Thermal Analysis of Polymeric Materials

__________________________________________________________________

Sects. 4.2–4.4. Intermediate between thermometry and calorimetry is differential thermal analysis, or DTA. In this technique transition temperature information is derived by the qualitative changes in heats of transition or heat capacity. As the instrumentation of DTA advanced, quantitative heat information could be derived from temperature and time measurements. The DTA has in the last 50 years increased so much in precision that its applications overlap with calorimetry, as is shown in the discussion of the different forms of differential scanning calorimetry, DSC (Sects. 4.3 and 4.4).

The basic measurement of length or volume is called dilatometry if it is carried out at constant pressure or stress. Details are given in Sect. 4.1. When measuring stress as well as strain, the technique is called thermomechanical analysis, TMA. This technique is described in Sect. 4.5. Measurements can be made at constant or variable stress or strain, including periodic changes as in dynamic mechanical analysis, DMA, also discussed in Sect. 4.5. Finally, the thermal analysis technique to measure mass as a function of temperature and time is thermogravimetry, TGA. The technique is treated in Sect. 4.6.

Several more complicated thermal analysis techniques are mentioned from time to time, but are not described in detail because they involve additional specialization. Of particular interest are also the many thermal analysis techniques that involve the addition of time and temperature measurements to well-established analysis techniques, such as the various types of microscopy, any kind of scattering or spectroscopy, chromatography, mass spectroscopy, and nuclear magnetic resonance. Coupling of more than one of the mentioned techniques is naturally of additional advantage for a more complete materials-characterization. Of particular importance is the addition of the macroscopic thermal analysis with microscopic analyses leading to information on molecular structure and motion, as emphasized throughout this book. Additional techniques which are recognizable from their names are evolved gas analysis, emanation thermal analysis, thermoacoustimetry, thermobarometry, thermoelectrometry, thermoluminescence, thermomagnetometry, thermooptometry, thermoparticulate analysis, thermophotometry, thermorefractometry, and thermosonometry.

An Advanced THermal AnalysiS scheme, the ATHAS scheme has been developed over the years and is described in Sect. 2.3.7. It was created to improve the quantitative aspects of thermal analysis and includes methods of data collection and evaluation (computation). Furthermore, it provides computer courses for distance learning, and a data bank of critically evaluated heat capacities of linear polymers and related compounds, both in downloadable information from the ATHAS website. An abbreviated data bank, of use in connection with the discussions in this book, is collected in Appendix 1.

2.1.4 Temperature

Temperature, as the intensive parameter of heat, could not be well understood until the cause of heat became clear (see Sect. 2.1.1). Early thermometers were based on the property of specific materials, such as their length or resistivity as described in Sect. 4.1. Figure 2.7 displays the mercury-in-glass thermometer, the most precise

80 2 Basics of Thermal Analysis

__________________________________________________________________

liquid-in-glass thermometer which has been in use for the last 300 years (see Sect. 4.1.2). Such thermometers are still used extensively. For convenience of measurement, however, secondary thermometers are employed, thermometers which need a calibration at a known temperature.

The development of absolute temperatures began when quantitative experiments were made on the properties of gases. Boyle’s law was long known. It states that for a given amount of gas the product of pressure and volume is constant for a given temperature, as summarized in Fig. 2.8. In 1802 Gay-Lussac found that gases expand linearly with temperature when measured with a mercury-in-glass thermometer. With these two observations it became possible to define a temperature that is independent

Fig. 2.7

of any particular material since all ideal gases behave identical. Today’s value for the expansivity of an ideal gas is 0.003661 or 1/273.15 K 1. With this knowledge one can define the gas temperature, T, as indicated in Fig. 2.8.

Even more interesting, one can relate the gas temperature easily to its microscopic description. Figure 2.9 illustrates that the translational kinetic energy of an ideal gas atom is the cause of pressure through the change of momentum of gas atoms on collision with the surface. Coupled with the ideal gas law, the temperature is connected to the average kinetic energy of the gas atoms (see also Sect. 2.1.6).

More fundamental is the thermodynamic temperature scale. It is based on the second law of thermodynamics which is discussed in Sect. 2.2. This temperature scale is defined such, to be identical to the gas temperature in the region where the gas temperature is measurable. The thermodynamic temperature scale is independent not only of any material property, but also of the state of matter. The zero of the thermodynamic temperature fixes the zero of thermal motion. For more details on thermometry, see Sect. 4.1.

__________________________________________________________________

Fig. 2.8

Fig. 2.9

2.1.5 Heat (The First Law of Thermodynamics)

To follow the discussion of the history of the understanding of heat in Sect. 2.1.1 and the identification of heat as a part only of the conservation law of energy in Sect. 2.1.2, it is necessary to establish a description of the total energy U, and then connect

82 2 Basics of Thermal Analysis

__________________________________________________________________

it to the description of heat. As pointed out above, heat becomes recognizable during energy transfer where dU is the sum of heat ( Q) and work ( w). The conservation of energy U is the content of the first law of thermodynamics, and the limits to the interconversion of heat and work are set by the second law of thermodynamics, as is discussed in Sect. 2.2.3.

Energy is a function of state (i.e., a state fixed by its variables of state). The change in energy dU is then the total differential that can be represented as a sum of the partial differentials, as given in Eq. (1) of Fig. 2.10. Each of the coefficients of the partial changes is accessible to measurement.

The description of heat, Q, in terms of the first law of thermodynamics and the ideal gas law is given by Eq. (2) of Fig. 2.10, written for the situation in which all work is the maximum volume work, pdV, where p and V are the equilibrium pressure and volume of the system. The negative sign results from the fact that for an increase in volume the system must do some work (lose energy). If other types of work are done, such as electrical or elastic work, additional terms must be added to Eq. (2). Note that the chosen volume work, pdV, is determined by the equilibrium pressure and volume, it is thus also fixed by the state of the material, i.e., dQ is in this case also a function of state. As soon as the work lost is less, as happens when the opposing pressure on expansion is less than p, as in a partially free expansion against a piston, the corresponding heat exchanged is also less and must be written as Q. For the evaluation of Q under such (nonequilibrium) conditions, information about the actual opposing pressure must be provided.

Combining Eq. (1) with (2) results in Eq. (3). One can see that in this case the change of heat, dQ, can be expressed in three terms: the first is due to the change in temperature, the second due to the change in volume, and the third due to the change of the number of moles of substance in the system. A full characterization by thermal analysis would, thus, require calorimetry, dilatometry, and also thermogravimetry.

If one can keep the number of moles and the volume constant, terms 2 and 3 are zero because dV and dn are zero. The change in heat, dQ, is then expressed by the first term alone. These conditions of zero dV and dn are used so often that ( U/ T)V,n has been given a new name, the heat capacity (heat capacity at constant volume and constant number of moles). Speaking somewhat more loosely, one can say that the heat capacity is the amount of heat, necessary to raise the temperature of the system by one kelvin at constant volume. The heat capacity that refers to one gram of the substance is the specific heat capacity. The older term specific heat is to be abandoned since it would refer to the integral quantity U. This is the type of error in nomenclature that led, for example, to the questionable interpretation of Count Rumford’s experiment in Sect. 2.1.1.

There is no difficulty in keeping the number of moles constant during the measurement of heat by thermal analysis, but to keep the volume constant is easy for gases only. Solids and liquids develop enormous pressures when heated at constant volume. To allow easier experimentation, it would, thus, be better to use the variables of pressure, p, temperature, T, and number of moles, n, instead of volume, temperature, and number of moles. With these new variables, Eq. (1) takes on the form given by Eq.(5) in Fig. 2.10. To combine Eq. (5) with Eq. (2), one needs to express volume in the same variables as U. This is done in Eq. (6). When one now

__________________________________________________________________

Fig. 2.10

inserts Eqs. (5) and (6) into Eq. (2), to come up with an expression for dQ, one gets to the somewhat unhandy expression of Eq. (7). It consists of six quantities, to be compared to the simpler Eq. (3). Each of the three terms in Eq. (7) is made up of two quantities that must be known for evaluation. With such a complicated correlation, it is not convenient to define and use a heat capacity at constant pressure and composition as expressed in part 1 of Eq. (7).

84 2 Basics of Thermal Analysis

__________________________________________________________________

To resolve this problem, one defines a new function of state out of the old one, the enthalpy, H [Eq. (8a) of Fig. 2.10]. Enthalpy is the internal energy, U, plus the product pV. The term pV represents the work needed to create a cavity of volume V to insert the sample in question at constant pressure p. If V changes during a process,U and H are different. Since solids and liquids change only a little in volume with temperature, the difference between U and H is usually small and can often be approximated (see Sect. 2.3.2). Liquids consisting of linear macromolecules need a somewhat special treatment since they may be rubbery and can take up large quantities of work on deformation. The state of extension is then of importance for the evaluation of U. As with volume, it is not experimentally easy to keep the length constant when changing temperature. For macromolecules it is therefore sometimes of value to define H as given by Eq. (8b), i.e., by subtracting a term for the work needed to extend the sample to length against a constant force f. Equation (8b) will be used for the discussion of rubber elasticity in Chap. 6. Note that the terms pdV and fd have opposite signs. Volume expansion leads to a loss of energy of the sample (work done by the sample), while on drawing a rubbery material, work has to be done on the sample (gain of energy by the sample).

Combining Eq. (8a) with Eq. (7), one can see that the first term in Eq. (7) changes to Eq. (9). The two p( V/ T)-terms cancel, and the first term of Eq. (7) is now as simple as the first term in Eq. (3), so that one can define a second, more convenient heat capacity, this time, a heat capacity at constant pressure (and if necessary at constant extensive force). The two different heat capacities can easily be related to each other by insertion of the definitions in, for example, Eq. (3). The result of the derivation of such relationship is given as Eq. (12), it contains, as expected, the extra term of part I of Eq. (7).

Adding heat to a sample can also have an effect that does not increase the temperature. This heat is called a latent heat. It has its origin in a change of the structure of the sample as in a chemical reaction (heat of reaction) or a phase transition (heat of transition). The measurement of a latent heat is done by direct calorimetry as described in Chap. 4. Note that the latent heat is only the heat of reaction or transition if temperature, T, is constant; if not, heat capacity effects must be considered separately.

2.1.6 The Future of Thermal Analysis [2]

In the past, nature was either studied by experiment and a link to a simple, allencompassing theory was sought. In the last three decades, however, methods evolved to simulate atomic motion with computers, as described in Sects. 1.3.6–8, and to predict properties by neural network techniques, as summarized in Appendix 4. Both techniques are neither theory nor experiment. The simulations let us see the changes in the microscopic structure in slow motion and give a base for the understanding of physical problems which at present are too complicated to fit into a simple theory. The neural network analysis, in turn, looks for a method to use implicit functional relationships between the properties of different substances which are too difficult to ascertain explicitly [3]. The method of Appendix 6, produces a computer program which then correlates the properties.

__________________________________________________________________

The molecular dynamics simulation for thermal analysis uses the classical equations of motion. Figures 1.43 and 44 illustrate the basic equations which introduce the model polyethylene crystals used for the simulation. A thermal analysis simulation with 192 chains of 50 CH2-units each is shown in Fig. 2.11 [2]. A fixed

Fig. 2.11

amount of random kinetic energy, as expressed in Fig. 1.44, corresponding to a given temperature, was distributed over the 9600 atoms and followed as the motion develops. A cut-off of the non-bonded interaction at 1.0 nm leads to a continuous loss of energy, so that the crystal cools. This loss of energy is sufficiently slow to keep the crystal in thermal equilibrium and simulate a cooling curve in thermal analysis. Next, the total energy, U, is shown in Fig. 2.12. Both figures show the fluctuations typical for microscopic systems.

Knowing the changes in temperature and energy, permits the calculation of heat capacity as illustrated in Fig. 2.13 by averaging data over 6 ps time intervals. The error bars are strictly due to the averaging procedure. Since the simulation is done using classical mechanics, one finds heat capacities close to the Dulong-Petit limit of 3R ( 25 J K 1 mol 1, see Sect. 2.3). The decrease from this value, seen at low temperature, is difficult to discuss since the heat capacity is known to follow quantum mechanics (see Sect. 2.3). For comparison, the known vibrational heat capacity of CH2 is drawn in Fig. 2.13 along with data for polyethylene.

More important is the increase in heat capacity above 3R which is also seen in the experimental data. Figure 2.14 suggests that this increase is due to the formation of conformational defects (gauche conformations in the otherwise all-trans crystal, see Sect. 5.3). The simulation is in this case in close agreement with infrared experiments which assess the gauche concentration from its changed local vibrations.

86 2 Basics of Thermal Analysis

__________________________________________________________________

Fig. 2.12

Fig. 2.13

Further, the number of double defects is listed in Fig. 2.15. Such double defects lead in Sect. 5.3 to an understanding of defect crystals and their deformation kinetics. The number of defects refers to the whole crystal. Assuming that the increase in heat capacity from 3 to 4.5 R in Fig. 2.14 is due to defect formation, it is possible to calculate the energy of defect formation as 44 kJ mol 1.