Thermal Analysis of Polymeric Materials

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Thermodynamic Functions of Paraffins [20]. Another example of heat capacity measurements by adiabatic calorimetry and DSC is given by the paraffins. Figure 4.49 illustrates heat capacities of some normal paraffins of short chain length CxHx+2. The drawn-out lines represent the computed heat capacities based on vibrational contributions fitted using the ATHAS system (see Sect. 2.3). Two observations can be made from these data. First, at higher temperature there is a constant increment in

Fig. 4.49

heat capacity for each additional CH2-group, second, the heat capacities increase beyond the computed vibrational contributions as they approach the melting and transition region. As in polyethylene, the increase is attributed to local, largeamplitude, conformational motion.

The heat capacity for a longer alkane, pentacontane, C50H102, is shown in Fig. 4.50 and integrated to the thermodynamic functions H, G, and S in Fig. 4.51. This paraffin approaches the vibrational characteristics and thermodynamic behavior of polyethylene (see Sect. 2.3). The skeletal vibrations continuously approach those of polyethylene, while the group vibrations are practically the same for CH2-groups in polyethylene and alkanes.

The use of the thermodynamic data for a discussion of stability of crystals is demonstrated in Fig. 4.52. The differences in crystal structure and melting temperatures of successive n-alkanes have been linked to the symmetries of odd and even CH2-sequences. Note, that planar zig-zag chains of odd-numbered sequences of carbon atoms in a molecular backbone point with their final bonds into the same direction, while even ones point into opposite directions. Odd paraffins have an orthorhombic, rectangular layer-structure, while the even ones up to C24H50 are triclinic with oblique layers. From structural analyses, orthorhombic crystals can accommodate even and odd chains without difference in packing density. The triclinic

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Fig. 4.50

Fig. 4.51

crystals, in contrast, can only accommodate even molecules, but with a higher heat of fusion than the orthorhombic crystals. Figure 4.52 represents the thermodynamic reasoning for this odd/even effect. Structural requirements force the odd n-paraffins into the orthorhombic phase, which, in turn, requires a lower melting temperature to reach the common melt. For the even paraffins there is the structurally more favorable

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Fig. 4.52

oblique packing, making the orthorhombic phase less stable, as shown in the figure. The skeletal vibration frequencies are lower in the orthorhombic crystal, but cannot overcome the heat of fusion effect.

4.3 Differential Scanning Calorimetry

4.3.1 Principle and History

Differential scanning calorimetry, DSC, is a technique which combines the ease of measurement of heating and cooling curves as displayed in Fig. 4.9 with the quantitative features of calorimetry (see Sect. 4.2). Temperature is measured continuously, and a differential technique is used to assess the heat flow into the sample and to equalize incidental heat gains and losses between reference and sample. Calorimetry is never a direct determination of the heat content. Measuring heat is different from volume or mass determinations, for example. In the latter cases the total amount can be established with a single measurement. The heat content, in contrast, must be measured by beginning at zero kelvin where the heat content is zero, and add all heat increments up to the temperature of interest.

In Fig. 4.53, a brief look is taken at the history of the DSC. Both, heating curves and calorimetry had their beginning in the middle of the nineteenth century. Progress toward a DSC became possible as soon as continuous temperature monitoring with thermocouples was possible (see Fig. 4.8), and automatic temperature recording was invented (see also Sect. 4.1). These developments led to the invention of differential thermal analysis, DTA. In Sect. 2.1.3 an introduction to thermal analysis and its instrumentations is given.

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Fig. 4.53

Le Chatelier1 seems to have been first to record temperature as a function of time in heating curves such as shown in Fig. 4.9. He used a mirror galvanometer to determine the thermocouple output and recorded the position of the galvanometer mirror on a photographic plate with a beam of light. Complete DTA, consisting of measurement of temperature difference, T, as a measure of the heat-flow rate, sample temperature, Ts, and time, t, was first performed at the beginning of the 20th century in the laboratories of Roberts-Austen, Kurnakov, and Saladin. The classical DTA setup of Kurnakov is shown in Fig. 4.53. The key elements of the DTA are, as today, the furnace for equal environment of reference and sample, a means to continuously measure Ts and T, and a recording device that also fixes the time, t, by being driven with a constant speed motor. The final output is a typical DTA plot of Ts and T versus t.

Very little further progress in instrumentation was made until the 1950s. A brief summary of the classical DTA and DSC operating system, instrumentation, rules of data treatment, and the differences between DSC and DTA are given in Appendix 9. The first milestone was reached by 1952 when about 1,000 research reports on DTA had been published. At that time, DTA was mainly used to determine phase diagrams, transition temperatures, and chemical reactions. Qualitative analysis (fingerprinting) was developed for metals, oxides, salts, ceramics, glasses, minerals, soils, and foods.

The second stage of development was initiated by the use of electronics in measurement and recording. It became possible to quantitatively measure the heat

1 Henry-Louis Le Chatelier, 1850–1936, Professor of Chemistry at the University of Paris, France. Besides his work on thermocouples, he is best known for the principle of Le Chatelier, which predicts the direction on a chemical or physical change caused by a variation of temperature, pressure, or concentration.

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flow, i.e., do calorimetry. The DTA used for quantitative measurement of heat was then called DSC, differential scanning calorimetry [21]. A higher accuracy in determining T and a decrease in heat losses through smaller sample sizes and faster heating rates were critical for the development of DSC. Next, the electronic data generation allowed the direct coupling with computers. Presently DSC is used in almost all fields of scientific investigation and has proven of great value for the analysis of metastable and unstable systems. By 1972 the annual number of research publications using DSC and DTA had reached 1,000, as many as were published in the first 50 years. The biannual reviews given by the journal Analytical Chemistry offer a quick introduction into the breadth and importance of DTA and DSC.

By now it is impossible to count the research papers dealing with DSC since it has become an accepted analysis technique and is not listed as a special research tool in the title or abstract. The main development of DSC since the 1990s has gone towards the improvement of computers and their software. Most recent was the invention of temperature-modulated DSC, discussed in Sect. 4.4 [22]. The effort spent on developing suitable computer software has, perhaps, slowed advances in the improvement of calorimeters by hiding instrumentation problems in data-treatment routines.

Key research papers in DSC are often presented in Thermochimica Acta and the

Journal of Thermal Analysis and Calorimetry. Progress in DSC is also reported in the Proceedings of the International Conferences on Thermal Analysis and Calorimetry, ICTAC. The annual Proceedings of the Meeting of the North American Thermal Analysis Society, NATAS, are also a useful source of information.

4.3.2 Heat-flux Calorimetry

The principle of heat-flux calorimetry is illustrated with a schematic of a classical DTA in Appendix 9. Accuracies of heat measurements by DSC range from 10% to 0.1%. Temperature can be measured to ±0.1 K. Typical heating rates vary between 0.1 and 200 K min 1. Sample masses can be between 0.05 and 100 mg. The smaller masses are suitable for large heat effects, such as chemical reactions (explosions), phase transitions, or when fast kinetics is studied. The larger masses are necessary for assessment of smaller heat effects as in studies of heat capacity or glass transitions. Sensitivities are hard to estimate, but effects as small as 1.0 J s 1 are observable.

A frequent confusion in reporting of data is that the direction of the endotherm may be plotted downwards or upwards. The first arises when plotting T = (Ts Tr) instead of T = (Tr Ts). To avoid confusion, the endotherm or exotherm direction must be marked on DSC traces. Next, several modern, commercial, DSCs are introduced. They were selected because of their differences in design and performance. All these instruments are usable between 150 and 1000 K.

Special instrumentation exists for measurements under extreme conditions and is summarized in Appendix 10. A brief summary with references is given of very low and very high temperature measurements, high-speed DTA, DTA at very high pressures, and microcalorimetry in form of a DTA coupled with atomic force microscopy, AFM. Note that the term microcalorimetry is also used when small amounts of heat are measured for larger samples, as in the calorimeter of Fig. 4.39.

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Slow calorimetry can be done isothermally by direct observation, as described in Sect. 4.2, and intermittently by checking the progress after appropriate time intervals, as in Figs. 3.98 and 4.41.

Figure 4.54 illustrates the measuring head of a modern heat-flux DSC of TA Instruments Inc., Type Q 1000 with the needed equation for data evaluation. The shown constantan body is solidly connected to the bottom of a measuring chamber

Fig. 4.54

made of silver which also permits a conditioned purge gas, usually nitrogen, to flow through. The internally heated furnace is connected to cooling rods and the controlled temperature is applied to the bottom of the chamber, producing a regulated temperature Tb. The sample and reference temperatures, Ts and Tr are measured separately with the chromel area detectors against the constantan reference wire shown in the figure. The heat-flow rate into the sample is proportional to T = Tr Ts and corrected by software for the asymmetry in thermal resistance and capacitance to the reference and sample platforms, as well as for the heating rate imbalance. Appendix 11 gives more information on the online corrections developed for this calorimeter (Tzero™ method).

The heat-flow rate of the sample calorimeter, consisting of a pan and the sample, and the reference calorimeter, consisting usually of an empty pan, is governed by the rate of temperature change, q (in K min 1), and the heat capacity, Cp (in J K 1). The heat capacity measured at constant pressure, p, and composition, n, can then be represented by Cp = ( H/ T)p,n, with H being the enthalpy and T, the temperature. The overall heat capacity of the sample calorimeter is written as Cs = (mcp + Cr), where m is the sample mass, cp is the specific heat capacity of the sample per unit mass, and Cr is the heat capacity of the empty calorimeter pan. The empty sample and reference pans are assumed to have the same heat capacity Cr so that in a differential measure-

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ment mcp can be established directly after calibration of K, the Newton’s law constant. Corrections for remaining asymmetry are made by a baseline run with the two empty calorimeters which is then subtracted from the sample and calibration runs. The measurements consist, thus, of three separate runs over the same temperature range, namely, baseline, calibration, and sample run. The details of the mathematical description of the measurements are given in Sect. 4.3.5.

On the left of Fig. 4.55 the changes of the temperatures are illustrated for a standard DSC experiment. The data were calculated with K represented by Cr /20 J K 1 s 1 and using the Fourier equation of heat flow (see Sect. 4.3.5). The temperature of the constantan body, Tb, which is controlled by the heater starts at time

Fig. 4.55

zero with a linear increase. After about 100 s, the reference and sample temperatures reach a steady state, i.e., both change with the same heating rate, q, as the T become constant. At steady state, the heat capacity can then be represented as shown by the left equation in Fig. 4.54. The second term on the right-hand side is a small correction term, needed since the sample and reference calorimeters change their heat capacities differently with temperature, i.e., Tr and Ts in Fig. 4.55 are not strictly parallel to Tb. This correction can be calculated and needs no further measurement. To have a negligible temperature gradient within the sample, also, is not a stringent condition as long as steady state is kept. A temperature gradient of 2.0 K across a sample of crystalline polyethylene, for example, will cause an error in the measurement of the magnitude of dcp/dT, of about 0.3% at 300 K. This is less than the commonly accepted error for such measurements.

Modulating the temperature (see Sect. 4.4, TMDSC [22]) the measurement of cp is just as easy as long as the condition of steady state and negligible temperature gradients within the sample can be maintained. In Fig. 4.55, the curves on the right

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illustrate a temperature-modulated DSC, a TMDSC in a sinusoidal mode. One observes that the sinusoidal modulation reaches a constant average after a few cycles, and that the sliding averages over one modulation period yield the same curves as seen in standard DSC, and lead to the so-called total heat capacity. Next, a simple subtraction of these averages from their instantaneous values yields the pseudoisothermal reversing signals, to be discussed in Sect. 4.4. A simple calculation of mcp, using the equation on the bottom right of Fig. 4.54 without consideration of the correction term in the square root, is possible only if steady state is not lost during modulation, the temperature gradient within the calorimeters is negligible, and K is calibrated at the given frequency. The modulation amplitudes A and ATs represent the reversing temperature difference and sample temperature and are obtained as the first harmonic of their Fourier-transform. The frequency, , is given by 2 /p, where p represents the modulation period in seconds. As long as the studied process is reversible, the total and the reversible Cp should be identical. Irreversible processes should not show in the reversing signal. The conditions for quantitative TMDSC are more stringent than for the standard DSC, because if even a small temperature gradient is set up within the sample during the modulation, each modulation cycle reverses this gradient and leads to smaller positive and negative heat-flow rates which depend on the unknown thermal conductivities. A negligible temperature gradient within the sample requires that the sample calorimeter oscillates in its entirety. An empirical solution to this problem is given in the right equation of Fig. 4.54 with representing a characteristic time constant. This method is described in Figs. 4.94 and 4.95, below. For high frequencies, becomes itself frequency dependent. With minor modifications and addition of the proper software, thus, any DSC can become a TMDSC. Even more important, any TMDSC can be used as a standard DSC by turning off the modulation. Most applications described in this Sect. 4.3 can usually be done better and faster by using standard DSC, and temperature modulation is the superior mode for most of the applications which are selected for discussion in Sect. 4.4.

Figure 4.56 shows a dual DSC cell with three measuring positions that can be used to make two measurements simultaneously. It is a further development of the DSC which is shown in Fig. A.9.2 as sketch A with only two measuring positions [23,24]. The temperature and temperature difference are measured with the chromel-alumel and chromel-constantan thermocouples. The heating block is made of silver for good thermal conductivity. The temperature range is 125 to 1000 K, the heating rate, 0.1 to 100 K min 1, the noise level is reported as ±5 W, and the sample volume can be up to 10 mm3. Even more important than the capability to measure two samples simultaneously for comparison purposes is the ability to measure heat capacity in a single run by measuring a reference material and the sample at the same time. Full details and software, as well as a number of applications are given in the list of references at the end of the chapter [25].

A somewhat different DSC is shown in Fig. 4.57. The heating rate is controlled through the furnace temperature, as shown. The two sets of 28 thermocouples measure the heat-flow rates between the furnace and the pan for the sample and reference sides (thermopiles, vapor deposited on ceramics). The two central terminals bring the average T signal from the 56 thermocouples out to the computer, ready to be inserted into the equation on the left of Fig. 4.54.

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Fig. 4.56

Fig. 4.57

4.3.3 Power Compensation DSC

A power compensation DSC uses a radically different measurement method. With its development in 1963, the name DSC was coined [21]. In Fig. 4.58 a schematic of sample and reference arrangement is drawn and some typical parameters are given.

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Fig. 4.58

The sample and reference thermometers and heaters are platinum resistance thermometers (see Sect. 4.1). Instead of relying on heat conduction from a single furnace, governed by temperature difference, reference and sample are heated separately as required by their temperature and the temperature difference between the two furnaces. The two calorimeters are each less than one centimeter in diameter and are mounted in a constant temperature block. This instrument is, thus, a scanning, isoperibol twin-calorimeter (see Sect. 4.2).

The operation schematic is shown in Fig. 4.59. A programmer provides the average-temperature amplifier with a voltage that causes it to heat sample and reference calorimeters equally so that the average temperature increases linearly. The dotted line in the diagram of Fig. 4.60 represents this linear temperature increase of the average temperature, TAV. If one assumes that initially both calorimeters are identical and the differential-temperature amplifier is not operating (open T loop), the temperatures of the reference, TR, and sample, TS, follow the solid, heavy line at the lower end of the diagram. If at time t1 the heat capacity of the sample calorimeter increases suddenly, the rate of change of the temperature of the sample calorimeter is reduced. At the same time, the rate of change of temperature of the reference calorimeter must increase to keep TAV constant, as is shown by the lower and upper thin lines, respectively.

Mathematically this situation without a T loop is expressed by the upper three boxed equations in Fig. 4.60, where dQR/dt and dQS/dt are the heat-flow rates into the reference and sample calorimeters, respectively. The measured and true temperatures are represented by Tmeas and T. For simplicity, one can assume that the proportionality constant K is the same for the sample and reference calorimeters. Differences are assessed by calibration. Both bottom equations are then equal to the power input from the average temperature amplifier, WAV (in W or J s 1).