Flory-Fox equation

The Flory-Fox equation (synonymous Fox-Flory equation ) is an equation for describing the viscosity of polymers as a function of the respective molar mass . She corrected the previously held assumption that the glass transition temperature of a linear polymer was the temperature with the highest viscosity, in that the glass transition temperature is the temperature at which the available space for molecular movement of a polymer becomes a given value.

properties

At the ends of a polymer, the repeat units (white) have more space available for molecular movement (gray)

The Flory-Fox equation is:

${\displaystyle [\eta ]=kM^{1/2}\alpha ^{3}}$

with as viscosity number (in dl / g), as physical constant , as undisturbed mean length of a polymer (in cm), as molar mass (in g / mol), as a universal constant called intrinsic viscosity parameter and with as expansion factor . In a - solvent with takes the value 1, and the Flory-Fox equation turns into: ${\displaystyle [\eta ]}$ ${\displaystyle k:=\Phi \ (r_{0}^{2}/M)^{3/2}}$ ${\displaystyle r_{0}}$ ${\displaystyle M}$ ${\displaystyle \Phi =2{,}6\cdot 10^{21}}$ ${\displaystyle \alpha :=r^{2}/r_{0}^{2}}$ ${\displaystyle \Theta }$ ${\displaystyle r^{2}=r_{0}^{2}}$ ${\displaystyle \alpha }$

${\displaystyle [\eta ]=\Phi {\frac {r_{0}^{3}}{M}}}$

Dependence of the glass transition temperature (in Kelvin) on the mean molar mass (in g / mol)

Another relationship was published by Flory and Fox:

${\displaystyle T_{g}=T_{g,\infty }-{\frac {K}{M}}}$

with T _{g, ∞}   as the maximum glass transition temperature that can be reached with a theoretically infinite molar mass, and K as the empirical constant that relates to the free volume. It is only suitable for large values ​​of M.

A modified version of this relationship was published by Toshio Ogawa:

${\displaystyle T_{g}=T_{g,\infty }-{\frac {K}{(M_{\mathrm {n} }M_{\mathrm {w} })^{\frac {1}{2}}}},}$

in which the inverse dependency of M is replaced by one of the square root of the product of the number average molecular weight M _n and the weight average molecular weight M _w .

Another modification was published by Thomas G. Fox and S. Loshaek:

${\displaystyle {\frac {1}{T_{g}}}={\frac {1}{T_{g,\infty }}}+{\frac {K}{T_{g,\infty }^{2}}}{\frac {1}{M_{\mathrm {n} }}}}$

However, the molecular weight is not necessarily a practical parameter for influencing the glass transition temperature, since the changeable range without changing the physical properties is small.

history

The Flory-Fox equation was published in 1950 by Thomas G. Fox and Paul J. Flory. In 1956, Thomas G. Fox also established the Fox equation .

literature

• A. Lederer, W. Burchard, A. Khalyavina, P. Lindner, R. Schweins: Is the universal law valid for branched polymers? In: Angewandte Chemie. Volume 52, number 17, April 2013, pp. 4659-4663, doi : 10.1002 / anie.201209228 , PMID 23512582 .

Individual evidence

1. ^ Hershel Markovitz: Thomas G. Fox 1921-1977. In: Rheologica Acta. 17, 1978, p. 207, doi : 10.1007 / BF01535056 .
2. ^ ^{A b} M. Alger: Polymer Science Dictionary. Springer Science & Business Media, 1996, ISBN 978-0-412-60870-4 , p. 201.
3. ^ ^{A b} Thomas G. Fox, Paul J. Flory: Second-Order Transition Temperatures and Related Properties of Polystyrene. I. Influence of Molecular Weight. In: Journal of Applied Physics. 21, 1950, p. 581, doi : 10.1063 / 1.1699711 .
4. ^ FR Schwarzl: Polymer Mechanics. Structure and mechanical behavior of polymers. Springer, Berlin / Heidelberg 1990, ISBN 978-3-642-61506-1 , p. 72. doi : 10.1007 / 978-3-642-61506-1 .
5. ^ Thomas G Fox, Paul J. Flory: The glass temperature and related properties of polystyrene. Influence of molecular weight. In: Journal of Polymer Science. 14, p. 315, doi : 10.1002 / pol.1954.120147514 . ( PDF ).
6. ↑ ^{a b} Paul Hiemenz, Timothy Lodge: Polymer Chemistry. 2007, CRC Press. ISBN 1-57444-779-3 .
7. ^ Toshio Ogawa: Effects of molecular weight on mechanical properties of polypropylene. In: Journal of Applied Polymer Science (1992), Volume 44, p. 1869, doi : 10.1002 / app.1992.070441022 .
8. ↑ TG Fox, S. Loshaek: Influence of molecular weight and degree of crosslinking on the specific volume and glass temperature of polymer. In: Journal of Polymer Science (1955), Volume 15, p. 371, doi : 10.1002 / pol . 1955.120158006 .
9. ^ Thomas G. Fox, Influence of Diluent and of Copolymer Composition on the Glass Temperature of a Polymer System. In: Bull. Am. Phys. Soc. (1956), Volume 1, p. 123.