J. Phys. Chem. B 1997, 101, 6243-6246
6243
Exact Coincidence Site Lattice in Ice Ih B. J. Gonzalez Kriegel, C. L. Di Prinzio,* and O. B. Nasello Facultad de Matema´ tica Astronomı´a y Fı´sica, UniVersidad Nacional de Co´ rdoba, Ciudad UniVersitaria, 5000 Co´ rdoba, Argentina ReceiVed: October 11, 1996; In Final Form: April 30, 1997X
In this work coincidence site lattices for ideal ice Ih are investigated. Values of Σ, Γ, Miller indices, and atom positions are presented for all CSLs with Σ < 50 corresponding to rotations about 〈112h0〉, 〈101h0〉, and 〈0001〉 axes. The present results are compared with those reported in the literature, and the differences found are remarked on.
Introduction In the study of grain boundary structures one of the geometrical models most commonly used is the coincidence site lattice (CSL).1 This lattice is characterized by Σ, the ratio of the CSL unit cell volume to the crystal unit cell, and Γ, the lateral and diagonal planes’ surface density of the CSL unit cell. In many materials, the correlation between lower Σ and higher Γ values and special properties of the grain boundaries is frequently striking. In ice Ih the CLS concept was first applied by Kobayashi and Furukawa (1975)2 (KF1). They studied snow crystals of 12 branches and twin prisms and found that they have boundaries with low Σ. In a later work (Kobayashi et al. (1976)3) (KFKU) the concept was extended to the analysis of other polycrystalline structures such as columns, bullets, and other kinds of snow crystals. KFKU and Kobayashi and Furukawa (1978)4 (KF2), considering that, in some cases, the simple hexagonal lattice is suitable to examine the ice grain boundary periodicity, found Σ values and Miller indices for twin planes of the CSL unit cell, from crystal rotations about the a ) 〈112h0〉, b ) 〈101h0〉, and c ) 〈0001〉 axes. Hondoh (1984)5 (H), also studying CSLs of the simple hexagonal structure, gave Σ and Γ values, types of CSL unit cells, and Miller indices for misorientations about the previously defined a, b, and c axes. A thorough analysis of the cited papers shows that some of the Σ values reported by different authors disagree. Furthermore, it may be noted that the simple hexagonal structure is only an early approach to the complex structure of ice Ih. So, having considered that for the study of ice grain boundary properties the knowledge of the corresponding CSLs is useful, ice Ih CSLs were investigated in the present work.
Method An ice Ih crystal structure with the ideal axial ratio c/a ) (8/3)1/2 was used and, for all misorientations about a, b, and c axes, the CSLs with Σ < 50 were analyzed. We obtained our results from a computer program that gives Σ and Γ values, a graphical representation of the CSL cell, and the Miller indices of the CSL planes. Lengths were expressed in a units (a ) 4.5169 Å 6) and Γ values were expressed in ac units. The ice a2, and b c and lattice was developed using primitive vectors b a1, b the atomic basis shown in Figure 1. The primitive vectors of a ′2, and b a ′3, were expressed in terms of the ice each CSL, b a 1′ , b X
Abstract published in AdVance ACS Abstracts, July 1, 1997.
S1089-5647(96)03275-0 CCC: $14.00
Figure 1. Orthogonal system (x,y,z) with primitive vectors of ice Ih (a1, a2, c) and its associated basis.
Figure 2. Sketch of a CSL cell. 0,1,2,3: vertices of the cell. 01 and 02: twin planes. 03 and 12: diagonal planes. The 01 plane is always bisecting the c axes of both crystals in rotation about a and b axes, and bisecting a axes in rotations about c.
primitive vectors as b a ′i ) ia b1 + ja b2 + kc b, with (i,j,k) being integer numbers. The positions of the CSL atomic basis m b i were b′1 + ma b′2 + na b′3. It was found that l, m, expressed as m b i ) la and n are rational numbers with a common denominator equal to 24. A sketch of a CSL cell is presented in Figure 2. Results and Discussion All the results obtained for ice CSLs with Σ < 50 are presented in Tables 1 and 2. In Table 1 the following CSL © 1997 American Chemical Society
6244 J. Phys. Chem. B, Vol. 101, No. 32, 1997
Gonzalez Kriegel et al.
TABLE 1: Angle, Σ, Miller Indices, and Γ Values of CSL for Rotations about the a, b, and c axes (01, 02, 03, and 12 Represent the CSL Plane with the Highest Density Γ)
parameters are presented for a, b, and c axes: rotation angle, Σ, Γ, and a pair of Miller indices, one viewed from the original crystal (upper) and the other from the rotated one (lower). To simplify the notation, we refer to the 00′1′1 plane shown in Figure 2 only as 01 and similarly for the others. In Table 2 we
report the components (i,j,k) of the CSL primitive vectors and the coordinates (l,m,n) of the CSL atomic basis for a, b, and c axes. Comparing the results presented in Tables 1 and 2 with those reported in the literature, we can conclude the following.
Exact Coincidence Site Lattice in Ice Ih
J. Phys. Chem. B, Vol. 101, No. 32, 1997 6245
TABLE 2: Angle, CSL Cell Primitive Vectors, and CSL Cell Basis for Rotations about the a, b and c Axes CSL Primitive vectors θ (deg)
i
j
k
26.53
2 4 1
-2 -4 1
9 -1 0
38.94
1 8 1
-1 -8 1
3 -3 0
50.48
4 2 1
-4 -2 1
9 -1 0
23.07
0 0 2
-1 -8 1
3 -1 0
34.05
0 0 2
-1 -16 1
2 -3 0
44.42
0 0 2
-2 -4 1
3 -1 0
57.12
0 0 2
-8 -3 1
9 1 0
13.17
5 2 0 6 -1 0
3 5 0 7 6 0
0 0 1 0 0 1
5 -1 0
6 5 0
0 0 1
3 1 0 3 -1 0
2 3 0 4 3 0
0 0 1 0 0 1
4 1 0
3 4 0
0 0 1
15.18
17.90
21.79 27.80
32.20
CSL atomic basis [1/24] l
m
n
CSL Primitive vectors θ (deg)
j
k
l
m
n
2 4 1
-2 -4 1
3 -3 0
8 1 1
-8 -1 1
9 -1 0
0 22 2 0 0 2 22 0 12 10 14 12 12 14 10 12 0 15 12 3
0 2 2 3 4 5 5 7 12 14 14 15 16 17 17 19 0 0 12 12
0 12 12 12 0 0 0 12 0 12 12 12 0 0 0 12 0 0 12 12
0 0 2
-1 -8 1
1 -3 0
0 0 2
-4 -2 1
3 -1 0
0 0 2
-8 -5 1
5 -3 0
0 12 12 0 12 0 12 0 0 12 9 21 21 12 9 0 0 3 12 15 0 3 12 15
0 0 3 3 12 12 15 15 0 0 0 0 12 12 12 12 0 0 0 0 12 12 12 12
0 20 20 0 12 8 12 8 0 8 20 12 20 0 12 8 0 0 8 8 20 20 12 12
2 -1 0
3 2 0
0 0 1
43.57
5 -3 0
8 5 0
0 0 1
46.83
3 -2 0
5 3 0
0 0 1
50.57
4 -3 0
7 4 0
0 0 1
60.00
1 0 0
1 1 0
0 0 1
0 0 16 16 0 0 8 8 0 0 16 16 0 0 16 16 0 0
0 0 16 16 0 0 8 8 0 0 16 16 0 0 16 16 0 0
0 15 3 12 0 15 3 12 0 15 3 12 0 15 3 12 0 15
a Axis 0 70.53 0 12 12 0 0 12 12 0 0 12 12 12 12 12 0 0 86.63 0 12 0 12 0 12 12 0 b Axis 0 62.96 20 20 0 8 12 8 12 0 78.46 12 12 0 8 20 20 8 0 88.83 8 20 12 8 0 20 12 0 8 8 0 12 20 12 20 c Axis 0 38.21 15
0 20 0 20 8 12 8 12 0 0 12 12 12 12 12 0 0 0 15 4 19 16 3 7 12
0 0 9 9 12 12 21 21 0 3 8 8 11 12 15 20 23 0 0 0 0 12 12 12 12
0 12 12 0 0 12 0 12 0 12 12 0 0 12 12 0 0 12 12 0 0 12 0 12 0 3 12 15 12 15 3 0
0 0 9 9 12 12 21 21 0 0 3 2 12 12 15 15 0 0 9 9 12 12 21 21 0 0 0 0 12 12 12 12
0 0
0 0
0 0 8 8 0 0 16 16 0 0
0 0 8 8 0 0 16 16 0 0
0 15 3 12 0 15 3 12 0 15
0 0 8 8 0 0
0 0 8 8 0 0
0 15 3 12 0 15
CSL atomic basis [1/24]
i
6246 J. Phys. Chem. B, Vol. 101, No. 32, 1997 (1) For misorientations about the a axis, a disagreement between our values and those of KF2 was found in all cases but in the 70.53° misorientation. The Σ values found by H agree with ours except for the 70.53°, 86.63°, and 50.48°. In this last case, he reported a value of Σ ) 48, but we found Σ > 50 for this angle. (2) For misorientations about the b axes we found a general agreement with the work of KF2. An exception occurred at 44.42°. In this case we found a Σ value that is double that reported by KF2. We found a total coincidence with the CSL results obtained by H. (3) For misorientations about the c axes, we found a general disagreement with the results obtained by KF1, KF2, and H. In some cases KF1 reported Σ values that are half of our results. However, KF2 reported the correct Σ value for 60.0°. The misorientations reported by H do not coincide in general with our results. We have not reported the angles 9.43° and 16.43° because they have a Σ > 50. (4) For all the studied misorientations no agreement was found between the Γ values reported in ref 5 and our results. In spite of this, the Miller indices are generally the same as those of the literature. The differences found with the CSL characteristic values Σ, Γ, etc., reported by H are, in general, a consequence of their use of a hexagonal lattice instead of that corresponding to ice. The differences found by KF1 and KF2, however, cannot be related to any known cause, because these authors did not specify clearly how they found the reported CSL. In the present work, the CSLs were found using the ice lattice with an ideal axial ratio which is very close to that experimentally deter-
Gonzalez Kriegel et al. mined.6 Therefore we consider that the CSL characteristic values Σ, Γ, etc., we have found are more adequate to study the relationships between the structure and properties of the grain boundaries in ice Ih than those of KF and H. These results, for instance, were satisfactorily applied by Di Prinzio and Nasello7 to analyze the migration rate of ice grain boundaries. In this work values of the product Mγ (M and γ, the mobility and the excess of free energy of the grain boundary, respectively) for different 〈101h0〉/θ tilt grain boundaries were obtained using the Sun-Bauer technique.8 It was found that Mγ depends markedly on the misorientation and the inclination of the grain boundaries. The observed dependencies were explained considering that the CSL grain boundaries that are parallel to planes reported in Table 1b for b axis and have high values of Γ and lower Miller indices have lower values of γ. Thus, we conclude that the planes reported in Table 1 are a good starting point to look at special grain boundaries in ice. References and Notes (1) Ranganathan, S. Acta Crystallogr. 1966, 21, 197. (2) Kobayashi, T.; Furukawa, Y. J. Cryst. Growth 1975, 28, 21. (3) Kobayashi, T.; Furukawa, Y.; Kikuchi, K.; Uyeda, H. J. Cryst. Growth 1976, 32, 233. (4) Kobayashi, T.; Furukawa, Y. J. Cryst. Growth 1978, 45, 48. (5) Hondoh, T. Ph.D. Thesis, Department of Applied Physics, Faculty of Engineering, Hokkaido University, 1984. (6) La Placa, L.; Post, D. J. Glaciol. 1966, 33, 1234. (7) Di Prinzio, C. L.; Nasello, O. B. To be published. (8) Sun, R. C.; Bauer, C. L. Acta Metall. 1970, 6, 635. (9) The present work was suported by CONICET, CONICOR, and Fundacio´n Antorchas.
