Chapter III: Thermodynamic stability and heat absorption of
3.1 Thermodynamics of nanoparticles
Both the enthalpy and entropy of Eq.3.1 can be determined from the heat capacity, Cp, with the relationships:
Hp(T) = H0+ Z T
0
Cp(T′)dT′ , (3.2) Sp(T) = S0+
Z T
0
Cp(T′)
T′ dT′ . (3.3)
The thermodynamic stability of a nanomaterial with respect to a bulk control material is determined by the difference ∆Gn−c (a positive ∆Gn−c makes the nanomaterial unstable with respect to the control material):
∆Gn−c(T) = Gn(T)−Gc(T)
= ∆Hpn−c(T)−T∆Spn−c(T), (3.4)
∆Hpn−c(T) = ∆H0n−c+ Z T
0
∆Cpn−c(T′)dT′ , (3.5)
∆Spn−c(T) = ∆S0n−c+ Z T
0
∆Cpn−c(T′)
T′ dT′ , (3.6)
where a notation such as ∆Cpn−c denotes the difference in heat capacity of the nanocrystalline material and the control bulk material at constant pressure.
The largerH0 of the nanocrystalline material originates from grain boundaries, internal stresses, and defects. Grain boundaries force atoms into irregular local coordinations, increasing their internal energies (the black atoms of Fig.3.1).
The weaker bonding of atoms near these grain boundaries has other macro- scopic consequences, such as lowering their melting temperature [1]. Metals with stronger bonding and higher melting temperatures are also expected to have grain boundaries with higher energies. From work on a number of grain
69 boundary structures in fcc metals, the grain boundary energy varies from ap- proximately 0.4 to 1.2 J/m2 in the sequence Au, Ag, Al, Pd, Cu, Co, Ni, Pt [6, 7]. For fcc Ni metal, however, grain boundaries between crystals with various orientations have energies that span much of this entire range [8]. We measured this enthalphy in nanocrystalline Ni3Fe by heating it up in calorimeter, and measuring the heat released while the grains grow and defects are recovered.
Sources of entropy
Entropy counteracts the enthalpy to lower the Gibbs free energy, and has configurational, magnetic, electronic and vibrational contributions. The large number of individual crystallites at different relative orientations give nanocrys- talline materials additional structural degrees of freedom. This results in a somewhat larger configurational entropy compared to materials with large grains. As derived in [5] however, this extra configurational entropy falls off ex- tremely fast with increasing grain size. For nanocrystals with 20 atoms across (which corresponds approximately to our materials), the configurational en- tropy should be lower than 0.01 kB/atoms, or about 0.3% of the total entropy at 300 K. On top of that, during our cryogenic calorimetric measurements, atom mobilities are suppressed. So we expect no changes in the atomic con- figurations, and therefore do not measure differences in the configurational entropy between the nanostructured material and the control samples.
As the temperature approaches absolute zero, all excitations are suppressed, but the structural disorder remains. Does this mean that nanostructured mate- rials have a residual configurational entropy (S0 in Eq.3.6). The answer is not necessarily clear, and has been the topic of heated debate in the case of amor- phous materials, which are also metastable with a large number of structural degrees of freedom [9–16]. From a thermodynamic perspective, what matters is if a materials is able to access and change its configuration. When the nanomaterial is cooled down, its atomic mobility is impaired and it becomes kinetically trapped in a specific configuration. This metastable nanomaterial cannot access all its degenerate configurational microstates as T →0 [12, 15, 16]. According to Boltzmann’s entropy,S =kBln Ω, the number of accessible microstates (or configurations) becomes Ω = 1, and the entropy vanishes at 0 K. We therefore adopt ∆S0n−c= 0.
Most of the entropy in materials stem from atomic vibrations. Nanocrys- tallinity changes the dynamical degrees of freedom, which affects the phonon spectrum. Vibrations of nanoparticles have been studied for some time, and the large literature on this subject is reviewed in the next section. The addi- tional degrees of freedom from individual nanocrystallites causes their phonon spectrum to have more low-energy vibrations than its bulk counterpart. This results in larger oscillatory amplitudes of atoms in the nanomaterial, and there- fore larger vibrational entropies. The Gibbs free energy of nanomaterials is lowered by this additional entropy, helping to stabilize against their extra en- thalpy from grain boundaries. Quantifying the thermodynamic consequences remains a challenge, however, and is the main goal of this study.
The spatial confinement also alters the bonding between atoms, and hence the electronic structure. Our low temperature heat capacity data show an enhancement of the heat absorbed by electrons in nanostructured materials by about 40%, which is reflected in a larger electronic entropy.
Lastly, the structural disorder in nanocrystalline materials also affects the magnetic ordering of spins. The magnetization of consolidated nanocrystals seem to become weaker with decreasing grain sizes in Fe and Ni [17–19] (see Fig.3.2a), and also in Ni3Fe [2,20]. This makes sense since magnetic ordering depends on exchange interactions between neighboring atoms, which can be disrupted at the grain boundaries. Additionally, the individual grains can act as different magnetic domains and reduce the net magnetization.
More importantly for the thermodynamics, the irregular local environments of atoms near grain boundaries cause their spins to be more susceptible to thermal excitations. This leads to a stronger dependence of the magnetism on temperature as shown in Fig.3.2b for iron nanoparticle [17]. This is reflected in the lower Curie temperatures observed in nanostructured materials. For Ni3Fe, nanocrystallinity can reduce the Curie temperature from 871 K, [21] down to 728 K [22]. In thin films of Ni, with two monolayers, the Curie transition can be suppressed down 325 K [23].
Most of the magnetic disorder, and hence most of the magnetic entropy, in- creases near the Curie transition. Only a small amount of magnetic entropy is expected below 300 K, where our cryogenic heat capacity measurements were carried out. Nonetheless, due the stronger temperature dependence of the
71
Figure 3.2:Size effects on magnetization.(a)Size dependence of the satruration magne- tization,Ms, of ferromagnetic nanoparticles [18].(b)Size and temperature dependence of Msfor iron crystallites (coated in Mg) [17].
magnetic spins in nanocrystalls, we do expect them to have a larger magnetic entropy than our control samples.