Effect of Seed Faces on Time-Dependent Crystal Morphology Jolanta Prywer* Institute of Physics, Technical University of Ło´ dz´ , Wo´ lczan´ ska 219, 93-005 Ło´ dz´ , Poland
CRYSTAL GROWTH & DESIGN 2004 VOL. 4, NO. 6 1365-1369
Received February 11, 2004
ABSTRACT: Crystal morphology is an important characteristic of any crystal, which depends on many internal and external factors. Some of them are size, shape, and orientation of the seed. The aim of this paper is to analyze the influence of the seed on the evolution of individual faces and, consequently, on the crystal morphology in a given growth time. The results show that seed faces, even if they disappear in the first stage of growth, influence the evolution of the directly neighboring faces. In particular, some of the seed faces, depending on the value of the interfacial angle, cause a very intense increase in the size of the neighboring faces. Some other seed faces, for other interfacial angles, cause a decrease of the size of the directly neighboring faces. Such changes in the rate of increase of various faces influence the evolution of these faces and modify the time-dependent crystal morphology. 1. Introduction Crystals, during their growth, take a variety of habits due to differences in the relative growth rates of the faces of which the crystal is composed.1 The changes in the relative growth rates are caused by the changes in the growth environment, for example, cooling rate, supersaturation, or temperature.2 On the other hand, crystal morphologies depend on size, shape, and orientation of the seed crystal.3 The orientation of the seed is particularly important in the case of seeded solution growth.4 The techniques commonly used are submergedseeded solution growth (SSSG; for example, refs 5 and 6) and top-seeded solution growth (TSSG; for example, refs 7 and 8). Research on the effect of seed orientation has been carried out in the case of both of these techniques;6,8 however, this is usually experimental research only. Independently of these techniques, the effect of the shape of the seed of a crystal has been studied from a theoretical point of view and compared with experimental studies.9 It is known that the crystal morphology is independent of the shape of the seed for infinite growth time. However, the aim of this paper is to analyze the effect of the shape of the seed on the time-dependent morphology of crystals. The shape of the seed is determined by the faces of which the seed is composed; therefore, the analysis is focused on the seed faces that influence the growth habit. The analysis is theoretical, and it is in connection with the crystallographic structure of the crystal, which is characterized by interfacial angles. 2. Predictions of the Theoretical Model The evolution of faces and the influence of the seed on this evolution are difficult to research on the basis of external crystal morphology. The details of growth behavior of crystal faces are best presented by internal morphology seen as growth bands in respective growth sectors and growth sector boundaries. For crystals whose growth process cannot be observed in situ, e.g., natural crystals or crystals synthesized by high tem* To whom correspondence should be addressed. E-mail: jolantap@ p.lodz.pl.
Figure 1. Cross-section of a hypothetical crystal illustrating the considered hkl face represented by the length lhkl and its neighboring h1k1l1 and h2k2l2 faces; Rhkl, Rh1k1l1, and Rh2k2l2 are the normal and constant growth rates of the respective faces; R and γ are interfacial angles; and l0hkl is the seed size of the hkl face. The seed size of the h′k′l′ face does not change (dlh′k′l′/dt ) 0), and the seed size of the h′′k′′l′′ face decreases (dlh′′k′′l′′/dt < 0). For the other faces, the rate dlhkl/dt is positive, and therefore, they increase.
perature solution or hydrothermal solution methods, the information on the growth behavior of crystal faces has to be inferred from such an internal morphology seen in a cross-section surface.1,2 The influence of the seed on the evolution of individual faces and, consequently, on the time-dependent crystal morphology is also best to research considering the cross-sections of a given crystal. Figure 1 presents a cross-section of a hypothetical crystal and shows that the hkl face in the crosssection is represented by the length lhkl. The crosssection size lhkl of the hkl face was derived in ref 10 and is given by
lhkl )
Rh1k1l1 sin γ + Rh2k2l2 sin R - Rhkl sin (R + γ) t+ sin R sin γ l0hkl (1)
where Rhkl, Rh1k1l1, and Rh2k2l2 are the normal growth rates of the hkl, h1k1l1, and h2k2l2 faces, respectively; R and γ are the appropriate interfacial angles illustrated in Figure 1; l0hkl is the initial size of the hkl face; and the growth time t corresponds to the change of the size
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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, as marked in Figure 1, or after some time of growth (after growth of the crystal layer). Generally, 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. However, in this paper, to analyze the influence of the seed on the growth morphology and to eliminate the influence of the other factors, we assume that the growth rates are constant during the whole growth process and we assume that the size l0hkl is the seed size of the hkl face. Consequently, the growth time t between growth bands (crystal layers) is the same. The instantaneous rate of changes in the cross-section size of a given hkl face, denoted by dlhkl/dt is given by11
dlhkl Rh1k1l1 sin γ + Rh2k2l2 sin R - Rhkl sin (R + γ) ) dt sin R sin γ (2) The physical meaning of the rate dlhkl/dt is schematically illustrated in Figure 1. The positive value of the rate dlhkl/dt corresponds to an increase in the size of the considered face. If the rate dlhkl/dt is equal to zero, the corresponding crystal face does not change. The negative value of the rate dlhkl/dt means that the given face decreases. Equation 1 shows that the final size lhkl of a given hkl face depends on the growth rate Rhkl of this face and the growth rates Rh1k1l1 and Rh2k2l2 of the directly neighboring h1k1l1 and h2k2l2 faces. It depends also on the seed size l0hkl of the hkl face, the growth time t, and the interfacial angles R and γ. This means that the final size lhkl depends on the face, which is the neighboring face to the hkl face. When the neighboring face changes, then the appropriate interfacial angle changes also, and according to eq 1, this influences the size lhkl. From this it results that during the growth, the crystal morphology changes depending on the faces of which the seed is composed. Furthermore, eq 2 shows that the rate dlhkl/dt also depends on the growth rates Rhkl, Rh1k1l1, and Rh2k2l2 and the interfacial angles R and γ. Consequently, the rate dlhkl/dt depends also on the faces of which the seed is composed. Our considerations concern a given growth time t and, hence, the so-called timedependent crystal morphology. For infinite growth time, the sizes lhkl of the faces proceed to the same value. 3. Verification of the Theoretical Model by Computer Experiment To verify the theoretical predictions, we consider potassium dichromate (KBC, K2Cr2O7) crystal chosen as a modeling crystal. The growth behavior of various faces, depending on the shape of seed, is analyzed using computer simulation made with the aid of the computer program SHAPE (version 6.0 professional12). SHAPE is a crystal drawing program whose basic concepts were published in ref 13. In the present work, this program is used as a graphic tool to illustrate the cross-sections of the KBC crystal, which is under consideration. The KBC crystal belongs to the 1 h point group, and the unit cell parameters are as follows:14 a ) 7.445 Å, b ) 7.376
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Å, c ) 13.367 Å, R ) 97.96°, β ) 96.21°, and γ ) 90.75°. The KBC crystals are of particular interest because of two facts. First, this crystal possesses a double-layered structure15 parallel to {001}. Second, this crystal is of interest because of its hypomorphism,16 first formulated by Shubnikov.17 Hypomorphism is related to the fact that the crystal demonstrates a lower symmetry than it results from its point group. Figure 2 illustrates the cross-sections of the KBC crystal. The cross-sections shown in Figure 2a,b (c,d) are obtained for the same constant relative growth rates. They differ from each other by the shape of the seed only. These relative growth rates are taken for modeling computations, and they are presented in Table 1. To perform an analysis of the seed faces influence on the time-dependent morphology and to eliminate the other factors, the growth rates of the individual faces are constant in the coarse of the growth. The growth time from seed up to the end of growth is 10 time units. The time distance between two adjacent crystal layers is considered as one time unit. Table 1 also presents the relative size lhkl/l(001h ) of a given hkl face, the morphological importance (MI) of faces estimated according to the relative size lhkl/l(001h ), and the rate dlhkl/dt. In ref 18, the time-dependent morphologies resulting from the growth on a seed of a given shape and by homogeneous nucleation were compared. Therefore, we do not consider this case in the present work. First, let us consider the cross-section shown in Figure 2a. At the first stage of growth, the (01 h2 h ) face neighbors to the (01h 1 h ) face. Then, the (01h 2 h ) face increases while the (01h 1 h ) face decreases and finally disappears. As a result, the (01h 2 h ) face starts to neighbor to the (01h 0) face. The change of the neighboring face from the (01h 1 h ) face to the (01 h 0) face causes the (01 h2 h ) face to start to decrease (dl(01h 2h )/dt ) - 0.26, Table 1). Let us analyze this situation in detail considering the dependence of the rate dlhkl/dt for the (01 h2 h ) face, illustrated in Figure 3a. Surface 1 presents the rate dlhkl/dt ) dl(01h 2h )/dt for the first stage of growth, when the (01 h2 h ) face neighbors to the (01 h1 h ) face (Figure 2a). The interfacial angles are then equal to R ) 16.54° and γ ) 38.36° [angles between normals to the pairs of faces: (01h 2 h ), (01h 1 h ) and (01h 2 h ), (001 h ), respectively]. Surface 2 presents the rate dl(01h 2h )/dt after the disappearance of the (01 h1 h ) face, when the (01 h2 h) face neighbors to the (01 h 0) face. Then, the interfacial angle R changes its value to 43.55° [angle between normals to the pair of faces: (01 h2 h ), (01 h 0)]. Figure 3b presents the same dependence of the rate dl(01h 2h )/dt but in a two-dimensional (2D) graph. The curves 1 and 2 are the intersection lines of the surfaces 1 and 2 presented in Figure 3a with the dl(01h 2h )/dt ) 0 plane. Below a given curve, the rate dl(01h 2h )/dt takes negative values and the (01 h2 h ) face decreases its initial size. For the interfacial angles R and γ lying exactly on the given curve, the (01 h2 h ) face does not change (dl(01h 2h )/dt ) 0). Above the given curve, the rate dl(01h 2h )/dt takes positive values, and consequently, the (01h 2 h ) face increases. The 2D graph presented in Figure 3b shows that when the (01 h2 h ) face neighbors to the (01 h1 h ) face, the rate dl(01h 2h )/dt takes values above the curve 1; therefore, the (01 h2 h ) face increases. However, when the (01 h2 h ) face neighbors to the (01 h 0) face (curve 2), then the rate dl(01h 2h )/dt takes
Effect of Seed Faces on Crystal Morphology
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Figure 2. Cross-sections of the KBC crystals illustrating the influence of the seed shape on the evolution of individual faces for 10 time units. Cross-sections shown in panels a and b (panels c and d) are obtained for the same constant growth rates, which are presented in Table 1. The numerals 1-8 near by the Miller indexes denote the MI of a given face. The plane of observation is (1 h 00). Table 1. Relative Size (lhkl/l(001h )) of Individual Faces, MI of Individual Faces Estimated Based on the Relative Size (lhkl/l(001h )), and the Rate (dlhkl/dt) for Individual Facesa Figure 2a hkl face
(lhkl/ l(001h ))
MI
(dlhkl/dt)
(001) (01 h 1) (01 h 0)
0.94 0.30 0.42
2 5 3
(01 h1 h)
0.00
-
(01 h2 h)
0.04
8
(001 h)
1.00
1
(011 h) (010) (011)
0.31 0.27 0.19
4 6 7
Figure 2b (lhkl/ l(001h ))
MI
2.84 0.92 1.59 (01h 1 h) 1.32 (01 h2 h) -0.65
0.91 0.30 0.43
2 5 3
0.17 (01h 1 h) -0.26 (01 h 0) 3.22
0.00
8
1.00 0.31 0.27 0.19
0.80 1.03 0.30
Figure 2a,b
Figure 2c
Rhkl((Rhkl/ R(001h )))
(lhkl/ l(001h ))
MI
1.00 1.40 1.42
1.03 0.28 0.26
1 4 5
1.83
0.15
7
-0.26
1.70
0.00
1
3.22
1.00
1.00
2
4 6 7
0.80 1.03 0.30
2.00 2.00 2.00
0.31 0.24 0.14
3 6 8
(dlhkl/dt) 2.84 0.92 1.32
Figure 2d
Figure 2c,d
(dlhkl/dt)
(lhkl/ l(001h ))
MI
(dlhkl/dt)
Rhkl((Rhkl/ R(001h )))
3.56 0.80 1.03
0.96 0.33 0.24
2 3 5
3.56 0.80 1.03
1.00 2.00 2.00
0.77 (01h 2 h) 0.30 (001 h) -0.62
0.07
8
0.30
2.00
3.71 (01h 2 h) 3.49 (01 h1 h) 0.89 0.97 0.26
1.00
1
3.49
1.00
0.29 0.23 0.13
4 6 7
0.89 0.97 0.26
2.00 2.05 2.05
2.00
a All of these parameters are related to the cross-sections of the KBC crystals presented in Figure 2. This table also shows the growth rates Rhkl, taken for the computer experiment, for which these cross-sections are obtained.
values below the curve 2. This means that in such a case, the (01 h2 h ) face decreases. The disappearance of the (01h 2 h ) face also influences the evolution of the (01h 0) face. First, when the (01h 0) face neighbors to the (01 h1 h ) face, the rate dl(01h 0)/dt is equal to 1.59 and the (01h 0) face increases quite intensively (wide growth sector). When the neighboring face changes to the (01 h2 h ) face, the increase in the size of the (01 h 0) face is less intensive. The seed shown in Figure 2b differs from the previous one, shown in Figure 2a, by one face only, namely, the (01 h1 h ) face. This face is absent in the seed shown in Figure 2b, and it does not show up during the growth process, although the growth rates of individual faces are the same as in the case shown in Figure 2a. Therefore, the (01 h2 h ) face directly neighbors to the (01 h 0) face and it decreases from the beginning of the growth. Despite the changes in the rate dlhkl/dt, which are caused by the seed faces, the MI of the faces in the final morphology does not change.
Furthermore, let us analyze the cross-sections presented in Figure 2c,d. The seeds presented in Figure 2c,d differ from each other by one face only, namely, the (01 h2 h ) face. This face exists in the seed in Figure 2c, and it is absent in the seed shown in Figure 2d. The relative growth rates of all faces, beginning from the seed till the end of the growth process, are the same (Table 1) for these two cross-sections. Let us concentrate on Figure 2c. Here, the (01 h2 h ) face decreases and, consequently, disappears causing the (01 h1 h ) face to start to neighbor the (001h ) face. The existence of the (01h 2 h) face in the seed and in the first stage of growth causes the (01 h1 h ) face to increase more intensively (dl(01h 1h )/dt ) 0.77, Table 1). This is seen in Figure 2c as a wide growth sector of this face. The disappearance of the (01 h2 h ) face causes the change of the interfacial angle R from the value of 16.54° to the value of 54.90°. In consequence, the (01h 1 h ) face increases in size more slowly (dl(01h 1h )/dt ) 0.30, Table 1) and its growth sector becomes narrower. The disappearance of the (01 h2 h ) face also influences the
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Figure 4. Cross-sections of KBC crystals illustrating the changes in evolution of individual faces depending on the shape of seed and growth time. Solid line, the morphology resulting from the growth of the crystal shown in Figure 2c; dashed line, the morphology resulting from the growth of the crystal shown in Figure 2d. (a) For 50 time units; (b) for 100 time units. The numerals from 1 to 8 near by the Miller indexes denote the MI of a given face in the final morphology. The plane of observation is (1 h 00). Figure 3. Dependence of the rate dl(01h 2h )/dt (eq 2) for the crosssection of the KBC crystal shown in Figure 2a. (a) Threedimensional graph: surface 1, the rate dl(01h 2h )/dt at the first stage of growth, when the (01h 2 h ) face neighbors the (01h 1 h ) face; surface 2, the rate dl(01h 2h )/dt, when the (01h 2 h ) face neighbors the (01 h 0) face. (b) Two-dimensional graph: the curves 1 and 2 are the intersection lines of the surfaces 1 and 2 presented in panel a with the dl(01h 2h )/dt ) 0 plane.
evolution of the (001 h ) face. After the disappearance of the (01 h2 h ) face, the rate dl(001h )/dt decreases (Table 1). The seed shown in Figure 2d does not possess the (01 h2 h ) face, and it does not appear during the growth process. Similarly, as in the previous case of the seed in Figure 2c, the (01 h1 h ) face shows up during the growth, although it was absent in the seed. However, its size in the final morphology is much smaller than in the case illustrated in Figure 2c. Consequently, the MI of the faces, in the final morphology (after 10 time units), changes. The (01h 1 h ) face now is the last face in this MI arrangement. Moreover, the MI of the (001) face and the (001 h ) face changes. These faces exchange places. The (001 h ) face in the seed shown in Figure 2d is much bigger than in this shown in Figure 2c. As a result, this face is the biggest face in the final morphology (Figure 2d), although its rate dl(001h )/dt is smaller, when it neighbors to the (01 h1 h ) face (Table 1). The rate of increase dl(01h 1h )/dt in size of the (01 h1 h ) face is much greater in the case when it neighbors to the (01 h2 h ) face. After the disappearance of the (01h 2 h ) face, the rate of increase of the (01 h1 h ) face is much smaller. As a result, the (01 h1 h)
face is much greater in the final crystal morphology. However, it should be remembered that our considerations are made for a given growth time; hence, this is time-dependent crystal morphology. Figure 4 illustrates the morphologies grown from seeds shown in Figure 2c,d but for growth time five and 10 times as long as this in Figure 2c,d (50 and 100 time units, respectively). Figure 4 shows that small differences in growth morphologies are observed even for very long growth times. For morphologies shown in Figure 4a, there are still differences in the MI sequence. The (01 h1 h ) face is seven for solid morphology (Figure 2c) and eight for dashed morphology (Figure 2d). This difference is due to the (01 h2 h ) seed face, which disappeared in the first stage of growth, but its existence in the seed and in the beginning of growth influences the MI of the faces even after a very long growth time. For growth morphologies shown in Figure 4b, the MI of the faces is the same but these morphologies are still not identical. The differences are particularly observed in the case of the (01h 1 h ), (001h ), and (01h 0) faces, whose growth processes were disturbed in the first stage of growth by the (01 h2 h) seed face. 4. Conclusions The analysis presented in this paper reveals that the seed faces may influence crystal morphology by modifying the evolution of directly neighboring faces. The scale
Effect of Seed Faces on Crystal Morphology
of such an influence depends, among other things, on the interfacial angles. In particular, there are seed faces with such interfacial angles, which cause a more intensive increase in the size of the neighboring face. There are also such seed faces with other interfacial angles that cause a decrease in the size of the directly neighboring face. Such changes in the rate of increase (decrease) of various faces influence the growth morphology and, consequently, the MI of the faces. If we compare the growth morphologies originating from two different seeds, the biggest differences in crystal morphologies are when the differences in the interfacial angles are big. From this, it follows that the geometry of the seed and, generally, the crystal geometry play key roles in the formation of crystal habit. The seed face’s influence on the growth morphology also depends on the growth time. For a short growth time, the influence of the seed faces is the most significant. However, seed faces may modify the crystal morphology for a very long growth time. Consequently, the changes in the MI arrangement are possible for a very long growth time also. References (1) Sunagawa, I. In Morphology of Crystals; Sunagawa, I., Ed.; Terra Scientific: Tokyo, 1987; Part A, p ix, p 361.
Crystal Growth & Design, Vol. 4, No. 6, 2004 1369 (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; p 367. (3) Iwasaki, H.; Iwasaki, F.; Yokokawa, H.; Kurashige, M.; Oba, K. J. Cryst. Growth 2002, 234, 711-720. (4) Kim, J. H.; Kang, J. K.; Chung, S. J. J. Cryst. Growth 1995, 147, 343-349. (5) Yokotani, A.; Miyamoto, A.; Sasaki, T.; Nakai, S. J. Cryst. Growth 1991, 110, 963-967. (6) Cheng, L. K.; Bierlein, J. D.; Ballman, A. A. J. Cryst. Growth 1991, 110, 697-703. (7) Cheng, G. C.; Qian, Z. Q.; Tang, G. K.; Song, W. B.; Tang, H. G. J. Cryst. Growth 1991, 112, 294-297. (8) Bolt, R. J.; van der Mooren, M. H.; de Haas, H. J. Cryst. Growth 1991, 114, 141-152. (9) Gadewar, S. B.; Hofmann, H. M.; Doherty, M. F. Cryst. Growth Des. 2004, 4, 109-112. (10) Kozlovskii, M. I. Kristallografiya 1957, 2, 760-769. (11) Prywer, J. J. Cryst. Growth 1999, 197, 271-285. (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. (14) Brandon, J. K.; Brown, I. D. Can. J. Chem. 1968, 46, 933941. (15) Plomp, M.; van Enckevort, W. J. P.; Vlieg, E. J. Cryst. Growth 2000, 216, 413-427. (16) Plomp, M.; Nijdam, A. J.; van Enckevort, W. J. P. J. Cryst. Growth 1998, 193, 389-401. (17) Shubnikov, A. Z. Kristallogr. 1911, 50, 19-23. (18) Prywer, J. Cryst. Growth Des. 2002, 2, 281-286.
CG0499360
