American Mineralogist, Volume 67, pages 676-685, 1982

Small angle X-ray scattering study of pre-nucleation behavior oftitanium-free and titanium-bearing diopside glasses

LuNc-CHueN Kuo aNo R. Jeuns Klnnpernlcr

Department of Geology, University of IllinoisUrbana, Illinois 61801

Abstract

The kinetics of nucleation in titanium-bearing and titanium-free diopside glasses has beenstudied at small undercoolings. Small angle X-ray scattering (sexs) techniques have beenused to determine the concentration and size of nuclei with radii less than the critical radiusin these glasses. Results ofthe saxs analysis indicate that the radius ofgyration (a measureof the average radius) of the subcritical nuclei ranges from about 200 to 650A, and increaseswith increasing time for each composition at each undercooling and with increasing TiO2content at a given time and undercooling. The number of subcritical nuclei (1.3 x l0l to 6.8x 103 cm-3) increases more rapidly than linearly at each undercooling and TiO2 contentand exhibits a trend similar to that of ordinary transient nucleation density variation. Thelong time (approaching steady state) rate of increase in number of these nuclei (42.1 to 137cm-3 hr-r) increases linearly with increasing TiO2 content, while the "induction time" (10to 47 hours) decreases with increasing TiO2 content at both undercoolings. These data arequalitatively consistent with the traditional physical picture of transient nucleation andyield a melt-crystal surface energy of the order of a few hundred ergs/cm2. The non-steadystate behavior can be described by the theory of transient nucleation developed byKaschiev /1969).

Introduction

The crystallization of a silicate melt occurs by theformation of a small volume of crystal in the melt(nucleation) and the increase in size ofthese nuclei(crystal growth). Although there have been a num-ber of recent theoretical and experimental investi-gations of nucleation and growth kinetics in geologi-cally important systems (see Lofgren, 1980; Kirk-patrick, 1981, for reviews), nucleation, particularlyat small undercoolings, is very poorly understood.The major experimental problems are that nucle-ation at small undercoolings is often sluggish, andthat the nuclei are submicroscopic, so that quantita-tive data on nucleation kinetics are difficult tocollect. It is also not possible to reheat the samplesat a lower temperature to grow the nuclei to a smallbut microscopically visible size without more nucle-ation taking place at that temperature. Thus, meth-ods of study other than direct microscopic observa-tion are desired.

One method to determine the average size andshape of small heterogeneities between about l0Aand a micron in size is small angle X-ray scattering(slxs). This method has been widely applied to0003-004x/82/0708-0676$02.00 676

phase separation studies in materials science (Ger-old and Kostorz, 1978), metallurgy (Gerold, 1967),biology (Kratky, 1963), and many other fields(Hosemann et al., 1970). Particle sizes in variousphase-separated silicate glasses have been deter-mined using this method (e.9., Williams et al., 1965:'Hench et al.. l97l: Neilson. 1972:'Patel et al., 1972:'Faber and Rindone, 1980).

The purpose of this study is to investigate thenucleation kinetics in diopside composition melt asa function of temperature and titanium content atsmall undercoolings. The number and radius ofgyration of subcritical nuclei are calculated fromsexs analysis and the rate of increase in number ofsubscritical nuclei is determined. These results areinterpreted in terms of nucleation theory and sili-cate melt structure.

Theory of slxs

Detailed treatment of the theory of sexs has beengiven by Guinier and Fournet (1955) and Hosemannet al./1970D. Several concise reviews of the theoret-ical and experimental aspects of s.q,xs are alsoavailable (e.9., Guinier, 1963; Kratky, 1963; Ger-

KUO AND KIRKPATRICK: PRE-NUCLEATION BEHAVIOR OF GLASSES 677

(6)

old, 1967; Gould, 1971). In this section only thetheory of sAXs pertaining to the present study isoutlined.

For a system in which identical, widely dispersedparticles of volume vp, and uniform electron densityp, are suspended in a matrix of electron density p',the intensity of X-rays scattered at small angles canbe written

r(e): n2 exp (- # t), (r)

where e is the scattering angle, )r is the wavelengthof the incident X-ray, R* is the radius of gyration ofthe particles , fr2 -- vzo (Adz I"Np, I" is the intensityscattered by one electron, Lp : fp', and No is thenumber of particles per unit volume (Guinier, 1963).Thus, a plot of log I vs . ez (referred to as the Guinierplot) tends to yield a straight line of slope o forsmall values of e:

4J"

: # 0.43$ R?. e)

For CuKa radiation

Re : 0'644 (-o)t' ' A- (3)

If there is a distribution of sizes of geometricallysimilar particles the measured radius of gyration isgiven by the relationship

*a: T ou*Eu (4)

I p*REn

(Guinier and Fournet, 1955). Thus the radius ofgyration is heavily weighted towards the largersizes. In fact, Guinier and Fournet (1955) haveshown that at very small angles, where the data ofthis work have been collected (see below), thescattering intensity is dominated by only the largestparticles. Thus, the radius of gyration determinedfrom our measurements is likely to be a goodmeasure of that of the largest particles.

Another important factor is the integrated intensi-ty of the scattered beam, Q, given by

f'

Q:4r I s2 t(s) ds : I"(Ap)2 V c(t - c) O (5)- J o

(Guinier and Fournet, 1955; Patel et al., 1972),where O : exp (-l.c^p^[), V is the volume ofsample, c is the volume fraction occupied by the

particles, O is the transmissivity of the samples, 4.-is the X-ray mass absorption coefficient, p- is themass density of the sample, and / is the thickness ofthe sample. The value of c may be determined if Q,I" and (Ap)z can be evaluated. The number ofparticles per unit volume is given by

No : c/vo

(Gerold, 1967).

Experimental procedure

The starting materials used in this study wereprepared from reagent grade CaCO3, MgO (Mal-linckrodt Inc.) and TiOz (Baker Co.) and ultrapureSiO2 (Johnson Matthey Co. and Aldrich Co.). Ho-mogeneous mixtures of appropriate amounts ofeach powder were fired to drive out COz and thenmelted in a Pt crucible in a Deltech MoSi2 resist-ance furnace at about 1470' C for 2 to 5 hours. Theresulting glass was reground and remelted twice.Three compositions were made: Pure diopside,diopside * 0.5 wt.Vo TiO2, and diopside + l.0wt.VoTio2.

Isothermal experiments at atmospheric pressurewere carried out at undercoolings from 10'to 50' C.Each run consisted of about 5 g of glass in a Pt foilboat (2.5 cm x I cm x 0.7 cm) and was melted at1471" C (80'C superheating) for I hour, cooleddirectly to the desired undercooling, and then treat-ed at that undercooling for the desired time. Carewas taken to avoid undershooting the run tempera-ture by setting the temperature about 5" higher thanthe run temperature and manually resetting thecontroller as the temperature fell. The time to reachthe run temperature was about 8 minutes. Thetemperature was calibrated against the meltingpoint of diopside using a PtlPt-l3VoRh thermocou-ple. At the end of each run, the charge wasquenched in air and then transferred into a Thermo-lyne 1300 electric furnace preheated to about 400' Cto release the residual stress in the glass. Thecharge was held at 400' C for about halfan hour andthen cooled to room temperature by turning off thefurnace. A glass sample for each composition wasobtained by direct quenching of the melt from1471" C followed by the annealing cycle.

Table I summarizes the experimental conditionsand results. Because several runs at 20" C under-cooling and almost all runs at undercoolings largerthan 20'C are fully crystallized, only glassy runs atundercoolings of l0'and 20" C were used for saxsstudy.

678 KUO AND KIRKPATRICK: PRE-NUCLEATION BEHAVIOR OF GZASSES

Table 1. List ofexperimental conditions and results

RunNo.

AT tlEe at( "c ) Ar (h rs ) Product*

the bottom and top of the charge were removed toeliminate crystals nucleated heterogeneously on thesurface. The slices were then observed using adisecting microscope to make sure that no bubblesor scratches were present.

The scattering data were collected stepwise at anangular increment of 1.524 x l0-3 degree 20 overthe range from about 0.072 to 0.094 degree 20 and of2.54 x 10-3 degree 20 over the range of 0.094 to0.102 degree 20. A Norelco X-ray generator usingCuKa radiation was operated at 40 kV and 12 mA.The X-ray beam was collimated with a Kratky sexscamera (entrance slit height: 73 p.m, exit slit height:50 pm) and partially monochromized with a Ni filterand a pulse-height analyzer. The sample was posi-tioned directly behind the collimating system withthe polished surface facing the X-ray source, theirradiated area being 0.33 cm2. The sample-to-detector length was 230 mm. The scattering intensi-ty was recorded with a proportional counter with adigital ratemeter output. Data for the quenchedglass sample were always taken on the same day asthose for the heat treated glasses of the samecomposition.

In order to obtain the real scattering of the heattreated samples, the following equation

I ( e ) : I s ( e ) - F I c ( e ) ( 7 )

was applied for data correction (see also Neilson,1972;Faber and Rindone, 1980). I is the true samplescattering, I5 and 16 the measured heat treated andglass sample scattering, and

F : IB"/I8r,

where IBe and Ifis are the direct beam intensities ofheat treated and glass samples taken at zero20 at 12kV and 13 mA. Similar methods of data correctionare given by, for example, Williams et al. (1965) andDwiggins (1979). Because our samples are essential-ly amorphous, data corrections for slit height andslit width were assumed to be negligible (Patel etal., 1972; Schmidt, 1976).

In order to confirm the reproducibility of thedata, scattering intensities of one set of samples(diopside + 1.0 wt.Vo TiO2, 10'C undercooling)were remeasured under the same operating condi-tions. The difference of log I values at a given angleis not more than about 2Vo relative, and that of log(AI) for any two angles is of the order of O.lVorelative. These lead to essentially identical Guinierplots and results.

crystalmorphoLogy

6

c

6

5

cxx

x

DG

D49U ) J

D55D58

D10D12D14D2LD30

D51D52D56

D59

1. D lops ide

2. D lops lde + 0 .5 wt7 . T10"

D r 5 1D r 5 3D ' | s 4D ' !55

D . I OD ' 1 1D ' 1 4D ' 1 7D r 1 8D ' t 9

3. Dlopslde + 1.0 wt7" TiO2

0

10 2270

I I O

7

20 206

45

L22 L . 57 l

0

I

r1620

16845702 79

z4

15

4

D'.G

D"46D"51D " 5 7D, ,58

D"8Dr '10D " 1 1D"16D, '17D"23D, '30

D"47D, ,59

0

10 48T J

I20

20 2027

t 62 l s

52

acicuLaracLcular

sphefrl lt lc

faceted

faceted

acicular

faceted

acicularacicular

c6

6

5

xx6

E

6

6

E

xb

x5

xx

(8)9

35 1524

* g : g lass ; x : fu l l y c rys taL l i zed .

Our potential difficulty in interpreting the sexsdata is that scattering could occur from gas bubblesin the glass. No bubbles were identified in theseveral pieces examined by TEM. To affect thesexs results the bubbles would be too small to beresolved in an optical microscope, although largerones were looked for and not found.

saxs analysis

The samples of sexs study were cut into rectan-gular slices of about 2 cm x 0.8 cm in area andabout 80 pm in thickness and polished on onesurface, finishing with 12.5 pm Al2O3 powder. Both

The main purposes of sexs study are to deter-mine the time dependence of the size and concen-tration of the subcritical nuclei in the glass. Theradius of gyration was found by quadratic fitting ofGuinier plots and then evaluating the slope of thenearly linear region in these plots at small angles(Guinier and Fournet, 195; Guinier, 1963; see Figs.I and 2). These slopes were used to calculate R,values using equation (3) and subsequently Q valuesin equation (5). The determination of (Ap)2 requiresthe absolute scattering intensity at zero angle, Ia(0)(Guinier, 1963; Kratky, 1963). Since this intensity isfar beyond the detection limit of the counter, valuesof Ia(O) were obtained by extrapolating Is(e) back tozero and then normalizing with the relationship

ra(0) : rs(0yF. (9)

It follows that

(Lp)z : Ia(o)/v3. (10)

The value of I" is 7.9 x l0-26lcm2 (Gerold, 1967).Therefore c and No can be calculated using equa-

679

tions (5) and (6) by assuming that the particles arespherical (see also Hench et al., l97l).

Results

Both raw and corrected scattering data for oursamples appear as steeply dome-shaped curves inintensity (I) vs. scattering angle (e) plots becausethe data were taken in the tail of the direct beamregion and the particles causing scattering are verysmall in volume. If a relatively larger volume ofparticles were present, the high scattering intensi-ties should appear atlarger angles, out ofthe directbeam region, resulting in a rather smooth intensitycurve covering a larger area in I-e plot (that is,higher Q value in equation 5). This is the case, forinstance, in the work of Hench et al. (1971) andFaber and Rindone (1980), who both studied meta-stable phase separation in the glasses. The presenceof large volumes of particles in their samples wasconfirmed by TEM study (Faber and Rindone,1980) and by X-ray diffraction (Hench et al.,l97l).

Therefore, our corrected data must represent the

KUO AND KIRKPATRICK: PRE-NTJCLEATION BEHAVIOR OF GZASSES

l l 6 h r s

7 3 h r s

2 O h r s

I hrs

D i o p s i d e +O.5 wf % T i O.

o l l 6 h r s

o T o h r so 2 2 h r sa T h r s

D i o p s i d e +l . O w i % T i O ,

o 7 3 h r s

o 4 8 h r s^ 2 O h r s

o t h r s

Qe)2 ( r od2 x l o6 )Fig. l. Guinierplots of heat treated samples at l0'C undercooling. Note the increasing slope at small angles (Rr) and increasing

area under the curve (No) with increasing time for each compositron.

4Eo3

o r "o

6E0

effect of scattering by a small volume of particles inthe glass. Because no bubbles are present in thesamples, and because the scattering intensitiesshow systematic variation with time (Figs. I and2),these particles are most likely clusters of atomswith radii smaller than the critical radius formedduring heat treatment at subliquidus temperatures.Any supercritical nucleus (i.e., those larger than thecritical nucleus at a given undercooling), onceformed, will grow fast enough to crystallize thewhole charge, as evidenced by the high growth rateof diopside (Kirkpatrick et al., 1976) and by ourtotally crystallized runs.

Size of subcritical nuclei

Table 2 lists the average radius of gyration ofsubcritical nuclei derived from Guinier plots (Figs.I and 2). The values ofRs, ranging from about 200to 6254, increase with increasing time for eachcomposition at each undercooling. At a given timeand undercooling R* is larger in the glass withhigher TiO2 content. These data are shown in theplot of R! (proportional to volume) ys. time (Fig. 3).All these plots are characterized by a high andvariable slope at short times followed by a period of

KUO AND KIRKPATRICK: PRE-NUCLEATION BEHAVIOR OF GLASSES

(? i lz ( ro dzx l06 )Fig.2. Guinier plots of heat treated samples at20'C undercooling. The results are similar to those in Figure 1.

lower and constant rate of increase. The slope in thestraight line portion is lower at the lower undercool-ings and at lower TiO2 contents and the timerequired to reach the straight line region is longer atsmaller undercoolings and lower TiOz content. Fig-ure 4a is a plot of the steady state slope ys. TiOzcontent.

Number of subcritical nuclei

The parameters used and the results of the calcu-lation of the volume fraction of subcritical nuclei, c,and the number of subcritical nuclei per unit vol-ume, No, are listed in Table 3. Figure 5 shows plotsof the calculated number of subcritical nuclei versustime of each undercooling and TiOz content. Thisnumber increases slowly at shorter times and thenmore rapidly at longer times. There is some indica-tion that a steady state is approached, althoughbecause no nucleation has taken place, this isprobably not a true steady state. This behavior isthe same as that for the time dependence of ordi-nary nucleation density (e.g., James, 1974). There-fore, the "steady state" increase rates (Il and"induction times" (z') were evaluated from theseplots assuming that the two longest time points fall

o 168 h r so 70 h rgo 45 h r sa 27 h r s. t h r s

. 7 l h r g

o 4 3 h r 8

o zohrsa 6 h r r

. 5 2 h r so 27 l r tso l 6 h r sa t h r s^ 3 h r s

Table 2. Average radius of gyration of subcritical nucleicalculated from Guinier plots

staltlng Eterlel t lre (hr8) Rs (i)

1. A t - 10oc

Diopside

Diopslde + 0.5 wt% TlO2

Diopslde + L.0 wt% TtO2

2. Lt = 2OoC

Dlopslde

Dlops{de + 0.5 st7" T1O2

Dlopslde + 1.0 wt% Tt02

on a straight line. These are presented in Table 4.Figures 4b and 4c illustrate the compositional de-pendence on the variations ofli and r'. Ii increaseslinearly with increasing TiOz content, whereas r'

decreases with increasing TiOzundercoolings and may apProachvalue at higher TiO2 content.

681

content at botha constant, low

Discussion

Cluster rsdius

The observed increase in the radius of gyrationwith increasing run time in these experiments isconsistent with the traditional physical picture ofhow the small clusters of atoms which we callsubcritical nuclei form. In this picture the clustersare assumed to grow by the addition of single atomsor molecular groups to the surface of the cluster. Amelt above its liquidus temperature has few crystal-like clusters and these are small. When the melt isbrought to below its liquidus temperature the num-ber and average cluster size begins to increase.Crystals do not appear instantaneously, however'because the free energy of a small cluster is largerthan that of the same atoms in the liquid state. Inthe classical theory this extra energy is assumed tobe describable by a surface energy term, althoughrecent work has shown that this is probably inap-propriate for small clusters (Hoare, 1979). No mat-ter how it is formulated, this extra energy is abarrier to the formation of crystals, and there is acritical cluster size with a maximum energy whichmust be exceeded before a cluster can grow into a

KUO AND KIRKPATRICK: PRE-NUCLEATION BEHAVIOR OF GIASSES

1167320I

1167022

734820I

554.76496.35383.90230.31

625.75550 .15458.133LO.27

625.L4574.24495.82385 .29

625.t5t83.17424.7L378 .68266.90

575 .00491.30395.76287.50

609.12499.69428.00370.39268.16

1687045279

43206

522 71693

AT= 2Ot

24.43

80 root ime ( h r s )

Fig. 3. Change of the radius of gyration of subcritical nuclei as a function of time for all three compositions at 10" C and 20" C

undercoolings.

l o

,'' S lFto

xn o l

E,

6040?oroo806040

A T = l O o C

o D i

o D i + O . S w t o / o T i O z

A D i r l . Q p f o l ' f i Q t

r 6 8

682 KUO AND KIRKPATRICK: PRE-NUCLEATION BEHAVIOR OF GTAssEs

increase in Ru in these experiments is of the order of3 to 5 angstroms per hour (about 10-13 cm sec-t).This is far less than the growth rates for macroscop-ic crystals from diopside melt, which are of theorder of 10-3 to 10-a cm sec-r in this undercoolingrange (Kirkpatrick et al., 1976). Even though R*represents an average radius, this difference is verylarge, and can only be due to some free energybarrier preventing the average rate of successfulatomic attachment to a subcritical cluster frombeing as high as it is for a macroscopic crystal.

The data for R, can be used to estimate themagnitude of this free energy barrier. It is arguedabove that R, is weighted towards the largest clus-ter present. The largest R* measured before theonset of macroscopic crystal growth is about 6254.Taking the critical radius to be somewhat largerthan this, say 700A, the extra energy of a cluster interms of the surface energy can be calculated fromthe standard relationship:

R * : ( l l )AG"

where Rx is the critical radius, o is the surfaceenergy, and AG, is the free energy per unit volumedriving the reaction (Christian, 1975). AG" can beestimated from the standard relationship

o . 5 r owl o/o TiO2

Fig. 4. Variations of AR:/AI. Ii and r, as a function of TiO2content in a diopside melt. Open squares, LT = 20" C, filledsquares, Af = l0'C.

crystal. During the time period after the melt isbrought below the liquidus temperature subcriticalclusters are constantly growing and dissolving. Onthe average, however, the number and size of theclusters increases with time. Eventually, through aseries of fortuitious, random attachments, one clus-ter exceeds the critical size and begins to grow.With increasing time progressively more crystalsnucleate. Eventually a steady state distribution ofsubcritical clusters is established, and the nucle-ation rate takes on the steady-state value character-istic of the temperature.

In the experiments described here we havelooked at the earliest stages of this process, beforeany crystals have formed. The increase in R* andfor that matter the number of clusters, is in goodagreement with this picture.

Although the use of surface energy in formulatingthe barrier to nucleation is probably inappropriate(Hoare, 1979) there must be some energy barrier tocrystal formation at small cluster sizes. The rate of

2o

@t o

x

-c

l.tc<

t , oE,

G|oEo

\ o

q

3

F

(t2)

where A.Fft is the latent heat of fusion, A? is theundercooling, and ft_ is the liquidus temperature.Taking AIli as 34 kcal/mole (Weill et al., 1980), LTto be 10o, and Ty to be 1664" K, AG is 204 callmole.Taking the density of diopside melt to be 2.7 glcm3(Bottinga and Weill, 1970), and its gram formulaweight to be 216, AG" is then 2.5 callcm3. Usingthese values, the extra energy in terms of o is about400 erglcmz, a value in good agreement with mea-sured crystal-melt surface energies for silicates (nodata are available for diopside). This value is at bestgood to a factor of two and should probably beconsidered only an order of magnitude estimate. IfAT were taken to be 20' instead of 10', o would be800 erg/cm2, and it is not entirely clear that R, is anaccurate measure of the average largest cluster.

Number of clusters

Figure 5, which is a plot of the number of clustersper unit volume vs. time, clearly indicates the non-

AG, : -O%rfr,

KI]O AND KIRKPATRICK: PRE.NUCLEATION BEHAVIOR OF GLASSES

Table 3. Parameters used and results of calculations of No

6E3

startingnaterial

time(hrs )

. 2 * r o - 5 e x t o 4 ( A p ) 2 * r o - 3 6 v * r o 3 c x 10- " N , /cm- x I0 -

p

1 . A T = I O o C

e r v P - r u E

Diopside +O.5 wt . t T iO2

Diopside +I .O wt t T iO2

2 . AT = 20 ' c

Diopside

Diopside +0.5 wt . t T iO2

Diopside +I .0 wt t T io2

l r67 32 0

U

l r67 02 2

7

482 0

I

1687 04 52 7

9

7 I4 32 0

5 22 7I6

93

L 7 - A 65 . 9 4 30 . 9 0 80 . 0 7 9

4 I . 3 99 . 7 9 5r . 8 5 3o . t o 4

4t .651 8 . 1 7

I . 8 7 80 . 1 3 9

4 7 . 5 I5 . t 6 12 . 6 2 00 . 9 6 30 . 0 6 0

2 1 . 3 56 , 3 2 6t . 0 3 90 . 0 6 3

3 5 . 4 67 . 6 5 02 . 9 6 41 . 0 7 0o . o 7 2

6 .9393 . 1 9 9r . 056o .426

1 1 . 0 23 . 4 7 2t . 2640 . 2 3 0

LL .226 . 3 1 61 . 0 1 40 . 1 5 9

1 2 . 8 02 , 9 7 02 . 2 4 II .T67o . 2 0 9

7 . r233 . 5 1 21 . O950 . 1 7 5

1 0 . 3 24 . 0 3 62 . 4 4 41 . 386o . 2 4 4

3 . 5 2 03 - 7 7 23 . 1 0 I4 . O23

4 . I O 73 -2694 . 4 4 33 . 5 2 0

4 . 4 4 34 . 2 7 53 . 0 1 84 . t o 7

4 - O 2 33 . 2 6 73 . 1 0 13 . I 0 82 . 9 3 4

4 . L 9 L4 . I O 74 . 2 7 53 . 7 7 2

4 . t o 73 . 9 4 0

3 . 9 4 03 . 1 0 1

0 . 1 9 00 . 1 6 9o . 2 4 20 . 1 5 0

o . L 4 4o . 2 r 40 . 1 2 30 . 1 9 0

o .L230 . I 3 3o . 2 4 Lo . r 4 4

0 . 1 5 0o . 2 t 4o . 2 4 20 . 2 3 I0 . 2 5 0

0 . 1 3 8o . r 4 4o . 1 3 30 . 169

o .L440 . 1 5 60 . 1 2 8U . I ] O

o . 2 4 2

1 0 . 1 71 5 . 8 07 4 . 9 4

L290

5 . 7 8 37 . 6 3 4

3L. '16252.7

6 . 4 1 07 . 3 9 3

1 2 . 5 15 2 , 6 9

5 . 4 3 0r ' 1 . 7 7, o a ?

6 3 . t 34 8 9 . I

6 . 9 2 r1 3 . 9 648.97

5 8 4 . 2

6 . 5 0 01 3 . 9 83 7 . 0 18 0 . 5 3

4 5 7 . 5

3 9 . t 3 2 . 5 4 2I 2 . 2 L 1 . 1 0 8

0 - 7 2 0 0 , 1 4 2o . 0 2 L 0 . 0 r 9

L Z 1 . O f . O r O

2 9 . 6 3 2 . 4 0 12 . 8 0 5 0 . 3 2 3o . o 5 2 0 , 0 1 9

L 2 2 . 9 5 . 5 8 05 ' 7 . 6 L 3 . 3 7 5

4 . 2 - 7 A 0 . 3 9 00 . 1 9 6 0 . 0 3 8

1 4 9 . 8 6 . 8 0 71 3 . 6 5 r . 3 4 2

3 . 9 1 9 0 . 5 5 7r . 2 L 6 0 . 2 0 10 . 0 2 3 0 . 0 I 3

7 r . 4 a 4 . L 7 3L 6 . 5 2 t . ) Z O

1 . 5 0 5 0 . 2 6 70 . 0 3 1 0 . 0 1 5

1 0 3 . 0 5 . 0 5 61 8 . 0 2 1 . 6 0 1

4 . 6 2 6 0 . 6 5 5I . 0 7 5 0 . 2 3 5o . o 2 ' l 0 . 0 1 6

steady state pre-nucleation behavior of these com-positions. As noted above the greater than linearincrease in the number of clusters per unit volume isin good agreement with traditional treatments ofearly stages of transient nucleation.

The most successful theory describing non-steady state nucleation in silicate compositions isthat of Kashchiev (1969) (James, 1974; see Kirkpat-rick, 1981, for a brief review), and it appears thatthis theory also describes our data for subcriticalclusters. In Kashchiev's theory the total number ofcrystals nucleated (in our case the total number ofclusters larger than the lower detectability limit) isgiven by the relationship

( 13)

where Io is the steady state rate of formation, t istime, and r is a time constant determined by extrap-olating the straight line portion of the data back tozero crystals.

Figure 6 presents our data plotted with Kash-

chiev's predicted relationship, along which all datashould lie. The agreement is very good. BecauseKashchiev's analysis is based on the classical viewthat subcritical clusters increase in size and numberbefore nucleation can occur, these data also supportthe traditional physical picture of the pre-nucleationprocess.

The present data show that at both undercoolingsthe number of subcritical nuclei in TiO2-bearingglasses is greater than the number in TiOz-freeglasses at the same run time. This indicates that

Table 4. Values of "induction time" (z') and steady state rates of

subcritical nuclei increase (Il) for the three compositions at l0'Cand 20'C undercoolings/ ? \

I * R l

No( t ) : I " l t - +1.- \ o l

start ingmater ial

Ar = rooc

T I I ;

- ? - l( h r s ) ( c n

- h r - )

Ar = 20oc

T l' - o- ? - l

(h rs ) (cn -h r - )

DiopsideDi + 0 .5 scr T io?Dj- + 1.0 wtt TiO;

4 72 5

1r0 r37

30

I O

684 KUO AND KIRKPATRICK: PRE-NUCLEATION BEHAVIOR OF GZAssEs

l ime (h rs )Fig' 5. Calculated number of subcritical nuclei per unit volume as a function of time. Note the greater than linear rate of increase.

rOt ^ 4

;roE()

CI

z

cluster formation is significantly aided by the pres-ence of TiO2 in the melt. Other investigators havealso found that TiO2 aids nucleation. Nikandrov(1964) has investigated the crystallization of a glasswith diopside as the liquidus phase, and found thatthe onset of heterogeneous nucleation takes 8-10hours at 960'C, whereas for the same glass with0.01 to 5 wt.Vo TiO2 added, nucleation occursalmost instantaneously even at temperatures ashigh as 1260" C. Leger and Bray (1966) also foundthat in a glass with a composition close to theanorthite-diopside join the extent of crystallizationof these two phases increases with increasing TiO2content in the glass. There are several possibleexplanations for this effect among which it is diffi-cult to choose.

One possible explanation is based on the structur-al role of TiO2 (and other polyvalent cations such asFe3*, Al3*, and P5*) in these melts. Mysen andVirgo (1980) and Mysen et al. (1930) have arguedthat Ti is 4-coordinated in silicate melts and that theaddition of TiO2 to metasilicate melts (e.g., wollas-tonite, diopside) results in a decrease in the ratio ofnon-bridging oxygen to tetrahedrally coordinatedcations and thus an increase in chain and sheet unitsrelative to monomers in the melt (i.e., increasedpolymerization). This effect was also observed bv

Kushiro (1975), who found that polyvalent cationscause the equilibrium liquidus fields of more polym-erized minerals to expand relative to those of lesspolymerized minerals. If this is true, it is possiblethat the increased number of chain-like units allowsmore rapid formation of clusters with a chain struc-ture.

Another possible effect of TiO2 in these melts isto decrease the surface energy of the nuclei (Nikan-drov, 1964). Assuming constant chemical free ener-

E- t/r 'Fig. 6. Comparison of Kashchiev's theory of non-steady state

nucleation (solid curve) with experimental data. Open symbols:LT = 20" C, filled symbols: AT: 10" C. Circles: Di; squares: DiI 0.5 wt.Vo TiO2; triangles: Di + 1.0 wt.Vo TiO2.

i' o 3H

+-o.2z

AT = lO"C AT = ZO"C

o 0i + O.swf ehTi0z

a D i+ l .Owlo / "T iOz

KUO AND KIRKPATRICK: PRE-NTJCLEATION BEHAVIOR OF GI,ASSES 685

gies this will lead to a lower total free energy for acluster and therefore a more rapid increase in thesize and number of clusters in the subcritical region.

It is also possible that TiO2 itself serves as thecenter of nucleation (Gutzow et al., 1977). Thisappears unlikely in the present study because TiO2is not a stable phase in our system and has not beendetected.

Acknowledgments

We wish to thank Prof. Haydn Chen and Mr. J. W. Park(Department of Metallurgy, University of Illinois) for theirvaluable suggestions in sexs analysis. Thanks are also due toProfessors John Hower and Donald M. Henderson for theircritical reviews of an earlier version of the manuscript. The finalmanuscript was greatly improved by a helpful review from Dr.Gordon E. Brown, Jr. We are indebted to Mrs. Carol Sandersonfor her competent help with the manuscript. This work has beensupported by NSF grant EAR 7m3923.

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Manuscript received, .Iuly 31, l98l;acceptedfor publication, March 17, 1982.