Effect of grain-boundary diffusion process on the geometry of the grain microstructure of Nd$-$Fe$-$B nanocrystalline magnets

TL;DR

This study investigates how the grain-boundary diffusion process alters the microstructure of Nd-Fe-B nanocrystalline magnets, revealing a transition from platelet to elongated particle shapes and increased coercivity.

Contribution

It provides detailed neutron scattering analysis showing the microstructural changes induced by GBDP in Nd-Fe-B magnets, highlighting the shape evolution of nanocrystals.

Findings

Change from platelet to elongated particle shape after GBDP

Smoothing of grain surfaces observed

Increase in magnetic coercivity after GBDP

Abstract

Hot-deformed anisotropic Nd$-$Fe$-$B nanocrystalline magnets have been subjected to the grain-boundary diffusion process (GBDP) using a $\mathrm{Pr}_{70}\mathrm{Cu}_{30}$ eutectic alloy. The resulting grain microstructure, consisting of shape-anisotropic Nd$-$Fe$-$B nanocrystals surrounded by a Pr$-$Cu-rich intergranular grain-boundary phase, has been investigated using unpolarized small-angle neutron scattering (SANS) and very small-angle neutron scattering (VSANS). The neutron data have been analyzed using the generalized Guinier-Porod model and by computing model-independently the distance distribution function. We find that the GBDP results in a change of the geometry of the scattering particles:~In the small-$q$ regime the scattering from the as-prepared sample exhibits a slope of about $2$, which is characteristic for the scattering from two-dimensional platelet-shaped objects, while the GBDP sample manifests a slope of about $1$, which is the scattering signature of one-dimensional elongated objects. The evolution of the Porod exponent indicates the smoothing of the grain surfaces due to the GBDP, which is accompanied by an increase of the coercivity.

Tables

Table 1. Table 1: Results of a weighted nonlinear least-squares fit using the generalized Guinier-Porod model [Eqs. ( 2 ) − - ( 5 )]. The values for the scaling parameters G 𝐺 G and D 𝐷 D are not listed; q 1 subscript 𝑞 1 q_{1} is not a fit parameter, but computed according to Eq. ( 4 ).

Sample	$s$	$R_G$ ($\mathrm{nm}$)	$n$	$q_1$ (${\mathrm{nm}}^{-1}$)
PrCu0	$1.80\pm 0.02$	$27.7\pm 0.9$	$3.53\pm 0.01$	$0.0369$
PrCu20	$1.60\pm 0.01$	$19.6\pm 0.3$	$4.06\pm 0.01$	$0.0669$
PrCu40	$0.94\pm 0.02$	$37.1\pm 0.5$	$4.09\pm 0.01$	$0.0485$

Equations

$$\frac{d\Sigma }{d\Omega }(\mathbf{q})=\frac{8{\pi }^3}{V}{b}_H^2(b_H^{-2}|\tilde{N}{|}^2+|{\tilde{M}}_x{|}^2+|{\tilde{M}}_y{|}^2{\cos }^2\theta$$

$$+|{\tilde{M}}_z{|}^2{\sin }^2\theta -({\tilde{M}}_y{\tilde{M}}_z^{\ast }+{\tilde{M}}_y^{\ast }{\tilde{M}}_z)\sin \theta \cos \theta ),$$

$$I(q)$$

$$q_1$$

$$D$$

$$p(r)=r^2{\int }_0^{\infty }\frac{d\Sigma }{d\Omega }(q)j_0(qr)q^{2}dq,$$

Full text

Nd$-$Fe$-$B; permanent magnets; neutron scattering; small-angle neutron scattering

I Introduction

Nd$-$Fe$-$B based nanocrystalline permanent magnets are of potential interest for electronic devices, motors, and wind turbines due to their preeminent magnetic properties such as high coercivity and magnetic energy product Coehoorn et al. (1988); Davies et al. (1989); Gutfleisch et al. (2002, 2011); Liu (2009); Bance et al. (2014). In this context, the effect of the grain-boundary diffusion process (GBDP) on the bulk magnetic microstructure of hot-deformed Nd$-$Fe$-$B based nanomagnets is currently extensively investigated (e.g., Ref. Liu et al. (2013); Sepehri-Amin et al. (2013); Liu et al. (2014); Sepehri-Amin et al. (2015); Salazar et al. (2018)). In the GBDP Sepehri-Amin et al. (2010), the Nd$-$Fe$-$B magnet is exposed at elevated temperatures to a fine powder or a vapor containing high-magnetic-anisotropy-inducing heavy-rare-earth elements such as Tb or Dy, which then diffuse (preferentially along liquid grain boundaries) into the bulk of the material, in this way locally increasing the coercivity.

The goal of the present work is to obtain microscopic information about the nanoscale structure of Pr$-$Cu infiltrated $\mathrm{Nd}_{2}\mathrm{Fe}_{14}\mathrm{B}$ nanocrystalline alloys by means of unpolarized small-angle neutron scattering (SANS) and very small-angle neutron scattering (VSANS). The SANS technique (see Ref. Mühlbauer et al. (2019) for a recent review) is ideally suited for monitoring the changing grain-boundary chemistry due to the GBDP Ueno et al. (2014); Yano et al. (2014); Saito et al. (2015); Périgo et al. (2016), since it provides—in contrast to electron-microscopy methods—statistically-averaged bulk information about both the structural and magnetic correlations on a mesoscopic length scale ($\sim 1-1000\,\mathrm{nm}$). This method has previously been applied to study the structures of magnetic nanoparticles Disch et al. (2012); Günther et al. (2014); Bender et al. (2015, 2018a, 2018b); Oberdick et al. (2018); Ijiri et al. (2019), soft magnetic nanocomposites Ito et al. (2007); Saranu et al. (2008), proton domains van den Brandt et al. (2006); Aswal et al. (2008); Noda et al. (2016), magnetic steels Bischof et al. (2007); Bergner et al. (2013); Pareja et al. (2015); Shu et al. (2018), or Heusler-type alloys Bhatti et al. (2012); Runov et al. (2006); Benacchio et al. (2019). Here, we aim to correlate the changes in the grain microstructure due to the GBDP with the coercivity.

II Experimental

We have studied a Pr$-$Cu-doped series of hot-deformed Nd$-$Fe$-$B nanocrystalline magnets. The specimens were prepared by the melt-spinning technique. The resulting melt-spun ribbons were crushed into powders of a few hundred micrometer sizes and then subsequently sintered at $1048\,\mathrm{K}$ under a pressure of $400\,\mathrm{MPa}$. The hot-pressing procedure results in the formation of shape anisotropic $\mathrm{Nd}_{2}\mathrm{Fe}_{14}\mathrm{B}$ grains which are stacked along the nominal $c$-axis (see the sketch of the microstructure in Fig. 1). The Pr$-$Cu-doped series contains an undoped reference sample and two samples which are doped with 20 wt.% and 40 wt.% of Pr70Cu30. In the following these samples are, respectively, referred to as PrCu0, PrCu20, and PrCu40. More details on the sample preparation can be found in Refs. Liu et al. (2013); Sepehri-Amin et al. (2013); Liu et al. (2014); Sepehri-Amin et al. (2015). Magnetization data were taken on a $14\,\mathrm{T}$ vibrating sample magnetometer.

The neutron experiment has been carried out at $300\,\mathrm{K}$ at the instrument SANS-I at the Paul Scherrer Institute (PSI), Switzerland, using unpolarized neutrons with a mean wavelength of $\lambda=5.0\,\mathrm{\AA}$ and $\Delta\lambda/\lambda=10\,\mathrm{\%}$ (FWHM) Kohlbrecher and Wagner (2000); Niketic et al. (2015). The external magnetic field $\mathbf{H}_{0}$ was applied perpendicular to the wave vector $\mathbf{k}_{0}$ of the incoming neutron beam ($\mathbf{k}_{0}\perp\mathbf{H}_{0}$); see Fig. 1 for a sketch of the neutron setup. This corresponds to the geometry where $\mathbf{H}_{0}$ is parallel to the nominal $c$-axis (pressing direction) of the textured samples. Neutron data were corrected for background scattering (empty sample holder), transmission, and detector efficiency using the GRASP software package Dewhurst . The measured neutron transmission of all samples was larger than $80-90\,\mathrm{\%}$ at all fields investigated. The SANS setup at PSI allowed us to access the following range of momentum transfers: $0.02\,\mathrm{nm}^{-1}\lesssim q\lesssim 1.0\,\mathrm{nm}^{-1}$. To significantly reduce the minimum momentum-transfer value to $q_{\mathrm{min}}\cong 0.003\,\mathrm{nm}^{-1}$, additional unpolarized runs were carried out at the very small-angle neutron scattering (VSANS) instrument KWS-3 operated by the Jülich Centre for Neutron Science (JCNS) at the Heinz Maier-Leibnitz Zentrum (MLZ), Garching, Germany Pipich and Fu (2015). In the VSANS experiments we used a mean wavelength of $\lambda=12.8\,\mathrm{\AA}$ [$\Delta\lambda/\lambda=16\,\mathrm{\%}$ (FWHM)] and a sample-to-detector distance of $9.8\,\mathrm{m}$.

III SANS cross section and correlation function

For $\mathbf{k}_{0}\perp\mathbf{H}_{0}$, the unpolarized elastic nuclear and magnetic differential SANS cross section $d\Sigma/d\Omega$ at momentum-transfer vector $\mathbf{q}$ reads Mühlbauer et al. (2019):

[TABLE]

where $V$ denotes the scattering volume, $b_{H}=2.91\times 10^{8}\,\mathrm{A^{-1}m^{-1}}$, $\widetilde{N}(\mathbf{q})$ represents the nuclear scattering amplitude, $\widetilde{\mathbf{M}}(\mathbf{q})=\{\widetilde{M}_{x}(\mathbf{q}),\widetilde{M}_{y}(\mathbf{q}),\widetilde{M}_{z}(\mathbf{q})\}$ is the Fourier transform of the magnetization $\mathbf{M}(\mathbf{r})=\{M_{x}(\mathbf{r}),M_{y}(\mathbf{r}),M_{z}(\mathbf{r})\}$, the asterisk “$\,{}^{\ast}\,$” marks the complex-conjugated quantity, and $\theta$ is the angle between $\mathbf{H}_{0}$ and the scattering vector $\mathbf{q}$.

In the literature on small-angle scattering there exist many approaches to analyzing anisotropic scattering patterns (see, e.g., Refs. Summerfield and Mildner (1983); Mildner (1983); Reynolds and Mildner (1984); Hammouda et al. (1986a, b); Saraf (1989); Svetogorsky (1990); Gu and Mildner (2016, 2018) and references therein). Here, the SANS data were analyzed in terms of the generalized Guinier-Porod model, which has been developed by Hammouda Hammouda (2010) in order to describe the $2\pi$ azimuthally-averaged scattering from nonspherical objects (such as rods or lamellae). The model is purely empirical and essentially decomposes the $I(q)=\frac{d\Sigma}{d\Omega}(q)$ curve into a Guinier region for $q\leq q_{1}$ and into a Porod region for $q\geq q_{1}$. Both parts of the scattering curve are then joined by demanding the continuity of the Guinier and Porod laws (and of their derivatives) at $q_{1}$; more specifically Hammouda (2010),

[TABLE]

where the scaling factors $G$ and $D$, the Guinier radius $R_{G}$, the dimensionality factor $s$, and the Porod power-law exponent $n$ are taken as independent parameters. From the continuity of the Guinier and Porod functions and their derivatives it follows that:

[TABLE]

where $n>s$ and $s<3$ must be satisfied. Note that $q_{1}$ is not a fitting parameter, but internally computed via Eq. (4).

The generalized Guinier-Porod model is commonly applied to orientationally-averaged microstructures. In the Supplemental Material to this paper smt we demonstrate that it can also be used to describe the small-angle scattering from oriented particles: the $2\pi$-averaged one-dimensional SANS cross sections from oriented cylinders and ellipsoids are similar to the corresponding SANS cross sections of the randomly oriented ensembles. Moreover, it is shown that synthetic data on oriented cylinders and ellipsoids with a distribution of sizes can be described by the generalized Guinier-Porod model.

In addition to the above analysis using the generalized Guinier-Porod model, we have model-independently calculated the distance distribution function Bender et al. (2017)

[TABLE]

where $j_{0}(z)=\sin(z)/z$ denotes the zeroth-order spherical Bessel function. This provides information on the characteristics (e.g., size and shape) of the scattering objects Svergun and Koch (2003); Fritz and Glatter (2006), and on the presence of interparticle correlations Lang and Glatter (1996); Fritz-Popovski et al. (2011).

IV Results and discussion

Figure 2 displays the effect of Pr$-$Cu infiltration on the room-temperature magnetic hysteresis of hot-deformed Nd$-$Fe$-$B. As expected and in agreement with data in the literature Liu et al. (2013); Sepehri-Amin et al. (2013); Liu et al. (2014); Sepehri-Amin et al. (2015), the coercivity $H_{c}$ increases significantly with increasing Pr$-$Cu content (from $\mu_{0}H_{c}=1.39\,\mathrm{T}$ for PrCu0 to $\mu_{0}H_{c}=2.03\,\mathrm{T}$ for PrCu20 to $\mu_{0}H_{c}=2.14\,\mathrm{T}$ for PrCu40) at the cost of lowering the saturation magnetization and the remanence.

The two-dimensional VSANS data are displayed in Fig. 3, whereas the one-dimensional (over $2\pi$ azimuthally-averaged) data are shown in Fig. 4. From the observation in Fig. 4(a) that the scattering curves at the remanent state and at an applied field of $\mu_{0}H_{0}=2.2\,\mathrm{T}$ (dashed lines) are basically identical, it follows that $d\Sigma/d\Omega$ is dominated by the field-independent nuclear scattering, i.e., $d\Sigma/d\Omega\propto|\widetilde{N}|^{2}$. Regarding the two-dimensional patterns in Fig. 3 this then implies that the origin of the pronounced angular anisotropy is related to an anisotropic (oriented) grain microstructure, which in turn is in line with the results of electron-microscopy investigations Liu et al. (2013); Sepehri-Amin et al. (2013); Liu et al. (2014); Sepehri-Amin et al. (2015); Michels et al. (2017). With increasing Pr$-$Cu content we see, in particular at the small momentum transfers $q$, that the two-dimensional $d\Sigma/d\Omega$ increase their horizontal and decrease their vertical elongation. This finding points towards a change of the geometry of the scattering objects due to the GBDP. While all the samples exhibit a similar power-law decay at the large $q$-values (essentially with a Porod exponent of $n\cong 4$, see below), there is a pronounced change in the slope of the scattering with doping in the small $q$-regime [compare Fig. 4(a)]. Such changes in the slope of $d\Sigma/d\Omega$ within the small-$q$ Guinier or the intermediate-$q$ Guinier regime indicate a change in the dimensionality of the scattering objects Hammouda (2010).

In order to gain further information on the changes in the geometry of the scattering particles, we have analyzed the SANS cross sections in the remanent state using the generalized Guinier-Porod model [Eqs. (2)$-$(5)]. The solid lines in Fig. 4(b) are the result of this analysis and the fitting parameters are listed in Table 1. The most remarkable observation is the change in the $s$-parameter, from $1.80$ for PrCu0 to $0.94$ for PrCu40. This suggests that the elements of the microstructure which give rise to $d\Sigma/d\Omega$ change their geometry (shape) Hammouda (2010): from two-dimensional platelet-shaped, corresponding to $s=2$, to one-dimensional rod-like ($s=1$). The change in $s$ is also observed when averages of $d\Sigma/d\Omega$ along the horizontal (easy $c$-axis) and perpendicular (hard axis) directions are taken. The effective Guinier radius varies between $R_{G}=27.7\,\mathrm{nm}$ (PrCu0), $R_{G}=19.6\,\mathrm{nm}$ (PrCu20), and $R_{G}=37.1\,\mathrm{nm}$ (PrCu40). This quantity is a measure for the smallest dimension of the scatterers, which sensitively depends on the particle-size distribution and on the particle shape, e.g., $R_{G}=R/\sqrt{2}$ for the cross section of a randomly oriented cylinder with radius $R$, while $R_{G}=T/\sqrt{12}$ for the cross section of a randomly oriented lamella with thickness $T$ Feigin and Svergun (1987); Hammouda (2010). Therefore, in view of this sensitivity, we cannot relate the variation of $R_{G}$ found from the fit to the actual changes in the real-space dimensions of the particles (see also the discussion in smt ). But the $R_{G}$-values are within the expected size range, as seen by electron microscopy Liu et al. (2013); Sepehri-Amin et al. (2013); Liu et al. (2014); Sepehri-Amin et al. (2015); Michels et al. (2017). The Porod exponent $n$ increases from $n=3.53$ for PrCu0 to $n\cong 4.1$ for the infiltrated samples. This indicates a smoothing of the surface of the particles as a result of the Pr$-$Cu infiltration Hammouda (2010).

The results for the distance distribution function $p(r)$ in Fig. 4(c) are consistent with the numerical fit analysis using the generalized Guinier-Porod model. The PrCu0 and the PrCu20 samples both exhibit a $p(r)$ which is typical for globular, but slightly anisotropic particles, whereas the $p(r)$ of the PrCu40 sample shows a maximum at small $r\cong 100\,\mathrm{nm}$ followed by a long tail at the larger $r$, suggesting that the scattering originates from shape-anisotropic elongated objects (compare Fig. 5 in the review by Svergun and Koch (2003)). The maximum of $p(r)$ at the small distances of the PrCu40 specimen corresponds to the shortest dimension of the particles. This behavior is consistent with the previously observed dimensional crossover ($s\cong 2\rightarrow s\cong 1$) using the Guinier-Porod model.

V Conclusion

Unpolarized small-angle neutron scattering (SANS) and very small-angle neutron scattering (VSANS) were used to monitor the effect of the grain-boundary diffusion process (GBDP) on the mesoscopic grain microstructure of Nd$-$Fe$-$B nanocrystalline magnets. The SANS data, which are predominantly of nuclear origin, reveal a pronounced effect of the GBDP, resulting in a change of the slope of $d\Sigma/d\Omega$ with increasing doping at small momentum transfers. Analysis of the scattering data in terms of the generalized Guinier-Porod model suggests that the GBDP results in a dimensional crossover of the geometry of the scattering objects: The SANS data from the as-prepared specimen are characteristic for two-dimensional platelet-shaped objects, while the doped samples manifest the signature of one-dimensional rod-like objects. This assessment is further supported by an independent model-free analysis in terms of the distance distribution. Moreover, based on the evolution of the Porod exponent, we find an indication for the smoothing of the grain surfaces due to the GBDP, which goes along with an increase of the coercivity.

Acknowledgment

Philipp Bender and Andreas Michels acknowledge financial support from the National Research Fund of Luxembourg (CORE SANS4NCC grant). Kotaro Saito has received funding from the European Union’s Horizon 2020 research and innovation programme under the Marie Skłodowska-Curie grant agreement No. 701647. This paper is based on results obtained from the future pioneering program “Development of magnetic material technology for high-efficiency motors” commissioned by the New Energy and Industrial Technology Development Organization (NEDO). The neutron experiments were performed at the Swiss spallation neutron source SINQ, Paul Scherrer Institute, Villigen, Switzerland and at the Heinz Maier-Leibnitz Zentrum, Garching, Germany.