Edge Dislocation Dipoles Research Articles

Work-hardening in sapphire (α-Al 2O 3)

Quantitative measurements of the densities of the several components of the dislocation substructure in pure sapphire deformed by basal glide—glide dislocations, dipoles, and prismatic loops—indicate that the accumulation of loops, and the resultant loop-glide dislocation interactions, make the principal contribution to work-hardening. A work-hardening model based on the interaction of loops and glide dislocations is proposed and the work-hardening rate is calculated to be μ 225, in good agreement with experiment. In this model, the loops are generated from edge dislocation dipoles by self-climb. The effects of recovery through loop annihilation by climb is introduced into the expression for the work-hardening rate, and the plateau flow stress for a given strain-rate and temperature is found to be inversely proportional to the cube root of the bulk diffusion coefficient for the rate-controlling species (oxygen). Diffusion coefficients calculated from experimental plateau flow stress data are in good agreement with experimental values. The actual shape of the theoretical stress-strain curve obtained from the model is also in good agreement with experimental curves. The maximum work-hardening rate decreases with increasing temperature, a trend attributed to the fact that, at the higher temperatures, recovery begins during the yield-point region before work-hardening becomes fully established and also loop-dislocation interactions may be less effective.

Climb dissociation of dislocations in sapphire (α-Al2O3)

Abstract Edge dislocation dipoles, formed in sapphire during basal glide at elevated temperatures, break up into strings of loops by self-climb. One stage in this process is the transformation of narrow dipoles into faulted dipoles. This is due to the climb dissociation of each dislocation in the dipole, according to the reaction 1/3[11 0]→1/3[11 0] + 1/3[01 0], with one pair of partials annihilating to form the faulted dipole. The stacking fault energy is estimated to be ∼400 erg/cm2, based on the maximum width of the faulted dipoles. A second stage, where the faulted dipoles rotate through 30° to the exact edge orientation, occurs to reduce the elastic energy.

THE ROLE OF DISLOCATIONS AND GRAIN BOUNDARIES IN TRANSPORT PROCESSES IN IONIC CRYSTALS

After a brief survey of the defects introduced by plastic deformation and of their removal by annealing, tlie influence of plastic deformation upon ionic conductivity is considered. Both enhancement and s~~ppression of the extrinsic conduction has been found experimentally in addition to an enhancement at very high temperatures in the intrinsic region 1'. These effects can be explained in terms of dislocations and their interactions during plastic flow. In polycrystalline material of near theoretical density, the grain boundaries appear to enhance only the extrinsic conductivity and to increase the observed activation energy in tliis region. 1 . Introduction. There is a growing volume of evidence that tlie presence of dislocations in ionic crystals may modify their transport properties in a number of different ways. The purpose of this paper is to review tlie experimental findings and to indicate what precautions lllust be taken to tnininiise the influence of dislocations upon measurements of ionic conductivity. Since most dislocations are generated by plastic defor~iiation, the paper starts with a brief survey of the defects introduced by plastic deformation and of the way in which these defects are removed by annealing. Subsecluently the influence of dislocations and grain boundaries on ionic conductivity is considered. 2. Defects introduced by plastic deformation. At room temperature plastic deformation is not homogeneous in ionic crystals, and i t is often difficult to compare the results of dilferent workers who strain their specimens plastically by tlie same amount but produce dilyerent defect densities and distributiolls by using dilferent specimen geometries. The shape of the stress-strain curve gives a good idea of the nature of the dcformation process, and where possible this should be measured in all deformation experiments. F o r specimens which arc tall compared with their cross-section, conipresscd along tlie [OOI] tall axis, three stages of work-hardening are fbu~ id with changes of slope after about 4 ;; and about 8 % plastic co~iipression [I]. Thesc three stages represent differelit modes of dcform:~tion, a n d diffcrent rates of accumulation o f dislocations ill thc crystal, with different effects upon the transport properties, and thus it is important to know whether a crystal has been deformed into stage I, stage I1 or stage 1II. By contrast, specimens which are squat compared with their cross-section, for example niost ionic conductivity specimens compressed along the squat axis, enter stage 111 deformation at very small plastic strains with a high rate of accumulation of dislocations and of generation of point defects. Most of the experiments on plastic deformation of NaCl have been carried out on tall crystals with dimensions of the order of 5 x 12 x 25 mm, having initial dislocation densities, inside the sub-grains, of between 10' and IOcm? The rate of increase of dislocation density is low in stage 1 and increases more rapidly in stage 11, to a total screw dislocation density of some 8 x 1O7/crn\af~er 8 O/;: compression. The moving screw dislocatiolls leave trails of debris, in the form of vacancy clusters and edge dislocation dipoles, oriented in [I001 directions in the slip plane. The density of edge dislocations and debris is about 10 tilnes that of screw dislocations during stage 11. By measuring the change in bulk density of the crystals during deformation the rate of accuniulation of point defects can bc estimated, although it is difficult lo dislinguisli between vavancies and interstitials in tliis way. Both species should be formed as a consequence of thc geometry of the deformation process, although in the alkali halides tlie life-time of' the intcrstitinls is likely to be very short at room temperature. Interpr-cling tlic long-term change of density produced by p l~~s t i c deformation in tcrms Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1973939

Short-Range Interactions Between Two Parallel Dislocations

An analytical model of the dislocation core is proposed. The center of a crystal dislocation in this model is represented by a continuous distribution of infinitesimal dislocations, whose distribution function incorporates the atomic configurations of dislocations as determined by Cotterill and Doyama using a many-body-crystal model of copper. The distribution functions used have the same form as the Peierls model except the width parameter, which is about twice that of the original Peierls model. Using the present model and methods of the continuum theory of dislocations, numerical calculations are made for the interaction energies between two parallel screw and two parallel edge dislocations. The results are significantly different from those predicted by the classical theory when the separation distance is less than ten atomic distances. Approximate expressions for the interaction energy between two coplanar edge dislocations and for that between two coplanar screw dislocations are obtained. The magnitudes of both the interaction energy and interaction force remain finite in the present model, enabling the unique determination of the self-energies of single dislocations and edge dislocation dipoles. Dislocation splitting in fcc metals is also discussed.

Notizen: On a New Theory of the Magnetization Curve of Plastically Deformed Nickel Single Crystals

Abstract A new theory of the magnetization process in plastically deformed nickel single crystals of single slip orientation is outlined. It is based on the interaction between the (101)-180°-domain walls and the three types of primary edge dis-location dipole and multipole arrangements dominating in stage I, in the transition region and in stage II of the stress-strain curve, respectively, i.e. the [121]-dipole bundles, the rope-like clusters of such bundles extended along [121], and the [121] -multipole walls, all having [T01]-Burgers-vectors. In increasing magnetic fields the primary edge dislocation arrangements are shown to be capable to glide to new ob-stacles under the pressure of the (101) -180°-domain walls which will follow them and whose segments at the same time start to vault themselves cylindrically or pillow-shapedly be-tween these dislocation barricades, finally overcoming them at the coercive force by Barkhausen jumps. In an earlier paper 1 first experimental results had been reported concerning the large decrease of num-ber and size of Barkhausen jumps observed on samples of plastically deformed nickel and on magnetite dur-ing simultaneously superposed intensive ultrasonic ir-radiation. To explain this unexpected effect in terms of domain wall motion, an attempt had been made to mo-dify the usual potential theory of the magnetization curve by introducting a new hypothetical magnetization process which should enable the domain walls also to move to some extent irreversibly and continuously but without Barkhausen jumps. In the meantime, the mentioned problems were sub-ject of a recently published extensive dissertation 2 which contains, in addition to a more detailed descrip-tion of the experiments and of the apparatus used, also the analysis of the main consequences of these experi-mental facts with respect to the usual potential theories of the magnetization curve of deformed nickel, showing the strong inconsistency between these theories on the one hand and the experimental results, reported in 1 and 2 , and certain other important empirical facts on the other. Last not least this dissertation contains a new consistent quantitative theory of the magnetization curve of plastically deformed nickel single crystals of single slip orientation. The aim of this note is to summarize in a qualitative manner the main statements of that new theory. Its basic assumptions which could be proved by quoting numerous experimental facts as well as by detailed quantitative estimations are the following ones:

Theory of High-Temperature Type Work-Hardening of Body-Centred Cubic Metals

A dislocation arrangement in cell boundaries is set up on the basis of experimental evidences that cell boundaries deviate from slip plane by a few degrees. The internal stresses associated with this cell structure are determined self-consistently from the dislocation arrangement in cell boundaries and the motion of secondary dislocations through the cell boundaries. In this model, the primary slip system activates discontinuously in one layer of the cell structure and it forms slip lines with finite shear strain. If no dislocation annihilation occurs in cell boundaries, the flow stress increases approximately linearly with strain. The decrease of work-hardening rate in the later stage of deformation is due to the annihilation of closely spaced edge dislocation dipoles in cell boundaries by conservative climb.

Kinetics of solute atom drift to dislocation dipoles

The time dependence of solute drift to an edge dislocation dipole is calculated for a dislocation-solute atom interaction characterized by the first-order size effect. It is found that the rate of solute arrival to dislocations at constant temperature is proportional to the square root of the recovery time, t, in contrast to the t 2 3 behavior characteristic of single straight edge dislocations as sinks. This effect can be a significant influence on solute segregation to extended dislocations, since an extended screw dislocation has a first order size effect interaction with a solute atom which is of the same form as the dipole considered. Thus t 1 2 kinetics are characteristic of solute drift to extended screw dislocations as well.

Limitation of the Phonon Heat Conductivity by Edge-Dislocation Dipoles, Single Dislocations, and Normal Processes

The scattering of phonons by edge-dislocation dipoles leads to deviations from the quadratic temperature dependence of the lattice thermal conductivity found when the latter is limited by the interaction of phonons with isolated dislocations. At temperatures where the dominant wavelength ${\ensuremath{\lambda}}_{d}$ of the phonons is large compared with the distance $R$ between the two dislocations of a dipole, it is found that the conductivity is almost independent of the absolute temperature $T$ and larger than the corresponding value for isolated dislocations. With increasing temperature ($R\ensuremath{\gg}{\ensuremath{\lambda}}_{d}$), the thermal conductivity due to dipoles approaches asymptotically the conductivity limited by isolated dislocations. The interaction of the phonons with the dipoles is treated by nonlinear continuum theory which includes also the normal three-phonon interaction. Numerical results, obtained with the material constants of copper, show that dipoles with distances $R$ smaller than $|\mathrm{b}|(370\ifmmode^\circ\else\textdegree\fi{}\mathrm{K}/T)$ (b=Burgers vector) can be distinguished from isolated dislocations by measuring the thermal conductivity.

The transient work hardening phenomena in crystals with edge dislocation dipole clusters

The formula for the flow stress and work hardening coefficient is deduced for the model crystal with homogeneous array of long edge dislocation dipole clusters. It is shown that some characteristics of the stress-strain curve, (i.e. the decrease of work hardening coefficient at the beginning of plastic deformation, the pre-yield plastic flow of the virgin crystal), can be understood on the basis of this model. Both cited facts are connected with the existence of dipoles and dipole clusters in crystals, part of which is not locked.

Stability of the Dislocation Dipole in Half‐Space

AbstractThe stability of an edge dislocation dipole of infinite length lying parallel to the free surface of an isotropic, semi‐infinite medium has been investigated by determining the dipole orientation for which the stress on both poles is less than a chosen dislocation friction stress. It is found that the effects of the surface are considerable, despite the fact that the stress field of the dipole is short‐ranged. In particular, the surface can induce orientations of stability which are not expected from the infinite body behaviour. Similar results are likely to arise for prismatic dislocation loops in bounded media. Some aspects of the effect of an applied shear stress on the dipole are also studied.

Dilatation in planes containing dipole accumulations

Abstract The dilatation at an arbitrary point in a plane containing the midpoints of an infinite array of edge dislocation dipoles is evaluated and its relation to the corresponding quantity in planes containing finite numbers of such dipoles is pointed out.

On the validity of the column approximation in electron microscopy

Abstract The authors develop a theory very slightly different from that of Takagi in the non-column case. They get numerical results in the case of 100 kev electron energy with a single edge dislocation and dislocation dipole close to the upper surface. Discrepancies between the column approximation and the non-column case are very small in the general case. If the Bragg angle becomes very large (≳4°) discrepancies may be important.

Interaction of Dislocation Dipoles with Surfaces

Using recent work by Conners on dislocation-image forces, it is shown that, in the isotropic elastic-continuum approximation, the forces of interaction between edge-dislocation dipoles and free or coated surfaces vanish to terms of order r−2, where r is the distance between the dislocation dipole and the surface. This is as would be expected from the response of dislocation dipoles to stress fields.

The effect of oxide films on dislocation-surface interactions in aluminum

Aluminum single crystals were deformed in cyclic strain, both in air and in vacuum, and with both 50 and 100 Å thick oxide films on the surface. The results show that deformation in vacuum produces a layer of edge dislocation dipoles trapped just beneath the surface, as well as reducing the amount of surface slip. The effect of increasing the oxide coating thickness is to reduce the amount of slip observable on the surface of the oxide. As an aid in interpretation, the mechanical properties of thin, amorphous aluminum oxide films were measured in air and in vacuum. Young's modulus of aluminum oxide, E c, was found to increase by a factor of 4 under vacuum, and the fracture strength by a factor of 1.5. These increases are most likely due to the removal of absorbed water vapor. The increase in E c can account for the presence of the trapped dislocation dipoles under vacuum by using the theory of Head and of Conners, which predicts a net repulsive force on near surface dislocations when the modulus of the coating is greater than the modulus of the substrate. The reduction in surface slip observed under vacuum is due to the increase in both E c and the strength of the oxide coating.

Annihilation of a dislocation dipole

The time dependence of annihilation and the life-time of an edge dislocation dipole at elevated temperatures is studied theoretically. The driving forces for the gradual approach of both dislocations composing the dipole are due to their elastic interaction. The two dislocations approach one another by climbing by the emission or absorption of vacancies; climbing is complemented by gliding.

APPLICATION OF X-RAY TOPOGRAPHY TO THE ANALYSIS OF THE DISLOCATION ARRANGEMENT IN DEFORMED COPPER SINGLE CRYSTALS

The paper deals with the application of Berg–Barrett X-ray topography to deformed copper single crystals. The imaged surfaces were sectioned from the deformed bulk crystals parallel to low-index planes.In stage I of the work-hardening curve, the braids of primary edge-dislocation dipoles, well known from electron transmission microscopy, have been observed over lengths of several tenths of a millimeter. Whereas in stage 1 the slip appears to be rather homogeneous and without strong interaction between primary and secondary slip systems, the transition region to stage 11 displays, as a characteristic feature, a systematic interaction of both types of slip systems, giving rise to layer-like dislocation structures about parallel to the primary slip plane. These layers, as in stage 11, are connected with a special type of lattice rotation. In stage II, dislocation walls or wall fragments perpendicular to the primary slip direction are observed. Generally, these walls are accompanied on either side by an extended region with an excess density of primary edge dislocation of one sign ("kink-walls"). As an example, the excess density at a flow stress τ ≈ 1–2 kg/mm2 is found to be of the order of 108/cm2.

The elastic interaction of a dislocation dipole with dilatation centres

The elastic interaction of a dilatation centre with edge dislocation dipole is investigated and the share of this elastic interaction in the time dependence of the diffusion of point defects to dislocations in plastically deformed crystalline materials is discussed.

The Equilibrium Position of an Orientation Junction of an Edge Dislocation Dipole

AbstractUsing a linear isotropic theory of elasticity, a calculation is made of the energy and the equilibrium position of an orientation junction of an edge dislocation dipole. The junction is approximated by segments of straight lines and the calculation shows that the junction is highly localized.

Observation of Dislocations in CaWO4 by Transmission Electron Microscopy

The distribution of dislocations in deformed calcium tungstate was studied by electron transmission microscopy. The photomicrographs confirmed earlier optical observations that (001) 〈100〉 is the slip system of this material between 500°–900°C. Screw dislocations, long edge-dislocation dipoles, quadrupoles, and tangles of dislocations were observed in foils prepared from specimens deformed at 500°C. Specimens deformed at 700°C showed similar features, except that many of the long dipoles had split into long strings of prismatic loops. At 900°C the loops were more isolated and sometimes grew to a diameter exceeding the thickness of the foil. Many single dislocations could also be seen with nearly pure edge or screw orientation. Two models are proposed for quadrupole formation and their relative stability considered. Strain hardening is thought to be due to the interaction of dislocations with multipoles and prismatic circular loops.

An electron microscope study of precipitation in MgO single crystals

Abstract An electron microscope study was made of precipitation in vanadium and titanium-doped MgO single crystals. In both crystals cubic spinel precipitates form in parallel orientation with the cubic matrix lattice in the temperature range 1100°C-1500°c. Precipitation of the vanadium spinel, Mg2VO4 is accompanied by the nucleation and growth of dislocation loops composed of edge dislocation dipoles with their axes parallel to the principal axes of the cubic MgO matrix lattice. It is concluded that dislocation loop nucleation occurs by the condensation of vacancies previously existing in the form of clusters with solute ions in the as-grown crystals. Similar loops grow from existing edge dislocations constituting sub-grain boundaries and those formed by prismatic punching from unidentified impurity particles. The equilibrium structure at 1200°c is that of 0·5 μ diameter spinel particles decorating irregular dislocation networks. No similar dislocation configurations are developed on precipitation of the titanium spinel. It is tentatively suggested that in the latter case, which involves an increase in oxygen ion lattice spacing on precipitation, precipitate nuclei form from solute ion-vacancy clusters which are too small for dislocation loop nucleation.