Ratelimiting processes

The rate of deformation of a crystal or of a polycrystalline aggregate depends on how rapidly dislocations can move. This, in turn, may depend upon factors such as the effectiveness of the mechanisms resisting motion, the ability of a dislocation to move from one atomic plane to another, and the orientation of the atomic plane in which the dislocation is moving. Usually, one process is significantly more important than the others, principally because it is more effective than the others in retarding dislocation motion. This process is called the rate-controlling or rate-limiting process.

It is, in general, not easy to identify the rate-limiting process. This is because it is different in different materials, and within any one material it changes with temperature and stress, and possibly also impurity content. Furthermore, the rate-limiting process may be different in single crystals and in polycrystalline aggregates.

In the next few paragraphs, we describe some possible rate-limiting processes in ice and present evidence for them.

Drag as the rate-limiting process

At the moderate stress levels normally found in glaciers, dislocations moving in a crystallographic plane are restrained in their motion by a number of drag mechanisms. The velocity of such a dislocation is given by:

where c is a constant of proportionality that depends, in part, on Boltzman's constant; a is the applied stress; b is the Burgers vector of the dislocations (a measure of the distortion of a crystal produced by a dislocation, and approximately equal to the atomic spacing in the crystal (Hull, 1969, pp. 19)); R is the gas constant; 0K is the temperature in Kelvins; and Q is the activation energy for the rate-limiting deformation mechanism (Weertman, 1983). Derivation of this equation, as well as some others in this chapter, is beyond the scope of this book. Noteworthy, however, is the dependence on very fundamental physical parameters.

A brief comment on activation energy is in order. The activation energy is the magnitude of an energy barrier that must be overcome for a kinematic process to occur. Each kinematic process has its own activation energy, so there is an activation energy for self-diffusion of hydrogen and oxygen in ice («60 kJ mol-1), an activation energy for creep of ice («79 kJ mol-1), and so forth.

The creep rate, e, resulting from movement of dislocations in a crys-tallographic plane is:

(Weertman, 1983), where a is a constant with dimensions of length and pd is the dislocation density (m-2); a depends on the orientation of the slip planes, but is —4.5 x 10-10 m. Of interest is the fact that the steady-state dislocation density, which reflects a balance between the applied stress and the internal stress from dislocations, is given by:

(Weertman, 1983;Alley, 1992), where n is the shear modulus. (The table on p. xiv gives definitions of parameters such as \x and values appropriate for ice.) Thus, combining Equations (4.1), (4.2), and (4.3) leads to:

Figure 4.7. Variation of total strain with time during deformation of a single crystal oriented so that glide occurs on the basal plane (easy ^

Figure 4.7. Variation of total strain with time during deformation of a single crystal oriented so that glide occurs on the basal plane (easy ^

Time, t

at constanttemperature. In other words, the velocity of dislocations varies with a, and their density varies with a2, leading to a cubic dependence of e on a.

A large volume of experimental data on ice deformed in the laboratory and on natural ice in glaciers and ice shelves yields such a cubic dependence, particularly for stresses above 0.1 MPa. This suggests that the theoretical model presented above is fundamentally sound, and thus that drag mechanisms which determine the velocities of dislocations may be the rate limiting factors.

When a single crystal of ice is stressed in such a way that there is an appreciable component of the stress on the basal (0001) plane, the deformation rate increases with time (Figure 4.7). This can be explained by the fact that the dislocation density is normally low in unstressed crystals, and increases gradually to its steady-state value, given by Equation (4.3), after the stress is applied. This further supports the suggestion that drag is the rate limiting process.

Climb as the rate-limiting process

There is, however, an additional dislocation process that must be considered. As dislocations multiply, they commonly begin to interfere with one another, resulting in gridlock. These dislocation tangles inhibit deformation. Under such conditions, application of a stress to a previously unstrained sample results in a deformation rate that initially decreases with time, a process called work hardening (Figure 4.8a). At sufficiently low temperatures this decreasing creep rate may continue indefinitely, but at higher temperatures recovery processes come into play. One such process involves diffusion of atoms away from (or vacancies toward) dislocations resulting in movement of the dislocation from one crystal-lographic plane to another. For example, if the atom at D in Figure 4.3b diffused away, the dislocation would move upward. This is called

C CC

Tertiary creep

Transient or primary creep

Work hardening

Secondary or steady-state creep

Time, t

g cc 00

\ /

Stress 1.02 MPa

Temperature -10 oC

Figure 4.8. (a) A typical strain-time curve for a sample of polycrystalline ice loaded in uniaxial compression. In early experiments, plots like this were used to identify the time span over which steady-state creep appeared to prevail. (b) As laboratory precision improved, it became possible to plot cumulative strain against strain rate, and thus more accurately identify the minimum strain rate. Note the increase in strain rate after about 1% strain. (Plot (b) is adapted with permission from Duval et al, 1983, Figure 6. Copyright 1983 American Chemical Society.)

dislocation climb if the dislocations are of the edge type, and cross slip if they are of the screw type. Both processes relieve tangles and, after an initial transient period, allow deformation to continue at a more-or-less steady rate (central part of curve in Figure 4.8a). As climb is the recovery process requiring more energy, it would be rate controlling.

Another recovery process, believed to be active particularly at low stresses, involves movement of grain boundaries as some crystals grow at the expense of their neighbors. The ice behind the moving boundary is unstressed, and thus relatively free of dislocations; pd is thus decreased by grain-boundary migration.

It is noteworthy that samples of polycrystalline ice deformed in compression invariably go through a transient phase of decelerating creep and then a period of nearly constant creep rate. (If the test is continued long enough, the creep rate normally increases again in what is called tertiary creep (Figure 4.8a); this is attributed to recrystallization, which is discussed further below.) This supports the suggestion that recovery processes are rate limiting. Furthermore, it may be significant that the activation energy for creep (79 kJ mol-1 )isclose to that for self-diffusion (60 kJ mol-1), again suggesting that climb (which is a result of diffusion) is the rate-limiting process. However, the difference between these two activation energies seems to be larger than can be explained by experimental error so other processes may also be involved.

Slip on other crystallographic planes

Deformation of one crystal in a polycrystalline aggregate necessitates deformation of neighboring crystals to preserve continuity of the medium. Thus, all crystals must take part in the deformation. Most crystals will be oriented so that the applied stress on that crystal will have a component parallel to the basal plane. However, a few crystals will not be so oriented, and these must therefore deform in some direction that is not parallel to the basal plane.

Slip on either the prismatic or pyramidal planes (Figure 4.9) is a possibility. However, on the prismatic planes the slip would probably be parallel to the basal plane (Figure 4.9), and so would not accommodate stresses normal to that plane. Thus, slip on pyramidal planes may be necessary, and because such slip is likely to be much harder than that on other planes, it could be rate limiting.

Inhomogeneous strain

The above considerations are based on the assumption of homogeneous strain. However, in polycrystalline ice, the stress on any single crystal is

Prismatic planes (0110) etc.

Basal plane (0001)

Prismatic planes (0110) etc.

Figure 4.9. Crystallographic planes in a hexagonal crystal.

Pyramidal planes (0111) etc.

Pyramidal planes (0111) etc.

not equal to the stress on the bulk sample, as crystals that slip easily in one direction but not in others will transmit the stress nonuniformly (Duval et al., 1983). If it is assumed, instead, that the strain is inhomogeneous, plastic strain is possible without slip on either prismatic or pyramidal planes. This is because processes such as climb of dislocations normal to the basal plane, grain-boundary migration, and grain translation and rotation can accommodate stresses normal to this plane and facilitate adjustments between crystals (Alley et al., 1997).

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