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Development of anisotropic particle morphology in an isotropically transforming matrix
Physica A 285 (2000) 279–294
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Development of anisotropic particle morphology in an isotropically transforming matrix Dunbar P. Birnie III, Michael C. Weinberg ∗ Department of Materials Science and Engineering, Arizona Materials Laboratory, University of Arizona, Tucson, AZ 85721-0012, USA Received 14 December 1999
Abstract In the present work we consider the growth of a very dilute concentration of anisotropic particles, nucleated with random orientations, in a matrix of growing spherical particles. It is assumed that the matrix particles obey the usual JMAK kinetics. We concentrate on describing the change in anisotropic particle grain morphology as the transformation process proceeds. An eccentric shaped particle will have certain growth directions that are rapid and others that are slower. Faster growing directions impinge upon the matrix particles much sooner than slower growing directions, and this feature leads to an e ective change in particle morphology as the transformation process progresses. We analyze such changes in particle morphology by deriving expressions for the probabilities that growth rays will travel a certain distance before encountering matrix particles and that the length ratio of fast-axis to slow-axis growth of the anisotropic particle will attain certain values. The changes in particle morphology are examined as a function of the relative speeds of growth of anisotropic particle growth rays to matrix particle growth rates. c 2000 Elsevier Science B.V. All rights reserved.
Keywords: Phase transformations; Crystallization; Anisotropic particles; Nucleation and growth
1. Introduction Transformation kinetics involving the nucleation and growth of anisotropic particles is a topic of practical importance due to its relevance to a variety of applications [1–14]. For example, materials consisting of plate-like particles or composite materials containing highly anisotropic embedded particles can exhibit superior mechanical toughness ∗
Corresponding author. Fax: +1-520-6218059. E-mail address: [email protected] (M.C. Weinberg).
c 2000 Elsevier Science B.V. All rights reserved. 0378-4371/00/$ - see front matter PII: S 0 3 7 8 - 4 3 7 1 ( 0 0 ) 0 0 2 6 4 - 8
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since cracks need to follow a tortuous path around or through the plate-like particles in order to propagate [15,16]. Also, interesting optical e ects such as birefringence and light guiding can be augmented by producing elongated particles within a transparent matrix [17,18]. In some situations it is desirable for the anisotropic particles to be randomly oriented within the matrix, while in other cases preferential or cooperative alignment of the particles is more bene cial for the envisioned application. However, in either situation the tailoring and control of the morphology of the transformed material is of great importance. Calculation of the kinetics of transformations involving anisotropic particles is a much more challenging problem than that for isotropic particles. The theory of the transformation kinetics for isotropic particles was developed by Johnson and Mehl [19], Avrami [20 –22], and Kolmogorov [23] many years ago. This formalism, referred to as JMAK theory, is a basic pillar of Materials Science that has been employed extensively for the computation of the rates of transformations that proceed by nucleation and growth mechanisms. In order to explain the complicating features associated with the transformation kinetics producing anisotropic particles, it is necessary to remember certain features in the derivation of the JMAK equation. The derivation rests on calculating the probability that a randomly chosen point in space (“O”) has not been transformed after a given time period, t. In order to satisfy this criterion a certain volume around O must be free of nuclei since any nucleus in the immediate neighborhood of the point could grow and transform the point O. Any nucleus which is close enough to O so that it has the potential to transform O in the time t is called an “aggressor” [24]. For transformations involving isotropic particles every aggressor is potent, meaning that all aggressors will transform the point O. Hence, in this case if the nucleation and growth rates are known, then the transformation kinetics can be computed in a straightforward manner [25]. However, for cases where anisotropic particles form, complications arise since not all aggressors are potent. Particle–particle interactions (i.e., particles growing to points of contact) can occur such that non-aggressors can block the path of aggressors rendering the aggressors impotent. Such non-aggressors, termed “blockers”, occur only for transformations involving anisotropic particles. Another e ect that can occur solely in the case of anisotropic particle formation is that of “phantom” particles. A phantom particle is one that nucleates in a region shadowed by the growth of another previously-nucleated anisotropic particle. Phantoms do not occur if all nucleation is completed prior to growth, but such particles arise if the number of particles nucleated is a function of time. Despite the diculties in describing the transformation kinetics of anisotropic particles, approximate and precise results have been obtained for several speci c cases [26 –30]. These studies have produced a qualitative understanding of the kinetics. It has been observed that growth velocity anisotropy generally leads to a retardation of the kinetics, but this slowing down is noticeable only if the probability for particle– particle interaction is suciently large (usually requires that about 30% of the volume has been transformed). This kinetic retardation is due to the e ects described above and leads to smaller Avrami exponents that are also time dependent [31].
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Although several works have been devoted to the discussion of the kinetics of transformations with anisotropic particle formation, we know of no study that examines the grain morphology that results for transformations involving the nucleation and growth of anisotropic particles. This is a dicult problem due to the occurrence of blockers and phantoms. Here we address the grain morphology problem for a model system for which the usual complicating elements are absent. Speci cally, we model the process by examining how an anisotropic particle grows into a matrix of isotropic particles that transform in such a manner that they obey the JMAK Theory. This approach has the distinct advantage that the statistical interactions between the primary particle and neighboring particles are included, but blocking by “third party” particles is eliminated. The basis for this simpli cation relies upon the fact that we only consider the limit of very dilute concentration of anisotropic particles in a matrix of isotropic particles. Under such circumstances blocking need not be considered since the matrix, which obeys JMAK kinetics, does not experience blocking e ects and the probability of particle interaction between two anisotropic particles is vanishingly small. In many systems, simultaneous crystallization of two or more phases can occur. A particularly interesting and yet mathematically tractable case is that for which one phase is highly anisotropic and the other is essentially isotropic. Thus, in the present work we consider the case where the anisotropic particles nucleate with random orientations and then grow with their intrinsically eccentric shapes until they impinge upon one another creating an interconnected brous or plate-like microstructure. Our main focus will be placed on describing several aspects of the grain morphology for this process. One observes that an eccentric-shaped particle will have certain growth directions that are rapid and others that are slower. The faster growing directions are anticipated to impinge upon the matrix particles much sooner than the slower growing directions. This feature is expected to lead to an e ective change in particle morphology as the transformation process progresses such that the nal particle morphology is less eccentric than the initial eccentricity due to the growth velocity anisotropy. One of the main goals of this work is the quantitative description of this process. We examine this phenomenon for di erent extents of initial particle eccentricity and for di erent ratios of anisotropic-particle to matrix-particle growth speeds. Our ultimate objective is the elucidation of the complete crystalline morphology produced by transformations involving two phases that grow with di erent geometric characteristics. The present study is a rst step towards that goal.
2. Model As mentioned above, we consider the growth of a very dilute solution of anisotropic particles embedded in a matrix of growing isotropic particles. In particular, we select the anisotropic particle to be ellipsoidal in shape, so that there is an axis of symmetry and only two characteristic growth speeds. Whether the ellipsoid is oblate or prolate is immaterial, but we take the growth velocities in the two primary directions as
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representations of the nal 3D shape. The matrix particles are taken to be spherical, as required by JMAK Theory. For the sake of simplicity we will concentrate on the particle growth behavior along these two characteristic directions, which are chosen to be the fast growth axis and the slow growth axis. The corresponding growth speeds are denoted by va and vb , respectively. One should note that for ellipsoidal particles the fast and slow growth directions are spatially orthogonal. This feature allows us to select conditions such that possible interactions or coupling of the probabilities involved in growth of either of these perpendicular growth rays is prevented. Limitations placed on applicable va and vb values are discussed in the appendix. We assume that nucleation of both types of particles occurs prior to particle growth and is completed before the particles begin to grow. This assumption is an excellent representation of events in a two-stage heating procedure used in experiments to measure nucleation rates in amorphous systems [32]. Also, we consider a 3-D system that is of in nite extent. The concentration of ellipsoidal particles is deemed suciently dilute so that anisotropic-particle=anisotropic-particle interactions can be neglected. Particle growth is presumed to be interface controlled, so that all growth rates are constant.
3. Approach The initial shape of ellipsoidal particles will be distorted at the end of the transformation due to their encounters with matrix particles. We utilize the e ective de ciency in particle lengths (along fast and slow axes) and alteration in particle eccentricity produced by particle interactions in order to gauge the nal morphology. Since the fast and slow growth rays associated with a particle are not coupled on average the faster rays will be blocked from advancing by matrix particles at a shorter time than those rays growing in the slow direction. This fact leads to the reduction in eccentricity of the nal particle. The essence of our approach will be a calculation of the probability that any particular growth ray grows unimpeded over a given distance or time period. In other words, we calculate the probability that either the fast or slow growth ray associated with the ellipsoidal particle will encounter a growing spherical matrix particle. We decompose ray growth into a sequence of incremental time steps. It is clear that for a ray to have grown for some time period, t, it must have successfully grown for all time increments, t, comprising t. Hence, to obtain the probability that such a ray will extend its growth for at least one more time increment, we must nd the conditional probability that during this next time interval no collisions will occur given the knowledge that the ray grew unimpeded up to that time t. The ability for a ray to grow freely depends upon the absence of matrix particles in the vicinity of the ray that could grow and block its growth. This implies that a certain volume of space around the ellipsoidal particle is free from nuclei, and an important part of our procedure will entail the calculation of such nucleus-free volumes.
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Fig. 1. Illustration of incremental exclusion zone of a growing ray comparing times t and t + t when v ¿ g. Shaded area represents new volume that must also be free from matrix nucleation events.
4. Governing equations Let us consider an arbitrary point in space (x0 ; y0 ; z 0 ) which is not transformed at some time t. We assume that this point is at the leading edge of a growth ray that is moving in the positive direction along the x-axis at a speed v. Let the growth rate of a matrix particle be denoted by g. Since the point (x0 ; y0 ; z 0 ) was untransformed by matrix particles, then a sphere of radius gt must exist around this point which is free from matrix nuclei. Similarly, if the ray grows to the next point, (x0 + vt; y0 ; z 0 ), in the time t, then a sphere of radius g(t + t) must be centered about this second point, too (see Fig. 1). These two spherical regions are “exclusion volumes” where no matrix nuclei are allowed. In constructing Fig. 1 it has been assumed that v ¿ g. Since the time increment is small, the volume of the spheres will overlap. Thus, in order to nd the conditional probability that the ray will advance to x0 + vt given that it already made it to x0 , it will be necessary to compute the volume of the second sphere that falls outside the volume of the rst sphere (represented by the shaded area in Fig. 1). Spheres 1 and 2 satisfy the following equations, respectively: (x − x0 )2 + (y − y0 )2 + (z − z 0 )2 = g2 t 2 ;
(1)
(x − x0 − vt)2 + (y − y0 )2 + (z − z 0 )2 = g2 (t + t)2 :
(2)
The points of intersection follow from the solution of Eqs. (1) and (2) and is given by 2g2 t + t(g2 − v2 ) : (3) 2v One observes that if v = g (i.e., when the ray’s speed is identical to growth rate of matrix particle), then the spheres touch at merely one point. Hence, if v ¡ g, then x = x0 −
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Fig. 2. Illustration of incremental exclusion zone of a growing ray comparing times t and t + t when v ¡ g. Shaded area represents new volume that must also be free from matrix nucleation events. In this case the larger sphere completely encloses the smaller one, resulting in growth times identical to the JMAK matrix growth process.
sphere 1 is totally engulfed by sphere 2, and the portion of volume of sphere 2 which does not intersect sphere 1 is just given by the di erence between the volumes of spheres 2 and 1 (see Fig. 2). However, if v ¿ g, a line of intersection points occurs, and calculation of the conditional probability, re ected by the non-overlapping portion of the volume of sphere 2, entails a bit more e ort. In order to ascertain the probability that a ray will grow for some time t 0 in an unimpeded fashion, one must then construct the sequence of conditional probabilities for all small incremental times which span the time from t = 0 to t = t 0 . Let us consider the cases v6g and v ¿ g in turn. In the former case each exclusion sphere (i.e., sphere where no matrix nuclei are allowed) totally engulfs all exclusion spheres corresponding to earlier times in the ray’s history. This result implies that the probability that this slow ray will grow for some time period t 0 is governed solely by the matrix particle growth speed and that the expected average growth time will be the same as that for any isotropic matrix particle. Since the matrix particle growth is governed by the usual JMAK Equation, this case is of somewhat less interest than case 2. However, when v ¿ g, the exclusion spheres only partially overlap and the boundary of the intersection area is given by Eq. (3). For v ¿ g, the union of sequentially overlapping exclusion spheres, in the limit of very small t, will produce a shape that can be described by an ice cream cone with a scoop of ice cream on top. A 2-D projection of this object is shown in Fig. 3. If the position of the tip of the cone is taken as (0; y0 ; z 0 ) and the center of the nal sphere (ice cream ball) is taken at (x0 ; y0 ; z 0 ) and using the fact that x0 = vt, then it is easy to show that the distance d, de ned in Fig. 3, is given by r g2 (4) d = gt 1 − 2 : v
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Fig. 3. Illustration of the cumulative ice-cream-cone-shaped exclusion zone of a growing ray for time t when v ¿ g. The cone volume is made up of the superposition of all successively larger spheres starting at (0; y0 ; z 0 ) and extending to time t, when the ray has grown to (x0 ; y0 ; z 0 ). Note that x0 = vt.
Hence, the volume associated with the cone portion is g2 t vt − d2 ; VCone = 3 v and the volume associated with the truncated sphere is g3 4g3 t 3 1 3g − + : VSpherepart = 3 2 4v 4v3
(5)
(6)
If Eqs. (4) – (6) are combined and Vsphere denotes the volume of a sphere of radius gt, then the total volume of cone plus partial sphere is as follows: v g 1 + + : (7) VTotal = VSphere 2 4g 4v If one follows the standard arguments used for the derivation of the JMAK Equation [33], then it is easy to show that the total probability that a ray grows for time t; Z(v; t), is v g 4g3 t 3 1 + + ; (8) Z(v; t) = exp(−VTotal ) = exp − 3 2 4g 4v where is the nucleation density (i.e., the average number of matrix particles that nucleated per volume). If we let w = v=g, then the probability that a ray will grow to at least a length x; Z(v; x), is as follows: Z(v; x) = exp(− x3 ) ; where ≡
1 4 1 w ; + + 3w3 2 4 4w
≡
4 ; 3w3
for 1¿w :
(9a)
for w ¿ 1 ;
(9b) (9c)
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Eqs. (9a) – (9c) are the basic governing equations which will be used to examine the morphological development of the ellipsoidal particle.
5. Progress of individual growth rays Here we analyze the growth characteristics of a single ray traveling from its original nucleation site. In the following section, we will apply the ndings to a comparison of growth occurring for rays traveling along two orthogonal directions at di erent velocities. The latter quantities will give some indication of particle shape distortion. If P(x) denotes the probability density that a ray reaches some length between x and x + d x, then P(x) = −
@Z(v; x) = 3 x2 exp(− x3 ) : @x
(10)
Therefore, the total probability, Ptot , that a ray travelling along the positive x-axis will grow to a size in excess of x∗ is simply Z x∗ P(x) d x = 1 − exp(− (x∗ )3 ) ; (11) Ptot = 0
and the average growth length, hxi, is Z ∞ 4 −1=3 ; xP(x) d x = hxi = 3 0
(12)
where is the gamma function. One observes from an inspection of Eqs. (9b), (11), and (12) that both the probability distribution of ray lengths and the average length of a ray depends not only upon the ratio of the ray speed to matrix growth speed but also upon the number density of nucleated matrix particles. In order to remove the latter dependence upon nucleation density and to focus solely upon the in uence of the speed ratio we inspect the ratio of the average length to the average length if the ray were to obey JMAK statistics (i.e., w = v=g = 1), denoted by hxi=hxiJMAK . From Eqs. (9a) and (9b) and (12) one nds −1=3 1 w 1 hxi = : (13) + + hxiJMAK 2 4 4w Also, precisely the same result would ensue if we examined the ratio Ptot (x∗ )=PtotJMAK . Table 1 illustrates the dependence of these ratios on the relative growth speeds of ray and matrix. As anticipated the average length traveled by a ray will decrease monotonically with w. Since faster growing rays explore more new volume than slower growing rays in the same time period, the former are more likely to come into the vicinity of a matrix particle that will stop the ray’s growth. This process of shortening the ray’s growth time will tend to “round out” the particle’s ellipsoidal shape. From Table 1, one notes that w must be about 5, or greater, to have a signi cant e ect.
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Table 1 Average ray length normalized relative to the average ray length found for JMAK kinetics w = v=g
hxi=hxiJMAK
2 5 10 25 50 100
0.96 0.82 0.69 0.53 0.43 0.34
6. Anisotropic particle development Now let us consider the progress of two growth rays from the same anisotropic particle that are traveling in the positive directions along two orthogonal axes (i.e., take the fast and slow growth directions). The quantity of interest will be the expected ratio of lengths of these two rays since this will provide a measure of the extent of distortion of the ellipsoidal particle from its ideal shape if there were no collisions with matrix particles. Let us assume that the faster ray grows along the x1 -axis with a speed va and the slower growth grows along the x2 -axis with a speed vb . For any time t, the maximum extensions of the rays along the x1 and x2 axes will be denoted by xa; max ; xb; max , respectively. Further, denote by r the actual ratio of the lengths of the rays along the two axes. The total probability, Ptot , that r does not exceed some speci ed value, called r ∗ , is given by the following equation: Ptot ≡ P+ + P− ; where
(14)
x1 x1 ∗ d x1 d x2 Pa (x1 )Pb (x2 ) 1 − −r −1 ; P+ = x2 x2 0 0 Z xb; max Z xa; max x2 x2 d x1 d x2 Pa (x1 )Pb (x2 ) 1 − − r∗ −1 : P− = x1 x1 0 0 Z
xa; max
Z
xb; max
In the above equation Pn (xi ), with n = a; b and i = 1; 2, is given by Eq. (10) and the subscript n speci es the value of and is the Heaviside function. Eq. (14) accounts for the fact that even though va ¿ vb , the length of the ray along the x2 -axis could be longer than the one along the x1 -axis due to ray blockage by a matrix particle. Thus, P+ and P− are the contributions to Ptot when the length of the ray in the x1 direction is larger than the one in the x2 direction, and vice versa, respectively. Eq. (14) also contains the implicit assumption that the probabilities Pa (x1 ) and Pb (x2 ) are independent (i.e., due to the random statistical distribution of matrix particles in two separate directions). Although this assumption would normally be valid, it is not always so. Therefore, the condition needed to justify this assumption is discussed as
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an appendix. If we introduce two new coordinates, x10 = x1 and r = x1 =x2 , into the P+ term, then it can be written as follows: Z rmax Z dr xa; max 0 0 d x1 x1 Pa (x10 )Pb (x10 ; r)[1 − (r − r ∗ )](r − 1) ; (15a) P+ = 2 r 0 0 Z P+ =
r∗
1
Z
dr r2
xa; max
d x10 x10 Pa (x10 )Pb (x10 ; r) :
0
(15b)
Also, one may transform P− in a similar fashion to obtain Z r∗ Z dr xb; max 0 0 d x2 x2 Pa (x20 ; r)Pb (x20 ) : P− = r2 0 1
(16)
Now, the probability density of nding a length ratio between r ∗ and r ∗ + dr ∗ is denoted by P d and is given by the partial derivative of Ptot with respect to r ∗ . Hence, Z ∞ 1 d d d d x10 x10 Pa (x10 )Pb (x10 ; r) P = P+ + P− = ∗ 2 (r ) 0 Z ∞ 1 d x20 x20 Pa (x20 ; r)Pb (x20 ) : (17) + ∗ 2 (r ) 0 If one employs Eq. (10) in Eq. (17) and performs the integrations, then one nds that d = P+
where s=
3s(r ∗ )2 [s + (r ∗ )3 ]2
d P− =
3s(r ∗ )2 ; [1 + s(r ∗ )3 ]2
(18a)
b a
(18b)
and therefore, P d = 3s(r ∗ )2
1 1 + [s + (r ∗ )3 ]2 [1 + s(r ∗ )3 ]2
:
(18c)
One observes from Eq. (18c) that the probability density depends upon a single parameter, s, de ned above. If one utilizes Eqs. (9b) and (9c) in the de nition of s, then one obtains the following expressions for s, depending on the magnitudes of wa and wb : [ 12 + w4b + 4w1 b ] wa 3 ; wa and wb both ¿ 1 ; (19a) s = 1 wa [ 2 + 4 + 4w1 ] wb a
1 1 wa + + s= 2 4 4wa s=
wa wb
3 =
va vb
−1
wa wb
3 ;
wa ¿ 1; wb ¡ 1 ;
(19b)
3 ;
wa ; wb ¡ 1 :
(19c)
Thus, in general, the probability density depends individually upon the speeds of the rays (relative to the matrix) in the two key orthogonal directions. However, in some
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Fig. 4. The probability density of nding a length ratio between r ∗ and r ∗ + dr ∗ as a function of r ∗ for three values of the parameter s.
special cases the probability density depends only upon the ratio of the speeds of the orthogonal rays. Clearly, this is the situation when wa ; wb ¡ 1 and also if wa ; wb ¿ 1 and wb 1. In the latter case it is easy to show that s → (wa =wb )2 = (va =vb )2 . If one takes the derivative of P d with respect to r ∗ , using Eq. (18c), and sets it equal to zero, then one observes that for certain values of s there will be extrema in the probability density function. This behavior is shown in Fig. 4, where P d has been plotted as a function of r ∗ for three values of s. From Eq. (18c) one may easily see that the r ∗ = 1 intercept is 6s=(1 + s)2 . The plots corresponding to the larger values of s exhibit extrema, but the s = 4 curve shows monotonic behavior. If one analyzes the equation for the extrema, then it is easy to show that a necessary condition for the presence of extrema is s ¿ 2. However, it is more dicult to nd the sucient condition, although one can show that for s 1 extrema are still expected. A quantity of particular interest is the average value of r ∗ ; hr ∗ i, since it is a good measure of the change in particle distortion brought about by collisions with matrix particles. With the aid of Eq. (18c) we obtain the following expression for hr ∗ i: ∗
hr i =
Z 1
∞
∗ d
∗
∗
r P (r ) dr = 3s
Z 1
∞
∗
∗ 3
dr (r )
1 1 + 2 ∗ 3 [s + (r ) ] [1 + s(r ∗ )3 ]2
: (20)
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Fig. 5. Normalized average value of ratio of orthogonal ray lengths vs. s.
One can perform the integrals analytically and if one de nes y = s1=3 , then one nds y 1 − C(y) √ ; (21a) hr ∗ i = 1 + B(y) y + y 3 where
(1 + y)2 1 ; B(y) = √ − log 1 − y + y2 2 3 6 1 2−y y(2 − y) √ + 2 tan−1 : C(y) = tan−1 √ y 3y 3
(21b) (21c)
In Fig. 5, Eqs. (21a) – (21c) have been utilized to construct a plot of hr ∗ i, normalized by hr ∗ is=1 which is the JMAK value as a function of s. One notes that the normalized average value of orthogonal ray lengths increases monotonically with s, as anticipated. Further, one observes that the collisions with matrix particles produce a dramatic blunting, or reduction in particle anisotropy. This results from the fact that the more rapidly moving rays will encounter matrix particles much sooner than slowly travelling rays, and thus they travel only a short portion of the distance in the collision-free case. A plot that is somewhat easier to interpret physically may be constructed in some limiting cases. For example, previously it was indicated that when wa ; wb ¿ 1 and wb 1; s → (wa =wb )2 = (va =vb )2 . Hence, a plot of hr ∗ i=hr ∗ is=1 vs. s1=2 will show how the normalized length ratio varies with the velocity anisotropy (see Fig. 6). One notices from Fig. 6 that the normalized average value of the orthogonal ray lengths
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Fig. 6. Normalized average value of ratio of orthogonal ray lengths vs. s1=2 .
increases continuously with velocity anisotropy. Further, one can observe the dramatic reduction in velocity anisotropy. For example, an initial particle anisotropy ratio of about 14 is reduced to less than 4 due to the e ect described above. 7. Summary and conclusions The ultimate goal of our studies is to be able to describe the evolution of the shape of anisotropic particles as they form, grow, and interact with other particles. A detailed description of this process would not only allow one to predict the nal morphology of a fully transformed sample, but also it might permit one to ascertain information regarding the transformation mechanism from an inspection of the morphology of a fully transformed sample. Here, we have presented an initial step to our nal objective by considering the blunting of an anisotropic particle in a sea of matrix particles. We have assumed that the concentration of anisotropic particles is extremely small so that only interactions between the completely symmetric matrix particles and anisotropic particles need be considered. Also, we only considered the situation where particle nucleation was completed prior to growth, a situation which can be attained to a good approximation by two-stage heatings in certain systems. First, we considered a single growth ray and derived an expression for the average distance it would travel prior to encountering a growing matrix particle. We computed
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Fig. 7. Illustration of the condition that the growth of the two rays be independent. At time t the faster ray (moving horizontally) has reached point (x0 ; y0 ; z 0 ) and has its exclusion volume shown. The slower ray (moving vertically) will be independent if its instantaneous exclusion volume does not intersect any sphere along the faster ray’s trajectory. The inequality test construction is highlighted.
this average distance as a function of the ratio of the initial ray speed to the matrix growth speed and compared it with the result found when the two speeds were the same. We concluded that as the ray grew faster relative to the matrix particle growth, the distance by which it would grow was increasingly reduced due to more probable collisions with matrix particles. Next, we examined the ratio of fast to slow axis growth distances as a function of the speeds of these rays relative to the matrix particle. We observed that there was a dramatic blunting of the initial particle anisotropy, and we concluded that this e ect was due to the greater likelihood that the fast ray would interact with a matrix particle in any given time period.
Appendix Here we derive a sucient condition for treating the growth probabilities of the orthogonal rays independently. In order to guarantee the independence of the probabilities, none of the exclusion spheres associated with one ray is permitted to intersect with the exclusion spheres of the second. In order to visualize the requirement more easily, we examine the geometrical conditions as shown in Fig. 7. We require that any exclusion circle along either axis does not intersect a circle on the other axis
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corresponding to any earlier time. To be speci c, we consider a circle on the y-axis centered at the point v b t, and we show an exclusion circle along the x-axis centered va t, with the condition that 0 ¡ ¡ 1: Also, we assume that va ¿ vb . It is clear from an inspection of Fig. 7 that the following inequality must be ful lled. ((vb t)2 + (va t)2 )1=2 ¿ gt(1 + ) :
(A.1)
If once again we introduce w = v=g, then Eq. (A.1) becomes (wb )2 + (wa )2 ¿ (1 + )2 :
(A.2)
If we, alternately, x a circle on the x-axis and look at all circles at earlier times on the y-axis, then by symmetry we get (wb )2 + (wa )2 ¿ (1 + )2 :
(A.3)
Since wa ¿ wb , it is easy to demonstrate that the inequality shown in (A.2) is more stringent than the one given by (A.3). So henceforth only (A.2) will be used. If we de ne r = wa =wb , then (A.2) can be written as wb ¿
1+ : (1 + r 2 2 )1=2
(A.4)
If we are merely interested in a sucient condition that wb must satisfy, then we can select the maximum value which the right-hand side of (A.4) attains on the unit interval. It is easily demonstrated that the maximum value will occur when = r −2 . Hence, we nd that a sucient condition is 1=2 1 : (A.5) wb ¿ 1 + 2 r Thus, even in the worst-case scenario when wa = wb , we only need wb ¿ 21=2 . Since we are primarily interested in the case where wb ¿ 1, one can see that this is a very weak constraint. References [1] A. Halliyal, A.S. Bhalla, R.E. Newnham, L.E. Cross, in: M.H. Lewis (Ed.), Glasses and Glass-Ceramics, Chapman & Hall, London, England, 1989. [2] S.-Y. Wu, IEEE Trans. Electron. Dev. ED21 (1974) 499. [3] J.F. Scott, C.A. Paz de Araujo, Science 246 (1989) 1400. [4] D. Bondurant, Ferroelectrics 112 (1990) 273. [5] R. Mozzami, C. Hu, W.H. Shepard, IEEE Electron Dev. Lett. 11 (1990) 454. [6] R.K. Singh, J. Narayan, A.K. Singh, C.B. Lee, J. Appl. Phys. 67 (1990) 3448. [7] J. Brasunas, B. Lakew, C. Lee, J. Appl. Phys. 71 (1992) 3639. [8] A. Yoshida, H. Tamura, H. Takauchi, T. Imamura, S. Hasuo, J. Appl. Phys. 71 (1992) 5284. [9] B. Dwir, D. Pavuna, J. Appl. Phys. 72 (1992) 3855. [10] H.F. Taylor, Ferroelectrics 50 (1983) 141. [11] T. Kawaguchi, H. Adachi, K. Setsune, O. Yamazaki, K. Wasa, Appl. Opt. 23 (1984) 2187. [12] C.H.-J. Huang, H.C.-K. Chui, B.A. Stone, T.A. Rost, T.A. Rabson, Proceedings of the 1990 IEEE 7th International Symposium on Applications of Ferroelectrics, IEEE, New York, NY, 1991, pp. 726–729. [13] A.B. Wegner, S.R.J. Brueck, A.Y. Wu, Ferroelectrics 116 (1991) 195.
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[14] V. Joshi, M.L. Mecartney, J. Mater. Res. 8 (1993) 2668. [15] Y. Abe, T. Kasuga, H. Hosono, K. de Groot, J. Amer. Ceram. Soc. 67 (1984) C-142. [16] Y. Abe, M. Hosoe, T. Kasuga, H. Ishikawa, N. Shinkai, Y. Suzuki, J. Nakayama, J. Amer. Ceram. Soc. 65 (1982) C-189. [17] M.J. Bloemer, J.W. Haus, J. Lightwave Techn. 14 (1996) 1534. [18] A. Nahal, V.K. Miloslavsky, L.A. Ageev, Opt. Commun. 154 (1998) 234. [19] W.A. Johnson, R. Mehl, Trans. AIME 135 (1939) 416. [20] M. Avrami, J. Chem. Phys. 7 (1939) 1103. [21] M. Avrami, J. Chem. Phys. 8 (1940) 212. [22] M. Avrami, J. Chem. Phys. 9 (1941) 177. [23] A.N. Kolmogorov, Izv. Akad. Nauk SSR, Ser. Fiz. 3 (1937) 355. [24] M.P. Shepilov, D.S. Baik, J. Non-Cryst. Solids 171 (1994) 141. [25] R.M. Bradley, P.N. Strenski, Phys. Rev. B 40 (1989) 8967. [26] D.P. Birnie III, M.C. Weinberg, Physica A 230 (1996) 484. [27] D.P. Birnie III, M.C. Weinberg, J. Chem. Phys. 103 (1995) 3742. [28] D.P. Birnie III, M.C. Weinberg, Physica A 223 (1996) 337. [29] M.C. Weinberg, D.P. Birnie III, J. Chem. Phys. 105 (1996) 5138. [30] M.P. Shepilov, Sov. Phys. Crystallogr. 35 (1990) 164. [31] M.C. Weinberg, D.P. Birnie III, J. Non-Cryst. Solids 202 (1996) 290. [32] P.F. James, in: M.H. Lewis (Ed.), Glasses and Glass-Ceramics, Chapman & Hall, London, England, 1989. [33] M.C. Weinberg, R. Kapral, J. Chem. Phys. 91 (1989) 7146.
www.elsevier.com/locate/physa
Development of anisotropic particle morphology in an isotropically transforming matrix Dunbar P. Birnie III, Michael C. Weinberg ∗ Department of Materials Science and Engineering, Arizona Materials Laboratory, University of Arizona, Tucson, AZ 85721-0012, USA Received 14 December 1999
Abstract In the present work we consider the growth of a very dilute concentration of anisotropic particles, nucleated with random orientations, in a matrix of growing spherical particles. It is assumed that the matrix particles obey the usual JMAK kinetics. We concentrate on describing the change in anisotropic particle grain morphology as the transformation process proceeds. An eccentric shaped particle will have certain growth directions that are rapid and others that are slower. Faster growing directions impinge upon the matrix particles much sooner than slower growing directions, and this feature leads to an e ective change in particle morphology as the transformation process progresses. We analyze such changes in particle morphology by deriving expressions for the probabilities that growth rays will travel a certain distance before encountering matrix particles and that the length ratio of fast-axis to slow-axis growth of the anisotropic particle will attain certain values. The changes in particle morphology are examined as a function of the relative speeds of growth of anisotropic particle growth rays to matrix particle growth rates. c 2000 Elsevier Science B.V. All rights reserved.
Keywords: Phase transformations; Crystallization; Anisotropic particles; Nucleation and growth
1. Introduction Transformation kinetics involving the nucleation and growth of anisotropic particles is a topic of practical importance due to its relevance to a variety of applications [1–14]. For example, materials consisting of plate-like particles or composite materials containing highly anisotropic embedded particles can exhibit superior mechanical toughness ∗
Corresponding author. Fax: +1-520-6218059. E-mail address: [email protected] (M.C. Weinberg).
c 2000 Elsevier Science B.V. All rights reserved. 0378-4371/00/$ - see front matter PII: S 0 3 7 8 - 4 3 7 1 ( 0 0 ) 0 0 2 6 4 - 8
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since cracks need to follow a tortuous path around or through the plate-like particles in order to propagate [15,16]. Also, interesting optical e ects such as birefringence and light guiding can be augmented by producing elongated particles within a transparent matrix [17,18]. In some situations it is desirable for the anisotropic particles to be randomly oriented within the matrix, while in other cases preferential or cooperative alignment of the particles is more bene cial for the envisioned application. However, in either situation the tailoring and control of the morphology of the transformed material is of great importance. Calculation of the kinetics of transformations involving anisotropic particles is a much more challenging problem than that for isotropic particles. The theory of the transformation kinetics for isotropic particles was developed by Johnson and Mehl [19], Avrami [20 –22], and Kolmogorov [23] many years ago. This formalism, referred to as JMAK theory, is a basic pillar of Materials Science that has been employed extensively for the computation of the rates of transformations that proceed by nucleation and growth mechanisms. In order to explain the complicating features associated with the transformation kinetics producing anisotropic particles, it is necessary to remember certain features in the derivation of the JMAK equation. The derivation rests on calculating the probability that a randomly chosen point in space (“O”) has not been transformed after a given time period, t. In order to satisfy this criterion a certain volume around O must be free of nuclei since any nucleus in the immediate neighborhood of the point could grow and transform the point O. Any nucleus which is close enough to O so that it has the potential to transform O in the time t is called an “aggressor” [24]. For transformations involving isotropic particles every aggressor is potent, meaning that all aggressors will transform the point O. Hence, in this case if the nucleation and growth rates are known, then the transformation kinetics can be computed in a straightforward manner [25]. However, for cases where anisotropic particles form, complications arise since not all aggressors are potent. Particle–particle interactions (i.e., particles growing to points of contact) can occur such that non-aggressors can block the path of aggressors rendering the aggressors impotent. Such non-aggressors, termed “blockers”, occur only for transformations involving anisotropic particles. Another e ect that can occur solely in the case of anisotropic particle formation is that of “phantom” particles. A phantom particle is one that nucleates in a region shadowed by the growth of another previously-nucleated anisotropic particle. Phantoms do not occur if all nucleation is completed prior to growth, but such particles arise if the number of particles nucleated is a function of time. Despite the diculties in describing the transformation kinetics of anisotropic particles, approximate and precise results have been obtained for several speci c cases [26 –30]. These studies have produced a qualitative understanding of the kinetics. It has been observed that growth velocity anisotropy generally leads to a retardation of the kinetics, but this slowing down is noticeable only if the probability for particle– particle interaction is suciently large (usually requires that about 30% of the volume has been transformed). This kinetic retardation is due to the e ects described above and leads to smaller Avrami exponents that are also time dependent [31].
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Although several works have been devoted to the discussion of the kinetics of transformations with anisotropic particle formation, we know of no study that examines the grain morphology that results for transformations involving the nucleation and growth of anisotropic particles. This is a dicult problem due to the occurrence of blockers and phantoms. Here we address the grain morphology problem for a model system for which the usual complicating elements are absent. Speci cally, we model the process by examining how an anisotropic particle grows into a matrix of isotropic particles that transform in such a manner that they obey the JMAK Theory. This approach has the distinct advantage that the statistical interactions between the primary particle and neighboring particles are included, but blocking by “third party” particles is eliminated. The basis for this simpli cation relies upon the fact that we only consider the limit of very dilute concentration of anisotropic particles in a matrix of isotropic particles. Under such circumstances blocking need not be considered since the matrix, which obeys JMAK kinetics, does not experience blocking e ects and the probability of particle interaction between two anisotropic particles is vanishingly small. In many systems, simultaneous crystallization of two or more phases can occur. A particularly interesting and yet mathematically tractable case is that for which one phase is highly anisotropic and the other is essentially isotropic. Thus, in the present work we consider the case where the anisotropic particles nucleate with random orientations and then grow with their intrinsically eccentric shapes until they impinge upon one another creating an interconnected brous or plate-like microstructure. Our main focus will be placed on describing several aspects of the grain morphology for this process. One observes that an eccentric-shaped particle will have certain growth directions that are rapid and others that are slower. The faster growing directions are anticipated to impinge upon the matrix particles much sooner than the slower growing directions. This feature is expected to lead to an e ective change in particle morphology as the transformation process progresses such that the nal particle morphology is less eccentric than the initial eccentricity due to the growth velocity anisotropy. One of the main goals of this work is the quantitative description of this process. We examine this phenomenon for di erent extents of initial particle eccentricity and for di erent ratios of anisotropic-particle to matrix-particle growth speeds. Our ultimate objective is the elucidation of the complete crystalline morphology produced by transformations involving two phases that grow with di erent geometric characteristics. The present study is a rst step towards that goal.
2. Model As mentioned above, we consider the growth of a very dilute solution of anisotropic particles embedded in a matrix of growing isotropic particles. In particular, we select the anisotropic particle to be ellipsoidal in shape, so that there is an axis of symmetry and only two characteristic growth speeds. Whether the ellipsoid is oblate or prolate is immaterial, but we take the growth velocities in the two primary directions as
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representations of the nal 3D shape. The matrix particles are taken to be spherical, as required by JMAK Theory. For the sake of simplicity we will concentrate on the particle growth behavior along these two characteristic directions, which are chosen to be the fast growth axis and the slow growth axis. The corresponding growth speeds are denoted by va and vb , respectively. One should note that for ellipsoidal particles the fast and slow growth directions are spatially orthogonal. This feature allows us to select conditions such that possible interactions or coupling of the probabilities involved in growth of either of these perpendicular growth rays is prevented. Limitations placed on applicable va and vb values are discussed in the appendix. We assume that nucleation of both types of particles occurs prior to particle growth and is completed before the particles begin to grow. This assumption is an excellent representation of events in a two-stage heating procedure used in experiments to measure nucleation rates in amorphous systems [32]. Also, we consider a 3-D system that is of in nite extent. The concentration of ellipsoidal particles is deemed suciently dilute so that anisotropic-particle=anisotropic-particle interactions can be neglected. Particle growth is presumed to be interface controlled, so that all growth rates are constant.
3. Approach The initial shape of ellipsoidal particles will be distorted at the end of the transformation due to their encounters with matrix particles. We utilize the e ective de ciency in particle lengths (along fast and slow axes) and alteration in particle eccentricity produced by particle interactions in order to gauge the nal morphology. Since the fast and slow growth rays associated with a particle are not coupled on average the faster rays will be blocked from advancing by matrix particles at a shorter time than those rays growing in the slow direction. This fact leads to the reduction in eccentricity of the nal particle. The essence of our approach will be a calculation of the probability that any particular growth ray grows unimpeded over a given distance or time period. In other words, we calculate the probability that either the fast or slow growth ray associated with the ellipsoidal particle will encounter a growing spherical matrix particle. We decompose ray growth into a sequence of incremental time steps. It is clear that for a ray to have grown for some time period, t, it must have successfully grown for all time increments, t, comprising t. Hence, to obtain the probability that such a ray will extend its growth for at least one more time increment, we must nd the conditional probability that during this next time interval no collisions will occur given the knowledge that the ray grew unimpeded up to that time t. The ability for a ray to grow freely depends upon the absence of matrix particles in the vicinity of the ray that could grow and block its growth. This implies that a certain volume of space around the ellipsoidal particle is free from nuclei, and an important part of our procedure will entail the calculation of such nucleus-free volumes.
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Fig. 1. Illustration of incremental exclusion zone of a growing ray comparing times t and t + t when v ¿ g. Shaded area represents new volume that must also be free from matrix nucleation events.
4. Governing equations Let us consider an arbitrary point in space (x0 ; y0 ; z 0 ) which is not transformed at some time t. We assume that this point is at the leading edge of a growth ray that is moving in the positive direction along the x-axis at a speed v. Let the growth rate of a matrix particle be denoted by g. Since the point (x0 ; y0 ; z 0 ) was untransformed by matrix particles, then a sphere of radius gt must exist around this point which is free from matrix nuclei. Similarly, if the ray grows to the next point, (x0 + vt; y0 ; z 0 ), in the time t, then a sphere of radius g(t + t) must be centered about this second point, too (see Fig. 1). These two spherical regions are “exclusion volumes” where no matrix nuclei are allowed. In constructing Fig. 1 it has been assumed that v ¿ g. Since the time increment is small, the volume of the spheres will overlap. Thus, in order to nd the conditional probability that the ray will advance to x0 + vt given that it already made it to x0 , it will be necessary to compute the volume of the second sphere that falls outside the volume of the rst sphere (represented by the shaded area in Fig. 1). Spheres 1 and 2 satisfy the following equations, respectively: (x − x0 )2 + (y − y0 )2 + (z − z 0 )2 = g2 t 2 ;
(1)
(x − x0 − vt)2 + (y − y0 )2 + (z − z 0 )2 = g2 (t + t)2 :
(2)
The points of intersection follow from the solution of Eqs. (1) and (2) and is given by 2g2 t + t(g2 − v2 ) : (3) 2v One observes that if v = g (i.e., when the ray’s speed is identical to growth rate of matrix particle), then the spheres touch at merely one point. Hence, if v ¡ g, then x = x0 −
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Fig. 2. Illustration of incremental exclusion zone of a growing ray comparing times t and t + t when v ¡ g. Shaded area represents new volume that must also be free from matrix nucleation events. In this case the larger sphere completely encloses the smaller one, resulting in growth times identical to the JMAK matrix growth process.
sphere 1 is totally engulfed by sphere 2, and the portion of volume of sphere 2 which does not intersect sphere 1 is just given by the di erence between the volumes of spheres 2 and 1 (see Fig. 2). However, if v ¿ g, a line of intersection points occurs, and calculation of the conditional probability, re ected by the non-overlapping portion of the volume of sphere 2, entails a bit more e ort. In order to ascertain the probability that a ray will grow for some time t 0 in an unimpeded fashion, one must then construct the sequence of conditional probabilities for all small incremental times which span the time from t = 0 to t = t 0 . Let us consider the cases v6g and v ¿ g in turn. In the former case each exclusion sphere (i.e., sphere where no matrix nuclei are allowed) totally engulfs all exclusion spheres corresponding to earlier times in the ray’s history. This result implies that the probability that this slow ray will grow for some time period t 0 is governed solely by the matrix particle growth speed and that the expected average growth time will be the same as that for any isotropic matrix particle. Since the matrix particle growth is governed by the usual JMAK Equation, this case is of somewhat less interest than case 2. However, when v ¿ g, the exclusion spheres only partially overlap and the boundary of the intersection area is given by Eq. (3). For v ¿ g, the union of sequentially overlapping exclusion spheres, in the limit of very small t, will produce a shape that can be described by an ice cream cone with a scoop of ice cream on top. A 2-D projection of this object is shown in Fig. 3. If the position of the tip of the cone is taken as (0; y0 ; z 0 ) and the center of the nal sphere (ice cream ball) is taken at (x0 ; y0 ; z 0 ) and using the fact that x0 = vt, then it is easy to show that the distance d, de ned in Fig. 3, is given by r g2 (4) d = gt 1 − 2 : v
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Fig. 3. Illustration of the cumulative ice-cream-cone-shaped exclusion zone of a growing ray for time t when v ¿ g. The cone volume is made up of the superposition of all successively larger spheres starting at (0; y0 ; z 0 ) and extending to time t, when the ray has grown to (x0 ; y0 ; z 0 ). Note that x0 = vt.
Hence, the volume associated with the cone portion is g2 t vt − d2 ; VCone = 3 v and the volume associated with the truncated sphere is g3 4g3 t 3 1 3g − + : VSpherepart = 3 2 4v 4v3
(5)
(6)
If Eqs. (4) – (6) are combined and Vsphere denotes the volume of a sphere of radius gt, then the total volume of cone plus partial sphere is as follows: v g 1 + + : (7) VTotal = VSphere 2 4g 4v If one follows the standard arguments used for the derivation of the JMAK Equation [33], then it is easy to show that the total probability that a ray grows for time t; Z(v; t), is v g 4g3 t 3 1 + + ; (8) Z(v; t) = exp(−VTotal ) = exp − 3 2 4g 4v where is the nucleation density (i.e., the average number of matrix particles that nucleated per volume). If we let w = v=g, then the probability that a ray will grow to at least a length x; Z(v; x), is as follows: Z(v; x) = exp(− x3 ) ; where ≡
1 4 1 w ; + + 3w3 2 4 4w
≡
4 ; 3w3
for 1¿w :
(9a)
for w ¿ 1 ;
(9b) (9c)
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Eqs. (9a) – (9c) are the basic governing equations which will be used to examine the morphological development of the ellipsoidal particle.
5. Progress of individual growth rays Here we analyze the growth characteristics of a single ray traveling from its original nucleation site. In the following section, we will apply the ndings to a comparison of growth occurring for rays traveling along two orthogonal directions at di erent velocities. The latter quantities will give some indication of particle shape distortion. If P(x) denotes the probability density that a ray reaches some length between x and x + d x, then P(x) = −
@Z(v; x) = 3 x2 exp(− x3 ) : @x
(10)
Therefore, the total probability, Ptot , that a ray travelling along the positive x-axis will grow to a size in excess of x∗ is simply Z x∗ P(x) d x = 1 − exp(− (x∗ )3 ) ; (11) Ptot = 0
and the average growth length, hxi, is Z ∞ 4 −1=3 ; xP(x) d x = hxi = 3 0
(12)
where is the gamma function. One observes from an inspection of Eqs. (9b), (11), and (12) that both the probability distribution of ray lengths and the average length of a ray depends not only upon the ratio of the ray speed to matrix growth speed but also upon the number density of nucleated matrix particles. In order to remove the latter dependence upon nucleation density and to focus solely upon the in uence of the speed ratio we inspect the ratio of the average length to the average length if the ray were to obey JMAK statistics (i.e., w = v=g = 1), denoted by hxi=hxiJMAK . From Eqs. (9a) and (9b) and (12) one nds −1=3 1 w 1 hxi = : (13) + + hxiJMAK 2 4 4w Also, precisely the same result would ensue if we examined the ratio Ptot (x∗ )=PtotJMAK . Table 1 illustrates the dependence of these ratios on the relative growth speeds of ray and matrix. As anticipated the average length traveled by a ray will decrease monotonically with w. Since faster growing rays explore more new volume than slower growing rays in the same time period, the former are more likely to come into the vicinity of a matrix particle that will stop the ray’s growth. This process of shortening the ray’s growth time will tend to “round out” the particle’s ellipsoidal shape. From Table 1, one notes that w must be about 5, or greater, to have a signi cant e ect.
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Table 1 Average ray length normalized relative to the average ray length found for JMAK kinetics w = v=g
hxi=hxiJMAK
2 5 10 25 50 100
0.96 0.82 0.69 0.53 0.43 0.34
6. Anisotropic particle development Now let us consider the progress of two growth rays from the same anisotropic particle that are traveling in the positive directions along two orthogonal axes (i.e., take the fast and slow growth directions). The quantity of interest will be the expected ratio of lengths of these two rays since this will provide a measure of the extent of distortion of the ellipsoidal particle from its ideal shape if there were no collisions with matrix particles. Let us assume that the faster ray grows along the x1 -axis with a speed va and the slower growth grows along the x2 -axis with a speed vb . For any time t, the maximum extensions of the rays along the x1 and x2 axes will be denoted by xa; max ; xb; max , respectively. Further, denote by r the actual ratio of the lengths of the rays along the two axes. The total probability, Ptot , that r does not exceed some speci ed value, called r ∗ , is given by the following equation: Ptot ≡ P+ + P− ; where
(14)
x1 x1 ∗ d x1 d x2 Pa (x1 )Pb (x2 ) 1 − −r −1 ; P+ = x2 x2 0 0 Z xb; max Z xa; max x2 x2 d x1 d x2 Pa (x1 )Pb (x2 ) 1 − − r∗ −1 : P− = x1 x1 0 0 Z
xa; max
Z
xb; max
In the above equation Pn (xi ), with n = a; b and i = 1; 2, is given by Eq. (10) and the subscript n speci es the value of and is the Heaviside function. Eq. (14) accounts for the fact that even though va ¿ vb , the length of the ray along the x2 -axis could be longer than the one along the x1 -axis due to ray blockage by a matrix particle. Thus, P+ and P− are the contributions to Ptot when the length of the ray in the x1 direction is larger than the one in the x2 direction, and vice versa, respectively. Eq. (14) also contains the implicit assumption that the probabilities Pa (x1 ) and Pb (x2 ) are independent (i.e., due to the random statistical distribution of matrix particles in two separate directions). Although this assumption would normally be valid, it is not always so. Therefore, the condition needed to justify this assumption is discussed as
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an appendix. If we introduce two new coordinates, x10 = x1 and r = x1 =x2 , into the P+ term, then it can be written as follows: Z rmax Z dr xa; max 0 0 d x1 x1 Pa (x10 )Pb (x10 ; r)[1 − (r − r ∗ )](r − 1) ; (15a) P+ = 2 r 0 0 Z P+ =
r∗
1
Z
dr r2
xa; max
d x10 x10 Pa (x10 )Pb (x10 ; r) :
0
(15b)
Also, one may transform P− in a similar fashion to obtain Z r∗ Z dr xb; max 0 0 d x2 x2 Pa (x20 ; r)Pb (x20 ) : P− = r2 0 1
(16)
Now, the probability density of nding a length ratio between r ∗ and r ∗ + dr ∗ is denoted by P d and is given by the partial derivative of Ptot with respect to r ∗ . Hence, Z ∞ 1 d d d d x10 x10 Pa (x10 )Pb (x10 ; r) P = P+ + P− = ∗ 2 (r ) 0 Z ∞ 1 d x20 x20 Pa (x20 ; r)Pb (x20 ) : (17) + ∗ 2 (r ) 0 If one employs Eq. (10) in Eq. (17) and performs the integrations, then one nds that d = P+
where s=
3s(r ∗ )2 [s + (r ∗ )3 ]2
d P− =
3s(r ∗ )2 ; [1 + s(r ∗ )3 ]2
(18a)
b a
(18b)
and therefore, P d = 3s(r ∗ )2
1 1 + [s + (r ∗ )3 ]2 [1 + s(r ∗ )3 ]2
:
(18c)
One observes from Eq. (18c) that the probability density depends upon a single parameter, s, de ned above. If one utilizes Eqs. (9b) and (9c) in the de nition of s, then one obtains the following expressions for s, depending on the magnitudes of wa and wb : [ 12 + w4b + 4w1 b ] wa 3 ; wa and wb both ¿ 1 ; (19a) s = 1 wa [ 2 + 4 + 4w1 ] wb a
1 1 wa + + s= 2 4 4wa s=
wa wb
3 =
va vb
−1
wa wb
3 ;
wa ¿ 1; wb ¡ 1 ;
(19b)
3 ;
wa ; wb ¡ 1 :
(19c)
Thus, in general, the probability density depends individually upon the speeds of the rays (relative to the matrix) in the two key orthogonal directions. However, in some
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Fig. 4. The probability density of nding a length ratio between r ∗ and r ∗ + dr ∗ as a function of r ∗ for three values of the parameter s.
special cases the probability density depends only upon the ratio of the speeds of the orthogonal rays. Clearly, this is the situation when wa ; wb ¡ 1 and also if wa ; wb ¿ 1 and wb 1. In the latter case it is easy to show that s → (wa =wb )2 = (va =vb )2 . If one takes the derivative of P d with respect to r ∗ , using Eq. (18c), and sets it equal to zero, then one observes that for certain values of s there will be extrema in the probability density function. This behavior is shown in Fig. 4, where P d has been plotted as a function of r ∗ for three values of s. From Eq. (18c) one may easily see that the r ∗ = 1 intercept is 6s=(1 + s)2 . The plots corresponding to the larger values of s exhibit extrema, but the s = 4 curve shows monotonic behavior. If one analyzes the equation for the extrema, then it is easy to show that a necessary condition for the presence of extrema is s ¿ 2. However, it is more dicult to nd the sucient condition, although one can show that for s 1 extrema are still expected. A quantity of particular interest is the average value of r ∗ ; hr ∗ i, since it is a good measure of the change in particle distortion brought about by collisions with matrix particles. With the aid of Eq. (18c) we obtain the following expression for hr ∗ i: ∗
hr i =
Z 1
∞
∗ d
∗
∗
r P (r ) dr = 3s
Z 1
∞
∗
∗ 3
dr (r )
1 1 + 2 ∗ 3 [s + (r ) ] [1 + s(r ∗ )3 ]2
: (20)
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Fig. 5. Normalized average value of ratio of orthogonal ray lengths vs. s.
One can perform the integrals analytically and if one de nes y = s1=3 , then one nds y 1 − C(y) √ ; (21a) hr ∗ i = 1 + B(y) y + y 3 where
(1 + y)2 1 ; B(y) = √ − log 1 − y + y2 2 3 6 1 2−y y(2 − y) √ + 2 tan−1 : C(y) = tan−1 √ y 3y 3
(21b) (21c)
In Fig. 5, Eqs. (21a) – (21c) have been utilized to construct a plot of hr ∗ i, normalized by hr ∗ is=1 which is the JMAK value as a function of s. One notes that the normalized average value of orthogonal ray lengths increases monotonically with s, as anticipated. Further, one observes that the collisions with matrix particles produce a dramatic blunting, or reduction in particle anisotropy. This results from the fact that the more rapidly moving rays will encounter matrix particles much sooner than slowly travelling rays, and thus they travel only a short portion of the distance in the collision-free case. A plot that is somewhat easier to interpret physically may be constructed in some limiting cases. For example, previously it was indicated that when wa ; wb ¿ 1 and wb 1; s → (wa =wb )2 = (va =vb )2 . Hence, a plot of hr ∗ i=hr ∗ is=1 vs. s1=2 will show how the normalized length ratio varies with the velocity anisotropy (see Fig. 6). One notices from Fig. 6 that the normalized average value of the orthogonal ray lengths
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Fig. 6. Normalized average value of ratio of orthogonal ray lengths vs. s1=2 .
increases continuously with velocity anisotropy. Further, one can observe the dramatic reduction in velocity anisotropy. For example, an initial particle anisotropy ratio of about 14 is reduced to less than 4 due to the e ect described above. 7. Summary and conclusions The ultimate goal of our studies is to be able to describe the evolution of the shape of anisotropic particles as they form, grow, and interact with other particles. A detailed description of this process would not only allow one to predict the nal morphology of a fully transformed sample, but also it might permit one to ascertain information regarding the transformation mechanism from an inspection of the morphology of a fully transformed sample. Here, we have presented an initial step to our nal objective by considering the blunting of an anisotropic particle in a sea of matrix particles. We have assumed that the concentration of anisotropic particles is extremely small so that only interactions between the completely symmetric matrix particles and anisotropic particles need be considered. Also, we only considered the situation where particle nucleation was completed prior to growth, a situation which can be attained to a good approximation by two-stage heatings in certain systems. First, we considered a single growth ray and derived an expression for the average distance it would travel prior to encountering a growing matrix particle. We computed
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Fig. 7. Illustration of the condition that the growth of the two rays be independent. At time t the faster ray (moving horizontally) has reached point (x0 ; y0 ; z 0 ) and has its exclusion volume shown. The slower ray (moving vertically) will be independent if its instantaneous exclusion volume does not intersect any sphere along the faster ray’s trajectory. The inequality test construction is highlighted.
this average distance as a function of the ratio of the initial ray speed to the matrix growth speed and compared it with the result found when the two speeds were the same. We concluded that as the ray grew faster relative to the matrix particle growth, the distance by which it would grow was increasingly reduced due to more probable collisions with matrix particles. Next, we examined the ratio of fast to slow axis growth distances as a function of the speeds of these rays relative to the matrix particle. We observed that there was a dramatic blunting of the initial particle anisotropy, and we concluded that this e ect was due to the greater likelihood that the fast ray would interact with a matrix particle in any given time period.
Appendix Here we derive a sucient condition for treating the growth probabilities of the orthogonal rays independently. In order to guarantee the independence of the probabilities, none of the exclusion spheres associated with one ray is permitted to intersect with the exclusion spheres of the second. In order to visualize the requirement more easily, we examine the geometrical conditions as shown in Fig. 7. We require that any exclusion circle along either axis does not intersect a circle on the other axis
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corresponding to any earlier time. To be speci c, we consider a circle on the y-axis centered at the point v b t, and we show an exclusion circle along the x-axis centered va t, with the condition that 0 ¡ ¡ 1: Also, we assume that va ¿ vb . It is clear from an inspection of Fig. 7 that the following inequality must be ful lled. ((vb t)2 + (va t)2 )1=2 ¿ gt(1 + ) :
(A.1)
If once again we introduce w = v=g, then Eq. (A.1) becomes (wb )2 + (wa )2 ¿ (1 + )2 :
(A.2)
If we, alternately, x a circle on the x-axis and look at all circles at earlier times on the y-axis, then by symmetry we get (wb )2 + (wa )2 ¿ (1 + )2 :
(A.3)
Since wa ¿ wb , it is easy to demonstrate that the inequality shown in (A.2) is more stringent than the one given by (A.3). So henceforth only (A.2) will be used. If we de ne r = wa =wb , then (A.2) can be written as wb ¿
1+ : (1 + r 2 2 )1=2
(A.4)
If we are merely interested in a sucient condition that wb must satisfy, then we can select the maximum value which the right-hand side of (A.4) attains on the unit interval. It is easily demonstrated that the maximum value will occur when = r −2 . Hence, we nd that a sucient condition is 1=2 1 : (A.5) wb ¿ 1 + 2 r Thus, even in the worst-case scenario when wa = wb , we only need wb ¿ 21=2 . Since we are primarily interested in the case where wb ¿ 1, one can see that this is a very weak constraint. References [1] A. Halliyal, A.S. Bhalla, R.E. Newnham, L.E. Cross, in: M.H. Lewis (Ed.), Glasses and Glass-Ceramics, Chapman & Hall, London, England, 1989. [2] S.-Y. Wu, IEEE Trans. Electron. Dev. ED21 (1974) 499. [3] J.F. Scott, C.A. Paz de Araujo, Science 246 (1989) 1400. [4] D. Bondurant, Ferroelectrics 112 (1990) 273. [5] R. Mozzami, C. Hu, W.H. Shepard, IEEE Electron Dev. Lett. 11 (1990) 454. [6] R.K. Singh, J. Narayan, A.K. Singh, C.B. Lee, J. Appl. Phys. 67 (1990) 3448. [7] J. Brasunas, B. Lakew, C. Lee, J. Appl. Phys. 71 (1992) 3639. [8] A. Yoshida, H. Tamura, H. Takauchi, T. Imamura, S. Hasuo, J. Appl. Phys. 71 (1992) 5284. [9] B. Dwir, D. Pavuna, J. Appl. Phys. 72 (1992) 3855. [10] H.F. Taylor, Ferroelectrics 50 (1983) 141. [11] T. Kawaguchi, H. Adachi, K. Setsune, O. Yamazaki, K. Wasa, Appl. Opt. 23 (1984) 2187. [12] C.H.-J. Huang, H.C.-K. Chui, B.A. Stone, T.A. Rost, T.A. Rabson, Proceedings of the 1990 IEEE 7th International Symposium on Applications of Ferroelectrics, IEEE, New York, NY, 1991, pp. 726–729. [13] A.B. Wegner, S.R.J. Brueck, A.Y. Wu, Ferroelectrics 116 (1991) 195.
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