Modulated temperature DSC uses a modulated temperature profile combined with a deconvolution procedure to obtain the same information as conventional DSC plus, at the same time, the response to the modulation. A sinusoidal modulation is most often used as illustrated below together with the resultant heat flow. Also included is one of the signals derived from the deconvolution procedure, the phase lag between the modulation in the heating rate and that in the heat flow. As the first step in the deconvolution process, the raw data is averaged over the period of one or more modulations to remove the modulation. This then gives the total signal, which is equivalent to conventional DSC. The averaged signal is subtracted from the raw data and the modulation is then analysed using a Fourier Transform procedure (discrete or Fast FT) to obtain the amplitude and phase difference of the heat flow response at the frequency of the imposed modulation.

The basic output from the deconvolution procedure is therefore:

<dQ/dt> = the average or total heat flow

(1)

Where <> denotes the average over one or more periods

A_HF = amplitude of the heat flow modulation

A_HR = amplitude of modulation in the heating rate

w = the phase difference between the modulation in the heat flow and the heating rate

One point that needs to be stressed is that this analysis assumes that the sample's response to the modulation can be approximated as linear. Where this cannot be said to be true, the above analysis fails because it assumes a linear response. Where a sawtooth modulation is used (square wave in dT/dt) the Fourier Transform can, in principle, be used to extract the response at a series of frequencies. However, current commercial products restrict themselves to using the first component of the Fourier series, which is then equivalent to using a single sinusoidal modulation. It is true that looking at the whole Fourier series, rather than just the first component, offers scope for significantly increasing the amount of information that can be obtained from a MTDSC experiment. This applies even to single sinusoidal modulations (because non-linearities are expressed as harmonics) as well as multiple simultaneous sine waves and sawtooth modulations. However, a discussion of these points is beyond the scope of this introduction.

The response to the modulation can be treated as follows:

A_HF/A_HR = the cyclic or complex heat capacity = C^*

(2)

C^*cosw = the in-phase or reversing heat capacity = C_pR

(3)

C^*sinw = the out of phase or kinetic heat capacity = C_pK

(3)

If we use complex notation then:

C* = C’ – iC"

(4)

Where i = the square root of -1

C_pR = C' = the real component

(6)

C_pK = C'' = the imaginary component

(7)

A simple theory for transitions other than melting

We can consider a process that gives rise to a heat flow and is governed by some general kinetic function that is dependent on time and temperature, viz, for this process:

dQ/dt = f(t,T)

(8)

f(t,T) denotes some sample response that is not rapid, i.e. it is kinetically hindered and takes a measurable amount of time to come to equilibrium when the temperature is changed (the alternative is to respond to any change in temperature instantaneously). One possible specific form for this function could be the Arrhenius equation described below. The contribution to heat flow from heat capaxcity can then be included viz:

dQ/dt = bC_p + f(t,T)

(9)

Where C_p = the heat capacity of the sample.

The heat capacity, also called the vibrational heat capacity, can be considered as the energy contained principally in the various vibrational, but also the translational etc., modes available to the sample. These processes are usually very fast and can normally be considered to be instantaneous when compared to the frequency of the modulation that typically has a period of several tens of seconds. The energy contained in these molecular motions is stored reversibly, which simply means that the amount of energy taken up by them when increasing the temperature by 1°C can be entirely recovered by reducing the temperature by 1°C. This can be contrasted with the enthalpy associated with, for example, a typical chemical reaction that will be either gained by the sample (endothermic) or lost (exothermic) irreversibly.

When we employ a modulated temperature programme given by:

T = T_o + bt + B sinft

(10)

Where T_o = the starting temperature

B = the amplitude of the modulation

f= the angular frequency of the modulation

We then obtain:

dQ/dt = (b + Bfcosft)C_p + <f(t,T)> + C sinft

(11)

Where <f(t,T)> = the average of the kinetic response over the period of one or more modulations

C = The amplitude of the changes in heat flow engendered by the changes in the rate of the kinetic process caused by the temperature modulation.

Note that:

C = BfC*sinw

(12)

We can rearrange equation 11 to give:

dQ/dt = bC_p + <f(t,T)> …… the underlying signal

+ C_p Bfcosft + C sinft …… the response to the modulation

(13)

We should note that:

<( C_pBfcosft + C sinft)> » 0

(14)

Therefore, for a modulated experiment:

< dQ/dt> = bC_p + <f(t,T)> » bC_p + f(t,T)

(15)

Thus, as stated above, simply by averaging the modulated signal we recover the signal we would have obtained had we not used the modulation.

For simplicity we will consider the case where f(t, T) is an irreversible process.

Therefore C_pb = the component of the average heat flow signal in a MTDSC experiment that derives from the heat capacity. Because it is rapidly reversible it is called the reversing heat flow . In a simple case, where the heat capacity is frequency independent we can write:

C_p = C_pR

(16)

We recall that we obtain C_pR from the deconvolution procedure (see equation 3 above). From this and equations 15 and 16 we can write;

<dQ/dt> - b C_pR = <f(t, T)>

(17)

This signal is called the non-reversing heat flow. By carrying out this deconvolution procedure we are therefore able to separate the two fundamentally different contributions to the total heat flow; the reversible contribution that derives from the heat capacity (called the reversing heat flow) and the irreversible contribution that derives from f(t, T) (called the non-reversing heat flow).

We can then calculate one further signal:

C_pKb = the kinetic heat flow

(18)

In summary, the four signals obtained from MTDSC are:

the average or total heat flow = <dQ/dt>

(19)

the reversing heat flow = C_pR b

(20)

the non-reversing heat flow = <dQ/dt> - C_pR b

(21)

the kinetic heat flow = C = C_pKb

(22)

It is also possible to express all of these quantities as heat capacities viz:

the average or total heat capacity = <dQ/dt>/b

(23)

the reversing heat capacity = C_pR

(24)

the non-reversing heat capacity = <dQ/dt>/b - C_pR

(25)

the kinetic heat capacity = C_pK

(26)

In the description given above, essentially represented in equations 9 to 17, the ‘reversing’ signal was truly reversible and the ‘non-reversing’ signal came from an irreversible process. However, as shall be discussed further below, the non-reversing signal can also be the heat from a crystallisation or from the loss of water. Both of these processes are reversible in the sense that, with large-scale temperature changes, crystals can be melted and, on cooling, moisture can be reabsorbed. For this reason the term non-reversing was coined to denote that at the time and temperature the measurement was made the process was not reversing but it might be reversible. Similarly, at the glass transition, the reversing signal does not include all the enthalpy that is reversible because this is frequency dependent, thus it comprises only that part that is reversing at the chosen modulation frequency. For this reason the terms ‘reversing’ was preferred over reversible.

In the diagram above, the four deconvoluted signals from the data presented earlier are given, in this case expressed as heat capacities. Note that the glass transition appears in all four signals, the cold crystallisation appears in all except the reversing signal, while the melt again features in all four signals. This will be discussed in more detail below.

The above analysis can be applied to a range of processes governed by Arrhenius type kinetic behaviour including crystallisation as seen in above over the range of the cold crystallisation. Because this is an exothermic process the kinetic heat capacity is negative as predicted. The reversing heat capacity is the vibrational heat capacity (equation 16) and this can be see to remain approximately constant (in fact there is a slight decrease) over the course of the crystallisation as would be expected. With modification, it can also be applied to the glass transition where it can separate the contribution due to annealing from that due to molecular scale motions as shall be discussed in the second part of this introduction MTDSC.

A fundamental consideration that always applies is the requirement that there be many modulations over the course of any transition. The deconvolution process only makes sense if <f(t, T)> approximately equals f(t, T) and for this to be true the modulation must be rapid with respect to the progress of the transformation represented by f(t, T). Generally, where the transition is a peak in dQ/dt, then there should be at least 5 modulations over the period represented by the width at half height. Where the transition is a step change, there should be at least 5 modulations over that part of the transition where change is most rapid.

It is very often the case for glass transitions, chemical reactions, solvent loss and crystallisation, that the kinetic heat flow is very small and can be neglected thus;

C^* » C_pR

(27)

Note that this is not the same as saying that the kinetic heat flow or heat capacity cannot be detected, it simply means that it is not necessary to use the phase angle to calculate C_pR. This observation is useful because it is often the phase angle that is the most difficult signal to correct for instrumental effects. Consequently C^* is often used to calculate the reversing heat flow viz;

C^* b » the reversing heat flow

(28)

This approximation is almost always incorrect during melting where a large phase lag is very often measured. Nevertheless it is often used in this case in spite of resultant inaccuracies. The reader should be aware of this and be careful to observe whether the phase lag has been used in calculating the reversing signal. Sometimes neglecting the phase lag is perfectly acceptable (the errors involved are typically less that 1% for most glass transitions for example) and sometimes it is not.

Melting Transitions

Some materials, usually polymers, produce a range of crystalline forms with different melting temperatures. Typically a semi-crystalline polymers will comprise a distribution of crystallites with differing degrees of perfection and thus different melting temperatures. The melting transition in these materials is broad as a succession of crystallite populations melts one after the other as the sample temperature reaches their melting temperatures. We can express this as;

dQt/dt = b(C_p + g(t,T))

(29)

Where g(t,T) = some function that models the contribution to the heat flow from the melting process.

When the melting is rapid with respect to the measurement, g(t, T) will be simply a function of temperature. This means that, in the case of the distribution of crystallites, the melting contribution to heat flow will scale with heating rate exactly like heat capacity if no other process occurs. In reality this simple case is rarely encountered. This point will be discussed in more detail below in the section on the use of temperature modulation.

The melting of pure materials that occur over a very narrow temperature range inevitably take place, either entirely or to a significant extent, over the course of only one modulation. This then means that the response to the modulation will not be linear and the deconvolution procedure we have described above cannot be used.

As stated above, semi-crystalline polymers consist of a distribution of crystallites of different sizes and different melting points (where the melting point is related to the size of the crystal). Adapting equation 29 for modulated conditions we obtain:

dQ/dt = b [C_p + <g(t, T)>] …… the underlying signal

+ Bfcosft (C_p + E) + C sinft …… the response to the modulation

(30)

If we assume that melting is rapid with respect to the modulation then C=0. Under these conditions it has been shown that melting will appear both in the total and the reversing signal. Where there are no significant temperature gradients in the sample the reversing and total signals should be equal, thus the non-reversing signal is zero. In practice this is rarely observed. In the deconvoluited data for PET it can be seen that, above the cold crystallisation, the reversing signal is greater than the average until very near the end of the melting peak This means that the non-reversing signal is exothermic over most of the melt region. This occurs because, as can be seen from figure 1, at the lower heating rates an exotherm is observed within the modulation along with an endotherm at the higher heating rates. This is symptomatic of a rearrangement process where the molten material produced by melting the crystallites with lower melting temperatures can crystallise to form more perfect crystals with a higher melting temperature. This is seen because, at the lower heating rate, the rate of melting is lowest thus the exothermic process can predominate. At the higher heating rates the reverse is true. In conventional DSC, which provides the same curve as the average signal, there is little or no indication that this rearrangement process is occurring as the exothermic and endothermic processes cancel each other out. Thus one benefit from using MTDSC is simply the qualitative one that it can make the occurrence of this phenomenon far more apparent.

We therefore have complex process that involves melting a population of crystallites with a range of melting temperatures to form molten material which then crystallises (following some kinetics, thus involving some f(t,T)) to form a further population of crystallites which then, in their turn, melt and possibly undergo further rearrangement. To complicate maters further, some workers have suggested that melting is often not rapid with respect to the frequency of the modulation (i.e. some superheating can occur) thus there is a time dependency in g(t, T) and C is not zero even without taking account of crystallisation. To allow for this complex range of possibilities it is convenient to define a composite kinetic function that includes all terms other than the heat capacity and models both melting and the kinetics of crystallisation viz;

f₂(t, T) = g(t, T)dT/dt + f(t,T)

(31)

Under modulated conditions with no cooling

f₂(t, T) = <f₂(t, T)> + Dsinft + E Bfcosf

(32)

Equation 33 now becomes:

dQ/dt = b C_p + <f₂(t,T)> …… the underlying signal

+ Bf( C_p + E)cosft + C sinft …… the response to the modulation

(33)

It is clear that the 'reversing' signal during the melt no longer has the same meaning as for Arrhenius process and the glass transition because it contains a contribution, E, from the melting of the crystallites which will typically not be fully reversible due to supercooling. This gives rise to the requirement that there be no cooling at any point during the modulation so that the response does not become asymmetric and thus non-linear. As discussed above, When the sample response in non-linear, it is no longer appropriate to use the deconvolution procedure described here.