Liquid fragility calculations from thermal analyses for metallic glasses

Journal of Non-Crystalline Solids 386 (2014) 46–50

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Liquid fragility calculations from thermal analyses for metallic glasses L. Hu, F. Ye ⁎ State Key Laboratory for Advanced Metals and Materials, University of Science and Technology Beijing, 30 Xueyuan Road, Beijing 100083, People's Republic of China

a r t i c l e

i n f o

Article history: Received 30 August 2013 Received in revised form 10 November 2013 Available online xxxx Keywords: Metallic glass; Liquid fragility; Kinetics; Differential scanning calorimetry (DSC)

a b s t r a c t The liquid fragility which describes the temperature dependence of relaxation can be derived from the kinetic characteristics of glass transition for metallic glasses (MGs). We summarize and assess the models to predict fragility index (m) by differential scanning calorimetry analyses for six selected MGs. It is found that the kinetic glass transition behaviors are valid to predict liquid fragility for MGs. Particularly, a concise model based on the activation energy for glass transition, which can be estimated from the heating rate dependence of glass transition temperature, is introduced and exhibits well prediction of m for the relatively strong MGs (m ≤ 52 in this paper). With some of the models, the calculations of m show the apparent heating rate dependence, and thus should be carefully considered. However, the approach derived from heat capacity and enthalpy of fusion seems to fail to predict m for MGs. © 2013 Elsevier B.V. All rights reserved.

1. Introduction Upon rapid cooling a melt below its liquidus temperature (Tl), the liquid is supercooled, albeit thermodynamically metastable with respect to the equilibrium crystalline phases. With temperature continuously decreasing, the supercooled liquid will flow more and more viscously, accompanied by the increase of relaxation time (τ) required for atomic rearrangement [1–3]. When the viscosity (η) reaches ~1012 Pa s or τ ~ 102 s, the supercooled liquid cannot undergo the enthalpy/volume of equilibrium state but be frozen into a glass [4–8]. On a reverse heating, the glass inevitably transits into the metastable supercooled liquid state. The transition process is strictly kinetic in origin but not a thermodynamically driven behavior [7,9]. The change of viscosity or relaxation time of liquid with temperature rising or dropping is highly material-specific, and often deviates from the Arrhenius law. Angell has proposed a useful classification of liquids as “strong” to “fragile” scale to describe the magnitude of such deviation [7,10]. The relaxation behavior of strong formers exhibits nearly Arrhenius fashion, whereas fragile liquids display marked non-Arrhenius behavior. Fig. 1 shows the normal Angell plots representing liquid viscosities vs. the temperature normalized by Tg⁎(the temperature where η ~ 1012 Pa s). SiO2 is the prototypical strong liquid, whereas o-terpheny1 is the representative fragile glass former. For metallic glasses (MGs), they generally behave in the intermediate dependence of viscosity or relaxation time on temperature [11–14]. Over the decades, liquid fragility of MGs has been extensively studied, and demonstrated to be closely correlated with physical/mechanical/

⁎ Corresponding author. Tel.: +86 10 6233 3899; fax: +86 10 6233 2508. E-mail address: [email protected] (F. Ye). 0022-3093/$ – see front matter © 2013 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.jnoncrysol.2013.11.023

processing properties, atomic structure and glass formation, etc. [15–22]. Therefore, the concise and correct approach to predict liquid fragility is necessary and significant. In fact, temperature dependence of equilibrium viscosity is the most reliable way to evaluate liquid fragility for MGs. Nevertheless, relaxation kinetics behavior detected by differential scanning calorimetry (DSC) will be more efficient, since it is effortless and easily accessible. In this letter, we intend to evaluate and compare the calorimetric models to predict liquid fragility for MGs. In this sense, a MG with composition of Ca65Mg15Zn20 was selected due to our recent detailed research on viscosity and kinetic glass transition of this alloy. Additionally, five typical MGs, i.e. Mg65Cu25Y10, Zr41.2Ti13.8Cu12.5Ni10Be22.5 (Vit 1), Zr46.75Ti8.25Cu7.5Ni10Be27.5 (Vit 4), Pt57.3Cu14.6Ni5.3 P22.8 and Pd43Cu27Ni10P20 were selected as well due to the availability of their reliable kinetic parameters. 2. Theoretical basis 2.1. Kinetic glass transition and relaxation time Upon continuous heating, a glass will experience the glass transition and relax into the supercooled liquid region. The elapsed time for glass transition at one heating rate is generally regarded as the structural (α−) relaxation time, τ, described as [12,15,23] τ ¼ ΔTg=φ;

ð1Þ

where ΔTg is the width of glass transition region and can be defined as the temperature range between the start (Tgs) and finish (Tgf) of the endothermic DSC event, and φ the heating rate. With increasing heating rate, the glass transition shifts clearly to the higher temperature that

L. Hu, F. Ye / Journal of Non-Crystalline Solids 386 (2014) 46–50

47

The m at a given Tgs can be calculated from [24,27] D  T g0  T gs m¼ : 2 T gs −T g0  ln10

ð6Þ

2.3. Activation energy ΔEg for glass transition From the fact that the glass transition temperature depends on the heating rate, which arises actually from the temperature dependence of the relaxation time (see model A), the activation energy ΔEg of glass transition or structural relaxation can be approximated as [30,31] ΔE dlnφ  ¼− g; R d 1=T gs

Fig. 1. Angell plots describing liquid viscosity vs. temperature normalized by T⁎g . The temperature dependence of viscosity for three liquid models, i.e. strong (solid line), intermediate (dash line) and fragile (dot line) liquid, and the corresponding typical glass-forming liquids are shown as guides to the eyes.

corresponds to a faster relaxation behavior, i.e. a reduction in τ. Then Tgs vs. τ is expected to follow the Vogel–Fulcher–Tamann (VFT) relation,    D  T0 τ ¼ τ0 exp ; T−T 0

ð2Þ

in which, τ0 is the theoretical infinite temperature relaxation time, while D⁎ and T0 are fitting parameters. D⁎ is commonly referred to as the fragility parameter of the material. The VFT temperature, T0, is the temperature at which the barriers with respect to flow approach infinity [7,12]. The fragility index, m, can be calculated by the definition formula [24], d logτ   m¼  T¼T g d T g =T

ð3Þ

where Tg⁎ is often replaced by Tgs at the heating rate of 1 K/min [15].

ð7Þ

where R is the gas constant, and Tgs is measured using heating rate within the range of 1–80 K/min as the significant temperature lags and deviation from the linear fitting are encountered beyond each extreme [9,30]. When conducting this approach, the glass must be cooled from well above to well below the glass transition region at a rate qc which is either equal or proportional to φ [31]. For MGs, researchers normally take no account of fictive temperature (Tf) and the effect of qc in the calculation of fragility using DSC [12,15,30,32–36]. Therefore, we employ the data obtained just by different φ scanning but overlook the qc constraint in this work [37,38]. Additionally, the non-Arrhenius behavior of fragile liquids near glass transition produces a larger activation energy for relaxation and exhibits the rapidly changed relaxation time upon temperature. Consequently, greater error will be introduced when evaluating ΔEg for fragile liquids by this model. Fortunately, the fragility index of MGs is mostly located in the range of 30 ≤ m ≤ 60, which demonstrates the intermediately strong nature of this category of glass-forming liquids [11,13]. The experimental results of most nonMGs have manifested the validity of this model to determine the activation energy of structural relaxation when compared with the activation energy for shear viscosity, ΔEg(η), in the glass transition region [9,31]. Hence, by using ΔEg instead of ΔEg(η) at one Tgs, the fragility index is written as ΔEg ðηÞ ΔEg d logη   ≈ m¼  ≈ ; T¼T  g R  T  ln10 R  T gs gs  ln10 d T g =T

ð8Þ

2.2. Heating rate dependence of glass transition temperature from which a small ΔEg generally implies a strong glass-forming liquid. For one thermally activated kinetic transition, the transition behavior is normally dependent on the heating rate. Lasocka pointed out that the glass transition exhibits obvious kinetic behavior, and the change of Tgs upon φ follows the relationship [25], T gs ¼ a þ b  lgφ;

ð4Þ

where a and b are the constants. However, the Lasocka's relation is not so convincing in describing the kinetic behavior over the extensive heating rates [26,27]. A VFT-type function is frequently used to fit this dependence, formulated as lnφ ¼ lnB−

D  T g0 ; T gs −T g0

ð5Þ

where D is the strength parameter, B is a parameter representing the time scale in the glass-forming system, and Tg0 is the asymptotic value of glass transition temperature in the limit of infinitely slow cooling and heating [26]. In most cases, Tg0 approximates to the Kauzmann temperature, Tk, which is widely considered to be the lower bound of the glass transition for thermodynamic reasons [26–29].

2.4. As a function of the width of glass transition The glass transition either on heating or cooling takes place over a range of temperatures and an observable glass transition width can be measured by DSC. The glass transition reflects the relaxation behavior from glassy state into metastable supercooled liquid upon heating, and thus the kinetic relaxation characteristic will determine the width of glass transition. A strong glass-forming liquid shows sluggish kinetics with temperature, which results in a wide glass transition region. Contrarily, a rapidly changed relaxation behavior upon temperature will lead to a narrow glass transition region [9,31,39]. Therefore, the DSC measured glass transition width, ΔTg, can be an indicator of liquid fragility. According to Moynihan's demonstration, the mean structural relaxation time and the shear viscosity both vary by a factor of about 102 over the glass transition region [39], viz.,     τ T gs η T gs   ¼   ≈102 : τ T gf η T gf

ð9Þ

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Thus a direct expression relating the reduced glass transition width, Tgs/ΔTg, with the fragility index, m, was obtained as m = 2·Tgs/ΔTg[40]. Recently, on the basis of the definition of fragility index and the relaxation nature of glass transition, we deduced a similar expression form between Tgf/ΔTg and m, m¼c

T gf ; ΔT g

ð10Þ

where c is constant. By fitting the data for a number of MGs, the obtained c values are all very comparable, with a mean of 2.17 [41]. It should be presented that these characteristic temperatures are mostly measured at the heating rate of 5–20 K/min. Calculations with Tgf/ΔTg at the heating rate beyond the above range by using Eq. (10) with the fixed value of c (=2.17) might introduce much error in the evaluation of m. 2.5. Kinetic liquid fragility correlated with thermodynamic properties It has been constantly proposed that the kinetic liquid fragility reflects nontrivial thermodynamic properties. Accordingly, different approaches to connect them have been explored [6,7,42–45]. In the framework of potential energy landscape, high density of minima results in the larger configurational component of the total heat capacity, hence the larger jump in Cp. Since the density of minima is always associated with the liquid fragility, the jump of Cp at Tgs is supposed to be an implication of fragility for glass-forming liquids [7,43,46]. In addition, it was lately shown that the kinetic fragility can be correlated with a scale quantity representing excess entropy, using data over entire fragility range. This successful connection requires much data, but these data are often unavailable [42,47]. Recently, Wang et al. proposed a simplification using readily available data to produce a convincing relation which correlated the m with a dimensionless combination of the jump of heat capacity at the glass transition temperature, ΔCp(Tgs) = Cliquid (Tgs) − Cglass (Tgs), and p p the enthalpy of fusion, ΔHf, for nonpolymeric glass-forming liquids, written as [45,48]

m ¼ 56 

  T gs  ΔC p T gs ΔH f

or m ¼ 40 

  ΔC p T gs ΔS f

:

ð11Þ

4. Results Fig. 2 shows the continuous heating DSC curves of Ca65Mg15Zn20 MG samples after merely eliminating thermal history and after isothermal annealing relaxation. Tgs marked as the intersection of extrapolated lines of Cp in Fig. 2 is commonly identified as the start of glass transition, which is closely related with the enthalpy/volume state [15,49]. Unlike Tgs, however, Tgf is almost independent on initial state [36,49,50]. Consequently, the measured Tgf after the two types of thermal treatments are consistent [47,50]. Particularly, upon slowly heating, the overshot of Cp is faint, which brings difficulty to detect and mark Tgf. In this case, Tgf measured after isothermal annealing can be taken as the finish of glass transition. Fig. 3 illustrates the VFT fit (dash line) for the experimental data using Eq. (2), and detailed fitting procedure can be found elsewhere (the errors from data reading and fitting processes, similarly hereinafter) [12,23]. The fitted line is very close to the Arrhenius relation, and the value of m is calculated to be ~36 by Eq. (3). The lnφ vs. Tgs data and fitted line are shown in Fig. 4. It can be clearly seen that Eq. (5) can well describe the heating rate dependence of glass transition (R2 = 0.99), and cover a wide range of heating rates [26]. The asymptotic temperature of glass transition, Tg0 = 299 K, is numerically close to Tk = 303 K. A plot of lnφ vs. reciprocal Tgs for Ca65Mg15Zn20 MG is presented in Fig. 5. As is evident from the result, the data is well fitted by Eq. (7) with R2 = 0.99. The slope of the fitted straight dash line gives the activation energy of the structural relaxation, ΔEg = 253 kJ/mol. Table 1 tabulates m values of Ca65Mg15Zn20 MG from models B, C and D (i.e. Eqs. (6), (8) and (10), respectively) calculated with the glass transition temperatures at different heating rates. For the calculation of fragility from thermodynamic properties, we only estimated m ≈ 47.2 at the heating rate of 20 K/min by Eq. (11) because of the limited Cp data. The data of heating rate dependence of glass transition of Mg65Cu25Y10, Vit 1, Vit 4, Pt57.3Cu14.6Ni5.3 P22.8 and Pd43Cu27Ni10P20 MGs are extracted from references [11,15,20,33,34,51–56]. The fitting procedures are in the similar manner with those of Ca65Mg15Zn20 MG, and are not shown here. It should be stated that, for Vit 1 and Vit 4, due to the insufficient data and extraction error, we fixed Tg0 = Tk when using Eq. (5) for fitting. Some evaluated m values of the above five MGs with Ca65Mg15Zn20 MG are listed in Table 2. 5. Discussion

Based on the high quality data, the calculated m was in well agreement with the measured m except for a few cases. Therefore, we take into consideration this correlation and assess its reliability in MGs.

For the Ca65Mg15Zn20 MG, model A predicts a single m from the tangent of Angell plot at Tg⁎, whereas the four other models give variable

3. Experimental Thermal analyses of the Ca65Mg15Zn20 samples were performed in a TA DSC Q2000 under flowing high purity argon with different heating rates of 1, 2.5, 5, 10, 20, 40, and 80 K/min. The DSC system was calibrated for temperature and enthalpy using indium and zinc standards for each heating rate. Al pans were utilized for the thermal analyses, and Al pans with and without samples were scanned in the identical thermal condition to strip the baseline. All samples were firstly heated to above the glass transition temperature to eliminate the effects of thermal history and assure the same initial state, and then rapidly cooled to 273 K. After the preheating, two types of thermal treatments were applied to scan the samples: (1) the pre-annealed samples were continuously heated to 650 K with various heating rates; (2) the pre-annealed samples were firstly heated to 363 K at the heating rate of 20 K/min and isothermally annealed for a certain amount of time, then rapidly cooled to 273 K, and subsequently heated to 650 K at different heating rates. The heat capacity measurements were made by DSC using sapphirereference model. The heat flows of the empty crucible, of a sapphire standard and of the sample were measured under the identical condition.

Fig. 2. The continuous heating DSC curves of Ca65Mg15Zn20 MG after two types of heating treatment at the heating rate of 40 K/min. The characteristic temperatures were marked as the intersection of dash lines.

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49

Fig. 5. A linear fitting plot of lnφ vs. reciprocal Tgs using Eq. (7). The slope of the fitted dash line gives the activation energy of glass transition.

Fig. 3. Angell plot (dash line) from the dependence of relaxation time τ on T⁎g /T fitted by Eq. (2). T⁎g is taken as Tgs at the heating rate of 1 K/min.

values of m in dependence of heating rates. It is worthwhile noting that the fragility is an intrinsic property of the equilibrium (metastable) liquid. As such, it cannot be heating rate dependent. The same phenomena lie in the five other MGs as well. These facts reveal the common features of these models to predict m. Therefore, some guides on how to apply these models and pick the trustworthy value of m will be essential and discussed below. Based on our viscosity measurements and kinetic glass transition analyses, m = 42 is given for Ca65Mg15Zn20 MG [57], which can be taken as the standard value of this glass. m calculated by model A is ~36, which is a little smaller than 42. Models B and D predict the values of m showing apparent heating rate dependence, and these values decrease when substituted into the formulas by the characteristic temperatures at faster heating rates. Especially for Mg65Cu25Y10 MG, m is approximated by model B as 82 and 26 at the heating rate of 1 and 80 K/min, respectively. So large difference of m further raises a caution when using model B to evaluate the fragility. Excitingly, m from model C changes a little with heating rate, giving a mean value of 35.7 with error

within ±1 for Ca65Mg15Zn20 MG. This value is nearly equal to 36 from model A. Model E gives a slightly larger m ≈ 47.2. For overall comparisons and analyses, Table 2 shows the obtained m values of the six MGs. Model A provides the more reliable m with the standard value than others. This fact indicates the temperature dependence of relaxation time is a more effective way to evaluate the fragility of glass-forming liquids. Secondly, model C can give the almost invariable and reliable fragility index upon various heating rates but only requires a simple linear fitting [9,15,31]. For Pd43Cu27Ni10P20, however, the calculated m (≈ 44.9) is significantly smaller than the standard m = 65. This is because the rapid change of relaxation kinetics results in the nonlinear relation between lnφ and 1/Tgs. Despite this, model C cannot be arbitrarily discarded to predict m for the relatively fragile MGs just because of the above one failure. The marking of Tgs is somewhat influenced by the initial state of MG samples (i.e. Cp overshoot), and eventually gives rise to fitting deviations. In consequence, the applicability of model C on fragile MGs still needs more assessments. Nevertheless, the rarely used model is indeed adequate to predict liquid fragility for relatively strong glass-forming liquids (m ≤ 52 in this paper), and the results yet demonstrate its validity. The m values calculated from models B and D at 20 K/min are shown in comparison with the standard values (see Table 2). It can be found that the popularly used glass transition temperature at 20 K/min seems inappropriate to be substituted into formulas for predicting fragility, which always produces the large errors and apparently indicates a stronger liquid behavior. The m much closer to the standard value and their corresponding heating rates are also listed in Table 2. It is revealed that, for both models B and D, the calculated m from Eqs. (6) and (10) at φ = 5 K/min are much more approximate to the standard values under the acceptable error range. This instruction is very important to predict

Table 1 Liquid fragility index (m) calculated using models B, C and D at different heating rate dependence of glass transition temperatures for Ca65Mg15Zn20 MG. The maximal errors for all m values are within ±0.5.

Fig. 4. Heating rate dependence of glass transition temperature, i.e. lnφ vs. Tgs, was fitted by Eq. (5) (dash line).

Heating rate, φ (K/min)

m from Model B

m from Model C

m from Model D

80 40 20 10 5 2.5 1

27.7 30.6 33.5 35.8 38.2 42.5 45.9

34.7 35.1 35.5 35.7 36.0 36.4 36.6

35.8 37.1 38.4 42.3 44 45.3 48.3

50

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Table 2 Fragility index (m) predicted by all models for the six MGs. The standard values of liquid fragility are listed under the glass compositions. The numbers shown in parentheses denote the heating rate. The maximal errors for the unspecified m values are within ±1.5. Glass

m from Model A

m from Model B

m from Model C

m from Model D

Ca65Mg15Zn20 (m = 42)

36

35.7 ± 1

Mg65Cu25Y10 (m = 49)

48

Vit 1 (m = 50) Vit4 (m = 44)

48

Pt57.3Cu14.6Ni5.3 P22.8 (m = 52) Pd43Cu27Ni10P20 (m = 65)

52

33.5 (20) 38.5 (5) 42.5 (2.5) 39.2 (20) 45.7 (10) 54 (5) 40 (20) 49 (5) 29.3 (20) 39 (5) 47.2 (2) 49.8 (15) 54.2 (6) 34.5 (20) 57.6 (5) 72.8 (2)

38.4 (20) 42.3 (10) 44 (5) 41.2(20) 44.8 (5) 52.4 (1) 42.9 (20) 48.2 (5) 34.9 (20) 40 (5) 41.4 (1) 50.5 (30) 54.4 (6) 50.5 (40) 67.3 (5) 70.4 (1)

44

45.5 ± 1.3

46.6 ± 1.2 41.5 ± 1.2

50.6 ± 0.9 44.9 ± 1

m from Model E 47.2 (20)

34.5 (20)

85 (20) 55 (20)

67.1 (20) 121.9 (20)

fragility when making use of these two models. However, it is necessary to explicitly point out the flaws of the two models. Although model B itself derived from the kinetic characteristics of glass transition, this model still lacks clear physical basis. Besides, when putting the heating rate dependence of Tgs into Eq. (6), the calculated m will not always be the slope of Angell plot at temperature (Tg⁎) where η ~ 1012 Pa s but a slope more or less deviate from Tg⁎. Thereby, the calculated m shows such a heating rate dependence. For model D, the empirical correlation (i.e. Eq. (10)) deduced from the linear fitting m vs. Tgf/ΔTg which are measured within the particular range of heating rate, therefore the calculation of m varies with a wide range of heating rate. Like the very recent reports that the calculated m using Eq. (11) showed large difference from the measured values, in this study, model E failed to evaluate liquid fragility for these MGs. This is explained by hidden transitions in the supercooled regime, variations in bonding and fragile-to-strong transition during the vitrification of MGs [47,53,58,59]. Throughout the above discussions, the calorimetric models are valid to predict liquid fragility for MGs, except from the thermodynamic properties. Only considering the ease of use, model D is undoubtedly the most direct and convenient way. Regarding to the reliability, models A and C provide the trustworthy fragility index, wherein model C is more concise and deserves to be extensively applied to relatively strong MGs though its applicability on the fragile extreme needs further investigation. However, model B gives rise to the large discrepancy when using the temperatures at different heating rates, and thus one must be cautious to employ it for evaluating fragility.

6. Conclusions In this paper, attention has been focused on the study of prediction of liquid fragility by calorimetric analyses. We summarized five models referring to kinetic glass transition and thermodynamic properties, and applied these models to six MGs. The results demonstrate that calorimetric measurements can provide effective values of fragility index for MGs, except from the thermodynamic properties. However, cautions must be taken when using models B and D to predict liquid fragility since the calculated m from them are strongly influenced by which Tgs was substituted into the formulas. In our study, m calculated at the heating rate of 5 K/min appears to be more reliable for models B and D. In addition to model A which describes the temperature dependence

of relaxation time, we recommend that model C from activation energy of glass transition be an efficient approach to predict m for relatively strong MGs (m ≤ 52 in this paper). Acknowledgments The authors gratefully acknowledge the financial support from the Fundamental Research Funds for the Central Universities (FRF-TP-09026B), the Program for Changjiang Scholars and Innovative Research Team in University, the Specialized Research Fund for the Doctoral Program of Higher Education (20100006110013), and the National Natural Science Foundation of China (No. 51010001). References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25] [26] [27] [28] [29] [30] [31] [32] [33] [34] [35] [36] [37] [38] [39] [40] [41] [42] [43] [44] [45] [46] [47] [48] [49] [50] [51] [52] [53] [54] [55] [56] [57] [58] [59]

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