branching units, n, is proportional 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 situation 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 solutions 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 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 [16].

6 a 983 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 Assuming the screening coefficient of the k-th order term to be N k - 2IN k -[16], we would then have 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, 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 >> always of excluded volume type. 1 is 8. Summary. 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 branching 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 : In summary, by a Flory approximation applied to non flexible polymers we find that in general the upper marginal dimension dc for the excluded volume 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 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 misconceptions during the course of this work.

7 984 References [1] KRATKY, O. and POROD, G., Rec. Trav. Chim. Pays- Bas 68 (1949) [2] FLORY, P. J., Statistical Mechanics of Chain Molecules (Wiley, New York) [3] SCHAEFER, D. W., JOANNY, J. F. and PINCUS, P., Macromolecules 13 (1980) [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) [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) [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) [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 concentration 03A6 : fint/rd 03A6 (B203A6 + B303A ), and assuming that the series within the parentheses be of the form g(03a6 / 03A60) where 03A60 B-12 a-d is the close-packing concentration (independent of p), Ai+1/Ai Bi+1/pBi (03A60p)-1 we obtain 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 a 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) [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 communication). The general expression should be Nk-2z /Nk-1w for the term proportional to 03A6k so long as the polydispersity exponent 03C4 ~ 3. For the definition of this exponent, see, e.g., STAUF- FER, D., Phys. Rep. 54 (1979) 1. [17] We thank a 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 a Flory expression.

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