Strengthening η texture (⟨001⟩//rolling direction) through secondary recrystallization can significantly improve the magnetostriction coefficient of Fe-Ga thin sheets and expand the use of Fe-Ga alloys in high-frequency applications. However, the secondary recrystallization of Goss texture in Fe-Ga thin sheets deteriorates significantly with the thinning of the sheet thickness under the current inhibitors of micron-sized NbC particles or precipitates. In this paper, a new composite inhibitor system, composed of nano-sized MnS and Nb(C,N) precipitates, is proposed to induce the secondary recrystallization Goss texture in Fe-Ga thin sheets with thicknesses less than 0.3 mm. The density and size of MnS and Nb(C,N) is designed based on the effect of alloy composition and hot rolling parameters on the precipitation kinetics of MnS and Nb(C,N), including the relative nucleation rate-temperature (NrT) and precipitation-time-temperature (PTT) curves. It is found that the stored deformation energy increases the maximum nucleation rate and the fastest precipitation temperature. A large number of lamellar and rectangular precipitates (MnS and Nb(C,N)) with a size of 5∼20 nm were precipitated after hot-rolling at a temperature of 800∼1,050 °C and a total thickness reduction of 80%. These nano-sized inhibitors provide a strong pinning effect for the primary recrystallized grains, and induce abnormal grain growth of Goss grains at an annealing temperature of 950∼1,050 °C. Resultingly, centimeter-sized Goss grains with a small deviation angle and maximum magnetostriction coefficient of 264 ppm are successfully achieved in Fe-Ga alloy thin sheets with a thickness of 0.25 mm.

## I. INTRODUCTION

Giant magnetostrictive Fe-Ga alloys, discovered in 2000, have attracted considerable attention due to the combination of large magnetostriction (3/2*λ*_{100} = 400ppm) in a low applied magnetic field (∼300 Oe) and excellent mechanical properties (*σ*_{s}≥350 MPa, *δ*≥2%).^{1–5} The deviation angle between the ⟨100⟩ direction and the axial direction of the single crystal rod (or the rolling direction of the thin sheet) significantly affects the saturation magnetostriction coefficient. It is reported that the magnetostriction of Fe_{81}Ga_{19} single-crystals increases from 254 ppm to 312 ppm as the deviation angle between the ⟨100⟩ direction and the axial direction decreases from 12° to 0.5°.^{6–8} Therefore, the angle between the ⟨100⟩ direction and the axial orientation should be minimized for high-performance Fe-Ga alloys.

Due to eddy currents in metallic magnetic actuators at high-frequency applied fields, the energy conversion efficiency is severely limited in applications with thick materials. The eddy current loss (*P*_{e}) of a thin sheet is expressed as:^{9,10}

Where *f* is the frequency, *t* is sheet thickness, *ρ*_{0} is the resistivity, *B*_{m} is the maximum flux density, *k* is the waveform factor (*k*=1.11 for sinusoidal waveform), and *γ*_{0} is the material density. To minimize eddy current losses in high frequency applications, Fe-Ga thin sheets with a sharp η-fiber (⟨001⟩//rolling direction) texture through optimized rolling and secondary recrystallization processes have attracted much attention in recent years.^{11–17} The NbC particles can change the grain boundary cohesion strength to improve ductility, so as to facilitate the fabrication of the thin sheets by rolling. In addition, the combination of micron-sized NbC particles and the change in surface energy from H_{2}S gas or sulfur used during annealing is used to pin normal grain growth and induce secondary recrystallization in Fe-Ga alloys, yielding centimeter-sized Goss grains with a magnetostriction coefficient exceeding 250 ppm with thickness over 0.3 mm.^{15–18}

In order to reduce eddy current losses and improve the conversion efficiency of Fe-Ga alloys, it is necessary to further thin the sheet, below 0.3 mm. However, when the thickness of the Fe-Ga alloy sheet is less than 0.3 mm, the secondary recrystallization of Goss textured grains and magnetostriction coefficient significantly deteriorates. The introduction of surface energy modifiers, like H_{2}S or S, and textured microstructures, consisting of columnar crystals by directional solidification, cannot guarantee the complete secondary recrystallization in Fe-Ga alloy sheet with the thickness less than 0.3 mm.^{19–22} The core problem for improving secondary recrystallization texture and the magnetostriction coefficient in thinner Fe-Ga sheet is the insufficient pinning force to normal grain growth from the micron-sized NbC particles.^{19,20}

The relationship between pinning force and precipitated phase is:

Where *γ* is grain boundary energy, *f* is volume fraction of the precipitation, and *r* is the average radius of the precipitates. The strength of the pinning force depends on the quantity and the size of precipitated phases. To obtain an effective inhibitor capable of inducing complete secondary recrystallization in a thin sheet, it is necessary to prepare a large number of dispersed and nano-sized precipitates.

It is reported that nano-sized sulfides and niobium carbonitrides can precipitate in Fe-Ga alloys and other ferrite steels.^{24} Precipitates with a size of 30-100 nm are considered sufficient to suppress the normal growth of matrix grains effectively.^{23} Therefore, we propose a nano-sized composite inhibitor to effectively inhibit the normal growth of primary grains in this paper. The strength of the pinning force, depending on the quantity and size of precipitates, determines the completeness of secondary recrystallization after final annealing. The effect of alloy compositions and deformation parameters on the thermodynamics and kinetics of precipitation can provide a basis for the regulation of the size and quantity of precipitates.

Based on the relative nucleation rate-temperature (NrT) and precipitation-time-temperature (PTT) curves of MnS and Nb(C,N), a composite inhibitor system, consisting of nano-sized MnS and Nb(C,N) precipitates, is prepared by a reasonably designed composition, hot-rolling temperature and reduction percentage. These nano-sized precipitates provide effectively pinning for the normal growth of the primary grains and induce abnormal grain growth of Goss grains at an annealing temperature of 950∼1,050 °C. Centimeter-sized Goss grains with a smaller deviation angle and maximum magnetostriction coefficient of 264 ppm are successfully achieved in Fe-Ga alloy thin sheets with a thickness of 0.25 mm.

## II. THERMODYNAMICS AND KINETICS CALCULATIONS

### A. Thermodynamics

To calculate the kinetics of precipitation, it is first necessary to analyze the thermodynamics of precipitates. The sulfide phase (MnS) is formed based on the reaction:

Where, [Mn] and [S] are the concentrations of Mn and S in solution, respectively.

The solubility product of MnS in ferrite is:^{24}

The Nb(C,N) is formed by niobium carbide (NbC) and niobium nitride (NbN):

Where, [Nb], [C], and [N] are the concentrations of Nb, C, and N in solution, respectively, and *x* denotes the site fraction of C in carbonitriding. The solubility product of NbC_{x}N_{1-x} is calculated by the solubility product of NbC and NbN. Therefore, the following expressions are obtained:^{24}

If the NbC_{x}N_{1-x} precipitation is assumed to be stoichiometric, then:

Where *A*_{Nb}, *A*_{C}, and *A*_{N} are the molar mass of Nb, C, and N; *Nb*, *C*, and *N* are the given initial chemical composition. According to Equations 6-9, the equilibrium constituent of NbC_{x}N_{1-x} at a given temperature can be determined, and the [*Nb*]_{A}, [*C*]_{A}, [*N*]_{A}, and *x* at a certain annealing temperature *T*_{A} are also determined. The corresponding solid solubility product of NbC_{x}N_{1-x} is:

Where, *A′* and *B′* are the new solid solubility product coefficients according to *x*.

### B. Kinetics

The free energy change (∆*G*_{d}) on forming a spherical nucleus in dislocation lines is:^{25}

Where *d* is the diameter of a nucleus, *σ* is the interfacial energy between precipitation and ferrite, *A* is the core energy of a dislocation line per unit and equals *Gb*^{2}/(4π(1-*v*)), where *G* is elastic shear modulus, *v* is Poisson's ratio, and *b* is the Burgers vector. ∆*G*_{V} is the volume free energy of phase transformation considering the effect of strain storage energy, and is expressed as:^{24,25}

Where *V*_{m} is the molar volume of NbC_{x}N_{1-x}, *R* is the Gas constant, ∆*G*_{S} is strain energy storage, and *A*′ and *B*′ are the solid solubility product coefficients in Equation 10.

By taking the derivative of Equation 11, the critical radius ($dd*$) is:

The relative nucleation rate (*I*_{d}/*K*) of the NbC_{x}N_{1-x} precipitated in the matrix is:^{25}

Where *ρ* is the dislocation density, *K* is a temperature-independent parameter, *Q*_{d} is the diffusion activation energy of Nb along the dislocation line, and *k* is the Boltzmann constant. The relative start time (log(*t*_{0.05}/*t*_{0})) of precipitation is obtained by the Avrami kinetics equation:^{24,26}

Equation 15-16 presents the relative nucleation rate-temperature (NrT) and precipitation-time-temperature (PTT) curves, respectively. According to Equation 14-16, we can calculate the NrT and relative PTT curves of MnS and NbC_{x}N_{1-x} precipitation. The detailed parameters used in Equation 14-16 can be found in Ref. 24.

In addition to generic thermodynamics and kinetics, if a steel has undergone a large amount of plastic deformation, the precipitation will exceed the equilibrium precipitation. This super-equilibrium precipitation phenomenon is caused by the existence of deformation storage energy (∆*G*_{S}). The effect of deformation storage energy can increase the change in the free energy of the system, which is equivalent to increasing the free energy of phase transformation (∆*G*_{V}). Strain energy storage is mainly determined by the temperature (T) and strain ratio (*ε*) during hot rolling, and is approximately expressed as:^{24}

Where *a*, *b* are the related constants. Therefore, the deformation storage energy can be used as a quantitative parameter for the hot rolling process, so that the hot rolling process parameters can be reasonably selected for the effects of deformation storage energy on the kinetics of precipitation. The deformation storage energy has been calculated as 2,500 J/mol, 5,000 J/mol, and 7,500 J/mol corresponding to the hot rolling reduction amounts of 40%, 60%, and 80% in this work, respectively. Using these values, the effect of deformation storage energy on the precipitation kinetics of MnS and Nb(C,N) in ferrite has been accounted for, and the fastest precipitation temperature and maximum nucleation rate temperature of MnS and Nb(C,N) precipitated in Fe-Ga alloy can be estimated for actual rolling conditions.

## III. EXPERIMENTAL

An Fe_{81}Ga_{19} ingot with a nominal chemical composition (in weight %) of 22.65% Ga, 0.01% C, 0.15% Mn, 0.02% S, 0.01% N, 0.10% Nb and balance Fe, was prepared in a vacuum arc furnace using high-purity iron (99.9%) and gallium (99.99%), and Fe-C, Fe-S, Fe-Nb-N, Fe-Mn master alloys. The ingots were heated at 1,230 °C for 30 minutes and hot-rolled to 2.1 mm with a finishing temperature of 800 °C; the total hot-rolling reduction of more than 80% occurred at a temperature of 800∼1,050 °C. The hot-rolled strips were then subjected to warm rolling with a reduction of 65% at 200 °C, intermediate annealing at 950 °C for 10 minutes, and then cold rolled to 0.25 mm with a reduction ratio of 65%. The cold-rolled sheets were annealed at 800 °C for 20 minutes for primary recrystallization, and then heated to 1,100 °C at a rate of 20 °C/h in an atmosphere of pure argon (99.999%). The annealed samples were removed from the furnace at 800 °C, 850 °C, 900 °C, 950 °C, 1,000 °C, and 1,050 °C, respectively, and then quenched.

The microstructure of specimens with different reductions was studied by electron backscatter diffraction (EBSD) using a JEOL 6500F scanning electron microscope. The orientation distribution functions (ODFs) of EBSD are represented by Bunge notation.^{27} Precipitation was studied via a transmission electron microscope (TEM) equipped with energy-dispersive X-ray spectroscopy (EDS). The density and size of the precipitated phases were measured using Digital Micrograph software through TEM photos, and included more than 45 random regions with a size of 3 μm × 4 μm). Magnetostriction values were measured under applied fields from 0 to ± 1,200 Oe using strain gauges positioned along the rolling direction. The saturation magnetostriction coefficient was calculated by (3/2)*λ*_{s}=*λ*_{//}-*λ*_{⊥}, where *λ*_{//} and *λ*_{⊥} are the maximum magnetostriction coefficients measured by the strain gauge method under the applied magnetic field parallel and perpendicular to the rolling direction (RD), respectively. The strain gauges were stuck on the surface of the sheet using adhesive, and the sheets were fixed in the magnetic field by plastic clamps. The magnetostriction value for each processing condition was taken as the average of three samples in order to increase the statistical reliability.

## IV. RESULTS AND DISCUSSION

### A. The thermodynamics and kinetics of precipitation

The effect of deformation energy storage on the NrT and PTT diagrams for MnS and Nb(C,N) precipitation is shown in Figure 1. The NrT curves show inverse “C” shaped curves, while the PTT diagrams show “C” shaped curves. The nose temperatures of MnS and Nb(C,N) in NrT diagrams, corresponding to the maximum nucleation rate temperature, are around 950∼1,050 °C and 750∼850 °C, respectively. The nose temperatures of MnS and Nb(C,N) in PTT diagrams, corresponding to the fastest precipitation temperature, are 950∼1,000 °C and 800∼850 °C, respectively. It can be seen that the nose point temperature, including the fastest precipitation temperature and maximum nucleation rate temperature, significantly increases with deformation energy storage. Increasing the deformation stored energy moves the NrT curve to the upper right, and moves the PPT curve to the upper left. The deformation energy storage of 7,500 J/mol increases the maximum nucleation rate temperature and the fastest precipitation temperature approximately 100 °C for Nb(C,N) in ferrite. It is found that the nose point temperature of MnS and Nb(C,N) precipitated in deformed ferrite is generally between 850 °C and 1,050 °C at a deformation energy storage of 7,500 J/mol. These curves reflect the temperature range of the rapid nucleation of different precipitates, which facilitates the selection of the deformation parameters to achieve the maximum number and minimum size of the precipitated phase.

The increase of the phase transformation driving energy by the deformation energy storage also contributes to the increase of relative nucleation rate. It is found that the relative nucleation rates of MnS and Nb(C,N) with deformation energy storage of 7,500 J/mol are about 1-2 orders of magnitude higher than those with deformation energy storage of 2,500 J/mol. The predicted size of MnS and Nb(C,N) particles (<5 nm) precipitated under higher strain is about 1∼1.5 orders of magnitude smaller than the size (>100 nm) precipitated under lower strain in the temperature range of 800∼1,000 °C. This predicted size does not consider the growth process of the precipitate, although it can be used to characterize the size of precipitates in hot rolled strips without annealing. Further, the modeled precipitates nucleate at homogeneously distributed dislocations due to the increased and uniformly distributed deformation storage energy, and are not affected by the impingement from each other.

### B. The characteristic of precipitates and microstructure

Figure 2 presents the characteristics of the precipitation in hot-rolled strips, as well as in the primarily recrystallized sheets after annealing at 800 °C for 10 minutes. A number of lamellar and rectangular precipitates with a diameter of 5∼20 nm are observed in the hot-rolled strips. According to EDS, the lamellar precipitates with a diameter of 5∼15 nm are mostly identified as Nb(C,N); the spherical and rectangular precipitates with a diameter of 5∼20 nm are identified as MnS. The distribution density of the precipitation is 2.5×10^{9}/cm^{2}. This size of MnS and Nb(C,N) precipitated in hot-rolling is close to the predicted size based on the NrT and PTT curves of MnS and Nb(C,N). The slight increase in the size of the precipitates can be attributed to the growth of the precipitates during the cooling process after hot rolling. After intermediate annealing and primary recrystallization, the precipitates are dispersed in the matrix with an average size of 20∼60 nm, which can inhibit the normal growth of matrix grains effectively.

Zenner factor (1.5*f/r*) is used to represents the inhibiting of normal grain growth by precipitation.^{28} The smaller and higher density of the precipitated phases usually results in a stronger pinning force for the matrix grains. It is calculated that the Zenner factor in primarily recrystallized Fe-Ga thin sheet is about 45/mm, very close to the Zenner factor in grain-oriented silicon steel with perfect secondary recrystallization Goss texture. This indicates that the nano-sized MnS and Nb(C,N) precipitations can provide efficient inhibiting force to the primary grains.^{29} Therefore, the growth of primary grains can be reasonably regulated by the designing of alloy composition and the rolling parameters based on the precipitation thermodynamics and kinetic calculations. The characteristics of precipitation under different hot-rolling conditions, as well as the comparison of precipitation calculations and experimental results, will need to be more systematically studied to provide a more direct and accurate selection of rolling parameters.

### C. The secondary recrystallization texture and magnetostriction

Figure 3 demonstrates the microstructure evolution of primarily recrystallized sheets during high-temperature annealing. It is found that the average grain size increases from 13 μm to 21 μm as temperature increases from 850 °C to 950 °C. Abnormal grain growth occurs at 950 °C and almost consumes the entire matrix by 1,050 °C, reaching several millimeters in size. This fact indicates that the nano-sized composite precipitates can inhibit the normal grain growth of matrix grains and induce abnormal grain growth. According to EBSD results, the abnormal grains at 1,000 °C are identified as Goss with a deviation angle of 10.9° (Figure 4).

The magnetostrictive curves versus applied magnetic field of annealed specimens removed from the furnace at different temperatures are shown in Figure 5. A polycrystalline saturation magnetostriction, (3/2)*λ*_{s}=*λ*_{//}−*λ*_{⊥}, of 78 ppm is obtained at 900 °C, and increases to 198 ppm at 1,000 °C with the appearance of millimeter-sized Goss grains. Maximum magnetostriction, as high as 264 ppm, is achieved under no prestress for the samples after complete secondary recrystallization. *λ*_{//}is a positive magnetostriction of 60 ppm, and *λ*_{⊥} is a negative magnetostriction of −204 ppm. It has been reported that the spontaneously magnetized magnetic domains in the nearly single-crystal samples by complete secondary recrystallization exhibit significant magnetic anisotropy.^{30} The movement of most 180° magnetic domain walls contributes little to the magnetostriction under a parallel magnetic field, but the rotation of the magnetic domains contributes the largest magnetostriction under a perpendicular magnetic field. Therefore, the magnetostriction coefficient is asymmetric after secondary recrystallization, and the larger negative magnetostriction mainly comes from the rotation of magnetic moments previously aligned along the rolling direction during the magnetization process as the magnetic field is applied perpendicular to the rolling direction.^{30}

The magnetostriction coefficient obtained in Fe-Ga sheets with a thickness of 0.25 mm in this paper is much higher than those reported in secondarily recrystallized Fe-Ga-based sheets with thickness less than 0.3 mm induced by both the inhibitors of micron-sized NbC particles and the surface energy effect of sulfur,^{19,20} and close to the cases with a thickness of 0.3-0.5 mm with the help of micron-sized inhibitors and the surface energy effect.^{14–18,31} Therefore, the Goss texture induced by nano-sized composite inhibitors results in much higher magnetostriction in Fe-Ga alloy thin sheets with a thickness of 0.25 mm. In the present study, benefitting from the processing procedure and inhibitor system, the critical requirements are satisfied so that sharp Goss texture is developed by the secondary recrystallization process, without the need for a surface energy modifying annealing atmosphere or element doping.

## V. CONCLUSIONS

In pursuit of the perfect secondary recrystallization of Goss texture in Fe-Ga alloy thin sheets, a nano-sized composite inhibitor system is proposed to achieve complete secondary recrystallization in thin sheets with a thickness of 0.25 mm. The quantity and size of precipitates are regulated by the reasonable design of composition and rolling process parameters based on the effect of stored deformation energy on the thermodynamics and kinetics of precipitation. A large number of nano-sized MnS and Nb(C,N) precipitates were precipitated after hot-rolling at a temperature of 800∼1,050 °C and a total thickness reduction of 80%. The nano-sized inhibitors effectively inhibit the normal growth of the primary grains, and induce abnormal grain growth of Goss grains at an annealing temperature of 950∼1,050 °C. Centimeter-sized Goss grains with a smaller deviation angle and maximum magnetostriction coefficient of 264 ppm are successfully achieved in Fe-Ga alloy thin sheets with a thickness of 0.25 mm. Therefore, this paper proposes a prospective route to enhance high-performance Fe-Ga alloy thin sheets by secondary recrystallization texture, which can greatly reduce the eddy current loss.

## SUPPLEMENTARY MATERIAL

See supplementary material for the raw data of the magnetostriction coefficients of Fe-Ga alloy thin sheet at different annealing temperatures.

## ACKNOWLEDGMENTS

This work was supported by the National Key Research and Development Program of China (2016YFB0300305), the National Natural Science Foundation of China (51671049, 51931002, 52004164), the Inner Mongolia Natural Science Foundation (2018ZD10), the Education department program of Liaoning Province (LQGD2020013), and Open Project of Key Laboratory for Anisotropy and Texture of Materials (Education Ministry of China) Northeastern University (NEU-ATM-2020-01).

## DATA AVAILABILITY

The data that support the findings of this study are available within the article and its supplementary material.