Polymer excluded volume exponent v : An experimental verification of the n vector model for n = 0

Transcription

1 Polymer excluded volume exponent v : An experimental verification of the n vector model for n 0 J.P. Cotton To cite this version: J.P. Cotton. Polymer excluded volume exponent v : An experimental verification of the n vector model for n 0. Journal de Physique Lettres, 1980, 41 (9), pp < /jphyslet: >. <jpa > HAL Id: jpa Submitted on 1 Jan 1980 HAL is a multidisciplinary open access archive for the deposit and dissemination of scientific research documents, whether they are published or not. 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 LETTRES Three J. Physique 41 (1980) L 231L234 1 er MAI 1980, L 231 Classification Physics Abstracts Polymer excluded volume exponent v : An experimental verification of the n vector model for n 0 J. P. Cotton Laboratoire LéonBrillouin, DPhG/PSRM, CEN Saclay, B.P. N 2, GifsurYvette, France (Re~u le 7 fevrier 1980, revise le 6 mars, accepte le 10 mars 1980) Résumé Trois ensembles de mesures (Y. Miyaki et al., Macromolecules 11 (1978) 1180; M. Fukuda et al., J. Polym. Sci. 12 (1974) 871 ; A. Yamamoto et al., Polymer J. 2 (1971) 799) de diffusion de lumière par des solutions diluées polystyrène benzène sont analysés ici. Ces mesures, très soignées, donnent les variations des valeurs du rayon de giration R 2 >1/2 et du deuxième coefficient du viriel A2 en fonction de la masse moléculaire du polymère dont la variation couvre trois décades de valeurs. L analyse présentée ici consiste à corriger les effets de polydispersité afin d obtenir la valeur expérimentale de l exposant de volume exclu v. La valeur obtenue : v 0,586 ± 0,004 à partir des mesures de R 2 > vérifie mieux la théorie critique (v 0,588) que celle de Flory (v 0,60). Ceci est confirmé par les mesures de A Abstract. light scattering experiments (Y. Miyaki et al., Macromolecules 11 (1978) 1180 ; M. Fukuda et al., J. Polym. Sci. 12 (1974) 871; A. Yamamoto et al., Polymer J. 2 (1971) 799) for polystyrene benzene dilute solutions are analysed. se very careful measurements give the variations of the radius of gyration R 2 >1/2 and of the second virial coefficient A2 as a function of the molecular weight (0.246 x 106 ~ Mw ~ ). se data are corrected here for polydispersity effects and analysed in order to determine the excluded volume exponent v. obtained value : v ± from R2 > is in better agreement with the critical theory (v 0.588) than with the theory of Flory (v 0.60). This result is confirmed by the A2 data. In dilute polymer solutions the excluded volume interactions between monomers introduce a swelling of the chain characterized [1] by an exponent v. This exponent, or various combinations of it, are found [2] in the relations which connect the observables of solutions to the practical parameters such as the concentration or the molecular weight M of polymers. As a particular case, the observables measured by light scattering from dilute solutions are the radius of gyration R 2 > 1~2 of a chain and the second virial coefficient A2 which are connected to M using the following relations [1, 31 : where K and K are constants. first theoretical value of v, VF 0.600, has been first given by Flory [1] using a controversial [4] mean field theory. More recently de Gennes has shown [5] that v can be calculated using the n vector model with n 0. Following this method and using a perturbative expansion of renormalization group functions of field theory in three dimensions, the v value (ve ± 0.001) has been obtained [6]. difference between VF and Ve are so small that the experimental v values found in the literature [7, 8, 9] have so far been not able to differentiate between them. Nevertheless it is of fundamental interest to know if the n vector model and the expansion, mentioned above, are valid for n 0. first problem encountered in obtaining v from eqs. (1) and (2) lies in the difficulty of having a sufficient number of precise data obtained with polymers of the same chemical species in the same solvent and corresponding to a wide range of well characterized molecular weights. This problem is now solved since a set of very careful light scattering experiments for solutions of polystyrene of very high molecular weights have recently completed [9] two others sets of similar data [10, 11]. data of these sets are summarized in table I. values of v Article published online by EDP Sciences and available at

3 se L232 JOURNAL DE PHYSIQUE LETTRES Table I. data are obtained from 3 sets of experiments choosen in the literature for the weak polydispersity of the polymers used. A set corresponds to experiments of Miyaki et al. [9], B set to these of Fukuda et al. [10] and C set to these of Yamamoto et al. [11]. values of M,, for the set A are deduced from an average of those ~~ measured [9] in benzene and in cyclohexane. obtained by these authors from eq. (1) are [10], [11] and [9] following the considered sets of data. Because the polydispersity effects are more important for the higher molecular weights than for the smaller ones, a better accuracy on v can be obtained from a simultaneous correction of these effects. treatment of these data is given in the following in order to obtain the exponents of eqs. (1) and (2) respectively. 1. radius of gyration. principal difficulty found in obtaining a precise value of v from eq. (1) is linked with the weight distribution f (M) inherent to the chemical preparation of a polymer. This distribution is generally characterized by the well known [12] different average molecular weight : which is obtained [13] while it is the average molecular weight MW which is measured [12]. In a first step, neglecting the polydispersity of the samples we have fitted the data of table I by a least square method. This treatment leads to in good solvent (benzene). It leads to in 0 solvent (cyclohexane 34.5 ~C) for which it is recognized [1, 3] that the excluded volume effects are compensated and the v value is 1/2. exponent value of suggests that the polydispersity has a non negligible influence. In order to take polydispersity into account one has to express ( R 2 >z as a function of MW" : This equation introduces [13] ( R 2 >c of the radius of gyration a corrected value where 1 f (M) dm. In a light scattering experiment it is the z average of the radius of gyration C ~ 2 >Z which is the ( 7~ ~ value for a monodisperse sample with a molecular weight MW. For the correction the usual [3, 13] Shulz and Zimm distribution [14] is adopted for f(m) :

4 Log Here EXPERIMENTAL DETERMINATION OF v FOR n 0 L 233 where r(k) is the gamma function and k and y are adjustable parameters. A straightforward calculation using eqs. (7) and (8) leads to [13] : S.Z. distribution for the correction since the exponent found is now 1/2. For the good solvent, the v value first adopted for g(v) was [8, 9] it leads to se equations have been used in order to deduce the R 2 ) c values given in table I from the measured ones R 2 )z. Use of S.Z. distribution [14] corresponds to an approximation which can introduce an error. This error would be of little importance if the parameters of this distribution were evaluated identically from the same type of data, for example from the Mw/Mn ratios. However this uniformity in the treatment cannot be achieve here. values of k for the set B have to be determined from the Mw/Mn ratios only [11]. But for the set A it is more appropriate to use the Mz/Mw ratios which the authors claimed [9] to be of better accuracy. This procedure is justified, a fortiori from the results obtained for 0 solvent and from the following test of the S.Z. distribution : For the case of the set A a comparison of the values of k, deduced from Mw/Mn (kl) and from Mz/Mw (k2), can be achieved. Following eq. (9) it leads to : for Mw 38.5 x 106 and similar differences are found for the other values of MW. agreement between these values of k is not excellent but it is not so bad [15]. (Let us recall the poor accuracy announced [9] on the Mw/Mn ratios.) refore the taken values for k in eq. (9) were deduced from A~/Af~ for the set A and for MW/Mn for the set B. This correction (with v 1/2) is first tested on the data registered for 0 solvent is leads to for the A set of data. A wider range of Mw values can be obtained by taking the Ro B values for the set A and the Ro >z values for the set C (for which the Mw/Mn values are unfortunately not given). fit of these inhomogeneously corrected data (denoted Ro > cz) leads to : Comparison between these two relations confirms that the correction is really important for polymers with the highest molecular weights only. On the other hand these results give us confidence in the use of the Fig. 1. Log plot of R2 ):/2 versus MW. straight line corresponds to Af~~ (see text). symbols 0 shows the data of Miyaki et al. (( R 2 );/2); x for these of Fukuda et al. «R 2 );/2) and 0 for these of Yamamoto et al. (( R 2 )i/2). using the corrected data of the 6th column of table I. v value found is now consistent with the v value used in g(v). By using these last data and the ( R 2 ~Z~2 values for the C set of which correction has been shown to be very weak for 0 solvent one finds : se v values are precise [16] and are in very good agreement [17] with the prediction [5, 6] of of the n vector model. 2. second virial coefficient. the problem is due to the errors for the A 2 values which as wellknown [18] are greater than those obtained for the R 2 > values. polydispersity effects have been shown (see Yamakawa s book [3], p. 220) to be negligible relative to the experimental errors for A2 i.e. ~2w ~ A2. fit of the three sets of A 2 values yields (see Fig. 2) which corresponds to a v value of ± This value is again in better agreement with the critical theory. As a matter of fact these sets of data taken two by two leads to different v values [0.573 (B + C) ; (A + B) and (A + C)]

5 Log L 234 JOURNAL DE PHYSIQUE LETTRES on the considered measurement is stressed. Its influence appears clearly on the v value discussed here. But it is greater on the values of the interpenetration function VI where Xa is the number of Avogadro. A fuller discussion of it has been given elsewhere [3, 19]. Since a power three of the radius of gyration is found here the polydispersity has to be corrected. In figure 2 the following values of t/1 are plotted : Fig. 2. Log plot of A2 versus Mw (low data). straight line corresponds to 103 M" upper curve is semilog plot of the interpenetration function ~ (see eq. (11)). symbols used correspond to those of figure 1. of which dispersion perhaps expresses the systematic errors of each set of experiments. 3. Conclusion. importance of polydispersity using the reasonable approximation ~4 ~~ ~ A 2 already mentioned. Figure 2 shows clearly (see also table I) that ~ is rather a constant for A and B set of data. Its means value is Acknowledgments. author wishes to thank Drs. J. des Cloizeaux and R. Oberthur for fruitful discussions about the correction of the data and for a careful criticism of the manuscript. He is indebted to Dr. G. Jannink for various discussions. References and notes [1] FLORY, P. J., Principles of Polymer Chemistry (Cornell U. Press, Ithaca) [2] DES CLOIZEAUX, J., J. Physique 36 (1975) 281 ; DAOUD, M. et al., Macromolecules 8 (1975) 804 ; DE GENNES, P. G., Scaling Concepts in Polymer Physics (Cornell U. Press) [3] YAMAKAWA, H., Modern ories of Polymer Solutions (Harper and Row N.Y.) [4] DES CLOIZEAUX, J., J. Physique 37 (1976) 431. [5] DE GENNES, P. G., Phys. Lett. 38A (1972) 339. [6] LE GUILLOU, J. C., ZINNJUSTIN, J., Phys. Rev. Lett. 39 (1977) 95; BAKER, G. A., NICKEL, B. G., MEIRON, D. I., Phys. Rev. B 17 (1978) [7] DOMB, C., Adv. Chem. Phys. 15 (1969) 229. [8] COTTON, J. P., DECKER, D., FARNOUX, B., JANNINK, G., OBER, R., PICOT, C., Phys. Rev. Lett. 32 (1974) [9] MIYAKI, Y., EINAGA, Y., FUJITA, H., Macromolecules 11 (1978) [10] FUKUDA, M., FUKUTOMI, M., HASHIMOTO, T., J. Polym. Sci. 12 (1974) 87. [11] YAMAMOTO, A., FUJII, M., TANAKA, G., YAMAKAWA, H., Polym. J. 2 (1971) 799. [12] BENOIT, H., HOLTZER, A. M., DOTY, P., J. Phys. Chem. 58 (1954) 635. [13] OBERTHÜR, R. C., Makromol. Chem. 179 (1978) [14] SCHULZ, G. V., Z. Phys. Chem. B 43 (1939) 25 ; ZIMM, B. H., J. Chem. Phys. 16 (1948) [15] difference between k1 7 and k2 20 yields a difference of 5 % on the value of the coefficient which connects R2 >1/2z to R2 >1/2c. [16] It may seem surprising to find an accuracy a little less than 1 % while the precision on R2 >1/2 and Mw is of the order of a few percents. But using eqs. (1) and (7) the relative incertitude of v is given by 0394v v 0394 Log R2 >1/2c Log R2 >1/2c R2 >c R2 >c 0394 R2 >z R2 >z + which leads to a 0394v 0394g(v,k) g(v,k) 0394 Log M Log M value of for one of the most inaccurate data [9] : Mw (23.4 ± 1).106 ; R2 >1/2Z ± 90 A ; Mz/Mw 1.05 ± [17] Use of the less accurate value Mw/Mn for the determination of k in set A leads to a smaller central value and a larger inaccuracy of v. Even then, the obtained results are in better agreement with the n vector model prediction than the earlier 3/5 value. [18] See for example Light scattering from polymer solutions M. B. Huglin Ed. (Academic, N.Y.) [19] WITTEN, T. A., SCHÄFFER, L., J. Phys. A 11 (1978) 1843.