Thermal Analysis of Polymeric Materials

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filling, one waits for a few hours to establish a stable density gradient. Once established, the gradient slowly flattens out by diffusion, but usually it is usable for several weeks. The calibration of the density gradient is shown in the left graph of Fig. 4.16. The points in the graph are established by finding the height of glass floats of calibrated density. The floats can be precise to ±0.0001 Mg m 3. The height of a floating piece of sample can easily be measured to ±0.5 mm, which means that fourdigit accuracy in density is possible.

4.1.7 Application of Dilatometry

A typical length measurement by thermomechanical analysis, TMA (see Sect. 4.5), is shown in Fig. 4.17 on the change in dimensions of a printed circuit board made of an epoxy-laminated paper. Measurements of this type are important for matching the expansivities of the electronic components to be fused to the board, so that strain and

Fig. 4.17

eventual fracture of the printed metal can be avoided. The measurement is made in this case under zero load, so that the bottom curve directly gives the change in length relative to a reference length. The derivative, simultaneously recorded, yields the expansivity after changing from time to temperature. The glass transition at 401 K is easily established, and quantitative expansivities are derived, as is shown.

Figure 4.18 illustrates volume dilatometry of an extended-chain, high-crystallinity polyethylene sample (see also Chap. 6). A close to equilibrium melting is observed by such slow dilatometry with equipment illustrated on the left of Fig. 4.15.

The dilatometry at different pressures leads to a full p-V-T phase diagram. Linear macromolecules in the liquid state can reach equilibrium and have then been successfully described by a single p-V-T diagram. The semicrystalline and glassy

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Fig. 4.21

Fig. 4.22

To allow a better understanding of the condensed phase, the volume of a sample can be divided into two parts: the van der Waals volume, Vw, which represents the actual volume of the molecules or ions in the hard-sphere approximation, taken from Fig. 4.23, and the total, experimental volume V. It must be remembered that the van der Waals radius depends somewhat on the forces that determine the approach of the

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to a coordination number, CN, of 12 nearest neighbors. The resulting crystal structure is a cubic, close-packed crystal or one of the various trigonal or hexagonal close packs. By placing the spheres randomly, but still packed as closely as possible, the packing fraction drops to 0.64. An irregular pack with a coordination number of three, the lowest possible coordination number without building a structure which would collapse, yields a very open structure with k = 0.22. Packing of spherical molecules in the condensed phase could thus vary between 0.22 and 0.74.

The packing fraction of rods is another easily calculated case. It could serve as a model for extended-chain, linear macromolecules. Motifs of other, more irregular shapes are more difficult to assess. The closest packing of rods with circular cross section reaches a k of 0.91 with a coordination number of six. Packing with coordination number four reduces k to 0.79. A random heap of rods which do not remain parallel can result in quite low values for k which should also depend on the lengths of the rods.

Making further use of packing fractions, one may investigate the suggestion that at the critical point the packing fraction is at its minimum for a condensed phase, and that at the glass transition temperature, packing for the random close pack is perhaps approached, while on crystallization closest packing is achieved via the cubic or hexagonal close pack. Unfortunately such a description is much too simplistic. An additional accounting for differences in interaction energies and the more complicated geometry of actual molecules is necessary for an understanding of the various phases of matter. Volume considerations alone can only give a preliminary picture. At best, molecules with similar interaction energies can be compared. Looking at the packing fractions of liquid macromolecules at room temperature, discussed in Sect. 5.4, some trends can be observed. A packing fraction of 0.6 is typical for hydrocarbon polymers. Adding >N H, O , >C=O, CF2, >S or >Se to the molecule can substantially increase the packing fraction.

It is also of interest to compare the expansivity, the derivative of the extensive quantity volume, to specific heat capacity, the derivative of the extensive quantity enthalpy. A detailed discussion of heat capacity is possible by considering harmonic oscillations of the atoms (see Sect. 2.3). A harmonic oscillator does not, however, change its average position with temperature. Only the amplitudes of the vibrations increase. To account for the expansivity of solids, one thus, must look at models that include the anharmonicity of the vibrations. Only recently has it been possible to simulate the dynamics of crystals with force fields that lead to anharmonic vibrations (see Figs. 1.44–48). Despite this difference, the expansivity and heat capacity for liquids and glasses behave similarly (see Sect. 2.3). The reason for this is the larger influence of the change in potential energy (cohesive energy) with volume.

This concludes the discussion of thermometry and dilatometry. The tools to measure temperature, length, and volume have now been analyzed. The tools for measurement of heat, the central theme of this book, will take the next three sections and deal with calorimetry, differential scanning calorimetry, and temperaturemodulated calorimetry. The mechanical properties which involve dilatometry of systems exposed to different and changing forces, are summarized in Sect. 4.5. The measurement of the final basic variable of state, mass, is treated in Sect. 4.6 which deals with thermogravimetry.

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4.2 Calorimetry

Calorimetry involves the measurement of the extensive quantity heat. Its name derives from the middle of the 18th century when heat was called the “caloric,” as described in Sects. 1.1.1 and 2.1.1. As the main thermal-analysis method, calorimetry is discussed in this and the following two sections, covering classical calorimetry in Sect. 4.2, differential scanning calorimetry (DSC) in Sect. 4.3, and the more recent temperature-modulated calorimetry (TMC) in Sect. 4.4.

4.2.1 Principle and History

The SI unit of heat, as well as of work and energy is the joule, J as summarized in Fig. 2.3. Its dimension is expressed in [kg m2 s 2]. Heat and work describe the energy exchanged between thermodynamic systems, as discussed in Sect. 2.1.5 with the equation q + w = dU. An earlier, empirical unit, the “calorie,” was based on the

specific heat capacity of water (1 calthermochemical = 4.184 J). Since the early 20th century, however, energy, heat, and work are more precisely determined in joules, making the

calorie a superfluous unit. The calorie is not part of the SI units and should be abandoned. All modern calorimetry is ultimately based on a comparison with heat generated by electrical work.

Almost all calorimetry is carried out at constant pressure, so that the measured heat is the change in enthalpy, as discussed in Chap. 2. The thermodynamic functions that describe a system at constant pressure are listed in Fig. 4.25. These functions contain the small correction caused by the work term w = pdV. The free enthalpy, G, also called Gibbs function or Gibbs energy replaces the Helmholtz free energy and is used as measure of thermodynamic stability (see Sect. 2.2.3). The added pV term in these functions represents the needed volume-work for creation of the space for the system at the given, fixed pressure p. As indicated in Fig. 4.25, the pV-term is small. A difference of 0.1 J raises the temperature of one cm3 of water by a little more than 0.02 K. In addition, many processes have only a small change in volume ( V). The force-times-length term, fl, provides a similar correction for work exchanged by tensile force during a calorimetric experiment as occurs on changing the length, or geometry in general, of a sample, which is rubber-elastic (see Sect. 5.6.5).

A calorimeter does not allow one to find the total heat content (H) of a system in a single measurement, such as one can for other extensive quantities like volume by dilatometry in Sect. 4.1 or mass by thermogravimetry in Sect. 4.6. A calorimeter is thus not a total-heat meter. Heat must always be determined in steps as H, and then summed from a chosen reference temperature. The two common reference temperatures are 0 K and 298.15 K (25°).

Three common ways of measuring heat are listed at the top of Fig. 4.26. First, the change of temperature in a known system can be observed and related to the flow of heat into the system. It is also possible, using the second method, to follow a change of state, such as the melting of a known system, and determine the accompanying flow of heat from the amount of material transformed in the known system. Finally, in method three, the conversion to heat of known amounts of chemical, electrical, or mechanical energy can be used to duplicate or compensate a flow of heat.

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The earliest reasonably accurate calorimetry seems to have been carried out in the 18th century. In 1760 Joseph Black1 described calorimetry with help of two pieces of ice, as sketched in Fig. 4.26. The sample is placed into the hollow of the bottom piece. A second slab is put on top. After the sample has acquired temperatureequilibrium at 273.15 K (0°C), the amount of water produced is mopped out of the

cavity and weighed (Wwater). Equations (1) and (2) show the computation of the average specific heat capacity, c, from the latent heat of water, L. One expects the

results are not of highest accuracy, although with care, and perhaps working in a cold room at about To, an accuracy of perhaps ±5% might be possible. This is a respectable accuracy compared to the much more sophisticated calorimeters used today which often does not exceed ±1%.

In 1781 de la Place published the description of a much improved calorimeter. A picture of it can be found in the writings of Lavoisier [4] and is shown in Fig. 4.27. The outer cavity, a, and the lid, F, are filled with ice to insulate the interior of the calorimeter from the surroundings. Inside this first layer of ice, in space b, a second

Fig. 4.27

layer of ice is placed, the measuring layer. Before the experiment is started, this measuring ice is drained dry through the stopcock, y. Then, the unknown sample in basket LM, closed with the lid HG and kept at a known temperature T1, is quickly dropped into the calorimeter, f, and the lid, F, is closed. After 8 12 hours for equilibration of the temperature, the stopcock, y, is opened and the drained water,

1 Joseph Black, 1728–1799. British chemist and physicist who discovered carbon dioxide, which he also found in air, and the concept of latent heat. He noticed that when ice melts, it takes up heat without change of temperature. He held a position as Professor of Chemistry and Anatomy at the University of Glasgow, Scotland and also practiced Medicine.