Journal of Non-Crystalline Solids 362 (2013) 56–64

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Journal of Non-Crystalline Solids

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Nonstoichiometric crystallization of lithium metasilicate–calcium metasilicateglasses. Part 1 — Crystal nucleation and growth rates

Vladimir M. Fokin a,⁎, Raphael M.C.V. Reis b, Alexander S. Abyzov c, Clever R. Chinaglia b, Edgar D. Zanotto b

a Vavilov State Optical Institute, ul. Babushkina 36-1, 193 171 St. Petersburg, Russiab Vitreous Materials Laboratory, Department of Materials Engineering, Federal University of São Carlos, UFSCar, 13565-905, São Carlos-SP, Brazilc National Science Centre, Kharkov Institute of Physics and Technology, Ukraine

⁎ Corresponding author. Tel.: +7 812 695 30 38.E-mail address: vmfokin@gmail.com (V.M. Fokin).

0022-3093/$ – see front matter © 2012 Published by Elhttp://dx.doi.org/10.1016/j.jnoncrysol.2012.11.020

a b s t r a c t

a r t i c l e i n f o

Article history:Received 11 August 2012Received in revised form 18 November 2012Available online xxxx

Keywords:Glass;Crystallization;Nucleation;Growth;Eutectic

The initial stages of the crystallization kinetics of lithium metasilicate (LS) in glasses of the Li2O·SiO2–

CaO·SiO2 join – which has a simple eutectic – was investigated at high undercoolings, somewhat abovethe glass transition temperatures. Calcium metasilicate crystals precipitate and grow only in the advancedstages of crystallization. The variation of glass composition from the eutectic (26.5 mol% Li2O) towardslithium metasilicate (50 mol% Li2O) results in a sharp increase of the internal nucleation rate of LS crystals,whereas the growth rates increase only weakly. This strong increase of the nucleation rate is primarily causedby a decrease of the thermodynamic barrier for nucleation – due to an increase of the thermodynamic drivingforce for crystallization and a decrease of the nucleus/liquid interfacial energy – as the glass compositionapproaches the crystal composition.

© 2012 Published by Elsevier B.V.

1. Introduction

The vast majority of articles aimed at testing crystal nucleation,growth and overall crystallization models using experimental studiesof the crystallization kinetics of glass-forming liquids address theirstoichiometric or polymorphic crystallization, i.e., the growing crystalsand supercooled liquid have the same composition. However, very fewglasses satisfy the above condition [1,2], and moreover, the true natureof the critical nuclei in such stoichiometric glasses remains (mostly) un-known [3]. In other words, some research indicates that, even in the caseof “polymorphic” crystallization, the composition of the critical nucleimight differ from the parent glass [2]. Most oxide glasses display morethan one crystalline phase after crystallization with different composi-tion from the parent glass, at least in the advanced stages of crystalliza-tion. Hence, a detailed study of the kinetics of non-stoichiometriccrystallization is quite important from both theoretical and practicalpoints of view.

For the present study, we have chosen glasses belonging to theLi2SiO3 (LS)–CaSiO3 (CS) pseudo-binary join that has a single eutecticbetween LS and CS. To the best of our knowledge, only data on thecrystallization kinetics of the stoichiometric end-member composi-tions (Li2SiO3 and CaSiO3 glasses) exist for this system. However,the nucleation rate in LS glass is so high that it is extremely difficultto prepare such glass, whereas, according to Ref. [4], the internal nu-cleation rate of CS glass is very low, which is corroborated by our in-ability to detect internal nucleation in small (mm) pieces of this glass

sevier B.V.

treated for several hours above the Tg. Therefore, a strong differenceobviously exists between the nucleation kinetics in these two glasses(LS and CS). This difference correlates to the reduced glass transitiontemperatures (Tgr=Tg/Tm; where Tg is the glass transition tempera-ture and Tm is the melting point of the crystal). It should be stressedthat, according to empirical findings [5], the higher the Tgr, thelower the maximum nucleation rate, Istmax. The maximum crystal nu-cleation rate in the stoichiometric LS glass is expected to be approxi-mately 1023–26m−3 s−1, i.e., close to the values reported for somemetals (e.g., Istmax~1029m−3 s−1 for tin [6]).

The aim of this systematic study is to shed some light on the nu-cleation and growth processes of two stoichiometric crystallinephases (CS and LS) in non-stoichiometric glasses from the LS–CSjoin. The relevance of this type of research is related to the crystalliza-tion of commercial glass-ceramics, which is never stoichiometric! Thispaper addresses the first part of the present study, the nucleation ki-netics and data on the very initial stages of crystal growth when, to afirst approximation, the growth rates depend entirely upon the tem-perature and not on the treatment time. The forthcoming paper, part2 [7], is devoted to the evolution of the residual melt compositionduring the more advanced stages of crystallization.

2. Materials and methods

The glasses were prepared from lithium and calcium carbonatesand fumed silica. The solid-phase reactions of the desired mixturesof these reagents were performed at 800–900 °C. The producedmaterialwas thenmelted for 4 h in an electrical furnace at 1400 °C in a platinumcrucible. These melts were poured on a steel (or brass) slab and pressed

200 400 600 800 1000 1200 1400 1600

0

2

4

6

8

10

12

T,oC

1050 11500.2

0.3

0.4

1000 1100

5.0

5.5

1100 1300

6.2

6.4

6.6

B67

B60

B50

B47

B35

TS

TLTg

TL

TL

TL

µV/m

g

Fig. 1. DSC curves for the different glasses of this study. The insets show a zoom of theeutectic melting. A heating rate of 10 °C/min and 30 mg glass pieces were used.

1400

1600

T,°

C

a

liquid

57V.M. Fokin et al. / Journal of Non-Crystalline Solids 362 (2013) 56–64

by a steel (or brass) plate. The nominal composition of the batches isshown in Table 1.

The heat treatment of the glass samples was performed in a stabi-lized vertical electric box furnace kept within approximately 1 °C ofthe chosen temperature.

The characteristic temperatures for the parent glasses and residualglassmatrices were estimated using a Netzsch 404 differential scanningcalorimeter (Netzsch, Selb/Bavaria, Germany) with a platinum crucibleat 10 °C/min.

A Leica DMRX optical microscope (Leica Microsystems, Wetzlar,Germany) coupled with a Leica DFC490 CCD camera was used toinvestigate the morphology, size and number density of crystals em-bedded within the glass interior. Both transmitted light and reflectedlight modes were used. Thin slab-sided plates were prepared fortransmitted light optical microscopy from heat-treated pieces ofglass by cutting, grinding and CeO2 polishing.

A Philips XL30 field emission gun SEM (Philips, Amsterdam, TheNetherlands) was used for both secondary electron (SE) and backscattering (BS) imaging as well as EDS.

X-ray diffractionmeasurementswere carried out on both powderedand monolithic samples using an Ultima IV X-ray diffractometer(Rigaku Corp., Japan) operating at 20 mA and 40 kV. CuKα (1.5406 Å)incident radiation was used.

The glass densities were measured with a Mettler Toledo Balance(Model AX204).

3. Results

3.1. Phase diagram, concentration dependency of the glass transitiontemperature, and glass densities

The phase diagram of the LS–CS join was constructed by A.R. Westin 1978 [8]. We used DSC analysis of 25–30 mg glass samples to con-firm and enhance the data in Ref. [8]. Fig. 1 shows a typical DSCheating curve performed at a rate of 10 °C/min. The first exothermicpeak corresponds to the formation of LS and the second, to the crys-tallization of the residual glass in both LS and CS. The solidus, TS,and liquidus, TL, temperatures were determined by the beginningand the end of the melting peak, respectively (see also Ref. [9]).

We also measured the glass transition temperature, Tg, and thearea, Aeut/m, of the part of the endothermic peak corresponding tothe eutectic melting. The values of TL and TS (points in Fig. 2a) corrob-orate the data in Ref. [8], which is denoted by the solid lines. It shouldbe noted that only some of the compositions close to the eutecticwere studied in Ref. [8].

By plotting the area, Aeut/m, versus the melt composition, we wereable to determine the so-called “Tammann triangles” (Fig. 2b). Be-cause the Aeut/m reaches a maximum at the eutectic composition,this procedure helped us detect and refine the position of the eutecticpoint.

Figs. 3a and b show the dependence of the glass transition temper-ature and glass density on the glass composition, respectively. Thelinear dependence of the latter provides indirect evidence for thesimilarities between the batch and actual glass composition. This

Table 1The nominal composition of the batches.

Glass CaO·SiO2 Li2O·SiO2

mol%

B20 20 80B35 35 65B47 47 53B50 50 50B60 60 40B66 66.7 33.3

dependence also allows us to suppose that in the framework of the“model of associated solutions” [12] the glasses belonging to theLS–CS join only contain structural units similar to the lithium andcalcium metasilicate crystals. This assumption is corroborated by thefact that the LS–CS join is binary at all temperatures [8], i.e., only twocrystalline phases (Li2SiO3 and CaSiO3) are observed.

3.2. Crystallization

3.2.1. Volume nucleationOnly glasses containing less than 50 mol% CaO·SiO2 revealed ho-

mogeneous internal nucleation at a reasonable laboratory time scale(hours). Wemeasured the nucleation rates of the lithiummetasilicatecrystals (LS) in glasses B50, B47, and B35 using the development meth-od [2] in which the nucleation heat treatments were followed by crys-tal growth, i.e., higher-temperature development to crystal sizes thatwere visible with an optical microscope. Calciummetasilicate crystals(CS) only appeared during the advanced stages of the phase transi-tion. The X-ray diffraction spectra of crystallized glass B50 are shownin Fig. 4 with micrographs for different double treatments. Accordingto the X-ray analysis, a glass sample subjected to a nucleation heattreatment at 484 °C for 15 h and 35 min plus a development treat-ment at 675 °C for 1.5 min contains only LS crystals (Fig. 4a). To con-firm this finding, we decreased the XRD scanning rate in the angularrange corresponding to the main diffraction peaks of CS by a factorof 20. The new XRD spectrum also did not show any trace of CS crys-tals. But prolonging the heat treatment at 675 °C to 60 min resultedin full crystallization; at this point, the X-ray spectrum reveals bothLS and CS crystal phases (Fig. 4b).

0 20 40 60 80 100-1.0-0.5

0

1000

1200

b

Li2SiO3 CaSiO3mole%

Li2SiO3

Li2SiO3 + liquid

Aeu

t/m,a

.u

+ CaSiO3

CaSiO3 + liquid

Fig. 2. CS–LS phase diagram (a) and the Tammann triangle (b).The lines in the top figure were taken from Ref. [7].

50 m

400 m

Fig. 5. Transmitted light optical micrograph of a plate of glass B50 treated at Tn=494 °Cfor 12 h and then at Td=675 °C for 3 min. The inset shows the details of the crystalmorphology.

0 20 40 60 80 1002.2

2.4

2.6

2.8

3.0

,g/c

m3

mole % Li2SiO3 CaSiO3

crystal CaSiO3

b

400

500

600

700T

g,°C

a

crystal Li2SiO3

800

Fig. 3. Glass transition temperature (a) and glass density, measured at room tempera-ture, (b) versus glass composition. The doted and dashed lines show the density of theappropriate crystalline phases [11].The density of the Li2SiO3 glass was taken from Ref. [10].

58 V.M. Fokin et al. / Journal of Non-Crystalline Solids 362 (2013) 56–64

Twodifferentmethodswere used to estimate the crystal number den-sity, NV, in the glass volume. Because the nucleation rates for glasses B50and B47 are extremely low, we polished the sides of two plane-parallelplates with thickness between 80 and 200 μm (depending on the valuesof NV) after the heat treatment. These samples were then analyzed bytransmitted light microscopy to estimate the number of crystals in agiven volume. Cross sections of the heat treated samples with high

40s/step

2s/step

0

200

400

600° CaSiO

3(43-1460)

*

*

**

Li2SiO

3(29-828)a

10 20 30 40 50 600

500

1000

1500

2000

°°°°

°°°°

°

°°

°

°

°

°

°°°

***

*

*

* **

b

cps

cps

CaSiO3

Li2SiO

3

°*

500 m

Fig. 4. X-ray diffraction spectra of partly and fully crystallized B50 glass after nucleationat Tn=484 °C for 15 h and 35 min plus growth at Td=675 °C for (a) 1.5 and (b) 60 min.The photos show the corresponding transmitted (a) and reflected (b) light opticalmicrographs.

nucleation rates (glass B35) were prepared, and their number densities,NV, were estimated using standard stereological methods.

During the initial stages of growth, the LS crystals in glasses B50and B47 are needle-like dendrites with a complex structure (Fig. 5),while glass B35 had spherulitic shaped crystals (Fig. 6). It should benoted that prolonging the isothermal nucleation treatment and de-velopment time of glasses B50 and B47 led to the transformation ofthese needle-like LS crystals into star-like LS crystals by creatingnew growth directions (see Fig. 7 and inset in Fig. 4b).

The number density, NV, of LS crystals (nucleated on coolingduring the glass preparation and/or on heating to the develop-ment temperature) in glass B35 was very high, approximately106crystals mm−3, because of this material's extremely high nu-cleation rate. In contrast, glasses B50 and B47 without any preliminarynucleation heat treatment primarily displayed surface crystallization.The small number (~1 mm−3) of crystals in these latter two glassescould only be detected after a long development time. These crystalshave spherulitic shapes and according to X-ray analysis (Fig. 8) containLS crystals and traces of CS (wollastonite)whichmost likely formswith-in the residual glass remaining inside the spherulites. In contrast toglass B47, the spherulitic crystals in the B35 glass consisted of onlylithium metasilicate (Fig. 9). This could be caused not only by thedifference in the parent glass compositions but also by the large differ-ence in the sizes of the spherulites, which have grown about 100 times

100 mμ

Fig. 6. Transmitted light optical micrograph of a plate of glass B35 treated at Td=560 °Cfor 30 min. Cross-polarized contrast.

m50

10 20 30 40 50 600

200

400

600

800

1000

1200 *

*

**

*

*

* *

*

Li2SiO

3(29-828)

cps

Fig. 9. X-ray diffraction spectrum of powdered glass B35 partly crystallized at T=560 °Cfor 20 min, and a reflected light optical micrograph of its cross section.

400 mμ

Fig. 7. Transmitted light optical micrograph of a plate of glass B47 treated at Tn=515 °Cfor 10 h. Cross-polarized contrast.

0

5.0·103

1.0·104

1.5·104

2.0·104

NV,m

m-3

NV,m

m-3

0

1·103

2·103

3·103

4·103

B47 Tn = 460oC

Tn= 460B50

a

oC

b

59V.M. Fokin et al. / Journal of Non-Crystalline Solids 362 (2013) 56–64

longer in the glass B47. The residual glass within the spherulites hasalready started to form CS.

Thus, the first crystalline phase formed via internal homogeneousnucleation at relative low temperatures in glasses containing over50 mol% LS was lithium metasilicate. Therefore, all measurements ofthe nucleation kinetics of this paper refer to LS crystals. Fig. 10 showsthe typical kinetic dependence of NV on nucleation time, t, for glassesB50, B47, and B35. The solid lines were plotted using Eq. (1) for thetime-dependence of the number of super-critical nuclei, taken fromRef. [13]:

NV tð Þ ¼ Istτtτ−π2

6−2

X∞m¼1

−1ð Þmm2 exp −m2 t

τ

� � !: ð1Þ

This equation includes two fundamental parameters, the steady-statenucleation rate, Ist, and the time-lag for nucleation, τ, which were esti-mated as fit parameters. The temperature dependences of Ist and τ areshown in Figs. 11 and 12, respectively.

3.2.2. Crystallization at temperatures close to the liquidus, TL, andsolidus, TS

In contrast to the previous chapter, we consider the result of fastcrystallization at temperatures close to the TS and TL in this chapter.The reflected light optical micrograph in Fig. 13 (top) shows a crosssection of the sample from glass B47 treated for 10 min at T=900 °Cimmediately after melting at T=1250 °C (for this compositionTL=TS=1026 °C) for 10 min. A fine mixture of alternating crystals(LS and CS) is typical for eutectic crystallization as it is the case of

10 20 30 40 50 600

200

400

600

800

1000

°°° ° °°°°

°°°

°°°°°°

°

° CaSiO3 (43-1460)*

****

*

*

**

Li2SiO3 (29-828)

cps 200 m

Fig. 8. X-ray diffraction spectrum of a powder taken from an internal part of a sampleof glass B47 crystallized for 60 min at T=620 °C. The inset shows a transmitted lightoptical micrograph of a spherulite in the glass interior.

the B47 glass (see Fig. 2). The Ca and Si concentration profiles (seeFig. 14) along the white line showed in the top SEM BS image pro-vide direct evidence for such alternations. The chemical contrast inthe BS image also corroborates the Ca profile; the bright and darkcrystals are CS and LS, respectively. It should be noted that the Siconcentration is the same in both crystals as the two crystallinephases are metasilicates.

A sample of glass B66 (see Fig. 13 bottom) heated for 20 min atT=1100 °C (i.e., at a temperature between the TL and TS) and cooledto a temperature below the Tg at approximately 200 °C/min shows a

120 160 200 240

NV,m

m-3

t, h0 40 80

0 2 4 6 8 10 12 14 160

1·107

2·107

3·107

4·107

t, h

B35 Tn= 440oC

c

Fig. 10. NV(t) plots for glasses B50 (a), B47 (b), and B35 (c) for several nucleation temper-atures, Tn. The lines were plotted using Eq. (1) with proper fitting parameters Ist and τ.

m

a

b

m

Fig. 13. Reflected light optical micrographs of cross-sections of glasses B47 (a) and B66(b) subjected to the following heat treatments: a — T=1250 °C, 10 min+T=900 °C,10 min; b — T=1100 °C, 20 min. Full crystallization occurred during cooling of thesamples at a temperature TbTS (1026 °C).

440 460 480 500 520 540105

106

107

108

109

1010

1011

1012

1013

1014

1015

B50

B47

I st,m

-3s-

1

T,oC

B35

Fig. 11. Steady-state nucleation rates versus temperature of lithium metasilicate crystalsin glasses B50, B47, and B35.

60 V.M. Fokin et al. / Journal of Non-Crystalline Solids 362 (2013) 56–64

very different morphology. According to the phase diagram (Fig. 2),only CS crystals formed (large faceted crystals in Fig. 13, bottom) atT=1100 °C (region of primary crystallization). Then, cooling to tem-peratures below the TS causes the rapid crystallization of the residualglass matrix – with a composition close to the eutectic – with forma-tion of a fine structure that is typical for eutectics. Similar results areshown in Fig. 15, which presents a DSC cooling curve for glass B66with a photo (inset) of the sample surface taken directly after theDSC run. By accounting for the temperature of the crystallizationpeaks, we may attribute the first peak to the formation of CS crystalsand the second one to the crystallization of the residual eutecticliquid. It should be noted that a delay was observed for the twocrystallization peaks relative to TL and TS.

3.2.3. Crystal growth ratesThe data presented in this section relate to the growth of the LS

crystals at relatively low temperatures (as for the nucleation datashown in Section 3.2.1). Once the change in the residual glass compo-sition can no longer be neglected, the dependence of the crystal sizeon time begins to deviate from linear. The evolution of the residualglass composition and associated phenomena will be presented anddiscussed in detail in a companion paper [7]. In the present paper,we restrict our considerations to the initial stage of crystal growth,i.e., when the growth rates can be considered time-independent to a

1.28 1.30 1.32 1.34 1.36 1.38 1.406

8

10

12

14

ln(

,s)

B50

B47

B35

1000/T, K

Fig. 12. Temperature dependencies of the nucleation time-lags for glasses B50, B47, andB35 in Arrhenius coordinates. The top solid line is a linear fit of data of the B50 glass. Inthe case of glasses B47 and B35 the solid lines are plotted parallel to the former.

0

100

200

0 5 10 15 20 250

100

200

300

h , m

CS

i, a.

u.C

Ca,

a.u

.

Fig. 14. SEM BSE image and concentration along the cross section of a sample of glassB47 treated at 1250 °C for 10 min and then at 900 °C for 10 min.

1200 1100 1000 900 8000

3

6

9

TL

μV/m

g

T, oC

TS

100 μm

Fig. 15. DSC cooling curve (10 °C/min) of B66 melt. The inset shows a reflected lightoptical micrograph of the surface of a sample after the DSC run. The arrows point toCS crystals and the crystallized residual (eutectic) melt.

32 34 36 38 400

2000

4000

6000

8000

10000

3:1

22:11:45

(130)

65%

(200)

45%

(131)

19%

(002)

20%

10 20 30 40 50 600

2000

4000

6000

8000

10000

(111)

* *****

**

c

b

*Li2SiO

3(29-828)

2θ

cps

a

(020)

a

bc

Fig. 17. a) X-ray diffraction spectrum of a powder of the B50 glass sample crystallized atT=675 °C for 10 min. b) and c) X-ray diffraction spectra of different cuts of the samesample: cross section parallel (c) and perpendicular (b) to the external surface of thesample (see scheme in Fig. 16). The ratio between the areas of the diffraction peaksare shown close to the spectra. The dashed vertical lines correspond to the positionsof the diffraction peaks taken from card 29-828.

61V.M. Fokin et al. / Journal of Non-Crystalline Solids 362 (2013) 56–64

first approximation. Such restriction allows us to study and comparethe temperature dependence of the crystal growth rates in glassesB50, B47, and B35.

For glasses B50 and B47, we measured the growth rates of the crys-talline surface layer, which consisted of needle-like LS crystals sepa-rated by the residual liquid, at different temperatures. Without apreliminary nucleation treatment, these glasses contain primarilysurface crystallization as previously mentioned. Fig. 16 shows an ex-ample of the reflected light optical micrographs and SEM imagesfrom the cross sections of the B50 glass sample after treating atT=675 °C for 10 min. These cross-sectionsweremade in two differentdirections, parallel (taken from the vicinity of the sample surface) andperpendicular to the external surface of the sample, as shown by ascheme in Fig. 16. The morphology of the surface nucleated crystals issimilar to that of crystals nucleated in the glass volume (please comparewith Fig. 5). It should be emphasized that there is no contact betweenthe numerous LS crystals (Fig. 16 bottom) due to the Li-depleted diffu-sion zones around each crystal! The study of the residual melt compo-sition between the needle-like LS crystals that formed both on thesurface layer and far from it will be presented in Ref. [7].

Fig. 17 presents the X-ray diffraction spectra of the cross sectionsshown in Fig. 16, both parallel (Fig. 17c) and perpendicular (Fig. 17b)

500 m

500 m

50 m

Fig. 16. Reflected light optical micrographs of cross sections perpendicular and parallelto the external surface (see scheme) of a B50 glass sample treated at T=675 °C for10 min. The inset is an SEM image.

to the external surface. The bottom curve (Fig. 17a) relates to a finelyground sample (b20 μm). A strong crystallographic orientation of thesurface nucleated LS crystals was observed. The reflection (002) wasgreatly enhanced for the sample cut parallel to the surface (Figs. 16bottom, and 17c) while all the other reflections have been stronglysuppressed. The opposite is seen for the sample cut perpendicular tothe surface, with enhanced peaks corresponding to (200) and (020).This result indicates that the crystals are highly oriented with their caxis perpendicular to the sample surface. It should be noted that thesamples have different areas and crystal fractions due to sectioning, sodirect comparison between the absolute intensities of peaks from dif-ferent curves is not possible.

Fig. 18 shows the dependence of the crystal layer thickness, h, ontime, t, for different temperatures for glasses B50 (a) and B47 (b). Allof the plots were nearly linear. This feature allows us to concludethat, at least up to the values of h in these experiments, the crystalli-zation front advances towards the inner parts of the sample withoutchanging the liquid composition inside the glass sample. In otherwords, only the composition of the inter-dendritic residual glass,which is trapped between the LS arms, changed. Thus, the estimated

0 60 120 180 240 3000

200

400

600

800

1000

675°C

640 C°

620 C°590 C°

h,

m

t, min

b

0 30 60 90 1200

250

500

750a675°C

640 C°

620 C°

590 C°

B50

B47

Fig. 18. Thickness of the crystalline surface layer of B50 (a) and B47 (b) glass samplesversus heat treatment time at different temperatures. The solid lines are linear approx-imations to the data.

1.1 1.2 1.3-4

-2

0

2

B50B47B35

1000/T, K-1

log(

,m

/min

)U

Fig. 21. Temperature dependencies of the LS crystal growth rates for B35, B47, and B50

glasses in Arrhenius coordinates.

0 20 40 600

20

40

L/2l/2

L/2

and

l/2, μ

m

t, s

10

30

L

l

Fig. 19. Sizes L/2 and l/2 (see inset) of LS crystals nucleated in the volume of glass B47 at470 °C for 45 h versus time of growth at 675 °C. The solid lines are linear approxima-tions to experimental data. The dashed line is the h~ t plot shown in Fig. 18b for675 °C and shifted by 26 s to higher times.

62 V.M. Fokin et al. / Journal of Non-Crystalline Solids 362 (2013) 56–64

crystal growth rate, U=dh /dt, is a function of just the temperature. Itshould be noted that, at any given temperature, the growth rate of thesurface nucleated crystals equals that of the internally nucleated crys-tals. Fig. 19 shows the half length and width of the needle-like crystalsnucleated within the volume of glass B47 (see inset of Fig. 19) versustheir growth time at 675 °C. According to this figure, 0.5dL/dt (slopeof the solid line) is very close to dh/dt (slope of the dashed line). The ap-parent induction period, approximately 26 s, of the L/2 and l/2 time de-pendencies likely refers to the time necessary for sample heating.

Fig. 20 shows the time dependence of the radius, R(t), of theLS spherulites in B35 at several temperatures. The plot of R(t) atT=520 °C shows a stronger deviation from linearity than at highertemperatures. This deviation is discussed in Ref. [7]. In the presentpaper we restrict our analysis to the initial linear portion of the R(t)dependence. The temperature dependencies of the growth rates esti-mated from the data presented in Figs. 18 and 20 are shown usingArrhenius coordinates in Fig. 21.

4. Discussion

The following equations for the steady-state nucleation rate, Ist,(see e.g., [14]), the time-lag for nucleation, τ ([13], see also Eq. (1)),

0 5 10 15 200

2

4

6

8

Rm

t, min

540 C°

590 C°

0 300 600 9000

0.5

1.0

1.5

520 C°

Fig. 20. Radii of the LS crystals in the B35 glass versus time of heat treatment at differenttemperatures.

and the crystal growth rate, U (screw dislocation model, [14,15])were used to analyze the crystal nucleation and growth kinetics:

Ist ¼ffiffiffiffiffiffiffiffiσkBT

rDI

a4exp

−W�kBT

� �; ð2Þ

W� ¼163

πσ3

ΔG2V; ð3Þ

τ ¼ 163

kBTσΔG2

Va2DI

; ð4Þ

U ¼ fDU

4a1− exp −1

2ΔGVa

3

kBT

!" #; ð5Þ

f ¼ 12π

TL−Tð ÞTL

; ð6Þ

where kB is the Boltzmann constant; T is the absolute temperature;a is a characteristic size parameter (size of the building units in themelt); σ is the specific energy of the liquid/critical nucleus interface;ΔGV is the thermodynamic driving force for crystallization per unitvolume of crystal;W* is the work of formation for a nucleus of criticalsize, i.e., the so called thermodynamic barrier;DI andDU are the effectivediffusion coefficients for the building units in the case of nucleation andgrowth processes respectively; and f is the fraction of screwdislocationson the crystal surface.

Strictly speaking, Eq. (2) was derived for single-component sys-tems. When the crystal composition is the same as that of the liquid(stoichiometric crystallization) and the possible dissociation of struc-tural units is neglected, the liquid could be considered to only consistof building units similar to the crystalline cell. The system is oftenconsidered a single component in such situations.

Two corrections should be made when employing Eq. (2) fornon-stoichiometric crystallization, i.e., for precipitation of crystallinephases with different composition from the parent liquid. First, thevalue of the molar fraction of the precipitating substance in the melt, x,has to be introduced into the pre-exponential term. Thus, the decreasein the number of building units participating in the nucleation of thenew phase relative to stoichiometric crystallizationwill be taken into ac-count. However, the effect of this factor is much less than that for thechange in the thermodynamic driving force for crystallization, ΔGV.This driving force was derived for binary (metallic) melts in Ref. [16]

0. .6 0. .8 0.5 0 7 0 9 1.00.18

0.20

0.22

0.24

0.26

J/m

2

x LS, mole fraction

B 35

B 47B 50

Fig. 24. Energy of the liquid/critical LS crystal interface versus glass composition. Theempty circles refer to the present glasses. The solid line is a linear fit. The star is theexpected value corresponding to the LS melt.

0.7.109

0.8.109

0.9.109

1.0.109

1.1.109

700 720 740 760 780 800T, K

Fig. 22. Thermodynamic driving force for crystallization of lithiummetasilicate crystalsin glasses B50, B47, and B35, and in the liquid with Li2O·SiO2 composition.

63V.M. Fokin et al. / Journal of Non-Crystalline Solids 362 (2013) 56–64

based on the regular solutionmodel for the liquid and solid. In the limit-ing case involving the absence of solid solutions, i.e., when the solubilityof a second component in the crystalline phase is zero, the following for-mula was proposed for the driving force [16]:

ΔGV ¼ TL−Tð Þ ΔSm−R lnxð Þ=Vm; ð7Þ

where TL is the liquidus, ΔSm is the entropy of fusion per mole of purecrystal, and Vm is the molar volume.

Using ΔSm=50 J·K−1 mol−1 [17] for lithiummetasilicate, we cal-culated the thermodynamic driving force versus the crystallizationtemperature for LS in several glasses belonging to the LS–CS join asshown in Fig. 22. For glass B50 the value of TL was taken from the ex-tension of the liquidus to the RHS of the phase diagram. This prolon-gation corresponding to the metastable equilibrium between LScrystals and liquid takes place up to the formation of the CS phaseand is discussed in detail in Ref. [7].

We plotted Ist(T) and U(T) for glasses B50, B47, and B35 in Fig. 23, tocompare the influence of the glass composition on the nucleation andgrowth rates. It is clear that the crystal growth rate increased no morethan 1–2 orders of magnitude due to the increase in LS content from0.50 to 0.65 molar fraction, whereas the nucleation rate increased byseven orders of magnitude. At high undercoolings the crystal growth

450 500 550 600 650 70010

5

106

107

108

109

1010

1011

1012

1013

1014

1015

B 50

B 47

I st,m

-3s-1

T ,°C

B 35

10-3

10-2

10-1

100

101

102

103

104

105

106

107

U,

m/

mi

n

B 50

B 47

B 35

Fig. 23. Crystal nucleation and growth rates in glasses B50, B47, and B35 as a function oftemperature.

rate is primarily determined by the effective diffusion coefficient (seeEq. (5)). Hence, assuming DU≅DI, we can conclude that the strong in-crease in the nucleation rate is caused primarily by the decreasedthermodynamic barrier for nucleation W*. Combining Eq. (4) withEq. (2) and correcting by using the molar fraction, x, of LS in theglass (see discussion above), one can rewrite the equation for thesteady-state nucleation rate as:

Ist ¼ x163

σ3=2 ffiffiffiffiffiffiffiffikBT

pΔG2

Va6

1τexp

−W�kBT

� �: ð8Þ

Eq. (8) was employed to estimate σ as a fit parameter. One shouldrecall that σ enters into the thermodynamic barrier W* (see Eq. (3)).In our calculations, we used a=(Vm/NA)1/3≅3.9⋅10−10m, and the re-sults are shown in Figs. 24 and 25 for 477 °C, close to the temperatureof the maximum nucleation rate in glasses B50, B47, and B35. As wasexpected, the thermodynamic barrier strongly decreases with in-creasing Li2O content in the glass (Fig. 25).

Fig. 26 shows the maximum nucleation rates versus the reducedglass transition temperature. The large squares relate to the glassesB50, B47, and B35, whereas the other points, taken from Ref. [2], relateto different silicate glasses. The Ist(Tmax) values for glasses B50, B47,and B35 confirm the strong correlation between the Ist(Tmax) andTgr=Tg/TL. Because the new data relates to the same crystal (LS)and the same glass system, the correlation between Ist(Tmax) and Tgr

0.50 0.55 0.60 0.65

24

26

28

30

32

34

36

38

B50

B47

W*/

k BT

max

CLS,mol fraction

B35

Fig. 25. W*/kBTmax versus the molar fraction of LS in the glass.

0.46 0.48 0.50 0.52 0.54 0.56 0.58 0.600

5

10

15

20

25

30

B47

B35

log(

I max

,m-3

s-1 )

Tgr

B50

LS

Fig. 26. Maximum steady-state nucleation rates in different silicate glasses versusreduced glass transition temperature [2,5]. Squares show the data of Fig. 11. Star — seetext.

64 V.M. Fokin et al. / Journal of Non-Crystalline Solids 362 (2013) 56–64

is quite clear, despite the relative narrow Tgr interval. According tothis correlation, the lower the reduced glass transition temperature,the higher the maximum nucleation rate [5]. The physical reason forthis trend is the similarity between the Tmax and Tg. Because Tgr re-flects the undercooling achieved at Tg~Tmax, decreasing the Tgr andthus increasing the undercooling increases the thermodynamic driv-ing force for crystallization.

The thermodynamic barrier for nucleation, and hence the nucle-ation rate, depends strongly on the specific surface energy of the criticalnucleus/melt interface. Therefore we revert to Fig. 24 presenting thedependence of σ on the melt composition. The empty circles refer tothe glasses studied here. The solid line is the linear fit of the σ(x):

σ xð Þ ¼ 0:3375−0:154x; ð9Þ

where σ is given in J/m2 and x is the LS content in mole fraction.According to the extrapolation of this linear dependence, the spe-

cific surface energy of lithium metasilicate in its own melt (x=1) is0.183 J/m2. Combining this value with τ=0.03 s and ΔGV (see the topline in Fig. 22) we get Ist=1.45·1025m−3 s−1 at T=477 °C, which isvery close to the Tmax for glasses B35, B47, and B50. This Ist value followsthe trend of the data in Fig. 26 (compare the empty star to the solidline), especially if one accounts for the approximate character of theseestimates. To refine the real form of σ(x) for a wide x range, measuringthe nucleation rates for glasses containingmore than 0.53 mole fractionof LS would be necessary.

Nevertheless we can conclude that shifts in the glass compositionaway from that of the precipitated crystalline phase increases theinterfacial energy. Thus, one could represent the nuclei surface energyas two parts: the surface energy of the crystal in its own melt, and a

second that is proportional to the difference between the crystal andmelt compositions.

5. Conclusions

The nucleation and growth rates of lithiummetasilicate crystals weremeasured for several glass compositions of the Li2O·SiO2–CaO·SiO2

pseudo-binary join with a simple eutectic at a 0.53 Li2O·SiO2 molefraction. Increasing the LS content in the glass from 0.50 to 0.65 molefractions increases the maximum steady-state nucleation rate by ap-proximately seven orders of magnitude, whereas the growth rate onlyincreases by one or two orders of magnitude.

This extreme increase in the nucleation rate was primarily causedby a decrease in the thermodynamic barrier for nucleation due to theincreased thermodynamic driving force for crystallization and de-creased nucleus/liquid interfacial energy, σ. In the studied compositionrange, σ linearly decreases when the melt composition approaches thecrystal composition and can be represented by two portions: the first isthe surface energy of the crystal in its own melt, and the second is pro-portional to the difference between the crystal and melt compositions.

Acknowledgments

Financial support from Brazilian funding agencies FAPESP and CNPqare fully appreciated.

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