Thermal Analysis of Polymeric Materials Part 3 pps

2 Basics of Thermal Analysis __________________________________________________________________ 106 if possible [12]. Furthermore, one must be careful to use the proper units, A o refers to one mole of atoms or ions in the sample, so C p and C v must be expressed for the same reference amount. But, the difference between C p and C v remains small up the melting temperature, as seen in Fig. 2.51, below, for the polyethylene example. A rather large error in Eq. (6), thus, has only a small effect on C v . For polyethylene, the difference becomes even negligible below about 250 K. For organic molecules and macromolecules, the equivalent of the atoms or ions must be found in order to use Eq. (6). Since the equation is based on the assumption of classically excited vibrators, which requires three vibrators per atom (degrees of freedom), one can apply the same equation to more complicated molecules when one divides A o by the number of atoms per molecule or repeating unit. Since very light atoms have, however, very high vibration frequencies, as will be discussed below, they have to be omitted in the counting at low temperature. For polypropylene, for example, with a repeating unit [CH 2 CH(CH 3 )], there are only three vibrating units of heavy atoms and A o is 5.1×10 3 /3 = 1.7×10 3 KmolJ 1 . Equation (7) of Fig. 2.31 offers a further simplification. It has been derived from Eq. (6) by estimating the number of excited vibrators from the heat capacity itself, assuming that each fully excited atom contributes 3R to the heat capacity as suggested by the Dulong–Petit rule [13]. The new A o ' for Eq. (7) is 3.9×10 3 KmolJ 1 and allows a good estimate of C V even if no expansivity and compressibility information is available [14]. 2.3.3 Quantum Mechanical Description In this section, the link of C v to the microscopic properties will be derived. The system in question must, of necessity, be treated as a quantum-mechanical system. Every microscopic system is assumed to be able to take on only certain states as summarized in Fig. 2.32. The labels attached to these different states are 1, 2, 3, and their potential energies are  1 ,  2 ,  3 , respectively. Any given energy may, however, refer to more than one state so that the number of states that correspond to the same energy  1 is designated g 1 and is called the degeneracy of the energy level. Similarly, degeneracy g 2 refers to  2 , and g 3 to  3 . It is then assumed that many such microscopic systems make up the overall matter, the macroscopic system. At least initially, one can assume that all of the quantum-mechanical systems are equivalent. Furthermore, they should all be in thermal contact, but otherwise be independent. The number of microscopic systems that are occupying their energy level  1 is n 1, the number in their energy level  2 is n 2 , the number in their level  3 is n 3 , . The number of microscopic systems is, for simplicity, assumed to be the number of molecules, N. It is given by the sum over all n i , as shown in Eq. (1) of Fig. 2.32. The value of N is directly known from the macroscopic description of the material through the chemical composition, mass and Avogadro’s number. Another easily evaluated macroscopic quantity is the total energy U. It must be the sum of the energies of all the microscopic, quantum-mechanical systems, making the Eq. (2) obvious. For complete evaluation of N and U, one, however, needs to know the distribution of the molecules over the different energy levels, something that is rarely available. To solve this problem, more assumptions must be made. The most important one is 2.3 Heat Capacity __________________________________________________________________ 107 Fig. 2.32 that one can take all possible distributions and replace them with the most probable distribution, the Boltzmann distribution which is described in Appendix 6, Fig.A.6.1. It turns out that this most probable distribution is so common, that the error due to this simplification is small as long as the number of energy levels and atoms is large. The Boltzmann distribution is written as Eq. (3) of Fig. 2.32. It indicates that the fraction of the total number of molecules in state i, n i /N, is equal to the number of energy levels of the state i, which is given by its degeneracy g i multiplied by some exponential factor and divided by the partition function, Q. The partition function Q is the sum over all the degeneracies for all the levels i, each multiplied by the same exponential factor as found in the numerator. The meaning of the partition function becomes clearer when one looks at some limiting cases. At high temperature, when thermal energy is present in abundance, exp[  i /(kT)] is close to one because the exponent is very small. Then Q is just the sum over all the possible energy levels of the quantum mechanical system. Under such conditions the Boltzmann distribution, Eq. (3), indicates that the fraction of molecules in level i, n i /N, is the number of energy levels g i , divided by the total number of available energy levels for the quantum-mechanical system. In other words, there is equipartition of the system over all available energy levels. The other limiting case occurs when kT is very much smaller than  i . In this case, temperature is relatively low. This makes the exponent large and negative; the weighting factor exp[  i /(kT)] is close to zero. One may then conclude that the energy levels of high energy (relative to kT) are not counted in the partition function. At low temperature, the system can occupy only levels of low energy. With this discussion, the most difficult part of the endeavor to connect the macroscopic energies to their microscopic origin is already completed. The rest is just mathematical drudgery that has largely been carried out in the literature. In order 2 Basics of Thermal Analysis __________________________________________________________________ 108 Fig. 2.33 to get an equation for the total energy U, the Boltzmann distribution, Eq. (3), is inserted into the sum for the total energy, Eq. (2). This process results in Eq. (5). The next equation can be seen to be correct, by just carrying out the indicated differentiation and comparing the result with Eq. (5). Now that U is expressed in microscopic terms, one can also find the heat capacity, as is shown by Eq. (6) of Fig. 2.32. The partition function Q, the temperature T, and the total number of molecules N need to be known for the computation of C v .Next, C v can be converted to C p using any of the expressions of Fig. 2.31, which, in turn, allows computation of H, S, and G, using Eqs. (1), (2), and (3) of Fig. 2.22, respectively. For a simple example one assumes to have only two energy levels for each atom or molecule, i.e., there are only the levels  1 and  2 . A diagram of the energy levels is shown in Fig. 2.33. This situation may arise for computation of the C v contribution from molecules with two rotational isomers of different energies as shown in Fig. 1.37. For convenience, one sets the energy  1 equal to zero. Energy  2 lies then higher by . Or, if one wants to express the energies in molar amounts, one multiplies  by Avogadro’s number N A and comes up with the molar energy difference EinJmol 1 . A similar change is necessary for kT; per mole, it becomes RT. The partition function, Q, is now given in Eq. (7) of Fig. 2.33. The next step involves insertion of Eq. (7) into Eq. (5) of Fig. 2.32 and carrying out the differentia- tions. Equation (8) is the total energy U, and the heat capacity C v is given by Eq. (9). The graph in Fig. 2.33 shows the change in C v for a system with equal degeneracies (g 1 =g 2 ). The abscissa is a reduced temperature—i.e., the temperature is multiplied by R, the gas constant, and divided by E. In this way the curve applies to all systems with two energy levels of equal degeneracy. The curve shows a relatively sharp peak at the reduced temperature at approximately 0.5. In this temperature 2.3 Heat Capacity __________________________________________________________________ 109 1 The lowest energy level is ½ h above the potential-energy minimum (zero-point vibration). Vibrators can exchange energy only in multiples of h ,sothatlevel0istheloweststate. Fig. 2.34 region many molecules go from the lower to the higher energy level on increasing the temperature, causing the high heat capacity. At higher temperature, the heat capacity decreases exponentially over a fairly large temperature range. At high temperature (above about 5 in thereducedtemperature scale), equipartition between the two levels is reached. This means that just as many systems are in the upper levels as are in the lower. No contribution to the heat capacity can arise anymore. The second example is that of the harmonic oscillator in Fig. 2.34. The harmonic oscillator is basic to understanding the heat capacity of solids and summarized in Fig. A.6.2. It is characterized by an unlimited set of energy levels of equal distances, the first few are shown in Fig. 2.34. The quantum numbers, v i , run from zero to infinity. The energies are written on the right-hand side of the levels. The difference in energy between any two successive energy levels is given by the quantity h , where h is Planck’s constant and  is the frequency of the oscillator (in units of hertz, Hz, s 1 ). If one chooses the lowest energy level as the zero of energy, then all energies can be expressed as shown in Eq. (1). 1 There is no degeneracy of energy levels in harmonic oscillators (g i = 1). The partition function can then be written as shown in Eq. (2). Equation (2) is an infinite, convergent, geometrical series, a series that can easily be summed, as is shown in Eq. (3). Now it is a simple task to take the logarithm of Eq. (3) and carry out the differentiations necessary to reach the heat capacity. The result is given in Eq. (5). It may be of use to go through these laborious steps to discover the mathematical connection between partition function and heat capacity. Note that for large exponents—i.e., for a relatively low 2 Basics of Thermal Analysis __________________________________________________________________ 110 Fig. 2.35 temperature—Eq. (5) is identical to Eq. (9) in Fig. 2.33, which was derived for the case of two energy levels only. This is reasonable, because at sufficiently low temperature most molecules will be in the lowest possible energy levels. As long as only very few of the molecules are excited to a higher energy level, it makes very little difference if there are more levels above the first, excited energy level. All of these higher-energy levels are empty at low temperature and do not contribute to the energy and heat capacity. The heat capacity curve at relatively low temperature is thus identical for the two-level and the multilevel cases. The heat capacity of the harmonic oscillator given by Eq. (5) of Fig. 2.34 is used so frequently that it is abbreviated on the far right-hand side of Eq. (6) of Fig. 2.35 to RE( /T), where R is the gas constant, and E is the Einstein function. The shape of the Einstein function is indicated in the graphs of Fig. 2.35. The fraction /T stands for h /kT, and h/k has the dimension of a temperature. This temperature is called the Einstein temperature,  E . A frequency expressed in Hz can easily be converted into the Einstein temperature by multiplication by 4.80×10 11 sK. A frequency expressed in wave numbers, cm 1 , must be multiplied by 1.4388 cm K. At temperature , the heat capacity has reached 92% of its final value, R per mole of vibrations, or k per single vibrator. This value R is also the classical value of the Dulong–Petit rule. The different curves in Fig. 2.35 are calculated for the frequencies in Hz and Einstein temperatures listed on the left. Low-frequency vibrators reach their limiting value at low temperature, high-frequency vibrators at much higher temperature. The calculations were carried out for one vibration frequency at a time. In reality there is, however, a full spectrum of vibrations. Each vibration has a heat capacity contribution characteristic for its frequency as given by Eq. (6). One finds that 2.3 Heat Capacity __________________________________________________________________ 111 Fig. 2.36 because of vibrational coupling andanharmonicity,theseparation into normalmodes, to be discussed below, is questionable. The actual energy levels are neither equally spaced, as needed for Eq. (6), nor are they temperature-independent. There is hope, however, that supercomputers will ultimately permit more precise evaluation of temperature-dependent vibrational spectra and heat capacities. In the meantime, approximations exist to help one to better understand C v . 2.3.4 The Heat Capacity of Solids To overcome the need to compute the full frequency spectrum of solids, a series of approximations has been developed over the years. The simplest is the Einstein approximation [15]. In it, all vibrations in a solid are approximated by a single, average frequency. The Einstein function, Eq. (6) of Fig. 2.35, is then used with a single frequency to calculate the heat capacity. This Einstein frequency,  E , can also be expressed by its temperature  E , as before. Figure 2.36 shows the frequency distribution '() of such a system. The whole spectrum is concentrated in a single frequency. Looking at actual measurements, one finds that at temperatures above about 20 K, heat capacities of a monatomic solid can indeed be represented by a single frequency. Typical values for the Einstein temperatures  E are listed in Fig. 2.36 for several elements. These -values correspond approximately to the heat capacity represented by curves 1–4 in Fig. 2.35. Elements with strong bonds are known as hard solids and have high -temperatures; elements with weaker bonds are softer and have lower -temperatures. Soft-matter physics has recently become an important field of investigation. Somewhat less obvious from the examples is that heavy atoms have lower -temperatures than lighter ones. These correlations are 2 Basics of Thermal Analysis __________________________________________________________________ 112 easily proven by the standard calculations of frequencies of vibrators of different force constants and masses. The frequency is proportional to (f/m) ½ , where f is the force constant and m is the appropriate mass. The problem that the Einstein function does not seem to give a sufficiently accurate heat capacity at low temperature was resolved by Debye [16]. Figure 2.36 starts with the Debye approximation for the simple, one-dimensional vibrator. To illustrate such distribution, macroscopic, standing waves in a string of length, L, are shown in the sketch. All persisting vibrations of this string are given by the collection of standing waves. From the two indicated standing waves, one can easily derive that the amplitude, 1, for any standing wave is given by Eq. (1), where x is the chosen distance along the string, and n is a quantum number that runs from 1 through all integers. Equation (2) indicates that the wavelength of a standing wave, identified by its quantum number n, is 2L/n. One can next convert the wavelength into frequency by knowing that ,the frequency, is equal to the velocity of sound in the solid, c, divided by ,the wavelength. Equation (3) of Fig. 2.36 shows that the frequency is directly proportional to the quantum number, n. The density of states or frequency distribution is thus constant over the full range of given frequencies. From Eq. (3) the frequency distribution can be calculated following the Debye treatment by making use of the fact that an actual atomic system must have a limited number of frequencies, limited by the number of degrees of freedom N. The distribution '() is thus simply given by Eq. (4). This frequency distribution is drawn in the sketch on the right-hand side in Fig. 2.36. The heat capacity is calculated by using a properly scaled Einstein term for each frequency. The heat capacity function for one mole of vibrators depends only on  1 , the maximum frequency of the distribution, which can be converted again into a theta-temperature,  1 . Equation (5) shows that C v at temperature T is equal to R multiplied by the one-dimensional Debye function D 1 of ( 1 /T). The one-dimensional Debye function is rather complicated as shown in Fig. 2.37, but can easily be handled by computer. Next, it is useful to expand this analysis to two dimensions. The frequency distribution is now linear, as shown in Eq. (6) of Fig. 2.38. The mathematical expression of the two-dimensional Debye function is given in Fig. 2.37. Note that in Eq. (7) for C v it is assumed that there are 2N vibrations for the two-dimensional vibrator, i.e., the atomic array is made up of N atoms, and vibrations out of the plane are prohibited. In reality, this may not be so, and one would have to add additional terms to account for the omitted vibrations. The same reasoning applies for the one- dimensional case of Eq. (4) of Fig. 2.36. For a linear macromolecule in space, a restriction to only one dimension does not correspond to reality. One must consider that in addition to one-dimensional, longitudinal vibrations of N vibrators, there are two transverse vibrations, each of N frequencies. Naturally, the longitudinal and transverse vibrations should have different  1 -values in Eq. (5). For a two-dimensional molecule, there are two longitudinal vibrations, as described by Eq. (7) in Fig. 2.38, and one transverse vibration with half as many vibrations, as given in Eq. (6). As always, the total possible number of vibrations per atom must be three, as fixed by the number of degrees of freedom. 2.3 Heat Capacity __________________________________________________________________ 113 Fig. 2.38 Fig. 2.37 To conclude this discussion, Eqs. (8) and (9) of Fig. 2.38 represent the three- dimensional Debye function. The mathematical expression of the three-dimensional Debye function is also given in Fig. 2.37. Now the frequency distribution is quadratic in , as shown in Fig. 2.38. The derivation of the three-dimensional Debye model is analogous to the one-dimensionalandtwo-dimensionalcases. The three-dimensional case is the one originally carried out by Debye [16]. The maximum frequency is  3 2 Basics of Thermal Analysis __________________________________________________________________ 114 Fig. 2.39 or  3 . At this frequency the total possible number of vibrators for N atoms, N 3 ,is reached. From the frequency distribution one can, again, derive the heat capacity contribution. The heat capacity for the three-dimensional Debye approximation is equal to 3R times D 3 , the three-dimensional Debye function of ( 3 /T). In Fig. 2.29 a number of examples of three-dimensional Debye functions for elements and salts are given [17]. A series of experimental heat capacities is plotted (calculated per mole of vibrators). Note that salts like KCl have two ions per formula mass (six vibrators) and salts like CaF 2 have three (nine vibrators). To combine all the data in one graph, curves I are displaced by 0.2 T/  for each curve. For clarity, curve III combines the high temperature data not given in curve II at a raised ordinate. The drawn curves represent the three-dimensional Debye curve of Fig. 2.38, Eq. (9). All data fit extremely well. The Table in Fig. 2.40 gives a listing of the - temperatures which permit the calculation of actual heat capacities for 100 elements and compounds. The correspondence of the approximate frequency spectra to the calculated full frequency distribution for diamond and graphite is illustrated in Fig. 2.41. The diamond spectrum does not agree well with the Einstein -value of 1450 K (3×1013 Hz) given in Fig. 2.36, nor does it fit the smooth, quadratic increase in '() expected from a Debye -value of 2050 K (4.3×10 13 Hz) of Fig. 2.39. Because of the averaging nature of the Debye function, it still reproduces the heat capacity, but the vibrational spectrum shows that the quadratic frequency dependence reaches only to about 2×10 13 Hz, which is about 1000 K. Then, there is a gap, followed by a sharp peak, terminating at 4×10 13 Hz which is equal to 1920 K. In Fig. 2.41 the frequency spectrum of graphite with a layer-like crystal structure is compared to 3-dimensional diamond (see Fig. 2.109, below). The spectrum is not 2.3 Heat Capacity __________________________________________________________________ 115 Fig. 2.40 Fig. 2.41 related to the 3-dimensional Debye function of Fig. 2.40 with  = 760 K. The quadratic increase of frequency at low frequencies stops already at 5×10 12 Hz, or 240 K. The rest of the spectrum is rather complicated, but fits perhaps better to a two-dimensional Debye function with a  2 value of 1370 K. The last maximum in the spectrum comes only at about 4.5×10 13 Hz (2160 K), somewhat higher than the diamond frequencies. This is reasonable, since the in-plane vibrations in graphite [...]... the Einstein -value of 1450 K (3 10 13 Hz) given in Fig 2 .36 , nor does it fit the smooth, quadratic increase in '() expected from a Debye -value of 2050 K (4 .3 10 13 Hz) of Fig 2 .39 Because of the averaging nature of the Debye function, it still reproduces the heat capacity, but the vibrational spectrum shows that the quadratic frequency dependence reaches only to about 2×10 13 Hz, which is about 1000... a direct copy of 36 runs of differently treated poly(oxymethylenes) The almost vertical approach of some curves to the liquid Cp is caused by the end of melting of crystallized samples All curves superimpose when the samples are liquid Fig 2.58 132 2 Basics of Thermal Analysis The addition scheme of group contributions helps to connect larger bodies of data for liquid... temperature range of measurement should be limited, as was suggested in Sect 2 .3. 1 Fig 2.60 Fig 2.61 134 2 Basics of Thermal Analysis 2 .3. 10 Examples of Application of ATHAS The application of the ATHAS has produced a large volume of critically reviewed and interpreted heat capacity data on solid and liquid homopolymers This knowledge is helpful in the determination of the integral... its real (reactive) part cp1(7) and the imaginary part icp2(7), one must use the following equation to make cp2(7), the dissipative part, positive: (3) The real quantities of Eq (3) can then be written as: (4) (5) where -eT,p is the Debye relaxation time of the system The reactive part cp1(7) of cp(7) is the dynamic analog of cpe, the cp at equilibrium Accordingly, the limiting cases of a system in internal... to the heat capacity per mole of vibrators (when excited, see Sect 2 .3. 3 and 2 .3. 4) Types (2–4) add only R/2, but may also need some additional inter- and intramolecular potential energy contributions, making particularly the types (3) and (4) difficult to assess This is at the root of the ease of the link of macromolecular heat capacities to molecular motion The motion of type (1) is well approximated... about 2×10 13 Hz is responsible for the leveling of Cp between 200 and 250 K, and the value of Cp accounts for two degrees of freedom, i.e., is about 16 17 J K 1 mol 1 or 2 R 124 2 Basics of Thermal Analysis All motions of higher frequency will now be called group vibrations, because these vibrations involve oscillations of relatively isolated groupings of atoms along... 2 .38 , and one transverse vibration with half as many vibrations, as given in Eq (6) As always, the total possible number of vibrations per atom must be three, as fixed by the number of degrees of freedom 2 .3 Heat Capacity 1 13 Fig 2 .37 To conclude this discussion, Eqs (8) and (9) of Fig 2 .38 represent the threedimensional Debye function The mathematical expression of. .. described in Sect 1 .3. 5 It is an internal rotation and can lead to a 36 0o rotation of the two halves of the molecules against each other, as shown in Fig 1 .37 For the rotation of a CH3-group little space is needed, while larger segments of a molecule may sweep out extensive volumes and are usually restricted 122 2 Basics of Thermal Analysis to coupled rotations which... of it as a motion in which the two hydrogens come out of the plane of the paper and then go back behind the plane of the paper The twisting motion (#6), finally, is the asymmetric counterpart of the wagging motion, i.e., one hydrogen atom comes out of the plane of the paper while the other goes back behind the plane of the paper The stretching of the C C bond (#9) has a much higher frequency than the... appropriate proportions of Cp from all poly(amino acid)s All transitions and possible segmental melting occur above the temperature range shown in the figure 128 2 Basics of Thermal Analysis Fig 2. 53 2 .3. 8 Polyoxide Heat Capacities Besides providing heat capacities of single polymers, the ATHAS Data Bank also permits us to correlate data of homologous series of polymers The . Einstein -value of 1450 K (3 10 13 Hz) given in Fig. 2 .36 , nor does it fit the smooth, quadratic increase in '() expected from a Debye -value of 2050 K (4 .3 10 13 Hz) of Fig. 2 .39 . Because of the averaging. a repeating unit [CH 2 CH(CH 3 )], there are only three vibrating units of heavy atoms and A o is 5.1×10 3 /3 = 1.7×10 3 KmolJ 1 . Equation (7) of Fig. 2 .31 offers a further simplification of degrees of freedom. 2 .3 Heat Capacity __________________________________________________________________ 1 13 Fig. 2 .38 Fig. 2 .37 To conclude this discussion, Eqs. (8) and (9) of Fig. 2 .38 represent