Thermal Analysis of Polymeric Materials

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and for the example of a shear experiment is a measure of the time needed to extend the spring to its equilibrium length, as shown in the graph. The rise of the strain caused by the application of the stress to the parallel spring and dashpot is retarded, but ultimately the full equilibrium extension is reached. For the Maxwell model the same ratio is called its relaxation time and is a measure of the time needed for the initially applied stress to drop to zero, as shown on the right of Fig. 4.158. Description of the macroscopic behavior of a material would be simple if the spring in the Voigt or Maxwell models could be identified with a microscopic origin, such as a bond extension or a conformational change in an entropy elastic extension, and the dash pot with a definite molecular friction. One can imagine, however, that there must be many different molecular configurations contributing to the viscoelastic behavior of the sample. For this reason the simple models are expanded to combinations of many elements “i” as shown in Fig. 4.159. These combinations of the elements of the model are linked to retardation and relaxation time spectra. Naturally, there may also be combinations of both models needed for the description.

Fig. 4.159

For most polymeric materials, viscoelastic behavior can be found for sufficiently small amplitudes of deformation. Rigid macromolecules and solids, in general, show relatively little deviation from elasticity and behave approximately as indicated in Fig. 4.144. The major application of DMA is to flexible, linear macromolecules from the glass transition to the rubbery state.

The analysis of DMA data is illustrated in Fig. 4.160 with the example of shear deformation. The periodicity of the experiment is expressed in frequency, either in hertz, , (dimension: s 1), or in (dimension: radians per second). Periodic experiments at frequency in DMA, as also in TMDSC, are comparable to a nonperiodic, transient experiment with a time scale of t 1/ = 0.159 s.

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Fig. 4.160

To describe the stress and strain caused by the modulation, as given by the instrument of Fig. 4.155, one defines a complex modulus G* as shown in Fig. 4.160. Analogous expressions can be written for Young’s modulus and the bulk modulus. The real component G represents the in-phase component, and G is the out-of-phase component. The letter i represent, as usual, the square roots of 1. In order to evaluate G*, the measured stress is separated into its two components, one in-phase with the strain, and one out-of-phase. The simple addition theorem of trigonometry links to the complex terms of the modulus. Also given are the equations for G , G , and the tangent of the phase difference . Since the in-phase portions of the stress are only a part of the measured total stress, there is a loss in the recovered mechanical energy in every cycle. This energy, dissipated as heat over one cycle, W , is given at the bottom of Fig. 4.160.

A strain-modulated DMA parallels the temperature-modulated DSC discussed in Sect. 4.4. Figure 4.161 shows a comparison to the results in Figs. 4.90, written for a common phase lag . Note that the measured heat-flow rate HF(t) lags behind the modulated temperature, while the measured shear stress advances ahead of the modulated strain. Besides modulation of strain, it is also easily possible to modulate the stress, and even temperature-modulation is possible and of interest for comparison of DMA to TMDSC, as was established recently [44].

The data are further analyzed mathematically. In particular, it is of interest to establish retardation and relaxation time spectra that fit the measured data using Voigt or Maxwell models. Adding the temperature dependence of the data leads to the interesting observation that time and temperature effects are often coupled by the timetemperature superposition principle. Effects caused by an increase in temperature can also be produced by an increase in time scale of the experiment. The ratio of modulus to temperature, when plotted versus the logarithm of time for different temperatures,

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Fig. 4.161

can be shifted along the time axis by aT and brought to superposition. The shift factor aT, is the ratio of corresponding time values of the modulus divided by the respective temperatures T and To. A great variety of amorphous polymers have in the vicinity of the glass transition a shift factor aT that is described by the approximately universal Williams-Landel-Ferry equation (WLF equation):

log aT = [17.44(T Tg)] / [51.6 + (T Tg)].

The mechanical spectra and temperature dependencies derived from DMA provide, as such, no immediate insight to their molecular origin. Qualitatively the various viscoelastic phenomena are linked to the energy-elastic deformation of bonds and the viscous effects due to large-amplitude movement of the molecular segments. The latter are based on internal rotation causing conformational motion to achieve the equilibrium entropy-elastic response.

4.5.5 Applications of DMA

The application of DMA to the study of the glass transition of poly(methyl methacrylate) is shown in Fig. 4.162. The graph illustrates the change in Young’s storage modulus E as a function of frequency. Measurements were made on an instrument as in Fig. 4.155 with frequencies between 0.03 and 90 Hz. The lowtemperature data (below 370 K) show the high modulus of a glass-like substance. At higher temperature this is followed by the glass transition region, and the last trace, at 413 K, is that of a typically rubber-elastic material (see also Fig. 4.145). With the proper shift aT, as described in Sect. 4.4.4, the master curve at the bottom of Fig. 4.163 can be produced. The shift factors aT are plotted in the upper right graph in a form

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Fig. 4.162

Fig. 4.163

which indicates a logarithmic dependence on the inverse of temperature as expected from the WLF equation. An activation energy of 352 kJ mol 1 can be derived from the plot. Figures 4.164 and 4.165 show the same treatment for tan . The glass transition is marked by the maximum in the plot of tan versus frequency or temperature. The reference temperature chosen for the shift is 393 K.

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Fig. 4.164

Fig. 4.165

Figure 4.166 shows a DMA analysis of an amorphous polycarbonate, poly(4,4'- isopropylidenediphenylene carbonate). These data were taken with an instrument like that seen in Fig. 4.156. Measurements were made at seven frequencies between 0.01 and 1 Hz at varying temperatures. Again, the glass transition is obvious from the change in flexural storage modulus, as well as from the maximum of the loss modulus.

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Fig. 4.166

Fig. 4.167

Figure 4.167 illustrates the analysis of the shift factors using the WLF equation as given in Fig. 4.163 with of the constants fitted for the polycarbonate. Finally, in Figs. 4.168 and 169 the master curves are generated by shifting the data for the loss and storage moduli, as given in Fig. 4.166. The master curves represent the full data set of Fig. 4.166.

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Fig. 4.168

Fig. 4.169

Another DMA analysis is shown in Fig. 4.170 for poly(vinyl chloride), [ CHCl CH2]x. The data for G , G , and tan are given as a function of temperature for one frequency. The glass transition occurs at about 300 K, as indicated by the drop in G and the peaks in G and tan . In addition, there is a broad peak in G and in tan , indicating a secondary, local relaxation in the glassy state. Semicrystalline

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Fig. 4.170

polymers show a more complicated DMA picture (see Fig. 4.145) because of the additional melting transition and a considerable broadening of the glass transition (see Chaps. 6 and 7).

4.5.6 Dielectric Thermal Analysis, DETA

The mechanism of dielectric effects of interest to DETA involves permanent dipoles1 which exist within the sample and try to follow an alternating electric field, but may be hindered to do so when attached to segments of molecules with limited mobility. The fundamental analogy of dielectric and mechanical relaxation has been pointed out already in 1953 [45].

An electric field in a material causes polarization,2 as summarized in Fig. 4.171. The polarization may have different origins: (1) Electron polarization, an interaction that shifts the electrons with respect to the center of the atom. (2) Atom polarization, caused by shift of relative positions of bonded atoms. (3) Dipole or orientation polarization, the effect to be discussed. The first two types of polarization involve fast displacement of positive and negative charges relative to each other within a timescale of about 10 15 s. Both are usually treated together as induced polarization and are related to the refractive index, n, by the equation of Maxwell (see also the discussions of light scattering, Sect. 1.4.2 and Appendix 3). The time scale of dipole polarization is 10 12 to 10 10 s for mobile dipoles, but may become seconds and longer

1 In SI-units the dimension of the dipole moment , is C m (C = coulomb or A s, see Fig. 2.3).

2 Polarization is the charge introduced per unit surface area, identical to the average dipole moment per unit volume. The SI-unit has the dimension C m 2, C = coulomb or A s.