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

  • Bernhard A. WolfEmail author
Chapter
Part of the Advances in Polymer Science book series (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.

Keywords

Modeling Mixed solvents Phase diagrams Polymer blends Polymer solutions Ternary mixtures Thermodynamics 

Symbols

a

Exponent of Kuhn–Mark–Houwink relation (29)

a

Intramolecular interaction parameter (47) for blend component A

A,B,C

Constants of (13)

A2, A3

Second and third osmotic virial coefficients

ai

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

KN

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

LCST

Lower critical solution temperature

M

Molar mass

Mn

Number-average molar mass

Mw

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)

Tm

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)

Greek and Special Characters

\(\overline \omega \)

Parameter quantifying strong intersegmental interactions (42)

[η]

Intrinsic viscosity

Φo

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)

πosm

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)

Subscripts

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

Superscripts

Molar quantity

=

Segment-molar quantity

E

Excess quantity

Res

Residual quantity (with respect to combinatorial behavior)

Infinite molar mass of the polymer

Notes

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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Copyright information

© Springer-Verlag Berlin Heidelberg 2010

Authors and Affiliations

  1. 1.Institut für Physikalische Chemie der Johannes Gutenberg-Universität MainzMainzGermany

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