Gordon-Taylor equation

The Gordon-Taylor equation is an equation for describing the glass transition temperature of mixtures with two components as a function of the respective mass fractions . It is used in polymer chemistry , glass production and food chemistry , among others .

properties

The Gordon-Taylor equation is used to predict the glass transition temperature of mixtures of different substances . This is the transition of the mixture of substances, due to weak intermolecular interactions , into a solid, non-crystalline state. The internal structure of the amorphous solid formed is not thermodynamically stable, but is in a kinetic equilibrium and is dependent on the manufacturing process. This transition plays an essential role in food chemistry. Substances dissolved or suspended in water are often measured. During evaporation, the dissolved or suspended substance molecules are brought into close proximity to one another and thus temporarily brought into a glass-like state below their melting point. This state is influenced by additives, which are called either vitrifiers or plasticizers , depending on whether they increase or decrease the glass transition temperature . A further increase in temperature leads to the melt as a result of the dissolution of the weak bonds. With decreasing viscosity, the tendency towards chemical and enzymatic reactions increases, which leads to faster deterioration of food. For a longer shelf life of a food it is therefore necessary to store it below the glass transition temperature . The texture of ready meals and the solubility of instant soups and other powdery foods can also be influenced with the help of this parameter.

The Gordon-Taylor equation is:

${\displaystyle T_{\rm {g}}={\frac {w_{1}T_{\rm {g_{1}}}+kw_{2}T_{\rm {g_{2}}}}{w_{1}+kw_{2}}}}$

with as the glass transition temperature of the mixture, and as the mass fractions of the two components, and as the glass transition temperature of the two components (that of water is −135 ° C). is the Gordon-Taylor constant, a dimensionless physical constant that is determined experimentally with different mass fractions. If the chemical properties of the two substances are not similar, there will be deviations in the calculation of the glass transition temperature. The Gordon-Taylor constant can also be described as follows: ${\displaystyle T_{\rm {g}}}$ ${\displaystyle w_{1}}$ ${\displaystyle w_{2}}$ ${\displaystyle T_{\rm {g_{1}}}}$ ${\displaystyle T_{\rm {g_{2}}}}$ ${\displaystyle k}$

${\displaystyle k={\frac {\Delta \alpha _{2}V_{2}}{\Delta \alpha _{1}V_{1}}}}$

with as expansion coefficient and as volume . ${\displaystyle \alpha }$ ${\displaystyle V}$

Alternatively, the Couchman-Karasz equation , the Fox equation and the Kwei equation developed for stronger intermolecular interactions are also used:

${\displaystyle T_{\rm {g}}={\frac {w_{1}T_{\rm {g_{1}}}+kw_{2}T_{\rm {g_{2}}}}{w_{1}+kw_{2}}}+qw_{1}w_{2}}$

with as a variable for intermolecular interactions. ${\displaystyle q}$

For mixtures of three components, the extension of the Gordon-Taylor equation applies:

${\displaystyle T_{\rm {g}}={\frac {w_{1}T_{\rm {g_{1}}}+k_{1}w_{2}T_{\rm {g_{2}}}+k_{2}w_{3}T_{\rm {g_{3}}}}{w_{1}+k_{1}w_{2}+k_{2}w_{3}}}}$

Examples

Glass transition temperatures and Gordon-Taylor constants of various mixtures:

mixture	${\displaystyle T_{\rm {g}}}$ in ° C	${\displaystyle k}$
Fructose / water	005	3.8
Glucose / water	031	4.5
Glucose / sorbitol	032	0.464
Lactose / water	101	6.7
Maltose / water	087	6.2
Trehalose / sucrose	114	0.56
Sucrose / water	057	5.4
Maltodextrin DE 20 / water	141	6.8
Maltodextrin DE 10 / water	160	7th
Maltodextrin DE 5 / water	188	7.7
Starch / water	250	5.2
Wheat flour / water	128	-
Milk powder / water	101	8.6
Tomato powder / water	055	5.5

history

The Gordon-Taylor equation was published in 1952 by Manfred Gordon and James S. Taylor.

literature

• PJ Skrdla, PD Floyd, PC Dell'Orco: The amorphous state: first-principles derivation of the Gordon-Taylor equation for direct prediction of the glass transition temperature of mixtures; estimation of the crossover temperature of fragile glass formers; physical basis of the “Rule of 2/3”. In: Physical chemistry chemical physics (PCCP). Volume 19, Number 31, August 2017, pp. 20523-20532, doi : 10.1039 / c7cp04124a , PMID 28730199 .

Individual evidence

1. ^ ^{A b} Benjamin Caballero, Paul Finglas, Fidel Toldrá: Encyclopedia of Food and Health. Academic Press, 2016, ISBN 978-0-123-84953-3 , Volume 1, keyword "Agglomeration", p. 76.
2. ↑ ^{a b c} Patrick F. Fox: Advanced Dairy Chemistry Volume 3. Springer Science & Business Media, 1992, ISBN 978-0-412-63020-0 , p. 316.
3. ↑ M. Alger: Polymer Science Dictionary. Springer Science & Business Media, 1996, ISBN 978-0-412-60870-4 , p. 228.
4. ↑ Thomas Sabu: Characterization of Polymer Blends. John Wiley & Sons, 2014, ISBN 978-3-527-64561-9 , p. 379.
5. ^ Dennis R. Heldman: Encyclopedia of Agricultural, Food, and Biological Engineering (Print). CRC Press, 2003, ISBN 978-0-824-70938-9 , p. 760.
6. ↑ Cristina Ratti: Advances in Food Dehydration. CRC Press, 2008, ISBN 978-1-420-05253-4 , p. 40.
7. ^ TG 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.
8. ^ Mark F. Sonnenschein: Polyurethanes. John Wiley & Sons, 2014, ISBN 978-1-118-73793-4 , pp. 155 f.
9. ↑ L. Weng, R. Vijayaraghavan, DR Macfarlane, GD Elliott: Application of the Kwei Equation to model the Behavior of Binary Blends of Sugars and Salts. ${\displaystyle T_{\rm {g}}}$In: Cryobiology. Volume 68, number 1, February 2014, pp. 155-158, doi : 10.1016 / j.cryobiol.2013.12.005 , PMID 24365463 , PMC 4101886 (free full text).
10. ↑ M. Shafiur Rahman: Food Properties Handbook. CRC Press, 1995, ISBN 978-0-849-38005-1 , p. 140.
11. ↑ Manfred Gordon, James S. Taylor: Ideal copolymers and the second-order transitions of synthetic rubbers. i. non-crystalline copolymers. In: Journal of Applied Chemistry. 2, 1952, p. 493, doi : 10.1002 / jctb.5010020901 .