J. Phys. Chem. 1983. 87,4074-4078
4074
on grain size,35probably because the extent of basal slip, important during the primary, is limited by grain boundaries. The recoverable part of the deformation, too, increases with grain size probably for the same reason. The onset of fracture is known to depend on grain size. "he work of Schulson and CurrieF is typical: the fracture strength uf in tension varies as d-lj2. The internal stresses which appear during creep of ice, at applied stresses of 1.5 MPa or more, are sufficient to nucleate cracks (Figure 6). The wavelength of the internal stress, and thus the crack size, is roughly equal to the grain size. A grain-sized crack will propagate when 0
> Klc/(ird/2)1'2
where K l c is the fracture toughness of ice. This appears to be the origin of the grain size effect in fracture. 6. Conclusions Basal-plane deformation of ice is much easier than deformation on nther planes. Because of this, nonuniform internal strs'.,. !s, with a wavelength about equal to the grain size, appear in polycrystalline ice when it is deformed plastically. The minimum creep rate, which is reached after a strain of about 1% (regardless of the stress levels and temperature), is largely controlled by processes occurring on nonbasal planes. Polycrystalline ice shows a marked normal transient on loading, during which the creep-rate falls by a factor of 100 or more. The hardening processes causing this transient (35) P. Duval and H. Le Gac, J. Glaciol., 25, 151 (1980). (36) S. J. Jones and H. A. Chew, J . Glaciol., 27, 517 (1981).
(37) E. M. Schulson and J. H. Currier, Acta Metall., 30,1511 (1982).
Velocity of Dislocations in Ice on {OOOl) and
are partly caused by the development of the long-range internal stress distribution (which gives kinematic or directional hardening) and partly by conventional shortrange dislocation interactions (which give isotropic or nondirectional hardening). The behavior of ice after small changes in stress can only be explained if both components of hardening exist. On unloading, polycrystallineice shows large recoverable (or anelastic) strains. These are caused by the long-range internal stresses which cause reverse strain when the load is removed. The peak internal stresses seem to be important in other ways also: they cause internal cracking at applied stresses above 1 MPa, and they may be responsible for the onset of dynamic recrystallization after a critical strain of about 1%. Grain size influences diffusionalflow in ice, and (perhaps by limiting the extent of easy basal slip) it influences the transient behavior. Recent studies show, however, that the minimum creep rate does not depend on grain size. The deformation of polycrystalline ice is much more complicated than that of single crystals. The two are related only in an indirect way, largely because of the requirements that at least four independent slip systems operate for compatible deformation of the polycrystal. The resulting distribution of internal stresses has a profound influence on almost every aspect of the deformation.
Acknowledgment. This study was supported by the Centre National de la Recherche Scientifique and the Delegation GBnBrale ii la Recherche Scientifique et Technique. We thank T. H. Jacka for sending us unpublished results. Registry No. Water, 7732-18-5.
{ l o l o ) Planes
R. W. Whltworth Department of Physics, University of Birmingham, Birmingham B 15 277, England (Received: August 23, 1982; I n Final Form: December 3, 1982)
Dislocations moving in ice must overcome the barrier presented by proton disorder, and a new model is developed for the velocity at which a flexible dislocation can move if limited only by this process. For {OOOl) slip it is already known that proton disorder will prevent dislocation motion on planes of the shuffle set. It is now shown that the observed velocities are not incompatible with slip on planes of the glide set, but it is not clear whether proton disorder is the rate-limiting process. The theory is applied to (1210) slip on {lOIO)planes; there are again two possible sets of planes between which slip may occur.
Introduction The structure of ice Ih consists of (OOO1) layers of water molecules stacked in the sequence AABBAABB... as shown in Figure 1. Dislocations of Burgers vector 1/3a(1190) can in principle glide either between pairs of widely spaced AA or BB planes (called planes of the shuffle set) or between pairs of more closely spaced AB or BA planes (the glide set). Dislocationsof this same Burgers vector can also glide on prismatic (lOT0)planes, and there are again two possible sets of planes between which slip may occur; these are labeled types I and I1 in Figure 1. Experiments have shown' that the stress required for basal slip is about 50 times less than that for prismatic slip. A satisfactory (1)A. Higashi,
Phys.Snow. Ice, Conf. Proc., 1966, 1, 277 (1967).
theory of dislocation mobility should show between which sets of parallel planes slip can occur most easily, and should explain why basal slip is so much easier than prismatic slip. Glen2 has pointed out that the disorder of the protons in ice presents an obstacle to slip, because a dislocation can only move forward by the breaking of one bond and the formation of another if the relevant bonds are appropriately oriented. At the time of the Cambridge con~,~ ference in 1977 the existing models of this p r o c e ~ spredicted that dislocations should not be able to move on the (2) J. W. Glen, Phys. Kondens. Mater., 7 , 43 (1968). (3) R. W. Whitworth, J. G. Paren, and J. W. Glen, Phil.Mag., 33, 409 (1976). (4) H. J. Frost, D. J. Goodman, and M. F. Ashby, Phil.Mag., 33,951 (1976).
QQ22-3654l83l2087-4074$01.5QIQ 0 1983 American Chemical Society
The Journal of Physical Chemistry, Vol. 87, No. 27, 1983 4075
Velocity of Dislocations in Ice
" -
100011
-'T
0 0 0 0 . 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 . 0 0 0 0 0 0 0 0 0 0
shuffle
-
- glide
[ioiol
Figure 1. Projection of ice structure onto (1210) plane showing the basal phnes of the shuh set and the glide set and the (1070)prismatic planes of type I and type 11.
Figure 2. Diagram showing the zone bounded by the broken lines within which a dislocation is free to move on its slip plane. The solid line indicates the mean position of the dislocation. Removal of the obstacle at P causes the forward boundary to advance over the area A' and the mean position to advance over area A .
T R S Figure 3. Example of small region of computer simulation of boundary to zone of dislocation movement. Each circle represents a lattice point, and the computer has randomly placed obstacles at the solid points with probability q = 1/16. Solid line represents boundary approached from below. Removal of barrier in row R produces no advance because of rule limiting flexibility. Removal of barriers at S and T produce advance to the broken lines.
mean position to move backward. Let the number of all such events which occur in a time t be CLt, where C is a constant, and let them result in a net displacement of the mean position over an area L6y. In the absence of a stress the average of the individual displacements A should be zero, but this is a random walk problem for which8
(L6y)2= Z C L t
(1)
We may now introduce a diffusion coefficient D such that basal planes at the speeds observed experimentally, and 6y2 = 2Dt (2) it seemed necessary to consider seriously the possibility that dislocations in ice might have a noncrystalline ~ o r e . ~ ~ ~ giving The weakness of this argument was that it assumed that D = $C/(2L) (3) slip occurred on planes of the shuffle set. It has subsequently been shown7that there is more freedom for bonds Under the action of a shear stress 7 directed parallel to to be rearranged and thus for dislocations to move if slip the Burgers vector (of magnitude b) this segment of the occurs on planes of the glide set. dislocation will experience a force 7bL and will drift forModels of kink motion along an otherwise straight ward with velocity ud. We may define the mobility p of dislocation have been developed both for slip on planes the segment by of the shuffle set3v4and for slip on the planes of the glide U d = p7bL (4) set,7 but these models are unable to yield dislocation velocities as high as those observed experimentally for any The quantities p and D are related by the Einstein relarealistic density of kinks. To obtain an upper limit for the tion8 velocity of a dislocation limited only by proton disorder p = D/kT (5) we will assume that the kink energy and line tension are so small that the dislocation is free to move over all conand therefore figurations permitted to it by the orientations of the bonds across its slip plane. This is the basis of the flexible dislocation model, which was developed in an approximate form for glide on the shuffle set,3but in that form is difThis result does not depend on L and is a valid expression ficult to adapt for other situations. The present paper for the dislocation velocity. describes a simpler formulation of this model and its apIf we ignore for the time being events which involve plication both to the basal planes of the glide set and to dangling bonds on the dislocation core, all events which prismatic planes. produce a change A in the mean position of the line arise from movements over an area A' = 2A of one or other of Flexible Dislocation Model the boundaries. Concentrating on one boundary there will, Consider the situation shown in Figure 2 which is by the principle of detailed balance, be equal numbers of slightly simpler than the real case of ice. A flexible disevents causing the boundary to move forward or backward location of projected length L is confined by obstacles on by any given amount. The total rate of annihilation of its glide plane to lie within the narrow zone bounded by barriers on one boundary is therefore C' = C/4, and we the broken lines. It can move easily between all the can calculate ud from (6) by considering only events inpossible configurations lying within this zone and its mean volving the outward movement of one boundary. In ice position is shown by the solid line. On a slower time scale the rate C'depends on the rate of reorientation of bonds obstacles are being annihilated and new ones created, and at the boundary, while A'2 can be determined by a comfor each such event the mean position of the dislocation puter simulation for a random array of obstacles of apwill be changed. In Figure 2 , for example, the removal of propriate density. an obstacle at P causes the forward boundary to advance Essentially the same reasoning can be applied to calover the shaded area A' and the mean position to move culate the velocity of a kink on an otherwise straight forward over A = A'/2. Similarly the creation of an obdislocation. In this case no approximations or simulations stacle at a point like Q within the zone but ahead of the are involved and the result is identical with the equivalent instantaneous position of the dislocation will cause the case analyzed previ~usly.~ (5) R. W. Whitworth, J. Glaciol., 21, 341 (1978). (6)J. Perez, C. Mai, and R. Vassoille, J. Glaciol., 21, 361 (1978). (7) R. W. Whitworth, Phil.Mag. A, 41, 521 (1980).
(8) F. Reif, "Fundamentals of Statistical and Thermal Physics", McGraw-Hill, New York, 1965, sections 12.5 and 15.6.
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30T,cocl L
The Journal of Physical Chemistry, Vol. 87,No. 21, 1983
of a given bond is 7b. It is therefore possible to express the dislocation velocity in the form ud
10
20
30
LO
&b3 2kT7b 7
- = a-
b
2000
1
Whitworth
where the dimensionless parameter Cy
m’
O0
Flgure 4. Histogram of values of m’ for advance of boundary for 9 = 1/16. 30000 lattice points along the boundary were examined. There were obstacles at 7401 of these points and histogram gives results for these 7401 cases. The number of cases of m‘ = 0 within the first column was 151 1.
Computer Simulation Figure 3 shows a set of lattice points on the glide plane. The dislocation can move freely over those marked by open circles, but solid circles represent obstacles. The probability of each point being an obstacle is q, and the figure shows the case of q = 1/16 required for basal planes of the glide set in ice.7 A straight dislocation starting at the lower edge of the diagram can move forward as far as the boundary shown by the solid line. As in ref 3 the degree of flexibility is assumed to be such that the dislocation can move forward or backward by one but not more than one lattice spacing per atomic step along its length. Some such restriction is necessary to prevent the dislocation wrapping round obstacles, which would be quite unrealistic for the separations and stresses considered. A computer has been programmed to simulate this situation for various values of q using arrays of 500 X 20 points at B time. Each obstacle along the boundary is then taken in turn, and the computer determines the number m’ of lattice points over which the boundary would move forward if this obstacle were removed. Examples in the figure are points R, S, and T for which m’ = 0, 3, and 12. In the case of q = 1/16 a total length of 30000 units was examined and in this length there were 7401 obstacles along the boundary. A histogram of values of ”for these obstacles is given in Figure 4 and yields 2 = 125 f 6. The precise result depends on the assumption made about flexibility. It looks in Figure 3 as if the boundary should be allowed to bow out further in the central section, but we have already assumed a generous degree of flexibility and the value of is not unreasonable when seeking an upper limit on ud.
Basal Slip in Ice For the basal plane in ice the areas A’ correspond to m’ unit - cells of area (v‘3/2)b2, and from the facts that 2 = mr2and C = 4C’ eq 6 can be written as (7)
For the diffusion theory to apply it is necessary that s i b 3
