Crystal Faces Existence and Morphological Stability from a

Jun 10, 2003 - Crystal Faces Existence and Morphological Stability from a Crystallographic Perspective. Jolanta Prywer*. Institute of Physics, Technic...
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Crystal Faces Existence and Morphological Stability from a Crystallographic Perspective Jolanta Prywer* Institute of Physics, Technical University of Ło´ dz´ , Wo´ lczan´ ska 219, 93-005 Ło´ dz´ , Poland Received January 31, 2003;

CRYSTAL GROWTH & DESIGN 2003 VOL. 3, NO. 4 593-598

Revised Manuscript Received May 21, 2003

ABSTRACT: The existence and the morphological stability of crystal faces in connection with their relative growth rates and crystal geometry characterized by interfacial angles has been analyzed. The analysis reveals that not only growth environment, which influences the relative growth rates, but also crystal geometry plays a key role in the formation of final crystal morphology. It is shown that the crystal faces with interfacial angles R + γ close to π are those with the greatest likelihood of existence in crystal morphology. Introduction Crystal habit is an important characteristic of any crystal. Investigation of the growth morphologies can supply valuable information related to important peculiarities of the crystal growth process. The preservation of crystal shape during crystal growth and the stability of faces is a complex and fundamental problem in crystal growth. Nowadays, it is well-known that the growth conditions are one of the main factors determining the so-called end-growth morphology. Growth conditions influence the relative growth rates of individual faces, and therefore, modify crystal habit.1 However, it is observed that for the same relative growth rates, various crystal faces behave differently. Therefore, the aim of this paper is to study the existence of various faces in crystal morphology depending on the relative growth rates and crystallographic structure of the crystal, which is characterized by interfacial angles. Materials and Methods Behavior of a given face during the growth process is quite difficult to investigate on the basis of external crystal morphology. The details of growth behavior of crystal faces are best presented by cross-sections of individual crystals. All changes in growth conditions are recorded in the crystal in a form of growth banding in respective growth sectors and growth sector boundaries (i.e., in a form of internal morphology). Internal morphology may be revealed after cleavage of crystal, in a cross-section surface. For crystals whose growth process cannot be observed in situ (e.g., natural crystals or crystals synthesized by high-temperature solution or hydrothermal solution methods), the information on growth behavior of crystal faces has to be inferred from such an internal morphology1,2 seen, for instance, in a cross-section surface. Therefore, to analyze the phenomenon of existence and stability of crystal faces, we apply the formula for the cross-section size lhkl of the hkl face derived in ref 3 and given by

lhkl )

Rh1k1l1 sin γ + Rh2k2l2 sin R - Rhkl sin(R + γ) sin R sin γ

0 t + lhkl

Figure 1. Cross-section of a hypothetical crystal illustrating normal growth rates Rhkl, Rh1k1l1, and Rh2k2l2 of the hkl, h1k1l1, and h2k2l2 faces, respectively; lhklscross-section size of the hkl face; R and γsinterfacial angles; GBsgrowth bands; GSBs growth sector boundaries; and GSsgrowth sectors. The figure illustrates also the physical meaning of the ratio Rhkl/Rcrit hkl : the h′k′l′ face increases between the m1 and m4 growth bands (Rhkl/Rcrit hkl < 1), it preserves the size between the m4 and m8 growth bands (Rhkl/Rcrit hkl ) 1), and the h′k′l′ face decreases between m8 and m11 growth bands (Rhkl/Rcrit hkl > 1). appropriate interfacial angles illustrated in Figure 1; l0hkl is the initial size of the hkl face (not marked in Figure 1); and growth time t corresponds to the change of the size of the considered face from l0hkl to lhkl. The initial size l0hkl may be considered as the size of a given face in the seed or after some time of growth (after growth of a crystal layer). The constant growth rates Rhkl, Rh1k1l1, and Rh2k2l2 are required not throughout the whole growth process but only during the growth of a layer of crystal that corresponds to the change of the size from l0hkl to lhkl. Further, the instantaneous rate of changes in cross-section size of a given hkl face, denoted by dlhkl/dt, was derived in ref 4 and is given by

dlhkl Rh1k1l1 sin γ + Rh2k2l2 sin R - Rhkl sin(R + γ) ) dt sin R sin γ

(2)

(1) where Rhkl, Rh1k1l1, and Rh2k2l2 are the normal growth rates of the hkl, h1k1l1, and h2k2l2 faces, respectively; R and γ are the * Corresponding author. Tel: 48 42 6313667. Fax: 48 42 6313639. E-mail: [email protected].

The positive value of the rate dlhkl/dt corresponds to an increase in size of the given hkl face, the negative value means that the hkl face decreases, and if dlhkl/dt ) 0, the given hkl face does not change. Evaluating the integral of the expression presented by eq 2, we find that the size lhkl takes the form of eq 1, where l0hkl is the integration constant.

10.1021/cg034015v CCC: $25.00 © 2003 American Chemical Society Published on Web 06/10/2003

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Prywer

Equating the numerator in eq 1 to zero, we evaluate the so-called critical growth rate5 denoted by Rcrit hkl . Critical growth rate Rcrit hkl takes the following form: crit Rhkl )

Rh1k1l1 sin γ + Rh2k2l2 sin R sin(R + γ)

(3)

It is easier to explain the physical meaning of the critical growth rate considering the ratio Rhkl/Rcrit hkl . Therefore, eq 3 should be inverted and multiplied by Rhkl. Hereby, we obtain the critical relative growth rate6 in the following form:

Rhkl crit Rhkl

)

sin(R + γ) sin R sin γ + Rhkl/Rh1k1l1 Rhkl/Rh2k2l2

(4)

6 The physical meaning of the ratio Rhkl/Rcrit hkl is as follows: (i) crit for Rhkl/Rhkl < 1, the size of the hkl face increases. If the hkl face does not exist in the habit, it appears and develops its size. This may be observed in the example of the h′k′l′ face crit visible in Figure 1. Here, the ratio Rh′k′l′/Rh′k′l′ , beginning from the seed up to the m4 growth band, is smaller than unity, and therefore, this face increases its initial size. (ii) For Rhkl/Rcrit hkl ) 1, the size of the hkl face is preserved. If the initial size of this face is equal to zero, the zero size is preserved, which means that this face does not appear in the habit. In the case crit of the h′k′l′ face shown in Figure 1, the ratio Rh′k′l′/Rh′k′l′ is equal to unity, beginning from the m4 to m8 growth band. This is why this face does not change its size at this stage of growth. (iii) For Rhkl/Rcrit hkl > 1, the size of the hkl face decreases that may lead, in consequence, to its disappearance. If the face does not exist in the habit, it still does not appear. In the case of crit the exemplary h′k′l′ face (Figure 1), the ratio Rh′k′l′/Rh′k′l′ changes its value becoming greater than unity beginning from the m8 growth band. It leads to the decrease in size of the h′k′l′ face. Hence, the presence, absence, decrease, or increase in size of the crystal face, and consequently, its stability or instability, depends on the value of the critical relative growth rate Rhkl/Rcrit hkl . On the other hand, the critical relative growth rate depends on the relative growth rates Rhkl/Rh1k1l1 and Rhkl/Rh2k2l2 and the crystal geometry represented by interfacial angles R and γ. Therefore, the analysis of the correlation between existence and morphological stability of crystal faces and their relative growth rates and crystal geometry is based on eq 4. It should be pointed out that all equations presented in this paper apply to the 2-D cross-section of crystal. However, they may be applied also to those faces of the 3-D crystal that belong to one and the same crystallographic zone. It is worth mentioning also that in these equations the growth rate of a given face is multiplied by the sine of the interfacial angle opposite to that face (i.e., Rh1k1l1 sin γ, not Rh1k1l1 sin R).

Results and Discussion Predictions of the Theoretical Model. The correlation between the critical relative growth rate Rhkl/Rcrit hkl and the presence and morphological stability of a given face in a crystal habit follows from the physical meaning of the ratio Rhkl/Rcrit hkl . The most stable faces are those faces for which the ratio Rhkl/Rcrit hkl is smaller than unity. Then, a given face increases its size. For the ratio Rhkl/Rcrit hkl greater than unity, a given face decreases its size, and consequently, it may disappear from the crystal habit. The faces with such values of crit the ratio Rhkl/Rcrit hkl are not stable. For the ratio Rhkl/Rhkl equal to unity, a given face preserves its size; hence, it exists in the habit. However, if the other faces increase their sizes, the morphological importance of the considered face decreases. By the arguments advanced above,

Figure 2. Exemplary cross-section of the KBC crystal illustrating different evolution of faces occurring in this section, in particular the evolution of the (001), (010), and (01h 2 h ) faces that are under consideration. The numeration m1 to m14 denotes the consecutive growth bands.

it results that all faces can be unstable for the value of the ratio Rhkl/Rcrit hkl close to unity. For example, if the ratio Rhkl/Rcrit is equal to 0.9, the given face, in rehkl sponse to the change of growth conditions, can easily exceed unity and decrease. In a case of a small face, it can even disappear. However, it can reappear for the value of the ratio Rhkl/Rcrit hkl just below unity. Therefore, the transition from increasing to decreasing (in a case of small faces, from appearing to disappearing) is the most probable growth behavior. In the case of small faces, such a growth behavior may lead to formation of wide sector boundaries7 (WSB). WSB are a result of disappearance and reappearance of faces with small growth sectors, and they are formed when neighboring boundaries overlap and the small sectors between them are not distinguishable. This phenomenon is analyzed in detail in ref 7. Besides these simple conclusions, it is interesting to know for which relative growth rates Rhkl/Rh1k1l1 and Rhkl/Rh2k2l2 and for which interfacial angles R and γ the ratio Rhkl/Rcrit hkl is smaller than, equal to, and greater than unity. Therefore, to analyze the correlation between existence and morphological stability of crystal faces, their relative growth rates, and crystal geometry represented by interfacial angles, we apply eq 4 for the potassium bichromate (K2Cr2O7sKBC) crystal chosen as an exemplary crystal. Computer Experiment. The KBC crystal belongs to the 1h point group, and the unit cell parameters are as follows8: a ) 7.445 Å, b ) 7.376 Å, c ) 13.367 Å, R ) 97.96°, β ) 96.21°, and γ ) 90.75°. KBC crystals are of particular interest because of two facts. First, this crystal possesses a double-layered structure9 parallel to {001}. Second, it is of interest because of its hypomorphism,9,10 first formulated by Shubnikov.11 Hypomorphism is related to the fact that the crystal demonstrates lower symmetry than it results from its point group. Let us consider the cross-section of KBC crystal shown in Figure 2, obtained by computer simulation using the program SHAPE (version 6.0 professional;12 the basic concepts of this software were published in ref 13). This

Crystal Faces Existence and Morphological Stability Table 1 a consecutive growth bands (GB) or induced striations (hkl) (IS) Rhkl/Rh1k1l1

Rhkl/Rh2k2l2

1

2

3

4

5

6

(001)

GB 0-1 1-2 2-3 3-4 4-5 5-6 6-7 7-8 8-9 9-10 10-11 11-12 12-13 13-14

R(001)/R(011) 0.52 0.65 0.50 1.14 1.14 1.20 1.25 1.05 1.25 1.25 2.00 2.25 0.89 0.98

R(001)/R(01h 1) 0.46 0.72 0.50 1.14 1.14 1.20 1.25 1.05 1.25 1.25 2.00 2.25 1.14 1.77

crit R(001)/R(001) 0.24 0.33 0.24 0.55 0.55 0.58 0.61 0.51 0.61 0.61 0.97 0.97 0.48 0.60

dl(001)/dt 4.63 2.93 3.15 2.05 2.05 1.92 1.81 2.26 1.81 1.81 0.13 -0.47 1.92 1.30

(010)

GB 0-1 1-2 2-3 3-4 4-5 5-6 6-7 7-8 8-9 9-10 10-11 11-12 12-13 13-14

R(010)/R(011h ) 1.00 1.14 1.14 1.20 1.25 1.05 1.25 1.00 1.00 1.40 1.05 1.05 0.89 0.98

R(010)/R(011) 0.72 1.14 1.14 1.20 1.25 1.05 1.25 1.00 1.00 1.40 1.05 1.05 1.14 1.77

crit R(010)/R(010) 0.73 1.00 1.00 1.05 1.10 0.92 1.10 0.88 0.88 1.23 0.92 0.92 0.88 1.12

dl(010)/dt 2.47 0.00 0.00 -0.42 -0.79 0.67 -0.79 1.03 1.03 -1.88 0.67 0.67 0.99 -1.28

(01 h2 h)

GB 0-1 1-2 2-3 3-4 4-5 5-6 6-7 7-8 8-9 9-10 10-11 11-12 12-13 13-14

R(01h 2h )/R(01h 1h ) 0.84 0.89 0.89 0.98 0.98 0.98 0.89 0.89 0.89 0.98 0.98 0.98 1.05 1.00

R(01h 2h )/R(001h ) 1.14 1.45 1.14 2.21 2.21 2.21 1.14 1.14 1.14 1.77 1.77 1.77 1.05 1.00

crit R(01h 2h )/R(01 h 2h ) 0.83 0.91 0.86 1.08 1.08 1.08 0.86 0.86 0.86 1.04 1.04 1.04 0.95 0.90

dl(01h 2h )/dt 1.52 0.69 1.17 -0.58 -0.58 -0.58 1.17 1.17 1.17 -0.26 -0.26 -0.26 0.52 0.99

(021 h)

IS 1-2 2-3 3-4 4-5 5-6 6-7 7-8 8-9 9-10

R(021h )/R(011h ) 1.02 1.12 0.97 1.25 0.98 1.10 1.07 0.95 1.13

R(021h )/R(010) 0.99 0.97 1.01 1.08 0.93 0.89 1.07 1.03 1.01

crit R(021h )/R(021 h) 0.97 1.01 0.96 1.12 0.92 0.95 1.03 0.95 1.03

dl(021h )/dt 2.55 -0.32 3.70 -10.05 5.45 3.57 -2.07 3.13 -1.75

Rhkl/Rcrit hkl

dlhkl/dt

a The theoretically assumed relative growth rates R /R hkl h1k1l1 and Rhkl/Rh2k2l2 taken for modeling computations to obtain the crosssection shown in Figure 2. In the case of the (021h ) face shown in Figure 4, these relative growth rates are measured as the distances between appropriate induced striations. The ratio Rhkl/Rcrit hkl and the rate of changes dlhkl/dt in the size lhkl of the hkl face are calculated based on eqs 4 and 2, respectively, for different faces of the KBC crystal shown in Figures 2 and 4.

cross-section is obtained for theoretically assumed relative growth rates, which are presented in Table 1. In

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this cross-section three faces, namely, (001), (010), and (01 h2 h ), are of our interest. First, let us concentrate on the (001) face. The neighboring faces to this face are the (011) and (01 h 1) faces, and the interfacial angles are as follows: R ) 54.90° (angle between normals to the pairs of faces: (001) and (011)) and γ ) 67.29° (angle between normals to the pairs of faces: (001) and (01 h 1)). The (001) face is one of the biggest faces in this crystal. Let us analyze its evolution on the basis of Figure 2. Up to the m3 growth band, the relative growth rates Rhkl/Rh1k1l1 ) R(001)/R(011) and Rhkl/Rh2k2l2 ) R(001)/R(01h1) are smaller than 1 (the (001) face grows more slowly than the neighboring facessTable 1, columns 3 and 4), and the (001) face behaves according to our expectation (i.e., it increases its size). The rate of increasing in size (evaluated based on eq 2) is shown in Table 1, column 6. Beginning from the m4 up to m11 growth band, the (001) face grows faster than the neighboring faces (Table 1, columns 3 and 4), and despite this, it still increases in size. It increases until it grows more than twice as fast as the neighboring faces (between the m11 and m12 growth bands). The rate of decreasing of this face dl(001)/dt is then equal to -0.47 (Table 1, column 6). When this face grows more slowly than the (011) face, and at the same time, faster than the (01h 1) face, it increases in size (m13 and m14 growth bands). From the above analysis, it follows that the relative growth rate of the (001) face may change in wide range, but it does not influence, to a great degree, the size of the (001) face. This face still remains one of the biggest and the most stable faces of this crystal. Further, let us analyze the behavior of the (010) faces Figure 2 (R ) 30.80° and γ ) 27.01°sinterfacial angles between normals to the respective pairs of faces: (010), (011 h ) and (010), (011)). First (up to the m1 growth band), this face grows more slowly than the (011) face, and at the same time, equally fast as the (011 h ) face (Table 1, columns 3 and 4), and it increases. The rate of increasing dl(010)/dt equals 2.47 (Table 1, column 6), and the considered face is relatively big. Then, between the m1 and m3 growth bands, the (010) face grows with relative growth rates Rhkl/Rh1k1l1 ) R(010)/R(011h ) ) Rhkl/Rh2k2l2 ) R(010)/R(011) ) 1.14, and it preserves its size (dl(010)/dt ) 0; cf. Table 1, column 6). It is worth noticing that the (010) face, between the m1 and m3 growth bands, grows with the same growth rate in relation to its neighboring faces (cf. values of relative growth rates Rhkl/Rh1k1l1 and Rhkl/Rh2k2l2 in Table 1) as the (001) face, between the m3 and m5 growth bands. However, the behavior of these two faces is different. The (001) face increases its size, while the (010) face preserves the size. Further, the (010) face grows faster (between the m3 to m7 growth bands) or equally fast (between the m7 to m9 growth bands) as the neighboring faces. For relative growth rates Rhkl/Rh1k1l1 ) Rhkl/Rh2k2l2 ) 1.20, the (010) face decreases its size (dl(010)/dt ) -0.42), but for these relative growth rates equal to 1.05, the (010) face increases its size (dl(010)/dt ) 0.67). As compared to the (001) face, we find that, for the same relative growth rates (1.05), the (001) face increases also, but the rate of increasing dl(001)/dt equals 2.26 (Table 1, column 6), hence it is much higher. It is worth noticing also that for relative growth rates Rhkl/Rh1k1l1 ) 0.98 and

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Rhkl/Rh2k2l2 ) 1.77, the (001) face increases in size (dl(001)/dt ) 1.30), while the (010) face decreases (dl(010)/dt ) -1.28). However, the absolute value of the rate of size changes is almost the same. As a result of such changes in relative growth rates, the (010) face either increases or decreases. Decreasing leads sometimes to its disappearance. As it was mentioned earlier, disappearance and reappearance of a growth sector of a given face may lead to the formation of wide sector boundaries (WSB). The formation of WSB in the KBC crystal is a common phenomenon described in ref 7. Finally, let us focus on the (01 h2 h ) facesFigure 2 (R ) 16.54° and γ ) 38.36°sinterfacial angles between normals to the respective pairs of faces: (01h 2 h ), (01 h1 h ) and (01 h2 h ), (001 h )). This face also behaves in different ways depending on appropriate relative growth rates. Growing more slowly than the (01 h1 h) face (Rhkl/Rh1k1l1 ) R(01h2h)/R(01h1h) ) 0.89; Table 1, column 3), and at the same time, faster than the (001h ) face (Rhkl/Rh2k2l2 ) R(01h 2h )/R(001h ) ) 1.14; Table 1, column 4), the (01 h2 h ) face increases the size (dl(01h 2h )/dt ) 1.17; Table 1, column 6)sbetween the m2 and m3 growth bands (Figure 2). Similarly, for such relative growth rates, the (001) and (010) faces behave in the same waysthey increase. For relative growth rates Rhkl/Rh1k1l1 ) 0.98 and Rhkl/Rh2k2l2 ) 1.77, the (01 h2 h ) face decreases as the (010) face and contrary to the (001) face that, for such relative growth rates, increases. For relative growth rates Rhkl/Rh1k1l1 ) Rhkl/Rh2k2l2 ) 1.05, all these three faces, (001), (010), and (01 h2 h ), increase. They differ from each other by the rate of increasing only. From the above considerations, it results that for the same relative growth rates Rhkl/Rh1k1l1 and Rhkl/Rh2k2l2, the (001), (010), and (01h 2 h ) faces behave differently. Some of them increase while the others decrease. Therefore, the question arises: on what does such a behavior of faces depend? It seems to be the key role, in this phenomenon, that the geometry of these faces plays. The geometry is characterized by the interfacial angles R and γ. Let us consider the dependence of the ratio Rhkl/Rcrit hkl on the relative growth rates Rhkl/Rh1k1l1 and Rhkl/Rh2k2l2 for the (001), (010), and (01h 2 h ) faces presented in Figure 3Assurfaces 1, 2, and 3, respectively. The surface 1, crit given by the ratio Rhkl/Rcrit hkl ) R(001)/R(001), is quite flat crit (in relation to the Rhkl/Rhkl ) 1 plane) and intersects crit the R(001)/R(001) ) 1 plane for the relative growth rates Rhkl/Rh1k1l1 ) R(001)/R(011) and Rhkl/Rh2k2l2 ) R(001)/R(01h 1) much greater than 1. It is easier to study this case considering Figure 3B, which presents the curves being intersection lines of the surfaces presented in Figure 3A with the Rhkl/Rcrit hkl ) 1 plane. For the relative growth rates Rhkl/Rh1k1l1 and Rhkl/Rh2k2l2 lying below a given curve, the ratio Rhkl/Rcrit hkl is smaller than unity. Therefore, for such relative growth rates, the hkl face increases its size. For the relative growth rates Rhkl/Rh1k1l1 and Rhkl/Rh2k2l2 lying exactly on a given curve, the size of the hkl face is preserved (Rhkl/Rcrit hkl ) 1). For relative growth rates Rhkl/Rh1k1l1 and Rhkl/Rh2k2l2 lying above a given curve, the ratio Rhkl/Rcrit hkl is greater than unity, and the hkl face decreases. Figure 3A,B shows that the region of the relative growth rates Rhkl/Rh1k1l1 ) R(001)/R(011) and Rhkl/Rh2k2l2 ) R(001)/R(01h 1) below the crit R(001)/R(001) ) 1 plane (below the curve 1), for which the (001) face increases, is very wide. We can find also that

Prywer

Figure 3. (A) Dependence of the ratio Rhkl/Rcrit hkl on the relative growth rates Rhkl/Rh1k1l1 and Rhkl/Rh2k2l2 (eq 4); 3-D graph presenting the surfaces defined by the ratio Rhkl/Rcrit hkl for different faces of the considered KBC crystal. (B) 2-D graph presenting the curves being intersection lines of the surfaces presented in Figure 3A with the Rhkl/Rcrit hkl ) 1 plane. Surface 1/curve 1sthe (001) face; surface 2/curve 2sthe (010) face; surface 3/curve 3sthe (01 h2 h ) face; and surface 4/curve 4sthe (021 h ) face.

the (001) face may increase its size growing much faster than the neighboring faces (i.e., for R(001)/R(011) ) 4.0 and R(001)/R(01h1) ) 1.2, the (001) face still increases). For faces such as the (001) face, even quite big changes of growth conditions, which can induce large changes in relative growth rates, do not cause the changes in its behavior. Therefore, we conclude that faces such as the (001) face belong to morphologically stable facessthe existence of such faces in crystal morphology is not menaced. From the dependences of the ratio Rhkl/Rcrit hkl for the (010) facessurface 2 and for the (001) facessurface 1, we find that for the (010) face the region of the relative growth rates for which this face increases (below the Rhkl/Rcrit hkl ) 1 plane) is much smaller. Additionally, Figure 3 shows that for this face, decreasing in size at a growth rate smaller than the growth rates of neighboring faces is possible (for values of the relative growth rate Rhkl/Rh1k1l1 ) R(010)/R(011h ) smaller than unity, the crit value of the ratio Rhkl/Rcrit hkl ) R(010)/R(010) may be greater than unity). Similarly, for the (01 h2 h ) face (surface 3), the region of the relative growth rates Rhkl/Rh1k1l1 ) R(01h 2h )/R(01h 1h ) and Rhkl/Rh2k2l2 ) R(01h 2h )/R(001h ), for which this face increases, is not wide. Also, for this

Crystal Faces Existence and Morphological Stability

face, decreasing the size at a growth rate smaller than the growth rate of one of the neighboring faces is possible. For faces such as the (010) and (01h 2 h ) faces, the relative growth rates for which the considered face increases is much smaller than for the (001) face. Moreover, the surfaces given by the ratio Rhkl/Rcrit hkl for the (010) and (01 h2 h ) faces are more inclined surfaces (in relation to the Rhkl/Rcrit hkl ) 1 plane). Therefore, for such faces, quite small fluctuations in growth conditions could cause such a large change in relative growth rates Rhkl/Rh1k1l1 and Rhkl/Rh2k2l2 that this face changes its growth behavior; for example, it starts to decrease while it has increased before. From the above analysis, we are able to draw a conclusion that from these three (001), (010), and (01h 2 h ) faces of the KBC crystal, the (001) face is the most stable. The geometry of the (010) and (01 h2 h) faces causes these faces to be much less stable. The geometry and the growth behavior of the (01h 2 h ) face presented in this paper in connection with the growth mechanism of the (001h ) face presented in ref 10 may be helpful in the explanation of the structure of coarse crystal surfaces. It should be pointed out that all relative growth rates (Table 1) are assumed theoretically for modeling computations to obtain the cross-section of the KBC crystal shown in Figure 2. The above considerations reveal that, generally, stable faces are those for which the surface given by the ratio Rhkl/Rcrit hkl is not inclined very much in relation to the Rhkl/Rcrit hkl ) 1 plane. The more flat the surface given by the ratio Rhkl/Rcrit hkl , the more stable a corresponding crystal face. The closer R + γ is to π, the more flat the surface given by the ratio Rhkl/Rcrit hkl . Moreover, for such angles, the surface given by the ratio Rhkl/Rcrit hkl intersects the Rhkl/Rcrit hkl ) 1 plane for very large values of the relative growth rates Rhkl/Rh1k1l1 and Rhkl/Rh2k2l2. This means that the ratio Rhkl/Rcrit hkl takes values smaller than unity for a very wide range of these relative growth rates. As a result, the given face can increase in size for a very wide range of the relative growth rate, even growing faster than the neighboring faces. Consequently, the faces with the interfacial angles R + γ close to π are the most stable faces. The smaller the value of the sum R + γ, the more inclined is the surface given by the ratio Rhkl/Rcrit hkl . Moreover, the surface given by the ratio Rhkl/Rcrit hkl intersects the Rhkl/Rcrit hkl ) 1 plane for small values of the relative growth rates Rhkl/Rh1k1l1 and Rhkl/Rh2k2l2, even smaller than unity. As a result, the given face may decrease its size growing more slowly than one of the neighboring faces. Therefore, such faces are considered to be the least stable crystal faces. In this way, the crystallographic structure of the crystal determines the existence or nonexistence of a given face in crystal morphology. Experiment. Figure 4 illustrates the cross-section of the KBC crystal grown from solution by the method of decreasing temperature. The growth bands seen in this figure were induced artificially by changing the hydrodynamic conditions in equal and known time intervals. The growth bands were induced artificially; therefore, they are called induced striations. These induced striations were revealed by etching the cleavage plane in HCOOH. To improve the legibility of this photograph,

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Figure 4. Cross-section of the KBC crystal grown from solution illustrating induced striations on growth sectors of various faces, in particular on the growth sector of the (021 h) face. Plane of observation (1h 00). The numeration n1 to n10 denotes the consecutive growth bands.

the lines indicated as the consecutive induced striations at the growth sectors of the (010) and (021h ) faces are added. Figure 4 presents growth sectors of three faces, namely, the (010), (021 h ), and (011 h ) faces. The (021 h) face appears in KBC morphology very rarely. It forms the following interfacial angles: R ) 14.84° (angle between normals to the (021 h ) and (011 h ) faces) and γ ) 15.96° (angle between normals to the (021 h ) and (010) faces). Both these angles are not big. The surcrit face given by the ratio Rhkl/Rcrit h )/R(021h ) is very hkl ) R(021 inclinedssurface 4 in Figure 3A (curve 4 in Figure 3B). Therefore, quite small changes in the relative growth rates Rhkl/Rh1k1l1 ) R(021h)/R(011h) and Rhkl/Rh2k2l2 ) R(021h)/R(010) can cause the change in the value of the ratio Rhkl/Rcrit hkl crit ) R(021h )/R(021 h ) below or above unity, and consequently, the (021 h ) face can change its way of behavior. Table 1 shows that, in the case illustrated in Figure 4, the ratio crit Rhkl/Rcrit h )/R(021h ) fluctuates around unity, and in hkl ) R(021 response, the (021h ) face once increases, once decreases. For example, between the n4 and n5 induced striations, crit the ratio Rhkl/Rcrit h )/R(021h ) equals 1.12, and the hkl ) R(021 (021 h ) face decreases while, for the further induced striations (between the n5 and n6), the (021h ) face increases. Therefore, we conclude that the probability of existence of this face in the crystal habit is not very high, and we include it to unstable faces. Conclusions Generally, the smaller the value of the ratio Rhkl/Rcrit hkl , the larger the corresponding crystal face. The ratio Rhkl/Rcrit hkl takes the smallest values for faces with interfacial angles R + γ close to π. Such faces can increase in size for a very wide range of values of relative growth rates. The smallest values of the ratio Rhkl/Rcrit hkl ensure preservation of a given face even at high growth rates. It can increase in size, even growing faster than the neighboring faces. Moreover, in the case of such faces, even big changes in relative growth rates do not cause the change in the way of behavior of a given face. Therefore, such faces are those with the greatest likelihood of existence in crystal morphology, and they belong to stable faces.

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The smaller the value of the sum R + γ, the higher the value of the ratio Rhkl/Rcrit hkl , and consequently, the smaller the corresponding crystal face. Additionally, such faces can decrease in size, growing more slowly than one of the neighboring faces. If the growth time is long enough, they disappear, and they are not represented in the final crystal morphology. In the case of such faces, even small fluctuation in growth conditions can cause big changes in relative growth rates, which may lead to a fast disappearance of a given face from crystal morphology. Therefore, such faces are considered to be unstable faces. All these conclusions are drawn based on the crystallographic structure of the crystal, which is characterized by interfacial angles. References (1) Sunagawa, I. In Morphology of Crystals; Sunagawa I., Ed.; Terra Scientific: Tokyo, 1987; Part A, pp ix, 361.

Prywer (2) Roberts, K. J. In Science and Technology of Crystal Growth; van der Eerden, J. P., Bruinsma, O. S. L., Eds.; Kluwer Academic Publishers: Dordrecht, 1995; pp 367-382. (3) Kozlovskii, M. I. Kristallografiya 1957, 2, 760-769. (4) Prywer, J. J. Cryst. Growth 1999, 197, 271-285. (5) Szurgot, M.; Prywer, J. Cryst. Res. Technol. 1991, 26, 147153. (6) Prywer, J. J. Phys. Chem. Solids 2002, 63, 491-499. (7) Szurgot, M. Cryst. Res. Technol. 1993, 28, 511-518. (8) Brandon, J. K.; Brown, I. D. Can. J. Chem. 1968, 46, 933941. (9) Plomp, M.; van Enckevort, W. J. P.; Vlieg, E. J. Cryst. Growth 2000, 216, 413-427. (10) Plomp, M.; Nijdam, A. J.; van Enckevort, W. J. P. J. Cryst. Growth 1998, 193, 389-401. (11) Shubnikov, A. Z. Kristallogr. 1912, 50, 19-23. (12) Dowty, E. SHAPE v. 6.0 professional; Shape Software, Kingsport, TN, 2000; http://www.shapesoftware.com. (13) Dowty, E. Am. Mineral. 1980, 65, 465-471.

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