When a cyclic temperature profile is applied to a sample the heat flow signal will oscillate as a result of the temperature program, and the size of the oscillation will be a function of the heat capacity of the sample. Therefore, the amplitude of the heat flow signal allows a heat capacity value to be obtained. This is similar to DMA where the amplitude of the oscillation allows a modulus value to be obtained. Whilst other methods already exist to provide heat capacity, the value of this method is that the heat capacity measurement is separated from other potentially overlapping events, such as reactions or stress relaxations and can also be obtained with increased sensitivity compared to the slow linear scan rates of traditional DSC (differential scanning calorimetry).

The sinusoidal approach by Reading introduced the terminology of reversing heat flow for what is essentially the heat capacity trace, total heat flow for the average of the modulated heat flow trace, which is the conventional DSC trace, and non-reversing heat flow for the kinetic response. Since the Tg is observed as a step in the heat flow trace the reversing heat flow signal has been used to make measurements of Tg. This curve should show events that are truly reversing in the sense that the same event can be observed upon reheating or recooling.

1. Modulated temperature methods

In many DSC (differential scanning calorimetry) traces, there is evidence of a ‘kinetic signal’ being present in addition to the usual changes in heat capacity such as those due to the glass transition and melting. This is often hinted at by the kinetic signal being sensitive to the scanning rate while other signals appear to be relatively unaffected Figure 1. Essentially, the kinetics of slow events such as reactions and recrystallisation events are influenced by the scan rate and so shifted to higher temperatures as a function of increasing scan rate. Modulated temperature DSC (MTDSC) techniques allow the reversing heat flow (heat capacity) signal which is measured over the short time period (the period of the modulation) to be separated from slower transitions that occur over a much longer time period. Thus, the glass transition can be separated from relaxation, recrystallisation and other events that might obscure it and make the measurements clearer. Relaxation and other non-reversing events should then appear on the non-reversing curve. With appropriate parameters the sensitivity of measurement can be enhanced by the increased rate of scanning that occurs during part of the cycle, which results in an increased heat flow signal. Hence difficult-to-find glass transitions can become more obvious when using modulated methods. MTDSC can also be used in quasi-isothermal mode such that no overall temperature ramp is induced and allows measurement of Cp values at a single point. This can allow improved resolution of events where a change in Cp is observed. During melting, energy is continually absorbed by a material and a steady state is not maintained, making analysis of the data more complex.

Figure 1 The recrystallisation event in this fat sample is shifted as a function of scan rate whereas the melting events remain unaffected.

For many applications a qualitative analysis of the resulting data will be all that is necessary, for example giving values for the glass transition temperature. If a fully quantitative analysis is required it is important to decide on the limits of applicability. Schawe and Hohne have given clear guidelines as to when the reversing and non-reversing signals may be accurately separated from each other. They have stated that only in the case where the reaction is fast, such that the system is always in equilibrium relative to the temperature change and the enthalpy is independent of temperature, will the reversing and non-reversing signals be reliably separated. In addition, the amplitude of temperature modulation must be sufficiently small that the term bT2 in the temperature-dependent expansion of the reaction rate can be neglected. This is equivalent to a reaction, which although clearly time dependent has a weak temperature dependence and so will be in equilibrium for small temperature steps. This can be summed up in the following equations.

The degree of reaction can be expanded as

where α is the degree of reaction, a, b and c are constants, and T is the temperature. The scanning rate is

where β is the linear rate of temperature increase, and W0 and Ta are the frequency and the amplitude of modulation respectively.

The heat flow into the sample= C×scanning rate, where C is a generalized heat capacity

Therefore, the heat flow into the sample is

Ta can be chosen such that the effect of modulation on the reaction is small and so a linear approximation to the effect of the temperature profile on the reaction can be used.

Therefore, the heat flow into the sample is [Cpβ₀ + Hr dα/dT β₀] + Cpw₀T_a cos W0t where H_r is the enthalpy of reaction. It can be seen that in this case the reversing signal can be obtained from the second term in the periodic heat flow, as there is no complication of other contributions such as the energy of reaction appearing in this term. The conventional DSC trace can then be used as a background for subtraction to obtain the energy of the kinetic transition, provided the frequency dependence of the reversing signal is weak relative to the modulation frequency. This is a rather stringent test, even the well-known cold crystallization of PET fails to completely meet these criteria, except at the very lowest of modulation amplitudes and underlying heating rates.

If the modulation amplitude is higher, such that Ta cannot be neglected or it causes the reaction to behave non-linearly, i.e. requires higher terms in the expansion for α(T), then a simple separation can no longer be made. Further, if the signal is delayed in time from the stimulus, in this case the temperature rate, then the treatment of the data cannot be analyzed in the previous way and needs to be treated in a linear-response framework along the lines of rheology and dielectric data to obtain in-phase and out-of-phase components. Interpretation of data can then become very complex and difficult to relate to events in the material.

2. Stepwise methods

Using a stepwise approach, the method of calculating thermodynamic (reversing) and isokinetic baseline (non-reversing) signals is different from the methods given in the previous section and the above equations do not apply directly. As Figure 2 shows, the ‘nonreversing’ signal can more correctly be labelled the ‘slow’ component, and is the power offset at the end of the isotherm, whilst the reversing trace is calculated from the amount of heat required to raise the sample temperature by a given amount – a conventional heat capacity measurement. This can be achieved using either amplitude of heat flow, or the area under the curve. In order to achieve a reasonable (if only approximate) separation into reversing or rapid and non-reversing or slow signals it is important that the isotherm length is chosen correctly. If it is too short, then an artificially high signal can appear in the non-reversing trace. In general, the isothermal period should be long enough for the system to reach stability, typically 30–60 s. Moreover, if the slow signal varies with time or the isothermal profile changes with temperature, as has been shown for the rapid transitions between polymorphic forms such as in tripalmitin or chocolate, then there will be unavoidable sources of error. Additionally, if the amplitude of the step is too large, the peaks will be widened due to averaging so reducing resolution. To a first approximation, the slope of the step is unimportant in determining the heat capacity; at least in this form of analysis.

Figure 2. A diagram of the temperature profile used in step-based methods and the heat flow signal that results. Heat capacity is calculated either from peak height as in the classical method, or peak area of the heat flow trace. The kinetic response is obtained from the isothermal portion.

One example of information obtained from step-based methods is shown in Figure 3. The heat capacity of the material is obtained, separated from the crystallization and relaxation phenomena which appear in the kinetic trace, leading to a clearer measurement of the glass transition. This is typical of the type of information available from modulated methods in general.

Figure 3. Step-scan trace of PET shown as a function of time. The step increase in Cp identifies the glass transition, whilst a slight reduction also occurs as the sample recrystallises. The IsoK curve matches the lower envelope of the heat flow data. Quantitative data of these types require a baseline subtraction in accord with the requirements of the Cp method. Qualitative data showing the events can be obtained if this is not performed.

There is also a step-based equivalent technique founded on linear-response theory. This has been presented by Schick and co-workers and is formally equivalent to the real and imaginary heat capacities generated by modulation at a single frequency. In principle, it is therefore possible to apply a Fourier transform to step-scan type data and break down the overall response into heat flow responses to a series of sinusoidal temperature modulations.

In this case, the slope of the initial step is important as this contains the high-frequency information. Any phase differences observed between stimulus and response and not caused by the DSC instrument itself will invalidate the simple reversing/non-reversing methods, but are coped with in linear-response theory.