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J. Phys. Chem. 1903, 87, 4040-4044
Formation and Annihilation of Stacking Faults in Pure Ice Takeo Hondoh,’ Tairo Itoh, Shlnya Amakal, Kumlko Goto, and Akira Hlgashl Department Of Applied physics, Faculty of Engineering, HokkaMo University, Sapporo 060, Japan (Received August 23, 1982; I n Final Form: April 11, 1983)
Both formation and annihilation processes of stacking faults in fresh ice crystals grown from high-purity water were studied by X-ray topography. Stacking faults with the fault vectors R = ‘/6(2023) and R = 1010) were observed in the dislocation loops of interstitial type formed during crystal growth. The formation mechanism of such stacking faults is discussed in terms of a segregation process of excess interstitial water molecules which are introduced at the interface and are generated by cooling of the grown crystals. Shrinkage rates of both faulted and unfaulted dislocation loops were measured. The variety of the shrinkage rate for the faulted loops can be attributed to the large Burgers vector of the loops confirmed by their strong contrast in X-ray topographs. The stacking fault energy for the fault of which R = ‘/6(2023) was determined to be 0.31 mJ/m2 by the use of shrinkage rates for both the faulted and unfaulted dislocation loops.
Introduction Secondary lattice defects in ice crystals formed by heat treatment have been studied extensively in past few years.’-3 Results obtained are as follows: (1)the stacking faults were formed by dissociation of preexisting dislocations with Burgers vector b = 1/3( 11%)and by generation of faulted dislocation loops with the fault vector R = ‘/6(2023) when the crystals were cooled from a certain high temperature, (2) both the dissociation and generation stated above were caused by precipitation of excess selfinterstitials produced by cooling, (3) the predominant point defects in ice were self-interstitials, not vacancies, at a temperature above -50 OC,and (4)the equlibrium concentration of self-interstitials estimated from the measured density of dislocation loops was approximately 3 ppm at the melting temperature with a formation energy of 0.4 eV. On the other hand, stacking faults of large area, ca. 1-30 mm2, were observed by X-ray topography in fresh single crystals of ice grown from the melt. The purpose of this paper is to clarify both the formation and the annihilation processes of the stacking faults generated in ice single crystals during crystal growth and to evaluate the stacking-fault energy from the shrinkage rates of both faulted and unfaulted dislocation loops. Experimental Procedures Single crystals of ice were grown from high-purity water by the modified Bridgeman m e t h ~ d .The ~ ~ dislocation ~ densities of the crystals used in the present experiments were of the order of lo6 m-2 and its impurity contents measured by several ions were less than the order of 0.1 PPm. The grown crystals were kept in a cold room at -10 OC after they were removed from glass tubes of the growth apparatus. Within a few days after the crystal growth, thin planar specimens of 2-5 mm thickness were prepared from these crystals by sawing and without chemical polishing. (1)T. Hondoh, T.Itoh, and A. Higashi, Jpn. J. Appl. Phys., 20,L737 (1981). (2) T. Hondoh, T. Itoh, and A. Higashi, ‘Point Defects and Defect Interactions in Metals”, J. Takamura, M. Doyama, and M. Kiritani, Ed., University of Tokyo Press, Tokyo, 1982, p 599. (3) K. Goto, T.Hondoh, and A. Higashi, “Point Defects and Defect Interactions in Metals”, J. Takamura, M. Doyama, and M. Kiritani, Ed., University of Tokyo Press, Tokyo, 1982, p 174. (4) M. Oguro, Phil. Mag., 24, 713 (1971). (5) A. Higashi, J . Crystal Growth, 24/25,102 (1974).
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The specimens were mounted immediately in a specially designed specimen chambeFa installed on a Lang camera (Rigaku LGL-3s). Temperature in the specimen chamber was lowered by nitrogen gas evaporated from liquid nitrogen in a large Dewar vessel, and it was controlled within f0.1 O C by the programmed regulation of an electric heater installed in the tube of the gas supply.6a The high-brilliance X-ray source with a rotating Mo target (Rigaku RU-200) was operated at 60 kV and 20 mA with an apparent focus size of 0.1 X 0.1 mm2. Topographs were taken on nuclear plates (Ilford L4, -50 wm) or on commercial X-ray films (Fuji Ix-150). The exposure time was roughly 1 h for the nuclear plate, but it was possible to reduce it to 20 min when X-ray film was used. All topographs in this paper were taken by scanning the crystal on the (0001) plane. The Burgers vector b of dislocations and the fault vector R for the stacking faults were determined by comparisons of X-ray topographs taken with various diffraction vectors g of 1010 and 11‘20. The direction of the g vector is indicated by an arrow in each topograph in the paper. Criteria for the determination of b and R are that the contrast of the line image of the dislocations and the areal image of the stacking faults vanish when g b = 0 for screw dislocations, g.1 = 0 for edge dislocations, and g R = n for stacking faults, where 1 is the line vector of a dislocation and n is an integer. Results and Discussion Formation of Stacking Faults in Fresh Ice Crystals. Stacking faults in fresh ice crystals are observed as areal dark images in Figures 1 and 2, which are topographs of ice crystals grown in (0001) and (lOT0) directions, respectively. A number of the faulted dislocation loops (peripheral dislocations around stacking faults) seen in Figure l a shrank or disappeared after aging for 12 days as shown in Figure lb, though many unfaulted loops were left. In crystals grown in the (lOT0) direction, stacking faults extended to the direction of growth as shown in Figure 2a. They also shrank or disappeared after aging for 10 days as shown in Figure 2b. No stacking fault was observed in specimens which were prepared from the crystals kept in the cold room for more than 1week. Therefore, the faults must be formed during growth of the crystals. This inference is supported by the (6) (a) T. Hondoh and A. Higashi in “X-ray Instumentation for the Photon Factory”,Reidel, Tokyo, to be published. (b) M. Oguro, Thesis, Doctor of Engineering, Hokkaido University, 1978.
@ 1983 American Chemical Society
The Journal of phvsicai Chemistry, Vol. 87, No. 21, 1983 4041
Stacking Faults in Pure Ice
(a)
(b)
Flgure 1. X-ray topographs of a fresh ice crystal grown in the direction of [OOOl] (a) after aging for 1 day at -20 OC and (b) after aging for 12 days at -20 OC. The specimen surface is (0001) in all topographs.
-re 2. X-ray topographs of a fresh ice crystal grown in the direction of ( lOiO), the vertical direction of the figure, (a) after aging for 1 day at -20 OC and (b) after aging for 10 days at -20 OC.
facts that the appearance of the faults depends on the growth direction and is similar to those found in in situ X-ray topographic observations of the ice growth process.6b Since most of the faults were found to be surrounded by dislocations with the Burgers vector b = 2023) as were those formed by cooling and the surrounding dislocation loops were of the interstitial type, the same formation mechanism of the faults as in the case of cooling can be proposed. That is to say, excess interstitials generated during the growth process are segregated into dislocation loops with or without stacking faults in them. Excess interstitials can be produced either by the same way as in the case of cooling or by specific ways associated with the crystal growth. According to the equilibrium concentrations of interstitials estimated as 3 ppm at the melting point with a formation energy 0.4 eV, excess interstitials of about 1ppm can be produced when freshly grown ice crystals are cooled from the melting temperature to -10
"C. This amount of excess interstitials is sufficient for the formation of the dislocation loops shown in Figure la. Two different specific mechanisms could be inferred associated with the crystal growth. One is the intrinsic mechanism based on a concept of structural diffusion proposed by Fletcher: in which pfismatic dislocation loops are formed under supersaturation of interstitials introduced from the liquid phase at the interface. Another is the local instability mechanism proposed by Oguro and Higashi.8 In the latter, dislocation loops are generated by freezing of minute water droplets which remained for a short time unfrozen in ice just behind the interface, because of the local instability or of the dilute impurities at the interface. We have no proof yet to determine which mechanism is dominant among the above three or how (7) N. H.Fletcher, J. Crystat Growth,28,375 (1975). (8)M.Oguro and A. Higashi, J. CrystaZ Growth,51,71 (1981).
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The Jownal of phvsicai Chemism, Vol. 87, No. 21, 1983
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Fmre 3. Stacking faults f and f' with the fault vector R = 1/3( 1070). They are included in the large dislocation loops of interstitial type OPQR and STUV. The perfect dislocation A of b = 'I3[l 1201 dissociates into the Shockley partials X and X' of b = 1/3[01iO]and '/,[lOiO], r e s m e l y . The Burgers vectors of individual dislocations are indicated by thick arrows.
they are combined in real processes of crystal growth. The segregation process of excess interstitials into a dislocation loop in ice crystals can be considered in a way similar to that proposed in hexagonal metal^.^ A t first a single interstitial disk is formed between adjacent basal planes, Le., a dislocation loop with b = R = 1/2[0001]is generated. When the loop diameter increases, the loop is swept by a Shockley partial as to transform the fault vector into the most stable one, '/6(2023). Preferential formation of a second interstitial disk on the adjacent layer turns the Burgers vector into 1/6(2026). When the loop is swept again by a Shockley partial the fault is eliminated completely, thus a perfect dislocation loop with b = [0001] is formed. Most of the faulted and unfaulted dislocation loops observed in fresh crystals as shown in Figures 1 and 2 are those in various stages of the segregation process of interstitials as described above. Stacking faults surrounded by the dislocations with the Burgers vector b = 1/3( 10x0) were occasionally observed in fresh crystals as shown in Figure 3. The Burgers vectors of the peripheral dislocations were determined by comparisons of the six topographs taken with three different diffraction vectors of 1010 and of 1120. Directions of the Burgers vectors of the individual dislocations are shown by arrows on a schematic drawings of the topograph (Figure 3b). The perfect dislocation A (b = '/3[llZO]) dissociates into two Shockley partials X (b = 1/3[OliO]) and X' (b = '/,[lOlO]) of which separation exceeds 2 mm. Such a wide extended dislocation is, of course, never expected in equilibrium. In fact, no extended dislocation was observed after long-term annealing. Since the perfect dislocation, A, and the partial dislocations, Y, Y', and Z, all joined with the large interstitial loops OPQR and STUV, the stacking faults f and 'f surrounded by Shockley partials were probably formed during the formation and/or annihilation processes of the faulted loop OPQR. Detailed analyses of such complicated configurationsof the faulted loops will be given elsewhere. (9) S.Amelinckx, "Dislocationin Solids",Vol. 2, F. R. N. Nabarro, Ed., North-Holland, Amsterdam, 1979, Chapter 6.
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ANNEALING TIME ( sec.fm5
Flgure 4. Shrinkage rates of the faulted loops ( 1 , 2 , 3 , and 4) and the unfaulted loops (inset) at -20 OC. Squares of the loop radii are plotted against annealing time in the case of the unfaulted loops. The values of the shrinkage rates for 1, 2, 3, and 4, are 2.1 X lo4, 0.43 X lo4, 0.22 X and 0.068 X 10" m/s, respectively. The average shrinkage rate for unfaulted loops is 1.0 X m2/s.
Determination of Stacking Fault Energy. Stackingfault energy of the ice crystal is determined from the measured shrinkage rates of both the faulted and unfaulted dislocation loops. To carry out such measurements precisely, use of isolated circular loops are desirable. Since too many dislocation loops appeared in fresh crystals as can be seen in Figures 1 and 2, it was difficult to select isolated ones in such topographs. Therefore, we used those generated by cooling in nearly dislocation-free ice crystals. As reported in previous papers,'~~ the number and size of the dislocation loops generated by cooling depend on both the thickness of specimens and the cooling conditions. In the present experiments, specimens of 2 mm thickness cooled from -5 to -20 "C at a rate of 100 "C/h were used. Since Burgers vectors and fault vectors of dislocation loops introduced by cooling were identical with those of loops generated during crystal growth, the measured results of the shrinkage rates could be applicable for those in fresh crystals. The radii of both faulted and unfaulted loops were measured from successive topographs taken during an-
Stacking Faults in Pure
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Ice -
my---
dt.2 e dt
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Flglw 5. x*ytopogaphsofthefaulted kopson WhiChthesMnkage rates were measured. (a), (b), and (c) correspond to the data 4,3 and 1 in Figure 4, respectively.
nealing at a constant temperature of -20 "C. Typical examples of the result are shown in Figure 4. Due to the reason to be described, data for the unfaulted loops are expressed in areal shrinkage rates (9 vs. time) as in the inset of Figure 5 whereas data for the faulted loops are expressed in radial shrinkage rates. As can be seen in this figure, the slopes of straight lines of the shrinkage rates are different from one experiment to another for faulted loops, while the areal shrinkage rates of the unfaulted loops are identical. The shrinkage velocity of the dislocation loops can be expressed as follows:loJ1
where r is the radius of the loop, D the coefficient of self-diffusion, b, the thickness of the interstitial disk or the component of the Burgers vector normal to the plane of the loop, R the molecular volume, F the driving force for shrinkage, and @ a constant. The driving force F is given by12
F=
/
\
(2) where b is the component of the Burgers vector parallel to the pf&e of the loop, p the shear modulus, v the Poisson ratio, p the cutoff parameter of the dislocation cores, and yf the stacking-fault energy. Since the value of FQ/kTb, is much smaller than unity for the present case, eq 1 can be approximated by DFR dr N -6(3) dt bn2kT For the faulted loop, the line tension term (the first term) in eq 2 is much smaller than the stacking-fault term yp Then eq 3 is converted to (4) where the primes on r, b,, and @ are used to distinguish the quantities from those of the unfaulted loops. For the unfaulted loops, the driving force is due to only the first term in eq 2. Then the time rate of the square of the radius is given by (10) D. N. Seidman and W. Balluffi, Phil. Mag., 13,649 (1966). (11) P. S. Dobson, P. J. Goodhew, and R. E. Smallman, Phil. Mag. 16, 9 (1967). (12) J. P. Hirth and J. Lothe, "Theory of Dislocations",McGraw-Hill, New York, 1968, Chapter 6.
where $ is the angle between the Burgers vector and the plane of the loop. Data plots for the square of radius r2 vs. time for the unfaulted loops in the inset of Figure 4 are now justified. We can assume 8' = p if the measurements of shrinkage rates are carried out with loops of the same order of size at the same depth in specimens for both faulted and unfaulted loops. Under this assumption, the stacking-fault energy yf is given by the ratio of two time rates (4) and (5)as Yf
=
(6) where both D and'@are eliminated in the equation. The variety of shrinkage rates for the faulted loops as indicated by numbers 1 to 4 in Figure 4 can be explained as follows. Topographs a, b, and c in Figure 5 exhibit the differences in image contrast of the peripheral dislocations of faulted loops used for the shrinkage experiments designated 4, 3, and 1 in Figure 4, respectively. Stronger image contrasts in a and b than c indicate that peripheral dislocations in a and b have large Burgers vectors. Since the shrinkage rate of the faulted loops is inversely proportional to the square of the normal component of the Burgers vector b, as seen in eq 4, peripheral dislocations of stronger contrast should shrink much slower than the one of weak contrast. The slower shrinkage rates in experiments 4 and 3 than in experiment 1 shown in Figure 4 conform to this expectation. As the structure of the loops with large Burgers vectors is not known yet, we cannot compare the data in Figure 4 quantitatively but large difference in the shrinkage rate reflects the inverse square relation to b, in eq 4. In the case of unfaulted loops, the shrinkage rate is much slower so that those of approximately 1 mm radius are annihilated after 4 months of annealing at -20 "C. This can be understood as resulting from the difference of the driving force in eq 4 and eq 5. To derive the stacking-fault energy using eq 6, we measured the shrinkage rate on faulted loops with weak contrast of peripheral dislocations, of which Burgers vector was '/6(2023) (b',, in eq 6 is c / 2 or 0.368 nm), as well as the areal shrinkage rate on unfaulted loops with a Burgers vector of 1/3( 1123). Caution was paid to select suitable loops to hold the condition p = p' as described before. The value of yf for the fault of the '/6(2023) fault vector is evaluated to be 0.31 mJ/m2 (erg/cm2)at -20 "C from the ratio of the experimentally determined dr'/dt and dP/dt for the above two kind of loops. Measurement errors and uncertainty of the cutoff parameter p may lead an error of about 10% for this value. Since the driving force owing to the first term in eq 2 is of the order of lo4 N/m for the measured faulted loops while that caused by the stacking fault is 3.1 X lo4 N/m as derived above, the approximation used to derive eq 4 is valid. Oguro once carried out similar measurements and he obtained a stacking-fault energy on the order of 0.1 mJ/ m2.6bUncertainty in his result came from a wide variety of shrinkage rates of faulted loops as was also observed in Figure 4 of our experiments. In the present work, however, we clarified the origin of the variety and obtained an exact
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J. Phys. Chem. 1983, 87,4044-4050
value of yf by careful experiments. The stacking fault with the fault vector of ' / e ( 2023) is accompanied by one layer of cubic structure, whereas that with a fault vector of lOT0) which appeared in Figure 4 as a partial in the extended dislocation of 1/3(1120) Burgers vector is accompanied by two cubic layers. Therefore, the stacking-fault energy yf for the fault of 1/3( l O i O ) is roughly equal to 0.6 mJ/m2. Thus the width
of the extended dislocation is roughly estimated to be in the range 20-50 nm, depending upon the angle between the dislocation line and the Burgers vector of the perfect dislocation. This value is larger than usual values for metals but is still much smaller than the resolution of X-ray topography. Registry No. Water, 7732-18-5.
Generation and Absorption of Dislocations at Large-Angle Grain Boundaries in Deformed Ice Crystals Takeo Hondoh' and Aklra Hlgashl Department of Applied Physics, Faculty of Engineering, Hokkaido University, Sapporo 060, Japan (Received: August 23, 1982; In Final Form: April 19, 1983)
X-ray topographic observations were carried out on the large-anglegrain boundary (GB) in ice bicrystals during deformation. Misorientations of the bicrystals were made as close as possible to the ( 10T0)/34.1' CSL. The section topographs of the GB in equilibrium after a long-term annealing revealed no long-range stress field associated with the GB but there appear in the topographs strong stress fields at the intersections of GB facets when the GB was subjected to shear stress. Traverse topographs were taken during compressive deformation of bicrystal specimens to observe the generation and absorption of lattice dislocations at the GB and their behavior under stress. It was found that dislocations are generated from the GB when it was subjected to a shear stress. When the GB was subjected to only a compressive stress, absorption of dislocations occurred at the GB. Generation and absorption mechanisms of lattice dislocations are proposed on the basis of the structure of the faceted GB composed of stable CSL lattice plane and intrinsic grain boundary dislocations (GBDs).
Introduction Large-angle grain boundaries (GBs) in artificially grown bicrystals of ice were studied by the X-ray diffraction topographic meth0d.l Line images like Figure ICappeared on the topograph of grain boundaries which approximately satisfy the condition of the coincidence site lattice (CSL: ( 10T0)/34.1°) in a bicrystal, when it was subjected to light deformation. Those images were interpreted to be caused by stress concentrations at intersections of the facets on the GB. Although a stepped structure composed of lowenergy facets was assumed in the first paper,' later the real geometrical structure of the GB was revealed by optical photographs taken by oblique illumination as shown in Figure la.2 Bright images in the photograph are facets reflecting light and the geometrical structure could be schematically drawn as in Figure lb. This figure indicates that the structure should be saw-toothed rather than stepped, but we call it "the faceted structure" for convenience of our common terminology. The orientations of the individual facets were measured with reference to crystallographic orientations by measurements of both laser beam reflection from the facet and X-ray Laue patterns of the crystals? The angle between adjacent facet planes IC. was aproximately 9' for the boundary at which the misorientation deviated slightly from the exact CSL: ( 10T0)/34.1°. Crystallographic orientations of these facet planes did not coincide with any low-index plane of both grains. (1)T.Hondoh and A. Higashi, J . Glaciol., 21,629 (1978). (2) T. Hondoh, J. Mater. Sci. SOC.Jpn., 17,131 (1980). (3) T.Hondoh, Bull. Jpn. Inst. M e t . , 22,130 (1983).
The purpose of this paper is to clarify the mechanisms of stress concentrations and the generation of dislocations when the GB is subjected to shear stress, both by new experimental techniques and by theoretical considerations using the grain boundary dislocation (GBD) model for GBs slightly deviating from high-density CSL planes. Absorption of dislocations at the grain boundary subjected to nonshear stress is also described. Reaction kinetics of dislocations on and near the grain boundaries and calculations with the CSL model for ice are successfully used for the considerations. Experimental Procedures Ice bicrystals having a misorientation angle of about 34' around a common axis (lOI0) (CSL: (10T0)/34.1°) were grown from high-purity water by the method described in previous papers.'~~Two single crystals of ice (the seed crystals in the Czochralski method) were fixed such that the (1010) axes of both crystals were parallel to the growth direction and the angle between two (0001) axes was as close as possible to 34.1' of the exact CSL. Specimens were cut from the grown bicrystals and were shaped to thin planar sections (10 mm X 45-55 mm and 4.7 mm thickness) by sawing and planing without chemical polishing. Two types of specimens were prepared: one was cut so that the grain boundary made a 45' angle with the specimen's vertical axis and was perpendicular to the plane surface (Figure 2a) and another was cut so that the grain boundary was perpendicular to both the vertical axis and the plane surface (Figure 2b). (4)
T. Hondoh and A. Higashi, P M O S .Mag., 39, 137 (1979).
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