Influence of the heating rates on the correlation between glass-forming ability (GFA) and glass stability (GS) parameters

Journal of Non-Crystalline Solids 390 (2014) 70–76

Contents lists available at ScienceDirect

Journal of Non-Crystalline Solids journal homepage: www.elsevier.com/ locate/ jnoncrysol

Influence of the heating rates on the correlation between glass-forming ability (GFA) and glass stability (GS) parameters Thiago V.R. Marques a, Aluisio A. Cabral a,b,⁎ a b

Postgraduate Program in Materials Engineering, Federal Institute of Maranhão — IFMA, 65030-001 São Luis, MA, Brazil Department of Physics — DEFIS, Federal Institute of Maranhão — IFMA, 65030-001 São Luis, MA, Brazil

a r t i c l e

i n f o

Article history: Received 11 September 2013 Received in revised form 5 February 2014 Available online xxxx Keywords: Glass-forming ability; Glass stability parameters; Silicate glasses; Heating rates; Internal nucleation

a b s t r a c t It is well known that some glass stability (GS) parameters which can be used to estimate glass-forming ability (GFA) are readily obtained by means of Differential Scanning Calorimetry (DSC) or Differential Thermal Analysis (DTA) experiments. In this work, the GS parameters proposed by Hrubÿ (KH), Weinberg (KW), Lu & Liu (KLL), Long et al. (KLX) and Zhang et al. (KZW) were determined for the following silicate glasses: Li2O·2SiO2, BaO·2SiO2, Na2O·2CaO·3SiO2 and 2BaO·TiO2·2SiO2. Monolithic specimens of these glasses were placed in Pt–Rh crucibles and heated in air in a DSC furnace at different heating rates (ϕ = 2, 5, 10, 15, 20, 25 and 30 °C/min) from room temperature up to their respective melting temperatures. The results indicate that all the parameters under study vary significantly as a function of the heating rate, and that KH, KW and KLL increase with ϕ while KZW and KLX decrease. To evaluate the reliability of these parameters in estimating GFA, they were correlated with the critical cooling rates (RC) calculated by the continuous cooling (RCC) and “nose” (RCN) methods using experimental data of steadystate nucleation and crystal growth rates, free energy per unit of volume, and viscosity of each vitreous material investigated. It was found that the correlation between the GFA (RCC or RCN) and all the parameters studied here persists, even if the changing of the GS parameters with the heating rates is taken into account. © 2014 Elsevier B.V. All rights reserved.

1. Introduction It is well known that all materials can become vitrified if they are cooled fast enough to prevent them from crystallizing. The cooling rate that produces the lowest detectable degree of crystallization, XC, (usually assumed to be 10−6), is called the critical cooling rate for glass formation, RC [1]. This parameter is a measure of the glass forming ability of the substance, and can be calculated by the “nose” and continuous cooling methods, hereinafter referred to as RCN and RCC, respectively [1,2]. Several studies have been conducted to measure RC of some glass systems, as reported by [3–6]. These studies combined heating and cooling experiments performed in electrical [4–6] or DTA (DSC) [3,4,6] furnaces. However, it is well known that upon cooling, any solid impurity or mechanical perturbation can induce fast crystallization of the melt. Considering that most of these experiments were carried out using platinum crucibles, heterogeneous surface crystallization induced by contact between the melts and the DSC crucibles cannot be disregarded. Therefore, these techniques can lead to irreproducible crystallization temperatures upon cooling, thus resulting in overestimated critical cooling rates.

⁎ Corresponding author at: Federal Institute of Maranhão – IFMA, Department of Physics – DEFIS, 65030-001 São Luis, MA, Brazil. Tel.: +55(98)81621964. E-mail addresses: [email protected], [email protected] (A.A. Cabral).

http://dx.doi.org/10.1016/j.jnoncrysol.2014.02.019 0022-3093/© 2014 Elsevier B.V. All rights reserved.

On the other hand, some empirical criteria have been formulated to correlate RC (glass forming ability — GFA) with glass stability — GS (on a given heating path) using quantities that are more easily measured by DTA or DSC experiments, since they are obtained from the crystallization (Tx), glass transition (Tg), and melting (Tm) temperatures. These GS parameters include those ones investigated in this study were the following:
 KH ¼ T x −T g =ðT m −T x Þ

Hrubÿ [7]

ð1Þ


 ¼ T x −T g =T m

Weinberg [8]

ð2Þ


 KLL ¼ T x = T g þ T m

Lu and Liu [9]

ð3Þ

KW



 KLX ¼ T g =T x −2:T g = T g þ T m

Long et al. [10]

ð4Þ


 KZW ¼ T g = 2  T x −T g −T g =T m Þ

Zhang et al. [11]

ð5Þ

Several works have demonstrated that there is a strong correlation between some glass stability (GS) and GFA parameters. For instance, Cabral et al. [12], using experimental nucleation rate and crystal growth data of Li2O·2SiO2·(LS2)·Na2O.2CaO·3SiO2 (NC2S3), 2Na2O·CaO·3SiO2

T.V.R. Marques, A.A. Cabral / Journal of Non-Crystalline Solids 390 (2014) 70–76

(N2CS3) and BaO·2SiO2 (BS2) glasses, concluded that the Hrubÿ parameter (KH) can be used to estimate the glass forming ability; since the higher the value of RCN the lower the value of KH and the better the glass former obtained. A few years later, Cabral et al. [13] confirmed the correlation between RC and KH parameters, where RC was measured according to a modified version of the method proposed by Barandiaran and Colmenero [3] to seven glasses with homogeneous nucleation: Li2O·2SiO2, Na2O·2CaO·3SiO2, 2Na2O·CaO·3SiO2, BaO·2SiO2, Li2O· 2SiO2·OH (LS2-OH), 2BaO·TiO2·2SiO2 (B2TS2) and 0.44Na2O·0.56SiO2 (NS). Nascimento et al. [14] found that, besides KH, the GS parameters proposed by Weinberg and Lu and Liu, hereinafter called KW and KLL, respectively, can also be used to estimate the glass forming ability. The experiments were carried out for six glasses that nucleate heterogeneously: GeO2 (G), Na2O·2SiO2 (NS2), PbO·SiO2 (PS), CaO·Al2O3·2SiO2 (CAS2), CaO·MgO·2SiO2 (CMS2) and 2MgO·2Al2O3·5SiO2 (M2A2S5), and two others that nucleate homogeneously: Li2O·2SiO2 (LS2) and Li2O·2B2O3 (LB2). More recently, Ferreira et al. [15] reassessed the parameters studied by [14] for glasses of the Li2O–B2O3 system with compositions of 20 to 66.7 mol% of Li2O. They found that KH, KW and KLL are suitable for estimating the glass-forming tendency as a function of composition. Besides these GS parameters, Long et al. [10] and Zhang et al. [11] recently proposed KLX and KZW, respectively, to estimate the glassforming tendency. In both cases, the correlations with RC were tested using critical cooling rates selected from the literature to the following metallic glasses: Au-Ca-Ce-Cu-Fe-La-Mg-Ni-Pd-Pr-Ti-Zr [10], and Cu-, Ca-, Mg-, Ti-, Pd-, La-, Gd-, Pr-, Y-, Co-, Zr-, Fe- and Ni [11]. According to them, both parameters are inversely proportional to RC, i.e., RC increases with an increasing KZW and KLX. Nevertheless, these analyses were basically supported on the best linear fits obtained from the plots of Log (RC) vs KLX or Log(RC) vs KZW. Therefore, as the factor r2 became closer to 1, the GS parameter should be better to estimate the GFA. As one can see from Eqs. (1) to (5), KH, KW, KLL, KLX and KZW are defined in terms of Tg, Tx and Tm. In order to evaluate their sensitivity to these temperatures, Kozmidis-Petrovic [16–18] rewrote each one of these parameters as a function of r = Tx / Tg and m = Tm / Tg, which correspond to the range between the crystallization and glass transition events, and melting and glass transition phenomena, respectively. She concluded that all of these parameters were more sensitive to changes in r than in m. Concerning the order of sensitivity, the results demonstrated that KH was the most sensitive, while KLL almost did not change. Using some experimental data of critical cooling rates in the literature for some glasses, the author also compared the correlation between RC and the GS parameters investigated. Finally, she pointed out that there is not one parameter that can be declared the best, since they have a very similar correlation coefficient (r2). There is a plethora of experimental results demonstrating that glass transition and crystallization temperatures are strongly dependent on the heating rate (e.g., [19,20]). Nevertheless, as one can see from this extensive review of the literature, no study so far has investigated the dependence of glass stability parameters on heating rates and verified if the existing correlation between the GFA and GS persists. Therefore, in this study, numerical calculations of RC were made for the LS 2 , BS 2 , NC 2 S 3 and B 2 TS 2 glasses, using the continuous cooling and “nose” methods. DSC runs were then performed in a wide range of heating rates (2, 5, 10, 15, 20, 25, 30 °C/min) to study the behavior of several stability parameters as a function of these heating rates, which have not previously been investigated. For the first time, calculations of RCC are presented using experimental data of crystal nucleation (Ist) and growth rates (U), viscosity (η) and thermodynamics (ΔG), which are available in the literature for each glass investigated here. Finally, the main objective of this study was to evaluate the persistence of the correlation between RC and the GS parameters KH, KW, KLL, KZW and KLX, even at very different heating rates.

71

2. Theory According to the Classical Nucleation Theory (CNT), if one neglects a possible breakdown of the Stokes–Einstein equation at deep undercooling (≈ 1.1–1.2 T g ) and assumes that the surface tension does not change with temperature and nucleus size [21], the steady-state crystal nucleation rate I st (T) may be expressed by [22]:  Ist ðT Þ ¼

!  KT K′σ 3 : exp − η TΔG2

ð6Þ

where T is the absolute temperature, η the viscosity, ΔG the bulk free energy difference between liquid and crystal, σ the liquid–crystal surface tension, and K and K′ are constants. Assuming spherical nuclei, K′ ≡ 16πV2m/3, where Vm is the molar volume of the crystalline phase). Plots of ln(Ist·η / T) versus 1 / (TΔG2), traced using experimental data for Ist, η and ΔG, should produce straight lines whose slope and intercept can be used to evaluate σ and K, respectively. As the temperature dependence of the nucleation rate is well described by Eq. (6), these fitting parameters can be used to obtain an equation to evaluate Ist at a given temperature. However, previous results of Cabral et al. [12] demonstrated that the “nose” temperature, Tn, of some silicate glasses (for instance, BS2 and NC2S3) occurs at values much higher than those where the experimental crystal nucleation rates are available. Therefore, it is required to estimate the crystal nucleation rates in a wide temperature range. The formula for representing the most common crystal growth mechanism, screw dislocation, is the equation proposed by Uhlmann [23]:

U ðT Þ ¼

   ″ K ðT m −T ÞT ΔG 1− exp − η RT

ð7Þ

where K″ is constant. According to the Johnson–Mehl–Avrami–Erofeev–Kolmogorov (JMAEK) [24–29] equation, where spherical crystals nucleate and grow simultaneously inside a glass sample treated isothermally at a temperature T, the evolution of the crystallized fraction over time, x(t), is given by: 8 2t 33 9 > > Z
 < 4π Zt = ″ ″ ′ Iðt Þ4 U t dt 5 dt xðt Þ ¼ 1− exp − > > : 3 ; t0

t

ð8Þ

0

where t0 is the initial time, and I(t) and U(t) are, respectively, the crystal nucleation and growth rates as a function of time. Assuming small volume fractions (XC b b 1), constant cooling rates (RC = T / t) and that I and U are not time dependent, Eq. (8) can be readily rewritten as:

RCC ¼

2 33 8 ZT f ZT f
 > > > ′ ′7 6 > > 4π: IðT Þ4 U T dT 5 dT > > > <

91=4 > > > > > > > > =

> > > > > > > > :

> > > > > > > > ;

Tm

T

3  XC

ð9Þ

which assumes that T = Tm at t = t0. Tf corresponds to the temperature at the end of cooling and RCC is the critical cooling rate calculated by the continuous cooling method [2].

72

T.V.R. Marques, A.A. Cabral / Journal of Non-Crystalline Solids 390 (2014) 70–76

a) LS2

b) BS2

3

1,0 1

DSC, µV/mg

DSC, µV/mg

2

0 -1 -2

0,5

0,0

-0,5

-3 -1,0 -4 400

500

600

700

800

900

1000

1100

T, °C

c) NC2S3

600

700

800

900 1000 1100 1200 1300 1400

T, °C

d) B2TS2 4

2

3 2

DSC, µV/mg

DSC, µV/mg

1 0 -1 -2

1 0 -1 -2

-3

-3

-4 500

600

700

800

-4 600 700 800 900 1000 1100 1200 1300 1400

900 1000 1100 1200 1300

T, °C

T, °C

Fig. 1. Part of the DSC curves obtained for the (a) LS2, (b) BS2, (c) NC2S3, and (d) B2TS2 glasses when heated at different rates (15, 20, 25 and 30 °C/min).

In addition, RC can be also calculated by the TTT (time–temperaturetransformation) diagrams [1]. Assuming the internal nucleation of spherical crystals, the time required to attain a given crystallized fraction, XC, at a temperature T, is given by:

XC ¼

π 3 4 IðT ÞU ðT Þt 3

RCN ¼

ðT m − T n Þ

h

π 3

i1=4 IðT n Þ:U 3 ðT n Þ

ð12Þ

X C 1=4

ð10Þ 3. Experimental procedures and calculations

By definition, RCN corresponds to the slope of the straight line that intersects the “nose” of a TTT diagram. Therefore, RCN is given by:

RCN ¼

Combining Eqs. (10) and (11), one obtains:

T m −T n tn

ð11Þ

where Tn and tn are the temperature and time at the “nose” temperature, respectively.

3.1. Preparation of the glasses Stoichiometric compositions of the chosen glasses were prepared by the conventional melting technique. The fresnoite glass (2BaO·TiO2·2SiO2—B2TS2) was melted from reagent grade barium carbonate, BaCO3, TiO2, and SiO2 (Merck, Darmstadt, quartz — Germany) in an induction furnace, as detailed by Cabral et al. [30]. Unmelted or crystalline particles in the as-quenched glass were not detected by X-ray diffraction (XRD) or scanning electron microscopy (SEM) experiments.

Table 1 Values of Tg, Tx (onset) and Tm obtained for LS2, BS2, NC2S3 and B2TS2 glasses under different heating rates. Errors of around 5% were detected. ϕ (°C/min)

2 5 10 15 20 25 30

LS2

BS2

NC2S3

B2TS2

Tg (K)

Tx (K)

Tm (K)

Tg (K)

Tx (K)

Tm (K)

Tg (K)

Tx (K)

Tm (K)

Tg (K)

Tx (K)

Tm (K)

713.80 720.50 726.35 729.40 731.15 734.75 741.25

881.85 913.55 937.60 953.25 960.05 971.40 974.90

1305.6 1310.6 1317.4 1322.3 1326.2 1330.7 1337

953.60 953.95 962.55 962.65 962.60 981.60 985.20

1083.60 1107.75 1131.2 1144.25 1156.35 1164.55 1171.05

1673.70 1683 1686.8 1689 1692.3 1695.2 1693

839.15 843.65 848.05 850.45 854.10 857.05 858.40

939.50 962.40 982.85 996.25 1007.45 1017.50 1026.45

1560 1564 1567.7 1572.4 1574.5 1577 1580.7

– 971.9 975.35 979.45 979.35 979.4 980.65

1022.40 1038.95 1050.70 1059.20 1067.70 1071.30 1072.85

1705.8 1708.34 1710.82 1708.82 1711.34 1715.54 1717.26

T.V.R. Marques, A.A. Cabral / Journal of Non-Crystalline Solids 390 (2014) 70–76

a) LS2

73

b) BS2 0,5

0,8

Glass stability

Glass stability

0,4 0,6

0,4

0,2

0,3

0,2

0,1

0,0

0,0 0

5

10

15

20

25

30

35

0

5

10

φ, °C/min

15

20

25

30

35

25

30

35

φ, °C/min

c) NC2S3

d) B2TS2

0,5 0,4

Glass stability

Glass stability

0,4

0,3

0,2

0,3

0,2

0,1

0,1

0,0

0,0 0

5

10

15

20

25

30

35

0

5

10

15

20

φ, °C/min

φ, °C/min

Fig. 2. Dependence of the GS parameters on the heating rates for the: (a) LS2, (b) BS2, (c) NC2S3, and (d) B2TS2 glasses.

The other silicate glasses were synthesized in platinum–rhodium (Pt–Rh) crucibles in an electric furnace. The chemicals used were analytical grade barium, sodium, lithium, and calcium carbonates (VETEC, Synth and Mallinckrodt) and ground Brazilian quartz from Vitrovita (N 99.99% SiO2). The temperatures and times used, as well the chemical analyses results, are described in detail in [31,32]. XRD and optical microscopy (OM) analyses showed no evidence of unmelted or crystalline particles in the as-quenched glass. 3.2. DSC experiments

The specimens were heated at different rates (ϕ = 2, 5, 10, 15, 20, 25 and 30 °C/min) from room temperature to their corresponding melting temperatures.

3.3. Calculation of RC The values of RCN and RCC for each glass were calculated from the experimental data of crystal nucleation, Ist, and growth rates, U, viscosity, η, and thermodynamic, ΔG, which are available in the literature for each glass investigated here, as one can see in [30,33–39]. The volume

To determine Tg, Tx and Tm, monolithic samples of each glass (3 × 3 × 2 mm) were placed in platinum crucibles and heat-treated directly in a NETZSCH STA 449C thermal analyzer (NETZSCH-Gerätebau GmbH, Selb, Germany) equipped with a computer interface for storing and analyzing thermal data. All the measurements involved samples weighing ~38–40 mg taken from the same batch. The device was calibrated for temperature prior to and periodically during the measurements at all the heating rates, using RbNO3, KClO4, CsCl, K2CrO4, and BaCO3 standards. Table 2 Differences (%) between the largest and lowest results of the GS parameters – KH, KLL, KW, KZW and KLX – in terms of percentage, and values of RCC and RCN obtained for each silicate glass under study. Glass

LS2 BS2 NC2S3 B2TS2

Difference (%)

Critical cooling rates, K/s

KH

KLL

KW

KZW

KLX

Rcc

80.05 71.78 94.44 45.70

9.08 7.19 8.31 3.05

40.94 43.83 67.28 37.77

67.91 38.62 37.51 14.11

66.83 34.10 31.34 12.73

0.02 0.12 0.2 0.47

Rcn ± ± ± ±

0.004 0.02 0.04 0.09

0.86 2.23 4.1 10.04

± ± ± ±

0.17 0.44 0.82 2

Fig. 3. Plot of ln(Iη / T) × 1 / (TΔG2) for the LS2 glass using Ist and ΔG from [33] and η from [34]. The best fit obtained was given by y = 108.36-3.02 × 1013x, while the correlation factor was r2 = 0.91.

74

T.V.R. Marques, A.A. Cabral / Journal of Non-Crystalline Solids 390 (2014) 70–76

fraction was chosen as 10−6. For the first time, calculations of RCC using these experimental data are shown. Using the fitting equation obtained from the ln(Ist.η / T) versus 1 / (TΔG2) curves plotted for the silicate glasses studied here, the Ist(T) values were calculated in the temperature intervals in which nucleation rates are unknown, as proposed by Cabral et al. [12]. The experimental values of U(T) are known within a wide temperature interval only for the LS2 glass [33]. In the other cases, the growth rates were determined by calculating K″ (Eq. (7)) in order to obtain a curve that would better fit the experimental U(T) curve.

4. Results Fig. 4. Crystal growth rates for LS2 glass obtained by different authors, as one can see [33]. ΔG and η were selected from [33] and [34], respectively. The lines correspond to the spiral growth model with the fitting parameter K″ = 1.54 × 10−11. The correlation factor was approximately r2 = 0.93.

Since the heating rates applied here involve a large interval, only the DSC curves obtained for each glass at ϕ = 15, 20, 25 and 30 °C/min are presented, as one can see in Fig. 1 (a–d). The corresponding values of Tg (onset) Tx (onset) and Tm (endpoint) are given in Table 1.

a

b

c

d

e

Fig. 5. RCC estimated by continuous cooling method versus GS parameters obtained at 5, 15 and 30 °C/min: (a) KH; (b) KLL; (c) KW; (d) KZW; and (e) KLX.

T.V.R. Marques, A.A. Cabral / Journal of Non-Crystalline Solids 390 (2014) 70–76

Fig. 2(a–d) shows the behavior of the GS parameters as a function of heating rates. In percentage terms, the differences between the values obtained for the samples heated at 2 °C/min and 30 °C/min are presented in Table 2. 5. Discussion As expected, Fig. 1 indicates that Tg and Tx shift to higher temperatures as the heating rate increases, due to the shorter time for nucleation and growth. Therefore, an increasing residual glass fraction is observed [19,40,41]. It is also observed that the endpoint for the melting temperature tends to increase with the heating rate. Due to the extensive overlapping between the crystal nucleation and growth curves of the B2TS2 glass, its Tg was not detectable at 2 °C/min, since during longer times, more critical nuclei are formed during this passage and are available for growth, as previously demonstrated by Cabral and colleagues [30,39,40]. Hence, its residual glass fraction is almost negligible at very slow heating rates. In Fig. 2(a–d), note that KH, KW and KLL increased with the heating rates while KZW and KLX decreased with ϕ in all the silicate glasses investigated here. However, as can be seen from the data in Table 2, the KH, KW, KZW and KLX parameters of these glasses varied considerably, while KLL varied by less than 10%. Therefore, one can conclude that these GS parameters change according to the following increasing order: KLL b KLX b KZW b KW b KH. This order of changing is in excellent agreement with those results reported by Kozmidis-Petrovic [17,18], who demonstrated that KH was the most sensitive to changes of r (=Tx / Tg), while KLL almost did not change. Given that the parameters studied here exhibited opposite tendencies and changed significantly with the heating rates, we decided to calculate the critical cooling rates of each glass by the “nose” and continuous cooling methods in order to ascertain if the correlation between GFA and GS parameters remains valid. Based on Eqs. (9) and (11), one finds that the values of RCC and RCN are strongly dependent on U(T). As mentioned earlier, among all the glasses investigated here, the experimental growth rates are known in a wide temperature range only for LS2. Therefore, calculations of RCC were initially carried out for this glass and compared with those values obtained by Cabral et al. [12]. To our knowledge, this paper presents for the first time calculations of RCC using experimental data of Ist(T), U(T), η(T) and ΔG(T). Figs. 3 and 4 illustrate the fitting curves obtained for the calculation of I(T) and U(T) in a wide temperature interval. Considering that the nose of the TTT curve occurs at T/Tm ≈ 0.6–0.7 [12], which corresponds to the dashed regions in these figures, one can observe that the fitted and experimental values of I(T) and U(T) in this temperature interval are in reasonable agreement. Therefore, the same procedure was adopted to calculate the nucleation and crystal growth rates of the other silicate glasses studied here. The last two columns of the Table 2 list the values of RCC and RCN obtained for each glass. A comparison of the RCN values obtained here for the LS2, BS2 and NC2S3 glasses and those obtained by Cabral et al. [12] clearly indicates that the results are in very good agreement. The minor differences are due to the fact that we used experimental Ist(T), U(T) and η(T) data obtained by different authors, especially in the case of LS2 and BS2. Moreover, our results clearly demonstrate that the nose method systematically overestimates the critical cooling rates by at least one order of magnitude, as it was theoretically predicted by Weinberg et al. [2]. The errors were calculated assuming a Student's t-distribution with 95% confidence interval. From Table 2, one can also see that, regardless of the method employed to calculate RC (“nose” or continuous cooling), the tendency for good glass-forming ability appears to remain unchanged, i.e., LS2 N BS2 N NC2S3 N B2TS2 [12,13]. Finally, to test the validity of all the parameters, we plotted the RCN and RCC versus KH, KW, KLL, KZW and KLX for each silicate glass. This

75

paper shows only the plots obtained for each GS parameter for some heating rates (5, 15 and 30 °C/min), which are depicted in Fig. 5 (a–e). Since the RCN and RCC values points out to the same glass-forming tendency, only plots of RCC versus each GS parameter are shown in Fig. 5 (a–e). From Fig. 5(a–e) one observes that GFA (RC) changes differently with the GS parameters, i.e., for the parameters KH, KW and KLL the best glass former (LS2, in our case) has the lowest value of GFA and the highest value of GS, while for KZW and KLX the lowest value of GFA corresponds to the lower value of GS. This result is in excellent agreement with the results reported by Cabral et al. [12,13], Nascimento et al. [14], Long et al. [10] and Zhang et al. [11]. Therefore, we conclude that the correlation between the GFA and all parameters persists, even when taking into account the change in GS parameters as a function of the heating rates. Moreover, our results clearly confirm that all glass stability parameters studied here can be used to estimate the GFA of silicate glasses that nucleate internally. 6. Conclusions For the first time, experimental values of nucleation and growth rates were used to calculate RCC. In addition, RCN was estimated using TTT diagrams. The results, which are consistent with earlier findings [12,13] and with our laboratory experience of the four glasses under study, indicate that LS2 is the best glass-forming system and B2TS2 the most recalcitrant one. Although the GS parameters change as a function of heating rates, the strong correlation between KH, KLL, KW, KZW and KLX parameters of glass stability, based on the values of Tg, Tx, and Tm and the glass-forming tendency, is still valid. Hence, all these GS parameters can be used to estimate the GFA of silicate glasses that exhibit internal nucleation. Acknowledgments The authors are indebted to the following Brazilian research funding agencies for their financial support of this work: CNPq (National Council for Scientific and Technological Development), Process # 2013-0/ 304542, CAPES (Federal Agency for the Support and Improvement of Higher Education), Process # 2008/2365, and FAPEMA (Maranhão Foundation for Scientific Research and Development), Process # 2012/ 002717. References [1] D.R. Uhlmann, J. Non-Cryst. Solids 7 (1972) 337. [2] M.C. Weinberg, B.J. Zelinski, D.R. Uhlmann, E.D. Zanotto, J. Non-Cryst. Solids 123 (1990) 90. [3] J.M. Barandiaran, J. Colmenero, J. Non-Cryst. Solids 22 (1976) 367. [4] C.S. Ray, D.E. Day, J. Am. Ceram. Soc. 67 (1984) 806. [5] W. Huang, C.S. Ray, D.E. Day, J. Non-Cryst. Solids 86 (1986) 204. [6] C.S. Ray, S.T. Reis, R.K. Brow, W. Holand, V. Rheinberger, J. Non-Cryst. Solids 351 (2005) 1350. [7] A. Hrubÿ, Czechoslov. J. Phys B22 (1972) 1187. [8] M.C. Weinberg, Phys. Chem. Glasses 35 (1994) 119. [9] Z.P. Lu, C.T. Liu, Acta Mater. 50 (2002) 3501. [10] Z. Long, G. Xie, H. Wei, X. Su, J. Peng, P. Zhang, A. Inoue, Mater. Sci. Eng. A 509 (2009) 23. [11] P. Zhang, H. Wei, X. Wei, Z. Long, X. Su, J. Non-Cryst. Solids 355 (2009) 2183. [12] A.A. Cabral, C. Fredericci, E.D. Zanotto, J. Non-Cryst. Solids 219 (1997) 182. [13] A.A. Cabral, A.A.D. Cardoso, E.D. Zanotto, J. Non-Cryst. Solids 320 (2003) 1. [14] M.L.F. Nascimento, L.A. Souza, E.B. Ferreira, E.D. Zanotto, J. Non-Cryst. Solids 351 (2005) 3296. [15] E.B. Ferreira, E.D. Zanotto, S. Feller, G. Lodden, J. Banerjee, T. Edwards, M. Affatigato, J. Am. Ceram. Soc. 94 (2011) 3833. [16] A.F. Kozmidis-Petrovic, Therm. Acta 499 (2010) 54. [17] A.F. Kozmidis-Petrovic, Therm. Acta 510 (2010) 137. [18] A.F. Kozmidis-Petrovic, Therm. Acta 523 (2011) 116. [19] C.S. Ray, D.E. Day, W. Huang, K.L. Narayan, K.F. Kelton, J. Non-Cryst. Solids 204 (1996) 1. [20] A.M. Rodrigues, J.L. Narváez-Semanate, A.A. Cabral, A.C.M. Rodrigues, Mater. Res. 16 (2013) 811. [21] M.C. Freitas, A.A. Cabral, J.M.R. Mercury, J. Non-Cryst. Solids 356 (2010) 1607.

76

T.V.R. Marques, A.A. Cabral / Journal of Non-Crystalline Solids 390 (2014) 70–76

[22] V.M. Fokin, E.D. Zanotto, N.S. Yuritsin, J.W.P. Schmelzer, J. Non-Cryst. Solids 352 (2006) 2681. [23] D.R. Uhlmann, in: L.L. Hench, S.W. Freiman (Eds.), Columbus: The American Ceramic Society, 1971, p. 91. [24] A. Kolmogorov, Izv. Acad. Sci. URSS (1937) 355 (in Russian). [25] W.A. Johnson, R.F. Mehl, Trans. AIME 135 (1939) 416. [26] M. Avrami, J. Chem. Phys. 7 (1939) 1103. [27] M. Avrami, J. Chem. Phys. 8 (1940) 212. [28] M. Avrami, J. Chem. Phys. 9 (1941) 177. [29] B.V. Yerofyeyev, Dokl. Akad. Nauk USSR 52 (1946) 511 (in Russian). [30] A.A. Cabral, V.M. Fokin, E.D. Zanotto, J. Non-Cryst. Solids 330 (2003) 174. [31] V.M. Fokin, A.A. Cabral, R.M.C.V. Reis, M.L.F. Nascimento, E.D. Zanotto, J. Non-Cryst. Solids 356 (2010) 358.

[32] L.A. Silva, J.M.R. Mercury, A.A. Cabral, J. Am. Ceram. Soc. 96 (2013) 130. [33] M.L.F. Nascimento, V.M. Fokin, E.D. Zanotto, A.S. Abyzov, J. Chem. Phys. 135 (2011) 194703. [34] M.L.F. Nascimento, E.D. Zanotto, J. Chem. Phys. 133 (2010) 174701. [35] , (MsC. Dissertation). L.A. Silva, Federal Institute of Maranhão, 2011. (in Portuguese). [36] E.D. Zanotto, P.F. James, J. Non-Cryst. Solids 74 (1985) 373. [37] M.C. Freitas, D.I.G. Costa, A.A. Cabral, A.R. Gomes, J.M.R. Mercury, Mater. Res. 12 (2009) 101. [38] C.J.R. Gonzalez-Oliver, P.F. James, J. Non-Cryst. Solids 38 e 39 (1980) 699. [39] A.A. Cabral, V.M. Fokin, E.D. Zanotto, J. Non-Cryst. Solids 343 (2004) 85. [40] A.M. Rodrigues, A.M.C. Costa, A.A. Cabral, J. Am. Ceram. Soc. 95 (2012) 2885. [41] L.S. Everton, A.A. Cabral, J. Am. Ceram. Soc. 97 (2014) 157.