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Polymer Thermodynamics pp 1–66Cite as

Making Floryr–Huggins Practical: Thermodynamics of Polymer-Containing Mixtures

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Part of the book series: Advances in Polymer Science ((POLYMER,volume 238))

Abstract

The theoretical part of this article demonstrates how the original Flory–Huggins theory can be extended to describe the thermodynamic behavior of polymer-containing mixtures quantitatively. This progress is achieved by accounting for two features of macromolecules that the original approach ignores: the effects of chain connectivity in the case of dilute solutions, and the ability of polymer coils to change their spatial extension in response to alterations in their molecular environment. In the general case, this approach leads to composition-dependent interaction parameters, which can for most binary systems be described by means of two physically meaningful parameters; systems involving strongly interacting components, for instance via hydrogen bonds, may require up to four parameters. The general applicability of these equations is illustrated in a comprehensive section dedicated to the modeling of experimental findings. This part encompasses all types of phase equilibria, deals with binary systems (polymer solutions and polymer blends), and includes ternary mixtures; it covers linear and branched homopolymers as well as random and block copolymers. Particular emphasis is placed on the modeling of hitherto incomprehensible experimental observations reported in the literature.

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Abbreviations

a :

Exponent of Kuhn–Mark–Houwink relation (29)

a :

Intramolecular interaction parameter (47) for blend component A

A,B,C :

Constants of (13)

A 2 , A 3 :

Second and third osmotic virial coefficients

a i :

Activity of component i

b :

Intramolecular interaction parameter (47) for blend component B

c :

Concentration in moles/volume

E :

Constant of interrelating α and ζλ (34)

G :

Gibbs free energy – free enthalpy

g :

Integral interaction parameter

H :

Enthalpy

K N :

Constant of the Kuhn–Mark–Houwink relation (29)

LCST:

Lower critical solution temperature

M :

Molar mass

M n :

Number-average molar mass

M w :

Weight-average molar mass

N :

Number of segments

n :

Number of moles

p :

Vapor pressure

R :

Ideal gas constant

S :

Entropy

s :

Molecular surface

T :

Absolute temperature

t :

Ternary interaction parameter (61)

T m :

Melting point

UCST:

Upper critical solution temperature

V :

Volume

v :

Molecular volume

w :

Weight fraction

x :

Mole fraction

Z :

Parameter relating the conformational relaxation to β (53)

\(\overline \omega \) :

Parameter quantifying strong intersegmental interactions (42)

[η]:

Intrinsic viscosity

Φ o :

Volume fraction of polymer segments in an isolated coil (27)

Θ:

Theta temperature

α :

Parameter of (23), first step of dilution

β :

Degree of branching (52)

χ :

Flory–Huggins interaction parameter

δ :

Parameter of (57)

ε :

Parameter of (57)

γ :

Surface-to-volume ratio of the segments in binary mixtures (24)

ϕ :

Segment fraction, often approximated by volume fraction

κ :

Constant of (30)

λ :

Intramolecular interaction parameter (23)

μ :

Chemical potential

ν :

Parameter of (23)

π :

Any parameter of (23)

π osm :

Osmotic pressure

ρ :

Density

τ :

Parameter of (44)

ξ :

Differential Flory–Huggins interaction parameter for the polymer

ζ :

Conformational relaxation (second step of dilution) (23)

1, 2, 3 …:

Low molecular weight components of a mixture

A to P:

High molecular weight components

B:

Branched oligomer/polymer

c:

Critical state

cr:

Conformational relaxation

fc:

Fixed conformation

g:

Glass

H:

Enthalpy part of a parameter

i, j, k :

Unspecified components i, j, k

L:

Linear polymer

lin:

Linear oligomer/polymer

m:

Melting

S:

Entropy part of a parameter

s:

Saturation

ο:

Quantity referring to a pure component, to an isolated coil, or to high dilution

–:

Molar quantity

=:

Segment-molar quantity

E:

Excess quantity

Res:

Residual quantity (with respect to combinatorial behavior)

∞:

Infinite molar mass of the polymer

References

  1. Flory PJ (1953) Principles of polymer chemistry. Cornell Univ. Press, Ithaca

    Google Scholar 

  2. Simha R, Somcynsky T (1969) On statistical thermodynamics of spherical and chain molecule fluids. Macromolecules 2(4):342–350

    Article  CAS  Google Scholar 

  3. Trappeniers NJ, Schouten JA, Ten Seldam CA (1970) Gas–gas equilibrium and the two-component lattice-gas model. Chem Phys Lett 5(9):541–545

    Article  CAS  Google Scholar 

  4. Sanchez IC, Lacombe RH (1976) An elementary molecular theory of classical fluids. Pure fluids. J Phys Chem 80:2352

    Article  CAS  Google Scholar 

  5. Dee GT, Walsh DJ (1988) A modified cell model equation of state for polymer liquids. Macromolecules 21(3):815–817

    Article  CAS  Google Scholar 

  6. Beret S, Prausnitz JM (1975) Perturbed hard-chain theory – equation of state for fluids containing small or large molecules. AIChE J 21(6):1123–1132

    Article  CAS  Google Scholar 

  7. Chapman WG, Gubbins KE, Jackson G, Radosz M (1989) SAFT – equation-of-state solution model for associating fluids. Fluid Phase Equilib 52:31–38

    Article  CAS  Google Scholar 

  8. Hino T, Song YH, Prausnitz JM (1994) Liquid–liquid equilibria for copolymer mixtures from a perturbed hard-sphere-chain equation of state. Macromolecules 27(20):5681–5690

    Article  CAS  Google Scholar 

  9. Fredenslund A, Jones RL, Prausnitz JM (1975) Group-contribution estimation of activity-coefficients in nonideal liquid-mixtures. AIChE J 21(6):1086–1099

    Article  CAS  Google Scholar 

  10. Heil JF, Prausnitz JM (1966) Phase equilibria in polymer solutions. AIChE J 12(4):678–685

    Article  CAS  Google Scholar 

  11. Vimalchand P, Donohue MD (1989) Comparison of equations of state for chain molecules. J Phys Chem 93(10):4355–4360

    Article  CAS  Google Scholar 

  12. De Sousa HC, Rebelo LPN (2000) A continuous polydisperse thermodynamic algorithm for a modified Flory–Huggins model: the (polystyrene plus nitroethane) example. J Polym Sci B Polym Phys 38(4):632–651

    Article  CAS  Google Scholar 

  13. Chang TS (1939) The number of configurations in an assembly and cooperative phenomena. Proc Cambridge Philos Soc 35:265

    Article  Google Scholar 

  14. Flory PJ (1941) Thermodynamics of high polymer solutions. J Chem Phys 9:660

    Article  CAS  Google Scholar 

  15. Flory PJ (1942) Thermodynamics of high polymer solutions. J Chem Phys 10:51

    Article  CAS  Google Scholar 

  16. Huggins ML (1941) Solutions of long chain compounds. J Chem Phys 9:440–440

    Article  CAS  Google Scholar 

  17. Huggins ML (1942) Theory of solutions of high polymers. J Chem Phys 46:151

    Article  CAS  Google Scholar 

  18. Horst R (1995) Calculation of phase diagrams not requiring the derivatives of the Gibbs energy demonstrated for a mixture of two homopolymers with the corresponding copolymer. Macromol Theory Simulat 4:449

    Article  CAS  Google Scholar 

  19. Horst R (1996) Calculation of phase diagrams not requiring the derivatives of the Gibbs energy for multinary mixtures. Macromol Theory and Simulat 5:789

    Article  CAS  Google Scholar 

  20. Horst R (1998) Computation of unstable binodals not requiring concentration derivatives of the Gibbs energy. J Phys Chem B 102(17):3243

    Article  CAS  Google Scholar 

  21. Koningsveld R, Stockmayer WH, Nies E (2001) Polymer phase diagrams. Oxford University Press, Oxford

    Google Scholar 

  22. Flory PJ, Mandelkern L, Kinsinger JB, Shultz WB (1952) Molecular dimensions of polydimethylsiloxanes. J Am Chem Soc 74:3364

    Article  CAS  Google Scholar 

  23. Wolf BA (2003) Chain connectivity and conformational variability of polymers: clues to an adequate thermodynamic description of their solutions II: Composition dependence of Flory–Huggins interaction parameters. Macromol Chem Phys 204:1381

    Article  CAS  Google Scholar 

  24. Eckelt J, Samadi F, Wurm F, Frey H, Wolf BA (2009) Branched versus linear polyisoprene: Flory–Huggins interaction parameters for their solutions in cyclohexane. Macromol Chem Phys 210:1433

    Article  CAS  Google Scholar 

  25. Bercea M, Eckelt J, Wolf BA (2008) Random copolymers: their solution thermodynamics as compared with that of the corresponding homopolymers. Ind Eng Chem Res 47:2434

    Article  CAS  Google Scholar 

  26. Eckelt J, Sugaya R, Wolf BA (2008) Pullulan and dextran: uncommon composition dependent Flory–Huggins interaction parameter of their aqueous solutions. Biomacromolecules 9:1691

    Article  CAS  Google Scholar 

  27. Wolf BA (2009) Binary interaction parameters, ternary systems: realistic modeling of liquid/liquid phase separation. Macromol Theory Simulat 18:30

    Article  CAS  Google Scholar 

  28. Wolf BA (2010) Polymer incompatibility caused by different molecular architectures: modeling via chain connectivity and conformational relaxation. Macromol Theory Simulat 19:36

    CAS  Google Scholar 

  29. Eckelt J, Wolf BA (2008) Thermodynamic interaction parameters for the system water/NMMO hydrate. J Phys Chem B 112:3397

    Article  CAS  Google Scholar 

  30. Tanaka TE (2000) Experimental methods in polymer science. Academic, New York

    Google Scholar 

  31. Petri H-M, Wolf BA (1994) Concentration dependent thermodynamic interaction parameters for polymer solutions: quick and reliable determination via normal gas chromatography. Macromolecules 27:2714

    Article  CAS  Google Scholar 

  32. Krüger K-M, Sadowski G (2005) Fickian and non-Fickian sorption kinetics of toluene in glassy polystyrene. Macromolecules 38:8408

    Article  CAS  Google Scholar 

  33. Ninni L, Meirelles AJA, Maurer G (2005) Thermodynamic properties of aqueous solutions of maltodextrins from laser-light scattering, calorimetry and isopiestic investigations. Carbohydr Polym 59(3):289

    Article  CAS  Google Scholar 

  34. Karimi M, Albrecht W, Heuchel M, Weigel T, Lendlein A (2008) Determination of solvent/polymer interaction parameters of moderately concentrated polymer solutions by vapor pressure osmometry. Polymer 49(10):2587–2594

    Article  CAS  Google Scholar 

  35. Gao ZN, Li JF, Wen XL (2002) Vapor pressure osmometry and its applications in the osmotic coefficients determination of the aqueous monomer glycol and polymer polyethylene glycol solutions at various temperature. Chin J Chem 20(4):310–316

    CAS  Google Scholar 

  36. Eliassi A, Modarress H, Nekoomanesh M (2002) Thermodynamic studies of binary and ternary aqueous polymer solutions. Iran J Sci Technol 26(B2):285–290

    CAS  Google Scholar 

  37. Schreiber HP (2003) Probe selection and description in Igc: a nontrivial factor. J Appl Polym Sci 89(9):2323–2330

    Article  CAS  Google Scholar 

  38. Voelkel A, Strzemiecka B, Adamska K, Milczewska K (2009) Inverse gas chromatography as a source of physiochemical data. J Chromatogr A 1216(10):1551–1566

    Article  CAS  Google Scholar 

  39. Bercea M, Cazacu M, Wolf BA (2003) Chain connectivity and conformational variability of polymers: clues to an adequate thermodynamic description of their solutions I: Dilute solutions. Macromol Chem Phys 204:1371

    Article  CAS  Google Scholar 

  40. Petri H-M, Schuld N, Wolf BA (1995) Hitherto ignored influences of chain length on the Flory–Huggins interaction parameter in highly concentrated polymer solutions. Macromolecules 28:4975

    Article  CAS  Google Scholar 

  41. Bercea M, Wolf BA (2006) Vitrification of polymer solutions as a function of solvent quality, analyzed via vapor pressures. J Chem Phys 124(17):174902–174907

    Article  CAS  Google Scholar 

  42. Wolf BA, Adam HJ (1981) Second osmotic virial coefficient revisited – variation with molecular weight and temperature from endo- to exothermal conditions. J Chem Phys 75:4121

    Article  CAS  Google Scholar 

  43. Schotsch K, Wolf BA, Jeberien H-E, Klein J (1984) Concentration dependence of the Flory–Huggins parameter at different thermodynamic conditions. Makromol Chem 185:2169

    Article  CAS  Google Scholar 

  44. Bercea M, Wolf BA (2006) Enthalpy and entropy contributions to solvent quality and inversions of heat effects with polymer concentration. Macromol Chem Phys 207(18):1661–1673

    Article  CAS  Google Scholar 

  45. Saeki S, Kuwahara S, Konno S, Kaneko M (1973) Upper and lower critical solution temperatures in polystyrene solutions. Macromolecules 6(2):246

    Article  CAS  Google Scholar 

  46. Stryuk S, Wolf BA (2003) Chain connectivity and conformational variability of polymers: clues to an adequate thermodynamic description of their solutions III: Modeling of phase diagrams. Macromol Chem Phys 204:1948

    Article  CAS  Google Scholar 

  47. Schuld N, Wolf BA (1999) Polymer–solvent interaction parameters In: Brandrup J, Immergut EH, Grulke EA (eds) Polymer handbook, 4th edn. Wiley, New York

    Google Scholar 

  48. Krigbaum WR, Geymer DO (1959) Thermodynamics of polymer solutions. The polystyrene–cyclohexane system near the Flory theta temperature. J Am Chem Soc 81:1859

    Article  CAS  Google Scholar 

  49. Wolf BA, Jend R (1977) Über die Möglichkeiten zur Bestimmung von Mischungsenthalpien und -volumina aus der Molekulargewichtsabhängigkeit der kritsichen Entmischungstemperaturen und -drucke. Makromol Chem 178:1811

    Article  CAS  Google Scholar 

  50. Stryuk S, Wolf BA (2005) Liquid/gas and liquid/liquid phase behavior of n-butane/1,4-polybutadiene versus n-butane/1,2-polybutadiene. Macromolecules 38(3):812

    Article  CAS  Google Scholar 

  51. Sanchez IC, Balazs AC (1989) Generalization of the lattice-fluid model for specific interactions. Macromolecules 22:2325

    Article  CAS  Google Scholar 

  52. Khassanova A, Wolf BA (2003) PEO/CHCl: crystallinity of the polymer and vapor pressure of the solvent – equilibrium and non-equilibrium phenomena. Macromolecules 36(17):6645

    Article  CAS  Google Scholar 

  53. Wolf BA (2005) On the reasons for an anomalous demixing behavior of polymer solutions. Macromolecules 38(4):1378

    Article  CAS  Google Scholar 

  54. Šolc K, Dušek K, Koningsveld R, Berghmans H (1995) “Zero” and “off-zero” critical concentrations in solutions of polydisperse polymers with very high molar masses. Collect Czech Chem Commun 60(10):1661–1688

    Google Scholar 

  55. Schäfer-Soenen H, Moerkerke R, Berghmans H, Koningsveld R, Dušek K, Šolc K (1997) Zero and off-zero critical concentrations in systems containing polydisperse polymers with very high molar masses. 2. The system water-poly(vinyl methyl ether). Macromolecules 30(3):410–416

    Article  Google Scholar 

  56. Meeussen F, Bauwens Y, Moerkerke R, Nies E, Berghmans H (2000) Molecular complex formation in the system poly(vinyl methyl ether)/water. Polymer 41(10):3737–3743

    Article  CAS  Google Scholar 

  57. Eckelt J, Wolf BA (2007) Cellulose/water: l/g and l/l phase equilibria and their consistent modeling. Biomacromolecules 8:1865

    Article  CAS  Google Scholar 

  58. Xiong X, Eckelt J, Zhang L, Wolf BA (2009) Thermodynamics of block copolymer solutions as compared with the corresponding homopolymer solutions: experiment and theory. Macromolecules 42:8398

    Google Scholar 

  59. Wolf BA (2006) Polymer–polymer interaction: consistent modeling in terms of chain connectivity and conformational response. Macromol Chem Phys 207(1):65

    Article  CAS  Google Scholar 

  60. Halary JL, Ubrich JM, Monnerie L, Yang H, Stein RS (1985) Polym Commun 26:73

    CAS  Google Scholar 

  61. Ben Cheikh Larbi F, Lelong S, Halary JL, Monnerie L, Yang H (1986) Polym Commun 22:23

    Google Scholar 

  62. Rätsch MT, Kehlen H, Wohlfarth C (1988) Polydispersity effects on the demixing behavior of poly(vinyl methyl ether)/polystyrene blends. J Macromol Sci Chem A25:1055

    Article  Google Scholar 

  63. Greenberg CC, Foster MD, Turner CM, Corona-Galvan S, Cloutet E, Butler PD, Hammouda B, Quirk RP (1999) Effective interaction parameter between topologically distinct polymers. Polymer 40(16):4713

    Article  CAS  Google Scholar 

  64. Hill MJ, Barham PJ, Keller A, Rosney CCA (1991) Phase segregation in melts of blends of linear and branched polyethylene. Polymer 32(8):1384–1393

    Article  CAS  Google Scholar 

  65. Hill MJ, Barham PJ, Keller A (1991) Phase segregation in blends of linear with branched polyethylene – the effect of varying the molecular-weight of the linear polymer. Polymer 33(12):2530

    Google Scholar 

  66. Stephens CH, Hiltner A, Baer E (2003) Phase behavior of partially miscible blends of linear and branched polyethylenes. Macromolecules 36(8):2733–2741

    Article  CAS  Google Scholar 

  67. Singh C, Schweizer KS (1995) Correlation-effects and entropy-driven phase-separation in athermal polymer blends. J Chem Phys 103(13):5814–5832

    Article  CAS  Google Scholar 

  68. Eckelt J, Eich T, Röder T, Rüf H, Sixta H, Wolf BA (2009) Phase diagram of the ternary system NMMO/water/cellulose. Cellulose 16:373

    Article  CAS  Google Scholar 

  69. Cowie JMG, McEwen IJ (1983) Polymer co-solvent systems.6. Phase-behavior of polystyrene in binary mixed-solvents of acetone with N-alkanes – examples of classic cosolvency. Polymer 24(11):1449–1452

    Article  CAS  Google Scholar 

  70. Wolf BA, Willms MM (1978) Measured and calculated solubility of polymers in mixed solvents: co-nonsolvency. Makromol Chem 179:2265

    Article  CAS  Google Scholar 

  71. Samadi F, Eckelt J, Wolf BA, López-Villanueva F-J, Frey H (2007) Branched versus linear polyisoprene: fractionation and phase behavior. Eur Polym J 43:4236–4243

    Article  CAS  Google Scholar 

  72. Francis AW (1961) Critical solution temperatures. Advances in Chemistry Series, vol 31. ACS, Washington, D.C.

    Google Scholar 

  73. Eckelt J, Haase T, Loske S, Wolf BA (2004) Large scale fractionation of macromolecules. Macromol Mater Eng 289(5):393

    Article  CAS  Google Scholar 

  74. Bercea M, Eckelt J, Morariu S, Wolf BA (2009) Islands of immiscibility for solutions of compatible polymers in a common solvent: experiment and theory. Macromolecules 42:3620

    Article  CAS  Google Scholar 

  75. Bercea M, Eckelt J, Wolf BA (2009) Vapor pressures of polymer solutions and the modeling of their composition dependence. Ind Eng Chem Res 48:4603

    Article  CAS  Google Scholar 

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Acknowledgments

The author is grateful to Dr. John Eckelt (WEE-Solve AG, Mainz, Germany) and to Prof. Spiros Anastasiadis (University of Crete, Greece) for their constructive criticism, which has certainly improved the readability of this contribution.

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Authors and Affiliations

  1. Institut für Physikalische Chemie der Johannes Gutenberg-Universität Mainz, 55099, Mainz, Germany

    Bernhard A. Wolf

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  1. Inst. Physikalische Chemie, Universität Mainz, Jakob-Welder-Weg 13, Mainz, 55099, Germany

    Bernhard A. Wolf

  2. Sekr. TK7, TU Berlin, Straße des 17. Juni 135, Berlin, 10623, Germany

    Sabine Enders

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Wolf, B.A. (2010). Making Floryr–Huggins Practical: Thermodynamics of Polymer-Containing Mixtures. In: Wolf, B., Enders, S. (eds) Polymer Thermodynamics. Advances in Polymer Science, vol 238. Springer, Berlin, Heidelberg. https://doi.org/10.1007/12_2010_84

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