2.1. Crystal growth and sample preparation

Large pure ice monocrystals were grown from deionized and distilled water (electrical conductivity at 25°C ≤ 10 ⁵ Ω^-1 m^-1, rising during growth due to dissolved air) by a modified Bridgman method (crystals with code K) or by a Stöber type method (crystals with code TK) (Noll, 1973)- The etch-pit density determined in the usual way (Kuroiwa and Hamilton, 1963) was 5 × 10³ to 1.5 × 10 ⁴ cm^-2 for K-crystals, depending on the crystallographic plane. The density for the TK-crystals was higher on all planes by a factor 4 to 6, the values ranging from 2.5 × 10⁴ to 6 × 10 ⁴ cm^-2. The modified Stöber method, originally supposed to produce crystals with lower dislocation density by avoiding stresses from the vessel, failed in this respect. The density of Tyndall figures was 0.1 to 0.3 cm^-3 for the K-crystals, ≤ I cm^-3 for the TK-crystals.

The specimens for the uniaxial compression tests were prepared by Ruepp's method 3 (Ruepp, 1973, p. 179). For the shear tests the crystals were simply frozen onto the electrode plates.

2.2. Deformation tests

The specimens were deformed either by shear using an arrangement of the Bausch-type (Bausch, 1935; Steinemann, 1958) or by uniaxial compression, their orientation always being for glide in the basal glide system (0001)[1120]. With the design for the shear tests described and illustrated by Noll (1973), a relatively uniform gliding could be achieved, but it was difficult to maintain a perfect ice/electrode interface during deformation. This allowed only slow deformation rates at temperatures near to the melting point.

Fig. 1.

Apparatus for uniaxial compression test.

The uniaxial compression avoids such difficulties and has therefore often been used for plastic mechanical investigation (Griggs and Coles, 1954) and in combination with electrical measurements (Brill and Camp, 1957, 1961, p. 67-69; Higashi, 1969; Ackley and Itagaki, unpublished). Figure 1 shows the apparatus used for our uniaxial compression tests. Thin plastic foils (polyethylene or polyfluor(chlor)ethylene 3.5 to 8 μm thick) with evaporated silver electrodes on the outside surface, frozen on the end faces of the cylindrical ice specimen (40 mm ×40 mm), provided blocking electrodes and avoided air gaps between ice and electrodes (Ruepp, 1973). The plastic foils also made the glide planes rotate more easily on the electrode plates, because of the reduced adhesion.

All tests were carried out under constant strain-rate, the resolved shear strain-rate ? _s in the basal glide system ranging from 2 × 10^-5 to 1 × 10 ^-3 s^-1, the resolved shear strain ? _s from 3 to 60%.

2.3. Electrical equipment and techniques

All specimens in the experiments had guard-ring electrodes forming a three-terminal capacitor. The impedance of the specimen was measured by a.c. methods described previously (Ruepp, 1973; Noll, 1973, unpublished). The known capacitance of the plastic foils in series with the ice was eliminated by calculation and then the frequency spectrum for the conductivity σ(f ) and dielectric permittivity ?(f) of the ice was derived. In addition to the a.c. methods a transient method was used to give a quick result over a wide frequency range; Figure 2 shows its circuit diagram.

A voltage step-function U is applied to the ice capacitor and the corresponding current transient I(t) is measured by an electrometer (Keithley 610C) as a function of time t. The transient signal is recorded after passing a logarithmic converter (Hewlett Packard 7562A) either on a storage oscilloscope (Tektronix 564) or on a signal analyser (Hewlett Packard 5480A). This improved the signal-to-noise ratio by averaging several successive signals. The signals can finally be plotted on an -X?-recorder. The guarded electrode is controlled at earth potential by the electrometer amplifier; therefore a good guard-ring measurement is guaranteed. The measurement has been limited by the rise time of the amplifier, depending on the current range. After Fourier transformation and elimination of the foil capacitance the frequency spectrum σ(f) and ?(f) could again he obtained. A variant of the transient method was used by Schenk (unpublished).

Fig. 2.

Circuit diagram for transient measurements.

Three techniques were used to measure the trend of the dielectric properties during and immediately after the end of deformation: (1) σ and ? were recorded at a fixed frequency as a function of strain and time respectively (Noll, 1973). (2) With the transient method the frequency range < 100 Hz could be observed by a succession of measurements. (3) The whole spectrum was taken by quick measurements at various frequencies.

3.1. Survey

Table I gives a survey of the samples, the deformation parameters resolved shear strain-rate ? _s and resolved shear strain y_s, the age of the crystals, the time of annealing after sample preparation, and the symbols used in the figures. The temperature for the tests was T = 270.2 K, except for test D8(1).

Table I.

Survey of samples and test conditions

3.2. Influence of deformation: qualitative description

First the influence of the deformation on the dielectric spectrum will be demonstrated, using the uniaxial compression test D₃ as example. Figure 3 shows the resolved shear stress plotted against resolved shear strain τ _s = τ_s(γ_s) for a deformation with constant resolved strain-rate γ_s- The changes in the electric properties reach a nearly steady value when the stress T_S is well over its maximum and approaches its steady value. The period during which the dielectric measurements shown in Figure 4 were taken is marked.

Figure 4 represents the dielectric spectrum in a double logarithmic plot; dielectric permittivity ? = ? and the conductivity σ = ω?₀?" both as functions of frequency f (?” is the dielectric loss, ω the angular frequency ω = 2πf and ? ₀ the permittivity of vacuum). The solid line shows the spectrum before deformation, after the sample has been annealed for 20 d at —3°C. The Debye dispersion D at frequencies above f= 10² Hz is well separated from the low-frequency space-charge dispersions R₁ and R₃ arising from blocking electrodes (f< 10² Hz). They all show the form of simple relaxation processes, each one characterized by one single relaxation time τ and one dispersion strength △? or, in terms of conductivity σ

(1)

(This holds even for the space-charge dispersions, because very weak electric fields have been applied and the dielectric response contains only the fundamental component. For these processes the time constant τ should be understood as a mean or basic relaxation time.)

Fig. 3.

Resolved shear stress τ _s against resolved shear strain γ_s, resolved shear strain-rateγ_s = const.

The space-charge dispersion has a high-frequency limiting conductivity σ_s = Σ Δσi, representing the contributions of all slow processes to the conductivity and which is identical with the d.c. conductivity σ₀.

The influence of the deformation is obvious in the low-frequency part of the spectrum: there the curves ? (f) and σ(f) are shifted to lower values. The frequency position of the process RI is hardly affected, the time constant τ, stays nearly constant. The dispersion strengths Δ? and Δσ are decreased, resulting in a lower value of σ₀. The changes remain at least partially after the deformation has ended and the shear stress relaxed to zero (this will be described later in Section 3.4).

The spectrum keeps its features during and after deformation and can still be characterized by the superimposed dispersions D, R₁, and R₃. The influence of the deformation will therefore be described in the following by the changes of τ, Δ?, Δσ, and σ_0, the emphasis being on σ0 and process R₁.

3.3. Influence of resolved strain-rate and resolved strain

3.3.1. d.c. conductivity

In Figure 5 the change δσ_b of the d.c. conductivity (as indicated in Fig. 4) is plotted as a function of resolved shear strain-rate γ _s and resolved shear strain γ, respectively. It depends almost exclusively on the rate of deformation and not on the size of the strain. For very slow resolved strain-rate the d.c. conductivity decreased only slightly but it drops strongly when the deformation becomes faster. For still higher rates, the difference goes on falling, but not as steeply as before: .

The amount of strain has almost no influence on the change of σ(Fig. 5(b)). For a given strain-rate, strains between 3 and 60% produce within the limits of experimental error the same changes, which are determined by the parameter γ _s of the test experiment.

Fig. 4.

Dielectric spectrum; frequency dependence of dielectric permittivity and electrical conductivity before (Eqn) during (Eqn) plastic deformation by uniaxial compression.

Fig.5.

Change of d.c. conductivity σ₀: (a) as a function of resolved shear strain-rateγs, as a function of resolved shear strainγ_s. Symbols given in Table I.

3.3.3. Space-charge process R₃

This very slow process could not often be measured down to sufficiently low frequencies. Therefore the results obtained arc only approximate. The relaxation frequency stays unaltered or again changes unsystematically to higher or lower frequencies. The dispersion strengths Δσ₃ and Δ?₃ also change in either positive or negative direction for small strain-rate γ_s. Both quantities fall if γ _s > 4 × 10^-1 s^-1; this value seems to be a threshold for systematic changes. In contrast to process R₁, the strain itself influences the behaviour in this case.

Fig.6.

Space-charge process R₁: change of dispersion strength Δσ₁: (a) as a function of resolved shear strain-rate γ’s, (b) as a function of resolved shear strain γ_s.Symbols given in Table I.

Fig. 7.

Space-charge process R₁: change of dispersion strength Δ?₁: (a) as a function of resolved shear strain-rate γ_s, (b) as a function of resolved shear strain γs. Symbols given in Table I.

3.4. Behaviour with time: recovery after deformation

The changes depend on the position of the frequency in the spectrum (cf. Fig. 4). In the space-charge region ? and σ alter steadily and monotonically with increasing strain. The alterations are proportional to the resolved strain as long as the resolved shear stress rises. The dielectric properties still change after deformation has ended. Figure 8 shows how the space-charge processes depend on time. After a small deformation (test D₇(l), γ_s = 3.2%) the perturbations of the parameters τ₁ Δσ₁ Δ? diminish rather quickly; some go beyond their original value. After a stronger deformation (D₇(2), γ_s = 10%) the parameters of process R₁ change a little, the parameters of process R₃ attain their final values after some hours. The recovery processes are essentially faster than the ageing processes observed before and to some extent after deformation.

Fig. 8.

Time dependence of the changes in the space-charge parameters after the end of deformation. Test D₇(1) γ_s = 3.2%, test D₇(2)γ_s = 10%, γ_s = 1.3 × 10^-3s^-1 for both tests.

4.1. General aspects

The conspicuous result of plastic deformation on the electrical properties is the reduction of the conductivity in the whole space-charge range, i.e. below 100 Hz. The d.c. conductivity σ₀ and the strengths of both dispersions are diminished. The position of the dispersion in the spectrum, however, does not change significantly.

The electrical conductivity depends basically on the concentration c_i of the charge carriers i involved, their mobility μi and their charge qi. A change of the charge qi can hardly be imagined. The decrease of σ must therefore originate in smaller ci or _μi. But should this happen during deformation, if the charge carriers have contributed to the conductivity and have already been mobile before? As the concentrations c_i includes only active charge

carriers, its value can decrease by trapping mobile charge carriers in lattice imperfections created by the deformation. Charge carriers whose concentration is in thermal equilibrium, i.e. intrinsic ions or defects, would be produced by the lattice anew. A decrease would only be possible if their enthalpy of formation increases. Otherwise it is more likely that the conductivity is due to extrinsic charge carriers. The second possibility is a diminished mobility: either additional obstacles hinder the transport of charges, or previously favourable travel paths are destroyed by the deformation.

4.2. Dependence on the resolved strain-rate

The distinct dependence of the deformation effects on the resolved strain-rate (Fig. 5(a) and 6(a)) has to be seen in connection with the behaviour immediately after deformation. We may assume that generating and degenerating processes are already active during gliding (dynamic recovery). At low strain-rates the perturbations produced anneal while the deformation is still going on and major deviations from the original values are prevented. This explains the small effects found for slow deformations. During fast deformations the effects produced cannot be removed sufficiently. If the fast deformation is short (small resolved strain) the crystal recovers after the deformation has ended: the changes are reversible. For long lasting and rapid deformations (large resolved strains), however, large and permanent alterations of the spectrum are found.

During plastic deformation of ice monocrystals the resolved shear strain τ_S is also determined by generating and degenerating processes. The resolved shear strain γ_s, given by the test conditions, is opposed by the movement of dislocations (density ρ, Burgers vector b, velocity v) with a term depending on time t(see Weertman, 1973):

(2)

or

(3)

The dislocation density ρ and the dislocation velocity v arc functions of γ_s and τ_S respectively. The second term on the right-hand side of Equation (2) and Equation (3) can be neglected initially, when τ_S α γ_s (see Fig. 3). The dislocation density is then very small. With continuing deformation more and more dislocations are produced. When their contribution predominates, the resolved shear stress passes its maximum and decays with increasing strain to a steady-state value. From τ _s = 0 it follows that

(4)

In the steady state the dislocation term pbv is directly determined by the resolved strain-rate γ_s; the size of the strain has no influence. The results described in Section 3.3 indicate that this term also determines the decrease of the electrical conductivity. The changes still remaining (at least partially) after the deformation stops, give a hint that the dislocation density plays the determining role, rather than the dislocation velocity. For slow deformations the product pbv is small anyway (Equation (4)). During a fast but small strain the velocity v rises, but the density ρ hardly changes. In these cases the product still remains small and its influence decays quickly after the end of the deformation.

The strong influence of the resolved strain-rate and the independence of the strain can be understood. However, a model for the interaction between dislocation and charge carriers is still missing. Possibly it is not the dislocations themselves that are active, but point defects produced with them which anneal later.

4.3. Information from X-ray topography

Perez and co-workers have investigated the dislocation distribution in connection with their measurements of the internal friction (Perez and others, 1976) and ultrasonic propagation (Tatibouёt and others, 1975). They used monocryslals, grown by a Bridgman method similar to ours and with a dislocation density of the same size as the etch pit density of our K-crystals, ≈ 10⁴cm^-2 (Tatibouёt and others, 1975, p. 169; private communication from J. Klinger in 1975). The X-ray topographs of freshly grown crystals show a usual dislocation distribution. During ageing, a network structure develops due to aggregation of dislocations which has still not reached its final slate, even after one year at T ≈ 250 K. By plastic deformation and subsequent annealing it becomes more pronounced, leading to a cellular structure and polygonization (Vassoille and others, 1977).

From these observations it may therefore be concluded that the change of the dielectric spectrum by plastic deformation and the recovery after deformation is not only determined by the increased dislocation density, but also by the changing character of the dislocation network. This is additionally supported by our observations of ageing in the space-charge region of the spectrum. The initially complicated dispersion for frequencies below l00 Hz develops into two well-separated dispersions R₁ and R₃ during about 20 d after the preparation of the sample. During this ageing period the dispersion R₁ shifts lo higher frequencies and its dispersion strength Δσ₁ and Δ?₁ decreases, while at the same time the dispersion R₃ builds up and increases its dispersion strength Δσ₃ and Δ?₃. This results in a gradual decrease of the plateau ?Γ the d.c. conductivity σ₀. It seems that after 500 h the dispersion R₃ has nearly reached its final stage shown in Figure 4, but the dispersion R₁ and σ₀ are still decreasing. The differential behaviour of the space-charge dispersions might be attributed to the development of the cellular structure of the dislocation network, the two dispersions reflecting regions of the crystal with different density and character. Thus the plastic deformation seems to produce similar effects on the dispersion R₁ as does the ageing.