Elemental and cooperative diffusion in a liquid, supercooled liquid and glassresolvedDaniel R. Cassar, Ricardo F. Lancelotti, Rafael Nuernberg, Marcio L. F. Nascimento, Alisson M. Rodrigues,Luiza T. Diz, and Edgar D. Zanotto

Citation: The Journal of Chemical Physics 147, 014501 (2017); doi: 10.1063/1.4986507View online: http://dx.doi.org/10.1063/1.4986507View Table of Contents: http://aip.scitation.org/toc/jcp/147/1Published by the American Institute of Physics

THE JOURNAL OF CHEMICAL PHYSICS 147, 014501 (2017)

Elemental and cooperative diffusion in a liquid, supercooled liquidand glass resolved

Daniel R. Cassar,1,a),b) Ricardo F. Lancelotti,1,b) Rafael Nuernberg,1,b)

Marcio L. F. Nascimento,2,c) Alisson M. Rodrigues,1,b) Luiza T. Diz,1,b)

and Edgar D. Zanotto1,a),b)1CeRTEV—Center for Research, Technology and Education in Vitreous Materials, Department of MaterialsEngineering, Federal University of Sao Carlos, 13565-905 Sao Carlos, SP, Brazil2PROTEC/PEI—Graduate Program in Industrial Engineering, Department of Chemical Engineering,Polytechnic School, Federal University of Bahia, Rua Aristides Novis 2, Federacao, 40210-630 Salvador,BA, Brazil

(Received 15 April 2017; accepted 5 June 2017; published online 5 July 2017)

The diffusion mechanisms controlling viscous flow, structural relaxation, liquid-liquid phase sepa-ration, crystal nucleation, and crystal growth in multicomponent glass-forming liquids are of greatinterest and relevance in physics, chemistry, materials, and glass science. However, the diffusing enti-ties that control each of these important dynamic processes are still unknown. The main objective ofthis work is to shed some light on this mystery, advancing the knowledge on this phenomenon. For thatmatter, we measured the crystal growth rates, the viscosity, and lead diffusivities in PbSiO3 liquid andglass in a wide temperature range. We compared our measured values with published data covering16 orders of magnitude. We suggest that above a certain temperature range Td (1.2Tg–1.3Tg), crystalgrowth and viscous flow are controlled by the diffusion of silicon and lead. Below this temperature,crystal growth and viscous flow are more sluggish than the diffusion of silicon and lead. Therefore, Td

marks the temperature where decoupling between the (measured) cationic diffusivity and the effectivediffusivities calculated from viscosity and crystal growth rates occurs. We reasonably propose that thenature or size of the diffusional entities controlling viscous flow and crystal growth below Td is quitedifferent; the slowest is the one controlling viscous flow, but both processes require cooperative move-ments of some larger structural units rather than jumps of only one or a few isolated atoms. Publishedby AIP Publishing. [http://dx.doi.org/10.1063/1.4986507]

I. INTRODUCTION

The diffusion mechanisms controlling key dynamicprocesses—viscous flow, structural relaxation, liquid-liquidphase separation, crystal nucleation, and crystal growth—incomplex glass-forming supercooled liquids, such as silicatemelts, have been a matter of great interest and relevance inthe physics, chemistry, and materials science fields for manydecades. However, the atomic entities (structural units) thatdominate and control each one of these processes are stillunknown. These structural units could be individual networkforming ions (such as oxygen and silicon) or a more complexunit involving one or more SiO4 tetrahedra. In theory, thisentity could even include network modifier cations.

One well established phenomenon regarding dynamicprocesses in glass-forming liquids is the so-called breakdownof the Stokes–Einstein–Eyring (SEE) equation.1 Briefly, thisis a decoupling phenomenon between crystal growth andviscous flow kinetics that occurs below a certain tempera-ture, the so-called decoupling temperature, Td , observed in

a)Authors to whom correspondence should be addressed: daniel.r.cassar@gmail.com and dedz@ufscar.br

b)URL: www.certev.ufscar.br.c)URL: www.protec.ufba.br.

the range of 1.1–1.3Tg. Tg is the laboratory glass transitiontemperature,2,3 usually taken as the temperature where theequilibrium viscosity reaches 1012 Pa s. Below Td , the natureor size of the diffusing species that control these two kineticprocesses is clearly different. However, as we already stressed,the details about these species are still unknown. To the best ofour knowledge, it is only in the case of the strong glass formerSiO2 that the decoupling phenomenon has not been detected.In this liquid, self-diffusion of Si4+ and O2�controls both theviscous flow and crystal growth rates4 all the way from abovethe melting point down to Tg.

A relevant article which is well within the scope of thiswork is that of Farnan and Stebbins.5 These authors carried outa series of in situ high-temperature 2D-NMR experiments forK2Si4O9. They showed experimentally that the time scale oflocal silicate species exchange, between the Q3 and Q4 species,in this composition was the same as that of the shear relaxationtime.5 As the characteristic shear relaxation time, τ, is directlyrelated to the equilibrium viscosity, η (τ = η/G∞, where G∞is the infinite frequency elastic shear modulus), this proposedmechanism is contrary to the widespread accepted assumptionthat cooperative movements of some “structural units” controlviscous flow in glass-forming liquids. The flow mechanismappeared to involve a change in the nature of oxygen in theliquid between bridging (BO) and non-bridging (NBO) roles.

0021-9606/2017/147(1)/014501/10/$30.00 147, 014501-1 Published by AIP Publishing.

014501-2 Cassar et al. J. Chem. Phys. 147, 014501 (2017)

According to Farnan,6 this is equivalent to a change in localsilicate species. Hence Stebbins proposed that the Q speciesexchange is closely related to the mechanism of viscous flow.7

He also suggested that if viscous flow is closely linked to therate of network bond breaking, then it should also be closelyrelated to the diffusivities of the network forming cations andanions.

In this paper, we focus on the lead metasilicate composi-tion (PbSiO3, abbreviated as PS). To the best of our knowledge,apart from pure SiO2, PS is the only other known stoichio-metric oxide glass-forming composition for which experi-mentally measured diffusion coefficients of network formercations (silicon and lead) are available in a wide temperaturerange,8–10 above and below Tg. The diffusivities of siliconand/or oxygen species could be used as proxies of melt vis-cosity in silicate systems.7 Our main objective is to computethe diffusion coefficients controlling viscous flow and crystalgrowth and compare them with published self-diffusion dataof Si and Pb for lead metasilicate in the liquid, supercooledliquid, and glassy states. This is an attempt to understandthe mechanisms controlling these two important kinetic pro-cesses. We hope that this comparison might shed some light onwhether elemental diffusion or cooperative rearrangements ofsome larger structural units control viscous flow and crystalgrowth.

II. LITERATURE REVIEWA. Structure of PbSiO3 glass

The structure of lead metasilicate glass has been exten-sively studied using different spectroscopic techniques,11–14

neutron and X-ray diffraction.15 Lee and Kim14 studied leadsilicate glasses (with lower SiO2 content than PbSiO3) andobserved that the mixing between Si and Pb deviates from arandom distribution. It has been shown by NMR that Q2 (40%–60%), Q3 (25%–35%), and Q4 (6%) units are predominant inthis material with very low fractions of Q0+Q1. In anotherstudy,13 only Q2 and Q4 were predicted by thermodynamicmodeling in this glass. All these results indicate an unbal-anced Qn distribution toward glass-forming units (Q3+Q4). Ina study using high-energy X-ray and neutron diffraction mea-surements and structural modeling, Kohara et al.15 concludedthat PbSiO3 can be categorized as a dual network-formerglass. Besides, they claim that the formation of the networkis governed by the interplay of SiO4 tetrahedra and PbOx

polyhedra as network forming units, which results in a glasswith an unusual amount of free volume (void space) that isnot observed in other binary systems, such as CaO–SiO2 andNa2O–SiO2. Hence, a fraction of the Pb2+ ions indeed play therole of network formers.

It is relevant to mention an experimental fact that corrob-orates the dual network former concept; PbO·SiO2 is likely tobe the only metasilicate glass that shows a very good glass-forming ability as it easily vitrifies on the cooling path fromthe liquid state without any signs of crystallization even atlow cooling rates. In contrast, for instance, other metasilicates,Li2O·SiO2, Na2O·SiO2, MgO·SiO2, and BaO·SiO2, are veryreluctant glass formers.

B. Crystal growth

Nascimento and co-authors3 have studied the temper-ature dependence of crystal growth rates in several super-cooled liquids, including lead metasilicate. They have testedseveral growth models and considered that the secondarysuperficial nucleation growth mechanism (also known as “2Dgrowth”) describes the experimental growth rates well for leadmetasilicate16 for temperatures above Td ∼ 1.3Tg.

Equation (1) shows the general formula for this growthmechanism. In this equation, parameters B and C depend onthe secondary crystal nucleation rate that occurs at the interfacebetween a primary crystal and the ambient phase. If this rate issufficiently small then one has the layer-by-layer case scenario(also known as “small crystal case”), with B equal to one andC given by Eq. (2). However, if this rate is sufficiently large,the polynuclei case scenario dominates (also known as “largecrystal case”), with B equal to 3 and C given by Eq. (3).

U = CDU

d20

exp

(−

πd0γ2

BkBT∆µV

), (1)

C = d0NSA0, (2)

C =1

Γ (4/3)

3

√πNSd5

0

3

[1 − exp

(−∆µ

RT

)] 23

. (3)

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

is the effective diffusion coefficient of the structural units thattake part in crystal growth; d0 is their size; γ is the interfacialenergy between the crystal and the supercooled liquid; kB isthe Boltzmann constant; T is the absolute temperature; ∆µV

is the driving force for crystallization per unit volume [seeEq. (4) 17,18]; NS is the number of possible nucleation sites perunit area; A0 is the critical area of a secondary nucleus at thecrystal interface; Γ is the gamma function; ∆µ is the drivingforce for crystallization per mole [see Eq. (5) below]; and R isthe gas constant.

The driving force for crystallization is given by Eqs. (4)and (5),

∆µV =∆µ

Vm, (4)

∆µ = ∆Hm

(1 −

TTm

)−

∫ Tm

T(Cp,l − Cp,c)dT

+ T∫ Tm

T

Cp,l − Cp,c

TdT . (5)

Here, Vm is the molar volume; ∆Hm is the meltingenthalpy; Tm is the melting temperature; and Cp is the molarheat capacity. The subscripts “l” and “c” refer to the liquid andthe crystalline phases, respectively.

We calculated Vm and NS using d0, as shown below andused the A0 expression deduced by Cassar.19 The deduction isreproduced in the Appendix. NA is the Avogadro’s number.

Vm = d30NA, (6)

NS =

(1d0

)2

, (7)

A0 =2πγd0

∆µV. (8)

014501-3 Cassar et al. J. Chem. Phys. 147, 014501 (2017)

C. Diffusion

As we mentioned in the Introduction, measured self-diffusion coefficients for silicon (DSi) and lead (DPb) ions8–10

are available for lead metasilicate liquid, supercooled liquid,and glass. Here we extend the range of available DPb usingimpedance measurements and the Nernst–Einstein relation,

Dσ =σkBT

e2Ni, (9)

where N i is the number density of mobile ions, e is theelemental charge, and σ is the conductivity.

In Eq. (9), the number density of Pb ions can be esti-mated from the glass density and composition. However, it isimportant to note that Dσ is not a diffusion coefficient that canbe calculated or explained by Fick’s laws. The ratio betweentracer diffusivity (Dt) and charge diffusivity (Dσ) is known asthe Haven ratio (Hr = Dt /Dσ) and gives information about thecorrelation between tracer and charge diffusivities related tothe atomic mechanism of ionic motion.

For the interstitial diffusion mechanism in crystals, wherethe current is due to the drift of interstitial ions under anapplied field, Dt and Dσ are equal, resulting in a Haven ratioof 1. For the vacancy diffusion mechanism in crystals, thecharge is effectively carried by the vacancies, giving a Dt

that depends on the ion-ion coordination number, and is notequal to Dσ. For the vacancy diffusion mechanism in glasses,there is a distribution of interstices with different effectivecoordination numbers, making it very complex to obtain Hr .However, there are reports20–22 for glasses that show Hr val-ues lower than those calculated from vacancy migration incrystals. In summary, the Hr values for glasses generallyfall between 0.1 and 1 and are usually around 0.3 when themolar concentration of the mobile ion is higher than 10%.21,22

For soda-lime glass, the Haven ratio measured by Mehreret al.20 is about 0.45 and is practically independent of thetemperature.

The diffusion coefficient for viscous flow (Dη) can beestimated by the Eyring equation, Eq. (10), where the jumpdistance d of the moving unit is taken into account. What iscommonly done in the literature—and also here—is to assumethat d = d0 (d0 was defined in Subsection II B),

Dη =kBTdη

. (10)

In Eq. (10), η is the equilibrium viscosity of the liquid.In this paper, we used the MYEGA equation, Eq. (11),23 toobtain an expression for η as a function of temperature. Thisequation has three adjustable parameters (η∞, Tg,12, and m)that are defined below:

log10 (η) = log10 (η∞) +(12 − log10 (η∞)

) Tg,12

T

× exp

[(m

12 − log10 (η∞)− 1

) (Tg,12

T− 1

)], (11)

η∞ ≡ limT→∞ η (T ) , (12)

η(Tg,12

)≡ 1012 Pa · s, (13)

m ≡d log10 (η (T ))

dTg,12

T

������T=Tg,12

. (14)

In the equations above, m is the fragility index24 and Tg,12

is the temperature where the viscosity is 1012 Pa s.Finally, the diffusion coefficient for crystal growth (DU ) is

part of Eq. (1). If all the other parameters are known or can becalculated, then DU can be readily obtained from experimentalcrystal growth rate data.

III. MATERIALS AND METHODSA. Glass making

The process of making a stoichiometric lead metasilicateglass was repeated six times trying different homogenizationand melting routes in order to produce reproducible, homoge-neous samples. One problem with this particular compositionis that lead, which is significantly heavier than silicon, tendsto accumulate at the bottom of the crucible during the meltingprocedure. Another is lead evaporation due to its high vaporpressure. These difficulties may result in a chemically inhomo-geneous glass, which could be an issue because, in that case,the crystallization kinetics measured in different specimenswould not be coherent. Here we report data for two reliablelead metasilicate glasses we made, named Batch 1 and Batch 2.We distinguish these two glasses even though both yielded sim-ilar crystal growth rates, which are also comparable to reportedliterature data.

The glass of Batch 1 was obtained from a mixture ofground quartz SiO2 (Vitrovita, Brazil) and Pb3O4 (Sigma-Aldrich, USA), both of analytical grade, in the required stoi-chiometric quantities to form PbSiO3+O2. 100 g of the powdermixture was thoroughly homogenized in a TURBULA T10Bmixer. The homogenous mixture was melted in a Joule effectfurnace in a platinum crucible with constant manual agitationwith a platinum blade. The temperature did not exceed 1273 K.The melting process took less than 10 min to minimizevolatilization of the lead. We just waited until the majorityof the bubbles were eliminated, then the melts were poured ona stainless steel plate and pressed with a stainless steel pistonto quench them.

The glass of Batch 2 was also prepared by melting a thor-oughly mixed homogeneous mixture of the same chemicalsin a platinum crucible. The melting temperature was 1173 K,with a hold time of 2 h. The melt was then cast between twocold steel plates, resulting in a yellowish glass, similar to thatof Batch 1.

B. Viscosity measurements

The viscosity for glass samples of Batch 1 was mea-sured in the temperature range of 964–752 K at the Instituteof Silicate Chemistry, Russian Federation. The penetrationviscosimeter was designed, constructed, and operated by thelate Dr. Valentin P. Klyuev. The description of the apparatuscan be found elsewhere.25 We also considered viscosity val-ues reported by different authors,26–30 including (very rare)isostructural viscosity below Tg reported by Koide et al.30

Isostructural viscosity, as the name suggests, is a measureof viscous flow in a condition for which the structure of the

014501-4 Cassar et al. J. Chem. Phys. 147, 014501 (2017)

material does not change during the time scale of the mea-surements. For more information on this topic, please refer toRef. 31.

C. Crystal growth rate measurements

We used 2 × 2 × 2 mm3 samples with polished facesto measure the sizes of the PS crystals as a function of theheat treatment time in isothermal conditions. We used verticalfurnaces for the heat treatments and measured the temperatureswith a type K thermocouple calibrated following the ASTME 207–00 standard. We observed the surface crystals and tookdigital micrographs using a Leica DMRX optical microscopewith a Leica DFC490 image acquisition camera.

The time required to nucleate the first crystals on thesample surface at temperatures equal to or lower than 725 Kwas quite large. Because of that, for these temperatures (T≤ 725 K), we “seeded” the sample surface by pre-formingsome crystals at 785 K for 34 min. This treatment was enoughto nucleate a few crystals that would later grow at the desiredgrowth temperatures. All the image analyses were done withthe free and open-source software ImageJ.32 Most of the PScrystals have an ellipsoidal habit; hence, we measured thegrowth rates of the fastest (longest) crystalline direction. Thechosen crystals were isolated in the glassy phase. At eachtemperature, a unique crystal was located and measured aftereach heat treatment. The size (half of the longest length of thechosen crystal) versus time plots for all the crystals at all tem-peratures were well described by a linear regression. We alsoobserved spherulitic crystal forms in samples treated at highertemperatures.

We corrected all crystal growth rate data analyzed inthis work by the increased liquid–crystal interface temper-ature using Eq. (15). This important correction is neededbecause the latent heat released due to crystallization cansignificantly increase the interface temperature if it is notdissipated fast enough. Herron and Bergeron33 measured thetemperature increase at the liquid–crystal interface during thecrystal growth process in several oxide glass-formers. Cassar19

reanalyzed their data and obtained the following empiricalexpression:

∆Ti (T ) = 100.64(U (T )∆Hm)0.58, (15)

where ∆T i is the liquid–crystal interface temperature increaseat a temperature T. In this expression, U is given in m/s and∆Hm in J/mol.

D. Impedance measurements

The ionic conductivity of samples of Batch 1 was obtainedvia electrochemical impedance spectroscopy (EIS). For EISmeasurements, gold was sputtered on parallel opposite facesof a glass specimen in order to ensure electrical contact. Thesample was 0.294 cm thick and had a contact area of 0.602 cm2.EIS measurements were conducted using a NovocontrolAlpha-A impedance analyzer over the frequency range10 MHz–0.1 Hz, with an applied root mean square AC voltageof 500 mV. These measurements were carried out at tempera-tures below Tg over the range of 473–673 K. The temperature

control was performed by using the Novotherm tempera-ture control system with maximum temperature variation of±0.1 K (if the variation is higher than 0.1 K, the equipmentstops the measurement, waits for stabilization, and then returnsto the measurement) during the EIS measurements.

To determine the ionic conductivity σ of PbSiO3 glass,we fitted the impedance data using Zview 3.5 and an appropri-ate equivalent circuit. The chosen equivalent circuit [shown inFig. 2(a)] comprises a parallel combination of a resistance (Rg)and a constant phase element (CPEg) taking into account theglass sample resistance and capacitance, in series with a CPEe

related to electrode polarization (blocking ion effect). The usedtype of data weighting was calc-modulus since the obtainedimpedance spectrum covers a wide range of impedances.34

E. Activation energy

We computed the activation energies, E, for three diffu-sional processes—crystal growth (below the decoupling tem-perature Td), lead diffusion, and isostructural viscosity—byfitting an Arrhenian expression,

D (T ) = D0 exp

(−

ERT

), (16)

where D is the diffusion coefficient and D0 is a constant.We obtained E using a robust linear regression on a

linearized plot of ln (D) versus 1/RT. The uncertainty wasobtained considering a confidence level of 95%. The algo-rithm used was the function “theilslope”35 that implements aTheil–Sen estimator.36,37 This algorithm is available in the freeand open-source SciPy module38 for the Python programminglanguage.

The effective activation energy for equilibrium viscos-ity depends on the temperature, and we obtained it by usingEq. (17).39 In this equation, we used the MYEGA regression[Eq. (11)] for log10 (η (T )),

Eη,eff (T ) = R ln (10)d log10 (η (T ))

d1/T. (17)

IV. RESULTS AND DISCUSSIONA. Ionic conductivity

Figure 1 shows the dependence of the real (Z ′) and imag-inary (–Z ′′) parts of impedance as a function of frequencyover a range of temperatures (473–673 K), measured for sam-ples of Batch 1. The inverse of electrical conductivity can bepromptly read on the left y axis at the low frequency plateaubecause the results were normalized by the shape factor ofthe sample (thickness over area). The dependence of electricalconductivity with temperature is logarithmic, as expected.

With respect to the imaginary part of impedance, the–Z ′′ peak shifts to higher frequencies as the temperature isincreased, since the relaxation frequency is increasing as thebulk resistance Rg of the glass decreases (f ∼ 1/RgC, C is thecapacitance). The most important result is that at higher tem-peratures (633–673 K)—where the used frequency range cancover the entire electrical behavior of the cell Au/Glass/Au—the –Z ′′ part starts to increase again in low frequencies. This

014501-5 Cassar et al. J. Chem. Phys. 147, 014501 (2017)

FIG. 1. Dependence of the real (half up filled symbol) and imaginary (halfdown filled symbol) parts of impedance on frequency, normalized by the shapefactor.

behavior is typical for an ion conductor, and it is caused by ablocking ion effect in the metallic electrode.

Conversely, in case of an electronic conductor, –Z ′′ shoulddecrease continuously, since the electrode used in the measure-ment (gold in this case) is several orders of magnitude moreconductive than the analyzed glass. On the other hand, theblocking ion effect occurs because the ions cannot pass throughthe electrode used. These results indicated that the resistivitymeasured here takes place by the motion of ions through theglass, most likely Pb2+. Indeed, Cohen et al.40 already reporteda similar result when they measured the resistivity of a PbSiO3

glass, and their results were in close agreement with publisheddata for Pb-ion diffusion.40

Figure 2(a) shows the complex impedance and the fit-ted curves. The glass capacitances C ranged between 2 and 3pF, resulting in a dielectric constant (εr) between 23 and 33,which is in good agreement with the expected dielectric con-stant for this type of material.40 The electrode capacitances,which were determined only at higher temperatures (633–673 K), ranged from 1 to 12 µF. At lower temperatures, the usedfrequency range could not cover the entire electrical behav-ior of the Au/Glass/Au cell; therefore, there is no electroderesponse.

We estimated a theoretical electrode capacitance of 4.4 µFby considering a monolayer of Pb2+ ions of ionic radius of

119 pm blocked at both sides of the electrode and using theelectrode area of the glass sample (0.6 cm2) and a reasonableεr ∼ 2 (this procedure is the same as discussed in Ref. 41).This result is in agreement with the experimental electrodecapacitances that we obtained, supporting that the electri-cal conductivity of the PS glass is caused mainly by Pb2+

motion.In addition, we calculated the activation energy (Ea) for

Pb ion conduction using an Arrhenius equation [Fig. 2(b)]. Thecoefficient of determination (R2), the activation energy (Ea),and pre-exponential term (σ0) of an Arrhenius-like equation(σT = σ0exp[�Ea/RT ]) are also shown in the plot. The Ea

= 1.267(9) eV is in close agreement with previous reportedresults of 1.12 eV.40 To obtain Dσ, we used Eq. (9) for theconductivity data. Here we used a Ni of 1.274 × 1028 Pb+2/m3

estimated by using a density15 of 6 g/cm3 and the nominalmolar weight of 1:1 PbSiO3 composition. Finally, to obtainDPb, we used the average value of 0.3 ± 0.2 for Hr (this is theorigin of the uncertainty bars in Fig. 6 that will be presentedlater in the manuscript).

B. Viscosity and crystal growth rates

Figure 3 shows the experimental viscosity data measuredin this work (Batch 1) together with the literature data overa wide range of temperatures. The agreement between thedifferent datasets is good enough and within the usual vis-cosity measurement errors. We fitted the MYEGA equation,Eq. (11), to all these data (except Koide’s isostructural vis-cosity30) and obtained Tg,12 = 679.9(3) K, m = 55.1(4), andlog10(η∞) = �1.76(6) (η∞ in Pa s, the uncertainty is one stan-dard deviation, computed by the regression algorithm). Wethen used the fitted viscosity curve in all of the followingcalculations.

Crystal growth rate data for lead metasilicate glassesbetween 723 and 998 K have been reported by Neimanet al.42 Figure 4 shows the crystal growth rates obtained inthis work plotted together with the aforementioned data. Thiscombined dataset spans six orders of magnitude, between10�6 and 10�12 m/s. In particular, the maximum growth ratewas observed at about 923 K (Tmax/Tm ∼ 0.90). Hence, wehave significantly extended the temperature range of avail-able experimental crystal growth rates for this composition

FIG. 2. Set of EIS analyses representedby (a) complex impedance plots of a PSglass sample in the temperature range of473–673 K (symbols) and their respec-tive fit (line) based on the shown equiv-alent circuit and (b) the Arrhenius plotfor the dependence of ionic conductivityon inverse temperature.

014501-6 Cassar et al. J. Chem. Phys. 147, 014501 (2017)

FIG. 3. Equilibrium and non-equilibrium viscosity data for samples fromBatch 1 combined with selected datasets obtained by different techniquesfor different batches of lead metasilicate glasses reported by differentauthors.26–30 The full line is a regression of all these data points using theMYEGA equation.23 The Td region will be discussed in the text.

toward the glass transition range. The ellipsoid shaped surfacecrystals that nucleate in this composition are shown inFig. 5.

Figure 4 shows two calculated crystal growth rate curvesconsidering the Secondary Superficial Nucleation growthmodel. The full line is for the layer-by-layer case, for which weused a value of 32 Å for d0 and 0.97 mJ/m2 for γ. The tracedline is for the polynuclei case with d0 = 11 Å and γ = 13 mJ/m2.

FIG. 4. Crystal growth rates measured for samples of the two studied batchestogether with data from Neiman et al.42 The two fitted curves are based on the2D growth model, which satisfactorily describes the high temperature data.The uncertainty bars for our data are smaller than the symbol size. The Tdregion is discussed in the text.

FIG. 5. Surface of a PS glass sample (Batch 1) heat treated at 785 K observedby optical microscopy with transmitted light using 50 × objective lenses.

These values are those that best fit the data. Only the data nearthe melting point could be successfully fitted. We found nocombination of these two parameters that satisfactorily fittedthe data to temperatures below 807 K.

The values for d0 are of the expected order of magni-tude (i.e., the size of a few unit cells), but the values forγ are much lower than those reported for primary crystalnucleation studies (which are usually in the range of 100–200 mJ/m2 for silicate glasses). This, however, is not an issue.For instance, Calvert and Uhlmann43 studied the differencebetween the γ value for primary crystal nucleation and forsecondary-nucleation driven crystal growth and also found thatthe latter is smaller. In their manuscript, they analyzed crystal-lization data for five compositions and systematically obtainedvalues for γ from growth mediated by secondary superficialcrystal nucleation that are typically one order of magnitudelower (i.e., ∼10 mJ/m2) than those for primary crystal nucle-ation. This is in good agreement with the value for γ that weobtained here for the polynuclei case. In any case, the valuesfor d0 and γ in our analysis are only approximations becauseall the uncertainties of the used parameters and experimen-tal data—as well as the uncertainties due to the assumptionsof the model—are accumulated in these two unconstrainedparameters.

It is important to state that two other popular crystalgrowth models—the screw dislocation and the normal growthmodel16—were not able to fit the high temperature data rea-sonably well, even with these two free parameters (d0 and γ).Another interesting detail is that Neiman measured the growthof the crystalline layer, whereas we measured crystals growingon the glass surface. Even with this difference, both datasetsare in very good agreement in the temperature range wherethey overlap.

Figure 4 shows that even the considered model (SecondarySuperficial Nucleation growth) was not sufficient to describethe crystal growth data in the whole temperature range. Thisfinding supports the well-known concept that this supercooledliquid also undergoes the classical decoupling between crystalgrowth kinetics and viscous flow at a certain temperature aboveTg. Previous data hinted that this could be the case for this

014501-7 Cassar et al. J. Chem. Phys. 147, 014501 (2017)

material, but now, with our extended growth data, the overallpicture is much clearer. In some figures, we delimited a “Td

region.” This is the temperature range where the crystal growthdecouples from viscous flow. That is, below this temperaturerange they are decoupled. We estimated the Td region as beingin the range of 807 ≤ T ≤ 893 K (1.2Tg–1.3Tg).

Our analysis is in agreement with a previous test for thedecoupling done by Nascimento et al., considering much lessdata points.3 This, however, is not the most important find-ing of this article; we are more interested in calculating thediffusivities controlling viscous flow and crystal growth andcomparing them with those of the individual elements.

C. Diffusion coefficients

The dynamic properties presented above are controlledand connected to diffusion processes and provide a uniqueopportunity for performing a comparative analysis to estimatethe effective diffusion coefficients that control viscous flowand crystallization. To boost this analysis, we used additionaldata on the ionic conductivity and self-diffusion coefficientsof the cationic species Si4+ and Pb2+, which were measuredin actual diffusion experiments and also by electrochemicalimpedance spectroscopy.

Figure 6 summarizes all the different diffusivities mea-sured or calculated in a wide temperature range from above themelting point to below the glass transition range. It is impor-tant to note that they include viscous flow and crystal growthmeasurements from different authors in a wide temperatureinterval. To the best of our knowledge, this study comprisesone of the most complete sets of diffusion processes ever col-lected, calculated, and analyzed for an oxide glass-formingsystem. It covers four different transport processes spanning

TABLE I. Activation energies for the diffusion processes studied here. Amole here is defined as a mole of PbSiO3.

∆T range Activation ReferenceSymbol Diffusion process (K) energy (kJ/mol) for data

EPb,1 Pb self-diffusion (Batch 1) 473–673 117(2) This studyEPb,2 Pb self-diffusion 613–807a 100(20) 44

Eiso Isostructural viscosity 484–514 120(40) 30

Eη,eff Equilibrium viscosity 646–1537 Figure 7 This studyand Refs.

26–30EU,1 Crystal growth (Batch 1) 688–798 370(90) This studyEU,2 Crystal growth (Batch 2) 703–807a 260(30) This study

aThe upper temperature limit was chosen to avoid data in the Td region [see Fig. 6(a)].

16 orders of magnitude in a wide range of temperatures, fromabove Tm to far below Tg! We believe that these plots can givea thought-provoking insight into the diffusion kinetics in thisnon-crystalline material.

D. Activation energy

Table I compiles the activation energy for all the diffusionprocesses studied here, together with the temperature rangewhere the computation was performed. For some calculations,we limited the range to allow only data below the Td region.These are marked with the letter “a” in the table. The reasoningfor this is that, for temperatures in the Td region and above, theviscosity starts to control these processes, which results in adeviation from an Arrhenian behavior showing a temperaturedependence of the activation energy.

The data of Table I readily show that EPb,1 is statisticallyequal to EPb,2 (within the uncertainty), which is expected and

FIG. 6. Temperature dependence of theeffective diffusion coefficients calcu-lated from crystal growth rates and vis-cosity, and measured ionic conductivity,as well as self-diffusion coefficients ofSi and Pb. (a) Zoom in the tempera-ture range where crystal growth data areavailable. (b) Overall picture that spans16 orders of magnitude.

014501-8 Cassar et al. J. Chem. Phys. 147, 014501 (2017)

FIG. 7. Effective activation energy for equilibrium viscosity as a function oftemperature plotted with the values for EU,1 and EU,2 (see Table I).

suggests that there is no mechanism change in lead diffusionfrom 473 up to 807 K. What is quite interesting is that leaddiffusion has the same activation energy as the isostructuralviscosity, Eiso. This result suggests that the mechanism thatcontrols non-equilibrium viscous flow is the same that controlsPb ionic diffusion.

The magnitude of lead and silicon diffusivities below Tg

is quite similar. Unfortunately, the scarcity of data for silicondiffusion below the Td region inhibits a more detailed analy-sis. It is known that lead (partially) and silicon (always) havethe role of glass-formers40 in this particular system, and thisfact would suggest that their diffusivities could be close. Inaddition, Schaeffer10 suggested that the oxygen self-diffusionhas the same order of magnitude as that of silicon in most sil-icate glasses, which could indicate a possible mechanism ofdiffusion by cooperation of these 3 species.

Finally, the activation energies for crystal growth mea-sured in samples of two batches, EU,1 and EU,2, are statisticallyequal. Moreover, they are comparable to the effective activa-tion energy for viscous flow in the Td region, as shown inFig. 7. This observation may be particular to this specific com-position, but we believe it might be useful to connect Eη,eff(Td)with EU. This, however, is not in the scope of the presentstudy.

E. Decoupling

Figure 8 complements our analyses by showing the ratioDU /Dη and DPb/Dη as a function of the inverse temperature.Looking at Fig. 6(a), one might think that the crystal growthdata below the decoupling temperature are somewhat parallelto the viscosity, suggesting that there is actually no decoupling;perhaps only a change of growth mechanism that resulted in ashift of DU . Visually this may seem to be the case, but Fig. 8shows that it is not. If the curves were parallel, then the DU /Dη

data would reach a plateau at some temperature below thedecoupling. A glimpse at Fig. 8 clearly shows that this doesnot happen.

FIG. 8. DU/Dη and DPb/Dη against the inverse of temperature. Observe thegradual deviation from the DU = Dη full line, showing the decoupling betweencrystal growth and viscous flow.

Lindner44 also pointed out that there was a sharp changein the lead diffusion coefficient at 806 K, showing twostraight lines, suggesting a transition of the diffusion mech-anism due to two widely different activation energies. Thiscould be seen as additional evidence for the decoupling,observed from diffusion experiments. It is interesting thatlead diffusivity seems to decouple from viscous flow in thesame temperature range where crystal growth and viscositydecouple.

F. Diffusing entities

The calculated values of DPb from conductivity dataclosely agree with the reported lead diffusivities, suggestingthat electrical conduction in lead metasilicate glasses takesplace mainly by the motion of Pb2+ ions. Moreover, silicondiffusion is clearly decoupled from viscous flow at lower tem-peratures, as shown in Fig. 6(b). It is hard to draw a conclusionabout silicon diffusion above the melting temperature becauseonly one data point is available. However, it is worth mention-ing that it is within an order of magnitude of that of viscousflow diffusion.

We believe that this combined dataset gives a stimulatinginsight into the microscopic nature of the diffusion processescontrolling crystal growth and viscous flow for this oxide glass-former and may be a good basis for further detailed studiesof kinetic phenomena. In the following paragraphs, we dis-cuss our hypothesis about the diffusing entities that controlthe studied kinetic processes in PbSiO3.

At temperatures above the Td range (i.e., above 893 K),viscous flow and crystal growth kinetics are coupled and theyboth seem to be limited by the mobility of the network form-ing cations (Si and Pb). In other words, they are governed bythe same mechanism. This means that DU , Dη, DSi, and DPb

are proportional to each other (activation energies are quite

014501-9 Cassar et al. J. Chem. Phys. 147, 014501 (2017)

similar). This finding provides further evidence to supportthe validity of the Eyring equation for temperatures that aresufficiently high, in agreement with previous studies in othersilicate systems.3,45

At temperatures below the Td range (i.e., below 807 K),the diffusivities controlling viscous flow and crystal growthare clearly different characterizing the classical decouplingphenomenon, which has been assigned by different authors todynamic heterogeneities.46,47 The Eyring equation no longerpredicts crystal growth dynamics, and at least three differentdiffusing entities seem to be present: one that controls the equi-librium viscosity (slowest), one that controls crystal growth(intermediate), and one that controls self-diffusion of the net-work forming ions, and perhaps also non-equilibrium viscosity(fastest). Hence, our results show that at deep supercoolings,the viscous flow and crystal growth kinetics are controlledby different diffusion entities. In addition, individual networkformers (Si and Pb) have a much faster diffusivity, and theiractivation energy is very similar to that of non-equilibriumviscosity.

The overall results suggest that, below the decouplingtemperature, both viscous flow and crystal growth indeedrequire a cooperative movement of some “structural units”or “molecules” instead of just jumps of one or a few isolatedatoms. However, these units have different chemical naturesor sizes for these two dynamic processes, resulting in differ-ent activation energies. This is the most original and relevantfinding of this research work.

V. SUMMARY AND CONCLUSIONS

We studied the diffusion coefficients controlling fourdifferent dynamic processes in a liquid, supercooled liquid,and glass with PbSiO3 composition: crystal growth rates,equilibrium viscosity, non-equilibrium viscosity, and cationicself-diffusion of Si and Pb. With this extensive dataset, wesearched for clues that could shed light on the mechanismunderlying the well-known decoupling between crystal growthand viscous flow.

Based on our results, we propose that the nature and/orsize of the diffusing units that control viscous flow are differ-ent from the diffusing units that control crystal growth andthey are not single cations (lead or silicon), but some slower(larger) structural units. On the other hand, at temperaturesabove Td , crystal growth and viscosity are controlled by thesame diffusing units.

Another interesting result is that lead diffusion belowTd has the same activation energy as isostructural viscosity,suggesting that the latter is controlled by lead atom hopping.

Taken in toto, these findings shed some light on the under-standing of transport properties, such as crystallization, self-diffusion, and viscous flow in liquid and glassy PbSiO3, and

may be useful for further studies with other glass-formingsystems.

SUPPLEMENTARY MATERIAL

See supplementary material for the experimental dataobtained from this study in table format.

ACKNOWLEDGMENTS

We are extremely grateful to the constructive insights ofProfessor J. F. Stebbins. We are also thankful for the sup-port of the Sao Paulo Research Foundation, FAPESP, GrantNos. 2013/07793-6 (CEPID) and 2015/20681-8 (R.F.L.), andthe Brazilian National Research Council, CNPq Grant Nos.150490/2015-1 (D.R.C.) and 140456/2014-7 (RBN). Specialthanks to Bruno B. de Lima and Leonardo S. Gallo for car-rying out (long ago) some initial crystallization experimentswith this glass in our lab.

APPENDIX: AN EXPRESSION FOR A0

The parameter A0 present in Eq. (2) is the critical areato form a secondary nucleus at the crystal–ambient interface.Figure 9 illustrates the secondary nucleation. When a sec-ondary nucleus with cylindrical shape nucleates, it has avolume V c,

Vc (r) = πr2h, (A1)

and it creates the additional area Ac (shown in gray in Fig. 9),

Ac (r) = 2πrh. (A2)

The change in the Gibbs free energy (∆Gc) due to thenucleation of a secondary nucleus is

∆Gc (r, T ) = Vc (r)∆GV (T ) + Ac (r) γ. (A3)

Taking the partial derivative of ∆Gc in respect of r, weobtain

∂∆Gc (r, T )∂r

= 2πhr∆GV (T ) + 2πhγ. (A4)

The critical radius, r∗2D, is the radius where Eq. (A4) iszero. It means that any increase in the radius above this criticalvalue will result in a decrease in the Gibbs free energy.

∂∆Gc (r, T )∂r

�����r=r∗2D

= 2πhr∗2D∆GV (T ) + 2πhγ = 0. (A5)

Rearranging the above expression and using the relation18

∆µV = −∆GV , we have

r∗2D (T ) =γ

∆µV (T ). (A6)

Substituting Eq. (A6) in Eq. (A2) and considering thath = d0, we have an expression for A0,

Ac

(r∗2D (T )

)≡ A0 (T ) =

2πd0γ

∆µV (T ). (A7)

FIG. 9. Schematics of the nucleation of a secondarynucleus.

014501-10 Cassar et al. J. Chem. Phys. 147, 014501 (2017)

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