Article pubs.acs.org/JPCC
Modeling the Size- and Shape-Dependent Surface Order−Disorder Transition of Fe0.5Pt0.5 Nanoparticles Yuan Li,† Weihong Qi,*,†,‡,§ Baiyun Huang,‡ and Mingpu Wang†,§ †
School of Materials Science and Engineering and ‡State Key Laboratory of Powder Metallurgy, Central South University, Changsha 410083, People's Republic of China § Key Laboratory of Non-ferrous Materials Science and Engineering, Ministry of Education, Changsha 410083, People's Republic of China ABSTRACT: On the basis of our previous work, the Debye model has been generalized for the size- and shape-dependent order−disorder transition and surface order−disorder transition of FePt nanoparticle (NPs) by considering the configuration entropy. The ordering temperature of FePt NPs decreases with a decrease of the particle size for a specific shape and decreases with a decrease of the shape factor at fixed size. The particle size is the dominant factor while the shape is the secondary one in affecting the transition. The ordering process starts from the surface and then propagates to the core, indicating the order−disorder transition is a surface-dominant process. To induce ordering, the annealing temperature should be lower than the surface ordering temperature, which suggests our calculation can be used to determine the highest annealing temperature. The calculation results also suggest that, at a small size and low temperature, the ordered NPs are stable but, at a large size and high temperature, the disordered NPs are stable. The present predictions agree well with the available literature data.
1. INTRODUCTION The ordered FePt nanoparticles (NPs) with L10 structure, regarded as a candidate material for the next generation of highdensity recording media, have attracted much attention due to their high magnetocrystalline anisotropy.1,2 However, asproduced NPs are typically disordered solid solutions with a face-centered cubic (A1) structure. To recover the ordered structure, an annealing technique must be used.3−6 Due to the high surface to volume ratio, the disordered FePt NPs with A1 structure transform into L10 at a temperature lower than the bulk transition temperature 1573 K, which has been proved experimentally and theoretically.7−10 Nandwana et al.3 annealed 2 nm FePt NPs at 923 K for 1 h to obtain the L10 ordered FePt NPs. Delalande et al.4 chose 773, 923, and 1023 K as the annealing temperatures for 3 nm FePt NPs, 1023 K for 4 nm FePt NPs,5 and 1273 K for 5 nm FePt NPs.6 Unfortunately, there is no model to determine how to choose the annealing temperature, and also no one knows whether there exists a limit for the annealing temperature. Yang et al.7 applied Monte Carlo (MC) simulation to investigate size effects on L10 ordering in FePt NPs, and the estimated transition temperature of 4.77 nm NPs is 1450 K. Muller et al.,8 using lattice Monte Carlo simulation, found that the transition temperature (TC) of 2.5−8.5 nm NPs is 380−400 K, lower than the bulk value. They also found that the depression of TC is roughly 80 K for the 5 nm NPs and 120 K for the 4 nm NPs using the analytic bond order potential.9 Miyazaki et al.10 identified the critical size of the ordering of © XXXX American Chemical Society
FePt NPs as 1.5−2 nm, below which no ordering occurs. There are also many other studies showing similar results for the order−disorder transition of bimetallic NPs.11−14 Experiments indicate that the ordering process of FePt NPs starts from the surfaces and propagates toward the center of the particles.4 Because of the reduced coordination number and surface effect, the ordering transition of FePt NPs is regarded as a continuous transition rather than an abrupt one as in the bulk FePt system.15 Yang et al.15 emphasized the continuous ordering transition process from A1 to L10 to be a surface-induced phenomenon by using Monte Carlo simulation. Muller et al.9 pointed out that, in equilibrium, the ordered core of FePt nanoparticles is surrounded by an outer disordered shell. Yang et al.15 pointed out that this disordered shell leads to a reduction of the magnetocrystalline anisotropy. Also as mentioned, the choosing of the annealing temperature is related to the surface ordering temperature. However, to the best of our knowledge, there is no available model discussing the surface ordering of nanoparticles. Our previous works focused on modeling the thermodynamic stability of pure elements.16,17 In the present paper, we generalize our previous model for pure elements into a bimetallic system aiming at explaining the size dependence of Received: July 2, 2012 Revised: November 15, 2012
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TC of FePt NPs and illustrating the surface ordering. Furthermore, the size-dependent free energy difference, ordering enthalpy, and ordering entropy will also be discussed in detail.
where zs and zb denote the coordination numbers of surface and bulk atoms, respectively. According to the bond order− length−strength (BOLS) model,24 the vibrational amplitude (x) of the surface atoms of an alloy NP is
⎛ z ⎞1/2 xs = ⎜ b ⎟ Cz m /2 + 1 xb ⎝ zs ⎠
2. MODEL Let N and n be the total and the surface atoms of bimetallic NPs with equiatomic composition, respectively. The Helmholtz free energy (HFE) (Fp) of a nanoparticle can be expressed as
where m is a parameter for the nature of the bond. For alloys and compounds m = 4. Cz = 2/{1 + exp[(12 − z)/8z]} = 0.9743.24 The vibrational energy (Esvib) and entropy (Ssvib) of surface atoms are expressed as
s s Fp = NEp + (N − n)(Evib − TSvib) + n(Evib − TSvib )
− TSconf
(1)
where Ep is the atomic cohesive energy of the NP, which can be computed by the bond energy model as in the form18 ⎛ αd ⎞ ⎟ E p = E b ⎜1 − ⎝ D⎠
s Evib
(2)
Evib
⎡ ⎛Θ⎞ ⎤ 3 Svib = 4kB⎢B⎜ ⎟ − ln(1 − e−(Θ / T ))⎥ ⎣ ⎝T ⎠ ⎦ 4
Sconf = kB ln Ωsub 4 = 4kB ln
(N /4)! (pN /4)! ((1 − p)N /4)! (10)
where Ωsub is the number of arrangements for each sublattice and p is the occupancy probability. For an ordered structure, p = 1 and therefore Sconf = 0; for a disordered structure, p = 0.5 and Sconf = NkB ln 2. Presently, only completely ordered and disordered configurations are considered. To discuss the surface properties of bimetallic NPs, we can compute the HFE (Fsp) of surface atoms in the first layer as follows
(3)
(4)
s s s Fps = nEps + n(Evib − TSvib ) − TSconf
(11)
Equation 11 comes from eq 1 by removing the contributions from the interior atoms. Esvib and Ssvib can be computed with eqs 8 and 9, respectively. The configuration entropy (Ssconf) can be computed with eq 10 by replacing N with n, but for mole configuration entropy, Ssconf = Sconf. The cohesive energy of a surface atom can be obtained from eq 2 by considering the effective coordination number,26 i.e., Esp = (Zs/Zb)Ep. We estimate the ratio Zs/Zb to be 0.71, which is the mean value of {100} (equals 2/3) and {111} (equals 3/4). Then the HFE per mole of surface atoms can be written as
(5)
where Θb denotes the Debye temperature of the bulk solid. Sun23 proposed that the lattice vibrational frequency of surface atoms depends on the order (z), length (dz), energy (Ez), and reduced mass [μ = m1m2/(m1 + m2); m1 and m2 denote the atomic masses of the two elements] of the dimer atoms. By introducing the reduced mass μ, the vibrational frequency of alloy atoms can be computed by the formula for pure atoms. Therefore, we get the relation between the vibrational frequency (ω) of alloy surface atoms (denoted by the subscript “s”) and that of the bulk atoms (denoted by the subscript “b”): ωs z = s Cz−(m /2 + 1) ωb zb
(8)
The configuration entropy is defined as Sconf = kB ln Ω, where Ω is the number of arrangements of atoms. For the L10 structure, there are four independent sublattices, and therefore, the entropy is25
where kB and Θ are the Boltzmann constant and Debye temperature, respectively. E0 = 9kBΘ/8 is the zero-point energy, 3 which is usually neglected. B(Θ/T) = 3(T/Θ)3∫ Θ/T 0 (x /(exp(x) − 1)) dx is the Debye function. The Debye temperature and the cohesive energy follow22 Θp/Θb = (Ep/Eb)1/2, and therefore, the size-dependent Debye temperature (Θp) of NPs can be expressed as 1/2 ⎛ αd ⎞ ⎟ Θp = Θ b⎜1 − ⎝ D⎠
⎛ xs ⎞2 = ⎜ ⎟ Evib ⎝ xb ⎠
⎡ ⎛Θ⎞ ⎤ 3 s = Svib + ΔSvib = 4kB⎢B⎜ ⎟ − ln(1 − e−(Θ / T ))⎥ Svib ⎣ ⎝T ⎠ ⎦ 4 ωb + 3kB ln ωs (9)
Eb is the atomic cohesive energy of the corresponding bulk solid. d is the mean atomic diameter of the two components and D the diameter of the NP. α is the shape factor,19 defined as the surface area ratio between nonspherical and spherical NPs of identical volume. For a spherical shape, α = 1, and for a regular tetrahedral shape, α = 1.49. In the range between 1 and 1.49, spherical, cubic, octahedral, cuboctahedral, truncated octahedral, etc. are all included. In the following, α = 1 and α = 1.49, the lower and the upper limits, will be considered. According to the Debye model,20,21 the vibrational lattice energy (Evib) and vibrational entropy (Svib) can be expressed as20 ⎛Θ⎞ = 3kBTB⎜ ⎟ + E0 ⎝T ⎠
(7)
⎛Θ⎞ Fps = 0.71NAEp + 4RTB⎜ ⎟ − 4RT ⎝T ⎠ ⎡ ⎛Θ⎞ ⎤ 3 −(Θ / T ) )⎥ − 0.9RT − TSconf ⎢B⎜⎝ ⎟⎠ − ln(1 − e ⎣ T ⎦ 4 (12)
By replacing N by the Avogadro constant (NA), the HFE per mole of atoms can be obtained as follows according to eq 1 (the relation n/N = αd/D has been used):
(6) B
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⎛ Θp ⎞ Fp = NAEp + 3RT ln(1 − e−(Θp / T )) − RTB⎜ ⎟ ⎝T ⎠ +
⎞ αd ⎛ ⎛ Θp ⎞ RT ⎜⎜B⎜ ⎟ − 0.9⎟⎟ − TSconf D ⎝ ⎝T ⎠ ⎠
(13)
The relation between the Gibbs free energy (G) and HFE is G = F + PV, where the term PV, the product of pressure (P) and volume (V), is very small compared with the HFE. Then we have G ≈ F.
3. RESULTS AND DISCUSSION We calculate the transition temperature of bulk FePt using the Debye model for comparison. In the Debye model, the Debye temperature is an important parameter. Siethoff demonstrated a formula for calculating the Debye temperature of binary alloys.27 For the A1 structure, the Debye temperature is given as Θ0 = Ccs
−1/6
Figure 1. Gibbs free energy as a function of temperature for disordered and ordered FePt bulk solids. The intersection point corresponds to the order−disorder transition temperature. EFe = −413 kJ/mol, and EPt = −564 kJ/mol.31 The bulk formation enthalpies of FePt with A1 and L10 structures are −16.11 eV/atom9 and −27.98 eV/atom,29 respectively.
1/2
(aGc /M )
where Cc is a further constant, s is the number of atoms in the crystallographic unit cell, a is the lattice constant, and M is the atomic weight. The elastic constants can be computed by the elastic modulus (Gc) as Gc = {c44[c44(c11 − c12)/2]1/2 (c11 − c12 + c44)/3}1/3
Gc may be understood as the geometrical average of the (lowenergetic) transverse acoustic phonon modes over the principal lattice directions ([100], [110], and [111]) of a cubic crystal. In the present calculation, Cc = 25.64 K (m kg N−1)1/2 and the elastic constants are taken from refs 28 and 29. The mean value of the calculated Debye temperature of A1-structured FePt bulk solids is 310.6 K. Then for the ordered L10 structure, the Debye temperature is given as Θ0 = C ts−1/6((a 2c)1/3 Gt /M )1/2
Gt = [c44c66(c11 − c12)/2]1/3
Ct here is taken as 25.68 K (m kg N−1)1/2. The elastic constants and lattice constants are taken from refs 9 and 28−30 Then the mean value of the calculated Debye temperature for the L10 structure is 331.3 K. Figure 1 shows the TC for bulk FePt is 1607 K, close to the experimental transition temperature 1573 K, so it is safe to say that the Debye model is valid to estimate the thermodynamics of the FePt order−disorder transition and can be reasonably generalized for alloy NPs on the basis of our previous work.17 Figure 2 shows the comparison between the calculated TC as a function of size in terms of eq 1 and the computer simulation results.7,8,10,15,32 It is found that the model predictions agree well with the computer simulation. TC decreases with a decrease of the particle size, especially at sizes below 2 nm. In the present prediction, the 4 and 5 nm spherical FePt NPs undergo an order−disorder transition at 1453 and 1483 K, matching well with the corresponding values of 1413 and 1453 K,9 respectively. The present estimated TC of 4.77 nm NPs is 1478 K, consistent with the value 1450 K from MC simulation.7 The small discrepancy may be due to the assumption that we simply consider the FePt NPs to be in completely disordered or ordered structures, ignoring the partial ordering case (the ordering parameter is smaller than 1). Alam et al.33 proposed that TC is linear with the square of the order parameter;
Figure 2. Order−disorder transition temperature of FePt nanoparticles as a function of the particle size with spherical and regular tetrahedral shapes. The upper solid and dashed red lines denote the particle ordering, and the lower solid and dashed lines are for the surface ordering temperature of the first layer in FePt nanoparticles. The blue star,7 triangle,8 circle,25 and square32 symbols indicate the simulated particle ordering temperature.
therefore, partial ordering may lead to a decrease of TC. The present variation tendency of TC also agrees with other predictions.11−14 Figure 3 shows the size dependence of the ordering enthalpy and ordering entropy for FePt NPs, while both the ordering enthalpy and ordering entropy decrease with a decrease of the particle size. However, the shape effect is different from the size effect; i.e., the ordering enthalpy increases but the ordering entropy decreases with an increase of the shape factor. The variation of enthalpy and entropy indicates that the ordering process can be regarded as a firstorder phase transition. By neglecting the small term PV, the enthalpy can be replaced by the cohesive energy. According to the general thermodynamics, the entropy is C
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Table 1. Comparison of the Experimental Annealing Temperature and the Present Predicted Surface Ordering Temperature size (nm) 2 1.8−2.2 2.7−3.3 2.5−3.7 3.8−4.2 3.6−4.4 2−7 4 5 4.5−5.5
present predicted surface ordering temp (TSC) (K)
87335 9233 923,3 773,4 923,4 1073,4 97336 107337 8731 9233 10235 90338 1273,6 100039 9233
936−1008 912−1020 992−1065 976−1075 1039−1085 1031−1088 936−1113 1044−1083 1069−1097 1057−1102
the predicted TSC (1069−1097 K). The discrepancy may be from the substrate effect. In ref 6, the FePt NPs were prepared on a carbon substrate. However, the substrate may affect the interface energy and thus increase the TSC. Therefore, the experimental TA may be higher than our predictions for unsupported NPs. The annealing process can be understood as follows: kinetically, if TA < TSC, ordering starts from the surface and propagates toward the center of the NPs, and lengthening the annealing time and increasing the annealing temperature are needed to overcome the increasing ordering barrier. Yang et al.15 identified TSC, where the inner core of the NPs remains an ordered structure while the outer shell begins disordering. According to the present calculation, TSC denotes the order−disorder transition temperature of the outer shell, which is lower than the transition temperature of the inner core (TCC). If TCC > TA > TSC, the surface shell will remain disordered but the inner core will change to an ordered structure at TA, which is just the partial ordering phenomenon explained by the present model. In thermodynamic equilibrium, different layers of the NPs exhibit different ordering degrees. As the present model predicts, the ordering degree at fixed temperature deepens with increasing layer thickness, which matches the report of Muller et al.9 They calculated the first layer of 5 nm FePt using the analytic bond-order potential (ABOP) model, indicating the NPs to be almost completed disordered at 1450 K with long-range order of 0.1. The present predicted TSC of 5 nm FePt NPs ranges from 1069 to 1097 K. Although the quantitative results of both models are different, the qualitative tendency is the same. We also find that there exists a critical size, 1.55 nm, below which no ordering can take place, agreeing with the reported critical size of 1.5 nm.25 Figure 4 shows the Gibbs free energy difference (ΔG) of FePt NPs between A1 and L10 structures as a function of the size and temperature. The plane denotes ΔG = 0. In the upper region of ΔG > 0, the ordered NPs are stable, while in the lower region of ΔG < 0, the disordered NPs are stable. Figure 5 shows that ΔG decreases with increasing temperature and increases with increasing particle size. It is clear that the smaller the particle, the more obviously ΔG changes with the size variation. This fact indicates that the ordering temperature changes more rapidly for small sizes (
