Introduction
Grain growth or grain coarsening in ice has been studied by a number of authors, Reference WakahamaWakahama (1960), Reference RoosRoos (1966), Reference Jellinek and GoudaJellinek and Gouda (1969), and Reference SuzukiSuzuki (1969), but few measurements have been made of the grain-boundary energy of ice in spite of the fact that boundary energy plays an important role in these phenomena. First, this paper presents an attempt made to measure the grain-boundary energy in ice as a function of the angles formed by c-axes and a-axes in grains, giving evidence on the validity of the Read–Shockley Equation for grain boundary energy when ice grains are used. Secondly, this paper aims at proposing a new technique for the measurement of the solid–vapour grain-boundary groove angle in ice.
I. Grain-boundary energy in ice
Experimental method and preparation of triple grain boundaries
When polycrystalline ice is left alone for a long period of time at a temperature near the melting point, the interfacial tension acting in each grain boundary will pull into equilibrium with that acting in the adjoining boundaries. When ice crystals are grown from the melt, three grains almost always meet together at a point forming three boundaries in between. If interfacial tensions acting in three grain boundaries are in equilibrium at a triple junction, the following relationship should be satisfied:
where γ 1, γ 2, γ 3. are the interfacial tensions or grain-boundary energies, and θ 1, θ 2, θ 3 the angles made by two boundaries. If one of these γ is known and three angles are measured, then the other energies can be obtained. In our experiments, for the sake of convenience, one of the triple boundaries was defined as the reference boundary and its interfacial energy was assumed as a unit.
In order to prepare specimens of ice having triple boundaries, three blocks of seed single crystals of ice were attached on the surface of a large cylindrical brass block by freezing, as shown photographically in Figure 1 and diagramatically in Figure 2. The centre part of the brass block S can be rotated coaxially against the centre part of the block S′ in such a way that their crystallographic axes could be oriented in the desired directions. The third seed crystal 3 was frozen onto S, but its relative crystallographic orientation against the seed crystals 1 and 2 was changeable by the rotation of S. The three seed crystals were dipped carefully in distilled water kept near the melting point of ice. When the brass block was gradually cooled in a regulated cold box, the three seed crystals began to grow slowly, maintaining their original orientations. In a short time, they met at one junction, forming triple boundaries. When a block of ice having triple boundaries had grown to reach an acceptable length, it was taken out from the growing device and cut off carefully, leaving the seed crystals on the surface of the brass block. These seed crystals could be used again to develop ice blocks having triple boundaries. A thin plate (20 mm × 15 mm × 0.5 mm) was sliced from the block and placed on a clean glass slide in such a way that the line of the junction of triple boundaries was normal to the slide surface. Thus prepared, specimens were annealed in the atmosphere saturated with respect to ice at −5° C for about 8 months.
Fig. 1. Horizontal view of the crystal-growing device. Three seed crystals are attached on the surface of a brass block.
Fig. 2. Bottom view of the crystal-growing device. The seed crystal attached on S can be rotated against S′.
In our experiments, two types of specimen were prepared. In the first type, the c-axes in the three grains were distributed horizontally in the same plane as shown in Figure 3, and the boundary between grains 1 and 2 was chosen as the reference boundary. The angle ϕ 1 made by the c-axes in grains 1 and 2 was set at 50°, and the angles ϕ 2, and ϕ 3, made by the c-axes in grains 1 and 3 and in grains 2 and 3 were changed to desired values in every specimen by the rotation of grain 3. Accordingly, the ratio of the grain-boundary energy against the energy of the reference boundary, γ 2/γ 1 or γ 3/γ 1, could be measured as a function of or ϕ 2 or ϕ 3.
Fig. 3. A schematic diagram of a specimen of ice having triple grain boundaries. The reference boundary is indicated.
The second type of specimen was prepared in order to measure grain-boundary energy as a function of mismatch angle made by the a-axes in grains. The c-axes in the three seed crystals were set perpendicular to the brass surface and the angle ϕ 1 made by the a-axes in seed crystals 1 and 2 was set in each case at 15° or 27°. The relative orientation of the a-axes of grain 3 against grains 1 and 2 was changed in each specimen from 0° to 30° by the rotation of grain 3.
In Figure 4, (A) and (B) show the first and second type of specimen. The reference boundary is indicated by an arrow in each photograph. Etch pits showing the crystallographic orientation of individual grains were produced by the evaporation of water molecules through a thin formvar film applied on the specimen surface (Reference HiguchiHiguchi, 1958). Note that in photograph (B), the boundary between grains 2 and 3 was less etched because of minor mismatching between the a-axes.
Fig. 4. (A) First type specimen (B) second type specimen. Thermal etch pits indicate the crystallographic orientations of each grain.
Experimental results
Figure 5 shows the relative grain-boundary energy measured as a function of mismatch angle between the c-axes in two grains in the first type of specimen. The ordinate indicates the ratio of the boundary energies, R = γ 2/γ 1 and γ 3/γ 1. As seen in this figure, the value of R scattered widely in the range of 0.7 to 1.4. When the angle made by the c-axes in two grains approached the limiting values 0° or 180°, the value of R tended to decrease as shown by dotted lines.
Fig. 5. The value of relative grain-boundary energy obtained as a function of mismatch angle between the c-axes in two grains in the first type specimens.
Figures 6(A) and (B) show the value of R obtained as a function of mismatch angle between the a-axes in two adjoining grains in the second type of specimen. As seen in this figure, the grain boundary bounded by grains 1 and 2 was taken as the reference boundary, and the angle ϕ 1 made by the a-axes in the grains 1 and 2 was set at 15° in specimen (A) and at 27° in specimen (B). In order to measure the value of R = γ 2/γ 1 and γ 3/γ 1 as a function of mismatch angle made by the a-axes in grains, the orientation of the a-axes in grain 3 was changed in the range between 0° and 30°. The measured value of R was found to equal approximately unity in (A) and slightly less than unity in (B) for mismatch angles less than 25°. This difference in R may be related to the difference of the energy of the reference boundary used in (A) and (B).
Fig. 6a.
Fig. 6. The relative boundary energy measured as a function of mismatch angle between the a-axes in two grains in the second type specimens.
Validity of the Read–Shockley equation for grain boundary energy in ice
According to Read and Shockley (1949), the energy E of a grain boundary which consists of an array of edge dislocations is given by
where
and
where G is the rigidity modulus, v the Poisson’s ratio, b the Burgers vector, B the core energy of the edge dislocation, and ϕ the mismatch angle made by two crystallographic axes in adjoining grains, lf ϕ is small, it is given by b/h, where h is the distance between two edge dislocations. Equation (2) suggests that if this equation is valid, there may be a linear correlation between E/ϕ and In ϕ. Dividing Equation (2) by γ 1, we obtain
where R = E/γ 1 and R 0 = E 0/γ 1.
To test Equation (3), plots were made of R/ϕ against In ϕ for the experimental data obtained. Figure 7 represents the relationship of R/ϕ against In ϕ for the first type of specimen. The plots of R/ϕ were made respectively against acute mismatch angles in (A) and the complementary angles of obtuse mismatch angles in (B). As seen in Figure 7(A) and (B), the linear correlation is maintained for the c-axes mismatch angles less than 20°. Figure 8 shows the correlation between R/ϕ and ln ϕ for the second type specimens. In this figure, plots were made separately for data obtained from the specimens in which the mismatch angle at the reference boundary was set at 15° and 27°, respectively. A linear relationship was found for the a-axis mismatch angles less than 15°, suggesting that the Read–Shockley Equation may be valid for boundaries tilted within 0° to 15°.
Fig. 7. Plots of R/ϕ versus ln ϕ, ϕ and 180° – ϕ show the mismatch angle made by the c-axes in grains in the first type specimens.
Fig. 8. Plots of R/ϕ versus ln ϕ made from the data obtained in the first type specimen. (A) for ϕ1 = 15°, (B) for ϕ1 = 27°.
II. Measurement of the Solid–Vapour Grain-Boundary Groove Angle in Ice
Experimental procedure
When a polycrystalline ice surface has been exposed to the atmosphere nearly saturated with vapour for a long time, grooves are formed where the grain boundaries meet at the surface. The grooving occurs as a result of the adjustment to equilibrium of the grain-boundary energy γ gb and the surface energy of ice γ sv. If γ sv is isotropic, and γ gb is independent of the boundary orientation and the boundary is exactly normal to the surface, the following relation should be satisfied:
where θ sv is the solid–vapour grain-boundary groove angle formed at the root of the groove. If θ sv is measured experimentally and γ sv is known, the value of γ gb is determined. Therefore, in this method, it is very important to make an accurate measurement of θ sv. In metallurgy, the grain-boundary groove angle has been measured by the use of an interference microscope. The interferogram obtained by a monochromatic light allows the angle of the grain-boundary groove to be estimated. However, it may be difficult to have direct observations of the interferogram of a grain boundary groove in ice because of the lower reflectivity of light at the surface of ice. Reference Ketcham and HobbsKetcham and Hobbs (1969) measured the solid–vapour grain-boundary groove angles of ice by the use of the replica method. After making a formvar replica of grain-boundary grooves, they applied silvering to its replicated surface to create the high reflection of the light. According to Reference Kuroiwa, Hamilton, Kingery and PressKuroiwa and Hamilton (1963), however, the surface of ice is etched by the ethylene dichloride, the solvent of the replica solution. Since it is desirable to measure grain-boundary groove angles of ice without any chemical modifications, we used an extremely thin metallic foil instead of the silvered replica. When a piece of thin brass foil (0.3 μm thickness) was put carefully on the surface of ice, it adhered tightly on the relief of ice. A slight pressure was applied on the foil surface with clean silk cloth to ensure the adhesion between the foil and ice surface. Figure 9 shows a typical interferogram of a foil that covers grain-boundary grooves. If the grain boundary is not exactly normal to the surface or θ sv is not isotropic, the asymmetric distribution of the interference fringes may be observed in the vicinity of the boundary groove.
Fig. 9. An interferogram observed near foil-covered grain-boundary grooves.
The following procedure (Reference Hilliard, Hilliard, Cohen and AverbachHilliard and others, 1960) was employed to estimate the angle of grain-boundary grooves in ice. Figure 10(A) shows a typical profile of a foil-covered grain-boundary groove obtained from the interferogram. In this figure, plots a, b, c, … f and g, h, i, … m were made by the measurement of the distance between fringes formed on the foil surface, and the nearest fringes to the grain boundary, f and g, were taken as the origin of the coordinates (X 1, Y 1) and (X 2, Y 2) respectively. The vertical value of Y 1 or Y 2 at X 1 or X 2 was calculated from the number of fringes and the wavelength of the monochromatic light. If we extrapolate the profiles which link a, b, c, … f and m, l, n, … g, they will meet at the point 0. If ψ 1, and ψ 2 are the angles which the side walls of the groove make with the vertical at the root of the boundary, the angle of groove θ sv is given by
Fig. 10. (A) Profile of a foil-covered grain-boundary groove obtained from interference fringes. (B) Derived plots of Y/X versus X.
Since the graphic determination of ψ 1 and ψ 2 may include experimental errors due to extrapolation of the profiles, the following procedure was used to estimate ψ 1, and ψ 2.
According to Reference MullinsMullins’ theory (1957), the profile of a grain-boundary groove can be expressed by the parabolic formula
where Y and X are coordinates respectively normal to and parallel to the specimen surface, and a, b, c are constants. Using this equation, if we denote the lateral distance between the origin and the root of the groove as δ 1, the angle ψ 1 will be given by
where c = 0.
Therefore, if we plot the value of Y 1/X 1, against X 1, measured from the origin, a linear correlation should be found between them. Thus cot ψ 1, and similarly cot ψ 2, can be determined by the construction shown in Figure 10(B).
Experimental results
Figure 11 shows a typical result of the solid–vapour grain-boundary groove angle θ sv measured by our foil method. The specimens used in this experiment were prepared at −5° C. The abscissa indicates mismatch angles ϕ made by the c-axes in two grains. The accuracy for measuring θ sv was within 5° in our method. As seen in this figure, the value of θ sv at first decreased with increasing ϕ and then remained approximately constant in the range between ϕ = 30° to 130°, and then increased again with ϕ.
Fig. 11. Solid–vapour grain-boundary groove angles obtained by both replica (open circle) and foil (solid circle) methods.
It is interesting to compare the value of θ sv measured by our foil method with those measured by the replica method. In order to do this, a half length of a grain-boundary groove was covered with the thin metalic foil and the other half of it was covered with 3% replica solution. After desiccation of the solution, the replica film (approximately 0.1 mm in thickness) was peeled off and the silvering was applied on the replicated surface. In Figure 11, A and B, and also C and D, indicate the value of θ sv measured by the replica and foil method at the same grain boundaries. Apparently, the values of θ sv obtained by the replica method were found to be larger than those measured by the foil method.
Figure 12 shows the experimental values of the angle of grain-boundary grooves formed on basal surfaces of ice. In these specimens, since the c-axes in grains were parallel to each other, ϕ indicated the mismatch angle made by a-axes in two adjoining grains. The solid and open circles represent the values of θ sv measured by the foil and replica methods at the same boundary grooves respectively. It was also found that all data for θ sv obtained by the replica method were larger than those measured by the foil method.
Fig. 12. Angle of grain-boundary grooves formed on a basal surface. The open and solid circles show data obtained by replica and foil methods respectively.
Concluding remarks
The relative values of grain-boundary energy in ice were measured as a function of mismatch angles made by the c-axes or a-axes in two neighbouring grains. The relative value of the grain-boundary energy was defined by the ratio of a given boundary energy to a reference boundary energy. The results are shown in Figures 5 and 6. The values of angle shown in the abscissa in Figure 5 indicated only the mismatch angle of the c-axes in two adjoining grains, because, in the first type specimens, it was very difficult to determine the relative orientation of the a-axes in grains owing to the small thickness of the specimen used. Figure 6(A) and (B) shows the value of boundary energy measured at triple boundaries formed on the basal surface as a function of the mismatch angle between the a-axes in grains. In these figures, the angle shown in the abscissa can be considered to be a true mismatch parameter, since the c-axis in each of the individual grains was previously oriented parallel to each other. Figures 7 and 8 indicate that the Read–Shockley equation for grain boundary energy may be valid for grain boundaries tilted within 0° and 15°.
Covering of grain boundary grooves with thin metallic foil allowed interferometry to be applied for the measurement of the angle of grain-boundary grooves in ice θ sv. In our experiments, the value of θ sv obtained by this method was 130° to 135° for c-axis mismatch angles of 15° to 160°, showing a good agreement with the value of θ sv of unclean grain boundary grooves obtained by Reference Ketcham and HobbsKetcham and Hobbs (1969). The value of θ sv of grain boundaries formed at the basal surface varied with the mismatch angle between the a-axes in grains as shown in Figure 12.
