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Estimation of critical cooling rates for formation of amorphous alloys from critical sizes
Journal of Non-Crystalline Solids 358 (2012) 2753–2758
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Estimation of critical cooling rates for formation of amorphous alloys from critical sizes Sheng Guo a,⁎, Yong Liu b a b
Department of MBE, City University of Hong Kong, Kowloon, Hong Kong, PR China State Key Laboratory for Powder Metallurgy, Central South University, Changsha, Hunan, 410083, PR China
a r t i c l e
i n f o
Article history: Received 29 February 2012 Received in revised form 19 June 2012 Available online 24 July 2012 Keywords: Metallic glasses; Aluminum alloys; Cooling rate; Heat transfer
a b s t r a c t The critical cooling rate for marginal glass formers, like Al-based alloys, is difficult to measure experimentally. In this work, we acquired the critical cooling rate for formation of amorphous alloys by estimating the cooling rates of the specimens with critical dimensions. Analytical solutions were given to estimate the cooling rates for the gas-atomized powders and melt-spun ribbons, and as an example the critical cooling rate to form an amorphous Al82Ni10Y8 alloy was estimated to be ~1.0 × 10 6 K s−1. The effect of melt temperature on the cooling rate was quantitatively evaluated and its effect on the glass forming ability was also discussed. © 2012 Elsevier B.V. All rights reserved.
1. Introduction Amorphous alloys, or metallic glasses have been developed rapidly and studied extensively during the past 30 years [1–3], due to their many unique properties over the crystalline counterparts. A large variety of amorphous alloy systems has been developed so far, including Zr-, Fe-, Pd-, Pt-, La-, Cu-, Mg-, Co-, Ni- and Ti-based alloys [4]. However, the preparation of bulk (typically above 1 mm in size) Al-based metallic glasses has been proved very difficult [5–9] and a breakthrough in the achievable size is yet to be reached. Considering the wide applicability of Al alloys and their light weight, the capability to make bulk Al-based metallic glasses is very attractive. The difficulty of making bulk Al-based metallic glasses is due to the fact that the critical cooling rate, Rc, to obtain amorphous Al alloys is fairly high, and such a demanding cooling rate limits the achievable sample size that can be fully amorphous in structure. To advance the development of bulk amorphous alloy with marginal glass forming ability (GFA), our understanding to the GFA and particularly to the improvement of the GFA need to be advanced greatly. In the meantime, the characterization of the GFA of Al alloys is also an issue. The critical cooling rate, which is the unambiguous criterion defining the GFA of various metallic glasses, can be experimentally measured for easy glass formers but not for marginal glass formers like Al alloys [10]. However, the critical size (Dc) of the amorphous Al alloys can be experimentally determined. If we can calculate the cooling rates (R) from these critical sizes, we can then get the critical cooling rates for Al alloys indirectly. Unfortunately, to the knowledge of the ⁎ Corresponding author. Tel.: +852 34428416; fax: +852 34420172. E-mail address: [email protected] (S. Guo). 0022-3093/$ – see front matter © 2012 Elsevier B.V. All rights reserved. doi:10.1016/j.jnoncrysol.2012.06.023
authors there lacks such an obvious link between the cooling rate and sample size in the literature, or at least not in a convenient way. In this work, we prepared amorphous Al82Ni10Y8 alloys in both ribbon and powder forms, and estimated their cooling rates by using simplified heat transfer models. The target alloy was chosen because it is a simple ternary alloy system and the alloying components well satisfy the three empirical rules proposed by Inoue to possess high GFA [3]. The melt treatment has been known to affect the GFA [11–15], but how the melt temperature affects the cooling rate is not quantitatively studied previously, and this issue is to be addressed here as well.
2. Material and methods The ribbon and powder specimens used in this work were prepared by melt spinning and gas atomization methods, respectively. Starting materials were pure Al (99.995 mass%), Ni (99.9 mass%) and Y (99.99 mass%). The master alloy was prepared by mixing the pure metals in a molar composition of Al82Ni10Y8 using arc melting (for melt spinning) or induction melting (for gas atomization) in the high-purity argon atmosphere. Repeated melting was carried out at least five times to ensure the chemical homogeneity of the alloy. For the gas atomization process, a close-coupled nozzle was used to promote the efficient atomization of the melt [16]. The master alloy was first remelted at ~ 1473 K in an alumina crucible in a vacuum of 10 −2 Pa. The melt was teemed through a guide tube, and then atomized by a jet of argon at ~ 3 MPa. The atomized powder was collected and sieved in a closed system filled with nitrogen. The ribbons were prepared by remelting the master alloy in a quart tube in argon
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and then ejecting the molten liquid through a nozzle onto a rotating Cu wheel with a rotation speed up to 2700 rotation per minute (rpm). The surface morphology of the powders with different size was observed using scanning electron microscopy (SEM). The amorphicity of ribbons was identified by X-ray diffraction (XRD) using Cu-Kα radiation.
3. Results The surface morphology of the gas atomized powder is shown in Fig. 1. It can be seen that the powders are almost perfectly spherical in shape (although a few ellipsoidal powders were also seen, not shown here). Those particles with diameter smaller than ~ 10 μm (Fig. 1a) are featureless in the microstructure, possibly having an amorphous structure; while in those particles with larger diameter hence slower cooling rate (~ 30 μm in Fig. 1b) dendrites appear, suggesting they are already crystalline. The XRD patterns for melt spun ribbons with thickness of 25 μm and 50 μm are shown in Fig. 2. The ribbons prepared using a faster Cu wheel rotation speed are thinner in thickness. Both ribbons are identified to be XRD amorphous. Compared to the critical powder size, the determination of the critical ribbon thickness to achieve the fully amorphous state is more time consuming, as the rotation speed has to be varied continuously. We will show later that the critical ribbon thickness is estimated to be ~ 85 μm for the Al82Ni10Y8 alloy.
(a)
Intensity, I/cps
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1500 rpm, 50 μm
2700 rpm, 25 μm
10°
20°
30°
40°
50°
60°
70°
80°
90°
Angle, 2θ Fig. 2. XRD patterns of the melt-spun Al82Ni10Y8 alloy ribbons with different thickness.
4. Calculation In this section, we will model the heat transfer during the gas atomization and melt spinning processes, and estimate the cooling rates for these two processes based on the models, from intrinsic properties of the materials and also properties of the cooling media (argon gas or the Cu wheel). The heat transfer for the gas atomization process is relatively more straightforward and we deal with it first. For the melt spinning process, the heat transfer is more complicated and we made many simplifications on it. We want to emphasize here that these treatments actually over-simplify the real heat transfer in the melt spinning process and the mathematical solution is hence not robust. However, the estimated critical cooling rates for the melt spinning process, out of apparently simplified treatments, agree well with that for the gas atomization process. It is hence our belief that the scheme we put forward here is still of practical value, to reasonably estimate the cooling rates for the melt spun ribbons. 4.1. Cooling rate calculation for the gas-atomized powders In the gas atomization process, the high-speed gas flow impinges on the liquid flow ejected from the nozzle and the liquid shatters into small melt drops. The melt drops release heat through their surfaces during the flight in the chamber, while the gas in the surrounding environment cools the melt drops via the convective heat transfer. A few assumptions are made here for the heat transfer process:
2 μm
(b)
1) The melt drops reach the thermal equilibrium state instantly. As such, their internal temperature could be assumed homogeneous; 2) The surrounding environment is assumed to be an infinite space. The heat released from the melt drops diffuses to the surrounding environment rapidly, and the environmental temperature is assumed to be constant. 3) Since there exists a huge difference between the melt drops and the environment, the heat transfer is assumed to be completely interface controlled and the cooling follows the Newtonian heat transfer mode. The thermal equilibrium condition is thus the heat release from the melt drops equaling to the heat conducted to the environment via the melt drop surfaces. Mathematically, it can be written as [16]: dT d hAðT d −T e Þ ¼− ρVC p dt
ð1Þ
5 μm Fig. 1. Surface morphology of the gas-atomized Al82Ni10Y8 alloy powders with different particle size. (a) b10 μm and (b) ~30 μm.
where V is the volume (m 3); ρ is the density (kg m −3); Cp is the specific heat capacity (J kg −1 K −1); A is the surface area of the melt drop (m 2); h is the coefficient of heat transfer of the gas (W m −2 K −1); Td
S. Guo, Y. Liu / Journal of Non-Crystalline Solids 358 (2012) 2753–2758
and Te are melt drop temperature (K) and environment temperature (K), respectively; and t is the time (s). For simplification, a spherical shape with a diameter of d (m) is assumed for the melt drop. This assumption is also a close reflection of the real powder shape, as shown in Fig. 1. Eq. (1) then becomes: dT d 6hðT d −T e Þ ¼− : ρC p d dt
ð2Þ
The critical parameter for calculating the cooling rate is the heat transfer coefficient, h. Between the inert gas and the molten drops, h has the form of [17,18] h¼
kg 1=2 1=3 2 þ 0:6Re P r d
ð3Þ
where kg is the thermal conductivity of the gas (W m −1 K −1), Re is the Reynolds number and Pr is the Prandtl number. Re can be further expressed as [19]
Re ¼
vd −vg ρg d
ð4Þ
μg
where vd and vg are the velocities for the melt drop and the gas (m s −1), respectively, ρg is the density of the gas (kg m−3) and μg is the kinetic viscosity of the gas (kg m−1 s−1). From Eqs. (2), (3) and (4) we have 0 1 11=2 0 dT d 6ðT d −T e Þ B2kg kg ρg vd −vg 1=3 A Pr C @ 2 þ 0:6 @ A: dt ¼ ρC p μgd d d
ð5Þ
The symbol ‖ indicates the absolute value and this makes the cooling rate a positive value. After the metal liquid is shattered into drops with different sizes by the flowing gas, the smaller drops get higher acceleration and they catch up with the gas speed easily and solidify rapidly; on the other hand, those larger drops get lower acceleration and they fail to catch up with the high-speed gas and hence break away from the gas flow before solidification. We can then assume the relative speed between the melt drop and the gas, (vd − vg), to be zero, and Eq. (5) now simplifies to the form of dT d 12kg ðT d −T e Þ : dt ¼ ρC d2
ð6Þ
p
The relevant parameters in Eq. (6) are listed in Table 1 and after substitution of these known parameters, Eq. (6) finally becomes: dT d 1 10−4 : dt ≈ d2
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−n spacing (SDAS), λ2, in the larger sized powder (λ2 ¼ A′ dT , A′ and dt n are constants [20]), where the slower cooling rate leads to crystallization (see Fig. 1 (b)). The two methods gave highly consistent results (2.1 × 10 4 K s −1 estimated from SDAS, and 2.4 × 10 4 K s −1 calculated from Eq. (7), for a 65 μm-sized powder particle) [21], and this certifies the validity of using Eq. (7) to estimate the cooling rate for the gas-atomized powder. For powders with diameters of 10 μm and 30 μm, their cooling rates are 9.8 × 10 5 and 1.1 × 10 5 K s −1, respectively according to Eq. (7). The critical cooling rate to obtain amorphous Al82Ni10Y8 alloy powder is thus ~ 10 6 K s −1.
4.2. Cooling rate calculation for the melt-spun ribbons There exist many discussions on the heat transfer for the melt spinning processes in the literature (for example, see [22–31]). However, a straightforward expression to give the cooling rate for the melt-spun ribbons as a function of ribbon thickness is rarely seen. The main reason is that the heat transfer in the melt spinning process is very complicated and many factors need to be taken into account [22–27]. To treat the heat transfer issue rigorously is not the target of this paper, and instead we aim at simplifying the mathematical description of the melt spinning process and generating an expression that can reasonably estimate the cooling rate of the melt-spun ribbons, even though some of our assumptions are not physically robust. As the critical cooling rate to form an amorphous alloy for the same composition shall be in principle an independent process, if the critical cooling rate estimated from the melt spinning process is comparable to that estimated from the gas atomization process as we discussed above, we then regard the scheme we use here for the melt spinning process as reasonable, at least for the sole purpose of estimating the cooling rate for the melt-spun ribbons. Here is our simplified treatment for the heat transfer in the melt spinning process. The molten liquid solidifies rapidly once it is ejected from the quartz nozzle and forms thin ribbons on the rotating Cu wheel. Compared to the thickness of the ribbons, their width and length can be assumed to be infinitively large and so are the width and diameter of the Cu wheel. The Cu wheel can be seen as a semi-infinite solid. If we ignore the radiation and convection of heat during the solidification process, the heat transfer between the liquid alloy and the Cu wheel can be assumed to be a one dimension non-steady heat transfer process. The cooling rate of the ribbons can be then estimated by solving the mature one dimension non-steady equations. The melt-Cu wheel contact is schematically shown in Fig. 3. The ribbon thickness is along the x direction and we set the origin at the
ð7Þ
The cooling rate of the gas-atomized powder estimated by Eq. (7) is compared with that estimated from the secondary dendrite arm Table 1 Parameters used for calculating the cooling rate of the gas-atomized powder [33,34]. Thermal conductivity, kga/W m−1 K−1 1.79 × 10−2
Density, ρb/kg m− 3
2385
Specific heat capacity, Cpb/J kg− 1 −1 K
Temperature, Temperature, Td/K Te/K
1080
1473
298
a
Thermal conductivity at 300 K. For simplicity, density and specific heat capacity for pure Al liquid were used for the Al82Ni10Y8 alloy. It is noted here that although the alloying effect and the temperature dependence of these parameters will certainly affect the estimated cooling rate, the variation shall be within one order of magnitude. This applies to the cited parameters in Table 2 as well. b
Fig. 3. Schematic of the simplified contact between the ribbon and the Cu wheel during the melt spinning process.
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interface between the ribbon and the Cu wheel. The one dimension heat transfer along the x direction can be mathematically described by [16]: 2
At the interface where x = 0, Fourier's law entails that: k1
∂T 1 ∂x
¼ k2
x¼0
∂T 2 ∂x
:
ð14Þ
x¼0
2
∂T ∂ T k ∂ T ¼α 2 ¼ C p ρ ∂x2 ∂t ∂x
ð8Þ
where T is the temperature (K); t is the time (s); x is the distance to the interface (m); α is the thermal diffusivity (W m 2 J −1); k is the thermal conductivity (W m −1 K −1); Cp is the specific heat capacity (J kg −1 K −1); and ρ is the density (kg m −3). The general solution to Eq. (8) is in the form of [16]
x pffiffiffiffiffiffi : 2 αt
T ðx; t Þ ¼ A þ B erf
ð9Þ x
erf() in Eq. (9) is the error function and erf ðxÞ ¼ p2ffiffiπ ∫e−y dy. A and
From Eq. (14) we obtain the interface temperature, Ti, as a function of T10 and T20: Ti ¼
pffiffiffiffiffiffi pffiffiffiffiffiffi k1 α 2 T 10 þ k2 α 1 T 20 : pffiffiffiffiffiffi pffiffiffiffiffiffi k1 α 2 þ k2 α 1
ð15Þ
To calculate the cooling rate of the ribbon, ∂T∂t1 , from Eq. (11), we still need to know the cooling time, t. Geller et al. [28] used the concept of temperature gradient to estimate t. It is assumed that ribbon cooling rate is constant (within a very short cooling time), and the temperature variation of the Cu wheel as a function of time follows:
2
0
B are constants in the infinite integral. It is noted here that the solution in the form of Eq. (9) is a solution for a semi-infinite solid [16], so basically it is suitable to be used for the Cu wheel but not for the ribbons. However, out of consideration for easy mathematical treatment, we use Eq. (9) as well to describe the heat transfer for the ribbons. We will see later that by doing this, the estimated critical cooling rate from the melt spinning is comparable to that estimated from the gas atomization process. We therefore think it is reasonable to use the scheme we put forward here to estimate the cooling rate of the melt-spun ribbons. We then treat the heat transfer function for the ribbon and the Cu wheel separately, both by solving Eq. (9) using the boundary conditions. For clarity, we use the subscripts 1 and 2 to denote the ribbon and the Cu wheel, respectively. For the ribbon (x b 0): T 1 ¼ A1 þ B1 erf
x pffiffiffiffiffiffiffiffi : 2 α1 t
ð10Þ
1) at x = 0 (t > 0): T1 = Ti, where Ti is the temperature at the interface between the ribbon and the Cu wheel; 2) at t = 0 (x b 0): T1 = T10, where T10 is the initial temperature of the ribbon when the ribbon just touches the Cu wheel. T10 can be assumed to the temperature of the molten alloy. From these two boundary conditions, A1 = Ti, B1 = Ti − T10. Eq. (10) now becomes:
x pffiffiffiffiffiffiffiffi : 2 α1 t
ð11Þ
Similarly, for the Cu wheel (x > 0): T 2 ¼ A2 þ B2 erf
x pffiffiffiffiffiffiffiffi : 2 α2 t
ð12Þ
ð16Þ
where β is the temperature gradient in terms of time. From Eqs. (13) and (16), we can derive: pffiffi t ∂T 2 ¼ −β pffiffiffiffiffiffiffiffiffi : πα 2 ∂x
ð17Þ
The boundary conditions at the interface are: T 1 ð0; t Þ ¼ T 2 ð0; t Þ ¼ T ð0; t Þ C p1 ρ1 δ1
ð18Þ
∂T ∂T ¼ −k2 : ∂t ∂x
ð19Þ
is β defined in In Eq. (19), δ1 is the thickness of the ribbon and ∂T ∂t Eq. (16). From Eqs. (17) and (19), the cooling time, t, can be given in the form of
t¼
The boundary conditions for Eq. (10) are:
T 1 ¼ T i þ ðT i −T 10 Þ erf
T 2 ð0; t Þ ¼ T 20 þ βt
2 δ1 2 π C p1 ρ1 k2 C p2 ρ2
:
ð20Þ
The relationship between the cooling time, t, and the ribbon thickness, δ1, as reflected in Eq. (20) has been verified by the experimental results [28]. According to Eq. (20), the time it takes to solidify a 25-μm thick ribbon is about 8.5 × 10 −6 s. This short time scale can justify the neglecting of the time dependence of the temperature for the boundary conditions. From Eqs. (11), (15) and (20) we can calculate the cooling rate of the ribbon, ∂T∂t1 , at any distance, x. For the easy comparison, we use the cooling rate at the outer surface (opposite to the surface that is in contact with the Cu wheel) to denote the cooling rate of the ribbon, i.e., ∂T 1 x¼−δ .
∂t
1
C k ρ pffiffiffiffiffi pffiffiffiffiffi − p2 2 2 k α 2 T 10 þk2 α 1 T 20 ffiffiffiffiffi pffiffiffiffiffi −α 1 e 4πCp1 k1 ρ1 −T 10 þ 1 k p ∂T 1 1 α 2 þk2 α 1 : ∂t x¼−δ1 ¼ C k ρ 3=2 2δ1 2 π2 C p1 k1 ρ1 p2 2 2
j
ð21Þ
The boundary conditions for Eq. (12) are: 1) at x = 0 (t > 0): T2 = Ti; 2) at t = 0, (x > 0): T2 = T20, where T20 is the initial temperature of the Cu wheel when the ribbon just touches it. T20 is assumed to the room temperature. From the boundary conditions we have A2 = Ti, B2 = T20 − Ti. Eq. (12) then becomes: T 2 ¼ T i þ ðT 20 −T i Þ erf
x pffiffiffiffiffiffiffiffi 2 α2 t
ð13Þ
Although Eq. (21) looks complicated, it is basically a simple function of the ribbon thickness (1/δ 2) once the experimental conditions (T10 and T20) are set, and all the other parameters are materials constants and are easily available in the materials handbooks. All the parameters needed for the calculation are listed in Table 2. From Eq. (21), ribbons with thickness of 25 μm and 50 μm have cooling rates of 1.5 × 10 7 and 3.7 × 10 6 K s −1, respectively. This magnitude of the critical cooling rate is comparable to that estimated from the gas-atomized powder, giving confidence to the scheme we use here to estimate the cooling rate for the melt-spun ribbons. If we take
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1010
Material
Thermal conductivity, k/W m−1 K−1 b
Al82Ni10Y8 94 melt1a Cu 398 wheel2a
Specific heat Density, Thermal Temperature, capacity, Cp/ ρ/kg m−3 diffusivity, T0/K α/W m2 J−1 J kg−1 K−1 1080
b
384
2385
b
8900
3.6 × 10
−5
1133
1.2 × 10−4
c
298
a
Subscripts 1 and 2 denote the Al82Ni10Y8 melt and the Cu wheel, respectively. b For simplicity, thermal conductivity, density and specific heat and density for pure Al liquid were used for the Al82Ni10Y8 alloy. c An overheat of 200 K was assumed.
Cooling rate, R/ K s-1
Table 2 Parameters used for calculating the cooling rate of the melt-spun ribbon [33,34].
dT 10 ¼ 2 dt R
ð22Þ
where R (m) is the typical dimension (thickness or diameter) of the sample. However, detail on how Eq. (22) is derived is not available. The cooling rates for 25 μm and 50 μm sized ribbons calculated according to Eq. (22) are 1.6 × 10 6 and 4 × 10 5 K s −1, about one order of magnitude lower than those given by Eq. (21). Comparing Eq. (7) with Eq. (22), for the same dimension in thickness or diameter, the former yields a cooling rate one order of magnitude lower than the latter, hence two orders of magnitudes lower than Eq. (21). For easy comparison, the estimated cooling rates for the ribbon and powder are plotted in Fig. 4.
ΔT=100K ΔT=200K
*
ΔT=300K ΔT=400K
108
ΔT=500K ΔT=540K
#
ΔT=600K
107 increasing ΔT
106
20
0
109
40
60
Ribbon thickness, δ /μm
(b)
100
80
ΔT=100K
108
Cooling rate, R/ K s-1
−3
(a)
109
105 1.0 × 10 6 K s −1 as the critical cooling rate for the Al82Ni10Y8 alloy, we can further estimate the critical ribbon thickness for the same alloy to be ~ 95 μm. Currently, when estimating the cooling rate of the amorphous alloys, the method proposed by Lin and Johnson [32] is most commonly used due to its simple form:
2757
ΔT=200K
*
ΔT=300K
107
ΔT=400K ΔT=500K
106
ΔT=540K
#
ΔT=600K
105 104 103 0
increasing ΔT
20
40
60
80
100
Powder diameter, d/μm Fig. 5. Effect of overheating on the cooling rates for (a) the melt-spun ribbons and (b) the gas-atomized powders. The symbol * in the legend denotes the overheating assumed for the ribbon and # denotes the overheating for the powder.
5. Discussion The effect of melt temperature on the GFA has been experimentally investigated [11–15], with the similar conclusion that increasing the melt temperature would increase the GFA. Does the melt temperature have an effect on the cooling rate? This is immediately obvious from Eq. (6) for the case of the gas-atomized powder: the higher Td (= T m þ ΔT, where Tm is the melting temperature, ΔT is the
1010
1010 Ribbon Powder
109
108
108
107
107
106
106
105
105
104
104
103
0
20
40
60
80
Cooling rate, R/ K s-1
Cooling rate, R/ K s-1
109
103 100
Ribbon thickness, δ/μm Powder diameter, d/μm Fig. 4. Comparison of the cooling rates for the melt-spun ribbons and gas-atomized powders.
overheating) of the alloy, the higher the cooling rate. The melt temperature effect on the cooling rate for the melt-spun ribbon is not immediately obvious from Eq. (21). For clarity, the cooling rates for both melt-spun ribbons and gas-atomized powders are plotted in Fig. 5, as a function of ΔT. It is clearly seen from Fig. 5 that in both cases, the cooling rate increases with increasing ΔT, and the increase is more noticeable at larger dimensions, i.e., thicker ribbons or larger powder particles. However, even at the most significant case the effect of ΔT on the cooling rate is limited: a difference of 500 K in ΔT would result in a change of the cooling rate within half an order of magnitude. Fig. 5 tells us that the benefit of the higher melt temperature on the GFA can almost not come from the increase in the cooling rate. Actually it has been reported that only a moderate ΔT can effectively enhance the GFA [15], not that the higher ΔT the better, again suggesting that the improvement of GFA is not due to the cooling rate effect. One plausible explanation for the benefit of the increased melt temperature on GFA is due to that the tendency for atoms to form clusters decreases at higher melt temperature [13], and the amount of pre-existing nuclei in the liquid reduces, hence inhibiting the crystallization or in other words enhancing the GFA. However, one needs to be aware that at higher melt temperature, the absorption of oxygen gets more serious and it would lead to easier nucleation in the liquid and hence deteriorate the GFA [15]. 6. Conclusions 1) The cooling rates for melt-spun ribbons and gas-atomized powders were estimated. The cooling rates ( dT ) for both ribbons and dt
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powders are inversely proportional to the square of their dimensions, thickness (δ) or diameter (d): dT ∝ δ12 or dT ∝ 12 . dt dt d 2) As an example, the critical cooling to form amorphous Al82Ni10Y8 6 −1 alloy was estimated to be 1.0 × 10 K s . 3) The effect of the melt temperature on the critical cooling rate was quantitatively evaluated and the higher the melt temperature the faster the cooling rate. However, such an enhancement of the cooling rate by increasing the melt temperature is not significant, and a difference of 500 K in the melt temperature would lead to the change of the cooling rate within half an order of magnitude. It was suggested that the enhancement of the glass forming ability by increasing the melt temperature does not come from the increase in the cooling rate. References [1] A. Inoue, T. Zhang, T. Masumoto, Production of amorphous cylinder and sheet of La55Al25Ni20 alloys by a metallic mold casting method, Mater. Trans. 31 (1990) 425–428. [2] W. Klement, R.H. Willens, P. Duwez, Non-crystalline structure in solidified gold– silicon alloys, Nature 187 (1960) 869–870. [3] A. Inoue, Stabilization of metallic supercooled liquid and bulk amorphous alloys, Acta Mater. 48 (2000) 279–306. [4] A. Inoue, A. Takeuchi, Recent development and application products of bulk glassy alloys, Acta Mater. 59 (2011) 2243–2267. [5] H.M. Fu, J. Mu, A.M. Wang, H. Li, Z.Q. Hu, H.F. Zhang, Synthesis and compressive properties of Al–Ni–Y metallic glass, Philos. Mag. Lett. 89 (2009) 711–716. [6] B.J. Yang, J.H. Yao, Y.S. Chao, J.Q. Wang, E. Ma, Developing aluminum-based bulk metallic glasses, Philos. Mag. 90 (2010) 3215–3231. [7] B.J. Yang, J.H. Yao, J. Zhang, H.W. Yang, J.Q. Wang, E. Ma, Al-rich bulk metallic glasses with plasticity and ultrahigh specific strength, Scr. Mater. 61 (2009) 423–426. [8] L.C. Zhuo, S.J. Pang, H. Wang, T. Zhang, Ductile bulk aluminum-based alloy with good glass-forming ability and high strength, Chin. Phys. Lett. 26 (2009) 066402. [9] J. Mu, H.M. Fu, Z.W. Zhu, A.M. Wang, H. Li, Z.Q. Hu, H.F. Zhang, Synthesis and properties of Al–Ni–La bulk metallic glass, Adv. Eng. Mater. 11 (2009) 530–532. [10] S. Guo, Z.P. Lu, C.T. Liu, Identify the best glass forming ability criterion, Intermetallics 18 (2010) 883–888. [11] H.C. Sheng, X.R. Zeng, H.X. Qian, D.J. Fu, Effect of melt treatment on the microstructure and magnetic properties of Nd2Fe14B/α-Fe nanocomposites, J. Non-Cryst. Solids 356 (2010) 19–23. [12] C. Fan, A. Inoue, Influence of liquid temperature on the crystallization behavior in Zr–Al–Cu–Pd amorphous alloy, Mater. Trans. 40 (1999) 1178–1180. [13] C. Fan, D. Chen, P.K. Liaw, H. Choo, C. Benmore, J. Siewenie, G.L. Chen, J.X. Xie, C.T. Liu, Influence of molten status on nanoquasicrystalline-forming Zr-based metallic glasses, Appl. Phys. Lett. 93 (2008) 261905.
[14] C. Fan, A. Inoue, Influence of the liquid states on the crystallization process of nanocrystal-forming Zr–Cu–Pd–Al metallic glasses, Appl. Phys. Lett. 75 (1999) 3644–3646. [15] J. Mao, H.F. Zhang, H.M. Fu, A.M. Wang, H. Li, Z.Q. Hu, The effects of casting temperature on the glass formation of Zr-based metallic glasses, Adv. Eng. Mater. 11 (2009) 986–991. [16] J.P. Holman, Heat Transfer, 8th Edition McGraw-Hill, New York, 1997. [17] J. Szekely, N.J. Themelis, Rate Phenomena in Process Metallurgy, Wiley-Interscience, New York, 1971.. [18] Y.Z. Zhou, Y. Wu, E.J. Lavernia, Process modeling in spray deposition: a review, Int. J. Non-Equilib. Pr. 10 (1997) 95–183. [19] J.L. Estrada, J. Duszczyk, Characteristics of rapidly solidified Al–Si–X powders for high-performance applications, J. Mater. Sci. 25 (1990) 886–904. [20] C.A. Muojekwu, I.V. Samarasekera, J.K. Brimacombe, Heat transfer and microstructure during the early stages of metal solidification, Metall. Mater. Trans. B 26 (1995) 361–382. [21] S. Guo, Fabrication of Non-equilibrium Al-based Alloys (in Chinese), Master Thesis, in, Central South University, 2005. [22] E.M. Gutierrez, J. Szekely, A mathematical model of the planar flow melt spinning process, Metall. Trans. B 17 (1986) 695–703. [23] H. Yu, A fluid mechanics model of the planar flow melt spinning process under low Reynolds number conditions, Metall. Trans. B 18 (1987) 557–563. [24] B. Karpe, B. Kosec, M. Bizjak, Modeling of heat transfer in the cooling wheel in the melt-spinning process, J. Achiev. Mater. Manuf. Eng. 46 (2011) 88–94. [25] J.K. Carpenter, P.H. Steen, Heat transfer and solidification in planar-flow melt-spinning: high wheel speeds, Int. J. Heat Mass Transf. 40 (1997) 1993–2007. [26] P.H. Steen, C. Karcher, Fluid mechanics of spin casting of metals, Annu. Rev. Fluid Mech. 29 (1997) 373–397. [27] A. Sundararaj, B.G. Thomas, Heat transfer during melt spinning of Al–7%Si alloy on a Cu–Be wheel, In: in: D.H. DeYoung (Ed.), Light Metals 2008, Wiley, New Orleans, 2008, pp. 793–809. [28] M. Geller, E. Brook-Levinson, V. Manov, Heat transfer during preparation of amorphous metallic alloy ribbon, In: in: M. Zinigrad (Ed.), Proceedings of the Second International Conference on Mathematical Modeling and Computer Simulation of Metal Technologies, Ariel, Isreal, 2002, pp. 419–424. [29] V.I. Tkatch, A.I. Limanovskii, S.N. Denisenko, S.G. Rassolov, The effect of the melt-spinning of processing parameters on the rate cooling, Mater. Sci. Eng., A 323 (2002) 91–96. [30] V.I. Tkatch, S.N. Denisenko, O.N. Beloshov, Direct measurements of the cooling rates in the single roller rapid solidification technique, Acta Mater. 45 (1997) 2821–2826. [31] M.J. Kramer, H. Mecco, K.W. Dennis, E. Vargonova, R.W. McCallum, R.E. Napolitano, Rapid solidification and metallic glass formation—experimental and theoretical limits, J. Non-Cryst. Solids 353 (2007) 3633–3639. [32] X.H. Lin, W.L. Johnson, Formation of Ti–Zr–Cu–Ni bulk metallic glasses, J. Appl. Phys. 78 (1995) 6514–6519. [33] J.H. Lienhard, J.H. Lienhard, A Heat Transfer Textbook, 3rd Edition Phlogiston Press, Cambridge, Massachusetts, 2003. [34] F. Cardarelli, Materials Handbook: A Concise Desktop Reference, 2nd Edition Springer Verlag, London, 2008.
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Journal of Non-Crystalline Solids journal homepage: www.elsevier.com/ locate/ jnoncrysol
Estimation of critical cooling rates for formation of amorphous alloys from critical sizes Sheng Guo a,⁎, Yong Liu b a b
Department of MBE, City University of Hong Kong, Kowloon, Hong Kong, PR China State Key Laboratory for Powder Metallurgy, Central South University, Changsha, Hunan, 410083, PR China
a r t i c l e
i n f o
Article history: Received 29 February 2012 Received in revised form 19 June 2012 Available online 24 July 2012 Keywords: Metallic glasses; Aluminum alloys; Cooling rate; Heat transfer
a b s t r a c t The critical cooling rate for marginal glass formers, like Al-based alloys, is difficult to measure experimentally. In this work, we acquired the critical cooling rate for formation of amorphous alloys by estimating the cooling rates of the specimens with critical dimensions. Analytical solutions were given to estimate the cooling rates for the gas-atomized powders and melt-spun ribbons, and as an example the critical cooling rate to form an amorphous Al82Ni10Y8 alloy was estimated to be ~1.0 × 10 6 K s−1. The effect of melt temperature on the cooling rate was quantitatively evaluated and its effect on the glass forming ability was also discussed. © 2012 Elsevier B.V. All rights reserved.
1. Introduction Amorphous alloys, or metallic glasses have been developed rapidly and studied extensively during the past 30 years [1–3], due to their many unique properties over the crystalline counterparts. A large variety of amorphous alloy systems has been developed so far, including Zr-, Fe-, Pd-, Pt-, La-, Cu-, Mg-, Co-, Ni- and Ti-based alloys [4]. However, the preparation of bulk (typically above 1 mm in size) Al-based metallic glasses has been proved very difficult [5–9] and a breakthrough in the achievable size is yet to be reached. Considering the wide applicability of Al alloys and their light weight, the capability to make bulk Al-based metallic glasses is very attractive. The difficulty of making bulk Al-based metallic glasses is due to the fact that the critical cooling rate, Rc, to obtain amorphous Al alloys is fairly high, and such a demanding cooling rate limits the achievable sample size that can be fully amorphous in structure. To advance the development of bulk amorphous alloy with marginal glass forming ability (GFA), our understanding to the GFA and particularly to the improvement of the GFA need to be advanced greatly. In the meantime, the characterization of the GFA of Al alloys is also an issue. The critical cooling rate, which is the unambiguous criterion defining the GFA of various metallic glasses, can be experimentally measured for easy glass formers but not for marginal glass formers like Al alloys [10]. However, the critical size (Dc) of the amorphous Al alloys can be experimentally determined. If we can calculate the cooling rates (R) from these critical sizes, we can then get the critical cooling rates for Al alloys indirectly. Unfortunately, to the knowledge of the ⁎ Corresponding author. Tel.: +852 34428416; fax: +852 34420172. E-mail address: [email protected] (S. Guo). 0022-3093/$ – see front matter © 2012 Elsevier B.V. All rights reserved. doi:10.1016/j.jnoncrysol.2012.06.023
authors there lacks such an obvious link between the cooling rate and sample size in the literature, or at least not in a convenient way. In this work, we prepared amorphous Al82Ni10Y8 alloys in both ribbon and powder forms, and estimated their cooling rates by using simplified heat transfer models. The target alloy was chosen because it is a simple ternary alloy system and the alloying components well satisfy the three empirical rules proposed by Inoue to possess high GFA [3]. The melt treatment has been known to affect the GFA [11–15], but how the melt temperature affects the cooling rate is not quantitatively studied previously, and this issue is to be addressed here as well.
2. Material and methods The ribbon and powder specimens used in this work were prepared by melt spinning and gas atomization methods, respectively. Starting materials were pure Al (99.995 mass%), Ni (99.9 mass%) and Y (99.99 mass%). The master alloy was prepared by mixing the pure metals in a molar composition of Al82Ni10Y8 using arc melting (for melt spinning) or induction melting (for gas atomization) in the high-purity argon atmosphere. Repeated melting was carried out at least five times to ensure the chemical homogeneity of the alloy. For the gas atomization process, a close-coupled nozzle was used to promote the efficient atomization of the melt [16]. The master alloy was first remelted at ~ 1473 K in an alumina crucible in a vacuum of 10 −2 Pa. The melt was teemed through a guide tube, and then atomized by a jet of argon at ~ 3 MPa. The atomized powder was collected and sieved in a closed system filled with nitrogen. The ribbons were prepared by remelting the master alloy in a quart tube in argon
S. Guo, Y. Liu / Journal of Non-Crystalline Solids 358 (2012) 2753–2758
and then ejecting the molten liquid through a nozzle onto a rotating Cu wheel with a rotation speed up to 2700 rotation per minute (rpm). The surface morphology of the powders with different size was observed using scanning electron microscopy (SEM). The amorphicity of ribbons was identified by X-ray diffraction (XRD) using Cu-Kα radiation.
3. Results The surface morphology of the gas atomized powder is shown in Fig. 1. It can be seen that the powders are almost perfectly spherical in shape (although a few ellipsoidal powders were also seen, not shown here). Those particles with diameter smaller than ~ 10 μm (Fig. 1a) are featureless in the microstructure, possibly having an amorphous structure; while in those particles with larger diameter hence slower cooling rate (~ 30 μm in Fig. 1b) dendrites appear, suggesting they are already crystalline. The XRD patterns for melt spun ribbons with thickness of 25 μm and 50 μm are shown in Fig. 2. The ribbons prepared using a faster Cu wheel rotation speed are thinner in thickness. Both ribbons are identified to be XRD amorphous. Compared to the critical powder size, the determination of the critical ribbon thickness to achieve the fully amorphous state is more time consuming, as the rotation speed has to be varied continuously. We will show later that the critical ribbon thickness is estimated to be ~ 85 μm for the Al82Ni10Y8 alloy.
(a)
Intensity, I/cps
2754
1500 rpm, 50 μm
2700 rpm, 25 μm
10°
20°
30°
40°
50°
60°
70°
80°
90°
Angle, 2θ Fig. 2. XRD patterns of the melt-spun Al82Ni10Y8 alloy ribbons with different thickness.
4. Calculation In this section, we will model the heat transfer during the gas atomization and melt spinning processes, and estimate the cooling rates for these two processes based on the models, from intrinsic properties of the materials and also properties of the cooling media (argon gas or the Cu wheel). The heat transfer for the gas atomization process is relatively more straightforward and we deal with it first. For the melt spinning process, the heat transfer is more complicated and we made many simplifications on it. We want to emphasize here that these treatments actually over-simplify the real heat transfer in the melt spinning process and the mathematical solution is hence not robust. However, the estimated critical cooling rates for the melt spinning process, out of apparently simplified treatments, agree well with that for the gas atomization process. It is hence our belief that the scheme we put forward here is still of practical value, to reasonably estimate the cooling rates for the melt spun ribbons. 4.1. Cooling rate calculation for the gas-atomized powders In the gas atomization process, the high-speed gas flow impinges on the liquid flow ejected from the nozzle and the liquid shatters into small melt drops. The melt drops release heat through their surfaces during the flight in the chamber, while the gas in the surrounding environment cools the melt drops via the convective heat transfer. A few assumptions are made here for the heat transfer process:
2 μm
(b)
1) The melt drops reach the thermal equilibrium state instantly. As such, their internal temperature could be assumed homogeneous; 2) The surrounding environment is assumed to be an infinite space. The heat released from the melt drops diffuses to the surrounding environment rapidly, and the environmental temperature is assumed to be constant. 3) Since there exists a huge difference between the melt drops and the environment, the heat transfer is assumed to be completely interface controlled and the cooling follows the Newtonian heat transfer mode. The thermal equilibrium condition is thus the heat release from the melt drops equaling to the heat conducted to the environment via the melt drop surfaces. Mathematically, it can be written as [16]: dT d hAðT d −T e Þ ¼− ρVC p dt
ð1Þ
5 μm Fig. 1. Surface morphology of the gas-atomized Al82Ni10Y8 alloy powders with different particle size. (a) b10 μm and (b) ~30 μm.
where V is the volume (m 3); ρ is the density (kg m −3); Cp is the specific heat capacity (J kg −1 K −1); A is the surface area of the melt drop (m 2); h is the coefficient of heat transfer of the gas (W m −2 K −1); Td
S. Guo, Y. Liu / Journal of Non-Crystalline Solids 358 (2012) 2753–2758
and Te are melt drop temperature (K) and environment temperature (K), respectively; and t is the time (s). For simplification, a spherical shape with a diameter of d (m) is assumed for the melt drop. This assumption is also a close reflection of the real powder shape, as shown in Fig. 1. Eq. (1) then becomes: dT d 6hðT d −T e Þ ¼− : ρC p d dt
ð2Þ
The critical parameter for calculating the cooling rate is the heat transfer coefficient, h. Between the inert gas and the molten drops, h has the form of [17,18] h¼
kg 1=2 1=3 2 þ 0:6Re P r d
ð3Þ
where kg is the thermal conductivity of the gas (W m −1 K −1), Re is the Reynolds number and Pr is the Prandtl number. Re can be further expressed as [19]
Re ¼
vd −vg ρg d
ð4Þ
μg
where vd and vg are the velocities for the melt drop and the gas (m s −1), respectively, ρg is the density of the gas (kg m−3) and μg is the kinetic viscosity of the gas (kg m−1 s−1). From Eqs. (2), (3) and (4) we have 0 1 11=2 0 dT d 6ðT d −T e Þ B2kg kg ρg vd −vg 1=3 A Pr C @ 2 þ 0:6 @ A: dt ¼ ρC p μgd d d
ð5Þ
The symbol ‖ indicates the absolute value and this makes the cooling rate a positive value. After the metal liquid is shattered into drops with different sizes by the flowing gas, the smaller drops get higher acceleration and they catch up with the gas speed easily and solidify rapidly; on the other hand, those larger drops get lower acceleration and they fail to catch up with the high-speed gas and hence break away from the gas flow before solidification. We can then assume the relative speed between the melt drop and the gas, (vd − vg), to be zero, and Eq. (5) now simplifies to the form of dT d 12kg ðT d −T e Þ : dt ¼ ρC d2
ð6Þ
p
The relevant parameters in Eq. (6) are listed in Table 1 and after substitution of these known parameters, Eq. (6) finally becomes: dT d 1 10−4 : dt ≈ d2
2755
−n spacing (SDAS), λ2, in the larger sized powder (λ2 ¼ A′ dT , A′ and dt n are constants [20]), where the slower cooling rate leads to crystallization (see Fig. 1 (b)). The two methods gave highly consistent results (2.1 × 10 4 K s −1 estimated from SDAS, and 2.4 × 10 4 K s −1 calculated from Eq. (7), for a 65 μm-sized powder particle) [21], and this certifies the validity of using Eq. (7) to estimate the cooling rate for the gas-atomized powder. For powders with diameters of 10 μm and 30 μm, their cooling rates are 9.8 × 10 5 and 1.1 × 10 5 K s −1, respectively according to Eq. (7). The critical cooling rate to obtain amorphous Al82Ni10Y8 alloy powder is thus ~ 10 6 K s −1.
4.2. Cooling rate calculation for the melt-spun ribbons There exist many discussions on the heat transfer for the melt spinning processes in the literature (for example, see [22–31]). However, a straightforward expression to give the cooling rate for the melt-spun ribbons as a function of ribbon thickness is rarely seen. The main reason is that the heat transfer in the melt spinning process is very complicated and many factors need to be taken into account [22–27]. To treat the heat transfer issue rigorously is not the target of this paper, and instead we aim at simplifying the mathematical description of the melt spinning process and generating an expression that can reasonably estimate the cooling rate of the melt-spun ribbons, even though some of our assumptions are not physically robust. As the critical cooling rate to form an amorphous alloy for the same composition shall be in principle an independent process, if the critical cooling rate estimated from the melt spinning process is comparable to that estimated from the gas atomization process as we discussed above, we then regard the scheme we use here for the melt spinning process as reasonable, at least for the sole purpose of estimating the cooling rate for the melt-spun ribbons. Here is our simplified treatment for the heat transfer in the melt spinning process. The molten liquid solidifies rapidly once it is ejected from the quartz nozzle and forms thin ribbons on the rotating Cu wheel. Compared to the thickness of the ribbons, their width and length can be assumed to be infinitively large and so are the width and diameter of the Cu wheel. The Cu wheel can be seen as a semi-infinite solid. If we ignore the radiation and convection of heat during the solidification process, the heat transfer between the liquid alloy and the Cu wheel can be assumed to be a one dimension non-steady heat transfer process. The cooling rate of the ribbons can be then estimated by solving the mature one dimension non-steady equations. The melt-Cu wheel contact is schematically shown in Fig. 3. The ribbon thickness is along the x direction and we set the origin at the
ð7Þ
The cooling rate of the gas-atomized powder estimated by Eq. (7) is compared with that estimated from the secondary dendrite arm Table 1 Parameters used for calculating the cooling rate of the gas-atomized powder [33,34]. Thermal conductivity, kga/W m−1 K−1 1.79 × 10−2
Density, ρb/kg m− 3
2385
Specific heat capacity, Cpb/J kg− 1 −1 K
Temperature, Temperature, Td/K Te/K
1080
1473
298
a
Thermal conductivity at 300 K. For simplicity, density and specific heat capacity for pure Al liquid were used for the Al82Ni10Y8 alloy. It is noted here that although the alloying effect and the temperature dependence of these parameters will certainly affect the estimated cooling rate, the variation shall be within one order of magnitude. This applies to the cited parameters in Table 2 as well. b
Fig. 3. Schematic of the simplified contact between the ribbon and the Cu wheel during the melt spinning process.
2756
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interface between the ribbon and the Cu wheel. The one dimension heat transfer along the x direction can be mathematically described by [16]: 2
At the interface where x = 0, Fourier's law entails that: k1
∂T 1 ∂x
¼ k2
x¼0
∂T 2 ∂x
:
ð14Þ
x¼0
2
∂T ∂ T k ∂ T ¼α 2 ¼ C p ρ ∂x2 ∂t ∂x
ð8Þ
where T is the temperature (K); t is the time (s); x is the distance to the interface (m); α is the thermal diffusivity (W m 2 J −1); k is the thermal conductivity (W m −1 K −1); Cp is the specific heat capacity (J kg −1 K −1); and ρ is the density (kg m −3). The general solution to Eq. (8) is in the form of [16]
x pffiffiffiffiffiffi : 2 αt
T ðx; t Þ ¼ A þ B erf
ð9Þ x
erf() in Eq. (9) is the error function and erf ðxÞ ¼ p2ffiffiπ ∫e−y dy. A and
From Eq. (14) we obtain the interface temperature, Ti, as a function of T10 and T20: Ti ¼
pffiffiffiffiffiffi pffiffiffiffiffiffi k1 α 2 T 10 þ k2 α 1 T 20 : pffiffiffiffiffiffi pffiffiffiffiffiffi k1 α 2 þ k2 α 1
ð15Þ
To calculate the cooling rate of the ribbon, ∂T∂t1 , from Eq. (11), we still need to know the cooling time, t. Geller et al. [28] used the concept of temperature gradient to estimate t. It is assumed that ribbon cooling rate is constant (within a very short cooling time), and the temperature variation of the Cu wheel as a function of time follows:
2
0
B are constants in the infinite integral. It is noted here that the solution in the form of Eq. (9) is a solution for a semi-infinite solid [16], so basically it is suitable to be used for the Cu wheel but not for the ribbons. However, out of consideration for easy mathematical treatment, we use Eq. (9) as well to describe the heat transfer for the ribbons. We will see later that by doing this, the estimated critical cooling rate from the melt spinning is comparable to that estimated from the gas atomization process. We therefore think it is reasonable to use the scheme we put forward here to estimate the cooling rate of the melt-spun ribbons. We then treat the heat transfer function for the ribbon and the Cu wheel separately, both by solving Eq. (9) using the boundary conditions. For clarity, we use the subscripts 1 and 2 to denote the ribbon and the Cu wheel, respectively. For the ribbon (x b 0): T 1 ¼ A1 þ B1 erf
x pffiffiffiffiffiffiffiffi : 2 α1 t
ð10Þ
1) at x = 0 (t > 0): T1 = Ti, where Ti is the temperature at the interface between the ribbon and the Cu wheel; 2) at t = 0 (x b 0): T1 = T10, where T10 is the initial temperature of the ribbon when the ribbon just touches the Cu wheel. T10 can be assumed to the temperature of the molten alloy. From these two boundary conditions, A1 = Ti, B1 = Ti − T10. Eq. (10) now becomes:
x pffiffiffiffiffiffiffiffi : 2 α1 t
ð11Þ
Similarly, for the Cu wheel (x > 0): T 2 ¼ A2 þ B2 erf
x pffiffiffiffiffiffiffiffi : 2 α2 t
ð12Þ
ð16Þ
where β is the temperature gradient in terms of time. From Eqs. (13) and (16), we can derive: pffiffi t ∂T 2 ¼ −β pffiffiffiffiffiffiffiffiffi : πα 2 ∂x
ð17Þ
The boundary conditions at the interface are: T 1 ð0; t Þ ¼ T 2 ð0; t Þ ¼ T ð0; t Þ C p1 ρ1 δ1
ð18Þ
∂T ∂T ¼ −k2 : ∂t ∂x
ð19Þ
is β defined in In Eq. (19), δ1 is the thickness of the ribbon and ∂T ∂t Eq. (16). From Eqs. (17) and (19), the cooling time, t, can be given in the form of
t¼
The boundary conditions for Eq. (10) are:
T 1 ¼ T i þ ðT i −T 10 Þ erf
T 2 ð0; t Þ ¼ T 20 þ βt
2 δ1 2 π C p1 ρ1 k2 C p2 ρ2
:
ð20Þ
The relationship between the cooling time, t, and the ribbon thickness, δ1, as reflected in Eq. (20) has been verified by the experimental results [28]. According to Eq. (20), the time it takes to solidify a 25-μm thick ribbon is about 8.5 × 10 −6 s. This short time scale can justify the neglecting of the time dependence of the temperature for the boundary conditions. From Eqs. (11), (15) and (20) we can calculate the cooling rate of the ribbon, ∂T∂t1 , at any distance, x. For the easy comparison, we use the cooling rate at the outer surface (opposite to the surface that is in contact with the Cu wheel) to denote the cooling rate of the ribbon, i.e., ∂T 1 x¼−δ .
∂t
1
C k ρ pffiffiffiffiffi pffiffiffiffiffi − p2 2 2 k α 2 T 10 þk2 α 1 T 20 ffiffiffiffiffi pffiffiffiffiffi −α 1 e 4πCp1 k1 ρ1 −T 10 þ 1 k p ∂T 1 1 α 2 þk2 α 1 : ∂t x¼−δ1 ¼ C k ρ 3=2 2δ1 2 π2 C p1 k1 ρ1 p2 2 2
j
ð21Þ
The boundary conditions for Eq. (12) are: 1) at x = 0 (t > 0): T2 = Ti; 2) at t = 0, (x > 0): T2 = T20, where T20 is the initial temperature of the Cu wheel when the ribbon just touches it. T20 is assumed to the room temperature. From the boundary conditions we have A2 = Ti, B2 = T20 − Ti. Eq. (12) then becomes: T 2 ¼ T i þ ðT 20 −T i Þ erf
x pffiffiffiffiffiffiffiffi 2 α2 t
ð13Þ
Although Eq. (21) looks complicated, it is basically a simple function of the ribbon thickness (1/δ 2) once the experimental conditions (T10 and T20) are set, and all the other parameters are materials constants and are easily available in the materials handbooks. All the parameters needed for the calculation are listed in Table 2. From Eq. (21), ribbons with thickness of 25 μm and 50 μm have cooling rates of 1.5 × 10 7 and 3.7 × 10 6 K s −1, respectively. This magnitude of the critical cooling rate is comparable to that estimated from the gas-atomized powder, giving confidence to the scheme we use here to estimate the cooling rate for the melt-spun ribbons. If we take
S. Guo, Y. Liu / Journal of Non-Crystalline Solids 358 (2012) 2753–2758
1010
Material
Thermal conductivity, k/W m−1 K−1 b
Al82Ni10Y8 94 melt1a Cu 398 wheel2a
Specific heat Density, Thermal Temperature, capacity, Cp/ ρ/kg m−3 diffusivity, T0/K α/W m2 J−1 J kg−1 K−1 1080
b
384
2385
b
8900
3.6 × 10
−5
1133
1.2 × 10−4
c
298
a
Subscripts 1 and 2 denote the Al82Ni10Y8 melt and the Cu wheel, respectively. b For simplicity, thermal conductivity, density and specific heat and density for pure Al liquid were used for the Al82Ni10Y8 alloy. c An overheat of 200 K was assumed.
Cooling rate, R/ K s-1
Table 2 Parameters used for calculating the cooling rate of the melt-spun ribbon [33,34].
dT 10 ¼ 2 dt R
ð22Þ
where R (m) is the typical dimension (thickness or diameter) of the sample. However, detail on how Eq. (22) is derived is not available. The cooling rates for 25 μm and 50 μm sized ribbons calculated according to Eq. (22) are 1.6 × 10 6 and 4 × 10 5 K s −1, about one order of magnitude lower than those given by Eq. (21). Comparing Eq. (7) with Eq. (22), for the same dimension in thickness or diameter, the former yields a cooling rate one order of magnitude lower than the latter, hence two orders of magnitudes lower than Eq. (21). For easy comparison, the estimated cooling rates for the ribbon and powder are plotted in Fig. 4.
ΔT=100K ΔT=200K
*
ΔT=300K ΔT=400K
108
ΔT=500K ΔT=540K
#
ΔT=600K
107 increasing ΔT
106
20
0
109
40
60
Ribbon thickness, δ /μm
(b)
100
80
ΔT=100K
108
Cooling rate, R/ K s-1
−3
(a)
109
105 1.0 × 10 6 K s −1 as the critical cooling rate for the Al82Ni10Y8 alloy, we can further estimate the critical ribbon thickness for the same alloy to be ~ 95 μm. Currently, when estimating the cooling rate of the amorphous alloys, the method proposed by Lin and Johnson [32] is most commonly used due to its simple form:
2757
ΔT=200K
*
ΔT=300K
107
ΔT=400K ΔT=500K
106
ΔT=540K
#
ΔT=600K
105 104 103 0
increasing ΔT
20
40
60
80
100
Powder diameter, d/μm Fig. 5. Effect of overheating on the cooling rates for (a) the melt-spun ribbons and (b) the gas-atomized powders. The symbol * in the legend denotes the overheating assumed for the ribbon and # denotes the overheating for the powder.
5. Discussion The effect of melt temperature on the GFA has been experimentally investigated [11–15], with the similar conclusion that increasing the melt temperature would increase the GFA. Does the melt temperature have an effect on the cooling rate? This is immediately obvious from Eq. (6) for the case of the gas-atomized powder: the higher Td (= T m þ ΔT, where Tm is the melting temperature, ΔT is the
1010
1010 Ribbon Powder
109
108
108
107
107
106
106
105
105
104
104
103
0
20
40
60
80
Cooling rate, R/ K s-1
Cooling rate, R/ K s-1
109
103 100
Ribbon thickness, δ/μm Powder diameter, d/μm Fig. 4. Comparison of the cooling rates for the melt-spun ribbons and gas-atomized powders.
overheating) of the alloy, the higher the cooling rate. The melt temperature effect on the cooling rate for the melt-spun ribbon is not immediately obvious from Eq. (21). For clarity, the cooling rates for both melt-spun ribbons and gas-atomized powders are plotted in Fig. 5, as a function of ΔT. It is clearly seen from Fig. 5 that in both cases, the cooling rate increases with increasing ΔT, and the increase is more noticeable at larger dimensions, i.e., thicker ribbons or larger powder particles. However, even at the most significant case the effect of ΔT on the cooling rate is limited: a difference of 500 K in ΔT would result in a change of the cooling rate within half an order of magnitude. Fig. 5 tells us that the benefit of the higher melt temperature on the GFA can almost not come from the increase in the cooling rate. Actually it has been reported that only a moderate ΔT can effectively enhance the GFA [15], not that the higher ΔT the better, again suggesting that the improvement of GFA is not due to the cooling rate effect. One plausible explanation for the benefit of the increased melt temperature on GFA is due to that the tendency for atoms to form clusters decreases at higher melt temperature [13], and the amount of pre-existing nuclei in the liquid reduces, hence inhibiting the crystallization or in other words enhancing the GFA. However, one needs to be aware that at higher melt temperature, the absorption of oxygen gets more serious and it would lead to easier nucleation in the liquid and hence deteriorate the GFA [15]. 6. Conclusions 1) The cooling rates for melt-spun ribbons and gas-atomized powders were estimated. The cooling rates ( dT ) for both ribbons and dt
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