Chapter III: Thermodynamic stability and heat absorption of
3.8 Contributions to the heat capacity at higher temperatures
Figure 3.13:Heat capacities of nanomaterial (+) and control samples (•) from calorimetry shown in Fig.3.10, superposed with phonon heat capacities calculated from the phonon DOS using Eq.3.10(solid lines) and with vibrational plus electronic heat capacities (dashed lines).
3.8 Contributions to the heat capacity at higher tempera-
89 ature electronic contribution (Cp,el =γT) to higher temperatures. The results for both the nanocrystalline and control materials are included in Fig.3.13.
At 300 K, the electronic heat capacities for the nanocrystalline and control samples are 0.20kB/atom and 0.14kB/atom, respectively.
Magnetism
The spin-wave model of magnetic fluctuations does not hold at elevated tem- peratures, so estimating the magnetic contribution at 300 K is not straightfor- ward. For instance the energy to create a magnon becomes dependent on the number of magnons already present in the material. This many-body aspect of spin waves, makes the concept of individual and independent magnons in- applicable at 300 K. Elevated temperatures can also enhances other scattering events with phonon for example. Since spin-waves can not propagate coher- ently at elevated temperatures, a magnetic contribution as Cp,mag =αT3/2 is not applicable at elevated temperatures.
Most of the change in magnetization occurs near the Curie temperature, so the magnetic thermal disorder is small at 300 K. For disordered bulk Ni3Fe the ferromagnetic transition occurs at 871 K [21]. Measurements show a decrease of only 5% in the spontaneous magnetization between 0 K and 300 K [70], so the magnetic contribution to the entropy at room temperature is small.
Nanostructured materials, however, have lower Curie temperatures [19] and a stronger temperature dependence of the magnetization [17], so we expect a larger entropy. Ball-milled nanocrystals have a lower magnetization than their bulk counterpart as measured by Mössbauer experiments [55]. Previous measurements on our nanocrystalline samples at 300 K showed that the mag- netization is 7% lower than that of the control samples [2]. In other work, at 4 K, the magnetization of a nanocrystalline sample (12.5 nm) was found to be 2% lower than that of a sample with larger grain sizes [71]. The Curie temper- ature reported for nanocrystalline Ni3Fe, ranges between 728 K [22] and 848 K [71]. Nanomaterials have, therefore, a larger contribution from magnetism to the heat capacity and entropy, even at 300 K and below.
The disordering the spins requires heat and contributes to the heat capacity as an endothermic signal. The total entropy associated with the disordering of spins can be quantified by from the magnetic heat capacity as Smagtot = R CP,mag/TdT. The magnetic heat capacity is reported in [72] up to 1400 K
for bulk Ni3Fe, givingSmagtot = 0.61kB/atom. At 300 K and below, the magnetic entropy is only a fraction of this total. To estimate this fraction we use the mean-field model derived previously in Eq.1.23, giving the entropy of ‘mixing’
spin-up and spin-down atoms on an Ising lattice:
Smix =−N kB 2
(1 +M) ln 1 +M
2 + (1−M) ln1−M 2
.
The maximum entropy in this model isN kBln 2 = 0.69kB/atom (whenM=0).
For Ni3Fe this has to be equivalent to the total measured magnetic entropy of 0.61kB/atom. For partial magnetization, the magnetic entropy becomes:
Smag(M) = Smix
N kBln 2Smagtot
= −Smagtot 2 ln 2
(M + 1) lnM + 1
2 + (1−M) ln1−M 2
. (3.11) The magnetization M(T) has been measured for bulk Ni3Fe [70], so we can compute the magnetic entropy from0−300K. At 300 K it isSmagc = 0.09kB/atom.We can also compute the magnetic contribution to the heat capacity, as plotted in Fig.3.14a together with the other contributions for the control samples. The magnetic contribution at 300 K (0.12kB/atom) is in good agreement with the magnetic heat capacity of Ni3Fe reported by Kollie (0.14kB/atom) [72] and the magnetic contribution calculated by Körmann et al. for Ni (0.11kB/atom) [73]. As seen from the small residual of Fig.3.14a, the sum of magnetic, elec- tronic and phononic contributions agrees well with the total heat capacity measured by calorimetry.
Magnetization data were available for the nanomaterial only at 4 K [71] and 300 K [2]. At 300 K Eq.3.11 gives Smagn = 0.15kB/atom. By assuming that it follows the same temperature dependence as the control material, we can estimate its magnetic heat capacity between0−300K (shown Fig.3.14b). As suggested by the larger residual heat capacity at 300 K, the nanocrystalline material has additional contributions, besides the electronic, magnetic and vi- brational heat capacities. This residual heat capacity is 0.22kB/atom at 300 K, corresponding to a residual entropy of 0.14kB/atom. Its origin is discussed be- low.
91
Figure 3.14: Individual contributions to the heat capacity of (a) the control and (b) nanocrystalline samples. The residual is the difference between the sum of the labeled contributions and the total heat capacity measured by calorimetry.
Anharmonic contributions
The vibrational contribution to the heat capacity was computed from the phonon DOS measured at 300 K using Eq.3.10, based on a harmonic model.
In reality, phonons are not harmonic: their vibration frequencies vary with temperature and their propagation has a finite lifetime. Here we discuss the effects of these two types of anharmonicity.
A general trend is the softening of phonons (lowering of the vibrational fre- quencies) with increasing temperature. This increases the phonon entropy and is often related an expansion of the lattice volume. Magnetism and electrons can also affect the volume, as will be explored in more depth in Chapter 5.
Their contributions and variations with temperature are usually much less im- portant than those from phonons, however. It is straightforward to estimate the contribution from thermal expansion to the heat capacity [5]:
∆Cth(T) =Cp(T)−Cv(T) = Bvβ2T . (3.12)
Using a bulk modulus of179GPa, molar volume of 6.75 cm3/mol, volume ther- mal expansion of 3.3·10−5K−1 (values for Ni3Fe at 300 K taken from [60]), gives
∆Cth=0.05kB/atom, about 1% of the total heat capacity at 300 K and about 20% of the residual heat capacity of the nanomaterial. Thermal expansion alone can not account for the excess residual heat capacity of Fig.3.14b.
Another possible contribution comes from the reduced lifetimes of phonons, resulting from their scattering at grain boundaries or other defects of nano- materials. Reduced lifetimes are compatible with the broadening the features of the DOS as shown in Fig.3.12a. According to the energy-time uncertainty principle (∆E∆t≥ℏ/2), when the lifetime is reduced, the energy uncertainty has to increase. Lifetime broadening can be modeled by damped harmonic oscillators. Introducing damping causes a shift of the vibrational spectrum to higher energies, giving an apparent reduction in the entropy when calculated with Eq.3.9. A correction for this effect, given in [74], predicts an increase of the phonon entropy by 0.15kB/atom for a damped harmonic oscillator with a damping factor ofQ= 6, which is consistent with the broadened phonon DOS of the nanocrystalline material. An anharmonic contribution of a fraction of onekB/atom is typical for metals at high temperatures [69]. Shortened phonon lifetimes are consistent with the larger amount of grain boundaries and defects of the nanomaterial, and account well for its residual entropy of 0.14kB/atom at 300 K.