Elastic constants of equiatomic Fe–Ni Invar alloy single crystal

Fe–Ni alloys (Permalloy) are known for their soft magnetic properties and low cost.¹ They are widely used in electronic devices and industrial applications, such as transformer cores, electrical motors, and magnetic recording heads.² The soft magnetic behavior of Permalloy is greatly influenced by the nickel content, with low coercivity obtained at about 80 at. % Ni, high saturation magnetization obtained at 50 at. % Ni, and reduced permeability but higher electrical resistivity obtained at about 35 at. % Ni.³ Fe–Ni alloys are also attractive as they possess an excellent combination of good strength, ductility, and corrosion resistance.^7,8 Equiatomic Fe–Ni alloys and alloys around this composition also exhibit zero or low thermal expansion and are commonly known as Invar alloys. Due to their excellent mechanical and dimensional stability, they are suitable for applications such as precision instruments, calibration equipment, and high-temperature structural components.^4–6 While the equiatomic Fe–Ni alloy in a disordered A1 structure is of great interest as a soft magnetic Invar alloy with a near zero or low thermal expansion coefficient, this alloy in the L1₀ ordered phase (tetrataenite phase observed in meteorites⁹) is currently also of great interest as a high-performance permanent magnet alloy for replacing rare-earth based magnets. However, no detailed studies of the structure and orientation-dependent behavior of this alloy using single crystals, in particular elastic constants and deformation behavior, have been reported.

Elastic constants are first-order derivatives of free energy with respect to strain and are related to the nature of atomic bonding. The absolute values of elastic constants impact a wide range of material properties that include force-constant-related properties, defect and defect interaction energies, dislocation dynamics, interatomic potentials, magnetoelastic response, thermal expansion, Debye temperature, Grüneisen parameter, and other thermodynamic characteristics.^10–12 The elastic constants of Fe–Ni alloys are strongly dependent on the crystallographic structure and orientation, and single crystals provide an ideal form for the determination of the direction-dependent elastic response. Understanding the elastic properties of Fe–Ni single crystals in conjunction with thermal properties is critical in establishing the equation of state in this alloy system, and the first detailed study of elastic constants in the Fe-50 at. % Ni alloy having a disordered fcc structure is carried out here using a well characterized [001]-oriented single crystal under a no applied magnetic field condition.

One of the early studies on the elastic constants of Fe–Ni alloys was conducted in 1960 by Alers et al., who used 10 MHz pulse-echo techniques to measure the elastic constant of Fe–30%Ni alloys.¹³ Several other investigations were carried out in the following decade using the same method but with different compositions and test conditions.^14–18 Later in 1969, Shirakawa et al.¹⁹ determined the elastic properties of Fe–Ni single crystals with compositions ranging from 35% to 100% of Ni using the resonant frequency measured by the force driving method. However, the elastic constants of equiatomic Fe–Ni single crystals determined by Hausch et al.¹⁷ and Shirakawa et al.¹⁹ were measured only under a sample condition of saturation magnetization under an applied magnetic field. More recently, the elastic properties of hexagonal close-packed iron alloys with less than 10% nickel under high pressure were theoretically examined using first-principles calculations.⁹ There is currently no complete study available on the elastic constants of equiatomic Fe–Ni single crystals under a no field condition.

Various techniques have been used for determining elastic properties, including impulse excitation (IE), nanoindentation (NI), resonant ultrasound spectroscopy (RUS), and four-point bending test (4PBT). Among these techniques, RUS and IE have demonstrated better precision than NI and 4PBT.²⁰ While both RUS and IE involve frequency measurements, IE uses a single impulse. In contrast, RUS utilizes a drive transducer to apply a continuously varying input frequency to the sample in order to measure the natural resonance frequency.²¹ As a result, RUS is a more accurate and versatile non-destructive test for determining all the single crystal elastic constants with high sensitivity using the same sample in a single measurement. Sample sizes can be as small as 1 mm, unlike large and multiple samples needed in other techniques. The RUS technique excels in precision, anisotropic material characterization, and versatility compared to other techniques, making it valuable for studying mechanical properties.

The RUS technique uses the absorption of sound waves at the natural resonance frequencies of a material to determine its elastic properties.²² Resonance in the solid is a function of elastic properties, density, structure, dimensions, and structural integrity of the solid,^23,24 and the knowledge of the relationship between these parameters is central to the RUS technique. It is commonly known that the resonance one hears in a cracked church bell is distinctively different from that in a structurally sound bell.²⁴ This idea has been used in some form or other in the quality control of industrial components. Frasier and LeCraw calculated the resonances in a freely vibrating spherical solid in 1946 and made the first widespread use of the RUS technique.²⁵ During a RUS measurement, the natural frequencies of a sample are measured under stress-free boundary conditions by holding the sample gently between surface transducers at two diagonally opposite corner points of the sample. One of the transducers drives the vibrations into the sample, and the second transducer measures the amplitude of the sound waves transmitted by the sample. As the drive frequency is varied, the resonance of the sample occurs at specific frequencies. This causes a reduction in the transmitted wave intensities and the peaks in the absorption spectrum. The peaks in the absorption spectrum occur at the natural frequencies (f_n) of the sample, from which elastic constants can be obtained.²⁶ Because this is controlled by a computer, the process proceeds in small frequency steps over a previously determined region of interest. During such a frequency sweep, the drive frequency typically brackets the resonance frequency. When the excitation frequency does not match the resonance frequency, minimal energy is transferred, resulting in negligible vibration. However, at resonance, the energy delivered to the part increases by a factor of Q, acting as a natural amplifier generating much larger vibrations. The quality factor, Q, defines the energy dissipation in the material and depends on stiffness.²³ Q for each resonance is given by f_n divided by the full width of a peak at its half maximum. The more rigid a part is, the higher the Q. The resonance frequencies of a part are determined by the dimensions, density, and elastic constants of the material.^24,27 RUS enables accurate modulus measurement with accuracies on the order of a tenth of a percent.²⁸ 

This work is the first detailed study of elastic constants in the Fe-50 at. % Ni alloy using a [001]-oriented single crystal. We used resonant ultrasound spectroscopy to accurately measure the three independent elastic constants C₁₁, C₁₂, and C₄₄ of the [100]-oriented of Fe–Ni single crystal. The [100]-oriented Fe–Ni single crystal sample having a rectangular parallelepiped shape was prepared from a single crystal grown using the vertical Bridgman technique. The resonant frequencies were measured over a frequency range of 100–900 kHz, and the elastic constants C₁₁, C₁₂, and C₄₄ were obtained from the resonant frequencies observed. Using these C₁₁, C₁₂, and C₄₄ values, Young’s modulus, shear modulus, bulk modulus, anisotropy factor, and Poisson’s ratio were calculated. The elastic constants determined in the study enhance our fundamental understanding of the structure and properties of Fe–Ni alloys of interest in soft and permanent magnet applications.

Fe-50 at. % Ni alloy ingots were prepared using a high-vacuum arc melting system. The Fe and Ni elements used to prepare the ingots were of high purity, 99.97+%. The alloy was doped with sulfur at a level of ∼100 ppm to enhance interdiffusion. To achieve a homogeneous composition, each alloy ingot underwent multiple re-melting cycles on a water-chilled copper mold. This process ensured the uniform distribution of the constituent elements within the ingots. Subsequently, the molten alloy was cast into cylindrical rods with a diameter of 12.5 mm and about 25 mm length. The cast cylindrical rods were loaded in a closed-one-end alumina tube. The other end of the alumina tube was connected using a three-way connector to a vacuum line and argon back-fill line. The growth of the Fe-50 at. % Ni alloy single crystal was performed using the vertical Bridgman technique described below. The alumina tube loaded with cast rods and connected to the vacuum/backfill line was positioned inside a MoS₂ resistance-heated two-zone vertical furnace. The alumina tube was evacuated and backfilled with ultra-high purity (UHP) argon gas. Argon flow was maintained using a bubbler with a flow of one bubble/second. The furnace temperature was increased to heat the alloy rods above the melting temperature. A controlled descent of the tubes down the temperature gradient at a translation rate of 4 mm/h was achieved using a stepper motor-drive, enabling the nucleation of the crystal at the bottom closed end of the tube and subsequent growth of the single crystal.

The directionally grown single crystal was then carefully sectioned using a diamond saw, and the sectioned piece was polished on 600 and then 1200 grit size SiC discs followed by polishing using 1 and 0.3 µm alumina polishing cloths. Orientation determination of the sectioned single crystal sample faces was performed using θ–2θ diffraction scans and an iterative combination of rocking curve, phi, and detector x-ray diffraction scans. X-ray diffraction scanning was carried out using a high-resolution Siemens^® D5000 x-ray diffractometer, employing Cu–Kα radiation. The (001) and (311) rocking curve data were analyzed using Carine software after each cycle of polishing to assess the crystal orientations. Different-angle polishing blocks on which the sample was mounted were used to grind and polish the samples to obtain the desired orientation to within ±0.5° off the (100) face. The resulting [100]-oriented single crystal sample was cut and polished into a rectangular parallelepiped of size 7.64 × 6.54 × 4.31 mm³ with each face normal to the 〈100〉 orientation. The sample was prepared for RUS study by final polishing of each of the faces to an optical surface finish using 50 nm colloidal alumina slurry in a Beuhler^® vibratory polisher for 6 h.

RUS measurements were conducted using a Dynamic Resonance System (DRS) Inc.^TM Modulus II system, comprising a transmitter and a receiver, as depicted in Fig. 1. Our experimental setup followed the approach developed by Fraser and LeCraw,²⁹ Demarest,³⁰ Ohno,³¹ and Migliori et al.,³² where the Fe–Ni sample with a rectangular parallelepiped shape having edges oriented along the [001], [010], and [001] directions was lightly clamped between two piezoelectric transducers. The transducer contacts applied very weak forces to the sample, allowing for minimal loading. The transmitter excited the specimen by applying vibrations across a specified range of increasing frequencies. During the RUS measurement, the Fe–Ni sample exhibited absorption of sound waves, with greater absorption observed at natural resonance frequencies of the material. At the receiver, the response of the sample to the transmitted signal was converted into an electrical signal, which was then fed to a computer for subsequent analysis.²³ 

To achieve accurate resonance frequency matching between the measured frequencies and the natural frequencies of the sample, it was crucial to minimize the force on the sample from the transducers. By carefully controlling the contact loads, it is possible to obtain a measurement accuracy of about 0.1%, with contact loads only slightly greater than the weight of the sample.²⁵ The absorption spectra of Fe–Ni alloys were obtained using the DRS Modulus system over a frequency range of 100–900 kHz.

For a material in a cubic system, there are only three important independent elastic constants: C₁₁, C₁₂, and C₄₄. These constants determine the material’s response to external forces.

To further examine the mechanical behavior of the material, several other mechanical parameters can be calculated using the Voigt–Reuss–Hill approximation.^33–35 These parameters include the bulk modulus (B), the shear modulus (G), and Young’s modulus (E), which provide insights into the material’s resistance to compression, deformation under shear stress, and overall stiffness, respectively.

In this study, an Fe-50 at. % Ni single crystal was grown and oriented in the [100] direction. A rectangular parallelepiped-shaped sample with dimensions of 7.64 × 6.54 × 4.31 mm³ prepared for RUS measurement had the edges of the sample aligned along the [001], [010], and [001] crystallographic directions (Fig. 2 left). As noted earlier, each face of the sample underwent a polishing process to achieve a surface finish of up to 0.05 µm. Figure 2 (right) displays an optical microscopy interference contrast image of one side of the sample after this polishing procedure. The image provides a visual representation of the sample’s surface quality, revealing the smoothness and uniformity achieved through the polishing process. The high-quality surface finish is crucial for accurate RUS measurements as it minimizes surface irregularities that could introduce unwanted noise or vibrations during the experimental analysis.

The (100) and (311) rocking curve scans were obtained to determine the orientation of each side of the sample. Figure 3(a) displays a (200) rocking curve scan on one of the sides of [100]-oriented sample, where the peak position indicates a deviation of the surface normal by 0.4° from the [100] direction of the crystal. Figure 3(b) exhibits the θ–2θ scans along the [001] direction, revealing a sharp peak corresponding to the (200) plane of the single crystal.

Figure 4 illustrates the resonance spectra obtained during the RUS measurement of the Fe–Ni single crystal at room temperature, covering a frequency range of 100–900 kHz. For the determination of elastic constants C₁₁, C₁₂, and C₄₄, the first 30 resonant peaks were utilized. The resonance spectrum in Fig. 4 showcases high-quality resonant peaks with an excellent signal-to-noise ratio, indicating the quality of the specimen and the RUS measurement.

To assess the elastic constants, the measured resonance frequencies were compared with the calculated frequency values from sample dimensions, density, and estimated initial elastic constant values. Data fitting was performed for the first 30 representative peaks to minimize the figure of merit, as detailed in Table I. A total of 30 RUS peaks were fitted, resulting in a root-mean-square (rms) error of 0.2351%. The rms value is obtained as the square root of the minimum figure of merit, which provides the estimation for the goodness to fit between the measured and calculated peaks. An optimal fit is achieved when the rms value falls within the range of 0.1–0.3 percent for the initial 20–30 resonance frequencies. This suggests good correspondence between the measured and calculated RUS peaks. The fitting results provided the following elastic constants: C₁₁ = 147.0, C₁₂ = 75.2, and C₄₄ = 118.1 GPa.

The C₁₁, C₁₂, and C₄₄ values determined were used to calculate the bulk modulus (B), the shear modulus (G), and Young’s modulus (E) to be 99.1, 73.5, and 176.7 GPa, respectively. Figure 5 compares the experimental results of this study with literature data for equiatomic Fe–Ni alloys. The data for C₄₄ are in agreement with the literature. However, the values of C₁₁ and C₁₂ in our study are lower than the literature values. Young’s modulus and shear modulus values exhibit agreement with the literature, while the bulk modulus is lower. It is noteworthy that only two previous studies by Shirakawa et al.¹⁹ in 1969 and Hausch et al.¹⁷ in 1973 measured the elastic constants of equiatomic Fe–Ni alloys, but under a saturated magnetization, and they used the force driving method and pulse echo technique, respectively. Our study was conducted at zero magnetic field. The differences could be attributed to the influence of the magnetic field as well as the less sensitive measurement technique used in Refs. 17 and 19, affecting the elastic constants.

Poisson’s ratio, which characterizes the material’s tendency to contract in the transverse direction when subjected to axial strain, was calculated to be 0.2029. This value suggests a moderate degree of transverse contraction relative to the applied axial strain. In addition, the anisotropy factor, a measure of the directional dependence of the material’s properties, was determined to be 3.288. The anisotropy factor indicates deviation from the isotropic behavior, with a value of 1 indicating perfect isotropy. In this case, the anisotropy factor being significantly higher than 1 suggests the presence of strong elastic anisotropy in the Fe–Ni single crystal, and the material’s mechanical properties will exhibit significant variations along different crystallographic orientations.

The first detailed measurements of elastic constants in a [001]-oriented Fe-50 at. % Ni alloy single crystal under a no magnetic field condition were successfully made using the resonant ultrasound spectroscopy technique. The single crystal sample used had a rectangular parallelepiped shape and was prepared from a single crystal grown using the vertical Bridgman technique. The elastic constants C₁₁, C₁₂, and C₄₄ were obtained from the resonant frequencies observed over a frequency range of 100–900 kHz. Using these C₁₁, C₁₂, and C₄₄ values, Young’s modulus, shear modulus, bulk modulus, and Poisson’s ratio were calculated to be 99.1 GPa, 73.5 GPa, 176.7 GPa, and 0.20, respectively. The large anisotropy factor of 3.29 highlights the strong directional dependence of the material’s mechanical behavior. These findings provide valuable insights into the mechanical behavior of Fe–Ni alloys of interest in various soft magnetic applications.

The authors are grateful for the support of this work through the award of the Graduate Innovation Fellowship from the College of Science, University of Utah, and the award of the Cooper Hansen Foundation Fellowship from the College of Mines and Earth Sciences, University of Utah to the first author R.S. Singh.

The authors have no conflicts to disclose.

Rahulkumar Sunil Singh: Conceptualization (equal); Data curation (equal); Formal analysis (equal); Funding acquisition (equal); Investigation (equal); Methodology (equal); Writing – original draft (lead); Writing – review & editing (equal). Sivaraman Guruswamy: Conceptualization (equal); Investigation (equal); Methodology (equal); Project administration (lead); Resources (lead); Supervision (lead); Writing – review & editing (equal).

The data that support the findings of this study are available from the corresponding author upon reasonable request.