Thermodynamic and experimental study on phase stability in nanocrystalline alloys

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Acta Materialia 58 (2010) 396–407 www.elsevier.com/locate/actamat

Thermodynamic and experimental study on phase stability in nanocrystalline alloys Wenwu Xu, Xiaoyan Song *, Nianduan Lu, Chuan Huang College of Materials Science and Engineering, Key Laboratory of Advanced Functional Materials, Ministry of Education of China, Beijing University of Technology, Beijing 100124, China Received 3 August 2009; received in revised form 4 September 2009; accepted 7 September 2009 Available online 12 October 2009

Abstract Nanocrystalline alloys exhibit apparently different phase transformation characteristics in comparison to the conventional polycrystalline alloys. The special phase stability and phase transformation behavior, as well as the essential mechanisms of the nanocrystalline alloys, were described quantitatively in a nanothermodynamic point of view. By introducing the relationship between the excess volume at the grain boundary and the nanograin size, the Gibbs free energy was determined distinctly as a function of temperature and the nanograin size. Accordingly, the grain-size-dependence of the phase stability and phase transformation characteristics of the nanocrystalline alloy were calculated systematically, and the correlations between the phase constitution, the phase transformation temperature and the critical nanograin size were predicted. A series of experiments was performed to investigate the phase transformations at room temperature and high temperatures using the nanocrystalline Sm2Co17 alloy as an example. The phase constitution and phase transformation sequence found in nanocrystalline Sm2Co17 alloys with various grain-size levels agree well with the calculations by the nanothermodynamic model. Ó 2009 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved. Keywords: Nanocrystalline materials; Thermodynamics; Phase transformations; Grain boundary energy; Grain boundary structure

1. Introduction In contrast to the coarse-grained polycrystalline alloys, nanocrystalline alloys have a much larger volume fraction of interfaces, e.g. grain boundaries, phase boundaries, and domain interfaces. Owing to their special structures, nanocrystalline alloys consisting of either single phase or multiphase exhibit distinctly different physical, chemical, and mechanical properties from those of the coarse-grained alloys with the same composition [1–3]. However, once the nanograin size increases to be larger than a critical value, the nanostructure may evolve unstably, and abrupt changes in thermal stability and phase stability of nanocrystalline alloys can occur [4]. As a result, the advanced properties of the nanomaterial may deteriorate due to the destabilization of the nanostructure. *

Corresponding author. Tel./fax: +86 10 67392311. E-mail address: [email protected] (X. Song).

In recent years, phase transformations induced by the change of grain size have been reported in a variety of nanocrystalline alloys synthesized by different methods, e.g. Fe–Ni [5], Mo–Si [6], Ti–Al [7], Fe–Cr–Ni and Fe– Mn [8], Cd–Se [9], and Sm–Co [10]. A typical example is that the room-temperature phase in the nanocrystalline Ni–Ti shape memory alloy prepared by high-pressure torsion transforms in a sequence of B2-austenite ? Rphase ? B190 -martensite with the increase of the nanograin size [11]. It shows that the grain-size effect plays a crucial role in the phase transformation behavior of the nanocrystalline alloy at a constant temperature. In our early work [4], it was proposed that the thermal stability and phase stability of nanomaterials are dominated by the energy and structure states of the materials. From the thermodynamic point of view, the disordered arrangement of the atoms at nanograin boundaries influences the configurational and vibrational entropies and also the enthalpy of the nanocrystalline material [12], hence the

1359-6454/$36.00 Ó 2009 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved. doi:10.1016/j.actamat.2009.09.016

W. Xu et al. / Acta Materialia 58 (2010) 396–407

fraction of nanograin boundary atoms has a significant effect on the Gibbs free energy of the nanocrystalline material. Therefore, it is of special significance to study quantitatively the effects of nanograin size (which is directly connected with the fraction of grain boundary atoms) on the energy state hence on the phase transformation characteristics of the nanocrystalline material. Due to the fact that nanocrystalline materials contain a large volume fraction of nanograin boundaries, the thermodynamic properties, particularly the Gibbs free energy, cannot be accurately described by conventional thermodynamic models in which the fundamental functions were derived neglecting internal boundaries, which certainly applies for coarse-grained polycrystalline alloys. In order to understand the thermodynamic properties and hence the phase transformation behavior of nanocrystalline alloys more profoundly, it is very necessary to develop the nanoscale thermodynamic model. As early as 1963, Hill pointed out in his book [13,14] that “the thermodynamic equations are no longer the same for small systems and macroscopic environments”. However, at that time due to the lack of experiments on the sufficiently small scale of the materials, the development of thermodynamics for small systems was very limited. With the emergence of nanomaterials and the rapid progress in the nanoscale characterization, a variety of theoretical approaches [15–17] and some important thermodynamicsrelated models have been developed for nanocrystalline materials [18,19]. In respect that “the study of sufficiently small systems at equilibrium requires a modification of ordinary macroscopic thermodynamics [20]”, the concept “nanothermodynamics” was proposed by Hill in his recent publications [20–22]. The nanothermodynamic principles have attracted increasing interest in the field of nanomaterials, although the nanothermodynamic descriptions of the phase stability and phase transformation behavior of nanocrystalline materials have been rarely reported in the literature. In this paper, we focus on the development of a nanothermodynamic model that describes the stability and transformation characteristics of phases in the nanocrystalline alloy, so as to understand the special mechanisms of phase transformation in nanocrystalline alloys with respect to the coarse-grained polycrystalline alloys. This paper is structured as follows: in Section 2, the development of a nanothermodynamic model for nanocrystalline alloys is introduced, in particular the correlation between the nanograin size and the excess volume of the nanograin boundary is for the first time deduced; in Section 3, the phase stability and phase transformation behavior of nanocrystalline alloy are predicted by the nanothermodynamic calculations using Sm2Co17 alloy as an example, moreover, the mechanism of grain-size induced phase transformation is proposed; in Section 4, the phase transformation behaviors of nanocrystalline Sm2Co17 alloy are investigated experimentally, and the verification of the nanothermodynamic model is presented.

397

2. Thermodynamic model for nanocrystalline alloy 2.1. Grain-size-dependence of nanograin boundary structure Nanocrystalline alloys can be considered as materials constructed by two kinds of components: the region of nanograin boundary and the nanoscale crystal (the nanograin interior) [23]. Atoms at nanograin boundaries and inside nanocrystals have different effects on the thermodynamic properties of the nanocrystalline alloy due to their different structure and energy states. To describe the structural characteristics of nanograin boundaries, Wagner [19] and Fecht [18] proposed a “dilated crystal” model, in which an important concept of “excess volume” at nanograin boundary DV was introduced to characterize the structural features of nanograin boundaries, which was defined as [19] DV ¼ ðV b =V 0 Þ  1

ð1Þ

where Vb and V0 are the primitive cell volumes at nanograin boundary and in nanograin interior, respectively. Apparently, DV characterizes the deviation of the nanograin boundary structure with respect to the perfect crystal [19]. The excess volume DV, which is directly related to the nanograin size d, is the most significant intrinsic parameter that affects the thermal properties of nanograin boundaries. Although it is recognized that the excess volume DV increases with the decrease of the nanograin size d [4,18,19], no quantitatively precise relationship between d and DV has been reported so far. In recent years, some models for estimating this relationship have been developed. The simplest approximation proposed by Chattopadhyay et al. [24] is 2

DV ¼

ðd 2D þ h=2Þ  d 22D d 2D

ð2Þ

In their model, DV equals the fraction of the excess surface area of the grain boundary, where d2D is the diameter of the equivalent circular area of the grain and h is the thickness of the grain boundary. Song [4] suggested that DV can be better estimated if the three-dimensional grain size, e.g. the diameter of the equivalent spherical volume, d, is used. For the space-filling nanograin structure, when the share of interface between neighboring nanograins is taken into account, the excess volume at the boundary of a nanograin can be expressed as h i. d 3  ðd  hÞ3 qb ð3Þ DV ¼ d 3 =qi where qi and qb are the spatial distribution densities of atoms in the nanograin interior and at the nanograin boundary, respectively. From the report that the atomic density at nanocrystalline interfaces is lower than that in the perfect crystal by 10–30%, the value qb/qi = 0.8 was used in early work [4]. However, it is not precise to take qb/qi as a constant, and the deviation may be increasingly larger with the decrease of nanograin size. Hence, a

W. Xu et al. / Acta Materialia 58 (2010) 396–407

function in Eq. (4) is proposed here to describe the change of qb/qi with the nanograin size: qb =qi ¼ A þ Bd C

ð4Þ

where A, B and C are the fitting coefficients. Once three pairs of values of DV and d are found, the relationship between qb/qi and nanograin size can be obtained by fitting, i.e. determining the parameters of A, B and C. A simple method is proposed here to deduce the fitting coefficients in Eq. (4). For a three-dimensional spherical nanograin with a diameter as d and the grain boundary thickness as h, the core crystal in the nanograin interior has a size of d  h. h is assumed to be equivalent to four atom layers and to remain the same when the nanograin size changes or the phase transformation takes place. The volume fraction of the nanograin boundary, fb, can be estimated as  2 , h ð5Þ d3 fb ¼ 3h d  2 The fraction of atoms at the nanograin boundary, xb, is then given by xb ¼

fb 1 þ DV  DVfb

ð6Þ

Thus, the molar volume of the nanocrystalline alloy, V nc mol , is expressed as V nc mol ¼ N A ½ð1  xb ÞV 0 þ xb V b 

ð7Þ

NA is the Avogadro constant. From another point of view, the molar volume of a nanocrystalline alloy can be calculated by the following equation [25], using the lattice constants that can be obtained from the X-ray diffraction (XRD) measurements on the prepared nanocrystalline sample: V nc mol ¼ N A

V CE Z

ð8Þ

Phase

Average nanograin size d (nm) (±102–101)

Lattice constants (nm) (±105–104)

Excess volume DV (±104–103)

a

c

b-Sm2Co17 b-Sm2Co17 a-Sm2Co17

15 25 80

0.87171 0.85362 0.83889

0.79347 0.80125 1.22312

0.3113 0.1743 0.0452

room temperature instead of a rhombohedral Th2Zn17-type crystal structure (a-Sm2Co17), which is well known as the stable room-temperature phase in the conventional polycrystalline Sm2Co17 alloys. Since the nanostructured single-phase Sm2Co17 is the basis for the nanocrystalline Sm2Co17-type magnets of various compositions [26,27], the understanding of phase transformation characteristics in nanocrystalline Sm2Co17 alloy will facilitate the development of nanocrystalline Sm2Co17-type alloys with superior magnetic properties for applications at room temperature and high temperatures. Based on the three pairs of values of DV and d as listed in Table 1, the fitting coefficients A, B and C in Eq. (4) can be determined. Therefore, the excess volume DV as a function of the nanograin size d in nanocrystalline Sm2Co17 alloy is deduced as h i 3 , d 3  ðd  hÞ DV ðdÞ ¼ ð9Þ ð0:3 þ 0:68d 0:14 Þ d3 In respect that for the extremely small grain sizes (e.g. d < 10 nm), the thickness of the nanograin boundary and the lattice distortion in the nanograin interior [28] may deviate obviously from the assumptions in the model, we propose that Eq. (9) is applicable to the nanograin sizes larger than 10 nm.

1.0

1.0 present

0.8

0.9

literature [4]

0.14

ρb/ρi=-0.30+0.68d

0.8

0.6

ρb/ρi

where VCE ispthe ffiffiffi unit cell volume (e.g. for hexagonal crystals, V CE ¼ 3 3a2 c=2, a and c are the lattice constants), Z is the number of atoms per unit cell. By combining the XRD analyses and the transmission electron microscopy (TEM) observations of the prepared nanocrystalline sample, the nanograin size of the alloy can be obtained. Then coupling Eqs. (3)–(7) with Eq. (8), one can calculate the excess volume corresponding to a given nanograin size. Based on this approach, we obtained three values of the excess volume corresponding to different average nanograin sizes of three samples of the prepared nanocrystalline Sm2Co17 alloy; the results are listed in Table 1. The reason for using the nanocrystalline Sm2Co17 as the example to study the phase stability and phase transformation behavior of nanocrystalline alloys is as follows. In our previous work [10], we discovered an abnormal phase stability in the nanocrystalline Sm2Co17 alloy, i.e. a hexagonal Th2Ni17-type crystal structure (b-Sm2Co17) exists stably at

Table 1 The measured average nanograin sizes and the lattice constants and the obtained excess volumes of the prepared nanocrystalline Sm2Co17 alloy samples.

ΔV

398

0.7

0.4 + experimental

0.6

0.2 0.0 0

20

40

60

80

0.5 100

d (nm) Fig. 1. The deduced relationship between the excess volume DV and the nanograin size d for the nanocrystalline Sm2Co17 alloy, with the relative distribution density of atoms at the nanograin boundary with respect to that in the nanograin interior qb/qi as a function of the nanograin size d.

W. Xu et al. / Acta Materialia 58 (2010) 396–407

The above relationship between excess volume and nanograin size is demonstrated in Fig. 1, in comparison with the function in Eq. (3) with qb/qi = 0.8 as used in the literature [4]. It can be seen that the deviation becomes larger when the nanograin size decreases to below approximately 30 nm. Therefore, it is more precise to introduce the dependence of qb/qi on the nanograin size. 2.2. Nanoscale thermodynamic functions Taking into account the relationship between the nanograin size d and the excess volume DV, the primitive cell volume Vb at the nanograin boundary, as a function of nanograin size, can be expressed as h i 1 0 3 d 3  ðd  hÞ =qb A ð10Þ V b ðdÞ ¼ V 0 @1 þ d 3 =qi Thus, the fundamental thermodynamic functions of nanograin boundaries, i.e. the excess enthalpy Hb, the excess entropy Sb, and the excess Gibbs free energy Gb per atom (subscript b denotes the nanograin boundary) are given as functions of the nanograin size d and the absolute temperature T [18]: H b ðd; T Þ ¼ EðdÞ þ P ðd; T ÞV b

ð11Þ

S b ðd; T Þ ¼ C V ðd; T Þcðd; T Þ ln½V b ðdÞ=V 0  ð12Þ Gb ðd; T Þ ¼ H b ðd; T Þ þ C V ðd; T ÞðT  T R Þ  T ½S b ðd; T Þ þ C V ðd; T Þ lnðT  T R Þ

ð13Þ

where  E is the binding energy of atoms at nanograin boundaries [29,30];  P is a negative pressure generated at nanograin boundaries due to the increase of excess volume with the decrease of nanograin size [31];  CV is the specific-heat capacity of nanograin boundaries at constant volume [18,19];  c is the Gru¨neisen parameter of the dilated crystal at nanograin boundaries [18,19];  Vb and V0 are the volumes of the Wigner–Seitz primitive cells [32,33]. Vb = 4prb3/3, V0 = 4pr03/3, rb and r0 are the Wigner–Seitz radii at the nanograin boundary and in the nanograin interior, respectively. For a binary alloy, the primitive cell volume can be taken as the sum of fractions of each constituent, i.e. V0 = x1V1 + x2V2, where x1 and x2 (equal to 1  x1) are the mole fractions of the two constituents, V1 and V2 are the corresponding primitive cell volumes of the constituents.  TR is the reference temperature. More details concerning the thermodynamic functions of nanograin boundaries were presented in our previous work [4,34]. Considering the different effects of atoms at nanograin boundary and in nanograin interior on the ther-

399

modynamic properties of nanocrystalline alloy, we introduce the atomic fraction at nanograin boundaries xb in the expressions of thermodynamic functions of the nanocrystalline alloy: H ðd; T Þ ¼ N A xb H b ðd; T Þ þ ð1  xb ÞH i ðT Þ

ð14Þ

Sðd; T Þ ¼ N A xb S b ðd; T Þ þ ð1  xb ÞS i ðT Þ

ð15Þ

Gðd; T Þ ¼ N A xb Gb ðd; T Þ þ ð1  xb ÞGi ðT Þ

ð16Þ

where subscript i represents the nanograin interior. The thermodynamic functions Hi, Si and Gi can be obtained from the known database, e.g. SGTE [35]. By the above model, the thermodynamic properties, as well as the phase stability and phase transformation characteristics of nanocrystalline alloys as a function of nanograin size and temperature, can be studied quantitatively. 3. Thermodynamic calculations and predictions 3.1. Input parameters for thermodynamic calculations To apply the nanothermodynamic model described in Section 2 to the nanocrystalline Sm2Co17 alloy, as a first step, the input parameters for thermodynamic calculations should be determined. The resources and the evaluation accesses for the related parameters are given as follows, and the obtained data are listed in Table 2. In the Sm2Co17 alloy, the parameters used in the calculations for a phase, including the equilibrium Wigner–Seitz radius r0, the volume expansion coefficient a0, the Debye temperature H0 and the bulk elastic modulus B0 at the reference temperature 298 K, were described in Ref. [34]. For the b-Sm2Co17 phase, the equilibrium Wigner–Seitz radius can be obtained by r0 = [3V0/(4p)]1/3, where V0 is the Wigner–Seitz cell volume of the alloy. The volume expansion coefficient of b phase can be estimated according to the thermal expansion limit equation of solids (Eq. (17)) proposed by Gru¨neisen [36]. T m a0 ¼

V T m  V 0K ¼C V 0K

ð17Þ

where Tm is the melting point, V T m and V0K are the volumes of the alloy at temperatures of melting point and 0 K, respectively. C is a constant related to the crystal structure of the alloy and has a value of 0.06–0.076 for the cubic and hexagonal crystals.

Table 2 Input parameters at the reference temperature (TR = 298 K) in thermodynamic calculations for hexagonal (b) and rhombohedral (a) phases of nanocrystalline Sm2Co17 alloy. Phase

Wigner–Seitz radius r0 (nm)

Volume expansion coefficient a0 (106 K1)

Debye temperature H0 (K)

Bulk elastic modulus B0 (GPa)

a-Sm2Co17 b-Sm2Co17

0.1461 0.1460

38.98 47.12

332 331

120.53 110.00

400

W. Xu et al. / Acta Materialia 58 (2010) 396–407

The Debye temperature of b-Sm2Co17 alloy can be obtained from Lindemann formula [36], H0 = 137.0{Tm/ [M(NAV0)2/3]}1/2, where M is the atomic mass. The bulk elastic modulus of b-Sm2Co17 alloy can be estimated from the Debye temperature using Eq. (18) [37,38]: B0 ¼

!2

2pk B H0 1=6 kðvÞhð48p5 Þ

M r0

As shown in Fig. 2a and b, the Gibbs free energies of b- and a-Sm2Co17 phases increase with the decrease of nanograin size at given temperatures, this implies that the degree of stability of b- and a-Sm2Co17 phases increases with decreasing the nanograin size at a constant temperature. Especially, when the grain size decreases to a few tens of nanometers, the degree of stability of b and a phases reduce obviously. Moreover, the relative phase stability between b and a phases at the give temperature changes as well, or in other words, phase transformation between b- and a-Sm2Co17 may take place. As indicated in Fig. 2a and b, the critical grain sizes for b M a phase transformations in the nanocrystalline Sm2Co17 at 300 K and 1000 K are 29.9 nm and 40.6 nm, respectively. It is shown that the phase stability and phase transformation in the nanocrystalline alloy depend on not only the temperature but also the grain size, which is the distinctly different characteristic of nanocrystalline alloys with respect to the coarse-grained polycrystalline alloys. Furthermore, the phase transformation temperature decreases obviously with the decrease of nanograin size, e.g. the temperature of phase transformation between b- and a-Sm2Co17 is reduced from 1460 K to 300 K when the nanograin size decreases from 95 nm to 30 nm (see Fig. 2c). It is worth noting that when the nanograin size decreases to below 30 nm (e.g. 15 nm, as indicated in Fig. 2c), the Sm2Co17 alloy will keep the hexagonal structure (b) in spite of the

ð18Þ

where h and kB are the Planck’s and Boltzmann’s constants, respectively. k(v) is a function of Poisson ratio v [38]. 3.2. Dependence of Gibbs free energy on temperature and nanograin size Based on the model calculations, the thermodynamic properties of nanocrystalline Sm2Co17 can be characterized, hence the phase stability and the tendency of phase transformation in the alloy can be predicted by the criterion of the minimum Gibbs free energy. The calculated Gibbs free energies of the two phases of b- and a-Sm2Co17, as a function of nanograin size at different temperatures, are shown in Fig. 2a and b. Fig. 2c shows the calculated Gibbs free energy difference of the two phases as a function of temperature at different nanograin sizes. -100

-283 -284

-250

29

30

29.9 nm

31

-1003.6

-900 -1004.0 40.4

20

40.8

-1000 α-Sm2Co17

0

40.6

40.6 nm

β -Sm2Co17

-300

β-Sm2Co17

-800

G (kJ/mol)

G (kJ/mol)

-1003.2

-282

-200

α-Sm2Co17

(b) T = 1000 K

β-Sm2Co17

-281

-150

-700

α-Sm2Co17

(a) T = 300 K

40

β -Sm2Co17

60

80

0

100

20

60

80

100

d (nm)

d (nm) 4

α-Sm2Co17

(c)

0

95 nm 75 nm 55 nm 35 nm 30 nm

β -Sm2Co17

-4

600

900

1200

1460 K

1387 K

-12 300

1223 K

-8 695 K

ΔGβ-α (kJ/mol)

α-Sm2Co17

40

1500

15 nm

1800

T (K) Fig. 2. Calculated Gibbs free energies of nanocrystalline b- and a-Sm2Co17 phases as a function of nanograin size at different temperatures of: (a) T = 300 K; (b) T = 1000 K, the insets showing local enlargement at the intersection points and (c) showing the calculated Gibbs free energy difference between the two phases as a function of temperature at different nanograin sizes.

W. Xu et al. / Acta Materialia 58 (2010) 396–407

3.3. Predictions on grain-size-dependence of phase stability 3.3.1. Room-temperature phase stability Based on the calculation results in Fig. 2a, the stability of different phases at room temperature in the nanocrystalline Sm2Co17 alloy can be studied. Fig. 2a indicates that at room temperature, the Gibbs free energies of both nanocrystalline b- and a-Sm2Co17 phases increase with the decrease of the nanograin size. The two Gibbs free energy curves intersect at a grain size of about 29.9 nm, indicating a critical state of phase transformation between b- and a-Sm2Co17. This means that the ultrafine nanocrystalline Sm2Co17 alloy (with the grain size smaller than 29.9 nm) will have a hexagonal structure at room temperature, while the coarser nanocrystalline (with the grain size larger than 29.9 nm) or the submicro-scaled and polycrystalline Sm2Co17 alloys will have a rhombohedral crystal structure at room temperature. In contrast to the conventional polycrystalline Sm2Co17 alloys which have the hexagonal crystal structure only at temperatures above 1520 K (which is the a-Sm2Co17 M b-Sm2Co17 transformation temperature at equilibrium state in the polycrystalline system), in the nanocrystalline alloy the b-Sm2Co17 phase can exist stably at room temperature when the grain size is reduced to below the critical value. Therefore, the nanocrystalline alloy exhibits distinctly different phase stability and phase transformation characteristics at room temperature as compared with the conventional polycrystalline alloy with the same composition. 3.3.2. High-temperature phase stability In order to study the phase stability and phase transformation in nanocrystalline Sm2Co17 alloy at high temperatures, we calculated the grain-size-dependence of the phase transformation temperature between b- and a-Sm2Co17 using the nanothermodynamic model, the results are shown in Fig. 3.

As seen from Fig. 3, the temperature of phase transformation between nanocrystalline b- and a-Sm2Co17 reduces with the decrease of nanograin size. Particularly, when the nanograin size is lower than approximately 50 nm, there is a great reduction of the phase transformation temperature in the nanocrystalline Sm2Co17 alloy. It is considered that when the grain size is smaller than 50 nm, with further decrease of the grain size, the volume fraction of nanograin boundaries increases drastically, thus the nanograin-size effect on the phase transformation temperature becomes remarkable. 3.4. Mechanism of phase transformation in nanocrystalline alloy A large number of experimental investigations show that phase transformations in nanocrystalline alloys are closely related to the nanograin size [5–11]. Nevertheless, there is no clear elaboration on the grain-size effect on phase transformation behavior of nanocrystalline alloys. In this work, we proposed a thermodynamic approach to describe the grain-size-dependence of phase transformation in nanocrystalline alloys (Section 2). This nanothermodynamic model demonstrated the mechanism for the abnormal phase stability discovered in nanocrystalline alloys. The schematic diagrams for allotropic phase transformation in a nanocrystalline alloy with different levels of nanograin size and certain grain size distributions is shown in Fig. 4. The a and b phases represent the low- and hightemperature phases in the coarse-grained alloy system. In the nanocrystalline alloy, there may exist different phase constitutions at different grain-size levels when the temperature is kept as a constant. When the sizes of all the grains in the nanocrystalline alloy are smaller than a critical value, the phase (e.g. b phase) whose Gibbs free energy (as can be calculated by the present nanothermodynamic model) is lower than the other is the single phase in the alloy (see Fig. 4a). In the nanocrystalline alloy with a grain size distribution, which has some grains larger than the critical value while the rest smaller than the critical value, the Polycrystalline Sm2Co17

1500

1520 K

1200

T (K)

changes of the temperature, due to that the Gibbs free energy of b phase is always lower than that of a phase. According to the thermodynamic analyses in the case of nanocrystalline Sm2Co17 alloy, it is shown that the degree of phase stability, as well as the phase transformation tendency, is dependent distinctly on the grain size of the nanocrystalline alloy. With the decrease of the nanograin size, especially when below a few tens of nanometers, the volume fraction of the nanograin boundaries is significantly increased. The nanograin boundary atoms, with the disordered or short-range ordered arrangement, play an increasingly important role in the changes of the entropy, enthalpy, and the free energy of the nanocrystalline alloy with the decrease of the grain size. Therefore, the thermodynamic properties of the nanocrystalline alloy are modulated by the variations of the nanograin size, i.e. the nanograin size affects the structure and energy states of the nanograin boundaries.

401

Nanocrystalline Sm2Co17

900

β -Sm2Co17

α-Sm2Co17

600

300 30

40

50

60

70

80

90

100

Critical grain size (nm) Fig. 3. Predicted grain-size-dependence of phase transformation temperature between nanocrystalline b-Sm2Co17 and a-Sm2Co17.

402

W. Xu et al. / Acta Materialia 58 (2010) 396–407

Fig. 4. Schematic diagrams for allotropic phase transformation in a nanocrystalline alloy with different levels of nanograin size and certain grain size distributions: (a) all the grain sizes smaller than the critical value; (b) some grain sizes larger than the critical value while the rest smaller than this value and (c) all the grain sizes larger than the critical value.

333R 244R 045R 600R

223R 303R

(c)

045R 600R

412H

223R 303R

113R

404R 413R 324R

220R 214R

α-Sm2Co17 β -Sm2Co17

220H 004H

600R 333R

306R 413R 045R

214R 223R

300R

412H

332H

300H

222H

(a)

220R 303R

113R 211R

021R

300R

Intensity (arb. units)

112H

302H

α-Sm2Co17 β -Sm2Co17

Firstly, the coarse-grained polycrystalline Sm2Co17 ingot was prepared by induction–melting Sm (99.99%) and Co (99.99%) in a vacuum furnace. The ingot was annealed for 24 h at 1320 °C in argon gas to obtain the homogeneous single Sm2Co17 phase, as identified by the X-ray diffraction (XRD) pattern shown in Fig. 5. Then the Sm2Co17 ingot was crushed into coarse powder, and sealed in a vial in the glove box filled with argon gas. The powder was subjected to the high-energy ball milling, from which the amorphous Sm–Co alloy powder was obtained. The concurrent consolidation and crystallization of the amorphous powder were carried out by the spark plasma sintering (SPS) technique at a temperature of about 1000 K in a system filled with highly purified argon gas. To obtain different levels of the average grain size, the as-SPSed sample was cut into three parts. Two of them were sealed in the quartz tubes filled with argon gas then annealed for 1 h at the temperatures of 973 K and 1073 K, respectively, and subsequently were cooled down naturally to room temperature. Through this procedure

112H

To verify the thermodynamic predictions on the phase stability and phase transformation behavior in nanocrystalline Sm2Co17 alloy, we prepared the nanocrystalline Sm2Co17 bulk samples with different average grain sizes.

4.1. Phase stability at room temperature

122R 300R

4. Experimental investigations

The phase transformations were investigated experimentally at room-temperature (Section 4.1) and high-temperature (Section 4.2) states for the nanocrystalline Sm2Co17 alloy.

Intensity (arb. units)

b and a phases will coexist in the microstructure (see Fig. 4b). When the sizes of all the grains are larger than the critical value, the Gibbs free energy of a phase will become lower than that of b phase, leading to a single a phase in the nanocrystalline alloy (see Fig. 4c). As described in the nanothermodynamic model (Section 2), the individual grains have their own grain-size-dependent energy state. Thus, an individual grain may have a different phase transformation tendency with respect to another grain having a different grain size. As marked in Fig. 4, a nanograin having the b structure (Fig. 4a) transforms to a structure (Fig. 4b) in the process of nanograin growth, and keeps the a structure with larger nanograin size (Fig. 4c). It is obvious that the allotropic phase transformation takes place along with the nanograin growth, which is dominated by the change of Gibbs free energy of different phases, as a function of nanograin size and temperature. This is one of the most significant phase transformation characteristics of nanocrystalline alloys with respect to the coarse-grained alloys.

(b)

ingot

20

30

40

50

60

2θ (deg.)

70

80

90

20

30

40

50

60

70

80

90

2θ (deg.)

Fig. 5. XRD patterns of Sm2Co17 samples with different treatments: the coarse-grained ingot: (a) as-SPSed sample; (b) annealed at 973 K for 1 h using the as-SPSed sample as the starting material and (c) annealed at 1073 K for 1 h using the as-SPSed sample as the starting material.

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the room-temperature phases with different nanograin sizes in the samples were obtained. The phase constitutions in the above three samples were detected by the XRD method, and the microstructures of the three samples were observed by transmission electron microscopy (TEM); the results are shown in Figs. 5 and 6, respectively. As shown by curve (a) in Fig. 5, together with the analysis by the selection area electron diffraction pattern (SADP) shown in Fig. 6a, the as-SPSed sample has a single Sm2Co17 phase with the hexagonal Th2Ni17-type crystal structure (b-Sm2Co17) at room temperature. By combing the estimation using the modified Scherer formula [39] based on the XRD data with the measurements on the TEM images (as an example shown in Fig. 6a), the average grain size of the as-SPSed Sm2Co17 sample was obtained as about 15 nm. The phase constitutions in the two annealed samples were determined by combining the XRD analysis and SADP indexing. Since the Scherer formula is not accurate to estimate the average grain size in a material containing multiple phases, instead, the linear intercept method was used to measure the grain size of the double-phase Sm2Co17 sample. It is found that the sample annealed at 973 K has the coexisting b- and a-Sm2Co17 phases at room temperature (as indicated by curve (b) in Fig. 5 and the SADP and its indexing in Fig. 6b), and the average grain size is about 45 nm. The sample annealed at 1073 K has

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the single phase of a-Sm2Co17 at room temperature (as indicated by curve (c) in Fig. 5 and the SADP and its indexing in Fig. 6c), and the average grain size is about 80 nm. As shown by the experimental results, at room temperature, the as-SPSed nanocrystalline Sm2Co17 with an average grain size of 15 nm has the single b-Sm2Co17 phase. Applying the newly developed method of characterizing the nanograin size distribution [40], we found that the asSPSed nanocrystalline Sm2Co17 sample has a grain size distribution in a range of 5–25 nm, as shown in Fig. 7a. This experimental finding validates the model prediction in Section 3.3.1, i.e. when the grain size is smaller than the critical value of 29.9 nm, the nanocrystalline Sm2Co17 has a stable b phase at room temperature instead of the a phase as in the conventional polycrystalline Sm2Co17 alloys. In the experiment, the nanocrystalline Sm2Co17 with an average grain size of about 45 nm and a grain size distribution of 15–80 nm (as shown in Fig. 7b, a relatively higher fraction of grains in a range of 20–40 nm) consists of b- and a-Sm2Co17 phases at room temperature. As referred to the model calculations (see Fig. 2a), while the nanograins with the grain size smaller than the critical value of 29.9 nm have a hexagonal structure, the grains larger than 29.9 nm have a rhombohedral structure. The appearance of the coexisting b- and a-Sm2Co17 phases is attributed to the nanograin size distribution, hence its effect on

Fig. 6. TEM images and SADPs of nanocrystalline Sm2Co17 samples with different treatments: (a) as-SPSed sample; (b) annealed at 973 K for 1 h using the as-SPSed sample as the starting material and (c) annealed at 1073 K for 1 h using the as-SPSed sample as the starting material.

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(a)

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Grain size (nm) Fig. 7. Grain size distributions of Sm2Co17 samples with different treatments: (a) as-SPSed sample; (b) annealed at 973 K for 1 h using the as-SPSed sample as the starting material; (c) annealed at 1073 K for 1 h using the as-SPSed sample as the starting material.

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In view of that the Sm2Co17-type magnet materials are expected to be able to operate continuously at high temperatures (up to about 800 K [41]), it is very important to study the phase stability and phase transformation behavior in nanocrystalline Sm2Co17 alloy at a high temperature. In our experiment, we examined the high-temperature phase stability in nanocrystalline Sm2Co17 alloy at 773 K and 873 K, respectively. The predictions of phase stability

2

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4.2. Phase stability at high temperature

in nanocrystalline Sm2Co17 at temperatures of 773 K and 873 K by the nanothermodynamic model are shown in Fig. 8. It is calculated that the critical grain sizes of phase transformation between b- and a-Sm2Co17 are 36 nm and 39 nm corresponding to 773 K and 873 K, respectively. Theoretically, at 773 K when the grain size is smaller than 36 nm, the grains in nanocrystalline Sm2Co17 alloy will have a single b-Sm2Co17 phase. While at 873 K, when the

ΔGβ-α (kJ/mol)

the phase stability in the nanocrystalline Sm2Co17 alloy. When the average grain size increases to 80 nm with a grain size distribution of 50–115 nm (as shown in Fig. 7c), the Sm2Co17 alloy has the single a-Sm2Co17 phase at room temperature. This confirms the thermodynamic calculations on the room-temperature phase stability in the nanocrystalline Sm2Co17 alloy with the grain size larger than the critical value. Therefore, the experimental results on the room-temperature phase constitutions in the three samples with different grain-size levels agree well with the model predictions demonstrated in Section 3.3.1. It is reasonably concluded that the nanograin growth due to the annealing treatment causes the changes in the energy and structure states of the nanocrystalline phases, hence results in the change of the phase stability in a coarser nanograin structure at room temperature.

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T (K) Fig. 8. Model predictions on phase transformation between nanocrystalline b-Sm2Co17 and a-Sm2Co17 at temperatures of 773 K and 873 K. Solid lines and dashed lines correspond to the states with critical grain sizes and the given grain sizes at the two temperatures.

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2θ (deg.) Fig. 9. XRD patterns of nanocrystalline Sm2Co17 samples annealed at different temperatures then quenched in liquid nitrogen: (a) T = 773 K and (b) T = 873 K.

grain size is larger than 39 nm, the grains will reveal the aSm2Co17 phase. In the experimental work, to keep the high-temperature phase structure in the nanocrystalline Sm2Co17 alloy, the samples were subjected to quenching treatment. Two samples of the as-SPSed nanocrystalline Sm2Co17 alloy were sealed in the quartz tubes filled with argon gas and annealed for half an hour at the temperatures of 773 K and 873 K, respectively. Subsequently, the annealed samples were quenched into the liquid nitrogen. The phase constitutions and the microstructures of the quenched samples were detected by XRD and TEM; the results are shown in Figs. 9 and 10. The grain size distributions in the quenched samples are shown in Fig. 11. It is found that the quenched sample from annealing at 773 K has an average grain size of about 25 nm and a grain size distribution ranging from 15 to 35 nm (Fig. 11a). This sample exhibits a single b-Sm2Co17 phase, as indicated by curve (a) in Fig. 9 and the SADP and its indexing in Fig. 10a. While in the quenched sample from annealing at 873 K, the average grain size is about 35 nm and the grain size distribution is in a range from 15 to 55 nm (Fig. 11b). This sample has the coexisting b- and a-Sm2Co17 phases, as indicated by curve (b) in Fig. 9 and the SADP and its indexing in Fig. 10b.

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Apparently, the actual phase constitution depends on the level of average grain size and grain size distribution. It can be seen that the experimental findings agree well with the thermodynamic predictions on the high-temperature phase stability in the nanocrystalline Sm2Co17 alloy. It should be noted that the hysteresis phenomenon, as well as the possible sources of barrier that prevent the phase transformation from taking place, has not been taken into account in the nanothermodynamic model. However, as the experimentally obtained correspondence of temperature, average nanograin size and phase constitution agrees quantitatively with the model calculations, it is reasonable to consider that the phase transformation hysteresis in nanocrystalline alloys is very low, and can be neglected with respect to that in the coarse-grained polycrystalline materials. 5. Conclusions In this paper, we developed a nanothermodynamic model to describe quantitatively the phase stability and phase transformation in nanocrystalline alloys. The nanocrystalline Sm2Co17 alloy was used as an example to apply the model for thermodynamic calculations and predictions. A series of experiments on preparation and characterization of the nanocrystalline Sm2Co17 alloys with various average grain-size levels were carried out to verify the nanothermodynamic model. The conclusions are drawn as follows: (1) The quantitative relationship between the excess volume at the nanograin boundary and the grain size was deduced, thus the dependence of the fundamental thermodynamic functions (the enthalpy, entropy and Gibbs free energy) on the temperature and the nanograin size was unambiguously determined for the nanocrystalline alloy. (2) The nanothermodynamic calculations indicate that, owing to the enhanced grain-size effect on the Gibbs free energy, the nanocrystalline alloy exhibits distinctly different characteristics of phase stability and

Fig. 10. TEM images, the corresponding SADPs and its indexing of nanocrystalline Sm2Co17 samples annealed at different temperatures then quenched in liquid nitrogen: (a) T = 773 K and (b) T = 873 K.

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Fig. 11. Grain size distributions of nanocrystalline Sm2Co17 samples annealed at different temperatures then quenched in liquid nitrogen: (a) T = 773 K and (b) T = 873 K.

phase transformation as compared with the coarsegrained polycrystalline alloys. It reveals that the phase transformation tendency in nanocrystalline alloys is dominated by the critical nanograin size at a constant temperature. (3) A theoretical quantitative relationship that the phase transformation temperature reduces with the decrease of the nanograin size was obtained. However, since there actually exist grain size distributions in the nanocrystalline alloys, the phase constitution of single phase or multiphase is dependent on the range of grain size distribution with respect to the critical nanograin size. (4) It is found in experiments that at room temperature, with the increase of the nanograin size, the phase transformation in the nanocrystalline Sm2Co17 alloy takes place in the sequence b ? b + a ? a. At the high temperature, the critical nanograin size becomes larger than that at room temperature, and the phase constitution in the nanocrystalline Sm2Co17 alloy changes from the single b-Sm2Co17 (T = 773 K) to the coexisting b- and a-Sm2Co17 phases (T = 873 K) in the testing temperature range. The experimental results of the phase constitutions corresponding to certain grain size distributions validate the predictions from the nanothermodynamic model. Acknowledgements This work was supported by the National Natural Science Foundation of China (Grant No. 50671001 & 50871001), the Program for New Century Excellent Talents in University (NCET-2006-0182), and the Doctorate Foundation of Chinese Education Ministry (20070005010). References [1] Sasaki TT, Ohkubo T, Hono K. Acta Mater 2009;57:3529. [2] Qu S, An XH, Yang HJ, Huang CX, Yang G, Zang QS, et al. Acta Mater 2009;57:1586.

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