The Polyurethane Glass Transition Temperature
Figure 4.13 defines the tensile properties of polyurethanes in relation to the phase separated hard and soft segment glass transition temperatures. The glass transition temperature is certainly one of the more difficult physical parameters to treat from a theoretical perspective due to its "nonunique" nature . This nonunique distinction is manifested in the value being a function of how it is measured (such as the rate of measurement), whether the transition is approached from low or high temperatures, and also by the instrumental technique. It is also problematic that the glass transition is not well defined by a single temperature but by a temperature region, with a finite and sometimes very substantial breadth (e.g., Fig. 5.24). In such cases, the glass transition temperature will be treated as a specific point in an instrumental response curve, but this is often more by convention than actual utility .
Theoretical approaches to the polymer glass transition temperature typically treat measured T values as kinetic manifestations of underlying thermodynamic interactions between polymer segments, end groups, and additives. The specifics of these interactions can vary widely. Furthermore, disagreement on the thermodynamic definition of glass transition temperature relative to discontinuities in thermodynamic quantities further complicates first principle theoretical treatment. Nevertheless there are numerous equations that are used in practice and in the scientific literature to estimate the glass transition temperature of polyurethanes. Many of these equations are semi-empirical, requiring input of measured data obtained from pure materials to make calculations based on interpolation. Others use measured reference values combined with fitting parameters that are defined in elaboration of fundamental principles. The choice of which equation a practitioner uses will depend on the individual's preferences and access to data that might be required for refined prediction of values. Another method has utilized group additively principles which can result in relatively good predictions if enough group factors are incorporated. Specific methods and values using group additively principles have been presented .
One of the most common equations used for predicting the glass transition temperature of binary systems of which polyurethanes might be considered is the Fox equation (4.24) . The exceptionally simple form and the sole need to obtain the glass transition temperatures of the pure components (which can normally be measured or even estimated with good precision) make it widely used. The simplicity of the equation is an expression of the numerous underlying assumptions. These assumptions are that the pure material glass transition temperatures are constant with blend composition; that the contributions of the polymer components are additive (a small molecule plasticizer or a polymer blended with a polymer have equivalent effects), symmetrical on their influence on the glass transition temperature; that the two polymers are miscible and amorphous; and that the heat capacity changes at the phase transition are the same for the two materials.
In Equation 4.24, Tg represents the glass transition of the miscible blend while w. represents the weight fractions of the respective polymer components and 7 the glass transition of the pure polymer component. The graphical representation of Equation 4.24 is given in Figure 4.20. It shows the weak Tg depression that results from mixing two materials due to the slight increase in free volume and entropy in the absence of specific interactions. This is visualized by comparison to the graphed function in which the blend Tg is represented by a function in which the Tgs are simply added in a mass weighted linear combination.
The Fox equation is in fact a special case of the Gordon-Taylor equation (Eq. 4.25) . This equation is very widely and successfully used containing the assumptions of the Fox equation but introduces a parameter (k). k depends on the specific volume of the components (V) and the thermal expansion coefficients a. In practice, k is usually treated as a fitting parameter but should be on the order of 1. Practically this parameter allows for asymmetry between the blend component contributions to the blend T. This would allow for a physical situation in which the contribution of blend component 1 to the blend glass transition temperature is weighted disproportionately
Figure 4.20 Graphical representation of the Fox equation (4.24) relative to a simple mass weighted linear combination of pure material Tgs. For illustration, Tg1 was chosen to be 273 K and 7g2 was chosen to be 373 K—relevant to many polyurethane systems.
Figure 4.21 Graphical representation of the Gordon-Taylor equation for varying values of k yielding either positive or negative deviations from a mass weighted linear combination of pure material glass transition temperatures.
to that of component 2, resulting in a stronger deviation (see Fig. 4.21) than predicted by the Fox equation.
The Fox equation can be derived from the Gordon-Taylor expression as a limiting case of the Simha-Boyer rule for k in Equation 4.25 where specific volumes of the components are treated as identical, that is, k ~ T /T 2 .
A special case of the Gordon-Taylor equation offered by Ginzburg and Sonnenschein  was developed specifically for polyurethanes reflecting microstructural details as described by Figures 4.6 and 4.17. Equation 4.26 specifies the T for the soft segment depending on the amount of dissolved hard segment in the soft segment (wH). The dissolved hard segment is a function of the microphase structure defined by Figure 4.6 and quantitatively Equation 4.3 by variable xNo.
In this variation of Gordon-Taylor formalism, if the hard segment weight fraction is below the order disorder transition (the D-S isomorph of Fig. 4.6) the fraction hard segment in the soft phase is the total hard segment mass in the polymer. However, if the polymer has a phase-separated microstructure, the amount of hard segment in the soft phase reflects the total hard segment minus the amount of hard segment that has precipitated into a separate phase.
An occurrence sometimes encountered in polymer blends is that two polymers interact through a specific and relatively strong interaction. Such an interaction would be, for instance, strong hydrogen bonding between blend components. In this instance, significant deviations from the previous equations may be observed. For this case, the Kwei equation 4.27 has been utilized for systems exhibiting both negative and positive deviations from the linear combination profile (Fig. 4.22) and "S" shaped profiles .
The functional versatility of the equation results from the quadratic function in the second half of the equation. While k and q are usually treated as fitting parameters, there has been significant investment in creating a fundamental justification and
Figure 4.22 Graphical representation of the Kwei equation keeping k at or near one (as in the Gordon-Taylor formalism) and varying q to demonstrate its effect on the functional form.
elaboration for the q variable as it relates to the intermolecular interactions responsible for the observed Tg behavior [73, 74].
Another equation employed in the case of significant deviations from a simple weighted linear combination of glass transition temperatures was offered by Lu and Weiss . It is also a modification of the Gordon-Taylor equation (as is the Kwei equation) and incorporates a quadratic dependence as the Kwei equation does. However, the q value is elaborated in terms of the Flory x value which can be obtained by a best fit routine (Eq. 4.28). In Equation 4.28, b is the ratio of amorphous densities of polymer 2 and polymer 1, ACp1 is the change in specific heat of polymer 1 at T , and the other variables are as previously defined for the Gordon-Taylor equation. In cases where the blend polymer interactions are quite strong, this equation has shown utility and resulted in reasonable values of x. In the case of weak or moderate interactions, the Gordon-Taylor equation has provided equally predictive results to the Lu and Weiss equation .
While all of these equations have been applied to polyurethanes, there are quite a few other approaches that have been tried or could be applied [63, 77-81].