Chapter V: Thermodynamic origin of the Invar effect
5.1 Anomalous thermal expansion
Metals usually have a pronounced thermal expansion. Especially those with lower bonding energies and melting points. The thermal expansion of glasses tends to be larger than of crystals, and liquids expand slightly more than solids. The conventional textbook description starts from the potential be- tween atoms, like atoms connected by springs. The potential in not symmetric and harmonic, however. Pauli exclusion principle prevents atoms from coming very close to each other, so the energy raises sharply at low atomic separations.
Think of a Lennard-Jones potential, for example (shown in Fig.5.1). When temperatures increase, most of the thermal energy goes into atomic vibrations, so the atoms explore larger amplitudes in their potential energy. Due to the skewed potential, the atomic separation tend to increase a little, causing the thermal expansion. Thermal expansion is inherently an anharmonic effect.
For actual computations of thermal expansion, the free energy G(V, T) must be minimized, giving an equilibrium volume V(T).
117 Negative thermal expansion
Thermal contraction is fairly rare, but we are all familiar with an example:
water. Ice floats on liquid water, because it is less dense (has a larger volume) than the warmer liquid. The volume of water therefore contracts when it melts. You should not forget a beer can in the freezer because the expansion of the freezing water might cause the can to explode and make a mess.
Negative thermal expansion (NTE) is found in several oxides, such as Si02 [2], ZrW2O8 [3], and many perovskites [4–6]. Geometric arguments are often used to explain the anomalous NTE. Transverse vibrations of the central atom in a linear chain like O-S-O (in SiO2 or CO2 [2]) can decrease the distance between the outer O atoms (as in a vibrating guitar string). Rigid unit modes of tetrahedra in SiO4 [7], and octahedra in ZrW2O8 [3] or layered perovskites [6], can distort the lattice causing NTE.
Geometric explanations sound appealing and may be intuitive, but they usu- ally focus on specific modes. The entire phonon spectrum has to be accounted for to quantify the thermal expansion, however. Cuprite, Cu2O, contracts when heated up to about 500 K. While the O atoms soften and oscillate more strongly, the phonon energies of the Cu increase with temperature, and they vibrate less. Details require anharmonic interactions between phonons, but the net effect is a thermal contraction [8]. Pure silicon also contracts at low temperatures. A full anharmonic picture, that includes the zero point motion of phonons, is necessary for understanding this anomalous thermal expansion behavior [9].
Invar behavior
Invar is a face-centered cubic (fcc) random solid solution of Fe and Ni, with 65 atomic % Fe and 35 atomic % Ni. Geometric and symmetry explanations have no chance of success for explaining its anomalous thermal expansion.
Pure Fe and Ni both have a well behaved positive thermal expansion. When alloyed, both show a similar phonon spectra [10]. But in a narrow range near a composition of 65% Fe, Invar shows an anomalously low thermal expansion (see Fig.5.2). Why does this happen? Phonons alone cannot explain it, we need to bring other excitations into the picture.
Figure 5.2:Linear thermal expansion coefficientαof Fe-Ni alloys measured by Guillaume.
The minimum, close toα= 0, corresponds to the Invar composition [11].
Fe-Ni Invar was discovered in 1895 by the Swiss physicist C.E. Guillaume. It was named Invar due to the invariance of its volume with temperature. Al- though the Invar effect of anomalously low thermal expansion was discovered in Fe65Ni35, it has since been found in Fe-Pd, Fe-Pt, Fe-Mn, Ni-Mn, Co-Mn, Fe-Cr and several other crystalline and amorphous alloys of iron with transi- tion metals [1,12]. Such a property can be exploited in numerous in engineered devices to have structural stability against temperature, and allow for tight tolerances [1]. Large structures such as pipelines or tanks that require cryo- genic cooling use Invar alloys. Invar has been used in composite molds in the aerospace industry to achieve tight dimensional tolerances in curing processes.
Invar is used for structural components in precision instrumentation, such as laser and optical measuring systems. Invar materials appear in microscopes, in support systems for mirrors in telescopes, and in orbiting satellites [13].
Guillaume himself presented applications for Invar and Fe-Ni alloys [11]. He demonstrated how Invar can be used for precision measurements of length in the late 1800s. By using a long Invar wire, he tracked the vertical motion of the Eiffel Tower, while it expands and contracts in the course of a day. He attached a wire from the ground up to the second platform of the tower and recorded its height variation as the iron structure, but not the Invar wire, expands and contracts with the changing temperature. Invar-like alloys also allow for precise measurement of time. A material with thermally-invariable elasticity modulus, namedElinvar, is achieved when the Ni content is changed to about 45%. Guillaume designed compensating springs for watches using Elinvar, reducing their error by five times. By 1927, over 100 million watches were made with Elinvar springs.
119 The discovery of Invar was so important that Guillaume was awarded the physics Nobel Prize in 1920 “in recognition of the service he has rendered to precision measurements in Physics by his discovery of anomalies in nickel steel alloy”. Because of Guillaume, Albert Einstein had to wait another year for his Nobel, awarded in 1921.
So why is the volume of Invar not affected by temperature? Guillaume knew it had something to do with the magnetic nature of Fe-Ni alloys. The Invar effect disappears when heating above the Curie transition. This is seen in Fig.5.3, where the coefficient of thermal expansion,α, is plotted against temperature.
For the classic Invar composition of 65% Fe, the thermal expansion is close to zero up to about 450 K (roughly 0.25/Tm). This is in striking contrast to the expected thermal expansion of the ‘lattice’ from the thermally enhanced phonon vibrations in an anharmonic potential. Above the Curie temperature, TC = 515K, the Invar behavior vanishes, and the measured thermal expansion matches that from the ‘lattice’.
The term Invar has since been generalized for all alloys that present a lower thermal expansion than expected from phonons. At 70% Fe the Invar be- havior is also observed at low temperatures, but above its Curie transition, the thermal expansion is larger than ‘normal’. This excessive thermal expan- sion became know as the anti-Invar effect. It is observed at all compositions above 70% Fe. It seems like the thermal expansion behaviour can be tuned by the alloy composition. At large Fe content, the plot is cut off at the Curie temperature because the magnetic transition is accompanied by a martensite transformation into the bcc phase.
The anomalous properties of Invar are by no means limited to the anomalous thermal expansion [1]. They include an anomalous softening of the elastic constants, affecting both the bulk and shear moduli. Invar has a strong tem- perature dependence of the magnetization, and an anomalous specific heat, with excessive magnetic contributions even well below the Curie transition.
And a Curie temperature that collapses near the Invar composition.