OR I G I N A L AR T I C L E

The diffusion coefficient controlling crystal growth in a silicateglass-former

Daniel Roberto Cassar1 | Alisson Mendes Rodrigues1 |

Marcio Luis Ferreira Nascimento2 | Edgar Dutra Zanotto1

1Department of Materials Engineering,CeRTEV, Federal University of S~aoCarlos, S~ao Carlos, S~ao Paulo, Brazil2Vitreous Materials Lab., Department ofChemical Engineering, PolytechnicSchool, Federal University of Bahia,Salvador, Bahia, Brazil

CorrespondenceDaniel Roberto CassarEmail: daniel.r.cassar@gmail.com

Funding informationFundac�~ao de Amparo �a Pesquisa doEstado de S~ao Paulo, Grant/AwardNumber: 2011/18941-0, 2013/07793-6;Conselho Nacional de DesenvolvimentoCient�ıfico e Tecnol�ogico, Grant/AwardNumber: 150490/2015-1, 201983/2015-0

AbstractOne of the most relevant open issues in glass science refers to our ignorance con-

cerning the nature of the diffusing entities that control crystal nucleation and

growth in non-crystalline materials. This information is very relevant because all

the existing nucleation and growth equations account for the diffusion coefficient

(DU) of these unknown entities. In this article, we measured the shear viscosity

(g) and the crystal growth rates of a supercooled diopside liquid (CaMgSi2O6) in

a wide temperature range. The well-known decoupling of viscosity and crystal

growth rates at deep supercoolings was detected. We tested and analyzed 4 differ-

ent approaches to compute DU, three existing and one proposed here. As

expected, the classical approach (DU ~ g�1) and the fractional viscosity approach

(DU ~ g�e) were not able to describe the crystal growth rates near the glass tran-

sition temperature. However, our proposed expression to calculate DU—gradually

changing from a viscosity-controlled to an Arrhenian-controlled process—was

able to describe the available data in the whole temperature range and yielded the

lowest uncertainty for the adjustable parameters. Our results suggest that viscous

flow ceases to control the crystal growth process below the so-called decoupling

temperature, corroborating some previous studies.

KEYWORD S

glass, supercooled liquid, crystal growth, viscosity, diffusion, diopside

1 | INTRODUCTION

Understanding and controlling crystal nucleation andgrowth in glass-forming liquids is a key issue either toavert crystallization1 to produce a glass or for the develop-ment and production of glass-ceramics.2 Crystallization issuch an important feature of glass-forming liquids that ithas been the most frequently used keyword in the past200 years of glass science,3,4 and crystal growth is a keypart of the crystallization process.

However, one faces some adversities when studying thephysics of crystal growth. One of the most relevant is our

ignorance of what the diffusing entities (or structural enti-ties5) are that control this phenomenon.6 The glass researchcommunity does not know if crystal growth kinetics is con-trolled by 1 chemical element (e.g. silicon, the network for-mer) or a set of atoms (e.g. one or more silica tetrahedral) inthe case of silicate glasses. We also do not know if the con-trolling entity is temperature-, pressure-, and/or structure-dependent. This information is relevant because the currentcrystal growth equations take into account the effective diffu-sion coefficient (DU) of these unknown units.

What seems to be a consensus7-14 is that viscous flow isan adequate property to estimate DU in a limited

Received: 30 June 2017 | Accepted: 29 August 2017

DOI: 10.1111/ijag.12319

Int J Appl Glass Sci. 2018;9:373–382. wileyonlinelibrary.com/journal/ijag © 2017 The American Ceramic Societyand Wiley Periodicals, Inc

| 373

temperature range, which spans from the melting pointdown to the so-called decoupling temperature (Td). Thistemperature has been reported for different substances11,12

to exceed the laboratory glass transition temperature by afactor of about 1.10-1.25.

In this article, we tested 4 different approaches to esti-mate DU, one of them is proposed here for the first time.The test was carried out for a glass with diopside composi-tion (CaMgSi2O6), for which we report a new dataset ofviscosity and crystal growth rates. This particular composi-tion was chosen because it has a plethora of availablekinetic and thermodynamic data that were necessary tocarry out the present analysis, and thus can be considered a“model” glass-former, similarly to what has been proposedfor lithium disilicate.13

2 | LITERATURE REVIEW ANDGOVERNING EQUATIONS

2.1 | Crystal growth kinetics

Crystal growth rates in oxide glass-forming systems are usu-ally analyzed by 1 of the 3 classical models: normal growth,screw dislocation growth or 2D secondary nucleationgrowth.15 Here, we focus on the screw dislocation growthmodel because it satisfactorily describes the experimentalcrystal growth data in supercooled diopside liquid in a widetemperature range.14 Readers are invited to study Gutzow &Schmelzer’s book1 for more detailed information on the dif-ferent crystal growth mechanisms, and Cassar’s thesis16 for anin-depth analysis of the normal and 2D secondary nucleationgrowth for diopside (briefly, these equations did not performas well as the screw dislocation growth for this material).

The normal and screw dislocation growth modes aredescribed by the same expression, Equation 1. The differ-ence between these 2 growth mechanisms is the fraction ofsites (f) on the crystal interface available for atomic attach-ment. Normal growth prevails if this parameter is tempera-ture independent and approximately unity (as proposed byWilson17 in 1900). Otherwise, if 0 < f < 1 and f is temper-ature dependent, the screw dislocation mechanism governsgrowth. This parameter can be estimated by Equation 2 ifone considers screw dislocations that form Archimedeanspirals18 at the liquid-crystal interface.

U ¼ fDU

d01� exp �Dl

RT

� �� �(1)

f ¼ d0Dl4prVm

(2)

In the above equations, U is the crystal growth rate, DU

is the effective diffusion coefficient controlling crystal

growth, d0 is the jump distance that some (unknown) dif-fusing entities or “structural units” must travel through theliquid to attach onto a growing crystal, r is the crystal-liquid interfacial energy, R is the universal gas constantand Vm is the molar volume. The driving force for crystal-lization, Dl, for a closed system in an isobaric conditioncan be calculated19,20 using heat capacity data:

DlðTÞ ¼ DHm 1� TTm

� ��Z Tm

TðCp;lðTÞ � Cp;cðTÞÞ

dT þ TZ Tm

T

Cp;lðTÞ � Cp;cðTÞT

dT

(3)

where DHm is the enthalpy of melting, Cp,l is the heatcapacity of the supercooled liquid, Cp,c is the heat capacityof the crystal and T is the absolute temperature.

Finally, the latent heat released at the liquid-crystalinterface during crystal growth can significantly increasethe interface temperature if it is not dissipated fast enough(oxide liquids are poor heat conductors). Fortunately, Her-ron & Bergeron21 measured the temperature increase at theliquid-crystal interface during the crystal growth process inseveral oxide glass-formers. Cassar16 reanalyzed their dataand obtained the following empirical expression:

DTiðTÞ ¼ 100:64ðUðTÞDHmÞ0:58 (4)

where DTi is the liquid-crystal interface temperatureincrease at temperature T. In this expression, U is given inm/s and DHm in J/mol. In this article, we corrected all theanalyzed data for this increased interface temperature usingEquation 4.

2.2 | The diffusion coefficient for crystalgrowth

As already stated, the mobility of the diffusing entities thatcontrol crystal growth is of major importance for crystalgrowth kinetics. The effective diffusion coefficient DU rep-resents the magnitude of this mobility, but the nature of themoving units in oxide glasses is not known. Authors9-14,22-24

have approached this problem differently, but our analysisindicates that they all use the following general phe-nomenological equation for DU:

DU ¼ ½x�n CTgn

� �þ ½1� x�q D0exp � Ea

RT

� �� �(5)

In the above equation, x is usually set to one (whichmakes the rightmost term of Equation 5 equal to zero) orset to be temperature dependent (see Subsection 2.2.3below), g is the viscosity and C, D0, Ea, ξ, n, and q areconstants. Please note that C = CE when Eyring’s diffusionequation25 is considered.

374 | CASSAR ET AL.

CE ¼ kBd

(6)

In the above equation, kB is the Boltzmann constant andd is the diameter of the moving entity that controls viscousflow, usually assumed to be equal to d0.

This complex form of DU (Equation 5, proposed in thisarticle) results from many attempts to describe DU in theliterature. These attempts will be described below, case bycase. In this article, we test 4 possible expressions of DU: 3published approaches (Cases 1-3) and a new method (Case4) suggested here. These DU expressions will be used to fitEquation 1 to experimental crystal growth rate data mea-sured for a supercooled liquid with diopside composition(CaMgSi2O6), the 4 DU expressions can be derived fromthe general Equation 5.

2.2.1 | Case 1: classical DU approach

In what we term the “classical” approach,9-11,13,14,22,23 themobility of the diffusional units controlling crystal growthis linked to the equilibrium viscosity at all temperaturesfrom the melting temperature (Tm) down to the laboratoryglass transition temperature (Tg) and below. This is thesimplest assumption from all the cases studied here, and iteffectively considers that n = ξ = x = q = 1, and g = geq

in Equation 5 (where geq is the equilibrium viscosity). Thefinal expression for DU in this case reads:

DU ¼ CTgeq

(7)

Turnbull and Cohen22 were some the first to proposeand use this method. More recently, however, severalreports11,13 indicated that the viscosity is indeed a goodparameter to calculate DU for temperatures near Tm, butdoes not correctly describe the transport part of crystalgrowth rates at deep supercoolings, i.e., temperatures nearTg. This observation is the basis of the so-called “decou-pling” or “breakdown” of the Stokes-Einstein/Eyring equa-tion that has been extensively discussed elsewhere.11,13,14,23

Various equations describe the temperature dependenceof the equilibrium viscosity of oxide glass-formers betweenTm and Tg well. Here, we use the MYEGA26 expression:

log10 geqðTÞ� � ¼ log10ðg1Þ þ Tg

T½12� log10ðg1Þ�

expm

12� log10ðg1Þ � 1� �

TgT

� 1� �� � (8)

where g∞, Tg, and m are taken as adjustable parameterswith the following definitions:

g1 ¼ limðT!1ÞgeqðTÞ (9)

geqðTgÞ ¼ 1012Pa.s (10)

m ¼ d log10 geqðTÞ� �� �

d Tg

T

� T¼Tg

(11)

where m is known as the liquid fragility.27

2.2.2 | Case 2: DU with fractional viscosity

In the fractional viscosity approach, Ediger et al24 consid-ered a parameter ξ in the DU expression that leads to a dif-ferent temperature dependence of DU on viscosity(Equation 12). Their aim was to restore the agreementbetween crystal growth kinetics and viscous flow in thewhole temperature range from Tm down to Tg. Accordingto their approach, there is no decoupling of the Stokes-Ein-stein/Eyring equation, as observed in the classical approach(Case 1). They found that the higher the fragility of the liq-uid, the higher the deviation of ξ from unity is. Thisapproach was recently re-tested for some oxide liquids andchallenged by Schmelzer et al.12

Based on the assumptions of this approach,24 we havethat n = x = q = 1, g = geq, and ξ becomes an adjustableparameter in Equation 5. The final expression for DU reads:

DU ¼ CT

gneq

(12)

2.2.3 | Case 3: DU with viscosity plus anArrhenian contribution

Schmelzer et al12 proposed another approach to calculate DU

by adding an additional Arrhenian diffusion term(D0 exp � Ea

RT

� �) in the “classical” DU equation. This is equiv-

alent to setting n = 0, ξ = q = 1, and g = geq in Equation 5.With these approximations, the final expression for DU is:

DU ¼ CTgeq

" #þ ½1� x� D0exp � Ea

RT

� �� �(13)

In their manuscript, Schmelzer et al12 used a hyperbolictangent function for x:

x ¼ 1þ tanhðaÞ2

(14)

with

a ¼ T � Td/Td

(15)

In Equation 15, / is a dimensionless parameter thatcontrols the smoothness of x and Td is the decoupling

CASSAR ET AL. | 375

temperature. The authors12 considered / as a fixed valueequal to 0.06.

Due to the [1-x] term on the right-hand side of Equa-tion 13, the Arrhenian diffusion term only becomes signifi-cant for temperatures near and below Td. For temperaturessufficiently above Td, Equation 13 yields similar values toEquation 7 (the classical approach).

A relevant property of Equation 13 is that the equilib-rium viscosity is used to evaluate DU at all temperatures.That is, no matter how deep a material is cooled below itsTd, the equilibrium viscosity will still be an importantparameter to calculate DU. It will be clearer later on thatthis is the main difference between this approach and theone that we propose in this manuscript.

3 | MATERIALS AND METHODS

3.1 | Experimental

The diopside glass used in this research was obtained bythe melt-quenching route. Melting was carried out in a bot-tom-loaded superkanthal electric furnace, model CMBloomfield. The diopside batch used in this study is thesame used by Reis.28,29 The chemical reagents used wereCaCO3 (J. T. Baker 99.9%, lot E06348), MgO (J. T. Baker99.9%, lot E25597) and colloidal silica (Aldrich 99.8%, lotMKAA0249). The powders were heated at 673 K (CaCO3

and MgO) and at 1173 K (SiO2) for twelve hours for dryingbefore weighing, and the mixture was homogenized in aFritsch Pulverisette 6 low rotation mill with 8 agate balls.The homogeneous mixture was heated at 1473 K for48 hours prior to melting to produce the diopside crystalphase via solid state reaction, and the diopside crystal wasindeed confirmed by XRD28 (JCPS card 75-1577). Fourhundred grams of this polycrystalline mass were melted ina platinum crucible at 1823 K for 15 minutes, cast into ametallic mold and annealed at 998 K (~Tg) for 240 min-utes. The analyzed composition of the glass28,29 was 25.8CaO, 18.6 MgO, and 55.4 wt% SiO2 (with 0.2 wt% impuri-ties). This result is very close to the nominal compositionof stoichiometric diopside.

The viscosity and the crystal growth rates were mea-sured isothermally at various temperatures using glass sam-ples of the same batch. The viscosity of our glass near theglass transition temperature was measured at the Instituteof Silicate Chemistry, Russian Federation, using a penetra-tion viscometer, which is described elsewhere.30 We mea-sured the viscosity using samples of the same batch nearthe melting point using a Brookfield Model DV-III rota-tional rheometer (Middleboro, MA).

We measured the advancement rates of the surface crystal-lized layer using cubic samples of polished faces having anedge size of approximately 2 mm. We used 3 cubes for each

studied temperature and all of them revealed a linear incre-ment of the crystal layer depth l as a function of heat treatmenttime16 (this particular glass only shows surface nucleation inlaboratory time/size scales). The slopes of l 9 time yieldedthe crystal growth rates, and for each temperature, we took theaverage values measured for the 3 cubes. For the isothermalheat treatments, the samples were dropped inside a verticalfurnace already stabilized at the desired temperature. Thisresults in the fastest possible heating rate between room tem-perature and the treatment temperature.

We confirmed that a wollastonite-like phase (WL)appeared after a certain crystallized layer depth of diopsidewas reached, as shown and discussed by Fokin et al31,32

Therefore, in this study, we only measured the growth ofthe diopside crystal layer before this WL phase appeared.The wollastonite-like phase nucleates in the sample interior,just underneath the diopside crystal-liquid interface when acertain level of elastic stress is reached due to the differentdensities between the crystalline and liquid phases.31,32

3.2 | Calculations and data analysis

We analyzed our crystal growth data near and above Tgalongside data measured by Reis et al29 (using samplesfrom this same diopside glass batch) and by Kirkpatricket al33 fortunately, the data from Kirkpatrick et al are wellabove the temperature where the wollastonite-like phasetransforms to diopside.

We calculated the thermodynamic driving force viaEquation 3, for which we used 334.57 J/K�mol for Cp,l,

34

137.7 kJ/mol for DHm,34 1670 K for Tm

34 and Equation 16for Cp,c

35 (with Cp,c in J/K�mol and T in Kelvin):

Cp;c ¼ 470� 0:1T þ 2:45T2 � 4820

ffiffiffiffiT

pþ 2:81� 10�5T2

(16)

We adopted the expression used by Reinsch et al,23

Equation 17 for the calculation of the jumping distance d0from the liquid to the growing crystal interface. Weobtained an equation for the molar volume of the liquidphase (Equation 18, with Vm in m3/mol and T in Kelvin)by a linear regression of the data reported by Lange36 andGottsmann & Dingwell.37 We assume that d = d0 andC = CE throughout this manuscript, in agreement with simi-larly published research.11-13,23

d0 ¼ Vm

NA

� �13

(17)

VmðTÞ ¼ 7:01� 10�5 þ 7:21� 10�9T (18)

We carried out all numerical analyses using the Pythonprogramming language with the Jupyter Notebook38

376 | CASSAR ET AL.

interactive computational environment equipped with thefree and open-source NumPy and SciPy modules.39 Allnonlinear regressions were performed using the moduleLMFIT40 using the Levenberg-Marquardt41,42 method. Theplots were also generated using Python with the aid of thefree and open-source matplotlib module.43 All the uncer-tainties reported in this manuscript are 1 SD.

3.3 | New approach to calculate DU (Case 4)

Here, we propose a new method to compute the effectivediffusion coefficient, considering that DU gradually changesfrom viscosity-controlled to Arrhenian-controlled.

As already mentioned, one important consideration ofthe approach of ref.12 (Case 3) is that the equilibrium vis-cosity (or whatever mass transport mechanism that controlsviscous flow) always takes some part in the diffusion pro-cess controlling crystal growth, even below Td. Here, wepropose and test another possibility: that the equilibriumviscosity gradually ceases to control the crystal growthkinetics near and below Td, effectively resulting in a “pure”Arrhenian behavior for temperatures sufficiently below Td.To account for this behavior, we have set n = ξ = q = 1and g = geq in Equation 5. The final expression for DU inthis context is:

DU ¼ ½x� CTgeq

" #þ ½1� x� D0exp � Ea

RT

� �� �(19)

We decided to use the same sigmoid function for x asemployed in ref.12 However, after various tests, we settledwith a lower value for / for this case: 0.02. As alreadymentioned, / controls the smoothness of the sigmoidfunction used for x. Using a lower value of / results in acurve that has a sharper transition between viscosity-con-trolled to Arrhenian-controlled DU. To be clear, we useda value of / = 0.06 for the Case 3 calculations for 2 rea-sons: the authors12 tested other values of / and reportedthis one to be the one that best fitted their data; and tobe able to compare our results with those published inref.12

4 | RESULTS AND DISCUSSION

4.1 | Experimental

Figure 1 shows the experimental viscosity data measuredfor the diopside liquid studied here. The continuous line isthe regression of Equation 8, from which we obtained thefollowing values: Tg = 1000.1(3) K, m = 64.5(3), andlog10(g∞) = �1.80(4) (g∞ in Pa.s).

Similarly, Figure 2 shows the experimental crystalgrowth rates obtained in this study together with data from

Reis28 (who used samples from the same batch used in thisstudy) and Kirkpatrick.33 All the experimental dataobtained from this research are reported in Tables S1 andS2.

4.2 | Crystal growth analysis

4.2.1 | Case 1

Figure 3 shows the regression of crystal growth rate datawhen DU is given by the considerations in Case 1 (seeSubsection 2.2.1). In this approach, we were not expect-ing a good fit for the data near Tg and this was pre-cisely what we obtained. The regression gives a smallerresidual near Tm when only data points to whichT ≥ 1.1Tg are considered, see Figure 3B in comparisonwith Figure 3A. The rationale to eliminate the datapoints below the suspected Td results from the fact thatdue to the suspected decoupling, the equilibrium viscositywould no longer describe the growth rates at such deepsupercoolings.

Only 1 adjustable parameter is needed in this approach,the surface energy, r. From the regressions, we obtainedr = 0.16 (2) J/m2 and RMS = 0.42 when all data pointswere considered (Figure 3A) and r = 0.22(1) J/m2 andRMS = 0.44 when only data points for which T ≥ 1.1Tgwere considered (Figure 3B). These are reasonable valuesof surface energy (comparable to values reported fromcrystal nucleation studies of silicate glasses that undergohomogeneous nucleation44,45).

FIGURE 1 Viscosity data and regression curve with Equation 8,which covers very wide ranges of viscosity and temperatures. Dataobtained for the same batch of diopside glass used for crystal growth

CASSAR ET AL. | 377

4.2.2 | Case 2

Figure 4 shows the regression curve of crystal growthrate data when DU is calculated by the considerations ofCase 2 (Subsection 2.2.2). Even though the data near Tmare reasonably well described in this approach, all theavailable data below 1300 K systematically deviate fromthe regression curve. Our observation is in agreementwith recent critiques of this approach. Two adjustableparameters are needed for this approach: r and ξ. Our

regression yielded r = 0.20(3) J/m2 and ξ = 0.95(1),with RMS = 0.31.

4.2.3 | Case 3

Figure 5 shows the regression curve fitted to crystalgrowth rate data when DU is calculated by the considera-tions in Case 3 (Subsection 2.2.3). It can be readily seenthat all crystal growth data are well described by the fit,including the data near Tg. Four adjustable parameters areneeded for this approach: r, Ea, D0, and Td. Our regres-sion yielded r = 0.26(4) J/m2, Ea = 640(60) kJ/mol, ln(D0) = 31(7), and Td = 1040(140) K, with RMS = 0.11(D0 in m2/s).

4.2.4 | Case 4

Figure 6 shows the overall results of the new approachsuggested here, i.e., the regression of crystal growth ratedata when DU is calculated following the considerations ofCase 4 (Subsection 3.3). Similarly, as in the previous case,all available data are well described by the regression. Thesame 4 adjustable parameters are needed for this approach:r, Ea, D0, and Td. Our regression yielded r = 0.223(9) J/m2, Ea = 650(50) kJ/mol, ln(D0) = 31(6), and Td = 1100(30) K, with RMS = 0.11 (D0 in m2/s).

4.3 | Comparison of DU expressions

From the 4 different expressions for DU studied here, 2were unable to describe crystal growth data near the glasstransition temperature: the classical approach (Case 1) andthe fractional viscosity approach (Case 2). These are

FIGURE 2 Experimental crystal growth rates (U) obtained fromthis study together with data from Reis28 (sample H from their study,using specimens of the same glass batch used here) and fromKirkpatrick33

(A) (B)

FIGURE 3 Regression curve of crystal growth rate data with DU considering Case 1 taking into account (A) all data points and (B) onlydata for T ≥ 1.1Tg [Color figure can be viewed at wileyonlinelibrary.com]

378 | CASSAR ET AL.

expected results due to the breakdown of the Stokes-Einstein/Eyring equation at Td, in agreement with formerpublications in the field.11,13 Both cases had a similar

performance for fitting data near the melting temperature,but Case 1 has fewer adjustable parameters. Our results forCase 1 show that viscosity can indeed describe the kineticpart of crystal growth at relatively low supercoolings forT > Td and it may be used when the available or studieddata is above the decoupling temperature.

We repeated the analysis for the approach of Sch-melzer et al,12 ie, DU expression with viscosity plus anArrhenian contribution (Case 3). As far as we know,this is the first independent study to use this expressionafter its publication, making this a consistency test.Within this framework, we obtained a Td value of 1040(140) K which is statistically equal to the value(1100 K) reported in their manuscript (they used thenotation ~Td for this parameter). However, we obtained avalue of 640(60) kJ/mol for Ea, which is higher than thereported12 value of 480 kJ/mol. Unfortunately, they didnot report the uncertainty for these 2 parameters. It ispossible that both Ea values are statistically equal withinthe uncertainty, but we cannot be sure. D0 and r werenot reported.

We also proposed and tested a new approach to thissame problem: DU undergoing a gradual change fromviscosity-controlled to Arrhenian-controlled (Case 4). Thisapproach fitted well all the available data, with very sim-ilar residuals and RMS as in Case 3. This approach alsohas the same 4 adjustable parameters as Case 3: r, D0,

FIGURE 4 Regression curve of crystal growth rate data with DU

considering Case 2 [Color figure can be viewed atwileyonlinelibrary.com]

FIGURE 5 Regression curve of crystal growth rate data with DU

considering Case 3 [Color figure can be viewed atwileyonlinelibrary.com]

FIGURE 6 Regression curve of crystal growth rate data with DU

considering Case 4 [Color figure can be viewed atwileyonlinelibrary.com]

CASSAR ET AL. | 379

Td, and Ea. However, we observed that Case 4 has aclear advantage over Case 3: the uncertainty in Td ismore than 5 times lower. This is important because somerecent analysis12 rely on computations that depend onthe value of Td (eg to obtain the viscosity at the decou-pling temperature) and high uncertainty in Td can jeopar-dize this type of analysis. We thus defend that the DU

given by Case 4 has the clear advantage of yielding alower uncertainty in Td than the value obtained in Case 3.

The main difference between these 2 cases is thatCase 3 assumes that equilibrium viscosity should beaccounted to compute DU, even for temperatures belowTd, whereas Case 4 assumes that viscosity ceases to be acontrolling factor in this process below Td. At this stage,we cannot conclude which one (Case 3 or Case 4) isthe best for the physical description of the decouplingphenomenon. In this context, Gillot et al5 studied a diop-side glass made via sol-gel and strongly defended thatcrystallization is not controlled by a bulk transport mech-anism at deep supercooling (close to the glass transitiontemperature).

Mathematically, the difference between cases 3 and 4lies in the value of n (see Equation 5), which is 0 for thefirst and 1 for the second. It is possible, however, to setthe value of n to any real number between 0 and 1. Thischanges the shape of the function xn as can be seen inFig. S1. From Equation 5 we have that when xn = 1 vis-cous flow is controlling DU and when xn = 0 viscous flowno longer controls it. If the shape of xn changes, then thetemperature range at which viscous flow ceases to controlDU also changes. We stress, however, that this “fine-tun-ing” of the parameter n is currently not possible or sup-ported by the available data.

An open and relevant question is to find out exactlywhat type of diffusing entities control viscous flow andcrystal growth below Td. For this task, computer simula-tions, in situ high temperature structural studies between Tgand Tm (e.g., by Raman spectroscopy or NMR), and pre-cise measurements of the diffusion coefficients of the slow-est species (Si and O in silicate glasses) are greatly neededto shed light on this very important, intriguingphenomenon.

5 | SUMMARY AND CONCLUSIONS

We carefully measured the viscosity and crystal growthrates in a wide temperature range (extending the availabledata and knowledge in this field) using samples from thesame batch of a stoichiometric diopside glass. Thisapproach is much cleaner and rigorous than using samplesfrom distinct batches, as is commonly done when literaturedata is used for analyses of kinetic processes. Also, we

avoided measuring the crystallization rates of the (spurious)wollastonite-like phase.

The well-known decoupling between viscosity andcrystal growth rates at deep supercoolings was detected.We tested and thoroughly analyzed 4 differentapproaches to compute the effective diffusion coefficientcontrolling crystal growth, a key parameter controllingthe crystallization kinetics that cannot be measured exper-imentally. As expected, the classical approach and thefractional viscosity approach were not able to describethe crystal growth rates at deep supercoolings near theglass transition temperature. Overall, the approach pro-posed here (Case 4)—which assumes that DU graduallychanges from a viscosity-controlled to an Arrhenian-con-trolled process—was able to describe the available crystalgrowth rate data in the whole temperature range andyielded the lowest uncertainty among the 4 cases. Thisresult supports the concept that viscous flow ceases tocontrol the crystal growth process below the decouplingtemperature.

ACKNOWLEDGMENTS

We thank Drs. A. J. Barbosa and R. C. M. V. Reis formaking the diopside glass used in this study and Dr. Vladi-mir M. Fokin for his critical and constructive insights. Weare grateful to the S~ao Paulo Research Foundation,FAPESP, grant numbers 2011/18941-0 and 2013/07793-6(CEPID) and The Brazilian National Research Council,CNPq grant no. 150490/2015-1, and no. 201983/2015-0 forfellowships to DRC and AMR.

ORCID

Daniel Roberto Cassar http://orcid.org/0000-0001-6472-2780Alisson Mendes Rodrigues http://orcid.org/0000-0003-0449-9219Marcio Luis Ferreira Nascimento http://orcid.org/0000-0001-5030-7500Edgar Dutra Zanotto http://orcid.org/0000-0003-4931-4505

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How to cite this article: Cassar DR, Rodrigues AM,Nascimento MLF, Zanotto ED. The diffusioncoefficient controlling crystal growth in a silicateglass-former. Int J Appl Glass Sci. 2018;9:373-382.https://doi.org/10.1111/ijag.12319

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