Glass Transition Temperature of Mixtures (Gordon-Taylor)
Reference ID: MET-0642 | Process Engineering Reference Sheets Calculation Guide
Introduction & Context
The Gordon-Taylor equation is a semi-empirical model used to predict the glass-transition temperature (\(T_{\text{g}}\)) of a homogeneous amorphous mixture. In process engineering, particularly in food, pharma, and polymer science, this equation is vital for quantifying the plasticizing effect of small molecules like water or solvents on a polymer matrix. Applications include:
Spray-Drying Optimization: Determining the maximum allowable outlet temperature to prevent particle stickiness and wall deposition.
Stability Assessment: Predicting the "critical water activity" at which a powder will transition from a stable glass to a reactive, rubbery state.
Formulation Design: Selecting plasticizers to lower the processing temperature of resins without inducing thermal degradation.
Methodology & Formulas
Convert all temperatures to Kelvin for thermodynamic calculations:
\[T_{\text{K}} = T_{^{\circ}\text{C}} + 273.15\]
Define the weight fractions of the dry solid (\(w_{1}\)) and the plasticizer/water (\(w_{2}\)):
\[w_{1} = 1 - w_{2}\]
Apply the Gordon-Taylor equation:
\[T_{\text{g,blend}} = \frac{w_{1}\,T_{\text{g1}} + k\,w_{2}\,T_{\text{g2}}}{w_{1} + k\,w_{2}}\]
where \(k\) is the Gordon-Taylor constant. Physically, \(k\) is related to the ratio of the changes in specific heat capacity (\(\Delta C_p\)) of the two components at their respective glass transitions: \(k \approx \Delta C_{p2} / \Delta C_{p1}\).
Convert the blend \(T_{\text{g}}\) back to Celsius for operational use:
\[T_{\text{g,blend}}(^{\circ}\text{C}) = T_{\text{g,blend}}(\text{K}) - 273.15\]
Typical Gordon-Taylor constants (\(k\)) for carbohydrate-based systems
System
\(k\) range
Physical Basis
Lactose–Water
5.2 – 7.0
High \(k\) due to water's large free volume and \(\Delta C_p\)
Starch–Water
4.5 – 6.0
Strong plasticization by water molecules
Lactose–Maltodextrin
0.8 – 1.2
Low \(k\) due to similar molecular structures and \(\Delta C_p\)
The most accurate theoretical estimate is the ratio of the change in specific heat capacity at the glass transition: \(k \approx \Delta C_{p2} / \Delta C_{p1}\). For water-carbohydrate systems, \(\Delta C_p\) for water is \(\approx 1.94 \, \text{J/g}\cdot\text{K}\) and for many sugars it is \(\approx 0.3 \, \text{J/g}\cdot\text{K}\), explaining why \(k\) values are typically between 5 and 8.
Water is a potent plasticizer because of its extremely small molecular volume and high free volume. In the Gordon-Taylor framework, \(k\) represents the relative "strength" of the plasticizer. A value of \(k > 1\) indicates that the plasticizer (component 2) has a disproportionately large effect on lowering the mixture \(T_g\) compared to its mass fraction.
The Gordon-Taylor equation assumes a single, homogeneous amorphous phase. Two \(T_g\) values indicate phase separation (immiscibility). In this case, the equation is invalid. You must either increase processing energy to achieve miscibility or treat the system as a composite using the Fox equation for each individual phase.
Extremely sensitive. In a 5% moisture system, miscalculating \(k\) as 0.5 instead of 5.0 results in a \(T_g\) error of over 30 °C. This is the difference between a free-flowing powder and a solid block of fused material in a silo. Always calibrate \(k\) using at least two moisture levels if high precision is required.
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Worked Example – Estimating the Glass-Transition Temperature of Spray-Dried Lactose
A food engineer is spray-drying lactose. The final powder has a residual moisture content of 4 wt %. To prevent caking during storage, the engineer must calculate the mixture \(T_g\) to ensure storage temperatures remain at least 20 °C below this value.
Knowns
Mass fraction water (component 2): \(w_2 = 0.04\)
Mass fraction amorphous lactose (component 1): \(w_1 = 0.96\)
\(T_g\) of anhydrous lactose: \(T_{g1} = 101\,^\circ\text{C}\)
\(T_g\) of water: \(T_{g2} = -137\,^\circ\text{C}\) (standard literature value)
Gordon-Taylor constant for lactose-water: \(k = 5.2\)
Convert the result back to °C:
\[
T_{g,\text{blend}}(^\circ\text{C}) = 331.77 - 273.15 = 58.62\,^\circ\text{C}
\]
Final Answer
The predicted glass-transition temperature of the 4% moisture lactose powder is 58.6 °C.
Note: Using an incorrect \(k < 1\) would have predicted a \(T_g \approx 98^\circ\text{C}\), leading to catastrophic product clumping at standard warehouse temperatures.
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