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Crystal growth kinetics in cordierite and diopside glasses in wide temperature ranges

Journal of Non-Crystalline Solids xxx (2008) xxx–xxx

ARTICLE IN PRESS

Contents lists available at ScienceDirect

Journal of Non-Crystalline Solids

journal homepage: www.elsevier .com/locate / jnoncrysol

Crystal growth kinetics in cordierite and diopside glasses in wide temperature ranges

Stefan Reinsch a,*, Marcio Luis Ferreira Nascimento b, Ralf Müller a, Edgar Dutra Zanotto b

a Federal Institute for Materials Research and Testing, 12205 Berlin, Germany bVitreous Materials Laboratory, Department of Materials Engineering, Federal University of São Carlos, 13595-905 São Carlos-SP, Brazil

a r t i c l e i n f o

Article history: Received 11 April 2008 Received in revised form 1 September 2008 Available online xxxx

PACS: 81.05.Kf 81.10.h 81.10.Aj

Keywords: Crystal growth Oxide glasses Silicates

0022-3093/$ - see front matter  2008 Elsevier B.V. A doi:10.1016/j.jnoncrysol.2008.09.007

* Corresponding author. Tel.: +49 30 6392 5950; fa E-mail address: stefan.reinsch@bam.de (S. Reinsch

Please cite this article in press as: S. Reins

a b s t r a c t

We measured and collected literature data for the crystal growth rate, u(T), of l-cordierite (2MgO  2Al2O3  5SiO2) and diopside (CaO MgO  2SiO2) in their isochemical glass forming melts. The data cover exceptionally wide temperature ranges, i.e. 800–1350 C for cordierite and 750–1378 C for diopside. The maximum of u(T) occurs at about 1250 C for both systems. A smooth shoulder is observed around 970 C for l-cordierite. Based on measured and collected viscosity data, we fitted u(T) using stan- dard crystal growth models. For diopside, the experimental u(T) fits well to the 2D surface nucleation model and also to the screw dislocation growth mechanism. However, the screw dislocation model yields parameters of more significant physical meaning. For cordierite, these two models also describe the experimental growth rates. However, the best fittings of u(T) including the observed shoulder, were attained for a combined mechanism, assuming that the melt/crystal interface growing from screw dislo- cations is additionally roughened by superimposed 2D surface nucleation at large undercoolings, starting at a temperature around the shoulder. The good fittings indicate that viscosity can be used to assess the transport mechanism that determines crystal growth in these two systems, from the melting point Tm down to about Tg, with no sign of a breakdown of the Stokes–Einstein/Eyring equation.

 2008 Elsevier B.V. All rights reserved.

1. Introduction nological models could be applicable only for limited temperature

Crystal growth kinetics in glass forming liquids has been exten- sively studied and reviewed elsewhere [1–5]. For isochemical or polymorphic crystallization, the crystal growth rates have been de- scribed in terms of standard models of interface-controlled growth (see, e.g., [6–8]) such as normal growth, screw dislocation growth or 2D surface nucleated growth. Detailed studies of crystal growth mechanisms are known for various glass forming compositions, such as SiO2 and GeO2 (normal growth) [9,10], Na2O  2SiO2 (screw dislocation) [11] and K2O  4B2O3 (2D nucleated growth) [12]. However, in most studies the inferred crystal growth mechanisms are typically restricted to temperature ranges near the melting point, Tm, or somewhat above the glass transformation tempera- ture, Tg, where crystal growth rates can be most easily measured.

In one of the first studies of crystal growth in glasses in wide temperature ranges, Burgner and Weinberg [13] analyzed the growth rates of internally nucleated lithium disilicate crystals in isochemical Li2O  2SiO2 glass forming melts between the glass transition temperature, Tg, and the melting point, Tm. Their analysis suggested that different governing growth mechanisms may be ac- tive for distinct temperature ranges, and that the usual phenome-

ll rights reserved.

x: +49 30 6392 5976. ).

ch et al., J. Non-Cryst. Solid

ranges. Meanwhile, crystal growth rate data for other two silicate

glasses, cordierite and diopside, have also been measured in simi- larly broad temperature ranges. Thus, the extensive studies of sur- face nucleated, isochemical crystallization of high-quartz solid solution crystals, denominated ‘‘l-cordierite” (2MgO  2Al2O3  5SiO2), in cordierite glasses reported in Refs. [14–21] allow a sim- ilar analysis. Some of these studies were part of a cooperative effort of the TC 7 Committee of the International Commission on Glass [22], which’s aim was to advance the understanding of surface crystallization phenomena. Herein, numerous glasses close to the stoichiometric cordierite composition were melted in different lab- oratories and crystal growth experiments were conducted over a long period of time covering various experimental conditions.

As another case of surface nucleated crystallization, the crystal growth rates of diopside (CaO MgO  2SiO2) in isochemical melts have also been comprehensively studied. Thus, crystal growth rate data are known for diopside glasses between 750–1155 C and 1277–1378 C [14,23–27]. In addition, thermodynamic and kinetic data, such as melting enthalpy and viscosity as a function of tem- perature are available for both systems, which facilitate quantita- tive comparisons between theory and experiment.

The objective of the present work is thus to summarize all mea- sured growth rate data for l-cordierite and diopside crystals in

s (2008), doi:10.1016/j.jnoncrysol.2008.09.007

2 S. Reinsch et al. / Journal of Non-Crystalline Solids xxx (2008) xxx–xxx

ARTICLE IN PRESS

their isochemical liquids in wide temperature ranges, then to ana- lyze and discuss their observed temperature dependencies in terms of the classical crystal growth models. We also present a proposal using combined growth mechanisms to explain the shoulder in the crystal growth curve of l-cordierite.

2. Experimental

2.1. Samples

2.1.1. Cordierite glasses Most of the data for the crystal growth rates of l-cordierite re-

fer to glasses having the nominal composition of cordierite (in wt%): 51.3 SiO2, 34.9 Al2O3 and 13.8 MgO. In this work and in Refs. [14,15,19] cordierite glasses were melted from reagent grade MgO and Al2O3 (both from Merck) and SiO2 (quartz sand, Walbeck GmbH, Weferlingen) at 1590 C in air for at least 8 h in Pt-cruci- bles. Meltings of 500 ml batches were carried out in conventional electric furnaces and in medium frequency inductive furnaces (4l batches). In the latter case, glass homogeneity was improved by stirring. Diaz-Mora et al. [20,21] used two cordierite glasses from Schott Glaswerke: one glass (B9455) had the nominal cordierite composition; another (GM30870) had 14.6 MgO, 33.2 Al2O3, 52.3 SiO2 wt%. Yuritsyn et al. [16,17] used stoichiometric cordierite glasses melted from chemically pure Al(OH)3, MgCO3 and SiO2  n- H2O in Pt/Rh crucibles at 1600 C for 4 h. All compositions are sum- marized in Table 1.

In this work and Refs. [14,15,19], glass plates of 10  15  1 cm3 were prepared by casting the melts onto steel plates and slowly cooling to room temperature from 750 C. Wet chemical analysis of the quenched glasses showed that no oxide component deviates more than 0.8 wt% from the nominal compo- sition. The water content determined by hot vacuum extraction and IR spectroscopy [28] for one of the glasses was 0.033 mol/l (226 wt ppm).

2.1.2. Diopside glasses The diopside glasses reported by several authors [14,23–27,29–

34] and that used in the present study were melted in air in Pt cru- cibles. In this work and Ref. [14], glass plates of 10  15  1 cm3 were splat cooled onto steel plates and slowly cooled to room tem- perature from 740 C. Briggs and Carruthers [23] made an X-ray fluorescence analysis confirming that the composition of their glass was very close to the nominal composition, and was free of iron, titanium and alkali metal oxides, but had 0.25 mol% Al2O3. Nascimento et al. [25] used the ICP technique for chemical analysis, which showed that his glass composition deviates less than 1 wt% from the diopside stoichiometry. Reinsch [14] used a diopside glass with addition of 1 wt% Al2O3. Chemical analysis of this glass (19.7

Table 1 Composition of all glasses used for crystal growth measurements in wt%

Cordierite glasses SiO2 MgO Al2O3 Impurities

Cordierite 51.3 13.8 34.9 This work and [13,14,18] 51.3 ± 0.8 13.8 ± 0.8 34.9 ± 0.8 Yuritsin [15,16] 51.85 12.82 34.62 0.39 CaO/0.33

Na2O Diaz-Mora [19,20] 52.3 14.6 33.2

Diopside glasses SiO2 MgO CaO Al2O3

Diopside 55.49 18.61 25.9 This work and Reinsch

[13] 54.55 19.72 25.80 1.02

Nascimento [24] 55.5 ± 1 18.6 ± 1 25.9 ± 1 Briggs [22] 55.49 18.61 25.9 0.25 mol% Zanotto [26] 54.96 18.4 25.64 1 Kirkpatrick [23] 56.2 19.1 26.4

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MgO, 25.8 CaO, 54.5 SiO2, 1 Al2O3 wt%) showed that no oxide com- ponent deviate more than 1 wt% from the nominal composition. The water content measured by hot vacuum extraction was 0.12 mol/l (758 wt ppm) [28]. Zanotto [27] used also a diopside glass with 1 wt% Al2O3. Kirkpatrick et al. [24] analyzed their glass with an electron microprobe, indicating a composition of 19.1 MgO, 26.4 CaO, 56.2 SiO2 wt%, thus having only a minor discrep- ancy from the nominal diopside composition. Unfortunately, how- ever, impurity and water contents were not always reported, but all these glasses have small departures from the stoichiometric diopside (18.61 MgO, 25.9 CaO, 55.49 SiO2 wt%). All compositions are summarized in Table 1.

2.2. Measurements

2.2.1. Viscosity In the present work, the viscosities of cordierite and diopside

melts were determined by complementary methods. Tg was deter- mined by a horizontal dilatometer (heating rate 5 K/min, Netzsch 402 E). Beam bending viscometry (heating rate 5 K/min, BAM) was used for the range log10(g/(Pa s)) = 12.3  9 and for log10(g/ (Pa s)) < 5 (T > 1000 C) rotational viscometry (BAM, measuring head Haake VT550) was applied.

The viscosity of the cordierite melt was also measured by Giess and Knickerbocker [35] at 900 and 920 C with a parallel plate vis- cometer. Yuritsyn et al. obtained Tg data by means of dilatometry [16]. The viscosity of the diopside melt was measured by Licko and Danek [29] in an oscillating viscometer using a platinum–rho- dium crucible and a cylinder with conical end. Nascimento et al. [25], Kozu and Kani [30] and McCaffery et al. [31] used a rotation viscometer. Sipp et al. [33] measured g by a compression method. Taniguchi [34] applied the counterbalanced method with Pt body and crucible, and the fiber elongation method. And finally, Neuville and Richet [32] did not disclose the technique used in their work.

2.2.2. Crystal growth rate 2.2.2.1. l-Cordierite. Crystal growth rate measurements were per- formed using bulk pieces of glass (5  5  5 mm3) with fractured, polished, or SiC ground surfaces in the present study as well as in Refs. [14–21]. Most of the thermal treatments were performed in air with 20% relative humidity (dew point 8.5 C). The influence of the ambient water vapor pressure was checked by crystal growth experiments in argon/air atmospheres of different humid- ity [14,36] (dew point = 60 to 25 C). Different techniques were used to measure the crystal growth rates of l-cordierite at low and high temperatures. Below 830 C we measured the growth of pre-existing crystals by electron or optical microscopy (increase of the maximum radius of selected surface crystals). Between 830 and 920 C the maximum radius of crystals grown during one-step crystallization treatments were measured. Above 1000 C we measured the thickness of the crystalline surface layer. Between 900 and 1050 C both methods were used.

Isothermal treatments at 830–1100 C were carried out in a conventional laboratory furnace. The samples were driven into the hot furnace using a platinum thermocouple as the sample holder. In other cases, annealing steps were made in a quartz glass tube furnace under controlled ambient conditions within a steel glove box. The accuracy of temperature measurement was ±10 K. Short time thermal treatments at high temperatures (1100– 1350 C) were performed in a specially designed vertical corundum tube furnace. The thermocouple and the platinum specimen holder were quickly moved along the tube axis where a linear gradient of 20 K/cm between 300 and 1500 C was maintained. Due to the small heat capacities of small samples and the platinum holder, very high heating rates, up to about 1200 K/min, at least in the vicinity of the glass surface were attained. An empty sample holder

s (2008), doi:10.1016/j.jnoncrysol.2008.09.007

1000 1200 1400 1600 1800 2000 -2

0

2

4

6

8

10

12

14

16 800 1000 1200 1400 1600 1800

lo g

10 η

(P as

)

T (K)

T (°C)

cordierite

diopside

Fig. 1. Viscosity curves of cordierite and diopside liquids. Cordierite: e: this work; D: Giess et al. [35]; .: Yuritsin et al. [16]. Diopside: }, : this work; o: Neuville et al. [32]; h: Sipp et al. [33]; +: Kozu and Kani [30]; : McCaffery et al. [31]; 5: Taniguchi [34]; /: Licko and Danek [29]. Lines: Vogel–Fulcher–Tammann–Hesse (VFTH) fits for cordierite and diopside according to Eqs. (1a) and (1b), respectively.

S. Reinsch et al. / Journal of Non-Crystalline Solids xxx (2008) xxx–xxx 3

ARTICLE IN PRESS

reached thermal equilibrium within 10 s. We measured the thick- ness of the crystalline surface layer between 200 and 1100 lm using annealing times between 1 and 3 min to ensure thermal equilibrium at least within the near surface layer of the sample. Nevertheless, for these short time measurements we have to as- sume a higher inaccuracy than for the other measurements. It is not possible to quantify this inaccuracy because of the unknown thermal conductivity of our samples and the resulting influence on the measured crystal layer thickness. But, the error is likely not larger than 50%, which is twice the size of the used symbols in Fig. 2(a) still fully embedded within the present data scatter.

2.2.2.2. Diopside. Numerous crystal growth rate data for diopside glasses between 750–1155 C and 1277–1378 C are known from literature. For instance, Briggs and Carruthers [23] measured crys- tal growth rates from 900 to 1150 C by hot stage microscopy. Zan- otto [27] measured the growth rate at 820 C. Reinsch ([14] and new measurements shown in this article) measured u from 750 to 1050 C by optical microscopy and SEM on polished and frac- tured surfaces. Fokin and Yuritsyn [37] measured growth rates be- tween 800 and 875 C by optical microscopy using polished and fractured surfaces. Nascimento et al. [25] measured diopside crys- tals on fire polished surfaces at 913, 923 and 950 C. Crystal growth rates at low undercoolings (DT = TmT 6 115 K) were determined by Kirkpatrick et al. [24] using a hot stage microscope.

2.3. Melting temperature

The melting temperature of diopside in its isochemical melt is Tm = 1397 C [38]. Tm of the metastable l-cordierite cannot be di- rectly measured. Therefore, we assume that its upper bound is the melting point of the stable high temperature polymorph of cordier- ite, denoted as indialite, h- ora-cordierite, at 1467 C [38]. The lower bound of Tm is assumed to be 1350 C sincemetastable l-cordier- ite is detectable as the primary crystal phase up to 1300 C [19].

3. Results

3.1. Viscosity

Viscosity data for cordierite melts are shown in Fig. 1 (upper curve). Our data are combined with data of Giess and Knicker- bocker [35] and Yuritsyn et al. [16]. The fitted VFTH curve for cor- dierite glass is given by Eq. (1a), where g is given in Pa s and T in K. The measured viscosity of our diopside melt and related literature data [25,27,29–34] are also shown in Fig. 1 (lower curve). The resulting average curve for diopside, by fitting all data, is given by Eq. (1b), with g in Pa s, and T in K. For any given temperature, the cordierite melt is more viscous than liquid diopside

log10ðg=ðPa sÞÞ ¼ 3:97þ 5316 K=ðT  762 KÞ ðcordieriteÞ; ð1aÞ log10ðg=ðPa sÞÞ ¼ 4:27þ 3961 K=ðT  751 KÞ ðdiopsideÞ: ð1bÞ

3.2. Crystal growth rates

Crystal growth rates of l-cordierite in its isochemical melt, measured in this work (open circles in Fig. 2(a)) and in previous studies [14–21], and similar data for diopside, measured here (open circles in Fig. 2(b)) and in Refs. [14,23–25,27,37], are shown in Fig. 2(a) and (b), respectively. These collected data cover an exceptionally wide temperature range, i.e. 800–1350 C (l-cordier- ite) and 750–1378 C (diopside). The maximum of u(T) occurs at about 1250 C for both glasses. A smooth shoulder is observed around 970 C for l-cordierite (arrow in Fig. 2(a)). For diopside, such a shoulder is much less pronounced and possibly hidden by

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the large data scatter in this temperature range. As expected from the viscosity curves, the crystal growth rates of diopside are some- what higher than of l-cordierite at any temperature. However, at their respective Tg (1012.3 Pa s) the growth rate of diopside is about one order of magnitude lower than that of l-cordierite.

3.2.1. Data scatter Whereas for diopside, u(T) data from different authors show

remarkable similarity, significant scatter is evident for l-cordierite. This scatter can be partially attributed to the decisive influence of humidity in different melts. This effect was measured in Refs. [14,36], where, e.g., u at 945 C increased from 0.2 to 0.6 lmmin1

for increasing air humidity (dew points between 60 and +25 C). The other important factor is that liquid diopside is much more depolymerised (has much more Q2 units in NMR notation) than cordierite melt, thus its highly broken structure is not so sensitive to a few percent more or less impurities.

Another source of data scatter could be impurities and small departures of stoichiometry and the different techniques used to obtain crystal growth rate data, i.e. by measuring the dimension of separate crystals or the thickness of the crystalline surface layer. However, these two types of data are not significantly different, especially in logarithmic scale. This finding is illustrated for l-cor- dierite in Fig. 3(a), where a large l-cordierite surface crystal is growing from a pristine, vacuum-fractured glass surface (parallel to the paper plane). Its radius is comparable to the thickness of the l-cordierite surface layer that grew perpendicularly from the sample surfaces (the latter surface was ground with SiC causing a high number density of surface crystals). Small deviations (a factor of two) due to the crystal orientation would not appear in the log- arithmic scale of Fig. 2(a). Fig. 3(b) shows diopside crystals on a polished glass surface annealed in air at 830 C for 210 min. Diop- side crystals mainly appear as separately grown squares.

4. Data analysis

4.1. Basic equations

In terms of the standard crystal growth models, the interface- controlled crystal growth rate u(T) is given by Eq. (2) [1,6], where

s (2008), doi:10.1016/j.jnoncrysol.2008.09.007

1100 1200 1300 1400 1500 1600 1700

800 900 1000 1100 1200 1300 1400 1500

TmTg

T (°C)

u (µ

m /m

in )

T (K)

1000 1100 1200 1300 1400 1500 1600 1700

700 800 900 1000 1100 1200 1300 1400

TmTg

T (°C)

T (K)

10-5 10-4 10-3 10-2 10-1 100 101 102 103 104

u( µm

/m in

)

10-5 10-4 10-3 10-2 10-1 100 101 102 103 a

b

Fig. 2. (a) Crystal growth rate, u, of l-cordierite in its isochemical liquid vs. temperature, T. Points: s: this work and [14,15,19]; h: Yuritsin [16,17]; }: Hanay [18]; D: Diaz-Mora [20,21]. (b) Crystal growth rate, u, of diopside in its isochemical liquid vs. temperature, T. Points: s: this work, [14]; }: Nascimento [25]; .: Zanotto [27]; D: Kirkpatrick et al. [24]; h: Briggs et al. [23]; r: Fokin [26]. Arrows indicate the dilatometric glass transition temperature, Tg, and the melting temperature, Tm. Possible upper and lower bounds for Tm according to 2.3 are indicated for metastable l-cordierite in (a). The maximum error is about twice the size of the used symbols.

Fig. 3. (a) l-cordierite crystal at a cordierite glass surface fractured and annealed in vacuum (980 C, 30 min). A compact crystalline surface layer of l-cordierite grew from SiC-ground surfaces of the sample. Transmitted light optical micrograph, top view, crossed nicols. (b) Diopside crystals at a polished diopside glass surface annealed in air (830 C, 210 min). Transmitted light optical micrograph, top view, crossed nicols.

4 S. Reinsch et al. / Journal of Non-Crystalline Solids xxx (2008) xxx–xxx

ARTICLE IN PRESS

d in m is the size of the molecular building unit, Z(T) is the attach- ment flow in m2 s1, f(T) is the fraction of preferred growth sites at the interface, DG(T) the bulk free energy change upon crystalli- zation, R the gas constant, and T the absolute temperature

uðTÞ ¼ d3  ZðTÞ  f ðTÞ  1 exp DGðTÞ RT

   ; ð2Þ

d is estimated from the volume of a building unit according to Eq. (2a), where this building unit is set to the formula unit, Vm is its crystal molar volume, and NA is the Avogadro number

d ¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Vm=NA

3 p

: ð2aÞ

Z(T) is given by Eq. (3a) [7], where k is the jump distance, c 6 1 is a steric factor, and kB is the Boltzmann constant. Eq. (3a) assumes that the molecular motion required for crystal growth is similar to that involved in shear viscous flow in the bulk liquid (Eyring assumption)

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ZðTÞ ¼ c kBT d3k2gðTÞ

: ð3aÞ

The first two terms in Eq. (2), d3  Z (T) in m s1, simplify to d  m, where m is the jump frequency in s1 defined by Eq. (3b) [2], when c = 1 and d = k are assumed. b in Eq. (3b) represents terms of weak temperature dependence

mðTÞ ¼ kBT d3gðTÞ

¼ b=gðTÞ: ð3bÞ

For f(T), the fraction of preferred growth sites, three phenomenolog- ical models are frequently used [1,2]. For normal growth, the crystal/ melt interface is rough on atomic scale, i.e. almost every site allows the attachment of new building units, and f(T) ffi 1.

According to the screw dislocation model, the crystal/melt inter- face is smooth on atomic scale but plagued by screw dislocations. Growth takes place at step sites provided by screw dislocations intersecting the interface, and fs (T) is expressed by Eq. (4) [1,39], where r is the specific surface energy of the melt-crystal interface, and cs is a constant

fsðTÞ ¼ cs dDGðTÞ 4prVm

: ð4Þ

In the 2D secondary surface nucleation growth model, the surface is atomically smooth and free from defects. Growth occurs by forma- tion of two-dimensional nuclei on the interface, followed by lateral growth, and f2D(T) is expressed by Eq. (5a) [1,7]

f2DðTÞ ¼ c2D exp  WðTÞ kBT

 

¼ c2D exp  ðpdVmr2Þ=eDGðTÞ

kBT

  ; ð5aÞ

where c2D is a constant and W(T) is the work of forming the critical 2D surface nucleus. Herein, e = 1 denotes the small crystal case (layer by layer growth) and e = 3 the large crystal case (multi-nu- cleus growth) [1,40]. The temperature dependence of f2D is illus- trated by Eq. (5b) representing all terms of weak temperature dependence by B according to Eq. (5c) [2]

f2DðTÞ / exp  B

TDGðTÞ

  ; ð5bÞ

B ¼ pdVmr 2

ekB : ð5cÞ

Furthermore, a combined mechanism is considered here. We assume that the melt/crystal interface, growing from screw dislocation de- fects, is additionally roughened by superimposed 2D surface nucle-

s (2008), doi:10.1016/j.jnoncrysol.2008.09.007

0.02 0.04 0.06 0.08 0.10 0.12 -8

-6

-4

-2

0

2

4

slope 2

slope 1

800°C 1000°C

1200°C 1300°C

ln u

η (u

η in

m s-

1 P as

)

106/TΔG (mol/JK)

Fig. 5. Estimation of the liquid/crystal interfacial energy, r, for l-cordierite. DG(T) was calculated by Eq. (7b). r was obtained from slope 1 and slope 2. Results are

S. Reinsch et al. / Journal of Non-Crystalline Solids xxx (2008) xxx–xxx 5

ARTICLE IN PRESS

ation, particularly at large undercoolings. This situation is schemat- ically illustrated in Fig. 4 and represented by Eq. (6). The constants cs and c2D were treated as freely adjustable with fs, f2D and fs + f2D < 1 is ensured. In this way, the normal growth model always yields the maximum values of u(T)

fcðTÞ ¼ fsðTÞ þ f2DðTÞ: ð6Þ

Finally, the bracket in Eq. (2) gives the occupation probability of growth sites, which depends on DG(T), the bulk free energy change upon crystallization. DG(T) can be approximated by Eq. (7a) for small undercoolings, DT = Tm – T, and by Eq. (7b) for wide tempera- ture ranges ½Tm < T < Tm, where Tm is the melting point of the crys- tal in its isochemical melt [1,7]. For binary alkali silicate glasses, Eqs. (7a) and (7b) were found to be accurate for small undercoolings only, giving for other cases only an upper and lower bound for DG, respectively,

DGðTÞ ¼ DHm  DT Tm

; ð7aÞ

DGðTÞ ¼ DHm  DT  T T2m

: ð7bÞ

summarized in Table 2.

4.2. Estimation of crystal/melt interfacial energy

Uhlmann and Uhlmann[2] proposed a simple method to esti- mate the crystal/melt interfacial energy, r. Expressing all the terms of weak temperature dependence in Eq. (2), by B (Eq. (5c)), b (Eq. (3b)), and C he wrote for the 2D surface nucleated growth model

uðTÞ ¼ Cb g

exp  B TDGðTÞ

  ð8Þ

or

ln ug ¼ ln Cb B TDGðTÞ : ð9Þ

Assuming B and Cb as constants, plots of ln(ug) versus (TDG)1

should yield straight lines of negative slope B. r can then be ob- tained from B according to Eq. (5c). r is often represented in terms of a used in the Skapski–Turnbull expression [41]:

a ¼ r DHm

ffiffiffiffiffiffiffiffiffiffiffiffi NAV

2 m

3 q

: ð10Þ

For l-cordierite, a plot of lnug versus (TDG)1 is shown in Fig. 5. Confirming previous u(T) studies in wide temperature ranges [2,13], the experimental data cannot be fitted by a single set of parameters. At least, two straight lines with widely different slopes, B, might be fitted. Each B yields two values of r according the small and large crystal case (e = 1 or 3 in Eq. (5c)). The calculated values of r and a are shown in Table 2. Due to the uncertain melting point of l-cordierite (Tm = 1350–1467 C), we considered DHm = 175 and 190 kJ/mol for the lower und upper bound of Tm, respectively, both

Fig. 4. Schematic illustration of the combined mechanism Eq. (6) assuming simultaneous crystal growth from screw dislocations (step) and from additional 2D surface nuclei at the spiral step planes.

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taken from [42]. The molar volume of the crystal building unit (one formulae unit) is Vm = 22.59  105 m3/mol [42]. Taking slope 1 (small undercoolings), a from the 2D nucleation-growth analysis is 0.06 < a < 0.2, which are lower than the well-known values de- rived from Classical Nucleation Theory (CNT) for several other sili- cate glasses (0.4 < a < 0.6 [43,44]). Taking slope 2, a ranges between 0.36 and 1.07, i.e. not far from the expected range esti- mated from CNT, and could thus corroborate the 2D surface nucle- ation as the operative growth mechanism for this system for large undercoolings.

For diopside, a plot of ln(ug) versus (TDG)1 is shown by Fig. 6. Surface energies r and a obtained from the fitted slopes are given in Table 2 (DHm = 138 kJ/mol [38], and Vm = 6.61  105 m3/mol). As for l-cordierite, the growth rate data for diopside can be fitted by two straight lines with widely different slopes. The a values from slope 1 are 0.04 < a < 0.07, which are far below the well-known values derived from CNT, while a values from slope 2, 0.24 < a < 0.75, are mostly within the expected range estimated from CNT.

4.3. Direct fitting

Crystal growth mechanism and r may also be inferred from a direct fitting, which can be performed by dropping the assump- tions Cb = constant, with d  k (used in Figs. 5 and 6). The best fit- ting results are shown in Figs. 7 and 8, for l-cordierite and diopside, respectively. We used Eqs. (1), (2), (3a) and (7b) with f = 1 for the normal growth model. For the screw dislocation model, f was calculated by Eq. (4). For the 2D surface nucleation model, f is set to Eq. (5a). In case of the combined mechanisms, both terms of f are superimposed according to Eq. (6). Due to the uncertain melt- ing point of l-cordierite Tm  1350–1467 C [42] it was set to the bound values. The corresponding melting enthalpies DHm = 175 and 190 kJ/mol were also taken from [42] assuming that the for- mulae unit (2MgO  2Al2O3  5SiO2; M = 585 g/mol) is the building unit. DHm for diopside is 138 kJ/mol (formulae unit; M = 216 g/ mol) [38].

For l-cordierite, normal growth does not fit the experimental data, except at the lowest temperatures (n, dotted line in Fig. 7). This is confirmed by Jackson’s treatment of the interface [45,46], which expects that materials with high entropy of melting, such as diopside (DSdm ¼ DHm=RTm ffi 10) and cordierite (DScm ¼ DHm=RTm ffi 13), have smooth crystal/melt interfaces.

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Table 2 Surface energy r (mJ/m2) and a obtained from different methods

System Method Mechanism e Tm (C) r1 r2 a1 a2

l-Cordierite Uhlmann 2D 1 1467 70 372 0.12 0.62 3 1467 122 644 0.2 1.07 1 1350 34 204 0.06 0.36 3 1350 58 353 0.10 0.63

r a

Direct fitting 2D 3 1467 140 0.25 Screw 3 1350 224 0.40 Combined 3 1350 280 0.50

Diopside Uhlmann 2D (solid) 1 1397 42 435 0.04 0.43 3 1397 73 754 0.07 0.75

2D (dashed) 1 1397 236 0.24 3 1397 408 0.41

r a

Direct fitting 2D 3 1397 70 0.07 screw 3 1397 400 0.40

e = 1 small crystal case, e = 3 large crystal case. If one changes e from 1 to 3, the calculated r and a change by a factor of 31/2 (Eqs. (5c) and (10)). ri and ai: obtained from slope i (1 or 2).

0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40 -12

-11

-10

-9

-8

-7

-6

-5

-4

-3

slope 1

slope 2

1200°C 750°C 1300°C 1350°C 1375°C

ln u

η (u

η in

m s-

1 P as

)

106/TΔG (mol/JK)

Fig. 6. Estimation of the liquid/crystal interfacial energy, r, for diopside. DG(T) calculated by Eq. (7b). r was obtained from slope 1 and slope 2. The dashed line (slope 2) indicates the uncertainty of slope determination. Results are summarized in Table 2.

1100 1200 1300 1400 1500 1600 1700

800 900 1000 1100 1200 1300 1400 1500 10-5 10-4 10-3 10-2 10-1 100 101 102 103

c

2D

n s

Tm TmTg

Temperature (°C)

u C (T

) ( µm

/m in

)

Temperature (K)

Fig. 7. Best fits of crystal growth rate data for l-cordierite: Normal growth: n, dotted line; screw dislocation: s, dashed line; 2D surface nucleated growth: 2D, dashed-dotted line, combined mechanism: c, solid line. Calculation parameters are summarized in Table 3. Experimental points: see Fig. 2(a).

6 S. Reinsch et al. / Journal of Non-Crystalline Solids xxx (2008) xxx–xxx

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Using the screw dislocation model, the experimental data above 1000 C can be fitted when T = 1350 C is assumed. The results (not shown) for higher Tm are much worse. For T < 1000 C the deviation from the experimental points is up to one order of mag- nitude. If Tm = 1467 C is used, the 2D surface nucleated growth model also fits the experimental points fairly well (2D, dashed-dot- ted line in Fig. 7). But in the low temperature range (T < 900 C) a deviation of about one order of magnitude occurs. The best fit is obtained using the combined mechanism. It is possible to fit the smooth shoulder in u(T) at 970 C and its temperature dependence below 900 C with r = 280 mJ/m2 (see Table 2) and Tm = 1350 C (c, solid line in Fig. 7). This fit required a high value of c2D compensat- ing the low value of the exponent in Eq. (5a) at the temperature range of the observed shoulder in u(T). Only this way, f2D  fs and a pronounced shoulder could be attained.

For diopside, all three growth models are more or less applicable when independently fitted. Thus, the normal growth model (n, dot- ted line in Fig. 8) fits the experimental points up to 1150 C. How- ever, we had to assume a small steric factor c = 0.04 notwithstanding the used values of c = 1 for all other models. Using c = 1 for normal growth results in a more than one order of magni-

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tude higher u(T). With the screw dislocation model (s, dashed line in Fig. 8) as well as with the 2D model (2D, dashed-dotted line in Fig. 8), the experimental data can be well fitted within the whole temperature range. However, for the latter model this fitting is only possible assuming a very low a (see Table 2). Fitting in terms of the combined mechanism was also possible requiring a high va- lue of c2D, too (not shown).

5. Discussion

5.1. Crystal growth mechanism

Two methods of assessment of the crystal growth mechanism were applied here for l-cordierite: direct fitting of crystal growth and estimation of crystal/melt interfacial energy. With direct fit- ting, both the 2D-surface nucleation and the screw dislocation growth model could be fitted with sufficient accuracy. These assessment uncertainties may arise from the unknown value of Tm or from the experimental difficulties to measure the very large

s (2008), doi:10.1016/j.jnoncrysol.2008.09.007

1000 1100 1200 1300 1400 1500 1600 1700

700 800 900 1000 1100 1200 1300 1400 10-5 10-4 10-3 10-2 10-1 100 101 102 103 104

s

2D

n

TmTg

u( T)

(µ m

/m in

)

Temperature (°C)

Temperature (K)

Fig. 8. Best fits of crystal growth rate data for diopside. Normal growth: n, dotted line, screw dislocation: s, dashed line, 2D surface nucleated growth: 2D, dashed and dotted line. The used parameters are summarized in Table 4. Experimental points: see Fig. 2(b).

Table 3 Parameters used for calculating crystal growth rates of l-cordierite (curves in Fig. 7)

Parameter l-cordierite

Normal growth

Screw dislocation

2D surface nucleated growth

Combined mechanism

M (g/mol) + 585 585 585 585 Tm (C) 1467 1350 1467 1350 k (Å) + 0.7 0.7 0.7 0.7 d (Å) Eq. (2a) 7.21 7.21 7.21 7.21 c + 1 1 1 1 a Eq. (10) – 0.4 0.25 0.5 r (J/m2) # – 0.224 0.140 0.280 cS # – 0.5 – 0.5 c2D # – – 25 3  109

‘+’ parameter generally set to a uniform value (e.g. M to one formulae unit) and ‘#’ parameter was adjusted.

Table 4 Parameters used for calculating crystal growth rates of diopside (curves in Fig. 8)

Parameter diopside

Normal growth

Screw dislocation

2D surface nucleated growth

M (g/mol) + 216 216 216 Tm (C) 1397 1397 1397 k (Å) + 4.79 4.79 4.79 d (Å) Eq.

(2a) 4.79 4.79 4.79

c # 0.04 1 1 a Eq.

(10) – 0.4 0.07

r (J/m2) # – 0.4 0.07 cs # – 1 – c2D # – – 0.05

‘+’ parameter generally set to a uniform value (e.g. M to one formulae unit) and ‘#’ parameter was adjusted.

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u(T) of l-cordierite in the high temperature range (small under- coolings). For small undercoolings, the value of r obtained from slope 1 in Fig. 5 (see Table 2), assuming the 2D mechanism, have little physical meaning. For large undercoolings, Fig. 5 more con- vincingly indicates the 2D surface nucleation mechanism since the resulting r has a more significant physical meaning. In the di- rect fitting (Fig. 7), the 2D-mechanism best fits u(T). Very best fits are obtained assuming that 2D-surface nucleation additionally roughens the spiral steps (combined mechanism).

A more defined situation is observed for diopside. For small undercoolings, Kirkpatrick [24] indicated the spiral growth mech- anism. Confirmingly, very small values of a (<0.07) of questionable physical meaning are found in Fig. 6 if the 2D surface nucleation mechanism is supposed. For large undercoolings, however, the 2D mechanism is more probable, since it yields physically mean- ingful r. Accordingly, Kirkpatrick [8] also indicated that growth oc- curs by a surface nucleation mechanism at large undercoolings. In the direct fitting, the 2D and the spiral growth model give almost the same result. Fitting in terms of the combined mechanism was also possible (not shown).

Direct fitting of crystal growth and estimation of crystal/melt interfacial energy for both cases indicate that the fraction of pre- ferred growth sites increases at large undercoolings, e.g. due to a change from the screw dislocation to the 2D nucleation growth mechanism.

5.2. Fitted parameters

The full and reduced energy parameters, r and a, respectively obtained from the two methods used here are summarized in Table 2. Uhlmann’s method assuming the 2D surface nucleation mecha- nism yields values which are too small for small undercooling (slope 1) if compared with well-known values from Classical Nucleation Theory for measurable homogeneous nucleation in sil- icate systems, which usually varies between 0.4 and 0.6 [43,44]. Slope 2 (large undercooling), which refers to temperatures near the respective shoulder and below yields reasonable a values for both systems. a values obtained from direct fitting for l-cordierite are plausible for the screw dislocation and the combined mecha- nism. This finding also holds for diopside.

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r and a summarized in Table 2 show a large scatter. This scatter may arise from the uncertain validity of adopted assumptions con- cerning the growth mechanism, the small or large crystal case, the constancy of Cb, the applicability of Eq. (7b), and the uncertain melting point of l-cordierite. Other small effects arise from the uncertain molar mass of the building unit, which is simply set to the formulae unit. At least for cordierite, which formulae unit con- tains five silica and four alumina tetrahedra, this is questionable [42] (please note that altering M will affect d, DHM and DG).

All these uncertainties indicates that one should not trust any of the such-wise obtained results for r and that the application of any method for estimating r, particularly from u(T) data restricted to narrow temperature ranges, could be misleading.

The mean jump distance, k, is frequently assumed to equal d. For l-cordierite, it was not possible to directly fit the experimental points with this assumption. Only for a significantly smaller k, an acceptable fitting was possible (see Table 3). Thus, we used the jump distance k as a free fitting parameter in Eq. (3a), whereas d was calculated with Eq. (2a). To perform a good fitting, even at low temperatures, near Tg, a somewhat lower k than the oxygen ion radius (k  0.13 nm) was necessary. For diopside, good fitting with k = dwas possible (see Fig. 8 and Table 4). But as we discussed for the estimation of r, the fitted k should only give an order of magnitude for the jump distance or the size of the jumping molecules.

5.3. Crystal growth rate shoulder

Most intriguing, a smooth shoulder appears at 970 C for l- cordierite and may be also present for diopside (but it is quite small and could be hidden by the data scatter). The best fitting of

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ARTICLE IN PRESS

u(T) including this shoulder was possible assuming that the melt/ crystal interface, growing from screw dislocation defects, is addi- tionally roughened by superimposed 2D surface nucleation, partic- ularly at large undercoolings (Fig. 4). The possibility that different crystal growth mechanisms operate in a wide temperature range has been discussed by Burgner et al. [13] in their study of LS2 glass. A possible change from screw dislocation to surface nucleated growth with increasing undercooling was also addressed by Gut- zow et al. [1] for systems with intermediate DSm/R. The value of this parameter is large for the present glasses if the building unit is equated to the formulae unit. Computer simulations showed that, depending on the crystallographic orientation, atomic rough- ness can increase significantly with increasing DG/RT, i.e., with increasing undercooling [47].

Thus, this supposition might be feasible, although no clear evi- dence can be claimed from the present study. Instead, several pos- sibilities must be considered to account for this phenomenon, which may also affect the assessed growth mechanism and fitted parameters.

5.4. Model applicability

5.4.1. Chemical composition The applicability of the applied standard model is restricted to

interface-controlled isochemical crystal growth, which we assume to be valid for both glasses studied here. And, indeed, the linearity of the crystal size versus time curves, expected for such case, was verified for l-cordierite between 920 and 1080 C [15], and for diopside from 750 up to 850 C [14]. The attained good fitting of u(T) in the wide temperature range covered here also indicates the applicability of standard growth models.

Nevertheless, small effects like the observed shoulder in u(T) might be affected by temporary changes of the chemical composi- tion of growing crystals with annealing time or temperature. Such compositional changes have been experimentally verified by Fokin et al. [48] and Roskosz et al. [49] in other silicate systems. Consid- ering diopside, however, preliminary results of quantitative EDX analysis by SEM (crystallized layer heat treated at 890 C for 16 h) showed no difference between the glass and crystal phase compositions [50]. Up to now, no such analysis was performed for cordierite. X-ray powder diffraction studies [51] have shown that the position of (101) peak of the X-ray powder diffraction trace of l-cordierite (JCPDF 14-249) is at 0.346 nm up to 1080 C. This observation indicates that the SiO2 content of l-cor- dierite is close to that of cordierite [52]. Beyond that temperature, however, compositional shifts cannot be excluded because l-cor- dierite readily transforms to indialite (JCPDF 21-549), the high temperature polymorph of cordierite [53]. The latter dominates the overall crystallization of cordierite glasses above 1000 C. Nev- ertheless, it was shown in Ref. [54] that on heating the glass first transforms to l-cordierite, and then to indialite. Thus, even the growth of a thick surface layer, which is mainly composed of indi- alite, is controlled by a thin ‘front-layer’ of l-cordierite. In Ref. [19] this mechanism was observed to occur up to 1300 C by X-ray experiments.

5.4.2. Crystal morphology For diopside, morphology changes have not been exhaustibly

tested, and to the best of our knowledge there is no published evi- dence about changes in crystal morphology with time or tempera- ture. Recently, however, Fokin [50] communicated two distinct morphologies, with different growth rates, growing on polished surfaces of a stoichiometric diopside glass at 800 C, but these data must yet be confirmed by more detailed experiments.

For l-cordierite, Diaz-Mora [21] observed a change in morphol- ogy from elongated hexagonal crystals to regular hexagons at

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T < 900 C. Changes in morphology were also reported in Ref. [51]. Small crystals <2 lm, grown at T < 900 C show cellular melt/crystal interfaces, whereas large crystals grown at T > 1000 C are dendritically structured. Nevertheless, the shape of isolated crystals does not deviate from hexagonal bi-pyramids and their growth rate does not explicitly depend on time (or crystal size). Hence, we believe that the influence of morphology may be neglected for the measured crystal growth rate data of l- cordierite.

5.4.3. Crystal surface roughness u(T) will increase by a transition from the small to the large

crystal case if crystal growth is governed by 2D surface nucleation (Eq. (5a)). This might explain the change of slopes in Figs. 5 and 6. For l-cordierite (Fig. 5), the ratio between slope 2 and slope 1 is 37 and 27 for Tm = 1350 C and Tm = 1467 C, respectively. For diop- side (Fig. 6), this ratio is 106 (solid line) or 31 (dashed line). According to Eqs. (5a) and (5c), however, this ratio should be only 3. Therefore, this enormous change of slope is not due to a transi- tion from small to large crystal growth. A transition from the small to the large crystal case might also explain the shoulder in u(T) of l-cordierite. Thus, we recalculated curve 2D in Fig. 7 (large crystal case) for the small crystal case (not shown in Fig. 7). The curves did not differ by more than  2.

5.4.4. Decoupling Another possible explanation for the shoulder might be attrib-

uted to a possible breakdown of the Stokes–Einstein/Eyring equa- tion (see Eq. (3b)). For instance, a recent study [55] demonstrates that below the glass transition temperature atomic motion through a metallic glass involves single-atom hopping, whereas motion above Tg is more collective. Similar findings have been re- ported for some organic liquids. For such fragile systems a substan- tial change in the diffusion mechanism seem to occur at temperatures about 1.2Tg, and this temperature has been denomi- nated decoupling temperature Td. But, this phenomenon has not yet been firmly demonstrated for oxide glasses. Here, we should stress that the weak shoulder in Fig. 7 occurs at 1.2Tg, where decoupling is expected. However, the inflection is too faint and uncertain to really demonstrate decoupling. For instance, one ex- pects that for real decoupling – such as those reported in Ref. [55] below Td, the Stokes–Einstein/Eyring equation would not de- scribe the diffusion coefficients any longer, but here they do. In other words, decoupling leads to a permanent change in slope and not only to a kink in the crystal growth rate curves.

5.4.5. Temperature at the liquid/crystal interface A further precondition for the accuracy of the crystal growth

analysis is that the temperature of liquid/crystal interface equals that of the bulk specimen. Herron and Bergeron [56] suggested and successfully tested empirical equations to estimate the li- quid/crystal interface temperature DTi for temperatures near the maximum crystal growth rate umax and at u < 0.67 umax [25]. In or- der to check the related potential influence on u(T), we used Her- ron and Bergeron’s equation,

DTi ¼ 17:12ðumaxDHmÞ0:486; ð12Þ

where umax is in (cm/s) and DHm in (cal/cm3), to correct our u(T) data with respect to temperature. For cordierite and diopside these corrections are at the most 5 and 50 C, respectively, in the range of maximum growth rate. Hence, they could, in principle, exert a sig- nificant effect on the kinetic analysis of diopside. Such corrected u(T) data were used to repeat the data analysis in Fig. 6, but calcu- lations performed with and without them demonstrate that the main conclusions would not change.

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6. Conclusions

We have analyzed crystal growth rate data of l-cordierite and diopside crystals in their corresponding isochemical glass forming melts in a wide temperature range, from about Tg to Tm, with u(T) varying by about 9 orders of magnitude. The maximum of u(T) is reached at about 1250 C in both systems. A weak shoulder in u(T) is observed at 970 C (1.2Tg) for l-cordierite.

For diopside, the experimental u(T) can be equally fit by the 2D surface nucleation growth mechanism and by the screw disloca- tion growth mechanism. However, the screw dislocation model yields parameters of more significant physical meaning.

For l-cordierite, these two models also describe the experimen- tal growth rates. However, the best fitting of u(T), including the ob- served shoulder, was attained for a combined mechanism assuming that the melt/crystal interface, growing from screw dis- locations, is additionally roughened by superimposed 2D surface nucleation at large undercoolings, starting at temperatures around the observed shoulder. Thus, this supposition might be feasible, although no clear evidence can be claimed from the present study.

Taken in toto, the experimental data could be fit within the whole temperature range by the standard dislocation growth mod- el. The good fittings indicate that viscosity can be used to assess the transport mechanism that determines crystal growth in these systems, from the melting point Tm down to about Tg, with no sign of a breakdown of the Stokes–Einstein/Eyring equation. Similar findings were recently reported for pure silica glass [57].

Acknowledgments

MLFN would like to acknowledge DAAD (Deutscher Akademi- scher Austauschdienst) and the Brazilian agencies CAPES for finan- cial support during his stay at BAM, and FAPESP for his post- doctoral fellowship (Grant 04/10703-0). EDZ thanks the Brazilian agencies Fapesp (Grant 07/08179-9) and CNPq for continuous sup- port in the last 30 years. The authors thank I. Schubart for u(T) measurements during his internship at BAM and Mark Ediger for a critical review of the manuscript.

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