Tg-conversion relationship

Figure 6 illustrates the Tg-conversion relationship for an epoxy-amine system cmed at several temperatures. All the data can be seen to collapse to a single cmve and are fit to the empirical DiBenedetto equation (44,45). Excellent theoretical treatises of the Tg-conversion relationship can be foimd in References 41-43. As can be seen from the increasing curvature of the data in Figure 6, Tg is a more sensitive measure of cure than is conversion, as measured by AH res, in the latter stages of cure, which are often the most critical. From the slope of the curve for the final 10% of cure [5.4°C/(% conversion)] and an ability to measure Tg to 2°C, control of degree of cure to 0.5% from measurement of Tg is readily feasible. For thermosetting systems in which weight loss is associated with cme or for which there is no simple or luiambiguous measure of cure, Tg is often the only practical means to monitor extent of cure.

The similarity of the Tg-time data in Figure 7 with the conversion-time data of Figure 5 is a consequence of the Tg-conversion relationship and illustrates... 

The focus of this section is to provide a description of the behavior of thermosets measurable by DSC. This consists primarily of cure and properties developed during cure, including cure kinetics. It also includes measurement of Tg and conversion to establish the important Tg-conversion relationship. 

Isothermal kinetic measurements fall into two categories method 1, in which the rate and extent of reaction at constant temperature are continuously monitored in the DSC and method 2, in which a partially cured sample is heated in the DSC to measure the residual heat of reaction. An advantage of method 1 is the simultaneous measurement of conversion and rate of conversion, which are necessary for some kinetic analyses. It should be noted that vitrification will occur during method 1 measurements if Tcure is less than Tg,. Method 2 has the advantage of simultaneous measurement of and conversion, from which the Tg-conversion relationship can be established. Both thermal and UV cure reactions can be measured by these methods. 

Figure 2.72 graphically illustrates the relationship between the glass transition temperature and conversion measured from the residual exotherm of the DSC curves of the epoxy-amine system shown in Fig. 2.70. Several workers have shown that for most thermoset systems there is indeed a unique relationship between the chemical conversion of a thermoset and its glass transition temperature, independent of the cure temperature and thermal history [see, e.g., Pascault and Williams (1990), Hale et al. (1991), Venditti and Gillham (1997), and, for a general overview. Prime (1997)]. It is good practice to establish the Tg-conversion relationship as part of a cure study. Conversion can be difficult or impossible to measure directly, for example, for systems where mass loss accompanies cure. In these cases this relationship may be invoked in order to use Tg as a measure of cure. As Fig. 2.72 suggests, may be preferred as a measure of cure for the final 5-10% of cure where it is usually more sensitive than the residual heat of reaction.

The empirical DiBenedetto equation was developed in the late 1960s to mathematically relate Tg and conversion (Nielson 1969 DiBenedetto 1987). Excellent theoretical treatises on the Tg-conversion relationship can be found in Pascault and Williams (1990), Hale et al. (1991), and Venditti and Gillham (1997). Venditti and Gillham (1997) developed an equation based on thermodynamic considerations put forth by Couchman and Karasz (1978) to predict Tg versus mole fraction of constituents of a linear copolymer ... 

The similarity of the Tg-time data in Fig. 2.73 with the conversion-time data of Fig. 2.71 is a consequence of the Tg-conversion relationship and illustrates the ability to monitor cure through measurement of Tg. Figure 2.73 also directly illustrates vitrification, defined as Tg increasing to r ure as a result of cure, and designated at each cure temperature by an arrow. Note that the progress of cure is significantly impeded shortly after vitrification, which marks the shift from chemical control to diffusion control of the reaction, as described at the beginning of this section. 

In selecting a criterion to be used for specifying Tg, the experimenter may take into account the major application for use of the Tg data for example, whether it is to be used as (1) a material property to measure material consistency (2) to evaluate the effects of processing, as in the curing of thermosets where Tg-conversion relationships are important or (3) as an engineering property where the Tg value has significance as a structural property. If mea-... 

In dealing with thermoset cure, Tg-conversion data should be generated as part of any cure characterization study. The easiest way to develop a Tg-conversion relationship is via DSC. Where degree of conversion cannot be measured directly, the Tg serves as a good means for evaluating cure advancement. This will be useful even if there is no direct proportionahty between and conversion. 

A fundamental property that determines the state of a reacting system is its extent of cure or chemical conversion (a). Several papers have shown that there is a unique relationship between the glass-transition temperature (Tg) and a that is independent of cure temperature and thermal history. This may imply that molecular structures of materials cured with different histories are the same or that the changes in molecular structure do not affect Tg. There are generally accepted to be two approaches to modelling glass-transition-conversion relationships, namely thermodynamic and viscoelastic approaches. These are summarized in Table 3.8. 

As seen in Figure 2.26, the Tg — x data obtained from the residual cure (MT)DSC experiments are well described by the optimised Tg— x relationship [Eq. (27)] of the diffusion-controlled cure model (Tg is the solid line and Tga is the dashed line). The departure of the experimental data from the continuous dashed line is due to the effect of increasing crosslinking beyond the gel point. The conversion at gelation correlates well with the value of 25% measured with dynamic rheometry (using the criterion G = G ). 

The assumption of a single or overall activation energy means that the only effect of temperature is to speed up or slow down the reaction. As illustrated in Fig. 2.71, when E is constant, conversion-time curves (or Tg-time curves through the Fg-conversion relationship see Fig. 2.72) will be parallel on a... 

The existence of a single relationship between Tg and x enables one to use the experimental measurement of Tg to follow the conversion of the thermosetting polymer. This is particularly important at high conversions where Tg is much more sensitive to follow small increases in conversion than the measurement of the residual reaction heat or the direct determination of free functional groups using IR spectroscopy. 

Usually, the Tg vs x relationship shows an upward curvature related to the shape of the curves describing the increase in the concentration of branching points and high-functional crosslinks as a function of conversion (Chapter 3). Theoretical equations relating the Tg increase to structural parameters of the polymer network have been proposed. But, in every case, adjustable parameters are required to fit experimental results. 

As discussed in Chapter 4, for many thermosetting polymers a unique relationship may be established between conversion and glass transition temperature, as was also verified for this particular diepoxide-diamine system (Wisanrakkit and Gillham, 1990). So, the left-hand side of Eq. (5.46) may be written as a unique function of Tg, F(Tg)... 

Most of these structural parameters can be derived from a statistical study (Chapter 3), but what is not obvious is the determination of their contribution to Tg. Surprisingly, very simple relationships are available to estimate the evolution of the glass transition temperature with conversion, Tg = f(x) (Chapter 4). 

Whereas the calculation of the time to gelation is relatively simple, the calculation of the time to vitrification (tyu) is not so elementary. The critical point is to obtain a relationship between T, and the extent of conversion at T, (Pvu)- Once the conversion at Tg is known, then the time to vitrification can be calculated from the kinetics of the reaction. Two approaches have been examined one calculates tyu based on a relationship between T, and Pyj, in conjunction with experimental values of Pvit the other approach formulates the Tg vs. pyj, relationship from equations in the literature relating Tg to molecular weight and molecular weight to extent of reaction... 

In many instances the ACp T-Tg) term in the denominator of Equation (8-49) is negligible with respect to the term, so that a plot of X vs. T will usually be linear, as shown in Figure 8-3. To remind us that the conversion in this plot was obtained from the energy balance rather than the mole balance it is given the subscript EP (i.e., Zgs) in Figure 8-3. Equation (8-49) applies to a CSTR, PER, PER, and also to a batch (as will be shown in Chapter 9). For g = 0 and Ws = 0, Equation (8-49) gives us the explicit relationship between X and T needed to be used in conjunction with the mole balance to solve reaction engineering problems as discussed in Section 8.1. 

Recent data indicate that PTX-2 is much less toxic orally than by intraperitoneal injection. Although early studies suggested that PTX-2 was orally toxic, these data are questionable because of the absence of a dose response-relationship. In a later study, no deaths or other changes were recorded with PTX-2 at a dose of 5000 [tg/kg. The low oral toxicity of PTX-2 may reflect poor absorption from the gastrointestinal tract or conversion to a less toxic material, such as PTX-2 seco acid, in the gut [26]. 

Because of the existence of a one-to-one relationship between Tg andx [13], it is often more adequate to use Tg instead of AH,es, especially towards complete conversion as the Ai/res is small and difficult to quantify in these conditions. It should be noted that any method that involves increasing the temperature of a partially cured sample may allow cure reactions to proceed Thus, Tg may increase while it is being measured. 

The extent of conversion at DF o.i, T)F o.s or DF qs decreases as the isothermal reaction temperature is lowered, which can be explained in terms of the one-to-one relationship between Tg andx. Close to complete conversion, only small changes in conversion result in large changes in Tg due to the influence of crosslink density. The conversion attained at... 

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