Thermodynamic Properties of Polymer Solutions

Transcription

1 etsu Sokutei Thermodynamic Properties of Polymer Solutions Takahiro Sato (Received May 22, 2003; Accepted June 19, 2003) Various statistical thermodynamic theories have been presented so far to understand thermodynamic properties of polymer solutions. The basic model and assumptions as well as deficiency of the theories are explained. The famous Flory-Huggins theory is favorably compared with experiment only at high polymer concentrations. The cluster expansion theory for polymer systems successfully calculates the second virial coefficient in dilute polymer solutions but not the higher virial coefficients. The renormalization-group theory makes possible the calculations of the higher virial coefficients, but it is not applicable to concentrated polymer solutions. A perturbation theory extended to polymer solutions is favorably compared with experiment except for high-molecular-weight flexible polymer solutions with low concentrations. 1) Flory-Huggins 1942 Flory-Huggins 1) 2) 3) Flory The Japan Society of Calorimetry and Thermal Analysis. 173

2 φ 0 n 0, φ 1 n 1 (2) n 0 n 1 n 0 n 1 Fig.1 Schematic diagram of the lattice model for polymer solutions. Huggins 4) Flory 5) Huggins 6) 2 Fig.1 S S R (n 0lnφ 0 n 1lnφ 1) (1) R n 0 n φ 0 φ (1) S (1) S Hildebrand 7) H H RTχn 0φ 1 (3) T χ Flory-Huggins χ Flory-Huggins (3) (1) (3) F F RT(n 0 lnφ 0 n 1lnφ 1 χ n 0φ 1) (4) Flory-Huggins Gibbs Helmholtz F µ 0 µ 0 µ 0 RT lnφ φ 1 χφ 2 1 (5) µ 0 P 0 Π P 0 P 0 exp[(µ 0 µ 0 )/RT] Π (µ 0 µ 0 )/V o [ ( ) ] P 0φ 0exp 1 1 φ 1 χφ 2 1 (6) RT lnφ φ 1 χφ 2 1 (7) V 0 [( ) ] [ ( ) ] P 0 V 0 H 0 ( ) H 0 RT 2 µ 0 µ 0 RTχφ1 2 (8) T RT φ 1 174

3 ( Π/ c) / RT / mol g 1 Fig.2 c / g cm 3 Comparison between theory and experiment of osmotic compressibility for toluene solutions of polystyrene with a molecular weight of 900,000: circles, data of oda et al. 8) ; dot-dash curve, the Flory-Huggins theory; solid curve, the perturbation theory; dotted curve, the renormalization-group theory. χ (8) χ c ( ) 1 Π 1 υ 2 1 χ c υ 3 RT c M V 0 2 3V 0 c 2 1 A2c A 3c 2 (9) M M υ A 2 A φ 1 υ c 2 3 A 2 χ Flory Flory-Huggins Flory-Huggins 4 Fig.2 8) 90 c Π/ c Flory-Huggins (7) c c υ χ g cm 3 Flory-Huggins Flory-Huggins 11,12) Flory-Huggins McMillan- Mayer 13) McMillan-Mayer Zimm 14) 1946 Flory-Huggins 175

4 (a) (b) Fig.3 (a) Schematic diagram of the bead model for polymer chains used in the cluster expansion theory; (b) cluster diagrams for the single, double, and triple intermolecular contact terms. 2 u 2 bare potential u potential of mean force Flory-Huggins Fig.3(a) 2 u w u w (Ri 1, i 2) (10) i 1 1 i 2 1 Ri 1, i 2 1 i 1 2 i 2 Ri 1, i 2 2 u (9) 2 A 2 2 Ursell-Mayer [ ] A 2 A β 2 β 2 P(0j 1,j 2)i 1,i 2 (11) 2M 2 i 1,i 2 1 j 1,j 2 1 A Avogadro β 2 Fig.3(a) 2 β V {1 exp[ w(r)/k BT]}dR (12) k B Boltzmann (11) Fig.3(b) i 1 i 2 j 1 j P(0j 1, j 2)i 1,i 2 R i 1,i 2 R j 1, j 2 Fig.3(b) A (11) 2 P(0j 1, j 2)i 1,i 2 Gauss 3 Gauss Barrett 15) A 2 2 A 2 16) A 2 2 Barrett A 2 Barrett Fig.4 17) 18) 8) A 2β 5 176

5 A 2 / mol g 2 cm 3 Fig de Gennes 19) M Molecular weight dependence of the second virial coefficient for polystyrene in toluene obtained by light scattering (unfilled circles), 17) by osmometry (triangles), 8) and by sedimentation equilibrium (filled circles); 18) solid and dotted curves, calculated by the theory of Yamakawa with and without considering the polymer-chainend effect, respectively; dot-dash curve, theoretical values using the single-contact approximation. Fig.2 90 Π/ c 2 β (12) c 0.03 g cm 3 (9) 2 Π/ c 3 (10) 2 u 2 Fig.5(a) u 2 r γ Fig.5(a) r u Fig.5(b) u r γ Fig.3(b) 2 177

6 (a) (b) (c) Fig.5 Schematic representations of interacting two cylinders, two short polymer chains, and two long polymer chains. ( Π/ c) / RT / mol g 1 r Fig.5(c) u u 11,12) F F F 0 i 1 (ε/k BT) i F i (13) F 0 F i i ε Barker Henderson 22) { (r d) u(r) ε (d _ r 1.5d) (14) 0 (r _ 1.5d) d ε 1 2 Monte Carlo 3 Padé ) Parsons 23) decoupling Barker-Henderson F 0 scaled particle theory 24) c / g cm 3 Fig.6 Comparison of osmotic compressibility for toluene solutions of low molecular-weight polystyrenes or oligostyrenes with the perturbation theory. 18) Fig ) d ε/k BT 0.56 nm 0.18 d (13) 2 Fig Fig.6 1 Π/ c 2 Π/ c 5 18) Fig

7 Fig.2 c 0.04 g cm 3 8,18) 24,25) 1) P. J.,, (1956). 2), (1 ), (1994). 3) H. G. Schild and D. A. Tirrell, J. Phys. Chem. 94, 4352 (1990); K. Yoshiba, T. Ishino, A. Teramoto,. akamura, Y. Miyazaki, M. Sorai, Q. Wang, Y. Hayashi,. Shinyashiki, and S. Yagihara, Biopolymers 63, 370 (2002). 4),, (1987). 5) P. J. Flory, J. Chem. Phys. 10, 51 (1942). 6) M. L. Huggins, J. Phys. Chem. 46, 151 (1942); Ann..Y. Acad. Sci. 41, 1 (1942); J. Am. Chem. Soc. 64, 1712 (1942). 7) J. H. Hildebrand and R. L. Scott, "The Solubility of onelectrolytes, 3rd Edition," Reinhold Publishing Corp., ew York (1949). 8) Y. Higo,. Ueno, and I. oda, Polym. J. 15, 367 (1983); I. oda, Y. Higo,. Ueno, and T. Fujimoto, Macromolecules 17, 1055 (1984). 9) H. Yamakawa, "Modern Theory of Polymer Solutions," Harper & Row, ew York (1971). 10) H. Yamakawa, "Helical Wormlike Chains in Polymer Solutions," Springer-Verlag, Berlin & Heidelberg (1997). 11) J. P. Hansen and I. R. McDonald, "Theory of Simple Liquids, 2nd Edition," Academic Press, London (1986). 12) C. G. Gray, and K. E. Gubbins, "Theory of Molecular Fluids, Vol.1: Fundamentals," Clarendon Press, Oxford (1984). 13) W. G. McMillan and J. E. Mayer, J. Chem. Phys. 13, 276 (1945). 14) B. H. Zimm, J. Chem. Phys. 14, 164 (1946). 15) A. J. Barrett, Macromolecules 18, 196 (1985). 16) H. Yamakawa, Macromolecules 25, 1912 (1992). 17) H. Yamakawa, F. Abe, and Y. Einaga, Macromolecules 26, 1898 (1993); Y. Einaga, F. Abe, and H. Yamakawa, Macromolecules 26, 6243 (1993). 18) R. Koyama and T. Sato, Macromolecules 35, 2235 (2002). 19) P.-G. de Gennes, "Scaling Concepts in Polymer Physics," Cornell Univ. Press, Ithaca, ew York (1979). 20) K. Freed, "Renormalization Group Theory of Macromolecules," John Wiley and & Sons, ew York (1987). 21) J. des Cloizeaux, and G. Jannink, "Polymers in Solution. Their Modelling and Structure," Clarendon Press, Oxford (1990). 22) J. A. Barker and D. Henderson, Rev. Mod. Phys. 48, 587 (1976). 23) J. D. Parsons, Phys. Rev. A 19, 1225 (1979). 24) T. Sato and A. Teramoto, Adv. Polym. Sci. 126, 85 (1996). 25) T. Sato, Y. Jinbo, and A. Teramoto, Macromolecules 30, 590 (1997). Flory-Huggins Flory- Huggins Takahiro Sato, Dept. of Micromolecular Science, Graduate School of Science, Osaka Univ., TEL.&FAX , tsato@chem.sci.osaka-u.ac.jp 179