Typical Results For Common Transformations Studied by MTDSC

We will consider the application of MTDSC to the following types of processes:

1. Transformations that follow Arrhenius type kinetics
2. Glass transitions.

Melting was dealt with briefly in part one of this introduction. There are benefits that derive from the ability of MTDSC to detect melting and rearrangement in the reversing signal and these will be dealt with in part 3.

Processes That Follow Arrhenius Type Kinetics

The Arrhenius equation is generally of the form

da/dt = f(a) A e^-E/RT

(34)

Where a = the extent of the reaction

t = time

f(a) = some function of extent of reaction

A = the pre-exponential constant

E = the activation energy

R = the gas constant

T = absolute temperature

This type of behaviour is associated with the well-known energy barrier model for thermally activated processes. In this model, a material changes from one form to another more thermodynamically stable form but must first overcome an energy barrier that requires an increase in Gibbs free energy. Only a certain fraction of the population of reactant molecules have sufficient energy to do this and the extent of this fraction and the total number of reactant molecules determine the speed at which the transformation occurs. The fraction of molecules with sufficient energy is dependent upon the temperature in a way given by the form of the Arrhenius equation, thus this must also be true for the transformation rate. The types of process that can be modelled using this type of expression include chemical reactions, diffusion controlled processes such as the desorption of a vapour from a solid and some phase changes such as crystallisation. There will be some constant of proportionality, H, such that the rate of heat flow can be directly related to the rate of the process viz.:

dQ/dt = H da/dt = Hf(a)Ae^-E/RT

(35)

Adding in the heat capacity contribution to the heat flow gives:

dQ/dt = bC_p + Hf(a)Ae^-E/RT

(36)

The curves below shows results for a chemical reaction, in this case a decomposition (the phase lag correction has not been used here as it can be neglected as discussed in part 1, see equation 27). By combining equations 10, 13 and 36 we obtain;

dQ/dt = b C_pR + <Hf(a)Ae^-E/RT > …… the underlying signal

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

(37)

Where C is often so small it can usually be neglected thus the response to the modulation is completely dominated by the heat capacity of the sample.

From equation 15 we obtain:

<dQ/dt> » b C_pR + <Hf(a)Ae^-E/RT >

(38)

Thus

<dQ/dt> - b C_pR » <Hf(a)Ae^-E/RT > = the non-reversing signal

(39)

In this way the two different contributions to the total heat flow from the heat capacity (the reversing signal = complex heat capacity in this case) and the irreversible reaction (the non-reversing signal) can be separated (deconvoluted) as clearly demonstrated above.

It can be shown that, for crystallisation and other processes that can be described by Arrhenius kinetics then:

C = d(Hf(a)Ae^-E/RT)/dT

(40)

Or if this expressed as a heat capacity:

C_pK = [d(Hf(a)Ae^-E/RT)/dT]/Bf

(41)

This ability to vibrational heat capacity from the enthalpy of chemical reactions is useful for studying cross-linking reactions such as epoxy cures. Typical results from this type of system are shown above. The non-reversing signal shows the progress of the chemical reaction while the complex heat capacity shows a glass transition as the reaction reaches a certain stage. This occurs because the T_g of the system increases as the cross-link density increases until it equals the cure temperature. When this happened the mobility of the system decreases dramatically and consequently the reaction rate decreases as it becomes diffusion controlled. This is seen as a decrease in the vibrational heat capacity and the system is said to have vitrified. As the temperature increases (because this is a rising temperature experiment) this tends to increase mobility thus increasing reaction rate thereby increasing the glass transition temperature. As this process continues it gives rise to a very extended period over which the reaction proceeds very slowly under quasi-diffusion control. This effect can be very important in determining the cure behaviour of a wide range of materials. Conventional DSC does not allow this phenomenon to be studied in any detail.

Glass Transitions

At first sight, the step change in C_p that occurs at the glass transition might be interpreted as a discontinuity that would mean that it is a second order transition. In fact the transition is gradual as it occurs over about 10 degrees or more. Its position also varies with heating rate (and with frequency in MTDSC) which reveals that it is kinetic phenomenon. The co-operative motions that enable large-scale movement in polymers have an activation energy in a way that is similar to (but not the same as) the energy barrier model discussed above for Arrhenius processes. Thus, as the temperature is decreased, they become slower until they appear frozen. There is some part of the heat capacity that is associated with these motions therefore, as they freeze, these large-scale motions are no longer possible and consequently the material appears glassy (rigid) and the heat capacity decreases. In reality, whether a polymer appears glassy depends on how rapidly the observer is attempting to deform it. Thus, if the polymer is being bent at a frequency of several times a minute, it may be springy and return to its original shape when let go. If it being bent at several times a year it may well behave like a pliable material that retains the shape it is given by the deformation when it is released. There is a parallel dependence of the heat capacity on how rapidly one is attempting to put heat into or take it out of the sample, thus the position of the Tg changes with heating and cooling rates.

The diagram above gives the enthalpy diagram for glass formation. The enthalpy gained or lost by a sample is determined by integrating the area under the dQ/dt or heat capacity curve. Above the T_g the sample is in equilibrium (provided no other processes such as crystallisation are occurring). Consequently, this line is fixed regardless of the thermal treatment of the sample and a given temperature corresponds to a unique enthalpy stored within the sample. As the sample is cooled there comes a point at which the C_p changes as it goes through the glass transition, thus heat capacity changes and so does the slope of the enthalpy line. At different cooling rates the temperature at which this happens changes, thus a different glass with a different enthalpy is created. Above the transition the sample is at equilibrium, below it is at some distance from this equilibrium line but is moving toward it very slowly, thus glasses are generally meta-stable. If the glass is annealed at temperatures near the glass transition then it looses enthalpy relatively rapidly and becomes a different glass as it moves toward the equilibrium line. At temperatures far below the T_g the rate of enthalpy loss becomes very slow and effectively falls to zero. In the diagram below, the results on heating are shown for a sample cooled at the same rate it was heated and a sample cooled much slower than it was heated. The characteristic relaxation peak is seen in the slow cooled, fast heated example. The same peak is observed for samples annealed so that they experience an enthalpy loss.

A very simple model for the increase in heat capacity at the glass transition is given by;

dh/dt = exp[(Dh^*/(RT_g²)(T-T_g)](TDCp - h)/t_g

(42)

Where h = d + TDCp:

d = the excess enthalpy relative to the equilibrium value

DCp = the heat capacity change at the glass transition

Dh^* = the apparent activation energy

T_g = the glass temperature

t_g = the relaxation time at equilibrium at T_g

It should be stressed that this is only an approximate model but it exhibits the main characteristics of the glass transition for moderate degrees of annealing (i.e. when the relaxation peak at the Tg is relatively small).

The figure above shows typical results for a glass transition for a sample that has been annealed for different lengths of time. It can be seen that, as expected, the total signal is the same as that observed for a conventional DSC experiment. As annealing increases, the characteristic endothermic peak at the glass transition increases. In contrast, the reversing and kinetic signals are largely unaffected by annealing thus the non-reversing signal shows an increasing peak with annealing time. The use of MTDSC then seems to eliminate the influence of annealing and enables the relaxation endotherm to be separated from the glass transition itself. To a first approximation this is true but this must be understood within the context of the frequency dependence of the glass transition. It is well know that the temperature of the glass transition is frequency dependent from measurements made with Dynamic Mechanical Analysis and Dielectric Thermal Analysis. This same frequency dependence is seen in MTDSC. The plots below show the results for polystyrene at a variety of frequencies. The cooling rate dependence for glass transition temperatures measured with conventional DSC is also well established, as explained above.

When we consider a cooling experiment with MTDSC we have both a cooling rate, b, and a frequency (the frequency of the modulation, f). The result seen in the reversing signal is largely independent of the cooling rate once it is slow enough to ensure many modulations over the transition region (where there are not many modulations the result is meaningless). Thus, as we keep the frequency of the modulation constant and vary the underlying cooling rate, the Tg seen in the average signal changes while the Tg seen in the reversing signal remains the same. Similarly, as shown above, if we keep the cooling rate the same and vary the frequency, the underlying signal remains constant while the reversing signal changes. In an MTDSC experiment, the average signal will always give a lower Tg than the reversing signal because the underlying measurement must, in some sense, be slower (i.e. on a longer time scale) than the reversing measurement. This is because of the requirement that there be many modulations over the course of the transition. As the cooling rates become slower, in other words as the time scale of the measurement becomes longer, the Tg moves to a lower temperature. Similarly, as the frequency decreases, the Tg moves to lower temperatures. As a consequence of this there is a peak in the non-reversing signal as the sample is cooled that is clearly not related to annealing but is a consequence the difference in effective frequency between the average measurement and that of the modulation.

On heating the non-reversing signal, is related to the amount of annealing and also must contain the effects of the different effective frequencies used in the measurement. We can consider these effects to be additive thus the non-reversing signal gives a measure of the enthalpy loss on annealing with an offset due to the frequency difference. This intuitively satisfactory as the enthalpy that is regained by the sample on heating cannot be again lost on a short time scale at the time and temperature the measurement is made. In this sense it is non-reversing in the same way that a chemical reaction or crystallisation is. This simple picture is only a first approximation but it will be adequate in many cases. In particular the non-reversing peak at the glass transition can be used to rank systems in terms of degree of annealing.

The average signal in MTDSC is given by equation 42. For the response to modulation it has been shown that:

C_{pR =} DCp(1 - exp(-Dh^* T/(RT_g²))/(1 + f² t_g² exp(-2 Dh^*/(RT_g²)(T-T_g)))

(43)

C_pK = DCp f t_g exp(-Dh^* T/(RT_g²)(T-T_g)).

(44)

It should be borne in mind that the above is based on a very simple model of the glass transition and is only approximately correct. However, it is adequate for our purposes here.

The fact that C_pR is time scale dependant in this case can be made explicit by using the notation C_pf for the response to the modulation and C_pb for the response to the underlying heating or cooling rate. Thus equation 13 becomes;

dQ/dt = b C_pb + <f(t,T)> .… the underlying signa1

(45)

Where C_pf = the reversing heat capacity at the frequency f

C_pb = the reversing heat capacity implied by the heating or cooling rate b

C = C_pKBf (from equation 26) where C_pK is given by equation 44.

The form of C_pK in equation 44 is very different from the case of an Arrhenius type processe. However, it is still basically a manifestation of the kinetics of the glass transition and thus the concept that this signal, and thus C from equation 45, is a measure of the kinetics of the transition, remains valid.

We can express the non-reversing heat flow as;

Non-reversing heat flow = b (C_pb - C_pf ) + <f(t,T)>

(46)

This then invites the question of how we define C_pb and relate it to C_pf. One approach is to define C_pb as the result obtained on cooling at b. This means there is no contribution from f(t,T) except on heating when it embodies the effects of annealing and reheating. This contribution would tend to zero as the annealing time was reduced. An approach to defining the relationship between C_pf and C_pb is to give a modulation frequency that gives the same glass transition temperature as a given cooling rate. This glass transition temperature equivalence could be defined as either the temperature at which the sample is 50% devitrified or at which the derivative of the heat capacity reaches a maximum or the intersection of the extrapolated enthalpy lines. Each measure would give a slightly different equivalent frequency. It has already been observed that the underlying signal on cooling is not the same in shape as the reversing signal thus a simple frequency equivalence is not strictly accurate and distribution of frequencies would be required as a function of extent of vitrification. Whether one of these alternatives or another approach is taken is largely a mater of convention.

In summary, the important concepts embodied in equation 45 are as follows; the glass transition, as measured by the reversing heat capacity, is a function of frequency. The T_g as measured by the total heat capacity (= <dQ/dt>/b) on cooling is a function of cooling rate. Broadly, there is an equivalence between these two observations because both changing the frequency of the modulation and the cooling rate changes the time scale over which the measurement is made. This means that there is always, in the non-reversing signal, a contribution from b (C_pb - C_pf ) which is present regardless of annealing (for example it is present when cooling). Ageing below the glass transition produces enthalpy loss that is recovered as a peak overlaid on the glass transition. However, this ageing does not have a great effect on the reversing signal and this is intuitively satisfactory as the ageing effect is not reversible on the time scale of the modulation. This means that the non-reversing signal includes a contribution from the different ‘frequencies’ of the cyclic and underlying measurements plus a contribution from annealing expressed as f(t, T) in equation 45. This implies that the relationship between the enthalpy loss on annealing and the area under the non-reversing peak should be linear. In reality, at higher degrees of annealing, the reversing signal is significantly affected thus this simple relationship breaks down giving rise to overestimates as high as 20%. However, the increase in the area underneath the non-reversing peak still increases monotonically with annealing thus ranking samples is still possible. While accepting that the analysis presented in equations 42, 43 and 44 is a greatly simplified model, it remains a useful way if interpreting the phenomenology of the glass transition in general applications. In reality, this analysis requires additional terms to accurately model the glass transition and there are complex interdependencies between the reversing, kinetic and total signals that have not yet been fully elucidated.

MTDSC has several significant practical advantages for studying glass transitions. The first is that the limit of detection is increased. The effect of using the Fourier Transform to eliminate all responses not at the driving frequency of the modulation is to decrease unwanted noise. The second is that it increases resolution. A high signal from the heat capacity is assured by a high rate of temperature change over the course of a modulation; a high resolution can be assured by using a low underlying rate of temperature increase. The third is that it makes the correct assignment more certain. When a glass transition is weak and set against a rising baseline due to the gradual increase in heat capacity of other components, the presence of a relaxation endotherm can give the impression of a melt or some other endothermic process rather than a glass transition. A clear step change in the reversing signal makes a correct assignment unequivocal in most cases. A fourth advantage is that quantification of amorphous phases is made more accurate. The increase in signal to noise already discussed above is obviously helpful in this respect. In addition, the elimination of annealing effects makes it easier to correctly quantify the increase in heat capacity. This is best done by looking at the derivative of the reversing heat capacity with respect to temperature as shown below.

This procedure then makes glass transitions appear as peaks, which can be fitted using standard Gaussian fitting routines to a good approximation. The area under these peaks is proportional to the amount of that phase. This then provides a method for studying the structure of polymer blends. When two polymers are blended but do not mix at the molecular level, they blend will give two distinct glass transitions that are the same temperature and shape as they would be in the pure materials. When they mix completely, a single glass transition will be seen at a temperature intermediate between the temperatures of the two constituents. When, as is most often the case, there is partial mixing, this can be characterised using MTDSC. This is illustrated for a polystyrene polyurethane interpenetrating network.

The pure materials can be seen to be completely different from the blend. The processing has increased the temperature of the glass transition for the polystyrene while, more typically, increasing that for the PU due to intermixing with the higher T_g phases. The complex structure of the blend can best be modelled assuming 4 Gaussian peaks corresponding to 4 phases. This type of detailed structural information can be difficult to obtain by other means.