J O U R N A L OF
Journal of Non-Crystalline Solids 151 (1992)81-87 North-Holland
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Non-isothermal surface nucleated transformation kinetics Michael C. Weinberg Department of Materials Science and Engineering, University of Arizona, Tucson, A Z 85721, USA Received 24 April 1992 Revised manuscript received 21 July 1992
A mathematical description of the transformation kinetics for surface initiated, non-isothermal, crystallization processes is presented. Previously derived equations for isothermal transformations are modified and used to compute the volume fraction crystallized as a function of time. The influence of cooling rate and surface nuclei density on the crystallization kinetics is studied. It is shown that, for slow cooling'rates, the fraction transformed is sensitive to the value of the surface nuclei density, unlike the situation for relatively large cooling rates where the fraction transformed is not very dependent upon the number of nuclei present on the surface. Finally, critical cooling rates, Qc, for the avoidance of a particular volume fraction are calculated and it is shown that computed Qc values are sample-size-dependent,by contrast with the case of bulk crystallization where Qc is independent of sample size and shape.
1. Introduction
Phase transformation kinetics proceeding by nucleation and growth mechanisms, in which surface nucleation plays an important role, were considered recently [1,2]. In particular, expressions were derived for the fraction of transformed material, X(t), as a function of time, for isothermal transformations. On the other hand, the ability to model phase change kinetics which occur under non-isothermal conditions is of paramount importance. For example, predictions of glass-forming ability are based upon computations of the crystallization rate during cooling of a liquid [3,4]. A key quantity, obtained from such calculations, is the critical cooling rate for glass formation [5,6]. Critical cooling rate expressions given in refs. [5,6] and elsewhere [7-9] are applicable for the case in which bulk nucleation occurs (usually Correspondence to: Dr M.C. Weinberg, Department of Materials Science and Engineering, University of Arizona, Tucson, AZ 85721, USA. Tel: + 1-602 621 6070. Telefax: + 1-602 621 8059.
homogeneous). However, in many marginal glass-forming systems, crystallization initiates on the surface, and devitrification proceeds by inward growth of crystals from the surface (either at the free surface or at a melt-container interface). Under such circumstances, the critical cooling rate is not uniquely defined since it depends upon the size of the sample. However, critical cooling rates for glass formation will be unambiguous if sample shape and size are specified. In this work the results given in refs. [1] are extended in order to describe the kinetics of surface initiated transformation kinetics. In particular, we investigate: (1) the influence of cooling rate upon transformation kinetics, (2) the change in kinetics with surface nucleation (seeding probability), and (3) the effect of particle size on critical cooling rate. Attention is restricted to a situation in which nucleation is completed prior to growth. Recent studies of surface crystallization [10,11] appear to indicate that the latter is a common occurrence, and hence the above assumption is physically realistic. In the following section, the model is presented, and a description is given of how we may
0022-3093/92/$05.00 © 1992 - Elsevier Science Publishers B.V. All rights reserved
82
M.C. Weinberg/ Non-isothermalsurfacenucleatedtransformationkinetics
compute the key quantities which describe non-isothermal transformation. In section 3, merical illustrations are presented to show influence of cooling rate and particle size on transformation kinetics. The conclusions stated, briefly, in the final section.
the nuthe the are
2. Transformation kinetics
2.1. Model For specificity, let us consider the transformation of glass to crystalline states during cooling at a constant rate. As mentioned above, it is assumed that the crystallization initiates solely on the surface, and that pre-existing nuclei (seeds) occur on the surface with some seeding probability. For sake of simplicity, the sample is assumed to be spherical. As a result of the above assumptions, the kinetics of the transformation depend upon the crystal growth rate, but the nucleation rate is irrelevant. For an isothermal transformation, with no change in glass composition, the crystal growth rate, g, is often interface-controlled. Hence, g may be taken as a constant and need not be specified. On the other hand, for a cooling experiment, the crystal growth rate is time-dependent as a consequence of its temperature dependence. Thus, a crystal growth model must be selected. Here, the normal growth model [12] is utilized. Several glasses with nearly Arrhenius temperature dependence of the viscosity seem to exhibit normal growth [13,14]. It should be noted that, although the precise form of the growth law which is selected will influence the details of the results, it will not affect the general trends of the findings. 2.2. Fraction transformed Since crystallization initiates only on the surface at any time, t, crystallization will be restricted to a spherical shell near the outer surface of the glass (see fig. 1). Thus, the volume fraction of the entire sample which has crystallized at t, X(t), is given by
X(t) =flXl(t).
(1)
Fig. 1. Plane section through a spherical sample, showing relevant lengths and regions.
In eq. (1), Xl(t) is the volume fraction crystallized in the region of volume A 1 and f l is the fractional volume. Thus, if V is the volume of the sample, then f l is fl=-Q- = 1 -
1-
The lengths d and R are shown in fig. 1. The fraction transformed in A 1 is given as [1,2,15]
fa dVIZl( t ) Xl(t )
(3)
A1
In eq. (3), Zl(t) is the probability that an arbitrary point, j, in A~ is transformed at time, t, and is given by Z l ( t ) = 1 - e x p - [PpA],
(4)
where Pp is the seeding probability/unit surface area and A is the surface area which contains nuclei which can contribute to the transformation of point j at time, t. Only those nuclei on the surface which can grow to j in time t are contained in A. By simple geometrical arguments, one can show that A is expressed by
A = 2arR2 1 - 1
-R +
r/R
J)
(5)
For an isothermal transformation, since the growth rate, g, is constant, d = gt. However, in
M.C. Weinberg/ Non-isothermalsurfacenucleated transformationkinetics
(6), which is written in reduced variables, then one finds
the case where the sample is cooled, then the growth rate is a function of time as a consequence of its temperature dependence. Hence,
So
u ( t ' ) dt =
Q(T'----~ dT'.
rl(T)
1 - exp
(
Io~1
,
(6)
where u(T r) = (Uo/~o)h(Tr). If Tr* represents the reduced temperature at which the growth rate is maximum, then eq. (9) can be written as
In eq. (5), Q(T') is the cooling rate, which in general can be a function of temperature, u(T') depends upon the assumed form of the temperature dependence of the crystal growth rate. If a normal growth law is utilized, then u(T) is
no I
Tr)dT,
d =
d=
83
d lal-lTmU(Tr *) -R = R I( Trl)'
(1Oa)
where T~ ]]"
(7) /(T'l)
Tr is the temperature divided by Tm (the melting temperature), ATr = 1 - T~, /3 is the entropy of fusion in units of the gas constant, and ~7 is the shear viscosity. The shear viscosity is taken to have Fulcher temperature dependence. Thus, rl ='00 exp (Tr -Tr0 ) "
_= f l
h(Tr)
]T,,h--~-~-,*) dT,.
(lOb)
For the problem of an isothermal transformation, the fraction transformed can be expressed in terms of a single variable, y = gt/R, which may be considered a reduced time. In the case of a non-isothermal transformation, X is a function of the single variable d / R . However, in order to be able to compare isothermal and non-isothermal transformation rates, a second parameter must be specified. If it is assumed that the isothermal
(8)
If the cooling rate is assumed to be constant, Q = - l a l , and eqs. (7) and (8) are used in eq.
Surface Crystallization of Sphere 1.0
~
~:|%%77,
0.8 i d i¸..l k
/
0.6 t-
tO
0.4 /
0.2
X .. / !["
.... i . . . . -.
, "..-j.. 0
o
~f:> "
~
0:2
o'.4
II. ¢
-
theta=8 theta=16
isothermal
0:6
y
Fig. 2. Volume fraction crystallizedas a function of scaled time (201 seeds on surface).
0.8
M.C. Weinberg / Non-isothermal surface nucleated transformation kinetics
84
transformation to which the non-isothermal transformation is compared occurs at Tr*, and a reduced inverse cooling rate, 0 = T m g / l a l R , is introduced, then one finds
d / R = OI( Trl ) 01(1 - y / O ) . (11) Equations (1)-(5), (10b) and (11) are used in the subsequent section to compute the crystallization kinetics and critical cooling rates for glass formation for surface initiated processes. =
3. Illustrations
In order to completely specify the crystal growth rate, the parameters /3, Tro, and b must be selected. Since G e O 2 and S i O 2 exhibit normal crystal growth, the latter parameters are chosen to be representative of these materials. The reduced entropies of fusion are 1.31 and 0.603 for G e O 2 and S i O 2 , respectively. Hence, for all calculation, /3 -- 1.0. The viscosity parameter, Tr0, is set to zero since the temperature dependence of the viscosity for both silica and germania show Arrhenius behavior. Finally, b is fixed at 30 since
b = 29.8 and 31.4 for GeO 2 and SiOz, respectively. In fig. 2, isothermal and non-isothermal crystallization kinetics are compared for several different cooling rates in the case of a relatively large seeding probability. It should be recalled that, since 0 ~ 1/I a l, large 0 means a slow cooling rate. This figure shows the fraction of the sample crystallized (for isothermal and non-isothermal paths) as a function of y, which is the reduced timescale of the isothermal transformation at Tr*. As a consequence of the choice g = u(Tr*), the isothermal curve provides an upper limit to the fraction transformed at any time. One observes that the transformation curves at different cooling rates intersect. There is a simple explanation for this finding. At long times, the more rapid the cooling rate, the smaller the volume fraction transformed. However, at short times, the situation is reversed since here, for any fixed time, the faster cooling rate path has probed more of the region of rapid crystallization. Thus, at shorter times the fraction transformed will be greater for the faster cooling path. If the two
Surface Crystallization of Sphere 1.0 # of seeds = 16 )( theta=4 .J. theta=8 -" theta=l 6
0.8
¢ "10 0
.,.~
~4
/~ -
/ . ~ -
i
0.6
0 ¢z
20
0.4
15 0.2
0.2
0.4
0.6
0.8
Y Fig. 3. Volume fraction crystallized as a function of scaled time (16 seeds on surface).
M.C. Weinberg / Non-isothermal surface nucleated transformation kinetics
85
The effect of the nucleation number upon the fraction transformed is shown in greater detail in fig. 4. In this figure, the fraction transformed after cooling over the entire path (i.e., from the liquidus to room temperature) is plotted against the reduced cooling rate (0 - t ) for two different seeding probabilities. One observes that, at large cooling rates, the fraction transformed is quite insensitive to the seeding probability. This behavior may be explained as follows. For the larger cooling rates, d/R values are small (d/R << 1). Therefore, from eq. (2), fx ~ 3(d/R)3 and f l << 1. Therefore, the 'propagation front' is shallow, regardless of the seeding probability, and thus X is small. On the other hand, if the cooling rates are slower, then the d/R values are not very small. In this case the pre-exponential term, f~, is not dominant. Thus, one observes from eqs. (1), (3)
cooling paths being compared are relatively slow (e.g., 0 = 8 and 0 = 16), then the 'cross-over' can occur at long times where the transformation is nearly complete. Also, one observes that, with the exception of the most rapid cooling rate (0 = 4), the non-isothermal paths result in fractions transformed comparable to the one produced isothermally at the maximum growth rate. This result is a consequence of the fact that a relatively large seeding probability (201 seeds) was used to produce this figure. If the seeding probability is smaller, then for identical cooling rates the amounts transformed will be suppressed more dramatically, and the 'cross-over' points will occur at shorter times. These features are illustrated by comparison of figs. 2 and 3, where in the latter figure the seeding probability used is about 8% of that used for construction of fig. 2.
Fraction Crystallized vs. Cooling Rate
0.6
0.4
0.2
"+.
015
--'+ . . . . . . . . . . . .
-t- . . . . . . . . . . . . . . . . . . .
1.0
~. . . . . . . . . . .
1.5
: ~
2.0
l/theta Fig. 4. V o l u m e fraction crystallized at c o m p l e t i o n of crystallization p a t h vs. 0 - 1 . _ • - - , s e e d s o n surface.
201 s e e d s o n surface; - - -
+---,
16
M.C. Weinberg / Non-isothermal surface nucleated transformation kinetics
86
and (4) that Pp plays an important role in determining the fraction of material crystallized. For bulk-nucleated crystallization processes, one can define a critical cooling rate for glass formation in a unique fashion. This rate is usually taken as that required to ensure that X < 10 -6. Further, if the sole source of time dependence of the nucleation rate, J, and growth rate, u, is via their temperature dependences, and the cooling rate is constant and uniform, then the critical cooling rate, Qc, (assuming spherical crystallites) is
]3 =
X 10 6.
Tm (12) For surface-nucleated transformation, however,
eq. (12) is invalid, and Qc is a function of sample size. In order to compute the cooling rate required to produce any particular volume fraction, the sample size and shape must be specified. This feature is illustrated in fig. 5, where the cooling rate l a I, divided by Tmg required to produce X = 0.05 for a spherical sample is plotted against R. One observes that the required cooling rate declines with increasing particle size in approximately a logarithmic fashion. For a fixed seeding probability, an increase in the sample radius has two effects which are somewhat counterbalancing. First, since the propagation of the crystallization front will penetrate to a smaller fractional depth in a larger sample, this feature tends to reduce the required (critical) cooling rate. On the other hand, since the surface area is larger, the total number of seeds on the surface will be
REQUIRED COOLING RATE AS A FUNCTION OF PARTICLE SIZE FOR x=0.05 0.30
0.25
0.20
0.15
0.10
0.05 t lOo
I
R
101
Fig. 5. Required (critical) cooling rate, z = (OR)-1, to transform X = 0.05 vs. sample radius.
M.C. Weinberg / Non-isothermal surface nucleated transformation kinetics enhanced. This increase in nuclei number tends to increase the required cooling rate. These offsetting effects tends to mitigate somewhat the dependence of critical cooling rate upon particle size.
4. Summary and conclusions Equations which were given previously for the description of isothermal surface initiated transformations were extended to treat the problem of non-isothermal crystallization kinetics. A normal crystal growth model was used, and the physical parameters were chosen to be representative of SiO 2 or G e O 2. Calculations were performed to elucidate the effects of surface nuclei density and cooling rates on the crystallization kinetics. Also, 'critical' cooling rate calculations were performed. It was found that, for large seeding probabilities, rapid cooling is required to substantially reduce the fraction crystallized at the end of the cooling path. For lowgr seeding probabilities, however, much more significant reductions can be produced using similar cooling rates. On the other hand, for sufficiently large cooling rates the fraction transformed was observed to be small, independent of the seeding probability. Finally, the required cooling rates for avoidance of a specific volume fraction crystallized was considered. It was shown that, unlike the situation for bulk nucleation, the critical cooling rate
87
for surface initiated transformations are size-dependent. The dependence of critical cooling rate upon sample radius was illustrated. The author wishes to express his gratitude to the Jet Propulsion Laboratory and the Division of Microgravity Science and Applications of N A S A for the financial support of this work.
References [1] M.C. Weinberg, J. Non-Cryst. Solids 134 (1991) 116. [2] M.C. Weinberg, J. Non-Cryst. Solids 142 (1992) 126. [3] D. Turnbull, Contemp. Phys. 10 (1976) 159. [4] D.R. Uhlmann, J. Non-Cryst. Solids 7 (1972) 337. [5] M.C. Weinberg, B.J. Zelinski, D.R. Uhlmann and E.D. Zanotto, J. Non-Cryst. Solids 123 (1990) 90. [6] D.R. Uhlmann, J. Non-Cryst. Solids 38&39 (1980) 693. [7] D.R. Uhlmann, J. Am. Ceram. Soc. 66 (1983) 95. [8] E.D. Zanotto and A. Galhardi, J. Non-Cryst. Solids 104 (1988) 73. [9] K.F. Kelton and A.L. Greer, J. Non-Cryst. Solids 79 (1986) 295. [10] E.D, Zanotto, J. Non-Cryst. Solids 129 (1991) 183. [11] E.D. Zanotto, in: Proc. 4th Int. Otto Schott Colloquium, 1990, J. Non-Cryst. Solids 129 (1991) 183. [12] D.R. Uhlmann, in: Advances in Ceramics, Vol. 4, Nucleation and Crystallization in Glasses, ed. J.H. Simmons, D.R. Uhlmann and G.H. Beall (American Ceramic Society, Columbus, OH, 1982) p. 80. [13] E.D. Zanotto and M.C. Weiltb~rg, Phys..Ch~:m. G!asscs 30 Q989) 186. [14] P.J. Vergano and D.R. Uhlmann, phys. Chem. ,Glasses ll (1970) 30. [15] M.C. Weinberg and R. Kapral, J. Chem. Phys. 91 (1989) 7146.