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 (Tg) 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. Because polymer chain mobility also governs viscosity, the temperature dependence of viscosity can be described by the Williams‑Landel‑Ferry (WLF) equation. For food engineers, understanding the glass transition temperature in frozen foods is essential for optimizing texture and stability during storage. 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.
🚀 Skip the Manual Math!
Use our interactive Glass Transition Temperature of Mixtures (Gordon‑Taylor) tool to compute these parameters instantly online, or download the offline Excel calculation; for a quick estimate of mixture \(T_{\text{g}}\) based on component weight fractions, see the Fox equation for glass transition temperature approximation.
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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