J. Phys. Chem. B 1997, 101, 6109-6112
6109
Effect of Bubbles on Grain Growth in Ice L. Arena, O. B. Nasello, and L. Levi* 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: March 31, 1997X
Grain growth is studied in artificial samples of pure ice. It is shown that the mean grain diameter D can be represented by the equation D2 ) D02 + 4kt, both in bubble free ice and in growth regimes where bubbles are present on boundaries. In the second case it is found k , ki, where ki is the intrinsic grain growth rate obtained for bubble free ice. This behavior is explained by considering that the drag effect Pb of migration bubbles on boundaries occurs in the low-velocity regime, where Pb ) ν/Mb with ν and Mb the migrate velocity and mobility of bubbles, respectively. From the values of ki, the free boundary mobility M is calculated and the values of Mb are derived from the equation k ) ki(1 + M/Mb)-1. The results obtained by different authors for k, both in artificial and glacial ice, are compared in an Arrhenius plot. It is shown that the results for glacial ice and for bubbly laboratory ice can be grouped in the same region where k , ki. These results are interpreted by assuming that also in glacial ice grain growth is affected by migrating bubble drag. It is noted that this interpretation is not in contradiction with the fact that in glacial ice bubbles gradually separate from boundaries, i.e., that the phenomenon would occur in the high-velocity regime. Correlation between the experimental and theoretical values of Mb, the latter calculated on the basis of molecular diffusion processes, is discussed.
1. Introduction It is well-known that polycrystalline ice is rarely free of bubbles. In glacial ice, these are generated by the progressive transformation into pores of the air trapped in deposited snow, but they also form in ice grown from bulk water due for instance to gas accumulation at the solid-liquid interface, resulting from the segregation by the solid of gas molecules dissolved in the liquid. Thus, as bubble-boundary interaction has been shown to reduce, in several cases, the grain growth rate in polycrystalline materials, research work on this effect could also provide useful information for a better understanding of polycrystalline ice evolution during annealing. Actually, the decrease of the grain growth rate with time reported by authors studying artificial samples of bubbly ice (Levi and Ceppi, 1982; Nasello et al., 1992) has been related to bubble accumulation on boundaries and to the consequent Zener drag that would counteract the boundary driving force. On the contrary, Alley et al. (1986b), studying glacial ice, noted that bubbles separate from more rapidly migrating boundaries and concluded that bubble drag could be, in this case, negligible. In the present work a new analysis of previous experimental results for grain growth in pure and bubbly ice will be carried out, in order to discuss the proposed interpretations of the process. Special attention will be given to the behavior observed by Nasello et al. (1992) in high-purity laboratory ice samples, containing different bubble concentrations. Results will be applied to discuss the bubble-boundary interaction in the case of glacial ice, as well. 2. Experimental Results As described in the previous paper (Nasello et al., 1992), polycrystalline samples were obtained by recrystallization of a pure ice single crystal. Two types of zones were observed, indicated as ZS and ZC. Zones ZS initially appeared substantially free of bubbles, whereas a non negligible bubble concenX
Abstract published in AdVance ACS Abstracts, June 1, 1997.
S1089-5647(96)03239-7 CCC: $14.00
TABLE 1: Experimental Values of Grain Growth Rate k (10-14 m2/s) and of Parameter NsO3 (10-6 m) for Zone ZS1 (without Bubbles) and Zones ZS2 and ZC2 (with Bubbles) zone ZS1
zone ZS2
zone ZC2
T (°C)
k
k
Nsφ3
k
Nsφ3
-2 -6 -10 -13 -20 -27
4.9 4.0 1.4 1.4 0.7 0.5
1.7 0.4 0.16 0.06 0.04 0.04
1.4 0.8 0.5
0.15 0.04 0.01
9 5 4
1.0
0.01
3
tration was always observed in zones ZC. In order to study grain growth, samples were annealed at different temperatures (T), between -2 and -27 °C, their structures were analyzed after annealing times of up to 4000 hs and zones ZS and ZC were separately studied. For each temperature, the results were represented in a D2 vs t plot (where D is the mean equivalent diameter of the crystals). The curves for zones ZS showed a sharp decrease of the growth rate, occurring after a few hundred hours annealing, so that two growth regimes (ZS1 and ZS2) could be distinguished, each one satisfactorily adjusted by a straight line. A regime change was also observed for zones ZC, but in this case the first regime (ZC1) was not clearly defined, while the second (ZC2) was also well-represented by a straight line. Thus, for the three regimes ZS1, ZS2, and ZC2 the typical grain growth equation
D2 ) D02 + 4kt
(1)
could be written out, though with different values of the grain growth rate k ) dR2/dt, where R ) D/2. In Table 1 we give the values of k obtained for the three regimes, at different temperatures. The table shows that k for regimes ZS2 and ZC2 can be up to 1 and 2 orders of magnitude lower than for regime ZS1. From the observation of surface replicas of the samples, information about their bubble structure as a function of the © 1997 American Chemical Society
6110 J. Phys. Chem. B, Vol. 101, No. 32, 1997
Arena et al. TABLE 2: Mobility Values for Different Temperaturesa zone ZS2
zone ZC2
T (°C)
Mgg
zone ZS1 Msb
Mbex
Mbth
Mbth/Mbex
Mbth/Mbex
-2 -6 -10 -20 -27
0.19 0.15 0.054 0.027 0.019
4 2 0.1 0.03
8 1.7 0.59 0.17 0.17
7 9 10 2
0.9 5 17 12
2 9 26 24
a Mgg and Msb (10-11 m2 s/kg) are the intrinsic grain boundary mobilities obtained from grain growth and single boundary migration, respectively. Mbex and Mbth (10-13 m2 s/kg) are experimental and theoretical mobilities of bubbles.
velocity regime, the grain growth velocity can be expressed as Figure 1. Experimental behavior of β ) k/ki as a function of T for bubbly artificial ice.
annealing time could also be obtained. It was found that, during the second regime, observable bubbles were present in both zones. In zones ZS, regime ZS2 was characterized by an average bubble concentration of about 20 bubbles/mm2 and mean diameter φ of about 10 µm. In zones ZC, regime ZC2, the concentration of bubbles reached 100 per mm2 and φ ≈ 20 µm. Results of replica analysis, for the mean values of the product Nsφ3, where Ns is the bubble concentration on boundaries, are also summarised in Table 1. As pointed out in the previous work (Nasello et al., 1992), replica analysis also showed that, during regimes ZS2 and ZC2, bubbles were mostly located on grain boundaries, this fact suggesting that the boundary driving force was sufficient to drag along bubbles. Thus, the results given for k in Table 1 could be interpreted by taking into account that the presence of bubbles attached to boundaries would slow down the grain growth rate in regimes ZS2 and ZC2 with respect to that observed in regime ZS1, which would represent the intrinsic grain growth rate k ) ki for the pure material. Accordingly, the value of k for bubbly ice could be written out as k ) βki, where β < 1 would represent the bubble effect on the grain growth rate. The experimental behavior of β as a function of T observed for regimes ZS2 and ZC2 is shown in Figure 1. It can be seen that, for T e -6 °C, β does not show a clear dependence on T and ranges between 0.1 and 0.01, the lower values corresponding to regime ZC2. When T increases above -6 °C, β increases with T, as especially shown by the diagram for regime ZS2. 3. Discussion 3.1. Relationships between Grain Growth Rate and Extrinsic Microparticle Drag. Alley et al. (1986a) observed that grain growth in polycrystals containing extrinsic materials may occur according to two different growth regimes: the lowvelocity regime, where impurities remain attached to boundaries, and the high-velocity regime, where impurity-boundary separation occurs. In the low-velocity regime, the grain growth behavior would be affected in a different way by the presence of solid microparticles and of migrating bubbles. In the first case, boundaries would be clamped by the extrinsic material and grain growth would vanish (Zener effect); in the second, bubbles would migrate under the action of the boundary driving force, so that a drag force depending on the bubble velocity would only reduce the grain growth rate below the free boundary value. On the contrary, in the high-velocity regime, particles of the extrinsic material would be left behind by advancing boundaries, regardless of their nature, so that the particle drag effect would be similar in both cases and depend on the force needed to bypass the particles. Following Alley et al (1986a), when migrating bubbles are present in the material and grain growth occurs in the low-
ν)
1 dD ) M(Pi - Pb) 2 dt
(2)
where M is the boundary mobility and Pi, Pb are the intrinsic boundary driving force and bubble drag force, respectively. They can be represented as a function of the grain boundary free energy γ and of the bubble mobility Mb, in the form
Pi ) ξγ/R
Pb ) ν/Mb
(3)
where ξ is a constant coefficient that depends on the used grain growth model and is ξ ) 16/81 ∼ 0.2, according to the stationary grain growth law proposed by Hillert (1965). We will use this value of ξ though we also point out it could be ξ ) 0.115, according to a two dimensional simulation model developed by Nasello and Ceppi (1985). Replacing Pi and Pb by their expressions given in eq 3, eq 2 becomes
dD 2ki M dD ) dt D Mb dt
(4)
where ki ) 2ξMγ. It can be immediately seen that eq 1 is a solution of eq 4, with
k ) ki(1 + M/Mb)-1
(5)
where k stands for the grain growth rate when migrating bubbles occur. Comparing with the experimental results described above, it can be seen that the grain growth rate in regimes ZS2 and ZC2, represented by k , ki, can be interpreted as occurring in the low-velocity regime, with
β ) (1 + M/Mb)-1
(6)
Notice that evidence of this event can also be derived from the observation that, in these regimes, all bubbles were located on boundaries. On the contrary, in regime ZS1, where visible bubbles were not observed, free boundary migration would occur, and the condition k ) ki ) 2ξMγ would be satisfied. 3.2. Evaluations of Boundary and Bubble Mobility for Laboratory Ice. According to the discussion in 3.1 the values of k listed for regime ZS1 in Table 1 can be used to evaluate M, considering that k ) ki ) 2ξMγ and deriving γ from the literature. The results obtained taking ξ ) 0.2 are given in column 2 of Table 2. In column 3, we present for comparison the values of M that could be derived from the Mγ mean values obtained by Di Prinzio and Nasello (1996) for fast migrating tilt boundaries (large-angle boundaries) in bicrystal studies. In both cases, γ ) 0.065J/m2 (Ketchum and Hobbs, 1969) has been used.
Effect of Bubbles on Grain Growth in Ice
J. Phys. Chem. B, Vol. 101, No. 32, 1997 6111
TABLE 3: Values of NsO3, the Grain Growth Rate k, and Related Parameters for Glacial Ice at Byrd Station k depth N sφ 3 (m) (10-6 m) (10-14 m2/s) 64 116 279
68 42 13
0.01 0.01 0.01
β 0.02 0.02 0.02
Mbex Mbth (10-13 kg s/m2) (10-13kg s/m2) 0.04 0.04 0.04
0.01 0.02 0.07
In Table 2, it can be seen that the values of M derived from the grain growth rate (Mgg) are, on the average, approximately 1 order of magnitude lower than those found for single boundary migration (Msb). Such disagreement could be justified, however, on the basis of different considerations. We notice for instance that the studied polycrystals contained a nonnegligible fraction of low-energy boundaries (γ < 0.065J/m2) and that, if the value of ξ derived from Ceppi and Nasello (1985) had been used, larger M values would have been obtained with the same γ. The values of Mb given in column 4 of Table 2 (Mbex) are derived from eq 6, with M ) Mgg, and with the values of β plotted in Figure 1 for regime ZS2. The values of Mbth in column 5 are derived from the expression (Shewmon, 1964; Alley et al., 1986a)
Mbth )
(
)
ν Ω DνFν 2Dsδs + 2Di (TNsφ3)-1 (7) ) + Pb 4πK Fi r
where Dν is the diffusion coefficient for water vapor in air and Ds and Di are those for water molecules at the solid-air interface and in the ice lattice surrounding the bubble, respectively; K is the Boltzmann constant, Ω is the volume of the water molecule, Fν and Fi are the water vapor and ice density, respectively, r is the bubble radius, and δs is the thickness of the layer where surface diffusion occurs. At glacial ice typical temperatures, Alley et al. (1986a) found that the only diffusivity term to be taken into account in eq 7 was that depending on Dν and the others can be disregarded, and they calculated this term for Dν ) 8.79 × 10-10T1.81 m2 s-1. However, in the present work, where experimental results obtained at temperatures up to near melting point were analyzed, the errors performed by neglecting the second diffusivity term in eq 7 had to be discussed. In fact, at these temperatures, the product Dsδs/r could not be negligible due to the formation of a liquid-like layer on the ice surface (Fletcher, 1968; Dash, 1995). Calculations of Dsδs/r at T ) -2 °C were carried out by considering Ds ∼ 10-9 m2 s-1 (Maruyama et al., 1992) and a liquid-like layer thickness of about 800 Å (Golecki and Jaccard, 1978). It was found that, in these conditions, surface and vapor diffusion would have similar values so that if both these processes had been taken into account, Mbth could have increased significantly, though by less than an order of magnitude. Thus, considering that the properties of the liquidlike layer are not well-known, we have calculated Mbth as a simple function of vapor diffusion, as proposed by Alley et al. (1986a), so that the values given in column 5 of Table 3 could rank somewhat below the actual values, especially at the highest temperatures. Notice that, for a constant bubble structure, Mbth should decrease slowly with temperature. Some wavering values, revealed in the table, depend on the variations of the product Nsφ3 shown in Table 1. The results of comparing Mbth with Mbex appear in columns 6 and 7 of Table 2, for regimes ZS2 and ZC2, respectively. It can be seen that both mobilities are of the same order of magnitude at -2 °C, but their ratio increases rapidly when T decreases down to -10 °C and remains approximately constant at lower temperatures. The observed discrepancy between Mbex and Mbth could depend partly on the possible underevaluation
Figure 2. Arrhenius plot of all of the results for k provided by the literature, both for artificial and glacial ice.
of Mgg, which propagates to Mbex, through the application of eq 6. However, it could also be ascribed to the exceedingly simplified model used to calculate Mbth. Actually, all bubbles are supposed to migrate attached to two grain boundaries, and their mobilities are calculated by assuming a spherical geometry (Shewmon, 1964). Thus, some actual features of the studied samples are not taken into account, which show that most bubbles depart markedly from the spherical shape and that there are bubbles located at three grain junctions where a lower migration rate can be expected (Spears and Evans, 1982). On the other hand, the rapid increase of Mbex observed when the temperature increases above -10 °C could be considered a result of a better correspondence between the actual and the theoretical migrating bubble distortion (Hsueh and Evans, 1983) occurring when a nonnegligible quasi liquid layer is formed on the ice surface. 3.3. Comparison of Laboratory and Glacial Ice Results. In the Arrhenius plot of Figure 2 all of the results for k provided by the literature, both for artificial and glacial ice, are given for comparison. The diagram shows that the experimental points can be grouped in two wide regions, A and B. Both regions meet near the melting point, but their limits diverge with decreasing T so that, at T < -10 °C, the values of k in region A can be up to about 2 or 3 orders of magnitude above those corresponding to region B. It must be noted that all points in region A are derived from laboratory experiments and correspond to samples substantially free of bubbles or to low annealing times. For instance, the experimental points for regime ZS1 are located in this region. On the contrary, we find in region B both the results obtained for glacial ice and those for the second regime of artificially polycrystallized samples (regimes ZS2 and ZC2), where bubble drag has been shown to markedly reduce the grain growth rate k. However, we have seen that, according to Alley et al. (1986b), little bubble drag should take place in glacial ice where the process would occur in the high-velocity regime. Now, in order to attempt a better understanding of the grain growth behavior in glacial ice, some results obtained by Gow (Gow and Williamson, 1976; Gow, 1968) for grain size and for bubble size and distribution, below pore close-off at Byrd station (T ) -28 °C), will be considered. It will be assumed
6112 J. Phys. Chem. B, Vol. 101, No. 32, 1997 that the values of ki and the correlated values of Mgg obtained for regime ZS1, at T ) -27 °C, also represent the intrinsic grain growth rate and boundary mobility for glacial ice. The experimental results obtained for k, for β ) k/ki and for Mbex are given in columns 2-4 of Table 3. In the last column, we also give the values of Mbth, as calculated from the experimental bubble size and concentration given by Gow (1968). Bubbles were assumed to be located on boundaries for samples at 64 m but uniformly distributed in the material for those corresponding to higher depths. Results in Table 3 show that, in glacial ice, it is β , 1, i.e., k , ki, as expected, and that the value of β is about the same as for that laboratory ice at the same T, despite the larger value of Nsφ3. Notice also that the effects of the Nsφ3 decrease with increasing depth, shown in the second column of the table, do not determine variations of k and related experimental parameters, though a nonnegligible variation of Mbth was calculated, as shown in the table’s last column. Notwithstanding, it is interesting to notice that the value of Mbex is now of about the same order of magnitude of Mbth. These results suggest that also at Byrd station grain growth was strongly affected by bubble drag, despite, as noted by Gow (1968), bubble-boundary separation occurs. This behavior can be explained by observing that, even when bubble-boundary separation occurred, i.e., in the high-velocity regime, boundaries always intersect a certain number of bubbles. As it results from the theory of bubble-boundaries separation developed by Hsueh and Evans (1983), in this case the distortion and velocity of bubbles would progressively increase while these are dragged by boundaries, as far as an unstable configuration is attained which determines bubble-boundary separation. According to this description of the phenomenon, the high-velocity regime would mainly differ from the low-velocity regime because, in the first case, both bubble separation from and collection by boundaries would occur, while, in the second, collected bubbles would not break away, so that they would accumulate on boundaries. As a consequence, for a given bulk concentration of bubbles, the drag effect of migrating bubbles should be less in the first case, but only in relation with their smaller concentration Ns on boundaries. Thus, bubble drag would still be represented by equations of the type eq 2-7, and calculations performed by Alley et al. (1986b) about bubble drag arising from bubble-boundary separation would only indicate that this second effect is negligible with respect to the first one. 4. Conclusions Samples of artificial ice have been studied where an evident drag effect of bubbles on boundary is observed. In this case, the phenomenon has been shown to occur in the low-velocity regime, and calculations permit us to derive evaluations of the boundary and bubble mobility, from the experimental results. The boundary mobility M, of about 5 × 10-13 m4/(Joule s) at
Arena et al. -10 °C, is about 1 order of magnitude lower than that derived from Di Prinzio and Nasello (1996) for fast migrating boundaries (large angle boundaries). This difference cannot be considered too large, if we take into account that the relationship between M and the grain growth rate ki depends on a grain growth model. Similar differences are also found between experimental and theoretical values of the bubble mobility, Mb, but again the simplifications inherent to the model used in the calculations make these differences acceptable. Previous authors have interpreted the grain growth behavior, observed in glacial ice samples of low impurity content, as evidence that the process occurred in the high velocity regime, where bubble drag was considered negligible. The analysis of the phenomenon performed in the present work shows that migrating bubble drag can strongly hinder grain growth even in this regime. Thus, we can conclude, from the comparison between the results obtained studying artificial and glacial ice that also in the second case bubbles should be the main responsible for the low grain growth rate, represented by the condition k , ki. Acknowledgment. The authors are grateful to CONICET and CONICOR for financial support. References and Notes Acha´val, E.; Nasello O. B.; Ceppi, E. A. J. Phys. Colloq. C1 1987, S3, 283-288. Alley, R. B.; Perepezko, J. H.; Bentley, C. R. J. Glaciol. 1986a, 32, (112), 415-424. Alley, R. B.; Perepezko, J. H.; Bentley, C. R. J. Glaciol. 1986b, 32, (112), 424-433. Azuma, N. E.; Higashi, A. J. Phys. Chem. 1983, 87, 4060-4064. Ceppi, E. Crecimiento de grano en hielo. Thesis, Facultad de Matema´tica, Astronomı´a y Fisica, National University of Co´rdoba, Argentina, 1985. Dash, J. G. Rep. Prog. Phys. 1995, 58, 115-167. Di Prinzio; Nasello, O. Study of the grain boundary diffusivity in ice bicrystals. To be published in J. Phys. Chem. B. Duval, P.; Lliboutry, L. J. Glaciol. 1985, 31, (107), 60-62. Fletcher, N. H. Philos. Mag. 1968, 18, 1287-1300. Golecki, I.; Jaccard, C. J. Phys. C 1978, 11, 4229-4237. Gow, A. J. Glaciol. 1968, 7, (50), 167-182. Gow, A. J. Glaciol 1969, 8, (53), 241-252. Gow, A.; Williamson. CREEL Report 76-35; Hanover, NH, 1976. Hillert, M. Acta Metall. 1965, 13, 227-238. Hsueh, C.; Evans, A. Acta Metall. 1983, 31, 189-198. Jellinek, H.; Gouda, V. Phys. Status Solidi 1969, 31, 413-423. Ketchan, W.; Hobbs, P. Philos. Mag. 1969, 19, 1961-73. Levi, L.; Ceppi, E. NuoVo Cimento Soc. Ital. Fis. 1982, 5C, 445-461. Maruyama, M.; Bienfait, M.; Dash, J. G.; Coddens, G. J. Cryst. Growth 1992, 118, 33-40. Mc Cappin, C.; Macklin, W. J. Atmos. Sci. 1984, 41, (16), 2437-2445, 2447-2455. Nasello, O.; Ceppi, E. Computer Simullation of Microstructure EVolution; Srolovitz, D., Ed.; Metallurgical Society: Warrendale, PA, 1985; pp 1020. Nasello, O.; Arena, L.; Levi, L. Physics and Chemistry of Ice; Maeno, N., Hondoh, T., Eds.; Hokkaido University Press: Hokkaido, 1992, pp 422427. Roos, D. J. Glaciol. 1966, 6, (45), 412-420 Shewmon, P. Trans. Metall. Soc. AIME 1964, 230, (5), 1134-1137. Spears, M. A.; Evans, A. G. Acta Metall. 1982, 30, 1281-1289.
