1. Introduction
The influence of plastic deformation on the electrical polarization and transport properties in ice, i.e. dielectric permittivity and electrical conductivity, has been further investigated since the symposium in Ottawa (Noll, 1973). Glen (1968) first pointed to the importance of the reorientation of H2O molecules on the plastic deformation, a mechanism originally only thought responsible for mechanical and electrical relaxation phenomena. Since then the interaction of various effects, initially treated separately, has attracted increasing attention. Experiments, showing the connection of internal friction with plastic deformation have been carried out by Perez and co-workers (Perez and others, 1976; Vassoille and others, 1978). The various aspects of the interaction between the reorientation of H2O molecules and dislocation movement is discussed by Whitworth (1978) in his review paper.
The measurements reported here, investigating the interaction between electrical properties and plastic deformation, have concentrated on the low-frequency range, the so-called space-charge dispersion, since the influence on the Debye dispersion seems to be very small and is difficult to measure.
2. Experimental
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. Measurements and results
3.1. Survey
Table I gives a survey of the samples, the deformation parameters resolved shear strain-rate ? s and resolved shear strain ys, 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 = τs(γs) for a deformation with constant resolved strain-rate γs- The changes in the electric properties reach a nearly steady value when the stress TS 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
(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 γs, resolved shear strain-rateγs = const.
The space-charge dispersion has a high-frequency limiting conductivity σs = 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 σ0, 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 δσb 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 strainγs. Symbols given in Table I.
3.3.2. Space-charge process R1
The frequency position of the process was only minimally affected by the deformation. In the majority of the tests, strains above a value of 4 to 10% resulted in a shift of a few hertz to higher frequencies (δτ1, < 0), but slight displacements to lower frequencies or no change at all also occurred. Since no systematic influence could be found, the time constant is assumed not to alter by deformation.
The changes of the dispersion strength Δσ1, with resolved strain-rate and strain shown in Figure 6 behave in a similar way to those of the d.c. conductivity σ0, (Fig. 5): Δσ1, falls increasingly with higher strain-rates and the alteration is largely independent of the strain. The similar behaviour of δ(Δσ1) and δσ0 reflects the predominant contribution of process R1 to the d.c. conductivity.
For tests S17(γ) and S18(EQN), δ(Δσ3) was positive but very small and of similar magnitude and negative δ(Δσ1), resulting in δσ0 ≈ 0. These two tests were the only ones with positive changes. This reflects the uncertainty for very slow deformations.
To derive Δ? from the measured Δσ1 according to Equation (1), the actual τ1-values and their changes δτ1 have been used. This results in a greater scatter of δ(Δ?1) and even produces one small positive value (X). For all other tests Δ?1 decreases with deformation (Fig. 7).
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 γs. 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 γs.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 γs, (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), γs = 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), γs = 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) γs = 3.2%, test D7(2)γs = 10%, γs = 1.3 × 10-3s-1 for both tests.
4. Discussion
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 ci 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 ci 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 τS is also determined by generating and degenerating processes. The resolved shear strain γs, 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):
or
The dislocation density ρ and the dislocation velocity v arc functions of γs and τS respectively. The second term on the right-hand side of Equation (2) and Equation (3) can be neglected initially, when τS α γs (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
In the steady state the dislocation term pbv is directly determined by the resolved strain-rate γs; 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.
Acknowledgements
I am grateful to Professor Dr. H. O. Kneser and Professor Dr. W. Eisenmenger, in whose institute these investigations were carried out, for their support; Professor S. Magun and my colleague Dipl. Phys. R. Ruepp for many helpful discussions; the technical and clerical staff, especially Mrs S. Rahfeld, for their assistance, and Dr J. W. Glen for his advice during- the preparation of the manuscript. The work was supported by the Deutsche Forschungsgemeinschaft.
Discussion
R. TAUBENBERGER: Could you explain what you mean by "d.c. conductivity", since σ0 is measured at about 102 Hz and below that one has a complicated spectral behaviour?
G. NOLL: The d.c. conductivity σ0 simply denotes the value of the plateau in the spectrum σ(f) at frequencies below the Debye dipole-rearrangement dispersion (f ≈ 100 Hz). I have followed the usual nomenclature, even if I do not feel happy with it. "Static conductivity σs" (corresponding to "static permittivity ?s", which refers to the corresponding plateau in the spectrum ?(f)) implies similar difficulties. I do not want to express a special underlying model by using this term.
TAUBENBERGER: Did you check if there is any correlation between the thickness of plastic foils and any of the strain-rate-induced changes of the parameters of R3? Could it be that you are seeing mainly electrode effects?
NOLL: At the very low frequencies of the space-charge dispersion R3, the foil contributes essentially to the impedance of the ice/foil combination. I have not sufficient experimental data in this frequency range to check the influence of the foil thickness and its change with deformation. In the frequency range of the dispersion R1 and the conductivity plateau σ0, the foil plays only a minor part. The effects found there are greater than the possible changes of the foil capacitance. I would further expect such a change to be proportional to the strain, rather than to the strain-rate as observed.
TAUBENBERGER: Did you make comparisons between the development of your spectra on plastic deformation and the ones on ageing, both alone and together with hydrostatic pressure, that I published?
INFLUENCE OF DEFORMATION RATE 289
NOLL: In this paper I could only indicate the relationship between the influence of plastic deformation and ageing. The observed ageing effects in the space-charge region of my crystals show a similar behaviour to yours (Taubenberger, 1973). With the extended measurement range the dispersions could be measured to very low frequencies. The dispersion R1 corresponds lo the process called by you "low-frequency dispersion", the dispersion R3 corresponds to your "blocking dispersion", which is just appearing at your lowest frequencies. These dispersions thus exist in crystals grown by different methods and in samples which are prepared in different ways and with different types of electrodes. The ageing behaviour of both dispersions is correlated. Hubmann has told me that he has investigated the ageing of the dispersion R1 over a much longer time and confirmed the trend which was indicated in my measurements in the first hundreds ?f hours. The result of this is finally a completely changed spectrum at frequencies below 1 000 Hz, with a much lower conductivity. I also think that the ageing results make doubtful the interpretation of the conductivity σ0 by intrinsic proton conductivity alone (Taubenberger, 1973).
TAUBENBERGER: It seems to me that the largest effect you have in your spectra is a tremendous shift of the lowest dispersion region R3 to lower frequencies, whereas process R1 stays stable. δσ0(D) then stems entirely from the decrease of Δσ(R3) ≈ ε0Δε(R3)/τ(R3), assuming that all the processes act in parallel. With this view, your changes of the spectra with strain resemble those I observed for R3 (blocking dispersion) on ageing. The central common feature then was that Δσ, = ε0Δε1/τ1 stayed largely constant. The relationships of your findings with ours on ageing should be further explored.
NOLL: The major contribution to the change of the d.c. conductivity σ0 comes from the change of Δσ 1, which can be seen from Figures 5 and 6. The change of the dispersion R3 may be impressive for the process itself, but contributes little to the conductivity σ0.
TAUBENBERGER: I think one has to bear in mind whether one wants to analyse the changes in the two relaxation regions as stemming from processes acting in parallel or in series. One has to find out which hypothesis for process superposition is more consistently described by the spectra and its changes, which in turn have to be linked to a hypothesis for the underlying microscopic process itself.
NOLL: I agree with you that an explanation for the complicated spectrum in the space-charge region and ils changes with deformation depends on the kind of interaction of the processes. To clarify this, further experimental data are needed, e.g. temperature dependence, and a further study of the ageing effects is essential too.
TAUBENBERGER: DO your values of Δ?(D) fit with or without an anisotropy (Δ?∥ — Δ?╧)/Δ? of the so-called Debye dispersion D?
NOLL: An anisotropy of the permittivity of the Debye dispersion could not be investigated with the crystal orientations used for the deformation tests.
