Heat capacity Debye model

The S(T) on the left-hand side of Eq. (39) could be an entropy 5vib,atp(7 ) determined from experiments or it could be the result of a detailed model calculation based on a certain E((o) in Eq. (27). There is a solution 0d to Eq. (39) for each temperature T. We shall call that 0d a Debye temperature. Because we represent the property of a system that has 3rL degrees of freedom with a single parameter 0d, it is obvious that we have to pay a price. In this case, the price is that 0d varies with T and also with the physical property that is modeled. (Here 0d refers to the entropy.) One may therefore introduce one Debye temperature 0 that describes the vibrational entropy, another Debye temperature 0c that describes the vibrational heat capacity, etc. The heat capacity Debye temperature 0c would be the solution to... 

Phonons are nomial modes of vibration of a low-temperatnre solid, where the atomic motions around the equilibrium lattice can be approximated by hannonic vibrations. The coupled atomic vibrations can be diagonalized into uncoupled nonnal modes (phonons) if a hannonic approximation is made. In the simplest analysis of the contribution of phonons to the average internal energy and heat capacity one makes two assumptions (i) the frequency of an elastic wave is independent of the strain amplitude and (ii) the velocities of all elastic waves are equal and independent of the frequency, direction of propagation and the direction of polarization. These two assumptions are used below for all the modes and leads to the famous Debye model. 

Its value at 25°C is 0.71 J/(g-°C) (0.17 cal/(g-°C)) (95,147). Discontinuities in the temperature dependence of the heat capacity have been attributed to stmctural changes, eg, crystaUi2ation and annealing effects, in the glass. The heat capacity varies weakly with OH content. Increasing the OH level from 0.0003 to 0.12 wt % reduces the heat capacity by approximately 0.5% at 300 K and by 1.6% at 700 K (148). The low temperature (<10 K) heat capacities of vitreous siUca tend to be higher than the values predicted by the Debye model (149). 

Figure 8.12 Experimental heat capacity of Cu at constant pressure compared with the Debye and Einstein Cv m calculated by using 0p = 244 K and p> = 314 K. The vibrational density of states according to the two models is shown in the insert.

The experimental constant-pressure heat capacity of copper is given together with the Einstein and Debye constant volume heat capacities in Figure 8.12 (recall that the difference between the heat capacity at constant pressure and constant volume is small at low temperatures). The Einstein and Debye temperatures that give the best representation of the experimental heat capacity are e = 244 K and D = 315 K and schematic representations of the resulting density of vibrational modes in the Einstein and Debye approximations are given in the insert to Figure 8.12. The Debye model clearly represents the low-temperature behaviour better than the Einstein model. 

Although the Debye model reproduces the essential features of the low- and high-temperature behaviour of crystals, the model has its limitations. A temperature-dependent Debye temperature, d(F), can be calculated by reproducing the heat capacity at each single temperature using the equation... 

The heat capacity models described so far were all based on a harmonic oscillator approximation. This implies that the volume of the simple crystals considered does not vary with temperature and Cy m is derived as a function of temperature for a crystal having a fixed volume. Anharmonic lattice vibrations give rise to a finite isobaric thermal expansivity. These vibrations contribute both directly and indirectly to the total heat capacity directly since the anharmonic vibrations themselves contribute, and indirectly since the volume of a real crystal increases with increasing temperature, changing all frequencies. The constant volume heat capacity derived from experimental heat capacity data is different from that for a fixed volume. The difference in heat capacity at constant volume for a crystal that is allowed to relax at each temperature and the heat capacity at constant volume for a crystal where the volume is fixed to correspond to that at the Debye temperature represents a considerable part of Cp m - Cv m. This is shown for Mo and W [6] in Figure 8.15. 

Kieffer has estimated the heat capacity of a large number of minerals from readily available data [8], The model, which may be used for many kinds of materials, consists of three parts. There are three acoustic branches whose maximum cut-off frequencies are determined from speed of sound data or from elastic constants. The corresponding heat capacity contributions are calculated using a modified Debye model where dispersion is taken into account. High-frequency optic modes are determined from specific localized internal vibrations (Si-O, C-0 and O-H stretches in different groups of atoms) as observed by IR and Raman spectroscopy. The heat capacity contributions are here calculated using the Einstein model. The remaining modes are ascribed to an optic continuum, where the density of states is constant in an interval from vl to vp and where the frequency limits Vy and Vp are estimated from Raman and IR spectra. 

First-order estimates of entropy are often based on the observation that heat capacities and thereby entropies of complex compounds often are well represented by summing in stoichiometric proportions the heat capacities or entropies of simpler chemical entities. Latimer [12] used entropies of elements and molecular groups to estimate the entropy of more complex compounds see Spencer for revised tabulated values [13]. Fyfe et al. [14] pointed out a correlation between entropy and molar volume and introduced a simple volume correction factor in their scheme for estimation of the entropy of complex oxides based on the entropy of binary oxides. The latter approach was further developed by Holland [15], who looked into the effect of volume on the vibrational entropy derived from the Einstein and Debye models. 

Entropies and heat capacities can thus now be calculated using more elaborate models for the vibrational densities of states than the Einstein and Debye models discussed in Chapter 8. We emphasize that the results are only valid in the quasiharmonic approximation and can only be as good as the accuracy of the underlying force-field calculation of such properties can thus be a very sensitive test of interatomic potentials. 

The heat capacity at constant volume in the Debye model is given by... 

The important message from Einstein or Debye models is that vibrations of atoms in a crystal contribute to Entropy S and to Heat Capacity C therefore they affect the thermodynamic equilibrium of a crystal by modifying both the Eree energy F, which... 

Figure 4.24 Molar heat capacity as a function of temperature, based on the Debye model. From K. M. Ralls, T. H. Courtney, and J. Wulff, Introduction to Materials Science and Engineering. Copyright 1976 by John Wiley Sons, Inc. This material is used by permission John Wiley Sons, Inc.

N is Avogadro s number and is the Debye temperature) and whose intercept at T = 0 is y. In the Debye model, the slope has a dependency (actually, r" for an n-dimen-sional solid) owing to the lattice or phonon contribution to the heat capacity. Of course, the heat capacity normally measured is Cp, the heat capacity at constant pressure. However, for solids the difference between Cp and Cy is typically less than 1 percent at low temperatures and thus can be neglected. 

The fact that a quantum oscillator of frequency u> does not interact effectively with a bath of temperature smaller than hu>/kg implies that if the low temperature behavior of the solid heat capacity is associated with vibrational motions, it must be related to the low frequency phonon modes. The Debye model combines this observation with two additional physical ideas One is the fact that the low frequency (long wavelength) limit of the dispersion relation must be... 

This result is called the Debye law. Figure A2.2.4 compares the experimental and Debye model values for the heat capacity C. It also gives Debye temperatures for various solids. One can also evduate Cy for the... 

Debye temperature (6,) - In the Debye model of the heat capacity of a crystalline solid, 0 = hvjk, where h is Planck s constant, k is the Boltzmann constant, and is the maximum vibrational frequency the crystal can support. For T 0, the heat capacity is proportional to P. 

Theoretical Estimates The use of the Debye model (Figure 3.2), which assumes that a solid behaves as a three-dimensional elastic continuum with a frequency distribution/(j ) = allows accurate prediction of the temperature dependence of the vibrational heat capacity C / of solids at low temperatures Cy oc r ), as well as at high temperatures (Cy = Wks). One may also use the same model with confidence to evaluate the temperature dependence of the surface heat capacity due to vibrations of atoms in the surface. 

For the thermal properties of solids, Einstein developed an equation that could predict the heat capacity of solids in 1907. This model was then refined by Debye in 1912. Both models predict a temperature dependence of the heat capacity. At... 

As mentioned, many experimental results have shown that the specific heat for composites increases sHghtly with temperature before decomposition. In some previous models, the specific heat was described as a Hnear function. Theoretically, however, the specific heat capacity for materials wiU change as a function of temperature, as on the micro level, heat is the vibration of the atoms in the lattice. Einstein (1906) and Debye (1912) individually developed models for estimating the contribution of atom vibration to the specific heat capacity of a sohd. The dimensionless heat capacity is defined according to Eq. (4.32) and Eq. (4.33) and illustrated in Figure 4.12 [25] ... 

The true specific heat capacity of a composite material was obtained by the mle of mixture and the mass fraction of each phase was determined by the decomposition and mass transfer model. The true specific heat capacity of resin or fiber was derived based on the Einstein or Debye model. The effective specific heat capacity was obtained by assembhng the trae specific heat capacity with the decomposition heat that was also described by the decomposition model. The modeling approach for effective specific heat capacity is useful in capturing the endothermic decomposition of resin and was further verified by a comparison to DSC curves. 

The quite complicated temperature dependence of the macroscopic heat capacity in Fig. 2.46 must now be explained by a microscopic model of thermal motion, as developed in Sect. 2.3.4. Neither a single Einstein function nor any of the Debye functions have any resemblance to the experimental data for the solid state, while the heat capacity of the liquid seems to be a simple straight line, not only for polyethylene, but also for many other polymers (but not for all ). Based on the ATHAS Data Bank of experimental heat capacities [21], abbreviated as Appendix 1, the analysis system for solids and liquids was derived. 

The first theoretical work providing information on the Debye temperature (Go) of intermetallic clathrates dates back to the year 1999 [33]. Molecular dynamics calculations for the carbon-framework of type-I and type-II clathrates used a Lennard-Jones potential (later on also for Si-based clathrates [34]). 0d for Ci36 [35] and for Siiae [34] were estimated from the calculated elastic constant Cn applying the empirical relation Qd = —11.3964 + 0.3475 x C — 1.6150 x 10 X Cj 1. Moriguchi et al. [36] used an empirical bond-order potential developed by Tersofif for the calculation of several thermodynamic properties, including the heat capacity, for the type-I and type-II Si networks. From the heat capacity data in the temperature range from 0 to 150 K 6d was extracted applying the Debye-model. The heat capacity, Cy, was calculated by the density functional theory (DFT),... 

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